Handbook of Magnetic Materials, Volume 17

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VOLUME

SEVENTEEN

HANDBOOK OF MAGNETIC MATERIALS

This page intentionally left blank

VOLUME

SEVENTEEN

HANDBOOK OF MAGNETIC MATERIALS Edited by

K.H.J. BUSCHOW University of Amsterdam

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

North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK

First edition 2008 Copyright ©2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material 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 Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53022-6 ISSN (Series): 1567-2719

For information on all North-Holland publications visit our website at books.elsevier.com Printed and bound in The Netherlands 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

Preface to Volume 17

The Handbook series Magnetic Materials is a continuation of the Handbook series Ferromagnetic Materials. When Peter Wohlfarth started the latter series, his original aim was to combine new developments in magnetism with the achievements of earlier compilations of monographs, producing a worthy successor to Bozorth’s classical and monumental book Ferromagnetism. This is the main reason that Ferromagnetic Materials was initially chosen as title for the Handbook series, although the latter aimed at giving a more complete cross-section of magnetism than Bozorth’s book. In the last few decades magnetism has seen an enormous expansion into a variety of different areas of research, comprising the magnetism of several classes of novel materials that share with truly ferromagnetic materials only the presence of magnetic moments. For this reason the Editor and Publisher of this Handbook series have carefully reconsidered the title of the Handbook series and changed it into Magnetic Materials. It is with much pleasure that I can introduce to you now Volume 17 of this Handbook series. Magnetic tunnel junctions form part of the exciting field of spintronics. In this field, nanostructured magnetic materials are employed for functional devices where both the charge and the spin are explicitly exploited in electron transport. Magnetic junctions offer a number of unique opportunities for investigating novel effects in physics and have led to several new research directions in spintronics. Equally important is the fact that magnetic junctions represent excellent materials for exploring novel and superior types of devices. The physics of spin-dependent tunneling in magnetic tunnel junctions is reviewed in Chapter 1, concentrating on ferromagnetic layers separated by an ultrathin insulating barrier. The tunneling current between the ferromagnetic electrodes in these junctions depends strongly on an external magnetic field and as such lends itself to novel applications in the fields of magnetic media and data storage. Followed by a short introduction on the background and the elementary principles of magnetoresistance and spin polarization in magnetic tunnel junctions, the author discusses basic and magnetic transport phenomena, emphasizing the critical role of the preparation and properties of the tunnel barriers. Later on, key ingredients to understand tunneling spin polarization are introduced in relation to experiments using superconducting probe layers. The author also discusses a number of crucial results directly addressing the underlying physics of spin tunneling and the role played by the polarization of the ferromagnetic electrodes. Apart from Al2 O3 , the successful use of alternative crystalline barriers such as SrTiO3 and MgO is discussed. With decreasing size of magnetic elements in magnetic storage media, read heads, and MRAM elements, the time and energy necessary for reading and writing

vi


magnetic domains have become of paramount importance and are studied intensively worldwide. A concept of substantial impact is that of spin-accumulation, i.e. a non-equilibrium magnetization that is injected electrically into a non-magnetic material from a ferromagnetic contact by an applied voltage. A breakthrough in magnetoelectronics is the observation of current-induced magnetization reversal in several types of layered structures. This effect finds its origin in the transfer of spin angular momentum by the applied current. On the other hand, magnetization dynamics induces spin currents into a conducting heterostructure. These novel effects couple the magnetization dynamics in hybrid devices with internal and applied spin and charge currents. The time-dependent properties become non-local, meaning that they are not a property of a single ferromagnetic element, but depend on the whole magnetically active region of the device. Recent progress in understanding the magnetization dynamics in ferromagnetic hybrid structures is presented in Chapter 2. Magnetic properties of 3d-4f intermetallic compounds have been reviewed in several previous Volumes of this Handbook. This includes reviews on magnetically hard materials and related compounds (Volumes 6, 9 and 10). In these materials, the magnetocrystalline anisotropy invariably plays a central role. Somewhat apart stands the literature on experimental studies of the crystal field effects in intermetallics of rare earths. Results obtained by means of inelastic neutron scattering have been reviewed in Volume 11. The separation between the topics of magnetic anisotropy and crystal field effects seems somewhat artificial. In view of the general acceptance of the single-ion model, little doubt remains about the intimate connection between the two phenomena. The origin of the apparent splitting between the two topics mentioned can most likely be found in the fact that the theoretical activity in the area has been lagging behind experiments ever since the appearance of the last major review written four decades ago by Callen and Callen, in 1966. However, one has to realize that theoretical advance on magnetic anisotropy and crystal field effects did not cease in the meantime. These topics just progressed in different directions, stimulated by the advent of the density functional theory (Volume 13). As regards the single-ion model proper, work on it proceeded at a rather slow pace. Nonetheless, a fair amount of new results has been published between the late 1960s and more recent times. Chapter 3 reviews the progress made in the theory, filling the gap in the literature between the anisotropy and the crystal field effects. In this Chapter the authors aim at reasserting the statement that magnetocrystalline anisotropy is the most important manifestation of the crystal field effects. Magnetocaloric effects in the vicinity of phase transitions were already discussed by Tishin in Volume 12 of this Handbook, published in 1999. Since then there has been a strong proliferation in research on magnetocaloric materials and their application, mainly dealing with the option of magnetocaloric refrigeration at ambient temperature. A comprehensive review dealing with this latter aspect is presented in Chapter 4 of the present Volume. The design of a refrigeration system involves many problems which are far from simple. Its design invariably requires a critical evaluation of possible solutions by considering factors such as economics, safety, reliability, and environmental impact. The vapor compression cycle has dominated the refrigeration market to date because of its advantages: high efficiency, low toxi-


vii

city, low cost, and simple mechanical embodiments. Perhaps this is because as much as 90% of the worlds heat pumping power; i.e. refrigeration, water chilling, air conditioning, various industrial heating and cooling processes among others, is based on the vapor compression cycle principle. However, in recent years environmental aspects have become an increasingly important issue in the design and development of refrigeration systems. Especially in vapor compression systems, the banning of CFCs and HCFCs because of their environmental disadvantages has opened the way for other refrigeration technologies which until now have been largely ignored by the refrigeration market. As environmental concerns grow, alternative technologies which use either inert gasses or no fluid at all become attractive solutions to the environment problem. A significant part of the refrigeration industry R&D expenditures worldwide is now oriented towards the development of such alternative technologies in order to be able to achieve replacement of vapor compression systems in a mid- to long-term perspective. One of these alternatives is magnetic refrigeration which is discussed in Chapter 4. In this chapter the author emphasizes the many novel experimental results obtained on magnetocaloric materials, placing them in the proper physical and thermodynamic background. Also measuring systems as well as demonstrators and prototypes for magnetic refrigeration are discussed. Intermetallic compounds in which 3d metals (particularly Mn, Fe, Co and Ni) are combined with rare earth elements exhibit a large variety of interesting physical properties. The magnetic properties of these intermetallics are a matter of interest for two main reasons: Firstly their study helps to elucidate some of the fundamental principles of magnetism. Secondly they are of technical interest, because several compounds were found to be a suitable basis for high performance permanent magnets. More recently the unique soft magnetic properties made amorphous metal-metalloid alloys to a further class of materials which has attained considerable importance with regard to industrial application. In Chapter 5 the hydrides of such compounds and alloys are discussed. In fact, this chapter can be regarded as an updating of Chapter 6 in Volume 6 of this Handbook, published in 1991. In order to reach a self-contained form of this chapter, the authors and the editor agreed to incorporate the most important results of the previous chapter into the present one. In this way the novel results can be viewed in the right perspective, not requiring the interested reader to go back to the previous chapter in Volume 6 at regular intervals. Here it should be mentioned that a large variety of novel techniques has been employed more recently in order to elucidate the mechanism and effects of hydrogen uptake which is particularly complex in intermetallic compounds. They can roughly be devided into surface sensitive methods such as photo emission and related spectroscopies, X-ray absorption (XANES, EXAFS), X-ray magnetic circular dichroism (XMCD), transmission electron microscopy, conversion electron Mössbauer spectroscopy and to some extent susceptibility measurements. The results of such investigations are discussed in Chapter 5 together with results of NMR and ESR and surface insensitive experiments, where only the bulk properties can be studied (magnetic measurements, neutron and X-ray diffraction, X-ray absorption, transmission Mössbauer spectroscopy).

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It is well known that there have been many new developments in the field of magnetic sensing and actuation, including new forms of magnetic material. Apart from this has been much progress in the development of microelectromechanical systems (MEMS). Hand in hand with this has gone the advance in density of electronic components on a chip, expressed by the so-called Moore’s Law, where areal density has doubled every eighteen months. Much of MEMS technology is silicon based, with three-dimensional structures being manufactured from a silicon platform by means of various lithographic techniques. It is common practice to include functionality in to MEMS, opening the possibility of sensing or actuation. Frequently piezoresistive materials are used which requires current and voltage connections to the sensor element, the measured quantity being the strain dependence of electrical resistivity in the active film. Of special interest is the incorporation of magnetic materials in to MEMS making it possible to use inductive coupling for sensing or activation. The major advantage to be gained from this is the possibility to avoid the requirement for connections, and that it allows packaging and deployment in remote or hostile environments. In Chapter 6 the authors address the integration of magnetic components into MEMS as a way of providing additional functionality. They present an overview of advances in thin film magnetic materials that make the use of MagMEMS a viable option. Volume 17 of the Handbook on the Properties of Magnetic Materials, as the preceding volumes, has a dual purpose. As a textbook it is intended to be of assistance to those who wish to be introduced to a given topic in the field of magnetism without the need to read the vast amount of literature published. As a work of reference it is intended for scientists active in magnetism research. To this dual purpose, Volume 17 of the Handbook is composed of topical review articles written by leading authorities. In each of these articles an extensive description is given in graphical as well as in tabular form, much emphasis being placed on the discussion of the experimental material in the framework of physics, chemistry and material science. The task to provide the readership with novel trends and achievements in magnetism would have been extremely difficult without the professionalism of the North Holland Physics Division of Elsevier Science B.V. VAN

K.H.J. B USCHOW WAALS -Z EEMAN I NSTITUTE U NIVERSITY OF A MSTERDAM

DER

Contents

Preface to Volume 17 Contents Contents of Volumes 1–16 Contributors

1.

Spin-Dependent Tunneling in Magnetic Junctions

v ix xi xv

1

H.J.M. Swagten 1. Introduction 2. Basis Phenomena in MTJs 3. Tunneling Spin Polarization 4. Crucial Experiments on Spin-Dependent Tunneling 5. Outlook Acknowledgements References

2. Magnetic Nanostructures: Currents and Dynamics

2 14 52 71 102 106 106

123

Gerrit E.W. Bauer, Yaroslav Tserkovnyak, Arne Brataas and Paul J. Kelly 1. Introduction 2. Ferromagnets and Magnetization Dynamics 3. Magnetic Multilayers and Spin Valves 4. Non-Local Magnetization Dynamics 5. The Standard Model 6. Related Topics 7. Outlook Acknowledgements References

3. Theory of Crystal-Field Effects in 3 d-4 f Intermetallic Compounds

124 125 127 135 139 142 144 144 145

149

M.D. Kuz’min and A.M. Tishin Foreword 1. Formal Description of the Crystal Field on Rare Earths 2. The Single-Ion Anisotropy Model for 3 d-4 f Intermetallic Compounds

149 150 166

x

Contents

3. Spin Reorientation Transitions 4. Conclusion References

4. Magnetocaloric Refrigeration at Ambient Temperature

210 228 229

235

Ekkes Brück List of Symbols and Abbreviations 1. Brief Review of Current Refrigeration Technology 2. Introduction to Magnetic Refrigeration 3. Thermodynamics 4. Materials 5. Comparison of Different Materials and Miscellaneous Measurements 6. Demonstrators and Prototypes 7. Outlook Acknowledgements References

5. Magnetism of Hydrides

237 237 239 241 247 270 274 280 281 281

293

Günter Wiesinger and Gerfried Hilscher 1. Introduction 2. Formation of Stable Hydrides 3. Electronic Properties 4. Basic Aspects of Magnetism 5. Review of Experimental and Theoretical Results Acknowledgement References

6. Magnetic Microelectromechanical Systems: MagMEMS

293 295 296 300 304 422 422

457

M.R.J. Gibbs, E.W. Hill and P. Wright 1. Introduction 2. MEMS Fabrication 3. Magnetic Materials for MEMS 4. Magnetoresistive Materials and Sensors 5. Magnetic MEMS Based Devices References Author Index Subject Index Materials Index

458 466 485 491 511 521 527 579 583

Contents of Volumes 1–16

Volume 1 1. 2. 3. 4. 5. 6. 7.

Iron, Cobalt and Nickel, by E. P. Wohlfarth . . . . . . . . . . . . . Dilute Transition Metal Alloys: Spin Glasses, by J. A. Mydosh and G. J. Nieuwenhuys Rare Earth Metals and Alloys, by S. Legvold . . . . . . . . . . . . . Rare Earth Compounds, by K. H. J. Buschow . . . . . . . . . . . . . Actinide Elements and Compounds, by W. Trzebiatowski . . . . . . . . . Amorphous Ferromagnets, by F. E. Luborsky . . . . . . . . . . . . . Magnetostrictive Rare Earth–Fe2 Compounds, by A. E. Clark . . . . . . .

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1 71 183 297 415 451 531

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1 55 189 243 297 345 381 509

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Volume 2 1. 2. 3. 4. 5. 6. 7. 8.

Ferromagnetic Insulators: Garnets, by M. A. Gilleo . . . . . Soft Magnetic Metallic Materials, by G. Y. Chin and J. H. Wernick Ferrites for Non-Microwave Applications, by P. I. Slick . . . Microwave Ferrites, by J. Nicolas . . . . . . . . . . . Crystalline Films for Bubbles, by A. H. Eschenfelder . . . . . Amorphous Films for Bubbles, by A. H. Eschenfelder . . . . Recording Materials, by G. Bate . . . . . . . . . . Ferromagnetic Liquids, by S. W. Charles and J. Popplewell . . .

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Volume 3 1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry and Technology, by U. Enz . . . . . . . . . . . . . . . . . . . . . 2. Permanent Magnets; Theory, by H. Zijlstra . . . . . . . . . . . . . . . . 3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R. A. McCurrie . . 4. Oxide Spinels, by S. Krupiˇcka and P. Novák . . . . . . . . . . . . . . . . 5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite Structure, by H. Kojima . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto . . . . . . . . 7. Hard Ferrites and Plastoferrites, by H. Stäblein . . . . . . . . . . . . . . . 8. Sulphospinels, by R. P. van Stapele . . . . . . . . . . . . . . . . . . . 9. Transport Properties of Ferromagnets, by I. A. Campbell and A. Fert . . . . . . . .

305 393 441 603 747

Volume 4 1. Permanent Magnet Materials Based on 3d-rich Ternary Compounds, by K. H. J. Buschow . . . . . 1 2. Rare Earth–Cobalt Permanent Magnets, by K. J. Strnat . . . . . . . . . . . . . . . . 131 3. Ferromagnetic Transition Metal Intermetallic Compounds, by J. G. Booth . . . . . . . . . . 211

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4. Intermetallic Compounds of Actinides, by V. Sechovský and L. Havela . . . . . . . . . . . . 309 5. Magneto-Optical Properties of Alloys and Intermetallic Compounds, by K. H. J. Buschow . . . . . 493

Volume 5 1. Quadrupolar Interactions and Magneto-Elastic Effects in Rare-Earth Intermetallic Compounds, by P. Morin and D. Schmitt . . . . . . . . . . . . . . . . . . . . . . . 2. Magneto-Optical Spectroscopy of f-Electron Systems, by W. Reim and J. Schoenes . . . . . 3. INVAR: Moment-Volume Instabilities in Transition Metals and Alloys, by E. F. Wasserman . 4. Strongly Enhanced Itinerant Intermetallics and Alloys, by P. E. Brommer and J. J. M. Franse . . 5. First-Order Magnetic Processes, by G. Asti . . . . . . . . . . . . . . . . . 6. Magnetic Superconductors, by Ø. Fischer . . . . . . . . . . . . . . . . . .

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1 133 237 323 397 465

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1 85 181 289 453 511

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Volume 6 1. Magnetic Properties of Ternary Rare-Earth Transition-Metal Compounds, by H.-S. Li and J. M. D. Coey . . . . . . . . . . . . . . . . . . . . . . . . . 2. Magnetic Properties of Ternary Intermetallic Rare-Earth Compounds, by A. Szytula . 3. Compounds of Transition Elements with Nonmetals, by O. Beckman and L. Lundgren . 4. Magnetic Amorphous Alloys, by P. Hansen . . . . . . . . . . . . . . . 5. Magnetism and Quasicrystals, by R. C. O’Handley, R. A. Dunlap and M. E. McHenry . . 6. Magnetism of Hydrides, by G. Wiesinger and G. Hilscher . . . . . . . . . . .

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Volume 7 1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann . . . . . . . . . . . 2. Energy Band Theory of Metallic Magnetism in the Elements, by V. L. Moruzzi and P. M. Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides, by M. S. S. Brooks and B. Johansson . . . . . . . . . . . . . . . . . . . . . 4. Diluted Magnetic Semiconductors, by J. Kossut and W. Dobrowolski . . . . . . . . . . 5. Magnetic Properties of Binary Rare-Earth 3d-Transition-Metal Intermetallic Compounds, by J. J. M. Franse and R. J. Radwański . . . . . . . . . . . . . . . . . . . . 6. Neutron Scattering on Heavy Fermion and Valence Fluctuation 4f-systems, by M. Loewenhaupt and K. H. Fischer . . . . . . . . . . . . . . . . . . . .

. . 139 . . 231 . . 307 . . 503

Volume 8 1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals, by J. J. Rhyne and R. W. Erwin . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in Rare-Earth Intermetallics with Cobalt and Iron, by A. V. Andreev . . . . . . . . . . . . . . . . . . 3. Progress in Spinel Ferrite Research, by V. A. M. Brabers . . . . . . . . . . . . . 4. Anisotropy in Iron-Based Soft Magnetic Materials, by M. Soinski and A. J. Moses . . . . . 5. Magnetic Properties of Rare Earth–Cu2 Compounds, by Nguyen Hoang Luong and J. J. M. Franse . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

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Volume 9 1. Heavy Fermions and Related Compounds, by G.J. Nieuwenhuys . . . . . . . . . . . . . 2. Magnetic Materials Studied by Muon Spin Rotation Spectroscopy, by A. Schenck and F.N. Gygax . .

1 57

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3. Interstitially Modified Intermetallics of Rare Earth and 3d Elements, by H. Fujii and H. Sun . . . . 303 4. Field Induced Phase Transitions in Ferrimagnets, by A.K. Zvezdin . . . . . . . . . . . . 405 5. Photon Beam Studies of Magnetic Materials, by S.W. Lovesey . . . . . . . . . . . . . . 545

Volume 10 1. Normal-State Magnetic Properties of Single-Layer Cuprate High-Temperature Superconductors and Related Materials, by D.C. Johnston . . . . . . . . . . . . . . . . . . . . . 1 2. Magnetism of Compounds of Rare Earths with Non-Magnetic Metals, by D. Gignoux and D. Schmitt . 239 3. Nanocrystalline Soft Magnetic Alloys, by G. Herzer . . . . . . . . . . . . . . . . . 415 4. Magnetism and Processing of Permanent Magnet Materials, by K.H.J. Buschow . . . . . . . . 463

Volume 11 1. Magnetism of Ternary Intermetallic Compounds of Uranium, by V. Sechovský and L. Havela . . . . 1 2. Magnetic Recording Hard Disk Thin Film Media, by J.C. Lodder . . . . . . . . . . . . . 291 3. Magnetism of Permanent Magnet Materials and Related Compounds as Studied by NMR, by Cz. Kapusta, P.C. Riedi and G.J. Tomka . . . . . . . . . . . . . . . . . . . . 407 4. Crystal Field Effects in Intermetallic Compounds Studied by Inelastic Neutron Scattering, by O. Moze 493

Volume 12 1. Giant Magnetoresistance in Magnetic Multilayers, by A. Barthélémy, A. Fert and F. Petroff . 2. NMR of Thin Magnetic Films and Superlattices, by P.C. Riedi, T. Thomson and G.J. Tomka 3. Formation of 3d-Moments and Spin Fluctuations in Some Rare-Earth–Cobalt Compounds, by N.H. Duc and P.E. Brommer . . . . . . . . . . . . . . . . . . . . 4. Magnetocaloric Effect in the Vicinity of Phase Transitions, by A.M. Tishin . . . . . .

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1 97

. . . . 259 . . . . 395

Volume 13 1. Interlayer Exchange Coupling in Layered Magnetic Structures, by D.E. Bürgler, P. Grünberg, S.O. Demokritov and M.T. Johnson . . . . . . . . . . . . . . . . . 2. Density Functional Theory Applied to 4f and 5f Elements and Metallic Compounds, by M. Richter 3. Magneto-Optical Kerr Spectra, by P.M. Oppeneer . . . . . . . . . . . . . . . . 4. Geometrical Frustration, by A.P. Ramirez . . . . . . . . . . . . . . . . . . .

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. 1 . 87 . 229 . 423

Volume 14 1. III-V Ferromagnetic Semiconductors, by F. Matsukura, H. Ohno and T. Dietl . . . . . . . 2. Magnetoelasticity in Nanoscale Heterogeneous Magnetic Materials, by N.H. Duc and P.E. Brommer 3. Magnetic and Superconducting Properties of Rare Earth Borocarbides of the Type RNi2 B2 C, by K.-H. Müller, G. Fuchs, S.-L. Drechsler and V.N. Narozhnyi . . . . . . . . . . . . 4. Spontaneous Magnetoelastic Effects in Gadolinium Compounds, by A. Lindbaum and M. Rotter .

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1 89

. . 199 . . 307

Volume 15 1. Giant Magnetoresistance and Magnetic Interactions in Exchange-Biased Spin-Valves, by R. Coehoorn . 1 2. Electronic Structure Calculations of Low-dimensional Transition Metals, by A. Vega, J.C. Parlebas and C. Demangeat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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3. II–VI and IV–VI Diluted Magnetic Semiconductors – New Bulk Materials and Low-Dimensional Quantum Structures, by W. Dobrowolski, J. Kossut and T. Story . . . . . . . . . . . . . . 289 4. Magnetic Ordering Phenomena and Dynamic Fluctuations in Cuprate Superconductors and Insulating Nickelates, by H.B. Brom and J. Zaanen . . . . . . . . . . . . . . . . . . . . . 379 5. Giant Magnetoimpedance, by M. Knobel, M. Vázquez and L. Kraus . . . . . . . . . . . . 497

Volume 16 1. 2. 3. 4.

Giant Magnetostrictive Materials, by O. Söderberg, A. Sozinov, Y. Ge, S.-P. Hannula and V.K. Lindroos . Micromagnetic Simulation of Magnetic Materials, by D. Suess, J. Fidler and Th. Schrefl . . . . . . Ferrofluids, by S. Odenbach . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic and Electrical Properties of Practical Antiferromagnetic Mn Alloys, by K. Fukamichi, R.Y. Umetsu, A. Sakuma and C. Mitsumata . . . . . . . . . . . . . . . . . . . . . . . . . 5. Synthesis, Properties and Biomedical Applications of Magnetic Nanoparticles, by P. Tartaj, M.P. Morales, S. Veintemillas-Verdaguer, T. Gonzalez-Carreño and C.J. Serna . . . . . . . . . . . . . . .

1 41 127 209 403

Contributors

Gerrit E.W. Bauer Centre for Advanced Study at the Norwegian Academy of Science and Letters, Drammensveien 78, NO-0271 Oslo, Norway; Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands Arne Brataas Centre for Advanced Study at the Norwegian Academy of Science and Letters, Drammensveien 78, NO-0271 Oslo, Norway; Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Ekkes Brück Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis SC, Brazil and Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands M.R.J. Gibbs Sheffield Centre for Advanced Magnetic Materials & Devices, Department of Engineering Materials, University of Sheffield, Sheffield, S1 3JD, UK E.W. Hill School of Computer Science, Information Technology Building, University of Manchester, Oxford Road, Manchester, M13 9PL, UK Gerfried Hilscher Institute for Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria Paul J. Kelly Faculty of Science and Technology and Mesa+ Research Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands M.D. Kuz’min Leibniz-Institut für Festkörper- und Werkstofforschung, Postfach 270116, D-01171 Dresden, Germany H.J.M. Swagten Eindhoven University of Technology, Department of Applied Physics, COBRA

xvi

Contributors

Research Institute and center for NanoMaterials (cNM), P.O. box 513, 5600 MB Eindhoven, The Netherlands A.M. Tishin Department of Physics, M.V. Lomonosov Moscow State University, 119992 Moscow, Russia Yaroslav Tserkovnyak Centre for Advanced Study at the Norwegian Academy of Science and Letters, Drammensveien 78, NO-0271 Oslo, Norway; Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Günter Wiesinger Institute for Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria P. Wright QinetiQ Ltd, Malvern Technology Centre, St Andrews Road, Malvern, WR14 3PS, UK

CHAPTER

ONE

Spin-Dependent Tunneling in Magnetic Junctions H.J.M. Swagten *

Contents 1. Introduction 1.1 From GMR to tunnel magnetoresistance 1.2 Elementary model for tunnel magnetoresistance 1.3 Beyond the elementary approach 1.4 Scope of this review 2. Basis Phenomena in MTJs 2.1 Basic magneto-transport properties 2.2 Oxidation methods for Al2 O3 barriers 2.3 Towards optimized barriers 3. Tunneling Spin Polarization 3.1 How to measure spin polarization? 3.2 Data on tunneling spin polarization 3.3 Ingredients of tunneling spin polarization 4. Crucial Experiments on Spin-Dependent Tunneling 4.1 The relevance of interfaces: using nonmagnetic dusting layers 4.2 Quantum-well oscillations in MTJs 4.3 Role of the ferromagnetic electrode for TMR 4.4 Towards infinite TMR with half-metallic electrodes 4.5 Role of the barrier for TMR 4.6 Coherent tunneling in MgO junctions 5. Outlook Acknowledgements References

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Abstract This chapter reviews the physics of spin-dependent tunneling in magnetic tunnel junctions, i.e. ferromagnetic layers separated by an ultrathin, insulating barrier. In magnetic junctions the tunneling current between the ferromagnetic electrodes depends strongly on an external magnetic field, facilitating a wealth of applications in the field of magnetic *

Eindhoven University of Technology, Department of Applied Physics, COBRA Research Institute and center for NanoMaterials (cNM), P.O. box 513, 5600 MB Eindhoven, The Netherlands E-mail: [email protected]

Handbook of Magnetic Materials, edited by K.H.J. Buschow Volume 17 ISSN 1567-2719 DOI 10.1016/S1567-2719(07)17001-3

© 2008 Elsevier B.V. All rights reserved.

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media and storage. After a short introduction on the background and elementary principles of magnetoresistance and tunneling spin polarization in magnetic tunnel junctions, the basic magnetic and transport phenomena are discussed emphasizing the critical role of the preparation and properties of (mostly Al2 O3 ) tunneling barriers. Next, key ingredients to understand tunneling spin polarization are introduced in relation to experiments using superconducting probe layers. This is followed by discussing a number of crucial results directly addressing the physics of spin tunneling, including the role of the polarization of the ferromagnetic electrodes, the interfaces between barrier and electrodes and quantum-well formation, and the successful use of alternative crystalline barriers such as SrTiO3 and MgO. Key Words: magnetic tunnel junctions, magnetoresistance, spin polarization, spin tunneling, spintronics

1. Introduction This review is focusing on the fundamental aspects of magnetic tunnel junctions or shortly MTJs. It will cover the preparation and experimental aspects of MTJs, and most of the crucial experiments that were performed to unravel their basic physics. In the last section, new promising directions for further research will be reviewed. In this introductory section the following subjects will be covered: • the breakthrough towards magnetoresistance in layered magnetic structures, more specifically in metallic multilayers and subsequently in magnetic tunnel junctions • phenomenology of magnetoresistance in MTJs using the Julliere model, including the concept of so-called tunneling spin polarization • the shortcomings of elementary models via an introduction to some crucial experimental observations and advanced theoretical approaches. It should be noted that several other reviews exist also partially covering the physics and applications of spin-polarized tunneling in tunnel junctions; see Meservey and Tedrow (1994), Moodera et al. (1999a, 2000), Moodera and Mathon (1999), Dennis et al. (2002), Ziese (2002), Maekawa et al. (2002), Miyazaki (2002), Tsymbal et al. (2003), Zhang and Butler (2003), Zutic et al. (2004), Shi (2005), and LeClair et al. (2005). In most cases, however, the focus is different as compared to the present paper, and some recent developments in this rapidly evolving field may not be included. To assist the reader, the last part of this introduction will briefly explain the scope of the present review.

1.1 From GMR to tunnel magnetoresistance Magnetic tunnel junctions are within the florishing field of magnetoelectronics or spin electronics, shortly spintronics. In this area, nanostructured magnetic materials are used for functional devices explicitly exploiting both charge and spin in electron transport, the so-called spin-polarized transport. As we will see later on, magnetic


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junctions are offering several unique opportunities for studying new, sometimes unexpected effects in physics, and, furthermore, they have opened up a number of new research directions within spintronics. Apart from that, magnetic junctions are superb materials for exploring novel device options, such as improved readhead sensors, magnetic memories or magnetic biosensors. Before a more detailed insight in the principles of magnetic tunneling will be given, it is instructive to first shortly review the field of spin-polarized transport and the ongoing increasing role of tunneling transport. In the mid-eighties the first crucial steps are made towards the exploitation of magnetic nanostructures for new electrical effects. These breakthroughs were strongly stimulated by the progress in ultra-high vacuum deposition and characterization techniques, enabling full control of layer-by-layer growth of metallic magnetic (multi-)layers. One of the first intriguing observations by Carcia et al. (1985) is the presence of perpendicular magnetic anisotropy in ultrathin magnetic (multi-)layers due to strong magnetic surface anisotropies, see, e.g., also Parkin (1994) and Johnson et al. (1996). Due to perpendicular anisotropy, the magnetization can be pointing out of the plane of a magnetic thin film, a novel way of engineering the direction of magnetization in ferromagnetic films. To illustrate the technological relevance, this phenomenon is now used in magnetic media to increase the data density as compared to in-plane magnetized (longitudinal) magnetic disks. The subsequent discovery of magnetic interaction across ultrathin nonmagnetic spacers has been critically important for the field of spin-polarized transport. It is shown by Grünberg et al. (1986) that this so-called interlayer coupling may favor an antiparallel, in-plane alignment of two neighboring magnetic layers separated by only a few atomic planes of a nonmagnetic element. It is now well accepted that the driving mechanism for the interaction is spin-dependent electron reflection and transmission at the interfaces between the magnetic and nonmagnetic layers (for a review, see Bürgler et al., 1999). The first observation of remarkable, unexpected electrical effects in these magnetic nanostructures is independently reported by the research groups of Fert and Grünberg (Baibich et al., 1988; Binasch et al., 1989). They have demonstrated that the resistance of a multilayered stack of magnetic layers separated by nonmagnetic spacers strongly depends on the mutual orientation of the layer magnetization. Due to the presence of antiferromagnetic coupling, the magnetization of these layers can be engineered between parallel and anti-parallel via an externally applied magnetic field. The enormous magnitude of the magnetoresistance at room temperature explains the term giant magnetoresistance or GMR used since then. The observation of GMR has initiated an intensive research effort. Fundamentally, the physics of the underlying spin-polarized transport is studied extensively using magnetic engineering tools, novel material combinations, and a variety of theoretical approaches (Coehoorn, 2003). Along with the fundamental interest, the application potential of this effect has been immediately recognized by the magnetic recording industries. As a well-known achievement in this area, the concerted scientific and industrial effort led to the introduction of a GMR read head already in 1997, just nine years after the pioneering, curiosity-driven experiments. A similar strong interplay between scientific discovery and subsequent device im-

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Figure 1.1 The development of room-temperature magnetoresistance in layered magnetic structures. Giant magnetoresistance (GMR) data are restricted to spin valves, where the active part is consisting of two ferromagnetic layers separated by a metallic spacer. The data for tunnel magnetoresistance (TMR) are shown since 1995 for tunneling across Al2 O3 barriers, as well as for MgO, showing a huge rise of TMR in recent years. Note that only a limited number of the available data have been collected in the graph just to give a representative illustration of the developments.

plementation can be observed in the field of magnetic tunnel junctions (MTJs). Although junctions were already studied for a long time (e.g. in the case of one superconducting and one metallic electrode), especially in the beginning of the nineties an increasing number of contributions are devoted to full magnetic junctions with two ferromagnetic electrodes. Although these experiments are certainly inspired by the original work of Julliere (1975) and Maekawa and Gafvert (1982) on Fe-Ge-Co and Ni-NiO-Ni(Co,Fe), respectively, the booming interest for GMR in metallic systems has also fuelled the renewed interest. For some of these pioneering experiments on MTJs in the beginning of the nineties, see Miyazaki et al. (1991), Nowak and Raułuszkiewicz (1992), Suezawa et al. (1992), Yaoi et al. (1993), and Plaskett et al. (1994). The final breakthrough in this field takes place in 1995 when unprecedented large magnetoresistance effects are discovered at room temperature. Moodera et al. (1995) as well as Miyazaki and Tezuka (1995a) are the first to show that a system of two magnetic layers separated by a very thin nonmagnetic oxide layer displays a huge tunnel magnetoresistance or TMR effect, substantially larger than GMR in a similar system with a metal spacer (for a review on exchange-biased spinvalves, see, e.g., Coehoorn, 2003). To illustrate the order of magnitude of GMR versus TMR, Fig. 1.1 shows the chronology of these developments. It is clear from the graph that the TMR data on Al2 O3 -based MTJs have shown a steady increase and are always well above GMR data. In more recent years, the use of MgO as a barrier (as well as other oxide and ferromagnetic material combinations) have un-


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Figure 1.2 The magnetoresistance (a), expressed as V /I , as a function of external magnetic field H at low temperature (T = 10 K) of an evaporated 80 Å Co/14 Å Al + oxidation/150 Å NiFe junction; see the schematics in (b). The arrows indicate the direction of magnetization of the two ferromagnetic electrodes. Antiparallel alignment between the layers is facilitated by different coercivities of the Co and NiFe layer; see also section 2.1.2. From Moodera (1997).

doubtedly demonstrated the record-high magnitude of TMR effects. In comparing these data, one should realize that the physics behind the magnetoresistance in tunnel junctions is completely different from that in all-metallic GMR structures, since quantum-mechanical tunneling is now the fundamental process governing the electrical transport. We will return to that in section 1.2. In Fig. 1.2 an experimental example of tunnel magnetoresistance is shown from the group of Moodera, using two magnetic layers of different coercivity separated by a thin alumina barrier. It clearly demonstrates a large resistance change when the two magnetic layers are switched from a parallel to an anti-parallel orientation by an external magnetic field. The magnetoresistance in MTJ’s can be exploited in a novel solid-state memory. It consists of (sub)micron-sized tunneling elements connected via word and bit lines in a two-dimensional architecture, a similar layout as in macroscopic ferrite core memories invented in the fifties; see Livingston (1997) and references therein. The fact that the electrical current flows perpendicular to the layers in an MTJ (due to the quantum-mechanical tunneling process across the insulator) rather than in the plane of the layers (as in GMR) allows for an efficient use of word and bit lines addressing individual bits. This, together with the huge magnetoresistances of MTJs paved the way to a fast implementation in memory applications. In fact, new nonvolatile solid-state memories based on magnetic tunnel junctions have entered the market in the beginning of the new millennium. In Fig. 1.3a a schematics is shown of one bit cell within a so-called magnetic random access memory or MRAM. It is shown how to use the magnetoresistance effect (as displayed in Fig. 1.2) to store information in a solid-state device. In this system one of the layers, the reference layer, is always pointing in one direction (in Fig. 1.3b to the right), which means that the applied magnetic fields created by the orthogonal word and bit line should never exceed its coercivity. On the other hand, the softer magnetic layer is used to

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Figure 1.3 (a) Schematics of a magnetic tunnel junction incorporated in a single cell of a magnetic random access memory (MRAM). Orthogonal word and bit lines create a magnetic field that is able to set the free layer magnetization direction of the MTJ. Semiconductor (transistor) elements are used as a switch for read-out. In (b) the memory function of an MTJ is illustrated by the magnetoresistance of a Co/Al2 O3 /NiFe junction (see Fig. 1.2 for the full curve of a similar MTJ). The arrows indicate the direction of the in-plane magnetization. To write a “0” or “1”, a magnetic field is applied by the word/bit line that is just large enough to switch the softest (storage) magnetic layer, but small enough not to switch the (magnetically harder) reference layer. To read a bit, the resistance is measured at zero magnetic field.

actually store the information, and is switched by a small magnetic field to create a zero-field state with low or high resistivity, corresponding to a logical “0” or “1”. The reader is referred to Tehrani et al. (2000, 2003), de Boeck et al. (2002), Parkin et al. (2003), DeBrosse et al. (2004), Shi (2005), and references therein, for papers on MRAM technology. Although the magnetoresistance effects in MTJs have been reproducibly reported by many groups, and applications are being developed since then, the fundamental issues in explaining the observed effects are far from fully understood, and need a careful introduction. In the following, it is explained how the existence of TMR can be predicted in the most elementary phenomenological model capturing some of the basic fundamental properties of these devices. This will serve as a starting point for a further exploration of the underlying physics, which is addressed later on in the review.

1.2 Elementary model for tunnel magnetoresistance In elementary textbooks on quantum mechanics, the tunneling current through a potential barrier is extensively treated, illustrating the finite probability for an electron to tunnel through energetically forbidden barriers. Within the WentzelKramers-Brillouin (WKB) approximation, which is valid for potentials U varying slowly on the scale of the electron wavelength, the transmission probability across


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Figure 1.4 The wave function in a metal-oxide-metal tunnel structure schematically shows the concept of quantum-mechanical tunneling for electrons with an energy close to the Fermi energy EF . The barrier height at the interface between metal and oxide is given by φ. A nonzero tunneling current is flowing when a bias voltage V is applied between the metallic electrodes. The grey areas in the metal regions represents the occupied density-of-states; in the barrier the energy gap of the insulator is indicated in white.

a potential barrier is in one dimension proportional to: t 2 T (E) ≈ exp –2 2me [U (x) – E]/h¯ dx

(1)

0

with E the electron energy, me the electron mass, and x the direction perpendicular to the barrier plane. This equation directly shows the well-known exponential dependence of tunnel transmission on the thickness t and energy barrier U (x) – E. Note that the electron momentum in the plane of the layers is assumed to be absent, i.e., k = 0. In fact, when electrons are impinging the barrier under an off-normal angle (k = 0), the tunneling probability rapidly decreases with increasing k since in that case the term 2m[U (x) – E]/h¯ 2 in the exponent of the transmission should be replaced by 2m[U (x) – E]/h¯ 2 + k2 . In an experimental situation, this tunneling process can be measured in a metaloxide-metal structure, a trilayered structure of two metals or electrodes separated by an insulating spacer. The thickness of the spacer is in the order of just 1 nanometer, a few atomic distances, otherwise the exponentially decaying tunneling current (proportional to the transmission in Eq. (1)) becomes immeasurably small. The metal-oxide-metal junction is drawn in Fig. 1.4 where the potential of the barrier U (x) is assumed to be constant across the barrier and located at an energy φ above the Fermi energy EF of the metals. Without a voltage difference between the metals layers, the Fermi levels will be equal on either side of the barrier, and the tunnel current is zero. When a finite bias voltage V is applied, the Fermi level is lowered at the right-hand side of the barrier, and electrons are now able to elastically tunnel from filled electron states (left) towards unoccupied states in the second (right) electrode. Note that in this case the electrode at right is at a higher electrical potential as compared to the left electrode, yielding a net electrical current from right to left. As a result, the amount of current will be proportional to the product of the available, occupied electron states on the left, and the number of empty states at the right electrode, multiplied by the barrier transmission probability. Therefore, the

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tunneling current is directly proportional to the density-of-states of each electrode (at a specific energy E) multiplied by the Fermi–Dirac factors f (E) and 1 – f (E) to account for the amount of occupied and unoccupied electron states, respectively. To analytically calculate the net tunneling current in the metal-oxide-metal structure, we first write the current due to electrons tunneling from left to right assuming an elastic (energy-conserving) electron tunneling process from occupied states on the left to empty states at the right (see the figure): IL→R (E) ∝ NL (E – eV )f (E – eV )T (E, V , φ, t)NR (E)[1 – f (E)].

(2)

As indicated by Eq. (1), the transmission T (E, V , φ, t) depends on the electron energy and barrier thickness and potential, but it is also affected by the bias voltage V that effectively reduces the barrier height φ. For the opposite current we write a similar equation, by which the total current I is obtained by integrating IL→R –IR→L over all energies: +∞ NL (E – eV )T (E, V , φ, t)NR (E)[f (E – eV ) – f (E)] dE. I∝ (3) –∞

For small voltages eV φ only the electrons at (or close to) the Fermi level EF contribute to the tunneling current, by which the transmission no longer depends on energy E. Moreover, in this limit also the density-of-states factors are in principle independent of E, which reduces the current to: +∞ I ∝ NL (EF )NR (EF )T (φ, t) (4) [f (E – eV ) – f (E)] dE. –∞

For low enough temperature (kB T eV ) the integral over the Fermi functions simply yields eV , by which we end up with a transparant expression for the tunnel conductance: G ≡ dI /dV ∝ NL (EF )NR (EF )T (φ, t).

(5)

It shows that in this simple model the tunnel conductance is proportional to the transmission probability and the density-of-states of the two electron systems. The explicit dependence of the density-of-states factors is originally proposed by the pioneering theoretical work of Bardeen (1961), now referred to the transferHamiltonian method (see Wolf, 1985). Note that usually in this method the probability T (φ, t) is written as |M|2 , which is the squared transfer matrix element that determines the tunneling transition rate between an initial and final state. Now we can proceed with evaluating the current in a magnetic junction, that is, two magnetic electrodes separated by a nonmagnetic insulator (see Fig. 1.5). The density-of-states of a ferromagnetic material is represented by a simple majority and minority electron band, shifted in energy due to exchange interactions. First, we consider two identical ferromagnetic electrodes with parallel magnetization orientations, separated by an insulating barrier. Assuming that the electron spin is conserved in these processes (Tedrow and Meservey, 1971a), tunneling may only occur between bands of the same spin orientation in either electrode, i.e., from a spin majority band to a spin majority band, and similar for the minorities. Using Eq. (5) and assuming equal transmission for both spin species, we write the


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Figure 1.5 Spin-resolved tunneling conductivity G for parallel (top panel) and antiparallel magnetization (bottom), as indicated at right, is proportional to the product of the density-of-states factors at the Fermi level EF . The total current in parallel orientation is governed 2 (E ) + N 2 (E ), in the antiparallel case by 2N by Nmaj maj (EF )Nmin (EF ). The voltage that F min F introduces a net tunneling current across the barrier (indicated by the grey bar) is negligible in this schematics.

conductance for parallel magnetization as: 2 2 GP = G↑ + G↓ ∝ Nmaj (EF ) + Nmin (EF ),

(6)

where G↑(↓) is the conductance in the up- (down-) spin channel, and Nmaj (EF ) (Nmin (EF )) is the majority (minority) density-of-states at EF . When we switch the magnetization orientation of one ferromagnetic electrode relative to that of the other ferromagnetic electrode, the axis of spin quantization is also changed in that electrode. Tunneling between like spin orientations now means tunneling from a majority to a minority band, and vice versa. The conductance for antiparallel aligned magnetization is then simply: GAP = G↑ + G↓ ∝ 2Nmaj (EF )Nmin (EF ).

(7)

It is immediately clear that conductances are different for parallel and antiparallel magnetizations. In other words, ferromagnetic tunnel junctions display a magnetoresistance when an external field is used to switch between these magnetic orientations. This tunnel magnetoresistance (TMR) is usually defined as the difference in conductance between parallel and antiparallel magnetizations, normalized by the antiparallel conductance, or, alternatively, as the resistance change normalized by the parallel resistance: GP – GAP RAP – RP (8) = . GAP RP Note that the equality of the two definitions for TMR is only valid for very small bias voltage, since in that case the inverse tunnel resistance R –1 = I /V is identiTMR ≡

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cal to the conductance dI /dV . In literature on MTJs, another, more pessimistic definition of TMR is used as well, normalizing the resistance change by the resistance in antiparallel instead of parallel orientation. However, throughout the review, Eq. (8) will be strictly applied to quantify the magnetoresistance ratio in magnetic junctions. Using Eqs. (6) and (7), it is easily derived that TMR is equal to [Nmaj (EF ) – Nmin (EF )]2 /[2Nmaj (EF )Nmin (EF )]. We can generalize this for two different magnetic electrodes, resulting in the well-known Julliere-formula for the magnetoresistance of MTJ’s (Julliere, 1975): 2PL PR , (9) 1 – PL PR where PL(R) is the tunneling spin polarization in the left (right) ferromagnetic electrode. The tunneling spin polarization of each electrode is defined as TMR =

P =

Nmaj (EF ) – Nmin (EF ) , Nmaj (EF ) + Nmin (EF )

(10)

and is simply the normalized difference in majority and minority density-of-states at the Fermi level. From these equations it is immediately seen that in the limit of zero polarization of one of the electrodes, no TMR is expected. On the other hand, for a full polarization of ±1, the TMR becomes infinitely high. These fully polarized materials (one spin channel is absent at the Fermi level) are referred to as being half-metallic, and have been intensively investigated in this field; see also section 4.4. In an experimental study, Julliere (1975) is the first to use Eqs. (9) and (10) for TMR in Fe-Ge-Co junctions, although in principal with a different interpretation of tunneling spin polarization. N(EF ) is defined as an effective number of tunneling electrons to stress the fact that the tunneling process is not only governed by the (static) density-of-states at EF . We will return to this crucial point later on. Nevertheless, it should be emphasized that the Julliere equation in its simplest form demonstrates the fundamental role of the tunneling spin polarization of the ferromagnetic electrode in understanding the observed TMR in magnetic junctions. The tunneling spin polarization of individual magnetic electrodes can be measured with a so-called superconducting tunneling spectroscopy (STS) technique that uses a superconductor (in most cases Al) to probe the spin imbalance in tunneling currents. In more detail, in a ferromagnetic-Al2 O3 -Al junction a magnetic field splits up the sharply-peaked density-of-states of the superconducting Al electrode, which leads to an asymmetry in the conductance G(V ) that reflects the amount of spin polarization. In section 3 this will be further introduced, here only a numerical example will be given. The tunneling spin polarization for Co is experimentally determined to be around +0.42, which via Eq. (9) corresponds to a TMR effect of more than 40% for Co-Al2 O3 -Co MTJs. This is only slightly above the observed (low-temperature) value. For the moment, it seems that we can use this formula as a phenomenological equation that nicely connects tunneling polarization P to the magnitude of the magnetoresistance. However, as we will see below, the physics of spin-polarized tunneling is much more complex and needs a dramatic reconsideration of these phenomena.


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1.3 Beyond the elementary approach Although the model we have introduced captures some of the basic physics in magnetic tunnel junctions and is rather illustrative on a tutorial level, it fails to predict a number of experimental observations. These observations for TMR include, for instance: • strong dependence of TMR on the applied bias voltage V and temperature T • sensitivity of TMR on the electronic structure of the barrier-ferromagnetic interface region, not just the bulk density-of-states (as suggested by Eqs. (9) and (10)) • relevance of the electronic structure of the barrier, in some cases even leading to an inversion of TMR. Here we will briefly introduce some of the advanced theories to better appreciate these observations, focusing at this point on the tunneling spin polarization for its fundamental role in the physics of magnetic tunnel junctions. A more detailed treatment will be postponed for sections 3 and 4. Later on in this review (Table 1.2 in section 3) we will show that the tunneling spin polarization of the 3d ferromagnetic metals are all positive, and in the range of 40–60%. According to the definition of Eq. (10), the positive sign of the polarization relates to a dominant majority density-of-states at the Fermi level. If one considers the band structure and density-of-states of the 3d metals, however, the situation is completely reversed. As an example, Fig. 1.6 shows the (calculated) density-of-states of Co and Ni, both having a surplus of minority states of the Fermi level. This would suggest a negative tunneling spin polarization, and completely contradicts the experimental observations. This dichotomy was recognized already in the seventies when pioneering experiments in the field of superconducting tunneling spectroscopy were reported on ferromagnetic-superconducting junctions (Tedrow and Meservey, 1971a, 1971b, 1975). Theoretically, Stearns (1977) has shown that the conductance in a tunnel junction is not simply determined by the electron density-of-states at the Fermi level, but should include the probability for them to tunnel across an ultrathin barrier. Especially the most mobile s-like electron states are able to tunnel with a much larger probability as compared to the d electrons due to their different effective mass. Based on this, Stearns could explain the positive spin polarization by considering the spin asymmetry of the s-like energy

Figure 1.6 Density-of-states of the elemental metals fcc Cu (a), fcc Ni (b), and hcp Co (c), obtained from self-consistent band-structure calculations using the Augmented Spherical Wave (ASW) method. From Coehoorn (2000).

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bands, thereby neglecting the contribution from the rapidly decaying d-like wave functions in tunneling experiments. More recently, another advanced aspect of spin-polarized tunneling is reported. Slonczewski (1989) emphasizes that spin-dependent tunneling is not a process solely related to the (complex) electronic properties of the ferromagnetic electrodes. He has analytically calculated the tunneling current between free-electron ferromagnetic metals within the WKB approximation (see Eq. (1)), assuming that tunneling electrons have a very small parallel wave vector, close to k = 0. By explicitly matching the electron wave functions at the barrier interfaces, the tunneling spin polarization is calculated as: P = P0 ×

κ 2 – kF ,maj kF ,min , κ 2 + kF ,maj kF ,min

(11)

where kF ,maj and kF ,min are the Fermi wave vectors, and κ the imaginary component of the wave vector of electrons in the barrier with k = 0 at the Fermi level, corresponding to κ = (2me φ /h¯ 2 )1/2 with φ the height of the barrier. The first term P0 is equal to the earlier result in Eq. (10). The second term, however, contains the properties of the barrier as well, and is due to the discontinuous change of the potential at the interface with the barrier. As a result of this interface factor, the polarization becomes greatly dependent on the band parameters in relation to the height of the barrier, with the possibility to even change the sign of P . This is in fact a first demonstration that tunneling spin polarization is not an intrinsic property solely determined by the ferromagnetic electrode. A similar conclusion is reached in free-electron calculations where the conductance is analytically obtained by matching the freeelectron wave functions (and its derivatives) at the two interfaces (MacLaren et al., 1997). In this free-electron calculation, also electrons with k = 0 are considered, although k is assumed to be strictly conserved upon tunneling. In Fig. 1.7 the freeelectron magnetoresistance calculated by Slonczewski (1989) and MacLaren et al. (1997) is plotted as a function of polarization P = (kF ,maj – kF ,min )/(kF ,maj + kF ,min ), which is equivalent to P0 in Eq. (11). For thick barriers, the solutions in the calculation of MacLaren et al. (1997) approach the model of Slonczewksi based on the WKB approximation, whereas no correspondence is found with the Julliere expression. However, it should be stressed that the predictability of this elementary, simplified free-electron model is rather poor. As already pointed out by Harrison (1961), this is related to the suspicious absence of density-of-states factors in the transport characteristics. MacLaren et al. (1997) and Zhang and Levy (1999) emphasize that, generally, these free-electron calculations (including the Julliere model) fail to predict the observed magnetoresistance behavior in magnetic junctions, and its dependencies on, e.g., barrier thickness, barrier height, and bias voltage. Nevertheless, there are some attempts to directly use Slonczewski’s or other free-electron calculations to investigate how TMR behaves as a function of the model parameters. For an example, see the work of Tezuka and Miyazaki (1998) on the variation of TMR with the Al2 O3 barrier height. After the work of Slonczewski (and free-electron calculations by others), a great number of advanced theoretical investigations have been published to further explore the physics of TMR and tunneling spin polarization; see for example the


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Figure 1.7 (a) Calculations of the magnetoresistance (RAP – RP )/RAP as a function of the tunneling spin polarization P = (kF ,maj – kF ,min )/(kF ,maj + kF ,min ). The Julliere curve is based on Eq. (9) although using the pessimistic definition of TMR = 2P 2 /(1 + P 2 ). Calculations within the model of Slonczewski are performed for a barrier height φ of 3 eV. In the free-electron calculations (labelled MacLaren), the thickness of the barrier t is 5 Å, 20 Å, and 200 Å (same barrier height). The inset schematically shows k conservation used in the free-electron model. (b) Energy versus density-of-states used in free-electron calculations, showing the parabolic bands for majority and minority electrons on each side of the insulating barrier. EF is the Fermi level. In the calculations the bias voltage is assumed to be small, eV φ. Adapted from MacLaren et al. (1997).

review paper of Zhang and Butler (2003). Along with that, experimental evidence has become gradually available that shows, e.g., the decisive role of the barrierelectrode combination for spin-polarized tunneling. Other exciting observations have been reported, such as the role of crystallinity and orientation of the magnetic electrode, oscillations in TMR due to the presence of nonmagnetic layers favoring quantum well states, and unprecedented, giant TMR in junctions when incorporating half-metallic electrodes or, more recently, crystalline MgO barriers. Especially in sections 3 and 4 these developments will be extensively addressed.

1.4 Scope of this review It is the purpose of this review to introduce the reader to the most important aspects of spin-polarized tunneling. We have just seen that spin polarization in MTJs is a complex parameter heavily dependent on the details of the potential the electrons experience when crossing the barrier region. This in turn strongly influences the fabrication process of MTJs, where obviously utmost care should be taken in designing and characterizing the barrier and the interface regions with the ferromagnetic metals. Barriers in MTJs are traditionally made out of oxidized Al for their relative ease to create superior coverage of the metallic electrode, together with the observation of large magnetoresistances. A huge research effort could be witnessed in the late nineties to optimize the oxidation process for enhancing the functionality and reliability of MTJs. Furthermore, a number of oxidation methods have been explored in great detail, in particular the use of an oxygen plasma to gradually

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oxidize a previously deposited Al layer. The physical properties and optimization of the barrier and adjacent ferromagnetic layers are the topic of section 2. Also in this section the basic design rules for a magnetic junction will be discussed together with elementary transport properties, such as the dependence of TMR on bias voltage, barrier thickness, and temperature. In section 3 we return to the physics of tunneling spin polarization. Details will be given on the experimental method involving superconducting probe layers, followed by a more in-depth discussion on the basic fundamental ingredients. Topics of interest are the relation between tunneling spin polarization and the ferromagnetic magnetization, the relevance of the barrier-electrode interface region including the local chemical bonding, and the relevance of the symmetry of the wave functions of tunneling electrons. Section 4 reviews a number of crucial experiments in the field of TMR in magnetic junctions. Especially those topics will be highlighted that have contributed to the understanding of the underlying physics of spin tunneling, e.g., addressing the role of the interfaces with the barrier, the (local) density-of-states of the magnetic layers, half-metallic or epitaxial ferromagnetic electrodes, and tunneling across crystalline barriers (such as MgO or SrTiO3 ). The review will be concluded by briefly considering some of the promising directions within this field of magnetic junctions or, in a wider perspective, the field of hybrid devices where tunnel barriers are often combined with new materials to create new physics or functionality. This includes, for example, the development of allsemiconductor MTJs, the use of magnetic semiconductors as (spin-filter) barriers, and the realization of three-terminal magnetic tunnel transistors.

2. Basis Phenomena in MTJs The fabrication of a properly operating insulating tunnel barrier, separating the magnetic electrodes, has developed as a wide and very active research field where many aspects on oxide growth, characterization, magnetism, and transport are being considered. Although there are several ways to fabricate barriers for MTJs, a clear distinction can be noticed between crystalline and amorphous barriers. The amorphous Al2 O3 barriers are most extensively studied due to the ability to serve as an excellent barrier with a sufficiently small density of pinholes (i.e., electrical shorts between top and bottom metallic electrode). Usually, alumina barriers are created by depositing a thin Al layer that is subsequently oxidized by thermal (natural) or plasma-enhanced oxidation. Figure 1.8 shows a prototypical example of the magnetic-field dependence of the resistance in a magnetic junction consisting basically of FeMn-Co-Al2 O3 -Co, with the alumina barrier formed by plasma-oxidizing an Al layer. The tunneling resistance or current across the Al2 O3 barrier is measured in the so-called 4-point geometry by contacting the bottom and top electrodes as indicated in Fig. 1.8c. Recently, there is an increasing amount of studies focusing on junctions with crystalline or even epitaxial barriers, such as the widely investigated MgO and SrTiO3 . In some cases, this yields magnetoresistance ratios superior to those with alumina barriers with the added advantage to be able to accurately


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Figure 1.8 In (a) the room-temperature magnetoresistance is shown of an 0.4 × 0.4 mm MTJ fabricated with UHV magnetron sputtering through metal shadow masks. The arrows indicate the orientation of magnetization. The structure is schematically shown in (b): Si(100)/SiO2 /50 Å Ta/50 Å Co/100 Å FeMn/35 Å Co/23 Å Al + oxidation/150 Å Co/50 Å Ta. The top-view layout of these junctions in (c) indicates the actual 4-point geometry for the resistance measurement. After LeClair (2002).

model the transport processes in these better defined systems. The discussion on other, crystalline barriers materials will be postponed for section 4. On the other hand, Al2 O3 -based junctions are a perfect playground to address a great number of basic physics in magnetic junctions, let alone the huge interest from industrial labs for the incorporation of these barriers in MTJ-based sensors and magnetic memories (see, e.g., Parkin et al., 2003). In this section, a number of basic phenomena in Al2 O3 -based MTJs will be reviewed, including the most relevant fabrication and characterization tools. The topics are: • basic properties of MTJs, emphasizing general tunneling transport characteristics, methods for switching the magnetization in junctions, and the basic behavior of TMR • oxidation of ultrathin metal layers, such as plasma and natural oxidation, in relation to the performance of TMR devices • optimizing barriers for TMR: under- and over-oxidation, pinholes, dielectric breakdown, thermal stability, and alternative amorphous barriers. In somewhat more detail, the first part (section 2.1) introduces the basic voltage dependence of tunneling current in relation to the thickness and electron potential of the insulating barrier, supplemented with a few experimental examples. We will also shortly focus on the magnetization reversal of the magnetic layers, aiming at the realization of two macroscopic, magnetically stable states of the ferromagnetic layers: antiparallel versus parallel. As we have seen in section 1, in these two states the total tunneling current (sum of spin-up and spin-down current) is essentially different in a magnetic junction. In the example of Fig. 1.8 the magnitude of the resistance change is more than 25% (using [RAP – RP ]/RP , Eq. (8)) when switching

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from the parallel to the antiparallel state, which is accomplished by using so-called exchange biasing, one of the most widespread magnetic engineering tools. The basic behavior of the magnitude of the magnetoresistance effect will be discussed next in section 2.1, for instance focusing on how TMR depends on oxide thickness, temperature, and bias voltage. The second part of this section is devoted to the oxide layer that is sandwiched between the ferromagnetic layers (section 2.2). In view of the fact that the current is exponentially dependent on thickness and height of the barrier (apart from many other details), the preparation and characterization of the oxide layers is the most critical step in the junction fabrication. The available oxidation procedures will be reviewed mainly in relation to the magnitude of TMR and the resistance R of a magnetic tunnel junction, both rather crucial when assessing device applications for MTJs (see for instance the introduction on MRAM in section 1). This is followed in section 2.3 by considering a number of key issues in this area, including overand underoxidation of ultrathin Al layers, the role of metallic shorts or pinholes and dielectric breakdown when the barrier becomes extremely thin, thermal stability of MTJs for processing or operation at elevated T , and the use of alternative barriers to further tune the device (magneto)resistance. Finally, it is worth mentioning that the use of a great number of experimental tools will be discussed in this section (in particular in section 2.3.1), such as X-ray photoelectron spectroscopy (XPS), Rutherford backscattering spectroscopy (RBS), transmission electron microscopy (TEM), ballistic electron emission microscopy (BEEM), and optical or ellipsometric characterization. All these tools have added considerably to the understanding of how physical or chemical properties of the barrier are related to the tunneling transport.

2.1 Basic magneto-transport properties This section reviews the basic experimental observations in electrical transport and magnetic behavior of magnetic tunnel junctions, and is aiming at explanations mostly on a phenomenological level. These observations can be summarized as follows: • tunneling current I is nonlinear in applied bias voltage V • conductance dI /dV is approximately parabolic in voltage V , expect for small bias • resistance R at low bias scales inversely with junction area A, and grows exponentially with barrier thickness • magnetization M of two ferromagnetic layers adjacent to the barrier can be switched independently by several magnetic engineering methods • TMR is rather independent of barrier thickness t, except for extremely thin barriers • TMR decays with temperature T and with applied bias voltage V . As mentioned before, more advanced approaches to address the underlying mechanisms for spin-polarized tunneling will be reviewed in section 3 and section 4.


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2.1.1 Tunneling transport in junctions It is shown in Eqs. (3)–(5) that a net tunneling current is induced across a tunnel junction when applying a finite bias voltage between the ferromagnetic electrodes of an MTJ. A straightforward I (V ) measurement is a useful tool to directly assess the existence and properties of the tunneling barrier. The Ohmic behavior as derived in Eqs. (3)–(5) is only valid for small applied bias voltage, and should be reconsidered for higher voltages where the I (V ) curve becomes essentially non-linear. For symmetric tunnel junctions with identical electrodes, Simmons (1963) has analytically calculated the tunneling current using the WKB approximation (see Eq. (2)) which is valid for thick and high barriers: αA eV eV I (V ) = 2 φ – exp –βt φ – t 2 2 αA eV eV – 2 φ+ (12) exp –βt φ + , t 2 2 with, α = e/(2πh), β = 4π 2m∗e /h (m∗e the effective electron mass in the barrier conduction band), V the applied voltage, t the barrier thickness, A the barrier area,

t and φ the average barrier height above the Fermi level 0 [V (x) – EF ] dx /t. Here we neglect the effect of the image charges on the shape of the barrier potential (Simmons, 1963), which, due to the tendency to round off the potential at the outer edges of the insulator, leads to an increase of tunneling current; see Hirai et al. (2002) for data on MTJs. The Simmons equation is later adapted by Brinkman et al. (1970) to include an asymmetry in the barrier potential, with φ the potential difference between right and left electrode. Generally speaking, the potentials the electrons experience when transported across a junction is not automatically symmetric in space. First of all, when employing two different metallic electrodes, their nonequal work functions will create an electrical field across the barrier, leading to an intrinsically tilted barrier potential. Apart from that, the barrier itself is often intrinsically asymmetric related to the preparation. For instance, when oxidizing an Al thin film by a post-growth oxidation process, the stoichiometry of the oxide may vary in the direction perpendicular to the layer planes due to over- or under-oxidation. Moreover, this oxidation procedure may create different interfaces with the electrodes, by which, even when using the same electrode materials, the asymmetry almost naturally arises. We will come back to this in section 2.3. The tunneling current for asymmetric barriers is approximated by a Taylor expansion to the third power (Brinkman et al., 1970): 2 2 2 βe tφ β e t –1 2 V3 . V + I = R0 V – (13) 3/2 48 φ 96 φ R0 is the Ohmic low-bias resistance of the junction given by: 2t exp(βt φ) R0 = , eAαβ φ

(14)

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which, as expected, scales inversely with area, and rapidly grows with thickness and height of the barrier. In the case of φ = 0, the current is cubic in applied voltage V , equivalent to a parabolic conductance, one of the basic properties of transport across tunneling barriers: 2 2 2 1 dI β e t V 2. = G≡ (15) + dV R0 32R0 φ The quadratic increase in conductance is in principle valid only for small V and simply reflects the fact that the effective barrier height becomes smaller when a voltage is applied across the junction. At higher voltages, however, higher order terms in Eq. (13) have to be included, and eventually at energies eV exceeding the barrier height, also the effect of a reduction of the effective barrier width (Rottlander et al., 2002). In experimental studies on MTJs, these formulas given by Simmons (1963) and Brinkman et al. (1970) have been extensively used to characterize the barrier characteristics, viz. the barrier height, including its asymmetry, and the thickness of the barrier. It should be kept in mind, however, that these formulas are based on free-electron-like calculations using single parabolic bands for the metallic electrodes. This means that the spin-dependence of the density-of-states of the magnetic electrodes is not explicitly incorporated, which is clear from the absence of these density-of-states factors in Eqs. (12)–(15). One can show that this is related to the fact that the group velocity of electrons at EF (which determines the rate of attempts to penetrate the barrier) decreases inversely proportional to the density-of-states at the Fermi level; see also the discussion by Harrison (1961). The widespread use of these equations can be explained by the possibility to at least compare the barrier parameters of junctions grown in different laboratories, and offers a first-order indication of the quality of the tunneling transport of an MTJ. One example out of the rich existing literature is given in Fig. 1.9a, showing the predicted parabolic conductance, in this case of a CoFe-Al2 O3 -CoFe junction (Oliver and Nowak, 2004). At low bias an additional anomalous conductance is observed especially at low temperatures, which we will discuss further in section 2.1.4. The extracted barrier thickness and barrier height are shown in Figs. 1.9b and 1.9c, respectively. The different parameters in parallel and anti-parallel case are directly related to the presence of spin-dependent tunneling, since, as we argued before, the Simmons or Brinkman equations do not contain density-of-states factors, and the conductance is entirely determined by t and φ. The deviations in t and φ when temperature is beyond 200 K are indicative for the presence of additional conduction processes, such as an inelastic, spin-independent hopping conductance that is dependent on both voltage and temperature (Oliver and Nowak, 2004). In another case (Dorneles et al., 2003), the fitted barrier thickness and area of a Al-Al2 O3 -Al junction is found to deviate considerably from the actual nominal values (e.g. obtained from X-ray diffraction or transmission electron microscopy). This hints to a tunneling process governed predominantly by so-called hot spots, small areas where the barrier thickness or barrier height is effectively much smaller than for the remaining part of the junction. Due to the exponential growth of tunnel resistance with t (see Eq. (14)), the slightest corrugation at the barrier-electrode interfaces


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Figure 1.9 (a) Parallel and anti-parallel conductance G = dI /dV at T = 5 K of a junction consisting of 50 Å Ta/250 Å PtMn/22 Å CoFe/9 Å Ru/22 Å CoFe/5 Å Al + oxidation/10 Å CoFe/25 Å NiFe/150 Å Ta. The solid lines are parabolic fits using the Brinkman expression (Eq. (13)). From these fits the temperature dependence is extracted of (b) the barrier thickness and (c) the average barrier height. After Oliver and Nowak (2004).

leads to lateral fluctuations in the barrier thickness, by which the current will be almost completely dominated by these hot spots. It was shown theoretically by Bardou (1997) that even when a barrier is controlled in the Ångstrom regime the tunneling transport can be governed by just a few probable paths due to statistical fluctuations. Moreover, when metallic shorts (pinholes) are present in junctions with extremely thin oxides, the barrier parameters are further obscured by a parallel metallic-like current shunting the true tunneling processes (Akerman et al., 2001). In that case extracting parameters by fitting to Eqs. (12) or (13) is clearly losing its physical significance (see also section 2.3.3). A clear demonstration of the ambiguities involved in extracting barrier parameters is facilitated by internal photoemission studies, from which the barrier height in thin-film tunneling structures can be adequately extracted. For early experiments in this direction see, e.g., Kadlec and Gundlach (1976), Nelson and Anderson (1966), and Crowell et al. (1962). Conceptually, the technique is rather straightforward, see Fig. 1.10b. One shines monochromatic light onto a junction structure, and measures the resulting photocurrent. The incident photons will excite electrons in the electrodes, gaining an amount of energy equal to the photon energy. When electrons are photo-excited to an energy higher than the internal barrier height φ, some of the electrons will be able to enter the conduction band of the insulator. After leaving the barrier at the other side, they will be responsible for a net photocurrent when the opposite contributions from the two electrodes do not cancel. From the onset of this current as a function of the photon energy, the barrier height can be accurately determined (shown in Fig. 1.10a), and is strongly deviating from the barrier potential as derived from fits to the Brinkman equation (Koller et al., 2003). Lateral fluctuations of the tunneling current can be adequately addressed by scanning probe microscopies. Costa et al. (1998) are the first to use the atomic force microscope with a conducting tip in contact with a naturally oxidized epi-

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Figure 1.10 (a) Photoconductance as a function of photon energy for a structure of glass/35 Å Ta/30 Å NiFe/100 Å IrMn/25 Å NiFe/15 Å CoFe/17 Å Al + oxidation/40 Å CoFe/100 Å NiFe/35 Å Ta, plasma oxidizing the Al for 200 sec. The light is incident on the top electrode; no additional bias voltage is applied. The inset shows the (average) barrier height as extracted from fitting to the Brinkman equation as well as from photoconductance, as a function of oxidation time. (b) Schematics of photocurrent generation, showing the energy across a tunnel junction. Electron excitation by light is indicated with hν. EF is the Fermi energy, φL,R is the barrier height for the left and right electrode. Adapted from Koller (2004).

taxial Co layer to map the strong fluctuations in tunneling current. Ando et al. (1999, 2000a) have used a more realistic junction without top electrode (Ta-NiFeIrMn-Co-Al2 O3 ), from which a wide distribution in barrier height can be directly determined, favoring tunneling only from a few hot spots in the barrier. In the atomic-force-microscope studies of Luo et al. (2001) on Co-Al2 O3 , the observed current fluctuations are attributed to thickness inhomogeneities on a nanometer scale. In a more advanced approach using ballistic electron emission microscopy or BEEM (Kaiser and Bell, 1988), the Al2 O3 barrier height can be directly measured on a local scale. Electrons emitted from a conductive tip are injected into the metal–insulator–metal system at variable energy as determined by the voltage between tip and surface. These injected, hot electrons can only pass the Al2 O3 barrier potential when their energy exceeds the barrier height (Rippard et al., 2001; Kurnosikov et al., 2002). Rippard et al. (2001) use structures containing Al2 O3 grown on top of Si substrates to create a well-defined Schottky barrier of typically 0.8 eV for selecting the hot electrons. In this work, the alumina barrier height (around 1.22 eV), is found to be rather independent of the deposition method (sputtering versus evaporation), the nominal Al thickness, and the oxidation conditions. In Fig. 1.11, BEEM images directly demonstrate that local barrier height fluctuations emerge upon thinning down of the Al thickness from 6.5 Å to around 4.5 Å (before oxidation). Note that BEEM is only sensitive to the local height of the barrier potential; lateral variations in the barrier thickness are not resolved. In follow-up studies, Rippard et al. (2002) combined BEEM with scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS). They have demonstrated that due to variations in the local atomic structure of ultrathin


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Figure 1.11 Ballistic electron emission microscopy (BEEM) image (a) of an evaporated junction consisting of Si(111)/75 Å Au/12 Å Co/6–7 Å Al + oxidation/12 Å Co/30 Å Cu. The grey scale is proportional to the collected current of hot electrons. (b) The BEEM current for a thinner barrier (4–5 Å Al + oxidation) shows much stronger variations due to an increase of barrier height fluctuations. Adapted from Rippard et al. (2001).

barriers, low-energy extended electron states may support conduction channels at energies below the alumina barrier height. This contradicts the common belief that for ultrathin barriers only metallic pinholes are an important issue for the collapse of spin-polarized transport properties (see section 2.3.3 for more details on the effects of pinholes). In another combined BEEM-STM study, Perrella et al. (2002) have found that mobile O–2 adsorbates are present on the surface of an oxidized Al layer, having localized energy states located 1–2 eV above the Fermi level. By thermal annealing (or by electron bombardment) it is possible to drive the adsorbates into the oxide thereby reducing the local transport via these low-energy channels (see Mather et al., 2005, and also the X-ray photoelectron spectroscopy results of Tan et al., 2005). This may be important for the fabrication of high-quality MTJs, since a thermal treatment of a full junction with two electrodes could homogenize the chemisorbed oxygen that is trapped close to the interface with the oxide layer. As a consequence, this could increase the effective barrier height and reduce the (unde-

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Figure 1.12 Junction resistance R versus the area A. (a) Results on Si(100)/200 Å Pt/40 Å NiFe/100 Å FeMn/80 Å NiFe/10–30 Å Al + oxidation/80 Å Co/200 Å Pt structured with e-beam and optical lithography. Adapted from Gallagher et al. (1997). (b) Structured junctions of glass/10 Å Si/100 Å Co/10–14 Å Al+oxidation/170 Å NiFe/40 Å Al by optical lithography and with shadow evaporation (Boeve et al. (1998)).

sired) oxidation of the top electrode. For optimization studies on MTJs, including the effects of over-oxidation and annealing; see section 2.3. We now return again to the Simmons and Brinkman equations (12)–(15), which also show that the current I in an MTJ is, obviously, linearly scaling with lateral area A of a junction. In other words, the resistance should be inversely proportional to the junction area. This is successfully tested by Gallagher et al. (1997) and Boeve et al. (1998) for Ni80 Fe20 -Al2 O3 -Co and Co-Al2 O3 -Ni80 Fe20 junctions, by varying the junction area over up to 5 orders of magnitude using micro-fabrication with ebeam or optical lithography, or during evaporation with the help of shadow masks. The area scaling is illustrated in Fig. 1.12. As a natural consequence of this, the resistance-area product R × A can be considered as an area-independent property of a magnetic junction, by which different junctions (from various laboratories) can be compared; see again Eq. (14). In the application of MTJ’s this product of resistance and area plays a crucial role, since it determines the resistance noise of the device. When devices are progressively reduced in lateral dimensions (smaller A), the resistance √ will naturally rise as well as the thermal or Johnson noise that is proportional to RT . Results on noise characterization in MTJ devices, including the role of low-frequency 1/f noise and the relation to the magnetic switching can be found in a number of publications; see Nowak et al. (1999); Ingvarsson et al. (1999, 2000); Smits (2001); Nazarov et al. (2002); Park et al. (2003); Jiang et al. (2004a). Apart from the noise issue in devices, it is also important that the read-out speed of a memory or sensor (as determined by the RC time) does not increase due to a further reduction of the junction area A (Tehrani et al., 2003; Das, 2003). Furthermore, for an optimal read-out of an MRAM cell, the resistance of the MTJ should match with the underlying transistor (see Fig. 1.3a). For future CMOS technology nodes, the required reduction of R × A is roughly scaling with the typical feature size within CMOS (Das, 2003). These considerations explain


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Figure 1.13 (a) Resistance times junction area R × A and (b) TMR as a function of nominal Al thickness. The tunnel junctions consist of 90 Å Ta/70 Å NiFe/40 Å CoFe/t Al + oxidation/30 Å CoFe/250 Å IrMn/30 Å Ta, in which the Al layer is optimally oxidized by a remote oxygen plasma. The junctions are patterned down to 1 × 2 µm. After de Freitas (2001).

the huge effort in reducing R × A, e.g. by either reducing the barrier width, or by exploring alternative (energetically lower) barriers; see section 2.3.5 and section 4. Another implication of the Simmons and Brinkman equation is the fundamental exponential decay of the current (or, equivalently, exponential growth in resistance) with the thickness of the tunneling barrier (Eq. (14)). This is experimentally demonstrated in Fig. 1.13a, where R × A is plotted against the barrier thickness for a large number of CoFe-Al2 O3 -CoFe junctions (de Freitas, 2001). Note that these junctions have also been annealed after the deposition process, which, however, does not considerable affect the junction resistance. The main purpose of the post-deposition anneal step is to enhance the TMR as seen in Fig. 1.13b. We will come back to this later on, see section 2.3.4. 2.1.2 Engineering and switching the magnetic constituents The central part of an MTJ is a sandwich of two ferromagnetic layers, separated by a barrier with a thickness usually below 20–30 Å. The ferromagnetic layers adjacent to the insulating barrier are typically a few nanometer in thickness, and should be backed with one or more layers to manipulate the magnetic switching. This is necessary to switch the magnetic orientation from a parallel state of the two magnetization vectors to an antiparallel state, which is the basic requirement to observe tunnel magnetoresistance in a ferromagnetic-insulating-ferromagnetic junction (see section 1.2). Creating an antiparallel magnetization state can be realized in several ways, as shown schematically in Fig. 1.14 for three important magnetic engineering schemes. The most straightforward realization is the use of two ferromagnetic materials having different magnetic anisotropy, of which an experimental example has been shown earlier in Fig. 1.2. The magnetization will be antiparallel in a field

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Figure 1.14 Magnetic engineering in magnetic tunnel junctions. In (a) two layers are used with different coercivities HC . Using an antiferromagnetic layer in (b) creates a wide range of antiparallel orientation governed by Hex . In (c) exchange biasing is combined with antiferromagnetic coupling across a metallic spacer to further improve the field range of antiparallel orientation, together with the magnetic and thermal stability. Note that the schematic behavior of M is shown over a much wider field range as compared to (a) and (b), to fully show the decoupling of the artificial antiferromagnet governed by the antiferromagnetic coupling strength JAF .

range HC1 < H < HC2 with HC1,2 the coercivities of the soft and hard magnetic layer, respectively (Fig. 1.14a). A serious drawback of this engineering scheme has been reported by Gider et al. (1999). When the magnetization of the softest magnetic layer is repeatedly reversed by magnetic field cycling, the other, magnetically harder layer is progressively demagnetized, equivalent to erasing the MTJ memory when used in an MRAM. Using Lorentz electron microscopy and micromagnetic simulations, the hard-layer magnetization decay is found to result from large fringe fields surrounding magnetic domain walls in the magnetically soft layer (McCartney et al., 1999). To avoid domain-wall formation and motion, the soft layer can be reversed by coherent rotation (Gider et al., 1999), which, in an MRAM architecture, can be simulated by subsequent switching with two current pulses from two orthogonal conduction lines below and above the junction cell (Schmalhorst et al., 2000a). In another study on Co80 Fe20 -Co-Al2 O3 -Ni80 Fe20 junctions, magnetic interactions between domains in the soft and hard magnetic lead to the effect of domain duplication, which in turn affects the magnetoresistance in these MTJs (Rottlander et al., 2004). Generally, however, in most cases the use of an antiferromagnet in direct exchange contact with one of the ferromagnetic layers is preferred above the hard-soft system. Due to unidirectional anisotropy induced by the antiferromagnetic layer, the hysteresis loop of the exchange-biased (pinned) magnetic layer will be shifted in field with respect to the free magnetic layer. This naturally creates an antiparallel field range between 0 < H < Hex , with Hex the strength of the exchange bias field


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(neglecting the coercivities of the two layers); see Fig. 1.14b. In Fig. 1.8 an example is given of an elementary exchange-biased MTJ, consisting of a stack of the following sequence: SiO2 -Ta-Co-Fe50 Mn50 -Co-Al2 O3 -Co-Ta. The Ta-Co layers grown directly on top of the substrate reduce the roughness and provide a proper (111) texture for the FeMn layer, by which an exchange bias field of more than 10 kA/m is established in this case. Usually these systems are additionally heated above the blocking temperature of the antiferromagnet, and subsequently cooled in the presence of an external magnetic field to enhance the interface interactions between ferro- and antiferromagnet. Other antiferromagnetic layers such as metallic PtMn or IrMn compounds are frequently used as exchange-biasing materials, in particular for their better thermal stability (higher blocking temperature). Also insulating, antiferromagnetic NiO films can be applied to exchange-bias one of the ferromagnetic electrodes (Shang et al., 1998a), in this case grown by reactive evaporation of Ni in an oxygen environment. Further details on these procedures as well as on the physics of magnetic engineering by exchange biasing can be found in review papers by Nogues and Schuller (1999) and Coehoorn (2003). In a third engineering method, the single exchange-biased magnetic layer is replaced by an antiferromagnetically coupled sandwich of two ferromagnetic layers separated by an ultrathin metallic spacer. This is the so-called artificial antiferromagnet (AAF) or synthetic antiferromagnet (Sy-AF) as proposed by Parkin (1995) and used, e.g., by Willekens et al. (1995); see also the review paper of Parkin et al. (2003). The magnetization behavior of the three ferromagnetic layers is schematically shown in Fig. 1.14c. Antiparallel orientation of the free and fixed layer on each side of the barrier is induced when the coupled, fixed layer and exchangebiased layer are approximately of equal thickness (Strijkers et al., 2000). In that case, not only the antiparallel field range is superior to the exchange biasing scheme in Fig. 1.14b, but also the two antiferromagnetically coupled layers are magnetically stable with minimal stray field that could affect the magnetization of the free layer. Especially when the lateral dimensions of MTJs become very small in sensor or memory applications, this magnetic rigidity is crucial. Moreover, when the free layer and the pinned layer are ferromagnetically coupled due to their correlated roughness (so-called orange-peel coupling, see Néel, 1962), the antiferromagnetically coupled sandwich of pinned and exchange-biased layer can be tuned (by layer thicknesses and coupling strength) to optimally control the switching of the free layer; see, for example, Vanhelmont and Boeve (2004). Although these advanced modifications in the junction stack are crucial for engineering the field sensitivity of MTJ-based sensors and memories (Engel et al., 2002; Parkin et al., 2003; Tehrani et al., 2003; Pietambaram et al., 2004), we will not further explain more details here. Issues in the magnetic behavior of (sub)micrometer MTJs are generally related to the size dependence of the switching (Gallagher et al., 1997; Lu et al., 1997; Koch et al., 1998; Kubota et al., 2003), the effect of boundary roughness of small magnetic elements due to the patterning process (Meyners et al., 2003), dipolar interactions between MRAM cells (Janesky et al., 2001), thermal stability of the magnetization (Pietambaram et al., 2004), and so on. Regarding the implementation of MTJs in MRAM technology, the dynamics of magnetization reversal is obviously also of utmost importance. Strategies to

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Figure 1.15 In (a) the Stoner–Wohlfarth astroid shows the switching stability as a function of the normalized hard-axis and easy-axis applied magnetic fields for coherent rotation of ellipsoidal particles. Measurements of the easy-axis switching field versus hard-axis applied field of 0.6 × 1.2 µm2 MTJ cells are shown in (b). Open circles are the average switching fields with applied fields swept quasi-statically. Closed circles are data on the same bit cells, but now with magnetic field pulses with 20 ns duration. Adapted from Slaughter et al. (2002).

switch the magnetization of the free layer by sending current through the word and bit lines of the MRAM array (see Fig. 1.3) are being widely developed by a number of research groups, see, e.g., Lu et al. (1999); Boeve et al. (1999); Sousa and Freitas (2000); Engel et al. (2002); Slaughter et al. (2002); de Boeck et al. (2002); Gerrits et al. (2002); Tehrani et al. (2003); Parkin et al. (2003). As a typical example within this research area, Fig. 1.15 shows the switching fields of micrometer-size patterned MTJ cells, using both quasi-static and fast current pulses (Slaughter et al., 2002). Apparently, in the regime of 20 ns pulses, coherent Stoner–Wohlfarth rotation is still applicable, without the need to consider more complex dynamical behavior (Koch et al., 1998). To improve the magnetic stability of switching one particular MRAM cell, without affecting the other cells along a row, a new scheme has been developed recently. This so-called toggle MRAM uses an artificial antiferromagnet (see earlier) as the free magnetic system, leading to a remarkably improved robustness of cell switching (Engel et al., 2005; Yamamoto et al., 2005). 2.1.3 Electrical measurement of TMR Usually the resistance or conductance of a magnetic junction is determined from a 4-terminal measurement. A power supply (or current source) is connected to the bottom and top electrode and one measures the tunneling current (or voltage difference) between the other two terminals; see Fig. 1.8c. An important possible pitfall of such a conductance measurement on a MTJ is related to a laterally inhomogeneous current flowing through the barrier when the resistance of the barrier


27

is too low as compared to the resistance of the electrode. This is first recognized by Moodera et al. (1996) demonstrating a strong artificial increase of TMR, and is later verified by van de Veerdonk et al. (1997b) using a finite-element approach to model the current crowding in the barrier region of a tunneling device. As discussed by Moodera et al. (1996), also the pioneering data by Miyazaki and Tezuka (1995a) are suffering from an apparent amplification of TMR. Sun et al. (1998a) report on geometrically enhanced TMR in mm2 -size junctions when the junction resistance is less than 5 times the resistance of the electrode over the junction area. In another regime, when the junction radius is much smaller than the width and length of the leads, Chen et al. (2002) have developed an analytical method to correct for the artificial changes of R × A and TMR in such a device-like geometry. It is interesting to mention that the tunnel conductance or resistance can be measured also without the need for two electrodes defining the tunneling area for electrical transport. By applying a voltage across in-line contacts touching the top of a planar (unpatterned) tunneling structure, a current will not only flow through the top conducting layers, but partially also via the tunneling barrier through the bottom part of the stack. It is shown by Worledge and Trouilloud (2003) that four micrometer-spaced probes can be used to reliably determine the (field-dependent) resistance of an MTJ, which is further refined to reduce the experimental errors involved in the positioning of the probes (Worledge, 2004). Especially for testing MTJ devices on a full wafer level, this method is believed to be extremely fast and convenient in assessing, e.g., the uniformity of the resistance or switching fields over a large area. 2.1.4 TMR: basic behavior, role of bias voltage and temperature In this part three subjects will be treated. First, some basic characteristics of TMR will be shortly reviewed on a phenomenological level, such as the experimental relation with the tunneling spin polarization P , the dependence of TMR on the thickness of the barrier, and the effect of annealing. The use of CoFeB compounds as an alternative magnetic electrode material will be discussed in some detail for its intriguing capability to considerably enhance TMR. Thereafter both the temperature dependence and the bias voltage dependence of TMR will be considered along with an introductory survey of the mechanisms proposed to explain these experimental data. Basic behavior of TMR, including the use of CoFeB In section 1 it is derived that the magnitude of the magnetoresistance in MTJs is directly determined by the tunneling spin polarization via TMR = 2P1 P2 /(1 – P1 P2 ). Although the physics behind the polarization P is far from understood (and will be further explored in section 3), it is clear that tunneling spin polarization of the electrodes at the interface with the barrier offers a direct way to tune TMR. For instance, Cox Fe1–x and Nix Fe1–x are frequently applied in actual devices because of their high values of polarization and TMR (see Kikuchi et al., 2000 for the effect of CoFe composition on TMR). In Fig. 1.13b it is shown that Co80 Fe20 -Al2 O3 -Co80 Fe20 junctions display TMR of 40% at room temperature for nominal Al thicknesses above approximately 8 Å (before oxidation). The rather constant TMR for increasing alumina thickness is a common observation in amorphous Al2 O3 -based MTJs, also when using

28

H.J.M. Swagten

other ferromagnetic electrodes. For thinner barriers it is generally observed that TMR is suppressed, see again Fig. 1.13b, most likely due to the increasing density of metallic shorts (see section 2.3.3). It is also observed that annealing of junctions, up to roughly 200–300°C, greatly improves the TMR (Parkin et al., 1999b; Freitas et al., 2000). This effect has been attributed to a redistribution of the oxygen in the alumina barrier (Sousa et al., 1998) possible combined with a change of the interface structure. We will get back to this later on (section 2.3.4). Recently, the magnetoresistance for Al2 O3 -based junctions is significantly improved by the use of CoFeB as a soft ferromagnetic electrode material, for instance by sputtering it from a target with composition Co73.8 Fe16.2 B10 (Cardoso et al., 2004; Ferreira et al., 2005a), or from a Co60 Fe20 B20 target (Wang et al., 2004; Dimopoulos et al., 2004a; Wiese et al., 2004). In the paper of Wang et al. (2004), a room-temperature TMR of 70.4% is achieved which would translate to a tunneling spin polarization of around 51%, probably even higher at low temperatures. Indeed, for Co72 Fe20 B8 , Paluskar et al. (2005b) measure a tunneling spin polarization of +53.5% using superconducting junctions which is above the polarization of all other 3d elements or compounds (for more details see section 3). Currently, studies are aiming at understanding these high TMR effects, which may stem from the as-deposited amorphous character of CoFeB, possibly reducing the roughness of the bottom electrode and improving the interface quality (Dimopoulos et al., 2004b; Bae et al., 2005). Upon annealing up to around 300–400°C, it is observed that these systems may become crystalline depending, e.g., on the composition (B content), film thickness (Wiese et al., 2004; Cardoso et al., 2005), or on the character of the adjacent layers (Bae et al., 2005). However, it is shown by Paluskar et al. (2005b) that the tunneling spin polarization of their thick CoFeB films remains almost unaffected by annealing in ultra-high vacuum conditions. It could be that the electronic structure of CoFeB is not very sensitive to (amorphous–crystalline) structural changes as suggested by first-principle calculations on Fe-B alloys (Hafner et al., 1994). Bae et al. (2005) have used MTJs with three different bottom pinned electrodes, Co32 Fe48 B20 , CoFe-Co32 Fe48 B20 -CoFe, and CoFe. In the former two B-containing electrodes, the TMR appears to be higher than for the electrode with only CoFe (after annealing). Given the fact that the tunneling spin polarization is determined by the interface with Al2 O3 only (section 3), the authors suggest that the surface flatness and interface quality may be rather important for obtaining high TMR with CoFeB electrodes. In section 4, junctions combining CoFeB electrodes with crystalline MgO barriers will be further discussed. These materials turn out to be superior for their enormous magnitude of TMR. The remainder of the data described in this section will be dealing with electrodes not containing these CoFeB electrodes, but rather traditional 3d elements or compounds such as Co, CoFe, and NiFe covering the majority of existing papers in this field. Temperature dependence of TMR The temperature dependence of the (magneto)resistance in MTJs has received enormous attention, both for fundamental interest as well as for applications in sensors and MRAMs operating almost exclusively at room temperature. In a similar way, also the transport behavior at a non-zero, finite applied bias voltage is extremely relevant, which will be the topic


29

Figure 1.16 Temperature and voltage dependence of TMR in a junction consisting of Si(100)/SiO2 /50 Å Ta/50 Å Co/100 Å FeMn/35 Å Co/23 Å Al + oxidation/150 Å Co/50 Å Ta. In (a) the low-bias normalized TMR (with respect to low T ) is shown as a function of temperature, together with the tunnel resistances in parallel and anti-parallel orientation. Panel (b) shows the voltage-dependence of TMR at T = 5 K. Both the resistance change (RAP – RP )/RP and conductance change (GP – GAP )/GAP are shown. V1/2 corresponds to the bias voltage where TMR has dropped to 50% of its zero-bias value. After LeClair (2002).

of the following subsection. Generally, three processes are believed to somehow contribute to the T dependence of TMR: • a thermal reduction of magnetic moment (polarization) at the barrier interface, directly affecting the magnitude of TMR • inelastic tunneling due to electron-magnon (spin-wave) scattering at the barrier interfaces • thermally-assisted hopping conductance via impurities or defect states located in the barrier region. In Al2 O3 -based magnetic junctions, the temperature dependence of the magnetoresistance is intensively studied, and it is generally seen that TMR gradually decreases with temperature. Figure 1.16a shows that TMR (for low bias voltages) is reduced by more than 25% when heating the junction from T = 5 K to T = 300 K, which is derived from the change in the parallel and antiparallel resistance (see the figure). To understand why TMR is reduced for higher T , one should first of all realize that an increasing temperature broadens the Fermi distribution of the tunneling electrons, which allows electrons with higher energies to tunnel across the barrier. As long as kB T φ, a criterion well fulfilled at room temperature, this leads to an increase of the low-bias tunnel conductance as G(T )/G0 = CT / sin(CT ), with C = (2π 2 kB t /h)(2me /φ)1/2 , and with G0 equal to 1/R0 in Eq. (15). This, however, corresponds to a resistance drop of only a few percent between 0 and 300 K for realistic values of barrier thickness and height, in contrast to experimentally observed changes in RP and RAP (see, for example, Fig. 1.16a). A first approach in further understanding the decaying TMR is proposed by Shang et al. (1998b). The

30

H.J.M. Swagten

zero-bias conductance of a magnetic junction is written as: GP,AP (T ) =

G0 CT [1 ± PL PR ] + Ginelastic (T ). sin(CT )

(16)

The prefactor of the first term G0 CT / sin(CT ) is the aforementioned enhanced tunneling conductance by smearing of the Fermi functions, PL,R is the tunneling spin polarization of the left and right electrode, where the “+” refers to parallel oriented magnetizations, “–” to antiparallel. The second term in Eq. (16) is representing a spin-independent inelastic contribution to the current, and is believed to originate from hopping conductance via imperfections in the Al2 O3 barrier. The role of scattering by impurities in the barrier is separately studied by Jansen and Moodera (2000) in artificially doped barriers, e.g. by plasma oxidizing an Al-Si-Al trilayer, with a Si thickness of 0.5– 2.0 Å. When using magnetic ions (Ni instead of Si), the inelastic nature of spin scattering is reflected in a more pronounced temperature dependence of TMR. Returning to the analysis of Shang et al. (1998b), it is instructive to calculate the magnetoresistance from Eq. (16) using the definition given in Eq. (8), yielding TMR = 2PL PR /[1 – PL PR + Ginelastic (T )/G(T )]. This explains the reduction of TMR with temperature whenever a nonzero inelastic tunneling term is present. Apart from that, also the polarization PL,R itself is depending on temperature which is shown theoretically by MacDonald et al. (1998). Due to the presence of thermally excited spin waves the polarization in the Julliere formula can be effectively written as P (T ) = P0 [m(T )/m0 ] with m(T ) the saturation moment at the interface of the ferromagnetic layer with the barrier, and P0 and m0 the zero-temperature polarization and magnetic moment, respectively. Using the approach captured by Eq. (16) including the polarization suppression by thermally excited spin waves, a good agreement with temperature-dependent experiments has been reported by Shang et al. (1998b). Another approach to model the temperature dependence of tunnel magnetoresistance is given by Davis et al. (2001). In this case, tunneling is treated purely elastically within a free-electron model (Slonczewski, 1989, see also section 1.3) without incorporating additional inelastic conduction channels. In a free-electron model the tunneling spin polarization of the (Fermi) electrons can be expressed as (kF ,maj –kF ,min )/(kF ,maj +kF ,min ), with kF ,maj,min = [2m∗maj,min (EF –Umaj,min )/h¯ 2 ]1/2 , m∗ the effective electron mass, and Umaj,min the bottom of the exchange-split parabolic bands. The temperature dependence of TMR now arises from the T dependence of the exchange splitting Umaj – Umin and is reported to be nearly proportional to M(T ) (Shimizu et al., 1966). From fitting the model calculations to experimental data (Davis et al., 2001), it is shown that a small drop in magnetization between 0 and 300 K may lead to a substantial variation of TMR in accordance with the experiments. This implies a rather prominent role of intrinsic band structure effects in understanding the T -dependence of transport in MTJs. In an alternative theoretical approach (Zhang et al., 1997a), the reduction of TMR with temperature is described in terms of inelastic magnon (spin wave) scattering. By the emission or absorption of magnons during the tunneling process


31

across the insulating barrier (involving a reversal of spin), TMR is more efficiently reduced with temperature than for elastic tunneling only. Using a detailed analysis of the conductance of exchange-biased junctions, Han et al. (2001) have found an excellent agreement with the magnon-assisted inelastic excitation model, which includes a proper description of the bias voltage dependence of TMR. We will return to the model of Zhang et al. (1997a) below. Bias-voltage dependence of TMR Since the discovery of magnetoresistance in alumina-based junctions, the significant suppression of TMR with increasing bias voltage V has been subject of a great number of experimental and theoretical studies. In Fig. 1.16b the typical reduction of TMR with applied bias voltage in a Co-Al2 O3 -Co junction is shown using two different representations, viz. as R /RP and as G/GAP . Obviously, the resistance and conductance change only coincide at sufficiently small bias voltage when R = V /I is identical to 1/G = dV /dI . The suppression of TMR with voltage is critically important when operating MTJs devices at finite voltage, and a huge research effort is seen in optimizing and understanding the decay of TMR. Usually the voltage where TMR = R /RP is reduced by 50%, indicated in the figure by V1/2 , is taken as a representative fingerprint of the bias-voltage dependence. From the huge amount of reports on the bias-voltage dependence of TMR, it is seen that V1/2 is typically in the order 0.3–0.6 V in Al2 O3 -based magnetic junctions. As to the explanations of the V dependence, several mechanisms have been proposed so far:

• spin-mixing due to electron-magnon scattering in the magnetic electrodes, at the interfaces with the barrier • additional tunnel conductance channels provided by defect and impurity states in the barrier region • intrinsic modification of the barrier shape, combined with the spin-dependent band structure of the magnetic electrodes. In Eq. (15), it is shown that in the WKB approximation the conductance of a tunnel junction is quadratic in voltage, as experimentally observed for high enough voltages (see Fig. 1.9). At low bias voltage, however, both the conductances in parallel and anti-parallel configuration strongly deviate from the parabolic law, and a quasi-linear, so-called zero-bias anomaly is universally observed in Al2 O3 containing MTJs. Zhang et al. (1997b) and Bratkovsky (1998) have shown that the excess energy of the tunneling electrons as provided by the applied voltage is capable of collectively excite magnons at the ferromagnet-barrier interface, thereby inducing an additional inelastic conductance contribution that is linear in bias voltage, viz. G(V ) ∼ V for voltages V kB TC /e with TC the Curie temperature of the magnetic electrode. Due to the reversal of the electron spin associated with the creation of a magnon, TMR is naturally decaying with voltage in this regime. For higher voltages, the lifetime of the magnons becomes too short and the additional inelastic conductance levels off upon further increase of V . Han et al. (2001) have carefully measured the bias dependence of exchange-biased Co75 Fe25 -Al2 O3 -Co75 Fe25 junctions, not only measuring I (V ) and dI /dV curves, but also measuring d2 I /dV 2 ,

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H.J.M. Swagten

so-called inelastic tunneling (IET) spectra. Using the magnon-assisted inelastic excitation model of Zhang et al. (1997b), their data are reasonable well captured by the calculations, provided that the wavelength-cutoff energy of the spin-wave spectrum is different for parallel and anti-parallel magnetization (Han et al., 2001). From a theoretical point of view, also mechanisms other than (interface) magnetic excitations have been proposed to explain the suppression of TMR with voltage. This includes the effect of the intrinsic band structure, and impurities in the barrier. The latter contribution is specifically addressed by intentionally adding a δdoped ultrathin layer within the Al2 O3 barrier (Jansen and Moodera, 1998, 2000). Only in the case of magnetic impurities, a stronger bias dependence has been observed and is attributed to spin-exchange scattering. Intrinsic band structure effects can be understood by realizing that already in elementary free-electron calculations the TMR is decaying with voltage (Zhang et al., 1997b); see section 1.3 and Fig. 1.7 for details on the free-electron model. This is due to the fact that the overall conductance is enhanced by applying a significant bias, simply due to an effectively reduced barrier height by tilting the barrier potential with voltage. On the other hand, the difference between conductance in the parallel and anti-parallel orientation is only slightly affected, since the voltage adds additional energy dependencies to the density-of-states (or spin polarization) thereby diminishing the imbalance between the number of majority and minority tunneling states. Assuming that tunneling in Fe-Al2 O3 -Fe is dominated by a single free-electron-like spin-resolved d band, Davis and MacLaren (2000) have found a fair agreement with the data of Zhang and White (1998), suggesting that the behavior of TMR with applied voltage has an intrinsic component resulting purely from the underlying electronic structure. This is corroborated by free-electron calculations and experiments on the bias dependence of Co-Al2 O3 -Co junctions (Xiang et al., 2002, 2003), in which a reasonable variation of the Co density-of-states over energy is required to describe the resistance and TMR over a broad range of bias voltages. Again this hints to the relevance of intrinsic electronic properties for the bias dependence of spin tunneling (see section 4.3 for a further discussion of other experimental results). Finally, it is expected that by applying a bias voltage across a magnetic junction, it should be possible to extract specific density-of-states features of the ferromagnetic electrodes from the conductance or TMR. However, this turns out to be far from trivial, and only a limited number of experimental studies are available. Excellent examples are reported for junctions containing, e.g., epitaxial La2/3 Sr1/3 MnO3 electrodes, or Fe combined with MgO barriers. This will be discussed in section 4. Another point of interest for the bias dependence is raised by the experiment of Valenzuela et al. (2005). They have produced a lateral double-barrier tunneling device basically consisting of CoFe-Al2 O3 -Al-Al2 O3 -NiFe, where the Al is laterally extended, separating the ferromagnetic electrodes and tunnel barriers over a distance between 1500 and 10 000 Å. Due to this, the spin-dependence of the electrons tunneling out of one electrode and tunneling into the other electrode can be disentangled. From their experiments it is suggested that tunneling into the empty states of the ferromagnetic electrode is dominating the reduction of TMR with increasing bias voltage, probably due to the intrinsically reduced polarization of the


33

hot electrons and the matching of the wave functions at the interfaces with the tunneling barrier (Valenzuela et al., 2005).

2.2 Oxidation methods for Al2 O3 barriers The breakthrough of high room-temperature magnetoresistance in MTJs as reported by Miyazaki and Tezuka (1995a) and Moodera et al. (1995) is strongly related to the successful fabrication of well-controlled, uniform tunneling barriers. Many investigations related to the search for MTJs with improved properties like high TMR, low RA product, large V1/2 and breakdown strength, and strong thermal stability, are intimately connected to improved control over the barrier region. As we have seen in the introduction, this is due to the physics of TMR and tunneling spin polarization, determined primarily by the barrier and the interfaces with the ferromagnetic electrodes. Consequently, a careful control over the barrier and interface regions is indispensable. Related to this, a wide variety of barrier oxidation and preparation techniques has been explored since the pioneering experiments, which includes: • • • • • •

plasma oxidation, using a DC or RF-generated O-plasma ion-beam oxidation thermal or natural oxidation in an O2 atmosphere UV-light assisted oxidation oxidation by ozone or by O radicals direct Al2 O3 deposition.

In this subsection, we will focus on these oxidation processes mainly in relation to the magnitude of TMR and R × A, since both are, among other properties such as electrical noise (section 2.1.1), decisive parameters for future device implementation of progressively down-scaled junctions. In Fig. 1.17, a compilation of some of the existing data is shown for Al2 O3 -based junctions prepared by these different oxidation techniques; see also Table 1.1. In this section we will restrict ourselves to alumina-type junctions since these are predominantly studied in this field. In later sections other barrier materials (such as SrTiO3 and MgO) will be discussed separately. With very few exceptions, it is clear that plasma oxidation, indicated by the solid symbols in Fig. 1.17, distinguishes itself from all other techniques by the highest values of TMR, but also with a characteristically high value of R × A. Thermally oxidized junctions, grouped mostly in the smaller circle, have in general a low RA product but also a low(er) magnetoresistance. Other techniques, which are explored for their potential of making junctions with both a high TMR and lower R × A, are situated between those extremes. Even with such a variety of techniques it currently appears to be practically impossible to enter the upper-left part (low R × A, high TMR) of Fig. 1.17 with alumina-based junctions. However, the results obtained with ion-beam oxidation (Ferreira et al., 2005a) are promising for their very low RA combined with reasonably high TMR ratios; see also section 2.2.1. Interestingly, junctions with crystalline MgO barriers are reported to exhibit much

34

H.J.M. Swagten

Figure 1.17 TMR versus the RA product (at room temperature) of junctions made by various barrier production techniques. The larger and small circle roughly indicate the plasma and thermally oxidized junctions, respectively. Note that ion-beam oxidation (resembling the use of a regular DC plasma) seems superior for their low RA and high TMR. For the underlying data including references, see Table 1.1.

higher magnetoresistances combined with a relatively low resistance-area product (section 4.6). Before discussing the results reported in literature in more detail, the reader must bear in mind that the observed spread in TMR or R × A as seen in Fig. 1.17 may naturally stem from lab-to-lab variations of the structure of the junctions, and, in particular, the barrier region. The structure and morphology of the unoxidized aluminum layer influences the oxidation process and therefore the quality of the resulting barrier. The oxide growth can easily be imagined to be affected when oxidizing an aluminum layer with a grain-like structure. Such a growth mode is intrinsically induced by the layer on which the aluminum is deposited and can vary with the bottom layer material and with deposition technique. For example, Ando et al. (2000b) have shown by atomic-force-microscopy measurements that the roughness of aluminum can be reduced with 80% by replacing the aluminum buffer layer under the bottom electrode by Pt. The deposition parameters and characteristics of the deposition facility can play a huge role. For instance, a small amount of surface contamination can induce a different growth mode of the aluminum layer. Fujikata et al. (2001) report a considerable improvement of TMR in junctions with an intentionally contaminated Ta buffer layer in their junctions. Furthermore, in the case of plasma oxidation and UV-oxidation, the exact lay-out and operation of the oxidation setup can be crucial for the quality of the barrier layer. Therefore, the comparison between oxidation techniques found in literature should be considered with great care.

35


Table 1.1 A selection out of the vast literature on room-temperature low-bias TMR and R ×A for alumina-based magnetic tunnel junctions. Oxidation methods are categorized in thermal oxidation, plasma oxidation, ion beam oxidation, UV-assisted oxidation, ozone-enhanced oxidation, oxidation by radicals, and reactive deposition. Only the electrodes next to the Al2 O3 barrier are indicated. In several cases the results are obtained after a post-deposition anneal. Data by Li and Wang (2002), indicated with †, are taken at T = 18 K. TMR with ‡ is measured at a 0.3 V bias voltage

Method

Electrodes

TMR (%)

R × A ( µm2 )

Reference

Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Thermal Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma

Fe–CoFe NiFe–NiFe Co–NiFe NiFe–NiFe NiFe–NiFe CoFe–CoFe CoFe–CoFe CoFe–CoFe CoFe–CoFe CoFe–CoFe Co–Co CoFe–CoFe CoFe–CoFe CoFe–CoFe CoFe–CoFe CoFe–CoFe Co–CoFe Co–NiFe NiFe–NiFe Co–Co CoFe–CoFe CoFe–CoFe CoFe–CoFe Co–Co CoFe–CoFe CoFe–CoFe NiFe–NiFe CoFe–CoFe Co–Co CoFe–CoFe CoFe–CoFe CoFe–CoFe CoFe–CoFe

5 13 20–23 16 14 32 18 25–30 14–17 29 20 22 23 18–25 11 29 20 6 17 19 32 25–27 15 28 31 50 30 32 20–25 26 59 48 48

2 × 103 2 × 103 60 230 × 103 14 30–40 140 30–70 10–12 34 68 × 106 8 580 8–14 4.4 60 150 × 106 200 × 106 60 × 106 200 × 106 11 × 106 (10–20) × 103 2 × 103 50 × 106 230 (1–10) × 103 (10–100) × 103 160 × 106 (1–100) × 103 6 × 103 1 × 106 (100–500) × 103 40 × 103

Tsuge and Mitsuzuka (1997) Matsuda et al. (1999) Parkin et al. (1999b) Chen et al. (2000) Ohashi et al. (2000) Sun et al. (2000a) Song et al. (2000) Zhang et al. (2001b) Zhang et al. (2001b) Moon et al. (2002) Diouf et al. (2003) Wang et al. (2003) Das (2003) Zhang et al. (2003b) Zhang et al. (2003b) Shang et al. (2003) Moodera et al. (1996) Nassar et al. (1998) Wee et al. (1999b) Gillies et al. (1999) Parkin et al. (1999b) Sun et al. (1999) Sun et al. (1999) LeClair et al. (2000a) Ando et al. (2000b) Ando et al. (2000b) Chen et al. (2000) Park and Lee (2001) Kuiper et al. (2001a) Dimopoulos et al. (2001a) Tsunoda et al. (2002) Tsunoda et al. (2002) Lohndorf et al. (2002) (continued on next page)

36 Table 1.1

H.J.M. Swagten

(Continued)

Method

Electrodes

TMR (%) R × A ( µm2 )

Plasma Plasma Plasma Plasma Plasma Plasma Plasma Plasma Ion beam Ion beam Ion beam Ion beam UV-assisted UV-assisted UV-assisted UV-assisted UV-assisted UV-assisted UV-assisted UV-assisted† UV-assisted Ozone Ozone Radicals Radicals‡ Reactive depo. Reactive depo.

NiFeCo–NiFeCo NiFe–NiFe Co–NiFe CoFe–CoFe Co–Co CoFe–CoFe CoFeB–CoFe CoFeB–CoFeB CoFe–CoFe NiFe–Co CoFeB–CoFeB CoFeB–CoFeB NiFe–NiFe NiFe–NiFe Co–NiFe NiFe–NiFe Co–NiFe NiFe–Co Co–Co NiFe–NiFe CoFe–CoFe CoFe–CoFe CoFe–CoFe Co–Co CoFe–NiFe NiFe–NiFe Fe(211)–CoFe

45 26 34 20 29 37 61 70 40 8–14 20 40–45 13 8 10–15 14–21 20 23 20 2–8 30 30 33 11–17 40 15–20 35–45

1 × 103 4.0 × 106 2.3 × 106 20 × 103 300 × 106 (5–10) × 103 25 × 106 24 × 106 (500–800) × 103 2 × 106 –2 × 109 2–15 60–150 2 × 103 300 51 × 103 102 –104 60 × 103 1 × 103 160 × 103 0.6–6 15 × 103 (8–24) × 106 11 × 103 350–200 × 103 (1–3) × 103 > 1 × 106 103 –109

Reference Engel et al. (2002) Song et al. (2003) Song et al. (2003) Das (2003) Koller et al. (2003) Kim et al. (2003) Wang et al. (2004) Wang et al. (2004) Cardoso et al. (1999) Roos et al. (2001) Ferreira et al. (2005a) Ferreira et al. (2005a) Song et al. (2000) Song et al. (2000) Girgis et al. (2000) Covington et al. (2000) Boeve et al. (2000) Rottlander et al. (2000) Rudiger et al. (2001) Li and Wang (2002) Das (2003) Park and Lee (2001) Park and Lee (2001) Shimazawa et al. (2000) Kula et al. (2003) Chen et al. (2000) Yuasa et al. (2000)

2.2.1 Plasma oxidation Plasma oxidation is currently the most widely applied method for producing aluminum oxide for MTJs with the highest values of TMR for amorphous barriers. As mentioned above, Moodera et al. (1995) are the first to reproducibly produce MTJs using plasma oxidation, and many groups have followed using numerous variations on plasma oxidation. A DC glow plasma is easy to set up (see Fig. 1.18b) and is therefore most commonly applied. Nassar et al. (1998) have applied an AC O2 /Ar rf-plasma for the production of MTJs. A TMR of 6% is found with an R × A of 200 M µm2 . The low TMR and high RA product suggest that the bottom electrode is oxidized in the process. An inductively coupled plasma (ICP) is generated without electrodes by Ando et al. (2002) and Song et al. (2003), which means that there is no contamination by sputtering of electrode material. This method is therefore thought to produce less impurities in the tunnel barrier. However, there is no


37

Figure 1.18 (a) Differential ellipsometry to in-situ monitor the amount of oxidizing metal as a function of time. The bottom two curves are taken on a Si/SiO2 /10 Å Al sample using natural oxidation followed by plasma oxidation (open symbols), and for plasma oxidation only (closed). The upper curve represents plasma oxidation of a full stack of Si/SiO2 /50 Å Ta/70 Å Co/100 Å FeMn/35 Å Co/23 Å Al. In (b) a picture of an oxidation chamber as taken from Knechten (2005) shows the DC glow discharge due to the high (negative) potential of the ring-shaped electrode. See Knechten et al. (2001).

conclusive evidence that impurities in the barrier due to sputtering of the electrode are causing a degradation of MTJ properties. A plasma generated by radio-frequency or microwave radiation has been successful in plasma oxidation of silicon and is also applied for aluminum oxidation, for example by Sun et al. (1999) and Yoon et al. (2001). Plasma oxidation is very fast as compared to many other oxidation methods. For example, Park and Lee (2001) optimally oxidize 18 Å of aluminum in approximately 40 seconds, and Kuiper et al. (2001a) need only 20 seconds of plasma oxidation to optimally oxidize 15 Å of aluminum. To monitor these dynamical processes, insitu characterization techniques have been developed. Wee et al. (1999a, 1999b) use the Van der Pauw method to in-situ measure the electrical resistance of the Al layer during plasma oxidation from which the tunneling barrier thickness can be estimated. Optical, ellipsometric techniques have been reported by LeClair et al. (2000c), Lindmark et al. (2000), and Knechten et al. (2001), using the extreme contrast between the dielectric constant of a metal and that of its oxide. In Fig. 1.18a it is illustrated that the growth of the oxide from a 10 Å and 23 Å Al layer can be monitored with high temporal resolution and with sub-monolayer sensitivity, offering the possibility to investigate the oxidation dynamics in great detail (for more details, see the thesis work of Knechten, 2005). Variations in plasma pressure and plasma composition can improve the performance of MTJs. Following the success of a krypton–oxygen mixture (97%:3%) in silicon oxidation (Sekine et al., 2001), junctions with a TMR of 59% have been obtained by Tsunoda et al. (2002). Lee et al. (2003) have found that the rough-

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ness of the barrier interfaces can be tuned by adding a small amount of Zr to the aluminum. At a doping level of 9.9% Zr, the barrier interfaces, as observed with transmission electron microscopy, are the smoothest and the TMR is highest. Tunnel junctions made by plasma oxidation with a low resistance-area product are, e.g., fabricated by Ando et al. (2000b). The barrier layer in their tunnel junctions is made by inductively-coupled-plasma oxidation of 8 Å Al. After annealing, the junctions with a 10 seconds oxidation time display an RA product of 230 µm2 combined with a TMR of 31%. When the oxidation time of the Al layer is longer (30–60 seconds) the maximal TMR is raised to roughly 50%, although now with a higher RA of typically a few k µm2 . Using CoFeB compounds as ferromagnetic electrodes (see also section 2.1.4) and a Ar/O2 plasma for Al oxidation, very large magnetoresistances of more than 70% are reported by Wang et al. (2004). However, the resistance of these junctions is very high, around 24 M µm2 . Other investigations are concentrating on further optimizing these CoFeB-based alumina junctions; see for example Pietambaram et al. (2004), Wiese et al. (2004), and Cardoso et al. (2005). Ionized atom-beam oxidation Roos et al. (2001) use an ionized oxygen atom beam of low energy (30–80 eV) to oxidize the aluminum. MTJs with a 8–14% TMR and a high R × A of 1–1000 M µm2 are produced. Cardoso et al. (1999) and Freitas et al. (2000) use an ion beam for the deposition of the layers and for the oxidation process as well. The ion beam is created by inserting a grid between a high-power (80 W) Ar/O plasma and the sample, and by applying a voltage (typically 30 V) over the grid, accelerating oxygen ions towards the sample. With this technique, junctions with a TMR of 40% and an RA product of around 500 µm2 were created. The high quality is explained by the better layer-by-layer growth as compared to the more frequently applied sputter deposition. The oxidation times for optimal TMR are comparable to plasma oxidation, in the order of 60 seconds. Recently, an even smaller RA product of around 2–15 µm2 has been established together with a TMR of around 20% (Ferreira et al., 2005a, 2005b). When an artificial antiferromagnet is present to engineer the top CoFeB electrode (see sections 2.1.4 and 2.1.2), R × A values of 60–150 µm2 are combined with a TMR of 40–45%. The ion-beam oxidation is performed in three steps with increasing plasma reactivity and leads to under-oxidized barriers from an initial 9 Å Al layer. As can be seen from the data points on ion-beam oxidation in Fig. 1.17, these junctions are very promising for device applications in sensors and memories due to the unique combination of low RA and high TMR (Ferreira et al., 2005a).

2.2.2 Thermal and natural oxidation In order to create junctions with lower specific resistances for device applications, the barriers in MTJs have become progressively thinner. For the oxidation of Al layers of 10 Å or less, plasma oxidation is thought to be too aggressive (due to the high-energy particles involved) and not well controllable, possibly resulting in damage to the interface with the bottom electrode. Therefore, for very thin layers often natural or thermal oxidation is chosen. We note that for oxidation in an oxygen atmosphere at room temperature normally the term natural oxidation is used,


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whereas thermal oxidation refers to oxidation in an oxygen atmosphere where the sample is usually but not necessarily at an elevated temperature. For most junctions reported in literature, oxidation at room temperature is used. Tsuge and Mitsuzuka (1997) and Matsuda et al. (1999) use pure natural oxidation, and a TMR of 13% is found with an RA product of only 1.5 k µm2 . The barrier is made by exposure of 20 Å Al to 0.27 bar of pure oxygen for one hour. The fact that a barrier made from an initial 20 Å Al results in an RA product that is three orders of magnitude lower than plasma-oxidized junctions starting with the same initial Al thickness, suggests that the barrier has an inhomogeneous thickness and the tunneling current runs through the thinnest parts of the barrier. The low TMR is probably due to unoxidized aluminum suggesting that the oxidation in this case is rather inhomogeneous. Generally speaking, it is observed that the resistance of plasma-oxidized junctions is much higher than their naturally oxidized counterparts. The explanation of this striking difference is not yet clear, and maybe directly related to the much higher oxidation rates for plasma oxidation, combined with the fact that the energetic O atoms in the plasma more easily oxidize the pinholes (Knechten, 2005). Natural oxidation is a slow process if the aluminum layer is typically thicker than 5 Å. For instance, the optimal oxidation of 10 Å Al takes 15 hours at room temperature, as reported by Das (2003). In order to reduce processing time, two cycles of deposition of Al and subsequent oxidation are used. The oxidation time resulting in optimum TMR is thereby reduced from 15 hours to 2 × 2 hours (Das, 2003). An identical technique is used by Moon et al. (2002) who report MTJs with 30% TMR and an RA product of only 140 µm2 . Extremely small values of R ×A are reported by Wang et al. (2003) (see also Zhang et al., 2001b, 2002, 2003b), and is set to 8 µm2 while these junctions have a TMR of 22%. Ohashi et al. (2000) have produced a functional low-resistance tunnel magnetoresistive sensor for use as hard disk read-head. The MTJ is made by natural oxidation of 8.5 Å Al in a pure oxygen atmosphere for 20 minutes, resulting in a TMR of 14% and an R × A value of 14 µm2 . 2.2.3 UV-light assisted oxidation Oxidation assisted by UV-light irradiation has been tried as a faster method of aluminum oxidation as compared to natural oxidation which is due to an increased reactivity by ozone generation. Since the damage due to high-energetic species is avoided as compared to plasma oxidation, UV-oxidized junctions possibly combine a low R × A with a high TMR. Boeve et al. (2000) and Girgis et al. (2000) are the first to report results on MTJs made with UV-assisted oxidation. They oxidize a sputter-deposited 13 Å Al layer for one hour in an oxygen atmosphere of 100 mbar, assisted by an in-situ ultraviolet lamp. They find an RA product of 60 k µm2 and a TMR of about 20%. Compared with their naturally oxidized junctions, UVoxidation results in higher TMR but also in higher R × A. Their plasma-oxidized junctions, which are identical except for the oxidation method, give a higher TMR but similar R × A values. Later experiments have resulted in junctions with a TMR of 10%, with a typical RA product of 1 k µm2 (Rottlander et al., 2000). Li and Wang (2002) have prepared junctions by UV-assisted oxidation of only 5 Å Al,

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resulting in a few percent TMR but an R × A as low as 3.2 µm2 . By oxidation of 4 Å Al, an extremely low R × A of approximately 0.6 µm2 is found. The TMR in these junctions has, however, dropped to 2%. Probably this is due to metallic shorts between the ferromagnetic layers. 2.2.4 Other oxidation and deposition processes Oxidation using ozone The high reactivity of ozone suggests that by using ozone for aluminum oxidation, shorter oxidation times with respect to natural oxidation are possible, and that possibly larger oxide thicknesses as compared to natural oxidation are attainable within reasonable oxidation times. An oxygen–ozone mixture to oxidize aluminum for MTJs is used by, for instance, Park and Lee (2001) and Park et al. (2002). In a comparison between ozone-oxidized and plasma-oxidized junctions, they report slightly higher TMR values (33%) for junctions oxidized by ozone, whereas the RA product of ozone-oxidized junctions is one order lower with a lowest value of 10.5 k µm2 . The process is still very slow; 50 minutes of oxidation are necessary to produce the junction described here. Junctions made by thermal oxidation have TMR values close to that of the ozone-oxidized junctions (30%), but with considerable lower R × A of 140 µm2 (Moon et al., 2002). Radical oxidation Shimazawa et al. (2000) report on experiments with oxidation using a beam of oxygen radicals, arguing that this can be an energetically low and slower process as compared to plasma oxidation, thus suitable for the oxidation of ultrathin Al layers. The radicals are produced by a microwave (electron cyclotron resonance) in approximately 10–3 mbar of oxygen. Junctions are created with R × A values of 350 µm2 , and with a TMR of about 11%. Kula et al. (2003) applied radical oxidation as well, resulting in junctions that have a TMR of 40% and a minimal R × A of only 2 k µm2 . In a comparison with natural and plasma oxidation, for radical oxidation a higher TMR is reported as well as a medium RA product in between thermal and plasma oxidized junctions. The possible advantage of radical oxidation over plasma oxidation might be the absence of more energetic particles such as ions. Direct deposition of Al2 O3 , atomic layer deposition As an alternative to oxidation processes, direct deposition of Al2 O3 layers has been tried in various forms. First of all, reactive sputtering has been applied to create Al2 O3 layers. By sputtering aluminum in an argon atmosphere which contains a few percent oxygen, in principle a homogeneous and stoichiometric Al2 O3 layer can be obtained as shown by Koski et al. (1999). The roughness of thick films (≈ 1 µm) grown on pure Si was reported to be about 7 Å. Chen et al. (2000) have tried to improve the method by preventing the oxidation of the bottom electrode by first depositing 7 Å of pure Al before adding 5 Å of reactively sputtered Al2 O3 . Since their method resembles plasma oxidation, the results are comparable to their plasma oxidation results, with a TMR of 18% and an RA product of 1 M µm2 . Yuasa et al. (2000) have produced reactively sputtered, amorphous alumina barriers on top of crystalline Fe electrodes (see section 4.3). The Al2 O3 is created by evaporation of Al at an O2 pressure of around 7 × 10–6 mbar. Due to the direct deposition method, it is possible to use wedges of variable alumina thickness, facilitating the study of magneto-transport as a function


41

of the thickness of the barrier layer in a single sample (Yuasa et al., 2000). Another method of direct Al2 O3 deposition is atomic layer deposition (ALD). In principle, this technique allows for deposition of very thin dielectric films with excellent conformality, uniformity, and atomic-level thickness control; see, for example, Paranjpe et al. (2001). Although tunneling transport has been demonstrated across sputtered exchange-biased magnetic junctions incorporating ALD-based Al2 O3 barriers, no appreciable TMR has been measured (Bubber et al., 2002).

2.3 Towards optimized barriers In the following subsections, experiments on the optimization of the oxidation process will be briefly outlined. In many studies, the optimization is performed mainly in relation to the magnitude of TMR, the resistance R of the junction, the temperature dependence of TMR and R, and, finally the bias voltage dependence. These properties have been reviewed in the previous sections. Here we will focus on optimization schemes in relation to the following prominent issues: • • • •

over- and under-oxidized barriers barrier pinholes and dielectric breakdown thermal stability upon junction annealing the use of alternative (amorphous) oxides.

We will start the discussion on these items with a short overview of the experimental tools that have been successfully applied in this area, focusing again on the properties of Al2 O3 barriers. 2.3.1 Tools for oxidation monitoring and optimization Apart from the direct measurement of (tunneling) transport characteristics, a number of diagnostic tools have been used to examine and further optimize the oxidation processes. Electrical and optical tools to in-situ monitor the oxidation dynamics have been mentioned in section 2.2.1, with which the transformation of Al into its oxide can be followed with submonolayer precision. Surface sensitive techniques are frequently applied to study the chemical composition of the junction and in particular the alumina barrier, e.g. using X-ray photoelectron spectroscopy (Mitsuzuka et al., 1999; LeClair et al., 2000c; de Gronckel et al., 2000; Kottler et al., 2001), Rutherford backscattering spectroscopy (Sousa et al., 1999; Gillies et al., 1999), and electron recoil detection (Gillies et al., 2000). In Fig. 1.19a the latter two techniques have been applied to measure the oxygen content in junctions of Ta-NiFe-IrMn-NiFe-Co-Al2 O3 -Co-NiFe-Ta for variable oxidation time. The observed ln(t) behavior hints to a simple logarithmic growth law which is proposed by Mott (1947) to describe the oxidation of thin metal films, typically below 40 Å. In this type of oxidation model, it is assumed that the aluminium ions are diffusing through the growing oxide and react with the oxygen at the outer interface. This has been confirmed by Kuiper et al. (2001a) using an isotope technique in which an Al layer is shortly oxidized with 16 O before continuing with 18 O; see Fig. 1.19a. In this study, secondary ion mass spectrometry depth profiles indicate that 16 O moves to larger depth with increasing 18 O oxidation

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Figure 1.19 (a) O content of junctions consisting of 35 Å Ta/30 Å NiFe/100 Å IrMn/25 Å NiFe/15 Å Co/15 Å Al + oxidation/40 Å Co/100 Å NiFe/35 Å Ta for variable oxidation time, as measured by Rutherford backscattering spectrometry (RBS) and elastic recoil detection analysis (ERD). Two-step oxidation using 18 O2 and 16 O2 oxygen isotopes is performed to establish that Al is the moving species during oxidation. (b) Oxide thickness for comparable junctions extracted from cross-section transmission electron microscopy (XTEM) as shown in panel (c) for a multilayer of 50 Å Co/15 Å oxidized Al. This suggests an intermediate oxidation step with increasing O content at constant oxide thickness. After Gillies et al. (1999) and Kuiper et al. (2001a).

time, while 18 O is incorporated close to the surface. This points to Al as the moving species during plasma oxidation. Cross-section transmission electron microscopy (XTEM), used for instance by Boeve et al. (2001) and Bruckl et al. (2001), is used to further characterize the growth and morphology of the films. For instance, XTEM on Co-Al2 O3 multilayers by Gillies et al. (2000) shows an excellent contrast between the metal and oxide layers, see Fig. 1.19c, from which the oxide thickness can be followed as a function of oxidation time as shown in Fig. 1.19b. When comparing this to the left panel of the figure, it is suggested that oxidation of the barrier is governed roughly by distinct steps. In the first stage the oxygen rapidly penetrates through the total Al layer, then a homogenization stage follows where the O content steadily increases at a fixed oxide thickness, until, at the third step, the Co electrode starts to oxidize. To continue the discussion on characterization tools, Shen et al. (2003) have combined XTEM with electron holography to directly measure the shape of the barrier and its interfaces. An ac-impedance technique is applied by Gillies et al. (2000) in order to characterize the dielectric properties of the barrier layer. Analyzing the data by modelling the tunneling across the oxide with an RC network, complementary information on the structure of the barrier and its evolution with plasma oxidation time has been extracted. Similarly, Landry et al. (2001) have used ac-impedance data on NiFe-Al2 O3 -NiFe junctions to determine an interfacial contribution to the capacitance and to extract the electron screening length in the NiFe electrodes. The complex capacitance of magnetic junctions has been measured also by Huang and Hsu (2004), in their case over a frequency range from 102 to 108 Hz in CoFe-Al2 O3 -CoFe junctions. From the analysis of the so-called Cole–Cole diagrams, a significant sensitivity to the oxidation process of the metallic Al layers is reported, being able to clearly discriminate between different stages in the oxidation process.


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2.3.2 Over- and under-oxidation In order to study the effect of the oxidation process or just to find the optimum oxidation parameters, often a series of samples at various stages of oxidation is produced. This is usually done in one of the following two ways: either by depositing a number of identical samples, oxidized with various oxidation times (see for example Sun et al., 1999; van de Veerdonk, 1999; Gillies et al., 1999, 2000; Song et al., 2000; Tehrani et al., 2000; Park et al., 2002), or by depositing a series of samples with a range of Al thicknesses, all oxidized at once (see Moodera et al., 1997; Song et al., 2000; Tehrani et al., 2000; Boeve et al., 2001; Freeland et al., 2003). The first method requires a rather time-consuming experiment due to the number of oxidation steps, since each oxidation includes a long pump-down stage and possibly sample transport. An example of such an optimization is shown in Fig. 1.20a. The second method, making one batch of samples with a range of Al thicknesses, usually allows for more rapid experiments because only one oxidation step is necessary, see Fig. 1.20b. The use of a wedge (a film with a lateral variation in thickness by linear displacement of a shutter during deposition) is even faster and yields detailed and accurate information (Fig. 1.20c), see for example the work of LeClair et al. (2000c), Song et al. (2000), and Covington et al. (2000). In another method, applied by Nowak et al. (2000) and Song et al. (2000), a series of samples at various stages of oxidation is made by a single deposition and a single oxidation step in a plasma that is not uniform over the wafer. The interpretation of the results in terms of oxidation conditions of this method is of course much more complicated. From the data shown in Fig. 1.20 it is evident that oxidation of the barrier is a critical process, where both over- and underoxidation are detrimental to the TMR effect. Underoxidation leaves the Al layer partially in its metallic state which effectively reduces the tunneling spin polarization of the carriers. The detrimental effect

Figure 1.20 (a) TMR as a function of oxidation time for as-deposited junctions comprised of Co90 Fe10 /17 Å Al + oxidation/Co90 Fe10 (Koller, 2004). (b) Results for Ni80 Fe20 /t Al + oxidation/Ni80 Fe20 (Moodera et al., 1997), and (c) for as-deposited and annealed (250°C) junctions of Ni80 Fe20 /t Al (wedge) + oxidation/Ni80 Fe20 (Covington et al., 2000). Note that the original data from Moodera et al. (1997) have been corrected to match the definition of TMR in Eq. (8). See the references for the composition of the full junction stack as well as for details of the Al oxidation.

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of foreign impurities has been addressed by Jansen and Moodera (1998, 2000) by δ-doping the Al2 O3 barrier with nonmagnetic and magnetic elements. Generally, a strong reduction of TMR is observed and interpreted via the effect of additional impurity-assisted tunneling channels combined with spin-flip processes when magnetic impurities are involved. Theoretically, in a number of papers the effects of impurity-assisted tunneling in magnetic junctions have been addressed; see for example Bratkovsky (1997), Zhang and White (1998), and Jansen and Lodder (2000). Parkin (1998) and LeClair et al. (2000d) alternatively demonstrate this effect of reduced TMR by adding a very thin nonmagnetic film at the barrier interface. As an example, the addition of 1 monolayer of Cu at the interface between the bottom electrode and the barrier reduces TMR by more than a factor of 2; see also section 4.1. The reduction of TMR is also observed for overoxidation, although now the quenching of polarization will be induced by the presence of antiferromagnetic oxides at the barrier interfaces (Moodera and Mathon, 1999). This leads to additional spin flip conductance channels, by which the spin polarization (and TMR) will be suppressed. Using the diagnostic techniques mentioned above, it is established by several groups that an asymmetry in the barrier potential, easily detected by fitting IV -data with the Simmons or Brinkman formula (Eqs. (12) and (13), respectively), is accompanied by a low TMR. Correspondingly, the maximum TMR is found when the junction has a minimal asymmetry, see the work of Sun et al. (1999), Covington et al. (2000), and Oepts et al. (2001). Both over- and under-oxidation cause this asymmetry by creating different electrode-barrier interfaces. An asymmetry in the barrier height is directly observed by Koller et al. (2003) in photoconductance measurements (see section 2.1.1). Due to the large sensitivity of the technique to the presence of Al close to the tunnel barrier, the disappearance of a negative contribution to the photocurrent is correlated to the complete oxidation of the barrier layer and the corresponding maximum in TMR. In order to increase the TMR by preventing such an asymmetry, some modifications to the simple oxidation process are often applied. In several cases, this is accomplished by creating a reservoir of oxygen at the interface with the bottomelectrode. In a later step, this reservoir is supposed to fill the Al-Al2 O3 from the bottom up in order to create a more homogeneous barrier. A first method in which such a reservoir is used is to first slightly oxidize the surface of the bottom electrode before deposition of Al. The subsequent deposition of Al will cause an unstable situation since generally the Gibbs free energy of ferromagnetic oxides is larger than that of Al2 O3 (Dean, 1992). Since Al2 O3 has a lower energy, the oxygen will naturally move into the aluminum. This is applied by Sun et al. (1999) and Kuiper et al. (2001b). In the latter case, the authors find that, when the Co bottom electrode is partially oxidized, up to 10 Å of Al can be completely oxidized by oxygen from this reservoir. A second method involves a small amount of intentional over-oxidation. A gradient of Al or O now exists in the barrier layer. After the top electrode and capping layers are deposited, the junction is subjected to an annealing step, i.e. the sample is brought to higher temperatures in a non-reactive argon (or high vacuum) atmosphere. This causes the oxygen to leave the bottom electrode and move into the


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barrier, often resulting in a more homogeneous barrier layer. This procedure was performed by, for instance, Song et al. (2000) and Dimopoulos et al. (2001b). In all cases, the barrier is homogenized and the TMR is increased. From photoconductance experiments on tantalum-oxide MTJs with variable oxidation time of Ta (Koller et al., 2005b), the shift of the maximum TMR with anneal temperature is accompanied by a similar shift of the oxidation time where the asymmetry of the barrier potential is absent. The experiments of Koller et al. (2005b) directly support the idea that for obtaining the highest magnetoresistance ratio one should anneal MTJs that would be characterized as slightly over-oxidized in the as-deposited state. It is suggested that this result can be understood by a homogenization of the oxygen distribution in the barrier, possibly combined with a change of the bottom barrier-electrode interface. Several groups have attempted to create more homogeneous barriers by applying a two-step process, each step being comprised of Al deposition and oxidation. Yoon et al. (2001) applied this technique with plasma oxidation, Das (2003) with natural oxidation, and Zhang et al. (2003b) with a slow ion-beam process. For all methods, it is reported that the TMR increases slightly in junctions with a two-step process with respect to a one-step process. The RA product increases, often by as much as one order of magnitude (see for example Zhang et al., 2003b). In a comparison between single-step and two-step oxidation experiments, Yoon et al. (2001) have found from X-ray photoelectron spectroscopy that indeed the concentration of O in the barrier is more homogeneous than from single-step plasma oxidation. The TMR of the two-step oxidized junctions is higher, although not much. 2.3.3 Pinholes and dielectric breakdown A pinhole is a path of relatively high conduction between the two electrodes, through the barrier layer. Often this is a metallic short due to inhomogeneous oxidation, or a very thin part of the barrier due to inhomogeneous deposition of the aluminum prior to oxidation. Generally, pinholes can decrease the TMR in two ways. First, for very thin barriers (or for barriers created from a too rough Al) a strong magnetic coupling between the two electrodes may be present in regions where the electrodes are in direct contact. In that case, the free layer no longer fully switches independently from the bottom electrode, and results in a decrease of TMR (Wang et al., 2003; Zhang et al., 2003b). The second, more generally observed, detrimental aspect of pinholes is that the largest part of the current through the junction will run via a normal metallic contact instead of the desired spindependent tunneling across the insulator. This shunting of the tunneling current via metallic shorts has been widely observed and is analyzed by simply modelling a tunnel resistor with an ohmic resistor in parallel (Oliver et al., 2002). Direct visualization of pinholes is usually difficult due to the extremely small dimensions, probably down to a few atomic distances. With the use of a nematic liquid crystal deposited on the sample, pinholes can be directly visualized (Oepts et al., 1998). Upon heating their liquid crystal above the clearing point (56.5°C), the material changes from the nematic state showing optical anisotropy, to the isotropic state. By operating the junction just below the clearing point, the power dissipation due to the pinholes are then visualized as black spots; see Fig. 1.21a. Schad

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Figure 1.21 (a) Polarized light picture of liquid crystal on top of a shadow-evaporated Co/20–22 Å Al2 O3 /Co50 Fe50 junction. The black spot in the middle of the junction surface is the location of a pinhole at a breakdown site (after Oepts et al. (1998)). In (b) a scanning-electron-microscopy image is made after the electrochemical growth of cauliflower-like Cu islands on an oxidized 12 Å Al layer grown on top of 125 Å NiFeCo. After Schad et al. (2000).

et al. (2000) have developed a method for pinhole imaging using electrodeposition of Cu. Selective nucleation at the metallic pinhole locations produces characteristic cauliflower-like structures that can be easily visualized. An example is shown in Fig. 1.21b. An indirect way to detect pinholes is based on the magnetic field generated by the large current density flowing through the pinholes. Due to the vertical direction of the current (⊥ to the plane of the layers), these magnetic fields are in the plane of the free magnetic layer and are thus able to shift or deform the switching behavior from which the location of the pinhole may be extracted (Oliver et al., 2002, 2004). Indirect indications for pinholes are widely reported. E.g., Zhang et al. (2003b) have found evidence for pinholes from resistance and TMR data, combined with the observation of magnetic coupling between the free and fixed magnetic layers. Whereas junctions with a barrier of nominal thickness tAl ≥ 6.5 Å are reported to be pinhole-free, thinner nominal Al layers are clearly suffering from the presence of pinholes. Han and Yu (2004) report on junctions in which the aluminum (9 Å) is underoxidized. The as-deposited junctions show a very low resistance (820 µm2 ) and practically no TMR (0.5%), consistent with transmission-electron-microscopy measurements showing pinholes with a diameter of a few nm. However, after an annealing step, both the junction resistance and the TMR increase enormously to 30 k µm2 and 43%, respectively, which is related to the reduction of pinholes and an improvement of the interfaces with the barrier. Moon et al. (2002) show that in otherwise identical conditions, the usage of a two-step oxidation process instead of a single-step process increases the TMR as well as the RA product. Together with the observation that the magnetic coupling between the bottom and top electrode is reduced in junctions fabricated by the two-step process, they conclude that twostep oxidation creates tunnel barriers with a much lower pinhole density. Pinholes in magnetic tunnel junctions have also triggered a reconsideration of the criteria formulated by Rowell and others (see, e.g., Brinkman et al., 1970) to discriminate true tunneling conductivity from other metallic-like current paths (Garcia, 2000; Jonsson-Akerman et al., 2000; Rabson et al., 2001;


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Figure 1.22 Ramped current stress measurements (a) and a constant voltage stress measurement (b) on junctions consisting of 30 Å Ta/30 Å NiFe/100 Å IrMn/100 Å CoFe/10 Å Al + oxidation/40 Å CoFe/60 Å NiFe/50 Å Cu. The arrows indicate the average breakdown voltage VBD and the moment of breakdown tBD . Adapted from Das (2003).

Akerman et al., 2001). According to these Rowell-criteria the conductance G (1) should vary exponentially with barrier thickness, (2) is parabolic in bias voltage, (3) scales with the junction area, and (4) displays a weak insulator-like temperature dependence; see also section 2.1. Although these are necessary criteria for tunneling, it is shown by Oliver et al. (2002) that they do not rule out the existence of pinholes, especially for junctions with ultrathin ( 0) – G/GAP (V < 0) is analyzed and shown in Fig. 1.41b. The original data are presented in the left panel of the figure. The strong minimum seen in the fcc data can be qualitatively explained by a modified elastic tunneling model using free-electron like bands de-

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Figure 1.41 (a) Conductance dI /dV versus voltage V in parallel (P) orientation for Si(100)/SiO2 /buffer + Co/23 Å Al + oxidation/150 Å Co/50 Å Ta. The open symbols refer to a buffer and magnetic bottom electrode composed of 50 Å Ta/50 Å Co/100 Å FeMn/50 Å fcc(111) Co. Closed symbols refer to 50 Å Ta/poly 50 Å Co where poly relates to polycrystalline and polyphase Co as determined by nuclear magnetic resonance (Wieldraaijer, 2006). (b) Odd part of G/GAP versus voltage for fcc Co and poly Co. The solid line is based on a calculation with the modified elastic tunneling model (Davis and MacLaren, 2002). Data are all taken at T = 5 K. In all cases V > 0 refers to electrons tunneling from the top to the bottom electrode. After LeClair et al. (2002b).

rived from ab-initio electronic-structure calculations. In this model, the conductance is dominated by the contribution of the highly dispersive, s-hybridized density-ofstates of fcc Co to reflect the fact that in these Al2 O3 junctions electrons with s character are decisive for spin-dependent tunneling (corresponding to positive tunneling spin polarization; see section 3). In particular, the presence of two sharp peaks in the s-derived density-of-states above and below EF , as well as a dispersive minority band just above EF , are key to the observed behavior (LeClair et al., 2002b; Davis and MacLaren, 2002). Inspired by these results, Hindmarch et al. (2005b) have measured the odd part of the conductance and TMR in junctions with a Cu38 Ni62 magnetic electrode having a Curie temperature of around 240 K. Due to the low TC of this alloy, the energy of the bottom of the minority spin bands close to the Fermi energy can be followed for temperatures up to the magnetic phase transition. From the odd part of G/GAP versus bias voltage, it is observed that slightly above the Fermi level the band minimum remains fixed in energy until the temperature is raised to around T = 190 K. Beyond this point it abruptly drops below the Fermi level, which is consistent with a Stoner-like collapse of the effective exchange splitting of energy bands responsible for tunneling.

4.4 Towards infinite TMR with half-metallic electrodes The implementation of electrodes with a (nearly) 100% tunneling spin polarization, the so-called half-metallic materials, is expected to yield infinite TMR as indicated by the Julliere formula 2PL PR /(1–PL PR ) with PL and PR equal to ±1. Experimentally


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as well as theoretically, an ongoing intensive research effort is devoted to these materials and their implementation; see, e.g., Pickett and Moodera (2001). Although many predictions of half-metallic behavior have been reported this is verified experimentally only in a few cases, including La0.7 Sr0.3 MnO3 (Park et al., 1998a; Soulen et al., 1998), NiMnSb (Ristoiu et al., 2000), and CrO2 (Ji et al., 2001). In the latter case of CrO2 , Parker et al. (2002) verified the near +100% tunneling spin polarization directly in a superconducting-tunneling-spectroscopy experiment on CrO2 -Cr2 O3 -Al and CrO2 -Cr2 O3 -Pb junctions (see Table 1.2). For La2/3 Sr1/3 MnO3 a polarization of +72% is measured using the same technique (Worledge and Geballe, 2000b). The use of these materials in ferromagnetic-insulator-ferromagnetic junctions is obviously extremely tedious due to the crucial control of two barrier interfaces. Indeed, for junctions employing one or two half-metallic Heusleralloy electrodes such as NiMnSb (Tanaka et al. 1997, 1999) and related MnSb (Panchula et al., 2003), Co2 MnSi (Kammerer et al., 2004; Schmallhorst et al., 2004; Nakajima et al., 2005), Co2 MnAl (Kubota et al., 2004), Co2 Cr0.6 Fe0.4 Al (Inomata et al., 2004), and Co2 FeAl (Okamura et al., 2005), the TMR remains relatively low and may result from oxidation at the Heusler-barrier interfaces or from sitedisordering and structural defects close to the barrier. More promising, Sakuraba et al. (2005a, 2005b) observe magnetoresistances of up to 70% at room temperature and 159% at T = 2 K in UHV-sputtered Co2 MnSi-Al2 O3 -Co75 Fe25 , which corresponds to a low-temperature tunneling spin polarization of +89%, closely approaching the theoretical prediction of half-metallicity. CrO2 -based junctions are not successful in terms of high TMR. As an example, Gupta et al. (2001) have grown CrO2 -Cr2 O3 -Co(Ni81 Fe19 ) tunnel junctions in which CrO2 is epitaxially grown on top of TiO2 , and Cr2 O3 (or a composition close to this) is stabilized by exposing the bottom electrode to an oxygen plasma. In this case, a TMR of only –8% and –2.3% has been achieved at T = 4 K for Co and permalloy, respectively. Magnetite (Fe3 O4 ) is also predicted to be half-metallic due to a gap for the majority band at the Fermi level. Junctions consisting of Fe3 O4 -MgO-Fe3 O4 show, however, only a very small TMR for all temperatures (Li et al., 1998), maybe related to a combination of spin scattering in a magnetically dead interface layer, a distorted spin structure due to a specific interface termination, or due to a reduced oxide such as antiferromagnetic Fe1–δ O present at the interface with the MgO barrier. Also in junctions consisting basically of NiFe-Al2 O3 -Fe3 O4 (with the magnetite fabricated by plasma oxidizing a thin Fe film) only a very small TMR has been reported (Park et al., 2005). The authors suggest that the observed negative sign of TMR is consistent with the expected gap for majority Fermi electrons in Fe3 O4 . On the other hand, Seneor et al. (1999) have reported a positive TMR of +43% at low temperature and +13% at room temperature in sputtered Co-Al2 O3 -Fe3–δ O4 -Al junctions where the iron oxide is sputtered from a Fe2 O3 target. This relatively large TMR is ascribed to the presence of a phase close to magnetite, although the data suggest that the TMR originates predominantly from conduction channels active only above and below the Fermi level. In epitaxial La0.7 Sr0.3 MnO3 -CoCr2 O4 -Fe3 O4 junctions a negative TMR of up to –25% is in qualitative agreement with the theoretically

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predicted negative spin polarization of Fe3 O4 (Hu and Suzuki, 2002). The observed maximum TMR at T ≈ 60 K is attributed to the paramagnetic to ferrimagnetic transition in the CoCr2 O4 barrier. Zhang et al. (2001a) have introduced an Feoxide layer at the barrier interface of CoFe-Al2 O3 -CoFe junctions to improve the thermal stability when annealing up to temperatures of around 400°C (see section 2.3.4). The large TMR measured after annealing is attributed to the formation of Fe3 O4 in the interfacial region, which is confirmed by a follow-up study using transmission electron microscopy combined with electron-energy-loss spectroscopy (Snoeck et al., 2004). In Co-Al2 O3 -NiFe junctions, it is shown that TMR is enhanced by roughly a factor of 1.25 due to δ doping the oxide barrier with an Fe layer with a thickness of less than 2 Å (Jansen and Moodera, 1999). Apart from other explanations, also in this case the possibility of half-metallic Fe3 O4 formation is hypothesized by the authors. Especially in the perovskite materials, a lot of progress has been witnessed as described in the review paper by Ziese (2002). Pioneering experiments are done by Sun et al. (1996, 1997, 1998) on junctions with La2/3 Ca1/3 MnO3 (LCMO) and La2/3 Sr1/3 MnO3 (LSMO) electrodes and SrTiO3 barriers, later also combined with ferromagnetic 3d transition metals (Sun et al., 2000b). Jo et al. (2000a, 2000b) use La2/3 Ca1/3 MnO3 as electrodes in LCMO-NdGaO3 -LCMO junctions reaching TMR magnitudes of more than 500% at low T . La2/3 Sr1/3 MnO3 is used by Lu et al. (1996) and Viret et al. (1997) in combination with oxide barriers such as SrTiO3 , yielding low-temperature magnetoresistances of more than 400%. This reasonably well corresponds to the measured La2/3 Sr1/3 MnO3 tunneling spin polarization of approximately +72% (Worledge and Geballe, 2000b) as discussed before (see also section 3). In the latter superconducting-tunneling-spectroscopy experiment, the authors use La2/3 Sr1/3 MnO3 -SrTiO3 -Al junctions with a thick layer of YBa2 Cu3 O7 grown as a buffer layer on the SrTiO3 substrate to prevent current crowding in the bottom electrode (see section 2.1.3). Junctions consisting of La0.7 Ce0.3 MnO3 -SrTiO3 -La0.7 Ca0.3 MnO3 exhibit a large positive TMR at low temperatures, whereas at intermediate temperatures below TC the sign of the observed TMR is dependent on the bias voltage, suggesting a high degree of tunneling spin polarization dominated by minority spins (Mitra et al., 2003). In the doubleperovskite Sr2 FeMoO6 the predicted half-metallicity has triggered spin-tunneling experiments in Sr2 FeMoO6 -SrTiO3 -Co junctions (Bibes et al., 2003a). The authors have reported a TMR of 50% at low temperature that corresponds to a tunneling spin polarization of more than 85% at the Sr2 FeMoO6 -SrTiO3 interface (see also section 4.5). To a great extent, this confirms the half-metallic character of this double-perovskite compound. Bowen et al. (2003) have convincingly demonstrated the impact of half-metals in MTJs. Epitaxial LSMO-SrTiO3 -LSMO junctions have been grown by pulsed laser deposition and careful post-deposition lithographic processing, yielding a TMR of 1850% at T = 4 K (see Fig. 1.42). This corresponds to a tunneling spin polarization of ≈ 95% when both LSMO-SrTiO3 interfaces are assumed to be equal. At higher temperatures though, the TMR is gradually suppressed and disappears at 280 K, below the Curie temperature of bulk LSMO, which is related to the interface structure and specifically the LSMO termination at the barrier interface


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Figure 1.42 Magnetoresistance measurements of 350 Å La2/3 Sr1/3 MnO3 /28 Å SrTiO3 /100 Å La2/3 Sr1/3 MnO3 epitaxial junctions using SrTiO3 substrates. On top a 150 Å Co layer is deposited and subsequently oxidized for magnetically pinning the top electrode. (a) Relative change in resistance ([R – RP ]/RP ) versus applied magnetic field H at T = 4.2 K and a bias voltage of 1 mV. In (b) and (c) the temperature dependence of TMR is shown for two junctions with different area, using V = 10 mV. Solid curves are guides to the eye. After Bowen et al. (2003).

(Pailloux et al., 2002). In a follow-up study by Garcia et al. (2004), the relatively low value of TC in LSMO could be exploited to measure how the temperature dependence of tunneling spin polarization is related to M(T ), a similar approach as followed by Hindmarch et al. (2005a) for ferromagnetic Cu38 Ni62 . In section 2.1.4 such a relation between tunneling spin polarization and the (surface) magnetic moment has been suggested to describe the temperature dependence of TMR for regular Al2 O3 -based MTJs. In Figs. 1.43a and 1.43b the TMR of La2/3 Sr1/3 MnO3 TiO2 -La2/3 Sr1/3 MnO3 and La2/3 Sr1/3 MnO3 -LaAlO3 -La2/3 Sr1/3 MnO3 junctions is plotted versus temperature. From this the tunneling spin polarization is calculated via the Julliere formula P (T ) = [TMR(T )/(2 + TMR(T ))]1/2 and is plotted in Fig. 1.43c and Fig. 1.43d for TiO2 and LaAlO3 , respectively. A close resemblance with the bulk magnetization of separately grown trilayers is observed, although the Curie temperature deduced from P (T ) is roughly 60 K lower than the temperature where M(T ) vanishes (350 K). Apparently, the magnetism of interfacial LSMO is well preserved at the interfaces with TiO2 , LaAlO3 , and SrTiO3 (not shown), certainly when comparing it with the polarization of a free surface of LSMO measured with spin-polarized photoemission (Park et al., 1998b). In that case a much stronger decay with temperature is observed, evidencing that free surfaces and embedded interfaces have strongly different properties in manganites (Garcia et al., 2004). The use of half-metallic electrodes is particularly attractive for directly extracting density-of-states or band-structure features from the bias dependence of the tunneling transport. When only majority electrons are tunneling from half-metallic LSMO, one is able to the directly probe the majority (minority) density-of-states

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Figure 1.43 Temperature dependence of TMR of epitaxial junctions containing (a) 350 Å La2/3 Sr1/3 MnO3 /32 Å TiO2 /100 Å La2/3 Sr1/3 MnO3 and (b) 350 Å La2/3 Sr1/3 MnO3 /28 Å LaAlO3 /100 Å La2/3 Sr1/3 MnO3 . The lines in (a) and (b) are guides to the eye. The normalized tunneling spin polarization deduced from TMR is shown as a function of temperature normalized to TC for the junction with TiO2 (c) and LaAlO3 (d). The solid line in (c) and (d) is the normalized magnetization measured on similar trilayers. After Garcia et al. (2004).

of the counter electrode when the magnetizations are (anti)parallel oriented. This idea is exploited by Bowen et al. (2005a, 2005b). They find a quantitative confirmation of the half-metallic band structure of La2/3 Sr1/3 MnO3 by measuring the conductance and TMR of LSMO-SrTiO3 -LSMO junctions for variable bias voltages. First of all, it is observed that the conductance dI /dV in one bias direction for parallel oriented magnetization of the LSMO layers shows a dramatic collapse at V ≈ 0.82 V, whereas the antiparallel conductance continues to increase; see Fig. 1.44a. The collapse in parallel conductance proves that no minority band is available at EF from which electrons can tunnel into the minority t2g band, demonstrating the half-metallic nature of the LSMO. It also proves that the majority electrons available at EF do not find any empty majority states at EF + Eg with Eg ≈ 0.82 eV; see the schematic diagram in Fig. 1.44c. This is consistent with a pseudo-gap in the majority density-of-states of the eg bands as predicted by Pickett and Singh (1998) for a distorted oxygen environment of Mn ions in manganites. In a related paper, the energy difference δ between the Fermi energy and the bottom of the minority t2g band is accurately extracted from TMR, conductance and conductance derivative measurements in these junctions (Bowen et al., 2005a). In Fig. 1.44b, d2 I /dV 2 reveals a sudden upturn of the antiparallel conductance at V ≈ 0.34 V. This marks the onset of a conduction channel for majority electrons tunneling into the minority t2g band at EF + δ (see the schematics in Fig. 1.44d). This turns out to be in good agreement with data obtained from spin polarized inverse photoemission experiments, yielding δ = 0.38 ± 0.05 eV. When the bias voltage across these LSMO junctions exceeds δ /e, the low-temperature TMR is seen to rapidly decrease with voltage (not shown). This is again due to the opening of a new conduction channel in the antiparallel orientation, corroborating


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Figure 1.44 (a) Conductance dI /dV versus applied bias voltage V of 350 Å La2/3 Sr1/3 MnO3 /28 Å SrTiO3 /100 Å La2/3 Sr1/3 MnO3 epitaxial junctions, for both antiparallel and parallel oriented magnetization. The conductance collapse in the parallel case (at ≈ 0.82 V) is due to the absence of conduction channels at V = Eg /e, as shown in the schematic band diagram in (c); adapted from Bowen et al. (2005b). In (b) the derivative of the conductance d2 I /dV 2 is shown for bias voltages below 0.5 V. The increase of d2 I /dV 2 in antiparallel orientation observed at V = δ/e ≈ 0.34 V marks the onset of tunneling into the minority (t2g ) spin band; see the schematics in (d). The lines in (b) are added to better visualize the conductance upturn. Adapted from Bowen et al. (2005a).

the predictions by Bratkovsky (1997) for the bias-voltage dependence of magnetic tunnel junctions with half-metallic electrodes.

4.5 Role of the barrier for TMR Now that we have seen that TMR may be tuned towards very large numbers by a proper choice of ferromagnetic materials, one should realize that the combined system of (magnetic) electrodes and barrier material is decisive for the magnitude of TMR and tunneling spin polarization (section 3). In a series of remarkable experiments on La0.7 Sr0.3 MnO3 -insulator-Co junctions, de Teresa et al. (1999a, 1999b) have used the full polarization of the half-metallic LSMO as a detector of the spin polarization of Co adjacent to tunnel barriers of a different character. When using traditional alumina in LSMO-Al2 O3 -Co, a positive TMR is found at temperatures well below room temperature, which, via the Julliere formula, reflects a positive spin polarization of Co. Although this is contrary to what is expected from the smaller density-of-states at EF for the Co majority spin channel, this is believed to reflect the positive polarization of s electrons that dominate the tunneling process (see also the more elaborate discussions in section 3). A striking sign reversal of TMR is observed when replacing the alumina by SrTiO3 or Ce0.69 La0.31 O1.845 . In this case, it appears that electrons with a d-like character are now preferentially transmitted at the Co-SrTiO3 or Co-Ce0.69 La0.31 O1.845 interfaces (Fig. 1.45a). Moreover, when using a double barrier in a LSMO-SrTiO3 Al2 O3 -Co junction the TMR is positive again, see Fig. 1.45b. Apparently, the electronic structure and chemical bonding at the Co-insulator interface is decisive for the tunneling spin polarization rather than the electron tunneling processes

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Figure 1.45 (a) TMR as a function of bias voltage V of SrTiO3 (001)/350 Å La2/3 Sr1/3 MnO3 /25 Å SrTiO3 /300 Å Co measured at T = 5 K and T = 30 K. The inset shows the resistance change ([R – RP ]/RP ) with applied magnetic field H at 5 K using a bias of –0.4 V. (b) TMR versus bias voltage at T = 40 K as in (a), but now with a composite barrier: SrTiO3 (001)/350 Å La2/3 Sr1/3 MnO3 /10 Å SrTiO3 /15 Å Al2 O3 /300 Å Co. (c) Relative position of the DOS in La2/3 Sr1/3 MnO3 and the d DOS at an fcc Co(001) surface for a bias around zero. The arrow indicates the high tunneling probability between the majority band of LSMO and the minority band of Co, when the magnetization is antiparallel (AP). For V < 0 electrons tunnel into the empty states of Co above the Fermi level EF . After de Teresa et al. (1999a, 1999b); note that in these papers TMR is alternatively defined as ([RP –RAP ]/RAP ).

in the full barrier. The dependence of TMR on bias voltage is another interesting aspect of the LSMO-SrTiO3 -Co junctions; see Fig. 1.45a. Since the conductance is determined by only one spin channel, the variations with bias are found to be easily correlated with the d-character density-of-states of a Co(001) surface; see Fig. 1.45c. At a negative bias voltage of around –0.4 V, the majority electrons are tunneling into the predicted peak in the (unoccupied) minority density-of-states of Co above the Fermi level, leading to a maximum in negative TMR. In a more general perspective, these experiments show that the interfacial bonding is of critical relevance for spin-dependent tunneling of electrons. When d–d bonding is allowed by using barriers with d-orbitals such as in SrTiO3 , it is possible to observe the d-dominated spin polarization of Co (P < 0). In the opposite case of alumina barriers, the absence of d orbitals apparently favors an s-dominated tunneling current (P > 0). Although these arguments are helpful to qualitatively understand the role of the barrier for tunneling spin polarization, it is evident that a more solid theoretical basis is required to substantiate this. This will be further discussed later on in this section. Thomas et al. (2005) have directly measured the tunneling spin polarization of Co-SrTiO3 -Al by superconducting tunneling spectroscopy, yielding a positive spin polarization of +31% (see Table 1.2) in striking contrast to the aforementioned results at low bias voltage (de Teresa et al., 1999a, 1999b). This could be explained by the thermal evaporation of the barrier on polycrystalline Co, leading


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to an amorphous SrTiO3 layer as seen by high-resolution transmission electron microscopy (instead of the epitaxial barriers in the work of de Teresa et al. (1999a, 1999b)). Correspondingly, also a rather small (positive) TMR of around +1% has been measured at low temperatures in Co-SrTiO3 -Co, Co-SrTiO3 -Ni80 Fe20 , and Co-TiO2 -Co-Ni80 Fe20 junctions (Thomas et al., 2005). The tedious role of the chemical structure of the barrier and the interfaces with the ferromagnetic layers in these LSMO-based junctions is also recognized in other experimental studies, see for example Sun et al. (2000b) and Hayakawa et al. (2002), showing both negative and positive TMR in CoFe-SrTiO3 -LSMO and Fe-SrTiO3 -LSMO, strongly and asymmetrically dependent on the bias voltage. In the work of Oleynik et al. (2002), first-principles density-functional calculations of the atomic and electronic structure of Co-SrTiO3 -Co(001) MTJs have established the key importance of the atomic arrangement at the barrier interfaces. It is found that the most stable structure represents the TiO2 -terminated interface with the O atoms lying on top of the Co atoms. At the interface with Co, an induced magnetic moment of 0.25 μB on the interfacial Ti atoms is aligned antiparallel to the magnetic moment of the Co layer, which may indeed lead to a negative tunneling spin polarization of the Co-SrTiO3 barrier (Oleynik et al., 2002; Oleynik and Tsymbal, 2003). Using ab initio transport calculations including firstprinciples band structure methods, Velev et al. (2005) predict a very large TMR (1000% and more) in Co-SrTiO3 -Co junctions with bcc Co(001) electrodes and barriers typically 7 to 11 monolayers in thickness. The complex band structure of SrTiO3 enables an extremely efficient tunneling of minority d electrons from the Co, causing the tunneling spin polarization to be negative. From the calculations it is estimated that a single Co-SrTiO3 interface carries a tunneling spin polarization of –50% that is rather independent of the barrier thickness. This is roughly a factor of 2 higher than P derived from the experiments of de Teresa et al. (1999a), and may be explained by effects of interface disorder, e.g. locally affecting the structure of bcc Co. It should be emphasized that these results show that a spin-polarized tunneling current across SrTiO3 is carried by minority d electrons. This is essentially different as compared to sp-bonded insulators such as Al2 O3 (sections 2 and 3) and MgO (section 4.6), where tunneling is dominated by electrons from majority bands. The argument of interface (chemical) bonding has also been used to explain the sign reversal of TMR observed in junctions with others barriers containing dtype ions. Experiments by Sharma et al. (1999) on Ta2 O5 are already discussed in section 3.3.3. Bibes et al. (2003b) have investigated junctions with TiO2 barriers. La2/3 Sr1/3 MnO3 -TiO2 -Co shows a negative TMR of around –3% at low temperature. Regarding the positive spin polarization of La2/3 Sr1/3 MnO3 (though against a SrTiO3 barrier by Worledge and Geballe (2000b)), the tunneling spin polarization of Co-TiO2 is negative, similar to the experiments by de Teresa et al. (1999a, 1999b) for interfaces of Co-SrTiO3 or Co-Ce0.69 La0.31 O1.845 . Also Co-Cr2 O3 and Ni81 Fe19 -Cr2 O3 interfaces display a negative spin polarization as determined from TMR in junctions with one half-metallic CrO2 electrode, the other electrode being Co or NiFe (Gupta et al., 2001). A related experiment has been performed using the ferromagnetic double perovskite Sr2 FeMoO6 with a TC of 415 K, and a predicted half-metallicity (Kobayashi et al., 1998). Bibes et al. (2003a) have obtained a +50%

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TMR at low temperature in a junction consisting of SrTiO3 -Sr2 FeMoO6 -SrTiO3 Co. Using Julliere’s formula and the experimental fact that the epitaxial SrTiO3 -Co interface carries a spin polarization at low bias voltage of about –25% (de Teresa et al., 1999a), this yields a very strong tunneling spin polarization of –80%. It is important to be aware of the fact that only in a few systems the presence of negative tunneling spin polarization has been confirmed straightforwardly by superconducting tunneling spectroscopy (section 3). In the case of Co-SrTiO3 interfaces, the expected negative tunneling spin polarization of around –25% (as deduced from the low-bias TMR data of de Teresa et al. (1999a) using LSMO as the second electrode) may be directly tested by STS on Co-SrTiO3 -Al superconducting junctions. However, the system of Co-SrTiO3 -Al is extremely difficult to realize epitaxially and may suffer from oxidation of either Al or Co, both reducing the spin polarization. STS data obtained by Thomas et al. (2005) using amorphous SrTiO3 indeed did not yield the anticipated negative spin polarization as mentioned earlier. A negative tunneling spin polarization of –9.5% is for the first time measured by Worledge and Geballe (2000c) using ferromagnetic SrRuO3 in a superconducting junction consisting of SrTiO3 (100)-YBa2 Cu3 O7 -SrRuO3 -SrTiO3 -Al (see Table 1.2). This negative sign is supported by theoretical calculations and emphasizes the crucial role of weighting the density-of-states factors in Eq. (20) with transmission probabilities for the tunneling processes. Also in Co1–x Gdx ferrimagnetic alloys for 0.2 x 0.75 a negative tunneling spin polarization has been observed directly from STS (Kaiser et al., 2005a). This is explained by the relative contribution of independent spin-polarized tunneling currents from the two sublattice magnetizations (see section 3.3.2). Via the Julliere formula TMR = 2PL PR /[1 – PL PR ], the negative tunneling spin polarization is in agreement with a negative magnetoresistance in junctions with one electrode of Co1–x Gdx (P < 0) and a counter electrode of Co70 Fe30 (P > 0).

4.6 Coherent tunneling in MgO junctions In the previous sections, it is emphasized that TMR is certainly not determined by the spin polarization of the individual ferromagnetic electrodes. Instead, it is sensitively dependent on the full system of ferromagnetic electrodes and the adjacent barrier, in which the electronic structure modifications at the barrier-electrode interface and the symmetry and matching of the electron wave functions are playing a crucial role. Based on this, it could be conceivable that certain electrode-barrier material combinations would allow for a highly efficient polarization of the spin currents, even with a bulk density-of-states displaying only a modest spin polarization. In Fe-ZnSe-Fe(001) junctions (MacLaren et al., 1999), it is theoretically shown that for thick enough barriers the conductance is dominated by slowly decaying s-states at k = 0 as provided by a 1 -band at the Fermi level of Fe(001). Together with the absence of a minority 1 -band at EF , this leads to a very strong asymmetry in the conductance and hence a large TMR. Experimentally, however, no such dramatic pseudo-half-metallic effects have been observed for ZnSe barriers. Gustavsson et al. (2003) report on a lowtemperature TMR of only 16% in a Fe-ZnSe-Co0.15 Fe0.85 junction, disappearing


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above T ≈ 50 K. Jiang et al. (2003b) have found magnetoresistance of less than 25% at low temperature and ≈ 10% at room temperature in ZnSe-based MTJ’s, which, although potentially relevant for low RA product MTJs, is again not in agreement with the promises given by theory. Although the interfaces between ZnSe(001) and Fe are reported to be very sharp without magnetically dead or modified interfacial regions even after annealing up to 300°C (Marangolo et al., 2002), it could be that significant modifications of the Fe spin-polarized band structure near EF as determined from spin-polarized inverse photoemission lead to a suppression of TMR (Bertacco et al., 2004). Also the presence of mid-gap localized states in the ZnSe barrier due to a small amount of disorder is shown to significantly suppress or even change the sign of TMR in epitaxial Fe-ZnSe-Fe junctions (Varalda et al., 2005). Similarly, the use of the II-VI compound ZnS has yielded magnetoresistances of only 5% at room temperature (Guth et al., 2001b; Guth et al., 2001a). In this case, it is suggested that the observation of an indirect ferromagnetic interaction across the insulating ZnS is mediated by the tunneling electrons (Dinia et al., 2003). Later on in this section, we will return to interlayer coupling across insulating spacers. Now we will concentrate on the spin-dependent transport properties when MgO barriers are employed. The experimental use of these barriers has also been triggered by theoretical predictions of pseudo-half-metallic behavior in Fe-MgOFe(001), and has resulted in a number of intriguing new observations, which will be extensively discussed below. 4.6.1 TMR of MgO-based junctions Using different theoretical approaches, both Butler et al. (2001b, 2005) and Mathon and Umerski (2001) come basically to the same conclusion for coherent tunneling in an Fe-MgO-Fe(001) magnetic tunnel junction, i.e. for electrons tunneling normal to the barrier in the [001] direction. For majority electrons, there are four Bloch states of different symmetry present around the Fermi level for k = 0, viz. a double-degenerate 5 state compatible with pd symmetry, 2 with d symmetry, and a 1 state with spd symmetry. However, for the minority spins the 1 state is replaced by a d-type 2 state. Due to its s-type character, only the Bloch states of 1 symmetry are able to effectively couple with the evanescent sp states in the MgO barrier region, which, at the Fermi level, is only available for majority electrons. This pseudo-half-metallicity of the band structure in the [001] direction is schematically shown in Fig. 1.46 and Fig. 1.47a by the absence of 1 minority states for tunneling electrons. For thick enough barriers, the majority conductance in parallel alignment of magnetization becomes fully dominated by these 1 -band contributions, and, correspondingly, extremely large TMR in these junctions (of 1000% and more) are expected to show up experimentally (Butler et al., 2001b; Mathon and Umerski, 2001). Early experiments using MgO as a barrier have only been partially successful. First of all, when the electrodes are polycrystalline and the MgO is amorphous (Moodera and Kinder, 1996; Platt et al., 1997; Smith et al., 1998; Kant et al., 2004c), only a modest TMR or tunneling spin polarization is found. In fully epitaxial systems grown by molecular beam epitaxy combined with pulsed

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Figure 1.46 Layer-resolved tunneling density-of-states (DOS) for k = 0 in Fe(100)/ 8 monolayers MgO/Fe(100) for majority electrons (a) and minority electrons (b) when the magnetization of the Fe layers is parallel oriented. Each curve is labelled by the symmetry of the incident Bloch state in the left Fe electrode, showing, for example, the absence of minority states with 1 symmetry, whereas the majority 1 states decay only very slowly in the MgO barrier. After Butler et al. (2001b).

Figure 1.47 (a) Calculated band dispersion of Fe in the [001] ( -H) direction. Solid and dotted curves represent majority and minority-spin subbands, respectively; as indicated, thicker lines are the 1 subbands. Adapted from Yuasa et al. (2004a). (b) Calculated local spin-polarized density-of-states for Fe at the bottom interface with MgO in Fe/MgO/Fe (grey) and Fe/FeO/MgO/Fe (black), the latter representing the presence of one complete O layer between Fe and MgO. EF is the Fermi level. Top panel is for majority electrons, bottom panel for minority electrons. After Tiusan et al. (2004).

laser deposition, the TMR is reported to be quenched by defects in the epitaxial MgO barrier (Klaua et al., 2001; Wulfhekel et al., 2001). Bowen et al. (2001) have reported a TMR of 60% at 30 K and 27% at room temperature in Fe-MgO-FeCo(001) by combining laser ablation and sputtering. In the case of


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Figure 1.48 Inner-loop resistance when switching the free magnetic layer, expressed as (R – RP )/RP , versus magnetic field strength H for MTJs with a crystalline MgO barrier. (a) Junctions consisting of 100 Å TaN/250 Å IrMn/8 Å Co84 Fe16 /30 Å Co70 Fe30 /29 Å MgO/150 Å Co84 Fe16 /100 Å Mg, annealed at TA = 120°C and 380°C (after Parkin et al. (2004)). (b) Junctions of 100 Å Ta/150 Å PtMn/25 Å Co70 Fe30 /8.5 Å Ru/30 Å Co60 Fe20 B20 /18 Å MgO/30 Å Co60 Fe20 B20 /100 Å Ta/70 Å Ru measured at T = 20 K and T = 300 K, after an anneal at 360°C (after Djayaprawira et al. (2005)).

Fe-MgO-Fe-Co grown by molecular beam epitaxy (Faure-Vincent et al., 2003), a TMR of 67% has been observed at room temperature, increasing up to around 100% at low T (see also the earlier work of Popova et al., 2002). Since the TMR is still far from the existing theoretical predictions, the authors attribute this to the growth-induced difference in topology of the two interfaces by which the required symmetric matching of the wave functions is affected. By first-principle calculations of the electronic structure of Fe-FeO-MgO-Fe, it is theoretically demonstrated that the chemical bonding between Fe and O strongly reduces the conductance in parallel orientation (Zhang et al., 2003a). The corresponding reduction in TMR could suggest that oxide formation at the barrier interfaces may be a common problem for epitaxial MgO-based junctions; see also the surface X-ray diffraction experiments by Meyerheim et al. (2001). On the other hand, Tusche et al. (2005) have shown that oxygen at the barrier interfaces may promote a fully coherent growth of Fe on top of the MgO spacer, leading to a coherent and symmetric MTJ structure characterized by FeO layers at both Fe-MgO interfaces. A considerably improved room-temperature magnetoresistance in MgO junctions has been found by Parkin et al. (2004). They have observed giant TMR values up to ≈ 220%, whereas at low T it rises towards 300%. In their approach, exchange-biased CoFe-MgO-CoFe(001) junctions are fabricated with regular sputtering deposition, the MgO being reactively magnetron-sputtered in an Ar-O2 mixture, and the full stack subsequently annealed at relatively high temperature (up to 380°C). In Fig. 1.48a an example curve for these junctions is displayed. Obviously, these films are not epitaxial but polycrystalline and (001)-textured (including the MgO barrier), which suggests that especially the well-defined crystalline orientation of the barrier and electrodes is key to the strong tunneling spin polarization. Separately, STS measurements on CoFe-MgO-Al junctions are used to directly

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measure the tunneling spin polarization. A positive P of 85% is found in optimized junctions in accordance with the dominance of majority electrons with 1 symmetry as indicated above. Via the Julliere formula TMR = 2PL PR /(1 – PL PR ) this relates to a magnetoresistance of ≈ 520% at low T , corresponding to a TMR effect of around 260% at room temperature when correcting for the T dependence of TMR, which is in close agreement with the magnetoresistance data (Parkin et al., 2004). An even higher TMR at room temperature is found when MgO is sandwiched between amorphous CoFeB ferromagnetic electrodes. Djayaprawira et al. (2005) have compared the magnetoresistance of magnetron-sputtered structures containing either Co70 Fe30 -MgO-Co70 Fe30 or CoFeB-MgO-CoFeB, where CoFeB is sputtered from a Co60 Fe20 B20 target. The barriers are deposited using rf sputtering directly from a MgO target. All junctions are annealed at 360°C. As shown by transmission electron microscopy, the structural quality of MgO in the CoFe junctions is very poor and the interfaces are rough. In that case, a TMR of only 62% at room temperature is observed. When growing MgO on top of the CoFeB, it shows after the anneal a good crystallinity with a preferred (001) orientation, most probably due to the amorphous nature of the underlying CoFeB (although some parts are crystallized). In the CoFeB-MgO-CoFeB junctions, the TMR ratio is now 230% at room temperature increasing to 294% at T = 20 K; see Fig 1.48b. It seems that for obtaining this very high TMR the correct structural symmetry of the MgO(001) barrier is crucial, although it is presently not clear how an amorphous magnetic electrode can give rise to giant TMR in view of the importance of the electrode band structure along the k = 0 direction. In this respect, the authors do not exclude the possibility that the annealed junctions show local (re)crystallization of a few monolayers of CoFeB at the electrode-barrier interfaces, beyond the detection limit of transmission electron microscopy. Indeed, Yuasa et al. (2005b) have shown that a sputtered, amorphous Co60 Fe20 B20 layer grown on top of e-beam evaporated Mg(001) crystallizes in a bcc structure with (001) orientation, after annealing at temperatures of around 360°C. In this reflective high-energy electron diffraction study, it is also demonstrated that a MgO layer grown on amorphous CoFeB initially has an amorphous structure as well, and begins to crystallize in the (001) orientation only when tMgO exceeds 5 monolayers. Hayakawa et al. (2005b) have shown from transmission-electron-microscopy images that annealing of Co40 Fe40 B20 -MgO-Co40 Fe40 B20 junctions at sufficiently high temperature results in the formation of highly-oriented crystalline CoFeB electrodes that were initially amorphous in the as-deposited state, a crystallization process that is initiated at both the interfaces with MgO. When annealing a junction with a MgO thickness of around 20 Å at 375°C, this yields an optimal TMR of 260% at room temperature and 403% at T = 5 K. By varying the Ar pressure for sputter deposition of MgO (Ikeda et al., 2005), their optimized annealed junctions exhibit a room-temperature TMR of 355%, and 578% at T = 5 K. An additional improvement of the crystalline orientation of the Mg(001) layer may be achieved by introducing an ultrathin ≈ 4 Å Mg layer between the bottom CoFeB electrode and MgO (Tsunekawa et al., 2005). Especially for MgO(001) layers in the range between 7 Å and 11 Å the addition of Mg during growth as suggested by Linn and


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Figure 1.49 TMR at T = 20 K and T = 293 K at low bias voltage as a function of the thickness of the MgO barrier tMgO . (b) Cross-sectional transmission electron microscopy of an MTJ with tMgO = 18 Å, using two different magnifications showing the excellent crystallinity of the layers. The junction stack consists of MgO(001)/MgO seed/1000 Å Fe/t MgO/100 Å Fe/100 Å IrMn. After Yuasa et al. (2004b).

Mauri (2005) leads to a considerable enhancement of TMR and is typically well above 100% (Tsunekawa et al., 2005). These huge TMR values are accompanied by extremely small RA products of only a few µm2 , an attractive combination never reached in alumina-based junctions (see section 2.2, Fig. 1.17). Comparable giant TMR values (180% at room temperature, 250% at T = 20 K) have been demonstrated in single-crystalline (001)-oriented Fe-MgO-Fe-IrMn junctions grown by molecular beam epitaxy (Yuasa et al., 2004a, 2004b). Apart from the large magnetoresistances, the variation of TMR with the thickness of the MgO barrier shows a number of interesting features, see Fig. 1.49. To start with, on the average the TMR increases with the thickness of the MgO, and saturates beyond tMgO ≈ 20 Å. This is in qualitative agreement with the experiments by Hayakawa et al. (2005b) on sputtered junctions, and is also in line with the aforementioned predictions (MacLaren et al., 1999; Butler et al., 2001b; Mathon and Umerski, 2001). When the barrier is thick, the conductance is dominated by electrons with the momentum vector normal to the barrier (k ≈ 0), by which electrons in the highly polarized Fe-1 band lead to giant TMR values. For thinner barriers the TMR effect is suppressed by the increasing probability of electrons tunneling with off-normal momentum vector. It is shown by Belashchenko et al. (2005b) from first principles that for small MgO thickness the minority spin bands at the interfaces make a significant contribution to the tunneling conductance in the antiparallel orientation of the Fe layers. In agreement with the data, this efficiently reduces the TMR in Fe-MgO-Fe for small tMgO . Additionally, in the experiments of Yuasa et al. (2004b) TMR versus tMgO is shown to clearly oscillate with a period of 3.0 Å over the full spectrum of barrier thicknesses (Fig. 1.49), not dependent on temperature or bias voltage. It is emphasized that the period is not corresponding to the thickness of one monolayer of MgO(001), i.e. 2.2 Å.

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Figure 1.50 Resistance at T = 20 K and T = 300 K expressed as (R – RP )/RP versus magnetic field strength H for fully-epitaxial MgO-based MTJs consisting of (a) MgO(001)/200 Å MgO(001)/1000 Å Fe(001)/23 Å MgO(001)/100 Å Fe(001)/100 Å IrMn/500 Å Au, and (b) MgO(001)/200 Å MgO(001)/1000 Å Fe(001)/5.7 Å bcc Co(001)/21 Å MgO(001)/100 Å Fe(001)/100 Å IrMn/500 Å Au. In (c) TMR is shown as a function of the composition x of the Fe1–x Cox bottom electrode. For the junction with Fe50 Co50 the thickness of the MgO barrier is 21 Å. In all experiments the bias voltage is 10 mV. After Yuasa et al. (2005a).

The apparent intrinsic origin of the oscillation is not yet understood. It could be related to quantum interferences in coherent tunneling processes across MgO, e.g. due to the difference in complex wave vectors for the 1 and 5 evanescent states in MgO (Butler et al., 2001b). On the other hand, this seems to be at odds with hot spots in momentum space with extremely high tunneling probability, which would result in a single-period oscillation (Butler et al., 2001b; Wunnicke et al., 2002). Using first-principles electronic-structure calculations, Zhang and Butler (2004) have predicted that the magnetoresistance can be further enhanced by using bcc Co and (B2-type) chemically ordered bcc Co50 Fe50 in MgO junctions. Again, for these systems the large magnetoresistance can be understood from the slowly decaying 1 states available for majority electrons only. However, in the case of bcc Co(Fe) the 1 band turns out to be the only majority band that crosses the Fermi level, whereas for Fe there are others crossing EF for k = 0. Yuasa et al. (2005a) have fabricated fully epitaxial bcc Fe1–x Cox (001)-MgO(001)-Fe(001) junctions to test these predictions. In agreement with the calculations, the bcc Co electrodes yield a higher TMR of 271% at room temperature and 353% at 20 K, as compared to the Fe electrodes with 180% and 247%, respectively; see Figs. 1.50a and 1.50b. However, for bcc Co50 Fe50 TMR is of the same magnitude as for the Fe bottom electrode (Fig. 1.50c). The authors suggest that the disordered character of their Co50 Fe50 layer explains the discrepancy with the calculations assuming perfect B2-type chemical ordering. Finally, it is worth mentioning that apart from the transport properties, the magnetic and electronic properties of thin 3d ferromagnetic elements (such as bcc Co) in contact with MgO are being studied both theoretically and experimentally. As an example, Sicot et al. (2005, 2006) have used X-ray absorption spectroscopy and X-ray photoemission spectroscopy to demonstrate a weak hybridization between epitaxial MgO(001) and Co(001) or Fe(001), and to


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rule out the formation of undesired magnetic oxides at the interfaces. Using X-ray magnetic circular dichroism, it is observed that the magnetic moments of Fe and Co are enhanced with respect to the bulk magnetic moments, ruling out the possibility of magnetically quenched regions at the interfaces with epitaxial MgO layers (see also Sicot et al., 2003, and the calculations for MgO-Fe by Li and Freeman, 1991). Similar experiments have also been reported by Miyokawa et al. (2005), although in this case exclusively focusing on the properties of 1 or 2 monolayers of bcc Fe(001) embedded in a Co(001)-Fe(001)-MgO(001) structure. 4.6.2 Bias-voltage dependence of MgO-junctions The bias-voltage dependence of the magnetoresistance of MgO-based junctions needs special attention. The epitaxially grown junctions of Yuasa et al. (2004b) show a remarkably small dependence on applied voltage, viz. V1/2 > 1.0 V for positive bias at room temperature (Fig. 1.51a). This is much better than usually reported for alumina-based junctions where V1/2 ranges roughly between 0.3 and 0.6 V (see also section 2.1.4). Interestingly, the bias voltage dependence is strongly asymmetric with respect to the sign of the voltage, see again Fig. 1.51a for a junction with a barrier of 20 Å MgO. However, the asymmetry becomes weaker upon an increase of the barrier thickness (up to 32 Å). It is hypothesized that the asymmetry may be explained by structurally unequal MgO-electrode interfaces (Yuasa et al., 2004a). The asymmetry is only slightly present in similarly grown Fe(001)-MgO(001)-Fe(001) epitaxial junctions (Nozaki et al., 2005). In this case the room-temperature V1/2 is around 0.7 V, combined with a TMR of almost 90%; see Fig. 1.51b. However, in double barrier junctions of Fe-MgO-Fe-MgO-Fe grown by the same group (see again the figure), the asymmetry with respect to the sign of the bias is also observed, yielding V1/2 = +1.44 V, and V1/2 = –0.72 V for opposite bias direction. Tenta-

Figure 1.51 TMR of epitaxial MgO junctions as a function of applied bias voltage V at room temperature. (a) MgO(001)/MgO seed/500 Å Fe/20 Å MgO/100 Å Fe/100 Å Co structures after Yuasa et al. (2004a). (b) Double-barrier junctions consisting of MgO(001)/MgO seed/500 Å Fe/20 Å MgO/15 Å Fe/20 Å MgO/200 Å Fe (solid curve), and regular single-barrier junctions of MgO(001)/MgO seed/500 Å Fe/20 Å MgO/15 Å Fe/100 Å Co (symbols). After Nozaki et al. (2005). (c) Structures of MgO(001)/500 Å Fe/25 Å MgO/50 Å Fe/100 Å Co junctions (open symbols) and Pd-seeded junctions of MgO(001)/400 Å Pd/20 Å Fe/25 Å MgO/50 Å Fe/100 Å Co (closed). After Tiusan et al. (2004). In all cases positive biasing (V > 0) corresponds to electron tunneling from the bottom into the top electrode.

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tively, this is ascribed to details of the band structure of the ultrathin 15 Å Fe layer sandwiched between two barriers (Nozaki et al., 2005). To shed more light on this bias dependence and the role of the electronic structure of the Fe layers in MgO junctions, Tiusan et al. (2004) have shown that a dramatic asymmetry in the bias dependence of TMR exists in their epitaxial FeMgO-Fe junctions, even leading to a sign reversal at negative bias voltages. In the case of V < 0, electrons are tunneling into the perfect bottom Fe(001) electrode where a so-called interface resonance for minority electrons is believed to strongly enhance the conductance in anti-parallel alignment; see Fig. 1.51c. The impact of these interface resonances has been treated in several theoretical papers (MacLaren et al., 1999; Butler et al., 2001b; Mathon and Umerski, 2001; Wunnicke et al., 2002) and is due to electron confinement between the bulk and the barrier region. For thin enough barriers, at particular discrete values of k in the two-dimensional Brillouin zone of the Fe minority electrons the tunneling transmission can become very high, leading to huge conductance spikes. In Fig. 1.47b the interface state is predicted from electronic-structure calculations on Fe-MgO-Fe (Tiusan et al., 2004), and is still present when a full monolayer of O is introduced between Fe and MgO, although shifted away with respect to the Fermi level. Note that the effect of bias voltage and interface resonances on TMR in Fe-FeO-MgOFe is treated from first principles by Zhang et al. (2004), although in that case the predicted small or even negative TMR at low bias is strongly enhanced for higher bias voltages (up to 0.5 V). Summarizing these remarks, it is suggested that for negative bias voltage across such an epitaxial junction (Tiusan et al., 2004), electrons are tunneling into the bottom epilayer at energies comparable to the location of the interface state leading to a strong enhancement of the antiparallel conductance. For large enough V this finally reverses the sign of TMR. Interestingly, the effect of the interface resonances is almost completely destroyed when the bottom electrode is backed with Pd (Tiusan et al., 2004). Mainly due to the absence of electron states starting 0.2 eV above the Fermi level in Pd, the interfacial resonant states of the Fe are no longer coupled to the electronic states of bulk Fe and drastically affect the propagation of Bloch states. In the case of backing the Fe layer with Pd, TMR becomes almost independent of bias voltage with V1/2 exceeding voltages of 1.5 V (Fig. 1.51c). Surprisingly, the mechanisms involved in the reduction of TMR as described in section 2.1.4 for Al2 O3 junctions are not active in these epitaxial junctions. As a reminder, these mechanisms include quenching of TMR by magnon creation, by intrinsic density-of-states effects, or by scattering at barrier imperfections. Note that a similar insensitivity of bias voltage has been observed by vacuum tunneling between a magnetic CoFeSiB tip and a clean Co(0001) surface, suggesting that defect scattering in the barrier of traditional alumina-based junctions is dominating the observed bias dependencies (Ding et al., 2003). Finally, it should be mentioned that the observation of the very strong bias dependence of TMR observed in Fig. 1.51c for V < 0 (Tiusan et al., 2004) is probably related to the presence of a small amount of carbon at the bottom interface between Fe and MgO incorporated during the growth of the sample (including the required annealing steps). Due to this, the conductance of propagating 1 states will be strongly suppressed and gets more


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Figure 1.52 Bias-voltage and temperature dependence in junctions consisting of 100 Å Ta/150 Å PtMn/25 Å Co70 Fe30 /8.5 Å Ru/30 Å Co60 Fe20 B20 /0–10 Å Mg/18 Å MgO/30 Å Co60 Fe20 B20 /100 Å Ta/40 Å Ru. (a) TMR and resistance V /I versus bias voltage V at T = 77 K. (b) Conductance dI /dV versus bias voltage V (77 K). (c) TMR and resistance V /I as a function of temperature. Resistances and conductances are shown for both parallel (P) and antiparallel (AP) magnetization directions. After Miao et al. (2005).

sensitive to d-like features in the band structure including the interface resonances, which otherwise would be negligible for the MgO thicknesses employed in these experiments. Indeed, in C-free junctions, only a very small asymmetry in TMR(V ) is reported, probably due to a residual asymmetry in the Fe-MgO interfaces related to roughness, defects, or lattice distortions (Tiusan et al., 2006). In the bias dependence of the conductance dI /dV and the derivative of the conductance d2 I /dV 2 also a number of unique features are present in epitaxial Fe(001)-MgO(001)-Fe(001) junctions, never observed in Al2 O3 -based magnetic junctions (Ando et al., 2005). In the antiparallel orientation of the Fe layers at low temperature (T = 6 K), a broad peak develops in d2 I /dV 2 at around ±1 V upon an increase of the MgO barrier thickness (varied between 25.5 Å and 31.4 Å). Tentatively, this is related to conduction channels between majority and minority spin 1 bands that open up at sufficiently high bias voltages. For parallel orientation of the Fe magnetizations, d2 I /dV 2 clearly oscillates with voltage and peaks at around ±0.1 V, ±0.35 V, and ±0.8 V, independent of the barrier thickness (again at 6 K). This excludes the possibility of quantum-well formation in the barrier region, and, moreover, these oscillations are not expected from tunneling dominated only by 1 bands at k = 0. It is suggested by Ando et al. (2005) that at higher bias voltage tunneling may be governed by minority electrons tunneling via complex interface resonant states at certain points in k-space with k = 0 (see Butler et al., 2001b and Mathon and Umerski, 2001). Also for sputtered MgO-based junctions with high TMR, data are reported on the bias dependence of the resistance, conductance dI /dV , and the derivative of the conductance d2 I /dV 2 . As an example, Miao et al. (2005) report on a decrease in conductance at a bias voltage of around 0.4 V in Co60 Fe20 B20 -MgO-Co60 Fe20 B20 junctions, indicating the emerging contribution from minority band states at these energies (see Fig. 1.52b). At lower voltages, dI /dV and d2 I /dV 2 contain features arising from magnon-assisted tunneling (see section 2.1.4) and phonon excitations in MgO; see also the work of Ando et al. (2005). An intriguing result for these junctions is seen in the bias voltage and temperature dependence of the resistance

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R = V /I when the magnetic layers have the same magnetization direction (parallel). This is illustrated in Fig. 1.52a and Fig. 1.52c. Contrary to the typical signatures for quantum-mechanical tunneling (section 2.1.4), there is hardly any variation of R with temperature, nor is there a variation of R with applied bias voltage V ; see also the I (V ) data obtained by Hayakawa et al. (2005b). This is probably related to the decay rate in the MgO barrier for states of 1 symmetry. At energies up to 0.5 eV away from the Fermi level (covering the voltages used in the experiments), the complex momentum vector varies extremely slowly with energy, typically less than 1%. In accordance with the experimental data for parallel magnetized Fe layers, this leads to a surprisingly small resistance (or conductance) change with bias voltage, typically less than 10% over this energy range (Miao et al., 2005). 4.6.3 Interlayer coupling across MgO-barriers As an intriguing spinoff to the use of epitaxial MgO barriers, the observed superior coherence of the tunneling electrons in these junctions could facilitate an interlayer coupling between the two ferromagnetic electrodes across the barrier. Due to the insulating character of the spacer, the well-known oscillatory Ruderman–Kittel– Kasuya–Yosida interaction (Fert and Bruno, 1994; Bürgler et al., 1999) is excluded to mediate the coupling. Slonczewski (1989) has derived an antiferromagnetic coupling from the torque produced by rotation of the magnetization of one layer relative to the other due to the tunneling electrons, which is further extended by Bruno (1995) to predict the temperature dependence. At T = 0 K, the coupling strength J across a barrier of thickness t and barrier height φ reads: φ e–2κt f (κ, kF ,maj , kF ,,min ), (21) 8π 2 t 2 where kF ,maj and kF ,min are the Fermi wave vectors of majority and minority electrons, respectively, and κ the imaginary component of the wave vector of electrons in the barrier with k = 0 at the Fermi level, corresponding to κ = (2me φ /h¯ 2 )1/2 . Note that Eq. (11) in section 1 is based on a similar free-electron model calculation by Slonczewski (1989), although in that case for the tunneling spin polarization. The function f (κ, kF ,maj , kF ,min ) in Eq. (21) can be either positive or negative depending on the Fermi wave vectors of the electrodes and the barrier height, and therefore determines whether J is ferromagnetic (parallel magnetization of the two layers) or antiferromagnetic (antiparallel magnetization). Faure-Vincent et al. (2002) have measured the magnetization loops of Fe-MgOFe-Co multilayers with thin MgO spacers (down to 5 Å) to detect the presence of coupling. As can be seen from the shift of the inner magnetization loop of the magnetically softer bottom Fe layer, an antiferromagnetic coupling is present for tMgO = 5.0 Å (see Fig. 1.53a). Upon an increase of the MgO spacer thickness the shift of the hysteresis curve is drastically reduced as illustrated for tMgO = 6.3 Å. In the right panel of the figure, the antiferromagnetic coupling is shown to be strongly suppressed for thicker spacers and becomes ferromagnetic beyond 7 Å due to a small ferromagnetic so-called orange-peel coupling (Néel, 1962). This is a dipolar type of interlayer coupling when two magnetic layers have a correlated periodic J =


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Figure 1.53 Interlayer coupling across MgO in layered structures composed of MgO(001)/500 Å Fe/4–25 Å MgO/50 Å Fe/500 Å Co/100 Å V. (a) Inner-loop normalized magnetization M/MSAT along the easy axis after a positive saturation magnetic field H , in a field range where the Fe/Co bilayer is magnetically rigid. The shift of the curve for tMgO = 5.0 Å towards positive H is a signature of antiferromagnetic coupling. (b) Interlayer coupling strength J versus thickness of the MgO spacer, tMgO , together with a fit using the model of Slonczewski (1989), see Eq. (21). After Faure-Vincent et al. (2002, 2003).

modulation of interface roughness, frequently observed across metallic spacers (see, e.g., Coehoorn, 2003) but also across insulating tunnel barriers; see section 2.1.2. Note that the presence of a small amount of C at the bottom interface between Fe and MgO as discussed earlier in the bias dependence of MgO-based junctions (Tiusan et al., 2006) does not affect these data on interlayer coupling. Despite the simplicity of the free-electron calculation of Slonczewski (1989), the authors show that a perfect agreement can be obtained using Eq. (21) with realistic effective parameters, as can be seen from the fit in the right panel of the figure (Fig. 1.53b). In conclusion, the results of Faure-Vincent et al. (2002) represent a clear signature for the existence of an intrinsic interlayer coupling due to spin-polarized tunneling of electrons between ferromagnetic layers. In the case of ZnS barriers (Dinia et al., 2003), qualitative fingerprints for interlayer coupling mediated by tunneling electrons have been reported, viz. the coupling strength varies monotonically and non-oscillatory with spacer thickness, and is increasing with temperature as predicted by Bruno (1995). However, since the sign of the coupling is positive in this case, it could be partially obscured by ferromagnetic orange-peel coupling (Dinia et al., 2003). In a calculation by Zhuravlev et al. (2005), an alternative explanation is presented to describe the interlayer coupling data of Faure-Vincent et al. (2002). When it is assumed that the barrier is not perfect but contains impurities or defects, the localized states within the gap of the insulator lead to a significant enhancement of the coupling. Moreover, for certain impurity energies a crossover from antiferromagnetic to ferromagnetic coupling with spacer thickness is predicted. This is in line with the experimental data shown in Fig. 1.53b, without the need to consider dipolar orange-peel coupling. As to the temperature dependence of the impurityassisted interlayer coupling, it is shown that for impurity levels in the vicinity of the

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Fermi level the coupling strength is suppressed when raising the temperature. This is opposite to the model of Bruno (1995) for perfect barriers, for which the thermal population of the electronic states above the Fermi level leads to an increase of the coupling strength with temperature. As a final remark to the interlayer coupling across MgO, it is suggested by Tiusan et al. (2006) that interface resonances related to the minority channel in Fe(001)-MgO(001)-Fe(001) could be crucial for the observed antiferromagnetic coupling without the need to include impurity bands in the MgO. As a reminder, interface resonances can strongly enhance the tunnel conductance when the Fe layers are aligned antiparallel (MacLaren et al., 1999; Butler et al., 2001b; Mathon and Umerski, 2001; Wunnicke et al., 2002), and are believed to play a crucial role in the bias dependence of MgO-based MTJs (Tiusan et al., 2004; Zhang et al., 2004). To substantiate this hypothesis, it would be required to study these electronic-structure effects by an ab-initio calculation of the coupling, which is beyond the approach of Slonczewski (1989) and Zhuravlev et al. (2005) using free-electron spin-split energy bands.

5. Outlook In the foregoing sections, the physics of spin-dependent tunneling is addressed focusing on a number of critical scientific breakthroughs in the field of MTJs. Since the first discoveries in the mid-nineties, the field is rapidly expanding in many other directions, related to various alternative hybrid material combinations often motivated by new application potential for solid-state devices. A few prominent examples are: • • • • • •

hybrid semiconductor magnetic tunnel junctions tunnel barriers used for spin injection into semiconductors magnetic tunnel transistors magnetic semiconductor spin-filtering barriers spin-torque effects in nanometer-scale ferromagnetic junctions spin-logic devices using magnetic tunnel junctions.

In the same order, these exciting research directions will be shortly introduced below. One of the first observations of TMR in all-semiconductor magnetic tunnel junctions is reported for Ga1–x Mnx As-AlAs-Ga1–x Mnx As (Tanaka and Higo, 2001). Ga1–x Mnx As is a p-type ferromagnetic diluted magnetic semiconductor with a TC of more than 100 K due to so-called carrier-induced ferromagnetism; see Story et al. (1986) and Dietl (2002). For reviews on diluted magnetic semiconductors, see, e.g., de Jonge and Swagten (1991) and Dobrowolski et al. (2003). The magnetic electrodes in the junctions of Tanaka and Higo (2001) are structurally well matched to the AlAs spacer, the latter with a thickness typically between 13 Å and 28 Å. A large TMR of more than 75% is observed for junctions with a thin (≤ 16 Å) AlAs barrier when the magnetic field is applied along the [100] axis in the plane of the film,


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see also Higo et al. (2001). Similarly, also in epitaxial ferromagnetic MnAs-AlAsMnAs junctions TMR has been observed, although in that case with a markedly smaller magnitude of typically 1.4% at T = 10 K persisting up to room temperature (Sugahara and Tanaka, 2002). Garcia et al. (2005) have improved this by measuring a TMR of up to 12% at T = 4 K in MnAs-GaAs-AlAs-GaAs-MnAs incorporating thin 10 Å GaAs layers to prevent Mn to diffuse towards the barrier region. Together with data on junctions with single GaAs barriers, it is found that resonant tunneling in these junctions may occur via a mid-gap defect band, which explains the rather low TMR (less than 2% for the GaAs barriers). However, by using a resonanttunneling model a large tunneling spin polarization at the MnAs-GaAs interface is deduced of around 60%. Ferromagnetic Cr1–δ with a TC in the range of 170-350 K has been used in Cr1–δ -GaAs-AlAs-GaAs-Cr1–δ junctions with δ ≈ 0.33, where the GaAs thin layers are used to prevent Mn, Cr, or Te atoms to diffuse into the barrier. A magnetoresistance of up to 15% at T = 5 K is reported, though rapidly decreasing with bias voltage and temperature (Saito et al., 2005a). Related to TMR in these semiconductor tunnel junctions, Gould et al. (2004) have surprisingly found a new magnetoresistance effect in GaAs(001)-Ga1–x Mnx As-Al2 O3 -Ti-Au using a single ferromagnetic layer only. This so-called tunneling anisotropic magnetoresistance is due to the large spin–orbit interaction in the valance band of Ga1–x Mnx As, which causes the density-of-states at the Fermi level, and therefore the conductance, to depend on the direction of magnetization; see also Brey et al. (2004). This observation may have important consequences for the interpretation of the aforementioned results in all-semiconductor MTJs, as shown by the extremely large tunneling anisotropic magnetoresistance of more than 150,000% in Ga1–x Mnx As-GaAs-Ga1–x Mnx As. In similar junctions with a ZnSe barrier, Saito et al. (2005b) have carefully disentangled genuine TMR of up to 100% from a 10% tunneling anisotropic magnetoresistance effect. To efficiently inject spin-polarized currents into semiconductor layers, one generally deals with the so-called conductivity mismatch between a ferromagnetic metal and the poorly conducting semiconductor (Schmidt et al., 2000). One way to circumvent the mismatch is to separate the layers by a nonmagnetic tunneling barrier. In this way, the spin-dependent resistance of the combined ferromagnetic-insulator can be better matched to that of the semiconductor, leading to efficient spin injection in semiconductors (Fert and Jaffres, 2001). As an experimental example, it is reported by Motsnyi et al. (2002) that electrons with more than 9% spin polarization can be injected in GaAs-based light-emitting diodes at T = 80 K using a CoFeAl2 O3 tunneling system. A small magnetic field under 45 degrees with the film normal is applied to manipulate the spins in the semiconductors via the so-called oblique Hanle effect (Motsnyi et al., 2003). This is necessary to optically asses the spin polarization in the light-emitting diode, for which a nonzero component of the electron spin normal to the sample surface is required. Efficient spin injection from Fe-Al2 O3 into a GaAs light-emitting diode is demonstrated by van ’t Erve et al. (2004), with a spin polarization in the GaAs of up to 40% at T = 5 K, in their case using large magnetic fields applied perpendicular to the film plane. A much higher efficiency in this geometry is reported when exploiting the extremely large tunneling spin polarization for crystalline MgO barriers (see section 4). In the experiment

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of Jiang et al. (2005), the spin injection efficiency is estimated to be at least 52% at T = 100 K and 32% at 290 K for a CoFe-MgO(100) tunnel injector. A fully electrical demonstration of spin injection into semiconductors is reported by Mattana et al. (2003). They use a Ga1–x Mnx As-AlAs-GaAs-AlAs-Ga1–x Mnx As double-barrier junction to inject spins from the Ga1–x Mnx As-AlAs into the GaAs, which is subsequently measured by the AlAs-Ga1–x Mnx As detector. Similar to the aforementioned all-semiconductor devices, thin GaAs films are incorporated between Ga1–x Mnx As and AlAs to avoid diffusion. It is shown that the transport properties in these junctions can be explained by sequential tunneling across the two barriers with a spin relaxation small enough to efficiently transmit spins across GaAs (Mattana et al., 2003). Al2 O3 tunnel barriers are also successfully employed in so-called magnetic tunnel transistors (Sato and Mizushima, 2001). Via an Al2 O3 layer, hot electrons are injected from a nonmagnetic emitter layer into a metallic base layer consisting of two ferromagnetic metals separated by a nonmagnetic metal. Subsequently, only those hot electrons are collected in a n-type GaAs collector that have sufficiently large energy to overcome a semiconductor Schottky barrier. The collector current strongly depends on the relative orientation of the two ferromagnetic layers due to spin-dependent filtering of hot electrons in the ferromagnetic layers. In a system of GaAs(001)-CoFe-Cu-NiFe-Al2 O3 -Cu, van Dijken et al. (2003) have determined that the relative change in the collector current is exceeding 3400% at T = 77 K (see also the discussion of Jansen et al., 2003 and Jiang et al., 2003a). Also alternative magnetic tunnel transistors with one ferromagnetic emitter and one ferromagnetic base layer are feasible with relative collector current changes of 64% at room temperature in GaAs(111)-CoFe-Al2 O3 -CoFe-IrMn-Ta (van Dijken et al., 2002). By tuning the bias voltage across the tunneling barrier, it is demonstrated that magnetic tunnel transistors can be used as a powerful tool to study hot-electron transport over a wide range of energies. Using a transport model similar as for ballistic electron emission microscopy (see section 2.1), spin-dependent inelastic electron scattering in the ferromagnetic base layer and electron scattering at the base-collector interface are included to describe the experimental data (Jiang et al., 2004b). As a final promising direction, magnetic tunnel transistors can also be used for efficient injection of spin-polarized carriers in a GaAs-based light-emitting diode. In samples of NiFe-CoFe-Al2 O3 -CoFe-Ta grown op top of (basically) a GaAs-InGaAs multiple quantum well, a spin polarization of around 10% is determined from analyzing the electroluminescence (Jiang et al., 2003b). As we have seen in section 4, the implementation of half-metallic electrodes or the use of crystalline barriers have yielded very large magnetoresistance ratios, corresponding to a tunneling spin polarization of up to 100%. Another route to full spin polarization is the use of magnetic spin-filter tunnel barriers. Due to the spin splitting of the conduction band of magnetic insulators such as EuS or EuO, their barrier height becomes spin-dependent and can act as a very efficient spin filter. This effect has been experimentally demonstrated by Moodera et al. (1988), Hao et al. (1990), and Santos and Moodera (2004) in STS experiments using one superconducting Al probe layer, the magnetic barrier, and a nonmagnetic layer as the counter electrode. Inspired by the proposed electrical device by Worledge and


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Geballe (2000d), a magnetoresistance effect can be obtained when combining a magnetic spin-filter barrier with one magnetic and one nonmagnetic electrode. In Al-EuS-Gd junctions, LeClair et al. (2002a) have reported TMR effects of more than 100%, provided that the temperature is below TC of the ferromagnetic EuS (≈ 17 K). However, the magnetic switching of the magnetic constituents is rather poor, probably related to details of the EuS-Gd interface; see Smits et al. (2004). Gajek et al. (2005) have used single-crystalline, insulating BiMnO3 as a spin filtering barrier having a much higher Curie temperature of 105 K. In La2/3 Sr1/3 MnO3 SrTiO3 -BiMnO3 -Au junctions a thin 10 Å SrTiO3 layer is used to magnetically separate LSMO from the spin filter. Although the observed TMR effects are only 50% at T = 3 K (corresponding to a filter efficiency of ≈ 22%), these experiments show the potential of using insulating oxides for spin filtering and spin injection. In this respect, NiFe2 O4 is considered as an extremely promising candidate for an insulating spin-filtering barrier with TC above room temperature (850 K in the bulk), although in thin films a metallic behavior is observed when growing it on singlecrystalline SrTiO3 (Lüders et al., 2005b). However, by tuning the conductivity via different growth conditions, NiFe2 O4 has been used as an insulating spin-filter in La2/3 Sr1/3 MnO3 -NiFe2 O4 -Au and La2/3 Sr1/3 MnO3 -SrTiO3 -NiFe2 O4 -Au junctions where a thin SrTiO3 layer in the latter structures is again used for decoupling the magnetic layers (Lüders et al., 2005a). Typically, a magnetoresistance of 52% is observed at T = 4 K, corresponding to a NiFe2 O4 spin-filter efficiency of around 23%. In a general perspective, calculations of Yin et al. (2005) have shown that a further enhancement of the magnetoresistance for this class of devices is feasible when a magnetic spin-filter barrier is combined with two instead of only one magnetic electrode. However, no experimental evidence is yet available to confirm this. When the current density within magnetic multilayers becomes sufficiently high, it has been experimentally demonstrated that the magnetic moment of the itinerant electrons may produce a so-called spin torque on the magnetization. Due to the torque, a rotation or even switching of a magnetic layer is feasible (Katine et al., 2000), as well as the possibility to create excitations of micro-wave frequencies (Kiselev et al., 2003). Especially for advanced MRAM applications, it is envisioned that such a novel switching scheme of the memory cell would no longer necessitate the traditional use of separate word or bit lines. In MTJs based on Co88.2 Fe9.8 B2 Al2 O3 -Co88.2 Fe9.8 B2 and with a very low RA product of < 5 µm2 , Fuchs et al. (2004) have demonstrated these effects when the junctions are laterally structured down to sub-micrometer dimension. Higo et al. (2005) have shown reproducible spin-torque switching of the free magnetic layer in 75 nm × 163 nm CoFe-RuCoFeB-Al2 O3 -NiFe junctions at threshold current densities of around 106 A/cm2 . A current density of 7 × 106 A/cm2 is reported for junctions based on a CoFeB free layer in CoFe-Ru-CoFeB-Al2 O3 -CoFeB (Huai et al., 2005). A low current density to switch the magnetization can be combined with the intrinsically large TMR in MgO-based magnetic junctions (section 4.6). In CoFe-Ru-CoFeB-MgO-CoFeB junctions of 100 nm×200 nm, threshold current densities of around 2×107 A/cm2 are reported Kubota et al. (2005a, 2005b). Hayakawa et al. (2005a) yield a current

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density of only 2.5 × 106 A/cm2 in similar MgO junctions having a TMR well above 100% at room temperature. Due to spin-torque effects, domain walls in magnetic materials are able to move by a (spin-polarized) electrical current passing across the domain wall. In a so-called shiftable magnetic shift register (Parkin, 2004), domain-wall movement is believed to create a new storage solution with many advantages as compared to solid-state memory and magnetic disks. An electric current is applied in order to move magnetic domains along a track, a strip of ferromagnetic material comprised of a large number of magnetic domains with a magnetization direction representing the state of the bit. Magnetic fields fringing from a domain wall in a strip-like writing device are used to set the magnetization within the track of the shift register. By sending a current through a magnetic tunnel junction that is part of the storage device, it is possible to read the stored magnetization direction of the domains. Another new device using magnetic tunnel junctions is a so-called spin-torque diode which may become relevant for applications in telecommunication circuits (Tulapurkar et al., 2005). It is again based on the torque of a spin-polarized current acting on small magnetic elements, leading, as mentioned before, to a highfrequency rotation of the magnetization (Kiselev et al., 2003). In their experiment, Tulapurkar et al. (2005) apply a radio-frequency alternating current to a nanometerscale magnetic junction, thereby generating a DC voltage across the device when the frequency is resonant with the spin excitations arising from the spin-torque effect. A final intriguing application of MTJs is emerging within the field of creating logic devices for computing and programming. Although many variations are feasible for these novel magnetic devices, such as the implementation of thin metallic layers at the barrier interface (You and Bader, 2000) or the combination of giant magnetoresistance with resonant tunneling diodes (Hanbicki et al., 2001), a simple concept is introduced by Hassoun et al. (1997) and later adapted specifically for MTJs by Richter et al. (2002) and Ney et al. (2003). In the latter configuration, a single MTJ cell offers the possibility to create nonvolatile output with basic logic operations such as (N)AND and (N)OR by addressing a number of additional current lines to predefine the magnetization directions; see also Moodera and LeClair (2003).

ACKNOWLEDGEMENTS Wim de Jonge, Corné Kant, Karel Knechten, Jürgen Kohlhepp, Bert Koopmans, Patrick LeClair, and Paresh Paluskar are acknowledged for many useful discussions and for critically reading this manuscript. Some of the results presented here are embedded in research programs of the Technology Foundation STW and the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, both financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”.

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CHAPTER

TWO

Magnetic Nanostructures: Currents and Dynamics Gerrit E.W. Bauer *,** , Yaroslav Tserkovnyak *,*** , Arne Brataas *,**** and Paul J. Kelly *****

Contents 1. Introduction 2. Ferromagnets and Magnetization Dynamics 3. Magnetic Multilayers and Spin Valves 3.1 Non-local exchange coupling and giant magnetoresistance 3.2 Non-equilibrium spin current and spin accumulation 3.3 Spin-transfer torque 3.4 Angular magnetoresistance of spin valves 4. Non-Local Magnetization Dynamics 4.1 Current-induced magnetization dynamics 4.2 Spin pumping 5. The Standard Model 5.1 Enhanced Gilbert damping and spin battery 5.2 Current-induced magnetization reversal and high frequency generation 5.3 Dynamic exchange interaction 5.4 Noise in magnetic heterostructures 6. Related Topics 6.1 Tunnel junctions 6.2 Domain walls 6.3 Spin transport by thermal currents 7. Outlook * ** *** **** *****

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Centre for Advanced Study at the Norwegian Academy of Science and Letters, Drammensveien 78, NO-0271 Oslo, Norway Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Faculty of Science and Technology and Mesa+ Research Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands



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Abstract The magnetization order parameter in magnetic nanostructures can be excited by torques from internal and external magnetic fields as well as electrically induced spin currents. Conversely, a time-varying magnetization emits spin currents that couple spatially separated magnetic elements in circuits and devices. Here we review the principles of timedependent magnetoelectronic circuit theory, a simple yet effective theoretical framework that captures the essential physics and in favourable limits allows quantitative description of phenomena such as Gilbert damping enhancement, coupled magnetization dynamics, and the spin battery effect.

1. Introduction Since the discovery of giant magnetoresistance (GMR) in ferromagnetic multilayers (Grünberg, 2001; Levy, 1994), i.e. the modulation of electron transport by magnetic-field-induced configuration changes of the magnetization profile, the use of ferromagnetic elements in electronic circuits and devices has mushroomed. The GMR effect is employed in read heads for mass data storage devices. Magnetic random access memories (MRAMs) are based on the related effect of tunnelling magnetoresistance (TMR) between two ferromagnets separated by a tunnel barrier. MRAMs have the advantage of being non-volatile, which means that an applied voltage is not needed to maintain a given memory state, and are therefore serious competitors for flash memories in processors, reprogrammable logic applications etc. These and other applications are reviewed by Parkin (2002) (see also the Whitebook on Innovative Mass Storage Technology, http://www.ex.ac.uk/IMST2002). With decreasing feature size of magnetic elements in magnetic storage media, magnetic read heads, and MRAM elements, the time and energy needed to read and write a magnetic domain are crucial parameters studied intensively by industry and academia. The magnetization dynamics of ferromagnetic films and particles under the influence of a magnetic field are reviewed by Miltat et al. (2002). Basic research in magnetoelectronics is concentrated on small hybrid structures and novel materials. The unifying concept is that of spin-accumulation, i.e. a non-equilibrium magnetization that can, e.g., be injected electrically into a nonmagnetic material from a ferromagnetic contact by applying a voltage (Johnson and Silsbee, 1988a, 1988b). A breakthrough in magnetoelectronics was the observation of current-induced magnetization reversal in layered structures fabricated into pillars with diameters down to about 50 nanometers (Kiselev et al., 2003; Krivorotov et al., 2005; Myers et al., 1999). This effect was predicted a few years earlier and arises from the transfer of spin angular momentum by the applied current (Bazaliy et al., 1998; Berger, 1996; Slonczewski, 1996). Current-induced magnetization switching has already been used to demonstrate a scalable MRAM concept, the spin-RAM (Hosomi et al., 2005) based upon MgO magnetic tunnel junctions (Parkin et al., 2004; Yuasa et al., 2004). Conversely, magnetization dynamics

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induces spin currents in a conducting heterostructure (Tserkovnyak et al., 2002) which in combination with the spin-transfer torque leads to crosstalk between different ferromagnets through conducting spacers: the dynamic exchange interaction (Heinrich et al., 2003). The relation to earlier ideas of Berger (1996) is discussed by Tserkovnyak et al. (2003). These novel effects arise from magnetization dynamics in hybrid devices whereby magnetic elements are coupled by spin and charge currents induced by either an applied bias or time-dependent magnetic fields. The dynamics therefore becomes non-local, i.e. it is not a property of a single ferromagnetic element, but depends on the whole magnetically active (or spin-coherent) region of the device. We believe that the basic physics is well understood by now and undertake a brief review in this contribution. For more technical expositions we refer to Brataas et al. (2006a), Stiles and Miltat (2006), and Tserkovnyak et al. (2005).

2. Ferromagnets and Magnetization Dynamics A ferromagnet has a broken symmetry ground state in which the spins of a majority of the electrons point in a certain common direction below a critical temperature which can be as high as 1388 K. The robustness of the magnetic order (the insensitivity of its magnitude and direction to elevated temperatures and external perturbations) is employed in applications as diverse as compass needles, refrigerator-door stickers, and memory devices. In spite of its apparent stability, ferromagnetism is neither rigid nor static. As a result of competition between exchange interactions, magnetocrystalline and shape anisotropies, uniform magnetic order is often unstable with respect to domain formation that lowers the magnetostatic energy. Thermal fluctuations reduce the macroscopic moment until it vanishes at the critical temperature Tc . At temperatures sufficiently below Tc , the internal dynamics of the ferromagnet are described by low-energy transverse fluctuations of the magnetization (spin waves or magnons), that are the magnetic equivalents of lattice vibrations (phonons). Classical coarsegrained computer simulations of the detailed position- and time-dependent magnetization (“micromagnetism”) describe these phenomena well (Miltat et al., 2002). When magnetic grains become sufficiently small, the exchange stiffness renders domain structures energetically unfavorable and a single-domain picture is adequate. At low temperatures, higher-energy spin waves freeze out and only the lowest-energy, zero-wavevector spin wave is excited, which is nothing but a rigid precession of the entire ferromagnetic order parameter. Restricting the ferromagnetic degrees of freedom to this mode is often referred to as the “macrospin model”. In thermodynamic equilibrium, the macrospin then points in a certain fixed direction with small thermal fluctuations around it. It can still be forced to change by applying external magnetic fields at an angle to the magnetization direction. The system then moves in response to minimize its Zeeman energy. The compass needle, a freely suspended single-domain ferromagnet with a sufficiently high anisotropy (coercivity), does this by mechanical alignment. Here we are interested

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in mechanically fixed magnets whose magnetic moments move in the presence of external and internal (exchange and anisotropy) effective magnetic fields. Viscous damping processes are required to achieve a reorientation (switching) of the magnetization under a suddenly applied magnetic field. Minimization of the switching time by engineering magnetic anisotropies as well as magnetization damping rates is an important goal in the design of fast magnetic memories. When the applied magnetic fields are large enough to surmount the anisotropies, the magnetization can be reversed, often by large amplitude and complex trajectories, even in the simple macrospin model. At finite temperatures, the magnetization reorientation becomes probabilistic and is described by a Fokker–Planck equation on the unit sphere (Brown, 1963). A traditional starting point in studying the transverse magnetization dynamics in a ferromagnetic medium is based on the phenomenological Landau–Lifshitz (LL) approach (Landau et al., 1980). The magnetization M(r, t) with direction m(r, t) = M(r, t)/Ms and (constant) magnitude Ms is treated in this approach as a classical position- and time-dependent variable, obeying equations of motion which are determined by the free-energy functional F [M] for degrees of freedom coupled to the magnetization distribution M (such as the electromagnetic field or itinerant electrons experiencing a ferromagnetic exchange field): ∂ m(r, t) = –γ m(r, t) × Heff (r), (1) ∂t where γ is (minus) the gyromagnetic ratio. For free electrons γ = 2μB /h¯ > 0. In transition-metal ferromagnets, it is usually close to this value. ∂F [M] (2) ∂M is the “effective” magnetic field. The magnetic free energy and effective field can be decomposed into applied external, demagnetization, magnetocrystalline anisotropy, and exchange fields. To lowest order in the frequency, dissipation can be described phenomenologically by an additional torque in Eq. (1) (Gilbert, 1955, 2004): Heff (r) = –

∂ ∂ m = –γ m × Heff + α0 m × m, (3) ∂t ∂t where α is the dimensionless Gilbert damping constant (it is sometimes convenient to work with a different Gilbert parameter G = α0 γ Ms ). As required, Eq. (3) preserves the local magnitude of the magnetization. For example, for a constant Heff obeying Eq. (2) and α0 = 0, m precesses around the field vector with frequency ω = γ Heff . When damping is switched on, α0 > 0 (assuming positive γ , as in the case of free electrons), the precession spirals down to a time-independent magnetization along the field direction, i.e., the lowest free-energy state, on a time scale of (α0 ω)–1 . For an axially symmetric effective field and close to equilibrium, the Landau–Lifshitz–Gilbert (LLG) equation (3) is obeyed by a small-angle damped circular precession. Equation (3) very successfully characterizes the dynamics of ultrathin ferromagnetic films as well as


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bulk materials in terms of a few material-specific parameters that are accessible to ferromagnetic-resonance (FMR) experiments (Bhagat and Lubitz, 1974; Heinrich and Cochran, 1993). In nanostructures of ferromagnets characterized by strong exchange interactions, spatial magnetization gradients cost a great deal of energy and may be disregarded. We then arrive at the “macrospin” model, in which (3) reduces to a non-linear differential equation in the unit vector m(t).

3. Magnetic Multilayers and Spin Valves The discovery that the energy of magnetic multilayers consisting of alternating ferromagnetic (F) and normal (N) metal films depends on the relative direction of the individual magnetizations (Grünberg et al., 1986) is perhaps the most important in magnetoelectronics. The existence of the antiparallel (AP) ground-state configuration at certain spacer-layer thicknesses was essential for the subsequent discovery of giant magnetoresistance (Baibich et al., 1988; Binasch et al., 1989). Adjacent ferromagnetic layers in such structures are coupled by nonlocal and, as a function of the thickness of the normal-metal layer, oscillatory (Parkin et al., 1990) exchange interaction that can be qualitatively understood using perturbation theory in analogy to the RKKY exchange coupling between magnetic impurities in a normal-metal host (Kittel, 2005). The different oscillation periods that can be resolved when the magnetization configuration (AP or P) is determined as a function of spacer thickness, are well explained in terms of Fermi surface (FS) spanning vectors of the normal metal in the growth direction. The magnetic ground-state configuration is, at least in principle, accessible to first-principles electronic-structure calculations in the spin-dependent version of density-functional theory (DFT), and that is basically the end of the story. However, in order to make a connection to the main topic of this review, we briefly discuss the formulation of the equilibrium exchange coupling in terms of scattering theory (Erickson et al., 1993; Slonczewski, 1989, 1993), that can also be formulated from first-principles and calculated using DFT (Bruno, 1995; Stiles, 2006). Another advantage of a scattering-theory formulation is that the effects of disorder can be understood employing the machinery of mesoscopic physics, such as random-matrix (Beenakker, 1997) or diagrammatic perturbation theory.

3.1 Non-local exchange coupling and giant magnetoresistance Let us consider a non-collinear N|F|N|F|N spin valve with angle θ between the magnetizations and an N-spacer thickness L. Suppose we can view the F|N|F trilayer as some spin-dependent scatterer embedded in a normal-metal medium. The total energy change induced by the scattering potential is given by an energy integral over the density of states that can be expressed by a standard formula as

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(Akkermans et al., 1991) 1 E(L, θ ) = 2πi

εF

ε –∞

∂ ln det s(L, θ, ε)dε, ∂ε

(4)

in terms of the energy-dependent scattering matrix s(L, θ , ε) of the trilayer F|N|F. Of special interest is the asymptotic dependence of energy E(L, θ ) for large L. In this limit, the coupling is governed by the lowest term in the expansion of the angular dependence in Legendre polynomials: E(L, θ )L → ∞ = cos θ

Jβ sin(qβ⊥ L + φβ ), L2 β

(5)

which is a sum over all FS cross-sectional extremal vectors of the normal-metal spacer labeled by β, and the parameters Jβ and φβ are model and material dependent (Stiles, 1999); qβ⊥ is the distance between a pair of critical Fermi points in reciprocal space in the layering direction. For configurations that are not in equilibrium, the derivative ∂ E(L, θ ) (6) ∂θ does not vanish. A finite τ is therefore interpreted as an exchange torque acting on the magnetizations, pulling them into the energetically-favorable configuration. Physically, this torque is a flow of angular momentum carried by the conduction electrons in the normal metal spacer. A spin valve that is strained by a relative misalignment of the magnetization directions from the lowest energy value therefore supports dissipationless spin currents. Essential for the existence and the magnitude of the nonlocal exchange coupling and the corresponding spontaneous persistent spin currents is the phase coherence of the wave functions in the normal spacer. An incoming electron in the spacer with information of the left magnetization direction has to be reflected at the right interface and interfere with itself at the left interface in order to convey the coupling information. This implies strong sensitivity to the effects of impurities, since diffuse scattering destroys the regular interference pattern required for a sizeable coupling. This qualitative notion has been formulated by Waintal et al. (2000) in the scattering-theory formalism invoking the “isotropy” condition for validity of the random-matrix theory. Isotropy requires diffuse transport, viz., that L is larger than the mean free path due to bulk and interface scattering. It can then be shown rigorously that the equilibrium spin currents vanish on average with fluctuations that scale like N –1 , where N stands for the number of transverse transport channels in the normal-metal spacer. In layered metallic structures, N is large and the static exchange coupling and spin currents can safely be disregarded in the diffuse limit. On top of the suppression by disorder, the absolute value of the coupling scales like L–2 even in ballistic samples, see Eq. (5). Experimentally, even the best Co|Cu|Co samples do not show any appreciable coupling beyond spacer-layer thicknesses of 20 atomic monolayers. τ =–


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The term giant magnetoresistance (GMR) stands for the reduction of the resistance of multilayers when the magnetic configuration is forced by an applied magnetic field from an antiparallel configuration of neighboring layer magnetizations to a parallel one. GMR was originally discovered in a configuration in which the current flow was in the plane of the film (CIP). More relevant in the present context is the configuration in which the current flows perpendicular to the planes (CPP) (Gijs and Bauer, 1997; Gijs et al., 1993; Pratt et al., 1991). Assuming diffusive transport, the CPP GMR is easily understood in terms of the two-spin-channel series-resistor model (Pratt et al., 1991; Valet and Fert, 1993). In ferromagnets, the difference between electronic structures of majority and minority spins at the Fermi energy gives rise to spin-dependent scattering cross-sections at impurities resulting in spin-dependent mobilities. The discontinuities of the electronic structure at a normal|ferromagnetic metal interface can also be very different for both spin species, corresponding to a large spin dependence of interface resistances. In the presence of applied electric fields and not too strong spin-flip scattering processes, a two-channel resistor model is applicable, according to which currents of two different spins flow in parallel. When the magnetizations of the ferromagnetic layers are parallel the charge current is short circuited by the low electrical resistance spin channel, explaining the reduction of the resistance under the magnetic field-induced configuration change.

3.2 Non-equilibrium spin current and spin accumulation The difference between spin-up and spin-down electric currents is called a spincurrent. It has a flow direction as well as a spin polarization, i.e. it is a tensor. In ferromagnets, the electron spin angular momentum states are good quantum numbers in the directions of the magnetization, sz = ±h¯ /2 for up and down spins. The spin current is then polarized along the magnetization direction. In normal metals there is no such preferred direction. Spin currents can flow without dissipation as in the strained equilibrium configurations discussed in Section 3. A non-equilibrium spin current excited by external perturbations is closely related to an imbalance in the electrochemical potentials that is called spin-accumulation. This is again a vector quantity that in a ferromagnet is parallel to the magnetization. Spin-related nonequilibrium phenomena have lifetimes that are usually much longer than all other relaxation time scales. Spin-flip scattering can originate from spin-orbit interaction effects in the band structure plus potential disorder (“intrinsic”). “Extrinsic” spin flips are caused by magnetic impurities or non-magnetic one with significant spinorbit interaction. Both depend strongly on the material, its chemical purity and crystalline order and destroy a non-equilibrium spin-accumulation (we disregard here the small spin accumulations that can be created by an electric bias in the presence of spin–orbit interaction, e.g. Edelstein, 1990). Spin-flip scattering can be very weak in clean metals with simple band structures, but is often significantly larger in 3d transition metals with a high density of states at the Fermi energy. In most theoretical approaches to magnetoelectronics (and also here) spin-flip scattering is treated phenomenologically, usually in terms of the spin-flip diffusion length, i.e. the length scale over which an injected spin accumulation loses its polarization, that

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Figure 2.1 The non-equilibrium spin accumulation injected from a ferromagnet (F) into a normal metal (N), which decays over a length scale given by the spin-flip diffusion length N sd . The spin accumulation in the ferromagnet is not shown, being small compared to the equilibrium magnetization.

is typically Fsd ∼ 5 nm (Permalloy, Py)–50 nm (Co). In bulk ferromagnets the spin accumulation vanishes at depths beyond Fsd although spin currents persist. Much of magnetoelectronics is based on the notion that even a normal metal such as copper can be magnetized. This magic is worked by applying a voltage over a normal|ferromagnetic metal contact (Fig. 2.1) so that a spin-polarized current is injected from the ferromagnet into the normal metal (Aronov, 1976; Johnson and Silsbee, 1987; van Son et al., 1987). The resulting spin accumulation is equivalent to a net ferromagnetic moment in the non-magnetic metal. The decay length of a spin accumulation injected into the normal metal is the spin-flip diffusion length Nsd that can be very large, e.g. about 1 µm in copper (Jedema et al., 2001), much larger than the smallest feature size created by microelectronic fabrication technology. The spin accumulation is a vector that in Fig. 2.1 is collinear with the ferromagnetic magnetization, i.e. parallel or antiparallel, depending on the spindependent interface and bulk conductances. In non-collinear (i.e. neither parallel nor antiparallel) spin valves, schematically F(↑)|N|F(), or other devices with two or more ferromagnetic contacts whose magnetizations are not parallel, the injected spin currents are also non-collinear, and (when the spacer is sufficiently thin) the resulting spin accumulation can be made to point in any direction. In contrast, in a ferromagnet the spin accumulation direction is fixed along the magnetization vector. The manipulation of electronic properties via the long range spin-coherence carried by the spin accumulation (Brataas et al., 2000) is the main challenge of modern magnetoelectronics. Typical magnetoelectronic structures are made from ferromagnetic metals like iron, cobalt or the magnetically soft permalloy (Py), a Ni/Fe alloy. The normal metals are typically Al, Cu or Cr, where the spin-density wave in the latter is usually disregarded in transport studies. These metals cannot be grown as perfectly as strongly bonded tetrahedral semiconductors; moreover, the Fermi wavelength is of the order of the interatomic distances. These systems are said to be “dirty”, meaning that size quantization effects on the transport properties may be disregarded (Waintal et al., 2000). In this limit, semiclassical theories on the level of the Boltzmann or diffusion equations most appropriately describe the physics. The spin accumulation is then just the difference in the local chemical potentials for up and down spin


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(Johnson and Silsbee, 1988a). Valet and Fert (1993) analyzed the GMR of magnetic multilayers in the perpendicular (CPP) configuration with collinear magnetizations. They used a spin-polarized linear Boltzmann equation to derive a diffusion equation including spin-flip scattering, that for vanishing spin-flip scattering reduces to the two-channel series resistor model (Pratt et al., 1991). The total current can then be interpreted as two parallel spin-up and spin-down electron currents, separately limited by resistors in series that represent interfaces and bulk scattering. When magnetization vectors and spin-accumulations are not collinear with the spin-quantization (z-)axis, the two-channel resistor model cannot be used anymore. The concept of up and down spin states must be replaced by a representation in terms of 2 × 2 matrices in Pauli spin space with non-diagonal terms that reflect the spin-coherence, analogous to the anomalous Green functions in superconductivity that reflect the electron–hole coherence in the superconducting state. A circuit theory for general non-collinear configurations can be derived from a given Stoner Hamiltonian in terms of the Keldysh non-equilibrium Green function formalism in spin space (Brataas et al., 2000, 2001) or simply by matching charge and spin distribution functions at interfaces with transmission and reflection probabilities. It reduces to a finite-element formulation of the diffusion equation with quantum mechanical boundary conditions between distribution functions on both sides of a resistor such as an interface. The initial step is an analysis of the circuit or device topology by dividing it into reservoirs, resistors and nodes that can be real or fictitious. The expressions are greatly simplified by assuming that the electron distributions in the nodes are isotropic in momentum space. This implies the presence of sufficient disorder (or chaotic scattering). The spin and charge currents through a contact connecting two neighboring ferromagnetic and normal metal nodes can then be calculated as a function of the distribution matrices on the adjacent nodes and the 2 × 2 conductance tensor composed of the (diagonal) spin-dependent conductances G↑ and G↓

e2 e2 nm 2 nm 2 Gs = |rs | = |t | , (7) M– h h nm s nm and the (off-diagonal) spin-mixing conductance

e2 nm nm ∗ rs (r–s ) , M– Gs,–s = h nm

(8)

where rsnm , tsnm are the reflection and transmission coefficients in a spin-diagonal reference frame with n, m the indices and M the total number of transport channels on the normal metal side of the contact. The expressions for the spin-up and down conductances are the Landauer–Büttiker formula in a two-spin-channel model (Bauer, 1992; Gijs and Bauer, 1997). Experimentally, these parameters have been obtained by extensive measurements on multilayers in the CPP (current perpendicular to the plane) configuration (Pratt et al., 1991). The complex spin-mixing conductance parameterizes the spin currents that are perpendicular to the magnetization, as discussed in the next section. The resistance of spin

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valves as a function of angle between the magnetizations is a sensitive measure of the spin-mixing conductance (Bauer et al., 2003b; Kovalev et al., 2006; Urazhdin et al., 2005). A requirement for the validity of the circuit theory are nodes with characteristic lengths smaller than the spin-flip diffusion length. When this criterion is not fulfilled, the diffusion equation has to be solved, with boundary conditions governed by the above conductance parameters (Huertas-Hernando et al., 2000). When the resistances are small, the distributions are not isotropic in momentum space anymore, reflecting the net electron flow. The assumption of isotropy can be relaxed, leading to the conclusion that the diagonal (Bauer et al., 2002; Schep et al., 1997) and the mixing conductances (Bauer et al., 2003b) in equations 7 and 8 contain spurious Sharvin resistances. A “drift” correction is essential for a meaningful comparison of ab initio calculations with experiments. The scattering matrices for various clean and disordered interfaces have been calculated from first-principles (Schep et al., 1997; Xia et al., 2001; 2006) and representative results are listed in the tables. The computed conductance parameters in general agree well with experiments (Bass and Pratt, 1999; Brataas et al., 2006a). In the present formulation, circuit theory cannot describe spin-flip scattering processes at interfaces addressed by Bass and Pratt (2007).

3.3 Spin-transfer torque A spin accumulation with polarization normal to the magnetization direction cannot penetrate deeply into the ferromagnet, but is instead absorbed at the interface on an atomic length scale, thereby transferring angular momentum to the ferromagnetic order parameter. A large enough torque overcomes the magnetic anisotropy and damping to switch the direction of the magnetization (Berger, 1996; Slonczewski, 1996). The spin transfer can be understood by analogy with the Andreev scattering at normal|superconducting interfaces (Beenakker, 1997). This is illustrated by the spin-transfer scattering process depicted in Fig. 2.2. Consider a spin-accumulation in the normal metal (that was injected optically or electrically by other contacts), polarized at right angles to the magnetization of the ferromagnet. No electric voltage or charge current bias is required at this stage. In the idealized case of a halfmetallic ferromagnet with an electronic structure that is perfectly matched to that of the normal metal for one spin direction, a straightforward angular momentum balance of incoming and scattering states shows that an incoming up-spin is flipped during reflection. The electrons therefore lose not only linear momentum (twice the electron wave vector normal to the interface when scattering at a specular barrier) but also angular momentum of 2 × h2¯ . The (transverse) angular momentum h¯ is conserved, however. Just as the linear momentum is absorbed by a wall from which a soccer ball bounces, the electron angular momentum change is transferred to the ferromagnet. This spin-flip reflection process is equivalent to a spin current that flows from the normal metal into the ferromagnet, where it is absorbed and exerts a torque on the magnetization. The Andreev analogy becomes clear by interpreting the ferromagnet as a condensate of angular momentum, just as a super-


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Figure 2.2 Illustration of the magnetization torque exerted by a spin current. The magnetization m of the ferromagnet is polarized at right angles to a spin accumulation in the normal metal. For metallic interfaces, the “anomalous” spin-flip scattering illustrated in this figure dominates the spin-conserving scattering. The angular momentum h¯ lost during this spin-flip is transferred as a torque to the magnetization order parameter.

conductor is a condensate of charge. The magnitude of the spin transfer is obviously proportional to the spin accumulation in the normal metal. Furthermore, for an ideal interface between metals whose bands match perfectly for one spin direction, each reflected electron is flipped with probability unity. The maximum spin current absorbed (and thus the maximum magnetization torque) is then N(μ↑ – μ↑ )/4π, proportional to the number of scattering channels in the normal metal. In more realistic interface models the qualitative picture remains the same, but a few corrections should be taken into account. The transverse spin accumulation is not absorbed on an exponential length scale, but by a destructive interference process that leads to an algebraic decay (Stiles and Zangwill, 2002). The penetration depth is still on an atomic length scale, viz. the magnetic coherence length π λc = F (9) , |k↑ – k↓F | F where k↑(↓) are characteristic Fermi wave numbers for majority and minority spin electrons. Spin conserving scattering at the interface, which becomes important in the case of a potential barrier between N and F, reduces the spin transfer efficiency. In the presence of conventional scattering processes, the number of channels N must be replaced by the spin-mixing conductance (8) divided by e2 /h. Re G↑↓ can therefore be interpreted as the material parameter governing spin transfer. Firstprinciples calculations predict (Xia et al., 2002; Zwierzycki et al., 2005) a small imaginary contribution that corresponds to a spin transfer torque in the direction normal to the plane in Fig. 2.2. It can be interpreted as an interface-generated non-local exchange field. The general equations for transverse and longitudinal spin and charge currents for arbitrary angles between spin accumulation and magnetization are a bit more complicated (Brataas et al., 2000). A complete theory of the spin-accumulation induced magnetization torque requires a quantitative treatment of the interface scattering but also a description of the whole device that allows computation of the spin accumulations. The magnetoelectronic circuit theory mentioned above (Brataas et

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Table 2.1 Spin-dependent interface resistances ARs = A/Gs in units of (f m2 ), for a number of commonly studied interfaces as calculated from first principles (Bauer et al., 2002). The drift correction is included (Schep et al., 1997)

Interface

Roughness

ARmaj

ARmin

Au/Ag(111) Au/Ag(111) Au/Ag(111)

Clean 2 layers 50-50 alloy Exp. (Bass and Pratt, 1999)

0.094 0.118 0.100 ± 0.008

0.094 0.118 0.100 ± 0.008

Cu/Co(100) Cu/Cohcp (111) Cu/Co(111) Cu/Co(111) Cu/Co(111)

Clean Clean Clean 2 layers 50-50 alloy Exp. (Bass and Pratt, 1999)

0.33 0.60 0.39 0.41 0.26 ± 0.06

1.79 2.24 1.46 1.82 ± 0.03 1.84 ± 0.14

Cr/Fe(100) Cr/Fe(100)

Clean 2 layers 50-50 alloy

2.82 0.99

0.50 0.50

Cr/Fe(110) Cr/Fe(110) Cr/Fe(110)

Clean 2 layers 50-50 alloy Exp. (Zambano et al., 2002)

2.74 2.05 2.7 ± 0.4

1.05 1.10 0.5 ± 0.2

al., 2000) contains all of the necessary ingredients. The first microscopic treatment explicitly addressing the spin torque in diffuse systems is the random matrix theory by Waintal et al. (2000). We later showed that the theories are completely equivalent for not too transparent interfaces (Bauer et al., 2003b), and can be generalized to arbitrary circuits. An example of the application of this circuit theory to a complex device is given by Bauer et al. (2003a).

3.4 Angular magnetoresistance of spin valves The perpendicular F|N|F spin valve is the prototype magnetic structure in which effects of the spin transfer torque can be observed. The (in theory) most simple observable is the angular magnetoresistance, i.e. the linear electrical resistance as a function of the angle θ between the magnetizations. For structurally symmetric spin valves p2 tan2 θ /2 G 1– = G(–θ ), G(θ ) = (10) 2 1 + gsf tan2 θ /2 + ζ where ζ =

(Re η + gsf )2 + (Im η)2 . (1 + gsf )(Re η + gsf )

(11)

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Table 2.2 Interface conductances in units of 1015 –1 m–2 (Brataas et al., 2006a). The drift correction is included (Bauer et al., 2003b)

System

Interface

G↑

G↓

Re G↑↓

Im G↑↓

Au|Fe (001)

clean alloy

0.40 0.39

0.08 0.18

0.466 0.462

0.005 0.003

Cu|Co (111)

clean alloy

0.42 0.42

0.38 0.33

0.546 0.564

0.015 –0.042

Cr|Fe (001)

clean alloy

0.14 0.26

0.36 0.34

0.623 0.610

0.050 0.052

Here G = G↑ + G↓ is the total conductance of one contact, p = (G↑ – G↓ )/G the polarization, η = 2G↑↓ /(G↑ + G↓ ) the relative mixing conductance and gsf = 2gsf /G the relative ‘spin-flip conductance’ gsf = De2 /(2τsf ) (Brataas et al., 1999) (D is the energy density of states). Equation (10) deviates from a simple 1 – cos θ dependence that reflects the projection of the two spin directions as expected for a tunnel barrier (Slonczewski, 1989). The additional dissipation of the injected spin accumulation at θ = 0 leads to an increase of the current for a voltage-based structure relative to the cosine form. In asymmetric spin valves this may lead to non-monotonic magnetoresistances and a sign change of the spin-transfer torque (Manschot et al., 2004a). By engineering the asymmetry, an enhancement of the spin-transfer torque can be achieved as well (Mancoff et al., 2004b). Using a single parameter fit to the experimental angular magnetoresistance (Urazhdin et al., 2005), Kovalev et al. (2006) found a mixing conductance for Cu|Py quite close to those in Table 2.2.

4. Non-Local Magnetization Dynamics In magnetic multilayer structures an applied current induces a spin transfer torque on the ferromagnets that may excite the low-energy degrees of freedom of a magnet and turn the static problem discussed above into a time-dependent one (Bazaliy et al., 1998; Berger, 1996; Slonczewski, 1996). After being spinpolarized by passing through a static ferromagnet a dc current can excite spin waves, precessional and more complicated motions, and at a critical current, even completely reverse the magnetization direction. The predictions have by now been amply confirmed by many experiments (Katine et al., 2000; Myers et al., 1999; Tsoi et al., 1998; Wegrowe et al., 1999) (see Stiles and Miltat, 2006). A resonant finite-wavevector spin-wave excitation can be excited by an applied dc bias even in a single ferromagnetic layer (Ji et al., 2003; Polianski and Brouwer, 2004; Stiles et al., 2004; Özyilmaz et al., 2004). The current-induced magnetization dynamics are interesting from a fundamental-physics perspective, requiring a grasp of

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the nontrivial coupling of non-equilibrium quasiparticles with the collective magnetization dynamics. It furthermore carries technological potential as an efficient mechanism to write information into magnetic random-access memories and to generate microwaves (Kiselev et al., 2003; Rippard et al., 2004).

4.1 Current-induced magnetization dynamics Slonczewski’s spin transfer (Slonczewski, 1996) arises when a spin-current with a polarization-component perpendicular to the magnetization is absorbed by the ferromagnet. Disregarding spin-orbit coupling (other than the crystal field anisotropy in Heff ) and other spin-flip processes, the total spin angular momentum is conserved. The angular momentum lost is transferred from the electron current to the coherent magnetization of the ferromagnet contributing to the magnetization equation of motion as a torque τ torque = m × (Is × m),

(12)

where Is is the spin current into the ferromagnet. To a very good approximation, the parallel component of Is does not affect the magnetization dynamics in transition metals since changes in the modulus of the magnetization is energetically very expensive. This is taken into account by the double outer vector product that only projects out the component of Is normal to the magnetization. In Section 3.3 we explained that the spin transfer torque can be written in terms of the spin accumulation s next to the interface and the real part of the spin-mixing conductance h¯ (13) Re G↑↓ m × (s × m). 2e2 The dynamics of a monodomain ferromagnet of volume V and magnetization Ms that is subject to the torque (12) is modified by an additional source term on the right-hand side of the LLG equation (3) (Slonczewski, 1996): ∂m γ (14) = m × (Is × m). ∂t MV τ torque =

torque

s

For a fixed current density, Eq. (14) is proportional to the cross section of the interface through the total spin current Is , and inversely proportional to the volume V of the ferromagnet. In layered structures this term therefore scales with the inverse of the ferromagnetic film thickness. The spin current Is is conveniently generated in perpendicular spin valves, i.e., current or voltage-biased F|N|F structures. In symmetric devices the spin transfer torques on both sides of the normal metal layer are identical in size and direction. The simplest solution in the presence of a current bias is then a constant rotation of both magnetizations. Usually one is interested in a situation in which the magnetizations move relative to each other. This can be achieved in asymmetric spin valves, in which the torque on one ferromagnet is suppressed by making it much thicker (a larger V in Eq. (14)). This magnet then behaves like a constant spin-current source, or polarizer, whereas the second, thinner dynamic ferromagnet is a magnetically “soft” analyzer easily excitable by the spin torque.


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Figure 2.3 (a) The spin transfer torque is a spin-current that drives magnetization motion somewhat like wind drives a windmill. (b) A moving magnetization excites a spin current when in contact with a conductor, in analogy with a moving fan that causes a flow of air.

4.2 Spin pumping When seeking a consistent theory of the magnetization dynamics in hybrid structures, the current-induced magnetization torque discussed above is only one side of the coin: a time-varying magnetization of a ferromagnet that is in electrical contact with normal metals emits (“pumps”) a spin current into its nonmagnetic environment (Tserkovnyak et al., 2002) as illustrated in Fig. 2.3b. Clearly, the magnetization motion will be determined by the competition between the driving and the braking torques. The loss of angular momentum due to spin pumping enters as another contribution to m, ˙ i.e. as an additional source term to the LLG equation. When the magnetization dynamics are induced by external magnetic fields the process is effective already in the absence of externally applied currents. When a bias is applied, the spin-pumping term is typically of the same order as the magnetization torque (14) and should be treated on the same footing, although it appears to not affect the current driven magnetization dynamics very strongly (Fuchs et al., 2007). Water streams through an elastic tube without external pressure by external “peristaltic” modulation (the tube is periodically squeezed, out of phase, at two different points). Similarly, electrons can be pumped through a scattering region when externally applied gates are activated by time-dependent signals that are out of phase. Such charge pumping is used in so-called single electron turnstiles. Formally, the effect can be expressed in terms of time-dependent scattering matrices that modulate the phases of transmitted and reflected electrons so that the net current in the leads does not vanish (Brouwer, 1998; Büttiker et al., 1994). The parametric spin-pumping formalism (Tserkovnyak et al., 2002), based on the low-frequency limit of the scattering theory of transport is a general, versatile and practical method to compute spin pumping and combine it with the spin-transfer torque as a source term for the time-dependent circuit theory in the adiabatic limit. The reflection of an electron with given spin at a normal metal|ferromagnet interface depends on the magnetization direction, implying that the scattering matrix is time dependent when the magnetization varies. When the magnetization precesses with frequency ω, a spin-up electron incident on the interface has a chance to pick up an energy quantum h¯ ω by reflecting into an unoccupied spin-down state. By this process the ferromagnet continuously loses angular momentum and energy to the normal metal, thus “pumping” spins. A detailed analysis (Tserkovnyak et al.,

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2002, 2005) reveals that the pumped spin current obeys the following equation: ∂m γ (15) =– Is,pump ∂t pump Ms V dm γ h¯ dm Re G↑↓ m × + Im G↑↓ . = (16) Ms V 4π dt dt Here we indicated that the (negative of the) pumped spin current is nothing but a torque on the ferromagnet, i.e. an additional source term in the LLG equation just as the spin-transfer torque, Eq. (14). The torque proportional to Re G↑↓ is dissipative and has the same functional form as the Gilbert damping in Eq. (3). Its effect can therefore be described by the modified Gilbert parameter: α = α 0 + α = α0 +

γ h¯ Re G↑↓ . Ms V 4π

(17)

Im G↑↓ acts like an effective magnetic field that can be taken into account by a renormalized gyromagnetic ratio. For intermetallic interfaces (Zwierzycki et al., 2005) this term can usually be disregarded (Table 2.2). When the pumped spin angular momentum is not dissipated sufficiently quickly to the normal-metal lattice, a spin accumulation builds up. However, as discussed in Section 3.3, the spin accumulation in proximity to a ferromagnet creates a reaction spin-transfer torque in the form of a transverse-spin backflow into the ferromagnet. In hybrid structures the magnetization dynamics and the non-equilibrium spin-polarized currents are clearly mutually interdependent. The conversion of magnetization movement into spin currents and vice versa at a possibly different location characterizes the “nonlocality” of the magnetization dynamics. Spin pumping can be also understood in terms of the linear response of the electron gas to a time-dependent magnetic perturbation (Barnes, 1974; Monod et al., 1972; Šimánek and Heinrich, 2003). Assume that a magnetic impurity at the origin in a normal metal perturbs the system with a localized and time-dependent vector exchange potential Vx (t) = x m(t). The effective field due to the induced non-equilibrium spin density is then given in terms of the linear-response magnetic susceptibility of the metal χsa sa (r, t) =

i (t) [sa (r, t), sa (0, 0)] , h¯

(18)

where (t) is the Heaviside step function, [· · ·] the commutator and · · · the ground state expectation value, as

∞ dt aˆ χsa sa (r, t – t )δma (t ), δHimp (r, t) = x (19) aa

–∞

where a, a ∈ {x, y, z} are the indices of the Cartesian axes and aˆ stands for the corresponding unit vectors. The leading terms for a slow and small-angle precession

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in a uniaxial system are (Mills, 2003):

dm dm + 2 m × , –γ m × δHeff (t) = γ Ms 2x 1 dt dt

where

dχsx sy (ω) , 1 = –i dω ω=0 dχsx sx (ω) 2 = –i dω

(20)

(21)

ω=0

are real numbers. We see that 1 is an effective field term that modifies the gyromagnetic ratio γ in Eq. (3) and 2 is a Gilbert-like damping parameter 2 dχsx sx (ω) . αeff = –iγeff Ms x (22) dω ω=0 When a precessing bulk ferromagnet is in contact with a normal metal, the spin pumping can be explained in the same way (Mills, 2003; Šimánek, 2003). This picture is physically appealing but cumbersome for material-specific calculations including disorder as compared to the scattering theory approach (Zwierzycki et al., 2005).

5. The Standard Model Combining the charge and spin coupling mediated by the spin transfer torque and spin pumping as explained above leads to “the standard model” for the charge and magnetization dynamics in magnetic nanostructures in which many phenomena can be discussed qualitatively and sometimes even quantitatively. The model is based on the macrospin model dynamics (3) for each magnetic element with additional surface torques due to spin pumping (15) and spin-transfer (14), where the latter is governed by the spin accumulation close to the F|N interfaces. Circuit theory (or any other semiclassical approach) can be used to compute the instantaneous spin accumulation for a parametrically constant magnetization configuration that changes slowly (adiabatically) compared to the electronic motion. In the following we briefly discuss some of the effects which can be understood with this standard model without going into details or fully representing the quite extensive literature on these topics, for which we refer to the technical reviews.

5.1 Enhanced Gilbert damping and spin battery The loss of energy and momentum of a magnetization varying in time effectively increases the damping with a functional dependence that is identical to that of the Gilbert phenomenology as explained in Section 4.2. However, when the ejected

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spins are not channelled away, a spin accumulation builds up close to the interface that in the steady state exactly cancels the spin pumping term. The dissipation of a spin accumulation in a normal metal layer is strongly material dependent; Pt is a strong spin-flip scatterer and even a few monolayers act as an efficient spin sink, whereas the spin lifetime in high-purity copper is very long. The magnetization damping in thin magnetic films can therefore be engineered by covering them with different normal metal films; α does not change when the ferromagnet is covered by Cu, whereas a maximized enhancement of the damping Eq. (17) is achieved with Pt. The excess enhancement due to spin pumping has been observed in several FMR experiments, starting with (Mizukami et al., 2001, 2002) on N|F|N trilayers. A second ferromagnet as in F|N|F spin valves can also be a good spin sink, thus enhancing the damping, as discussed in Section 5.3. When the normal metal in an F|N bilayer is a weak spin-flip scatterer, a significant spin accumulation suppresses the excess Gilbert damping. This spin accumulation can be considered as a source of spin motive force generated by the magnetization dynamics (Brataas et al., 2002). The spin-battery spin-voltage can be probed non-invasively by a magnetic tunnel junction attached to the normal metal at a distance not exceeding the spin-flip diffusion length. In the presence of spinflip scattering in the ferromagnet, part of the spin accumulation can return into the ferromagnet as a spin current polarized parallel to the magnetization. Due to different resistances in both spin channels the spin current is transformed into a voltage signal (Wang et al., 2006) that has recently been measured (Costache et al., 2006).

5.2 Current-induced magnetization reversal and high frequency generation As indicated in the introduction, the most important consequence of the spin transfer torque is the possibility to reverse the magnetization of a free layer relative to that of a fixed layer (and back again) when the applied electrical current exceeds a critical value. The experiments have by now become rather standard and the effect has already been utilized in prototype memory devices (Hosomi et al., 2005). When the magnetization becomes frustrated by conflicting spin-transfer and applied magnetic field torques, stable oscillations can be generated that, by means of the GMR, cause periodic resistance oscillations in the GHz regime, that are tunable by e.g. the applied current bias (Kiselev et al., 2003). The standard model appears to represent the experiment better than one might have expected considering the very high current densities. A reliable quantitative modelling of the effect is difficult and presumably requires a consistent treatment of the spatially varying magnetization and (spin) current distributions. The excess Gilbert damping is relevant since the free layer is usually very thin and becomes dependent on the instantaneous configuration during the reversal process (Tserkovnyak et al., 2003). The competition between homogeneous (macrospin) and spatially dependent (spin wave) excitations at criticality has been studied by Brataas et al. (2006b). As in the case of current-induced excitation of a single ferromagnetic layer (Polianski and Brouwer, 2004) it is essential to treat the inhomogeneity of the magnetization damping selfconsistently with that of the spin


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Figure 2.4 Single (left) and collective (right) ferromagnetic resonance in F1|NM|F2 spin valves under a static applied field HDC and an oscillating field hRF . The collective resonance can occur for different resonance frequencies of the isolated layers as a consequence of the dynamic coupling, as explained in the text.

current dynamics. The challenge of integrating detailed micromagnetic modelling with a realistic description of the electron transport for magnetic heterostructures remains.

5.3 Dynamic exchange interaction The combination of spin-pumping current and spin-transfer torque induces a dynamic crosstalk between moving magnetizations in magnetic heterostructures. This has been studied in quite some detail for the FMR of planar F|N|F spin valves (Tserkovnyak et al., 2005). The physics observed in experiments by Heinrich et al. (2003) can be understood intuitively as follows. In Fig. 2.4(left), layer F1 is resonantly exited and its precessing magnetic moment acts as a spin pump which creates a spin current propagating away from the F1|NM interface in the direction of the NM|F2 interface (dotted blue arrow in NM). The purple arrow indicates the instantaneous direction of the spin angular momentum of the spin current. Layer F2 is assumed to be detuned from its FMR, and therefore its rf magnetization component is negligible compared to that in the layer F1. A darker green area in F2 around the NM|F2 interface represents the region (of thickness λc < 1 nm) in which the transverse spin momentum is absorbed by the F2 layer. F2 acts as a spin sink since the transverse momentum from the spin current is transformed into a magnetization torque for the layer F2. The complete absorption of the spin current by F2 causes an additional damping of the magnetization dynamics of F1 as discussed above. In the right drawing, F1 and F2 resonate at the same external field. In this case both layers act as spin pumps and spin sinks. Consequently, the spin currents in the NM layer propagate towards both the NM|F2 and NM|F1 interfaces (see blue dotted arrows). An additional magnetic damping in F1 and F2 vanishes with the net spin flow through the interfaces. A dynamic locking between the magnetization dynamics has been observed and computed even when the FMR resonance frequencies of F1 and F2 are somewhat detuned. This phenomenon should be distinguished from the dynamic locking of magnetization dynamics by spin waves in ferromagnets and AC charge currents (Kaka et al., 2005; Mancoff et al., 2005). The mathematics of the coupled LLG equations for this system has been worked out by Tserkovnyak et al. (2005), reproducing the measured line width narrowing

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and frequency locking close to the common resonance for parameters determined at off-resonance conditions.

5.4 Noise in magnetic heterostructures Noise in magnetic devices can seriously limit their performance. The increased noise in spin valve read heads was ascribed to the spin torque by Covington et al. (2004). Indeed, both spin pumping and spin-transfer torque have strong effects on the noise in magnetic nanostructures. Noise in ferromagnets such as the Barkhausen noise due to discrete domain reorientation, has a long history. In nano-particles at finite temperature T the magnetization fluctuates as a whole, due to thermal fluctuations in the effective magnetic field h(t). These magnetization fluctuations are intimately related to the magnetization damping by the fluctuation dissipation theorem (Brown, 1963): (0) α0 δij δ(t – t ). hi (t)h(0) (23) j (t ) = 2kB T γ Ms V Foros et al. (2005) showed that the enhanced damping due to spin pumping is associated with enhanced fluctuating magnetic fields caused by transverse spin current fluctuations in the normal metal spin-sink that exert a stochastic spin-transfer torque on the ferromagnet. The enhanced thermal magnetization noise has a white correlation function identical to Eq. (23) with α0 replaced by the increased damping α, see Eq. (17). In spin valves, the dynamic coupling by the exchange of fluctuating transverse spin currents depends on the configuration. The total magnetization noise turns out to be larger in the antiparallel than parallel configuration (Foros et al., 2007).

6. Related Topics This article focuses on the spin and charge current and magnetization dynamics of magnetic multilayers and spin valves. The general concepts are relevant for a number of related phenomena, a few of which are very briefly introduced below.

6.1 Tunnel junctions The spin-transfer torque through tunnel junctions has been addressed theoretically by Slonczewski (2005). Especially since the discovery of single-crystalline MgO as a superior tunneling barrier material (Parkin et al., 2004; Yuasa et al., 2004) this is an important technological issue for MRAM applications (Hosomi et al., 2005). The physics is quite similar to that of the spin-transfer torque in spin valves, but without the complications of the spin accumulation in the normal metal. The spin transfer torque and angular magnetoresistance are therefore well described by simple geometric functions governed by the scalar product of the two magnetization vectors (Slonczewski, 2005). A significant effective field contribution to the torque


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has been found (Tulapurkar et al., 2005). Even relatively small currents induce large voltage drops over tunnel junctions so “hot” electrons may become important (Levy and Fert, 2006; Theodonis et al., 2006).

6.2 Domain walls A magnetic domain wall is the region between two magnetic domains whose magnetizations point in different directions. The study of current-driven domain walls in thin film nanostructures is a very lively field in recent years. By applying a magnetic field, the energy of one domain can be lowered with respect to the other, leading to a thermodynamic force that can accelerate the domain walls up to considerable velocities. An electric current sent through a ferromagnetic wire can also move the domain wall by the spin transfer torque (Berger, 1984; Tatara and Kohno, 2004; Yamaguchi et al., 2004). Many experiments have since confirmed this picture for ferromagnetic metal wires, but domain walls in diluted magnetic semiconductor (Ga,Mn)As can also be moved by electric currents (Yamanouchi et al., 2006, 2004). Parkin (2004) has suggested a novel memory concept in the form of ferromagnetic wire loops in which domain walls are collectively moved by current pulses between pinning sites in a nanoscale shift register. According to the two-spin-channel model, an electric current in a metallic ferromagnet is polarized since the lower resistance spin channel carries a larger current than the higher resistance one. When the current is passed through a domain wall, the spin current has to accommodate its polarization to it. The angular momentum lost by the electrons in this process is transferred to the magnetization in the domain wall region. The strong exchange interaction in transition metals renders the magnetization very stiff so that domain walls can extend over hundreds of nanometers. It is then safe to assume that the transfer of angular momentum occurs adiabatically, which leads to the simple expression for the local transfer torque ∂m(r,t) h¯ (24) ≈ P (j · ∇r )m, ∂t 2e torque where j is the charge current vector and P = (σ↑ – σ↓ )/(σ↑ + σ↓ ) its polarization in terms of the conductivities σs . For sufficiently large currents the torque will set the domain wall into motion. When the domain wall has very large gradients (Tatara and Kohno, 2004) or in the presence of spin-flip scattering (Zhang and Li, 2004), a “non-adiabatic” torque arises with a vector component normal to Eq. (24). There is still some controversy concerning the correct equation of motion for the magnetization in current-driven ferromagnets, in the presence of realistic spin dephasing processes (due to spin-orbit interaction and/or magnetic disorder) that correspond to a leakage of angular momentum into the lattice. In particular, there are still some loose ends in understanding basic quantities like the critical depinning current and domain-wall terminal velocity (Barnes and Maekawa, 2005). Microscopic first-principles calculations should help to understand the origin of the non-adiabatic torque and the role of Gilbert damping in current-induced domain wall motion (Kohno et al., 2006; Skadsem et al., 2007; Tserkovnyak et al., 2006).

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6.3 Spin transport by thermal currents Increasing data storage density and access rate is a continuing challenge for the magnetic recording industry. The relatively high-current densities and voltages that are required to operate magnetic random access memories will cause problems in further downsizing device dimensions due to heating effects that complicate modeling and deteriorate device stability and lifetime. Controlled heating can also be beneficial as, for example, in the case of recording by thermally assisted magnetization reversal. In magnetic nanostructures Peltier effects have been reported (Fukushima et al., 2006; Gravier et al., 2006). Recently Hatami et al. (2007) predicted a strong coupling of thermoelectric spin and charge transport with the magnetization dynamics in nanoscale magnetic structures. The thermal spin-transfer torque is believed large enough to realize a magnetization reversal by pure heat currents.

7. Outlook The dynamics of magnetic nanostructures in the presence of currents is a fastmoving field. It attracts many researchers since it combines interesting physics with immediate practical relevance. We expect that the trends to ever small structures will continue for some time. Topics to watch are (i) Semiconductor spintronics. This field has the promise of integrating the metalbased magnetoelectronics with semiconductor microelectronics. Very recently important progress has been made by demonstrating non-local electrical detection of electrically injected spin currents in semiconductors (Lou et al., 2007). Unique to semiconductors are the optical creation and detection of spins. Current-induced spin accumulations can be created in the presence of spinorbit interaction simply by an applied bias (Edelstein, 1990; Inoue et al., 2003; Kato et al., 2004). (ii) Molecular spintronics. High effective spin-transfer fields have been observed in C60 (Pasupathy et al., 2004). The magnetoresistance observed in spin valves with single-wall carbon nanotubes (Sahoo et al., 2005) and single-sheet graphene (Hill et al., 2006) is very promising. (iii) Magnetic nanoelectromechanical systems. The combination of small mechanical structures such as cantilevers with ferromagnetic particles have been employed to detect single spins by magnetic resonance microscopy (Rugar et al., 2004). Adding currents to such systems could lead to, e.g., current driven spin-transfer nanomotors (Kovalev et al., 2007).

ACKNOWLEDGEMENTS We acknowledge the collaboration and helpful discussions with A. Kovalev, M. Zwiercycki, K. Xia, M. Manschot, X. Wang, B. van Wees, J. Foros, H.-J. Skadsem, D. Huertas-Hernanod, Yu. Nazarov,


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B.I. Halperin, and A. Hoffmann. This work has been supported by NanoNed and the EC Contracts IST-033749 “DynaMax” and NMP-505587-1 “SFINX”. G.E.W.B. and Y.T. are grateful for the hospitality of the Centre of Advanced Study, Oslo.

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THREE

Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds M.D. Kuz’min * and A.M. Tishin **

Contents Foreword 1. Formal Description of the Crystal Field on Rare Earths 1.1 The single-ion approximation 1.2 Equivalent operator techniques for various subspaces: 4fN configuration, LS term, J multiplet, Kramers doublet 1.3 Local symmetry and the exact form of Hˆ CF 2. The Single-Ion Anisotropy Model for 3d-4f Intermetallic Compounds 2.1 Macroscopic description of magnetic anisotropy 2.2 The notion of an exchange-dominated RE system 2.3 The single-ion model for 3d-4f intermetallics 2.4 The high-temperature approximation 2.5 The linear-in-CF approximation: main relations 2.6 Properties of generalised Brillouin functions 2.7 The linear-in-CF approximation (continued) 2.8 The low-temperature approximation 2.9 J -mixing made simple 3. Spin Reorientation Transitions 3.1 General remarks 3.2 SRTs in uniaxial magnets 3.3 Spontaneous SRTs in cubic magnets 4. Conclusion References

149 150 150 155 161 166 166 170 176 177 183 188 193 199 203 210 210 214 222 228 229

Foreword Magnetic properties of 3d-4f intermetallic compounds have been reviewed on numerous occasions in recent times (Kirchmayr and Burzo, 1990; Franse * **

Leibniz-Institut für Festkörper- und Werkstofforschung, Postfach 270116, D-01171 Dresden, Germany Department of Physics, M.V. Lomonosov Moscow State University, 119992 Moscow, Russia



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and Radwański, 1993; Liu et al., 1994; Andreev, 1995; Buschow and de Boer, 2003). Especially extensive is the literature on magnetically hard materials in general (Buschow, 1991; Coey, 1996; Skomski and Coey, 1999) as well as on specific classes of such materials (Buschow, 1988; Strnat, 1988; Herbst, 1991; Li and Coey, 1991; Suski, 1996; Burzo, 1998), the magnetocrystalline anisotropy playing invariably a central role. Somewhat apart stands the literature on experimental (mainly by means of inelastic neutron scattering) studies of the crystal field (CF) in intermetallics with ‘normal’ (Moze, 1998) and ‘anomalous’ (Loewenhaupt and Fischer, 1993) rare earths. The separation between the subjects of magnetic anisotropy and CF seems something artificial. Nowadays, when the single-ion model has gained general recognition, little doubt remains about the indissoluble connection between the two phenomena. Perhaps the true cause of this split is that theoretical activity in the area has been lagging the experiment ever since the appearance of the last major review four decades ago (Callen and Callen, 1966). Of course, theoretical advance on magnetic anisotropy and CF did not cease in the meantime, it just took a different direction (Bruno, 1989; Richter, 2001), stimulated by the advent of the density functional theory. As regards the single-ion model proper, work on it proceeded at a rather slow pace. Nonetheless, a fair amount of new results has been published between the late 1960s (e.g. Kazakov and Andreeva, 1970) and more recent times (Magnani et al., 2003). This Chapter is to review the progress in theory in the post-Callens era, filling the gap in the literature between the anisotropy and the CF. We aim at reasserting the statement that magnetocrystalline anisotropy is the most important manifestation of the CF.

1. Formal Description of the Crystal Field on Rare Earths This section has an introductory character. We shall discuss the physical foundations of the approach that enables quantitative treatment of CF effects in RE-based hard magnetic materials—the single-ion approximation—and introduce the basics of the mathematical apparatus required for that treatment. Admittedly, this section contains mostly standard material, extensively covered in a number of monographs published in the 1960–70s (Griffiths, 1961; Ballhausen, 1962; Hutchings, 1964; Wybourne, 1965; Dieke, 1968; Abragam and Bleaney, 1970; Al’tshuler and Kozyrev, 1974). Hence the brief style of our exposition. Like in the rest of the Chapter, the approximate nature of the approach is emphasised and the bounds of its validity are set out.

1.1 The single-ion approximation The main experimental fact underlying the single-ion approach to 3d-4f intermetallics rich in 3d elements is the approximate non-interaction of the RE magnetic moments therein. Of course, the single-ion model as such is not restricted to

Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds

151

Figure 3.1 A two-dimensional view of a crystal of a 3d-4f intermetallic compound rich in the 3d element. The dark circles are the 3d atoms and the hatched areas are the RE ones.

intermetallic compounds (see e.g. Kolmakova et al., 1996), but it is in the 3d-4f intermetallics that it takes a particularly simple form. We shall therefore limit ourselves to this special case. The non-interaction of RE moments should be understood as relative weakness of the RE–RE exchange interaction in such compounds. That is, if we consider a crystal consisting of atoms of 3d and 4f elements, as shown schematically in Fig. 3.1, the 4f shell of each RE atom (hatched) interacts directly (or rather, via the 5d states) only with the 3d electrons of the transition element, but not with the 4f shells of neighbouring RE atoms. It is unimportant at this stage to specify if the 3d electrons are regarded as localised, belonging to individual atoms, or as itinerant electrons (dashed lines in Fig. 3.1). The weakness of the 4f-4f interaction should not be mistaken for its nonexistence or indetectability. Thus, the compound GdNi5 , where nickel is nonmagnetic, orders ferromagnetically at TC = 32 K (Gignoux et al., 1976) entirely due to the Gd–Gd exchange. The truth, however, is that in typical hard magnetic materials the 4f-4f exchange coexists with much stronger 3d-3d and 3d-4f interactions and that the former one is negligible in comparison with the latter two. This statement has been recently put to a direct test. The influence of the exchange on the 4f shell of Gd in GdCo5 was probed by two different techniques: (i) by applying a very strong magnetic field that breaks the antiparallel orientation of the Gd and Co sublattices (Kuz’min et al., 2004) and (ii) by inelastic neutron scattering on the exchange-split 4f states (Loewenhaupt et al., 1994). In the first case only the 3d-4f exchange is relevant, while in the second situation the 3d-4f and the 4f-4f interactions produce a combined effect on the 4f shell. Now, the intensity of the exchange interaction determined from these two kinds of experiment turned out to be the same within the estimated uncertainty bounds. This is a direct proof of the weakness of the Gd–Gd exchange in GdCo5 . In the other compounds of the RECo5 series the 4f-4f interaction can be neglected with even more reason, since its intensity decreases in proportion to the square of the total 4f spin. Another corner-stone of the single-ion approach is the weakness of the 3d-4f interaction in comparison with the 3d-3d one. Hence, on the one hand, the 3d4f exchange is all-important for the 4f subsystem. On the other hand, its action back on the 3d electrons, which are under the dominant influence of the intrasublattice 3d-3d exchange, is relatively insignificant. Once again, it does not mean

152


that the 3d electrons are completely unaware of the state of magnetisation of the 4f subsystem. The presence of a magnetic RE may result in a noticeable shift of the Curie point in an intermetallic compound with a magnetic RE as compared to its counterpart with Y or Lu. Insignificance in this instance means that the effect of the 3d-4f exchange on the 3d sublattice can be reduced to a renormalisation of the TC , without changing the dependence of the 3d sublattice magnetisation M3d on reduced temperature, T /TC . The explicit form of M3d (T /TC ) is given by Eq. (2.43) below. Thus, a peculiar hierarchy of exchange interactions takes place in 3d-4f hard magnetic materials, which can be schematically expressed as (3d-3d) (3d-4f) (4f-4f) ≈ 0. This fact enables us to regard the 3d subsystem as something external with respect to the RE, as something given, which orders magnetically largely due to its internal forces. Then, from the viewpoint of the 4f electrons the 3d4f exchange can be presented as the action of an exchange field produced by the ordered 3d subsystem. Therefore, the 4f subsystem can be regarded as a conjunction of non-interacting RE atoms (ions) immersed in several fields: the 3d-4f exchange field, applied magnetic field and the CF. The latter is not literally an electric field in the crystal, so we shall avoid terming it “crystal electric field” as misleading. Rather, CF is a combination of anisotropic time-even interactions involving the 4f electrons, presented as a fictitious electrostatic potential seen by the 4f shell. The so formulated single-ion approximation is an enormous simplification: all itinerant electron states have been eliminated and the attention has been concentrated on the 4f shell of one RE atom, which to a good approximation can be considered localised. This explains the origin of the term “single-ion”, widely applied to non-ionic solids. Of course, there are no charged ions in metals. Just the magnetic behaviour of RE’s in metallic systems is determined by the properties of the ground configuration 4fN , which in most solids is the same as in trivalent RE ions. Quantitatively this behaviour can be described by means of the following Hamiltonian: ˆ –e Hˆ 4f = Hˆ Coulomb + Hˆ s-o + 2μB Bex · Sˆ + μB B · (Lˆ + 2S)

N

VCF (ri ).

i=1

(1.1) Here the first two terms describe the isotropic (Coulomb and spin-orbit) interactions within the 4f shell; the third term presents the 3d-4f interaction by means of N the exchange field Bex acting on the 4f spin, Sˆ = i=1 sî ; the fourth one describes the (Zeeman) effect of the applied magnetic field B. The last term in Eq. (1.1) contains the CF potential VCF . Formally similar to an ordinary electrostatic potential, VCF (r) is a function of coordinates in real space, which can be expanded over a suitably chosen basis. It is convenient to use for the purpose spherical coordinates and to choose the so-called irreducible tensor operators (functions) Cm(n) (θ , φ) as the angular basis functions (Wybourne, 1965): VCF (r, θ, φ) =

n

n=2,4,6 m=–n

Vnm (r)Cm(n) (θ , φ).

(1.2)


153

Following a long-standing tradition, one may assume in addition that this anisotropic potential is created by electric charges situated entirely outside the region where the 4f shell is located. Then, within that region, VCF (r) must satisfy the Laplace equation. The radial functions Vnm (r) would then turn into simple power laws, Vnm (r) = Anm r n ,

(1.3)

where the coefficients are known as CF parameters. The approximation (1.3) is neither physically justifiable nor really useful. However, it causes no formal difficulties, as long as configuration mixing is neglected. Within the 4fN configuration the radial functions reduce to their expectation values and Eq. (1.2) is equivalent to Hˆ CF =

n N

Bnm Cm(n) (θi , φi ).

(1.4)

i=1 n=2,4,6 m=–n

The quantities Bnm = –eVnm (r)4f = –eAnm r n 4f are also called CF parameters. Generally speaking, Anm (as well as Bnm ) are complex numbers such that A*nm = (–1)m An,–m . Indeed, the potential VCF must be a real quantity and Cm(n) (θ , φ) are (n) complex-valued functions satisfying the condition [Cm(n) (θ , φ)]* = (–1)m C–m (θ , φ) (Varshalovich et al., 1988). Terms with odd n’s have been omitted from the expansion (1.2), because for n odd all matrix elements of Cm(n) (θ , φ) between any two 4f orbitals are nil1 for parity reasons. Likewise, we have left out all terms with n > 6, not complying with the triangle condition, n ≤ 2l. Finally, the isotropic n = m = 0 term has been omitted as well; its only effect is shifting all energy levels by the same amount, NB00 . For each n, the variable m may take 2n + 1 values, therefore the expansion (1.2) may contain maximum 27 terms. Thus, there are 5 basis functions with n = 2: 1 C0(2) (θ , φ) = (3 cos2 θ – 1), 2 12 3 (2) cos θ sin θ e±iφ , C±1 (θ , φ) = ∓ 2 12 3 (2) sin2 θ e±2iφ , C±2 (θ , φ) = 8

(1.5)

et cetera. Note that all C0(n) (θ , φ) possess cylindrical symmetry, in fact they are just the nth -order Legendre polynomials of cos θ : C0(n) (θ , φ) = Pn (cos θ ). Another (n) simple particular case is C±n (θ , φ) = (∓1)n [(2n – 1)!!/(2n)!!]1/2 sinn θ e±inφ . (n) The choice of Cm (θ , φ) for the basis is by no means unique—any complete set of functions can be used instead. For example, the spherical harmonics Ynm (θ , φ). 1 This is not to say that A nm must generally be nil for n odd. They genuinely vanish only when the RE occupies a crystallographic site with a centre of inversion. But even when they are nonzero, these Anm have no effect on the eigenvalues or eigenvectors of Hˆ 4f , as long as configuration interaction is neglected.

154


There is a simple connection between the two sets (Edmonds, 1957): 1 2n + 1 2 (n) Ynm (θ , φ) = Cm (θ , φ). 4π

(1.6)

So the five functions Y2,m (θ , φ) are obtained just by adding a prefactor (5/4π)1/2 to Eqs. (1.5). Extensive collections of explicit expressions for the spherical harmonics were compiled by Varshalovich et al. (1988; all Ynm with n ≤ 5) and by GörllerWalrand and Binnemans (1996; n ≤ 7). An alternative basis set, favoured particularly by experimentalists, is the one that uses the so-called Stevens normalisation (a term introduced by Newman and Ng, 1989). It is obtained by replacing the complex-valued functions Cm(n) (θ , φ) by (n) (θ , φ), and omitting all cumbersome their real combinations, ∝ Cm(n) (θ , φ) ± C–m numerical prefactors. The result is simple-looking trigonometric expressions. Thus, one gets instead of Eq. (1.5) the following 5 real functions: O20 = 3 cos2 θ – 1, O21 = cos θ sin θ cos φ, O22

= sin θ cos 2φ, 2

12 = cos θ sin θ sin φ,

22

(1.7)

= sin θ sin 2φ. 2

Also worth of mention are Racah’s unitary operators Um(n) (Racah, 1942). The connection to Cm(n) (θ , φ) is provided by a factor which depends on n and also on the orbital quantum number l: 1 2 (2l + n + 1)! 2 (n) (n) . n/2 [(n/2)!] (l – n/2)! Cm (θ , φ), n even. Um = (–1) n!(2l + 1)(l + n/2)! (2l – n)! (1.8) Within the 4f shell (l = 3) the corresponding relations are: 12 15 (2) . Cm(2) (θ , φ), Um = – 28 12 11 (4) . Cm(4) (θ , φ), Um = (1.9) 14 1 429 2 (6) (6) . Cm (θ , φ). Um = – 700 It should be emphasised that the introduced various sets of basis functions for expanding VCF (r) differ only in normalisation. From the standpoint of the physical approximations involved, all the above bases are absolutely equivalent and, if used properly, must yield identical results. One should just be consistent with notation and not allow indiscriminate use of CF parameters related to distinct normalisations. To avoid confusion, it is advisable to write out expressions of type (1.2) explicitly or at least to include references to such explicit expressions. The choice of a specific basis set is merely a matter of convenience. For example, Wybourne’s irreducible tensor operators Cm(n) (θ , φ) transform most simply


155

under rotations of the coordinate axes and are therefore particularly suited for serious theoretical work on magnetic anisotropy. The spherical harmonics Ynm (θ , φ) are normalised to unity on a sphere, hence they are a natural choice for the angular part of atomic wave functions (Condon and Shortley, 1935), being somewhat less convenient for expanding the CF potential. The ‘simple-looking’ basis functions in the Stevens normalisation are useful only for very simple calculations in high-symmetry cases. In particular, they are suitable for work in Cartesian coordinates (Hutchings, 1964), which is seldom done nowadays. Finally, Racah’s operators Um(n) possess the important property of having unitary reduced matrix elements within any atomic shell (hence their name). The significance of this can be learned e.g. from the book of Wybourne (1965). In this Chapter we shall not deal with problems necessitating the use of Um(n) or Ynm . So we shall use mostly the irreducible tensor operators Cm(n) (in various representations) and less often the operators Onm normalised according to Stevens.

1.2 Equivalent operator techniques for various subspaces: 4fN configuration, LS term, J multiplet, Kramers doublet Introducing equivalent operators in this section, we shall make a clear distinction between the following two aspects of the method: (i) reduction of the dimensionality of the space of available states at different stages of the method and the underlying physical approximations; (ii) the (formally exact) algebraic techniques facilitating the calculation of the matrix elements of VCF within those reduced subspaces. The need and the possibility of the approximate treatment of Hˆ 4f (1.1) are due to the fact that its individual terms describe interactions of grossly disparate intensities. On the other hand, to describe thermodynamic properties in a limited temperature interval, we only require a few low-lying eigenstates of Hˆ 4f (to about kT above the ground state), while the total number of states in a 4fN configuration can be a few thousand. From the point of view of physics, everything is determined by the intensity relations between the individual terms entering in Hˆ 4f . Several distinct situations are possible, these are considered in subsections 1.2.1–1.2.5. Subsection 1.2.6 will then be devoted to the algebraic aspect of the method. 1.2.1 No restrictions2 ECoulomb ∼ Es-o ∼ Eex ∼ EZeeman ∼ ECF .

(1.10)

This is a trivial case included here for the sake of completeness. No new approximations are possible (apart from the already mentioned neglect of the configuration interaction). Hˆ 4f has to be diagonalised within the full 4fN configuration. 1.2.2 The Russel–Saunders approximation ECoulomb Es-o ∼ Eex ∼ EZeeman ∼ ECF . 2

(1.11)

The energies in the symbolic relations (1.10)–(1.15) are of the order of the overall splitting of the relevant manifold by the respective terms in Eq. (1.1). For example, Es-o Eex can be understood as |λ|(2L + 1) 4μB Bex or even as |λ| μB Bex .

156


The dominant Coulomb interaction is isotropic both in the coordinate and in the spin spaces. Its eigenstates are (2L + 1)(2S + 1)-fold degenerate LS terms, i.e. sets of states with given total orbital moment and total spin quantum numbers. The construction of the |LML |SMS wave functions from one-electron orbitals |lm↑,↓ is in principle a purely algebraic problem having an exact solution. The explicit expressions, however, may be extremely cumbersome and will not be required herein. They have been described in full detail elsewhere (Sobel’man, 1972). The approximation here consists in restricting the space of states to those of the ground LS term [subject to the usual Hund’s rules: S = 12 (2l + 1 – |2l + 1 – N|) and L = S(2l +1–2S)]. Within that space Hˆ Coulomb reduces to a constant which will be omitted. The remaining four terms of Eq. (1.1) are projected on the ground term in the first approximation (i.e. their matrix elements on the |LML |SMS states are computed) and diagonalised. Validity of the Russel–Saunders approximation is determined by the strong inequality ECoulomb Es-o . Since it involves only intra-atomic interactions it has been investigated in a rather exhaustive manner. Though generally inaccurate for the RE’s, the Russel-Saunders approximation holds surprisingly well for their ground LS terms, which are all more than 96% pure (Dieke, 1968). This suffices for our purpose in this Chapter. In what follows we shall regard the Russel–Saunders approximation as valid in all cases. 1.2.3 The single-multiplet approximation (within the Russel–Saunders coupling scheme) ECoulomb Es-o Eex ∼ EZeeman ∼ ECF .

(1.12)

This is a particular case of the previous one. The added approximation here is the one expressed by the inequality Es-o max(Eex , EZeeman , ECF ). It is generally not a very good approximation, particularly for the light RE’s and most notoriously for samarium. On the other hand, this approximation is vital for the analytical tractability of many important problems. Its validity is hard to estimate a priori, since it depends on the characteristics of the solid, in particular on the relation between Eex and ECF . We shall dedicate a special section (2.9) to the question of validity of the single-J approximation in exchange-dominated systems (tentatively defined by Eex ECF ; see Section 2.2 for more a detailed definition). Technically the approximation is straightforward: Hˆ Coulomb + Hˆ s-o reduces to a constant and is omitted; Hˆ ex + Hˆ Zeeman + Hˆ CF is projected on the ground J manifold comprising the 2J + 1 states of type |LSJ M, where J = L ± S (3rd Hund’s rule). These are constructed from the states of the ground LS term according to the following simple relation (Condon and Shortley, 1935):

JM |LSJ M = (1.13) CLM SM |LM |SM , M ,M JM where CLM SM are the so-called Clebsch–Gordan coefficients (CGC)—exactly known functions of the quantum numbers J , M, L, M , S, M . The CGC have been extensively tabulated (Varshalovich et al., 1988; Rotenberg et al., 1959). More


157

information about the CGC can be found in Chapter 8 of Varshalovich et al. (1988), JM including explicit expressions for CLM SM in the practically important particular case of a ground multiplet obeying the 3rd Hund’s rule, J = L ± S. 1.2.4 CF-dominated RE systems ECoulomb Es-o ECF Eex ∼ EZeeman .

(1.14)

A special case of the situation considered in the previous subsection. It has little relevance to hard magnetic materials and is included here just for completeness. The subspace of accessible states is shaped by the CF, so it depends essentially on the symmetry of the crystallographic site occupied by the RE and also on the value of J . The analysis is particularly simple at low temperatures, where the magnetic behaviour is determined by the ground CF level. This may typically be either a singlet (J is an integer) or a Kramers doublet (J semi-integral). Within the ground CF level Hˆ CF as well as Hˆ Coulomb + Hˆ s-o reduce to irrelevant constants and can be omitted. Of course, the CF still governs the properties dependent on the wave functions of the ground state, e.g. the effective g-factor of a doublet. Often a pair of closely situated singlets make an accidental doublet, or a quasidoublet. The presence of the CF is then manifest in the zero-field splitting of such a quasi-doublet, as well as determining the g-factor. 1.2.5 Exchange-dominated RE systems ECoulomb Es-o Eex ECF ∼ EZeeman .

(1.15)

This special case is most closely relevant to RE-based hard magnetic materials, it will be investigated in greater detail in subsequent sections. Either part of the double inequality Es-o Eex ECF may fail under certain circumstances, which imposes substantial limitations on the applicability of this approximation. These will be considered separately for each half of the inequality (Sections 2.2 and 2.9). 1.2.6 Chains of equivalent operator representations In the previous sections we have analysed several possible intensity relations between the individual interactions within Hˆ 4f . In each case the quantitative description consisted in projecting some of the terms of Eq. (1.1) onto one of a hierarchically organised chain of subspaces within the 4fN configuration: LS term, J multiplet, Kramers doublet. . . Thus, the problem in each case reduces to computing the matrix elements of VCF within one of those subspaces.3 The equivalent operator technique makes this task easier. Let us consider the following chain of equivalence 3 The rather straightforward handling of the remaining terms of Eq. (1.1) has been described in many standard texts on quantum mechanics (e.g. Van Vleck, 1932; Condon and Shortley, 1935; Schiff, 1949). We shall make use of these well-known results according as we need them.

158


relations: N

. . ri2 Cm(2) (θi , φi ) = r 2 4f Cm(2) (θi , φi ) = αl r 2 4f Cm(2) (lî ) N

i=1

N

i=1 4fN

all space

i=1 4fN

configuration

configuration

. . . ˆ = = αL r 2 4f Cm(2) (L) αJ r 2 4f Cm(2) (Jˆ) = LS

term

J

multiplet

(±) const. singlet, (quasi)doublet

Here Cm(2) (Jˆ) denotes the following operator expressions4 : 1 C0(2) (Jˆ) = 3Jˆz2 – J (J + 1) , 2 12 3 ˆ ˆ (2) ˆ C±1 (J ) = ∓ Jz Jx ± i Jˆy + Jˆx ± i Jˆy Jˆz , 8 12 2 3 ˆ (2) ˆ C±2 Jx ± i Jˆy (J ) = 8

(1.16)

(1.17)

ˆ and Cm(2) (L) ˆ are the same expressions, but with lˆ or Lˆ substituted while Cm(2) (l) ˆ for J . Each operator in the chain (1.16) is defined in a distinct space of states, ˆ operate within an LS term. Each subsequent space is a subspace of e.g. Cm(2) (L) the previous one. Any matrix element of an operator standing on the right of the . equivalence sign ‘=’ between any two states belonging to the space where that operator is defined equal the corresponding matrix element of the operator on the . left-hand side of ‘=’. The opposite is not necessarily true. For example, ˆ M = LSJ M |αJ Cm(2) (Jˆ)|LSJ M. LSJ M |αL Cm(2) (L)|LSJ

(1.18)

However, if J = J , then ˆ M = 0 = LSJ M |αJ Cm(2) (Jˆ)|LSJ M. (1.19) LSJ M |αL Cm(2) (L)|LSJ That is, the matrix elements of the two operators coincide only within a subspace where both of them are defined. Hence the use of a special sign of equivalence in Eq. (1.16) instead of the usual equality sign. The reason for having so many different representations for the CF is mere convenience—each one is ideally suited for computing matrix elements within the corresponding space of states. The choice of that space is not arbitrary; it constitutes an approximation and is dictated by the physical situation under study, examples 4

These can be obtained from Eqs. (1.5) using the following simple rules: (n)

(i) convert r n Cm (θ , φ) to Cartesian coordinates, replacing r 2 with x 2 + y 2 + z2 ; (ii) symmetrise each monomial, e.g. xy = 12 (xy + yx); (iii) substitute Jˆx , Jˆy , Jˆz for x, y, z, respectively. The most complete list of explicit expressions for Cm (Jˆ), with 0 ≤ n ≤ 8, was compiled by Lindgård and Danielsen (1974). (n)


159

being given in the preceding sections. For example, the J -representation is particularly convenient for a J multiplet, the matrix elements of the second-order operators (1.17) are then given by 1 LSJ M |C0(2) (Jˆ)|LSJ M = [3M 2 – J (J + 1)]δM ,M , 2 12 3 (2) ˆ LSJ M |C±1 (J )|LSJ M = ∓(M + M) (J ± M )(J ∓ M) δM ,M±1 , 8 (1.20) (2) ˆ LSJ M |C±2 (J )|LSJ M 12 3 (J ± M )(J – 1 ± M )(J ∓ M)(J – 1 ∓ M) δM ,M±2 . = 8 Similar explicit expressions can be also written for the matrix elements of the higher-order operators Cm(4) (Jˆ) and Cm(6) (Jˆ). The equivalence coefficients in Eq. (1.16), αl , αL , αJ , are known rational numbers. In general, αl is given by αl = –

2 (2l – 1)(2l + 3)

(1.21)

that is, αl = –2/45 for all RE’s. A general explicit expression for αL is quite complicated, but for a ground term obeying Hund’s rules, S = 12 (2l + 1 – |2l + 1 – N|), L = S(2l + 1 – 2S), it simplifies to (Bleaney and Stevens, 1953): 2l + 1 – 4S (1.22) 2L – 1 where the upper sign is required for a shell less than half full, and the lower sign for a shell more than half full. Expressions for the Stevens coefficients αJ are generally extremely cumbersome (Wybourne, 1965). Of interest to us here are only those for the ground multiplets of the RE’s, J = L ± S. These are given by ⎧ (L + 1)(2L + 3) ⎪ ⎪ , if J = L – S (light RE) ⎨ (J + 1)(2J + 3) α J = αL × (1.23) L(2L – 1) ⎪ ⎪ ⎩ , if J = L + S (heavy RE). J (2J – 1) αL = ±αl

The values computed using Eq. (1.23) are collected in Table 3.1. Chains of equivalence relations similar to (1.16) can be also written for higherorder operators Cm(4) and Cm(6) , the coefficients therein being βl,L,J (n = 4) and γl,L,J (n = 6). By analogy with the second-order case, βl and γl are given by general expressions similar to Eq. (1.21); for all RE’s βl = 2/495 and γl = –4/3861. Formulae similar to (1.22) exist for βL and γL in the case of a Hund’s ground term (Bleaney and Stevens, 1953). The fourth- and sixth-order Stevens factors, βJ and

160 Table 3.1

N


The Stevens factors for the ground multiplets of the rare earths

Prototype RE ion

Ground multiplet

αJ

βJ

γJ

1

Ce3+

2F

2

3H 4

3

Nd3+

4I 9/2

4

Pm3+

5I 4

5

Sm3+

6

Eu3+

–

–

–

7

Gd3+

6H 5/2 7F 0 8S 7/2

2 32 ·5·7 –22 32 ·5·112 –23 ·17 33 ·113 ·13 23 ·7·17 33 ·5·113 ·13 2·13 33 ·5·7·11

0

Pr3+

–2 5·7 –22 ·13 32 ·52 ·11 –7 32 ·112 2·7 3·5·112 13 32 ·5·7

–

–

–

8

Tb3+

7F

9

Dy3+

6H 15/2

10

Ho3+

5I 8

11

Er3+

4I 15/2

12

Tm3+

3H 6

13

Yb3+

2F

–1 32 ·11 –2 32 ·5·7 –1 2·32 ·52 22 32 ·52 ·7 1 32 ·11 2 32 ·7

2 33 ·5·112 –23 33 ·5·7·11·13 –1 2·3·5·7·11·13 2 32 ·5·7·11·13 23 34 ·5·112 –2 3·5·7·11

–1 34 ·7·112 ·13 22 33 ·7·112 ·132 –5 33 ·7·112 ·132 23 33 ·7·112 ·132 –5 34 ·7·112 ·13 22 33 ·7·11·13

5/2

6

7/2

24 ·17 34 ·5·7·112 ·13 –5·17·19 33 ·7·113 ·132 23 ·17·19 33 ·7·112 ·132

0

γJ , for the ground multiplets of the RE’s are obtainable from equations similar to (1.23) and are presented in Table 3.1. Equations (1.20) are generalised (Smith and Thornley, 1966) to (2J + n + 1)! 1 (n) ˆ M CJJMnm LSJ M |Cm (J )|LSJ M = n (1.24) . 2 (2J + 1)(2J – n)! ˆ between the states |LML or those of Cm(n) (l) ˆ beThe matrix elements of Cm(n) (L) tween |lm are obtained from Eq. (1.24) through the obvious substitution of L or l for J etc. A less obvious fact with rather far-reaching consequences is that the matrix element of Cm(n) in any representation between |LSJ M is proportional to the same M , the proportionality factor being independent of the ‘projection’ CGC CJJMnm quantum numbers m, M, M . This statement is known as the Wigner–Eckart theorem (Edmonds, 1957; Varshalovich et al., 1988). It is the foundation-stone of the method of equivalent operators, since it directly follows that in order to compute such matrix elements, one only needs (apart from the standard CGC) a set of coefficients for n = 2, 4, 6, that is α, β and γ . Choosing one or another representation for Cm(n) is thus a matter of convenience. On the contrary, the choice of a correct set of basis states for Hˆ 4f is very important,


161

it involves approximations and should be based upon intensity relations of type (1.10)–(1.15) Likewise, it is a matter of convention whether to use Cm(n) or Ynm or Um(n) . The Wigner–Eckart theorem is valid for all of them. The chains of equivalences for Ynm and Um(n) look exactly the same as those for Cm(n) , including the values of the coefficients α, β and γ . Moreover, the same equivalences hold for the operators Onm normalised according to Stevens, which do not obey the Wigner–Eckart theorem. Indeed, there is a fairly straightforward connection between Eqs. (1.5) and (1.7): 12 1 (2) (2) 0 1 O2 = 2C0 , O2 = (1.25) C–1 – C1(2) , etc. 6 These relations are valid in any representation: coordinate, l, L or J . Hence equations of type (1.16) hold for Onm as well. It was in fact for Onm that the principle of equivalence was first stated (Stevens, 1952). Combining Eqs. (1.25) with the Wigner–Eckart theorem (1.24), one can express matrix elements of Onm in terms of linear combinations of CGC. These expressions are less convenient for analytical work and are seldom used because the coefficients therein depend not just on n, but also on m (a high price to pay for the apparent simplicity of the Stevens normalisation). More common are direct tabulations of the matrix elements of Onm (Stevens, 1952; Hutchings, 1964; Abragam and Bleaney, 1970; Al’tshuler and Kozyrev, 1974).

1.3 Local symmetry and the exact form of Hˆ CF Towards the conclusion of this introductory section, let us take a closer look at what exactly determines the number of relevant terms in the expansion of the CF potential (1.2). As long as configuration mixing is neglected, this number cannot exceed 27, but is usually far less than that—just 2 in the highest-symmetry case (cubic point groups Td , O or Oh ):5 Hˆ CF = b4 O40 + 5O44 + b6 O60 – 21O64 . (1.26) Not seldom one comes across an erroneous assertion of Eq. (1.26) being characteristic of the ‘cubic symmetry’ in general (Lea et al., 1962, to give just one example). In reality however, if the local symmetry of the RE site is described by either one of the cubic point groups T or Th (cf. Table 3.2), Hˆ CF must contain an extra sixth-order term: Hˆ CF = b4 O40 + 5O44 + b6 O60 – 21O64 + b6 O62 – O66 . (1.27) This expression is clearly distinct from Eq. (1.26), so the need to take this matter further should raise no doubts. (Many of such misstatements originate from the 5 For compactness we use the Stevens normalisation for the CF operators, since we do not intend calculating their matrix elements in this subsection. In this context the operators Onm should not be hastily identified with their J -representation, i.e. with Onm (Jˆ). The symmetry considerations determining the form of Hˆ CF are quite independent of the chosen m representation. Therefore, depending on the situation, Onm in Eqs. (1.26)–(1.32) may be understood as N i=1 On (θi , φi ) m ˆ etc. The loosely defined CF parameters are denoted with lower-case letters, to avoid confusion with properly or as On (L) specified CF parameters.

162 Table 3.2


The 32 point groups

No.

Label

Triclinic 1 2

C1 Ci

1 1¯

Monoclinic 3 4 5

C2 Cs C2h

2 m 2/m

Orthorhombic 6 7 8

D2 C2v D2h

222 mm2 mmm

Tetragonal 9 10 11 12 13 14 15

C4 S4 C4h D4 C4v D2d D4h

4 4¯ 4/m 422 4mm ¯ 42m 4/mmm

No.

Label

Trigonal 16 17 18 19 20

C3 C3i D3 C3v D3d

3 3¯ 32 3m ¯ 3m

Hexagonal 21 22 23 24 25 26 27

C6 C3h C6h D6 C6v D3h D6h

6 6¯ 6/m 622 6mm ¯ 62m 6/mmm

Cubic 28 29 30 31 32

T Th O Td Oh

23 m3 432 ¯ 43m m3m

old CF theory, aimed exclusively at d-ions and therefore limited to fourth-order terms.) Now, with full rigour one can say that the form of Hˆ CF is uniquely determined by the point group describing the local symmetry of the crystallographic site occupied by the RE. The traditional combinations of point groups called syngonies, or crystal systems (cubic, tetragonal etc., Table 3.2) provide no valid basis for judgement in this question.6 The form of Hˆ CF also depends on the orientation of the coordinate system in relation to the crystallographic directions. In the above example (1.26) all three coordinate axes were set parallel to 4-fold symmetry axes. A rotation through ±π /4 around z would correspond to a simultaneous change of sign of the coefficients of O44 and O64 in Eq. (1.26). Setting the z axis along a 2- or a 3-fold crystal axis would lead to Eqs. (6.14) or (6.15) of Hutchings (1964). Note that the number of independent CF parameters (in this case, two) is independent of the choice of the 6 Sometimes the name of a crystal system is used as a synonym for the most symmetric (holohedral) point group of that crystal system (listed last under each of the headings in Table 3.2). Such liberty with the terms should be avoided, as it only causes confusion.


Figure 3.2

163

A two-dimensional lattice with RE atoms shown as dark squares.

coordinate system. In the examples that follow the z axis will always be set along a high-symmetry crystal direction. Local symmetry is the symmetry of the whole crystal seen from the standpoint of the RE. (If RE atoms occupy several non-equivalent sites in a crystal, there are generally as many distinct symmetries.) It should not be confounded with the ‘type of coordination’, or the shape of the polyhedron made by the nearest neighbours of the RE. Figure 3.2 illustrates this point in two dimensions: the nearest neighbours of the central atom form a perfect square, described by the point group D4h . The true local symmetry is however lower, C4h . The corresponding expressions for Hˆ CF are distinct, cf. Eqs. (1.31) and (1.32) below. On the other hand, the local symmetry of the RE site should be distinguished from the crystallographic class, or the point group describing the crystal as a whole, regardless of the viewpoint. Thus, the famous permanent magnet materials 14 RE2 Fe14 B (space group P 42 /mnm—D4h ) belong to the tetragonal crystallographic class D4h . The RE atoms occupy sites of two kinds, 4f and 4g, the symmetry of both being described by the orthorhombic point group C2v . A definitive reference in this matter is the International Tables for Crystallography (Hahn, 1983). For the space group P 42 /mnm one reads there, on page 459 of volume A, that the 4f (xx0) and the 4g(xx0) sites both have the symmetry of m.2m, that is mm2, or C2v . When comparing atomic positions described in the original literature and in the International Tables, one should be aware of the possible multiple choices of the origin and axes orientations. Having determined the local point group, we are ready to formulate the main principle governing the form of Hˆ CF . It sounds surprisingly simple: Hˆ CF may only contain terms invariant under all symmetry operations of the local group. Alternatively, in terms of the theory of representations: only the terms which belong to the identity, or totally symmetric irreducible representation of the local point group may enter in Hˆ CF . Let us demonstrate this principle for the point group Cs (m). It is convenient to rewrite Eqs. (1.7) in Cartesian coordinates:

164


3z2 – 1, r2 xz yz O21 = 2 , (1.28)

12 = 2 , r r x2 – y2 2xy 2 ,

= . O22 = 2 r2 r2 The group Cs contains a single non-trivial symmetry element—a mirror plane perpendicular to the z axis. Being invariant in this case is equivalent to being even in z. The allowed second-order terms thus are O20 , O22 and 22 . This analysis is extended in a natural way to higher-order operators. The result is that all Onm and

mn with both n and m even enter in Hˆ CF (those with n > 6 do not affect the 4f shell and need not be included). A convenient collection of explicit expressions for Onm and mn can be found in Appendix V of Al’tshuler and Kozyrev (1974). c ∝ Onm and Alternatively, one may use the extensive list of tesseral harmonics, Znm s m Znm ∝ n , compiled by Görller-Walrand and Binnemans (1996, Appendix 2). The orthorhombic point group D2h (mmm) contains three mutually perpendicular mirror planes, as well as combinations thereof. The invariant terms are those even in all three coordinates, i.e. O20 and O22 from Eqs. (1.28), and generally Onm with n and m even. One need not perform this analysis every time anew. Exhaustive results for all 32 point groups have been obtained and tabulated (Bradley and Cracknell, 1972; Altmann and Herzig, 1994; Görller-Walrand and Binnemans, 1996). For example, for the simplest hexagonal point group C6 one finds on page 64 of Bradley and Cracknell (1972) the following table: O20 =

6 (C6 ) A B 1 E1 2 E1 1 E2 2 E2

m mod 6 0 3 4 2 1 5

The relevant information is in the first line, corresponding to the totally symmetric irreducible representation A. It reads: allowed are all spherical harmonics Ynm (or Cm(n) , or Um(n) ) with n arbitrary and m = 0 mod 6, i.e. 0, ±6, ±12 etc. (note that those with n odd are not forbidden, cf. footnote 1). Turning to the Stevens convention and limiting ourselves to n = 2, 4, 6, we arrive at

66 Hˆ CF = b20 O20 + b40 O40 + b60 O60 + b66 O66 + b66

(1.29)

where all the coefficients are real numbers. Equation (1.29) is related to the coordinate system whose z axis is parallel to the 6-fold crystal axis [001] and whose x axis is set along an elementary translation vector in the basal plane [100]. The last term in (1.29) can be eliminated by rotating the coordinate system around the


165

z axis through an (unknown a priori) angle φ0 , such that tan 6φ0 = b66 /b66 . Equation (1.29) is also valid for the hexagonal point groups C3h and C6h . Let us consider another hexagonal group, D6 . The corresponding table is on page 65 of Bradley and Cracknell (1972). The top part of it looks as follows:

l m mod (+6) φ-dep 0 0 c 7 6 s 6 6 s A2 ............................................... 622 (D6 ) A1

Once again, what we need is in the first line of the table (the second line, though related to the identity representation A1 , contains information on odd harmonics of order seven or higher). According to the convention adopted by Bradley and Cracknell (1972), l in the header is an abbreviation of l mod (+2). Thus, we read: allowed are the harmonics with l, in our notation n = 0 mod (+2) and m = 0 mod (+6). The last column further specifies: when m = 0, only symmetric combinations of type (Ynm + Yn–m ) ∝ cos mφ are admissible, i.e. Onm as opposed to mn . One therefore has Hˆ CF = b20 O20 + b40 O40 + b60 O60 + b66 O66 .

(1.30)

The same expression is obtained for the hexagonal groups C6v , D3h and D6h . A further example—tetragonal point groups C4 , S4 and C4h :

44 + b60 O60 + b64 O64 + b64

46 . (1.31) Hˆ CF = b20 O20 + b40 O40 + b44 O44 + b44

Note that no rotation in the basal plane can reduce this expression to Eq. (1.32), i.e. and b64 simultaneously. (This was only possible in the old CF theory, eliminate b44 aimed mainly at spectroscopic properties of d-ions, hence the misconception of the ‘united tetragonal symmetry’.) The expression for the remaining tetragonal groups (D4 , C4v , D2d , D4h ) is as follows: Hˆ CF = b20 O20 + b40 O40 + b44 O44 + b60 O60 + b64 O64 .

(1.32)

Summarising this subsection, the precise form of the CF Hamiltonian cannot be inferred from vague general categories like cubic (hexagonal, tetragonal etc.) symmetry. Rather, it is uniquely determined by the local point group of the crystallographic site occupied by the RE and can be inquired about in widely available tables (Bradley and Cracknell, 1972; Altmann and Herzig, 1994). Finally, one can consult the nearly complete list (missing are the cubic groups T , Th and O) of explicit expressions for Hˆ CF in terms of Cm(n) (Görller-Walrand and Binnemans, 1996, Appendix 3).

166


2. The Single-Ion Anisotropy Model for 3d-4f Intermetallic Compounds 2.1 Macroscopic description of magnetic anisotropy Let us consider a macroscopic system held at temperature T in an applied magnetic field B. These external parameters may vary only quasi-statically, so that at all times the system remains at thermal equilibrium. The standard thermodynamic description of such a system is afforded by specifying its free energy F (T , B, . . .), which is a characteristic function of T , B and perhaps further external parameters. The equations of state are then obtained by taking partial derivatives of the free energy with respect to its variables: ∂F S(T , B, . . .) = – (2.1) , ∂T B ∂F M(T , B, . . .) = – (2.2) . ∂B T Here S, M, . . . are the system’s internal parameters: entropy, magnetisation, etc. External and internal parameters make pairs of conjugate thermodynamic variables: T –S, B–M, etc. In this Chapter we shall deal mainly with the magnetic equation of state (2.2). This is not to say that the caloric equation of state (2.1) is less important. The entropy plays a central role in magneto-thermal properties, such as specific heat and magnetocaloric effect (Tishin and Spichkin, 2003). Alternatively to using the equilibrium free energy F (T , B) one can take as a starting point the non-equilibrium with respect to magnetisation thermodynamic potential (T , B, M). Unlike the usual equilibrium potentials, depends on both conjugate variables, B and M, and the corresponding equation of state is obtained by minimising it with respect to the internal parameter M: ∂ (2.3) =0 ∂M T ,B and (∂ 2 /∂M 2 )T ,B > 0. The so minimised thermodynamic potential is the equilibrium free energy, min (T , B, M) = F (T , B). M Both approaches are of course equivalent, as long as they lead to the same magnetic equation of state, either in the form of Eq. (2.2) or of Eq. (2.3). The free energy is the preferred route when the system’s partition function Z(T , B) can be computed; then F is given by the well-known relation of the statistical mechanics, F = –kT ln Z. In turn, is advantageous in phenomenological theories since its dependence on M and B can be inferred from the rather general considerations of symmetry. For example, a ferromagnet near its Curie point is described by (Landau and Lifshitz, 1958): 1 1 (T , B, M) = 0 + aM 2 + bM 4 + · · · – B · M (2.4) 2 4


167

where the quantities 0 , a, b, . . . depend on temperature and other external parameters, e.g. pressure, but not on magnetic field or magnetisation. The only assumptions used in the construction of the expansion (2.4) are the time-evenness of and the smallness of M. It is however sufficient to obtain a number of useful predictions regarding the thermodynamic behaviour of ferromagnets near the Curie point. Up to this point we have not paid much attention to the fact that both B and M are vectors, which means that in effect one has to deal with three pairs of conjugate magnetic variables. The system under consideration was tacitly assumed isotropic (therefore at equilibrium M||B), so setting one of the coordinate axes along B reduced the description to a single pair of scalar variables. Now we turn to the anisotropic case, which is both more general and more interesting, since it demonstrates to full extent the advantages of the symmetry approach based on the use of the non-equilibrium thermodynamic potential (T , B, M). The coefficients in the Landau expansion (2.4) now become tensor quantities. For example, 12 aM 2 is replaced with 12 aαβ Mα Mβ , where aαβ is a tensor of rank two, the sum being taken over α, β = x, y, z. Many of the components of aαβ , bαβγ δ etc. may be equal to each other or vanish for symmetry reasons. More strictly, this depends on the point group describing the symmetry of the crystal as a whole (since we are dealing with macroscopic properties), also known as the crystallographic class. Thus, for any point group of the cubic crystal system (see Table 3.2) the expansion (2.4) takes the following form: 1 1 (T , B, M) = 0 + aM 2 + bM 4 + b (Mx2 My2 + My2 Mz2 + Mz2 Mx2 ) 2 4 + · · · – B · M. (2.5) In this case the anisotropy makes its first appearance in the terms of fourth order. However, in lower-symmetry crystals it affects second-order terms as well. In certain classes of phenomena the magnitude of the magnetisation vector varies little, while its direction may change significantly. In such a situation it is convenient to use as internal thermodynamic parameter the direction of M, rather than its Cartesian components. Direction cosines or spherical angles can be used for the purpose. A specific example is the behaviour of a ferromagnet well below the Curie point in a weak to moderate magnetic field. The energy associated with a noticeable change of |M| is of the order of TC ∼ 103 K per atom (∼102 K for ferromagnets less suitable for applications). The anisotropy energy is usually much smaller, reaching ∼101 K per atom in YCo5 —one of the most strongly anisotropic 3d magnets (Alameda et al., 1981). The RE contribution to the anisotropy energy may exceed 100 K per RE atom at low temperature (Radwański, 1986), which upon averaging over all atoms in an iron- or cobalt-rich compound would yield ∼2 × 101 K per atom. This is significantly less than the 3d-3d exchange energy. At any rate, it should be kept in mind that the formalism of magnetic anisotropy is an approximate one and that its validity is limited by the requirement that |M| should be essentially constant. It does not apply near TC , in very strong magnetic fields or when |M| is itself strongly anisotropic. The latter restriction concerns e.g. YCo5 , where |M| changes by as much as 4% upon reorientation (Alameda et al., 1981). In general, the

168


approach might fail in RE compounds at low temperatures. From now on we limit ourselves to such phenomena where |M(T , B)| = Ms (T ), i.e. the magnitude of the magnetisation is practically independent of applied field and equals the spontaneous magnetisation. The anisotropic version of Landau’s expansion then becomes

1 1 = 0 + Ms2 (T ) aαβ nα nβ + Ms4 (T ) bαβγ δ nα nβ nγ nδ 2 4 α,β α,β,γ ,δ + · · · – Ms (T )n · B

(2.6)

where n ≡ M /|M| is a unit vector in the direction of the magnetisation. The fact that only quadratic and quartic terms have been written out explicitly does not imply rapid convergence of the expansion (2.6) or that its truncation after the quartic term is legitimised in any way. The situation here is radically different from that in a ferromagnet near TC —the subject of Landau’s theory of second-order phase transitions (Landau and Lifshitz, 1958). The truncation of Eq. (2.4) was based on the obvious fact that Ms → 0 as T → TC . Conversely, Ms is not at all small in Eq. (2.6). It is to be assumed that the series (2.6) diverges, unless proven otherwise. Equation (2.6) is too general to be useful. Let us rewrite it for some commonly encountered special cases. The key point here is the invariance of each term of the expansion under all symmetry operations of the crystallographic class. Thus, for the cubic classes Td , O and Oh Eq. (2.6) becomes = 0 + Ea – Ms (T )n · B

(2.7)

where 1 1 0 = 0 + aMs2 (T ) + bMs4 (T ) + · · · 2 4 and

Ea = K1 n2x n2y + n2y n2z + n2z n2x + K2 n2x n2y n2z + · · ·

(2.8)

The quantity Ea —the anisotropic part of the thermodynamic potential in the absence of magnetic field—is known as anisotropy energy, while K1 , K2 etc. are called anisotropy constants. The latter may depend on temperature and other external parameters, but not on magnetic field. The dependence of on B is limited to the last, Zeeman term of Eq. (2.7). Equation (2.8) was first obtained by Gans and Czerlinsky (1932). For the cubic crystallographic classes T and Th the anisotropy energy contains an extra sixth-order term in addition to that in Eq. (2.8): K2 n2x n2y n2x – n2y + n2y n2z n2y – n2z + n2z n2x n2z – n2x . (2.9) This term is invariant under the rotations around the 3-fold axes (equivalent to cyclic permutations within the triplet n2x , n2y , n2z ), but is not invariant with respect to rotations through 90° about the 4-fold axes (pair permutations of the type n2x ↔ n2y ). Therefore, it is allowed in the lower-symmetry cubic groups T and Th , containing 3-fold axes only, and forbidden in the higher-symmetry cubic groups Td ,


169

O and Oh , which include 4-fold axes as well. A direct parallel can be drawn here to Eqs. (1.26), (1.27) describing the CF for the same point groups. For non-cubic point groups it is customary to describe the direction of M by means of the conventional spherical angles θ and φ. The polar axis is conveniently set along a high-symmetry crystallographic direction, whose choice presents no difficulty except for the triclinic groups C1 and Ci . The anisotropy energy, Ea (θ , φ), is then expanded over a suitably chosen basis, e.g. over the irreducible tensor operators: Ea =

n ∞

κnm Cm(n) (θ , φ).

(2.10)

n=2 m=–n even

Note that there is no generally valid reason to truncate this expansion after any finite number of terms. An added advantage of presenting Ea as Eq. (2.10) is the possibility to exploit the formal analogy with the CF potential (1.2). A particular point group dictates the same form of both Ea (θ , φ) and VCF (r, θ, φ).7 Therefore, the expressions obtained in Section 1.3 for specific point groups can be simply taken over. Alternatively, one can consult the tables recommended therein (Bradley and Cracknell, 1972; Altmann and Herzig, 1994; Görller-Walrand and Binnemans, 1996). Thus, by analogy with Eq. (1.30), we write for the hexagonal crystallographic classes D6 , C6v , D3h and D6h : Ea = κ20 P2 (cos θ ) + κ40 P4 (cos θ ) + κ60 P6 (cos θ ) (6) (θ , φ) + · · · + κ66 C6(6) (θ , φ) + C–6

(2.11)

It will be remembered that C0(n) (θ , φ) ≡ Pn (cos θ ), Pn (x) being the Legendre polynomials. Another conventional form of this expression is due to Mason (1954): Ea = K1 sin2 θ + K2 sin4 θ + K3 sin6 θ + K3 sin6 θ cos 6φ + · · ·

(2.12)

The fairly straightforward relations between the anisotropy constants Ki and κnm can be found e.g. in Appendix B to the review of Franse and Radwański (1993) in Volume 7 of this Handbook. The corresponding expression for the hexagonal classes C6 , C3h and C6h contains an extra sixth-order term, K3 sin6 θ sin 6φ, ∝ 66 in Eq. (1.29). Finally, the anisotropy energy of tetragonal crystals is given by Ea = K1 sin2 θ + K2 sin4 θ + K2 sin4 θ cos 4φ + K2 sin4 θ sin 4φ + K3 sin6 θ + K3 sin6 θ cos 4φ + K3 sin6 θ sin 4φ + · · · K2

(2.13)

K3

and are nonzero if the crystallographic class is C4 , S4 where the constants or C4h , but must vanish if it is D4 , C4v , D2d or D4h . 7 This does not mean that the local symmetry of the RE site and the symmetry of the crystal as a whole are necessarily described by the same point group. When these point groups are distinct, so are the expressions for Ea and VCF (cf. the example of RE2 Fe14 B in Section 1.3).

170


Recapitulating, magnetic anisotropy energy Ea is that part of the nonequilibrium with respect to M thermodynamic potential (T , B, M)|B=0 which depends on the direction of M in a situation when |M| is known not to depend on applied magnetic field B. The dependence Ea (M /|M|) is usually presented as a series in powers of the direction cosines of M, whose form is dictated by the point group describing the symmetry of the crystal as a whole—the crystallographic class. As regards the convergence of the series, the formal theory is unable to make any positive prediction in this respect. Such predictions, leading to truncated expansions, can only be obtained in specific microscopic models, when the coefficients (anisotropy constants) prove proportional to growing powers of a small parameter. In the absence of a valid model, no custom or convention can justify the use of expressions truncated after the terms of 2nd , 4th or 6th order. Throughout this subsection we have been dealing with standard thermodynamic potentials, fit to describe macroscopic systems, containing very large numbers of atoms. The introduced concepts of ‘anisotropy energy’ and ‘anisotropy constant’ are inapplicable to nanoscopic systems, just as does not apply to them the notion of temperature.

2.2 The notion of an exchange-dominated RE system Let us consider a RE–transition metal compound satisfying the validity conditions for the single-multiplet approximation. We assume for simplicity that the applied magnetic field is nil. Following Section 1.2.3, the properties of the RE subsystem in this compound are described by a single-ion Hamiltonian, Hˆ ex + Hˆ CF , projected on the ground J multiplet: Hˆ 4f = 2(gJ – 1)μB Bex · ˆJ +

n

Bnm Cm(n) (Jˆ).

(2.14)

n=2,4,6 m=–n

Here B ex is the exchange field on the RE produced by the ordered 3d sublattice, Bnm are CF parameters incorporating the Stevens factors. Despite all simplifications, the Hamiltonian (2.14) is still very complicated, since it contains in the general case a large number of free parameters. Our consideration in this section will therefore be limited to a special case of the so-called exchange-dominated RE system. On the one hand, this will greatly simplify the calculations. On the other hand, the approximation, if not taken too far, is likely to apply to real hard magnetic materials. It is clear from the outset that the CF can be regarded neither as infinitesimally small, nor as negligible in comparison with the 3d-4f exchange. A strong CF is indispensable to a good permanent magnet performance. At the same time, many of these materials feature low-temperature RE magnetic moments close to the freeion value gJ J , as if the strong CF were not there at all. So, how exactly weak should the CF be to account for this behaviour? To answer this question, let us first consider a fictitious ‘training’ RE whose ground multiplet has J = 1. This is the simplest system displaying non-vanishing CF effects. (Being time-even, the CF does not split the simpler J = 1/2 multiplets). When J = 1, the triangle rule dictates that only second-order CF terms are


171

Figure 3.3 Normalised energy level pattern of a didactic RE with J = 1. The curved portions of the middle level are hyperbolae described byE = 2/(1 – 32 η), η < – 23 , and E = 1 – 2/(1 + 32 η), η > 23 . The hatched interval around the origin is the location of broadly exchange-dominated systems.

relevant. Our hypothetical system is supposed to be a permanent magnet material, i.e. a uniaxial crystal in which the magnetisation of the 3d sublattice is parallel to the high-symmetry axis z (consequently, B ex is antiparallel to z). We assume in addition that the local symmetry of the RE site is uniaxial as well, i.e. that it is described by a point group belonging to either of the three syngonies: tetragonal, trigonal or hexagonal. Then a single CF term is allowed—that in C0(2) —and Eq. (2.14) turns into 3 Hˆ 4f = 2(1 – gJ )μB Bex Jˆz + B20 Jˆz2 – 1 . (2.15) 2 The matrix of this Hamiltonian is obviously diagonal in the |J M basis, the three eigenvalues being obtainable through the substitution of M = 0, ±1 for Jˆz in Eq. (2.15). Of interest to us here is the energy level pattern, rather than the eigenvalues as such. So we set the ground state energy to zero and normalise the overall multiplet splitting to unity. The so defined level pattern (Fig. 3.3) is fully determined by a single dimensionless parameter—the CF-to-exchange ratio, B20 η= (2.16) ex where ex = 2|gJ – 1|μB Bex (2.17) is the exchange splitting between two adjacent levels of the multiplet. The converse is generally not true—knowing the pattern does not enable one to establish the value of η even for such a simple system (unless the levels are unambiguously labelled). For example, an equidistant spectrum could correspond to either η = 0 or 2 or –2, the three situations being physically quite distinct. One finds by inspecting Fig. 3.3 that from the standpoint of the level sequence the entire η axis is split into three domains separated by the points η = ± 23 , where two levels cross over.

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We now define that a J = 1 RE system is exchange-dominated in the strict sense if its locus in Fig. 3.3 lies near the origin, that is if |η| is much less than the size of the central segment, |η| 23 . The same system will be called exchangedominated in the broad sense if its η lies somewhere within the central domain, |η| < 23 , but not too close to its boundaries. (The latter clause is to exclude the nearly degenerate case, |η| = 23 – ε, 0 ≤ ε 23 .) The domain corresponding to broadly exchange-dominated systems is shown in Fig. 3.3 by hatching. Everywhere within the hatched area (in fact, also everywhere left of it) the low-temperature magnetic moment is constant and equals gJ μB , or rather gJ μB · sign(1 – gJ ). Before attempting a generalisation of these definitions for arbitrary J , let us consider one more particular case. This time it is a ‘nearly real’ RE with J = 5/2 in a hexagonal CF. Were it not for the extremely strong J -mixing that makes the singlemultiplet approximation fail, this example would be fully relevant to e.g. SmCo5 . Our goal however is not so much to develop an accurate quantitative approach to Sm-based magnets, as to demonstrate the concept of exchange-dominated RE systems on something more realistic than the above example of plain J = 1. Thus, within the single-multiplet approximation, the Hamiltonian of a RE with J = 5/2 in an exchange field B ex antiparallel to z and in a hexagonal CF has the following form: 3 ˆ2 35 ˆ ˆ H4f = 2(1 – gJ )μB Bex Jz + B20 Jz – 2 8 35 ˆ4 475 ˆ2 2835 J – J + . + B40 (2.18) 8 z 16 z 128 The eigenvalues EM are obtained by substitution of M = ±1/2, ±3/2, ±5/2 for Jˆz : 1 E±1/2 = ± ex – 4B20 + 15B40 , 2 3 45 E±3/2 = ± ex – B20 – B40 , (2.19) 2 2 5 15 E±5/2 = ± ex + 5B20 + B40 , 2 2 where ex is the exchange splitting (2.17) and it has been assumed for definiteness that, like in Sm3+ , gJ < 1. The energy level pattern is determined by two dimensionless parameters: η, defined according to Eq. (2.16), and ξ , given by B40 . (2.20) ex The latter describes the strength of the 4th-order CF in relation to the 3d-4f exchange. The ηξ plane can be divided into 53 domains, each one of them characterised by a certain sequence of the energy levels (Fig. 3.4). Of primary interest to us is the central domain, which contains the origin. There, the level sequence (i.e. the dependence of EM on M) is monotonic, just as it would be without any CF. ξ=


173

Figure 3.4 Character of the splitting of the ground sextet of a RE with J = 5/2, depending on the intensities of second- and fourth-order CF in relation to the exchange. Within each of the domains delimited by the oblique lines the eigenvalue sequence EM , arranged in ascending order, defines a permutation of the six values of the quantum number M characteristic of that domain. Level cross-over takes place at the domain boundaries. The domains within each one of the three sectors separated by the bold lines have the same ground state: M = –5/2 (west), –3/2 (north), –1/2 (south-east). The hatched area near the origin marks the location of broadly exchange-dominated systems. The dark diamond corresponds to SmCo5 .

At the domain boundaries the levels cross over. Let us demonstrate this for the levels with M = –5/2 and –3/2 assuming, by analogy with Sm3+ , that gJ < 1. Then, in the absence of CF, the level with M = –5/2 is the ground state and the one with M = –3/2 is the first excited state. The cross-over condition is obtained by equating the last two of the equations (2.19), in which the lower signs must be taken. Upon division by ex 6η + 30ξ = 1.

(2.21)

This equation corresponds to a straight line in the ηξ plane, namely, to that which makes the north-east border of the central domain in Fig. 3.4. Taking the other three combinations of upper/lower signs in the last two Eqs. (2.19) yields three more equations similar to (2.21), but with different right-hand sides. The four equations describe the set of four negatively sloping parallel lines in Fig. 3.4. Two further sets of four parallel lines are obtained in a similar fashion from the cross-over conditions E±1/2 = E±3/2 and E±1/2 = E±5/2 . The four lines delimiting the central parallelogram correspond to the crossings of the adjacent levels of the monotonic spectrum, i.e. those with M = ±1. There are five pairs of adjacent levels in a sextet, however no line is generated by the equation E1/2 = E–1/2 . Likewise, the conditions E3/2 = E–3/2 and E5/2 = E–5/2 produce no extra lines in the ηξ plane. The physical reason of this is the time-even character of the CF, on which grounds the latter does not affect the splitting of pairs of Kramers-conjugate states. These are split by the exchange interaction alone, the level sequence within each pair depending on the sign of the difference 1 – gJ . In the case of Sm3+ ,

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gJ = 5/7 < 1, the levels with negative M lie always below their positive counterparts.8 Consequently, only these three levels, M = –1/2, –3/2, –5/2, can claim the privilege of becoming ground state. On this principle, the entire ηξ plane can be divided into three sectors. These are separated by the bold lines in Fig. 3.4. We shall concentrate primarily on the west sector (containing the central parallelogram), where the ground state is M = –5/2. The north sector corresponds to M = –3/2 being the ground state, while in the south-east sector it is M = –1/2. A strictly exchange-dominated J = 5/2 system can now be readily defined as such a system whose locus in the ηξ diagram is close to the origin, the proximity being understood relative to the dimensions of the central domain. This definition is equivalent to a pair of strong inequalities |η| 1/6, |ξ | 1/30. A J = 5/2 system is exchange-dominated in the broad sense if its locus lies inside the central parallelogram of Fig. 3.4, excluding the regions close to its boundaries, as shown by the hatching. The hatched area belongs to the west sector of the drawing, where the low-temperature magnetic moment is gJ J = gJ 5/2. The reason for the duplicate definition is that typical RE-based hard magnetic materials are broadly exchange-dominated systems, without being such in the strict sense. For example, the dark diamond, corresponding to the archetypal permanent magnet material SmCo5 , is situated half way between the origin and the boundary of the central domain of Fig. 3.4.9 The above definitions can now be easily generalised for an arbitrary J > 5/2. In addition to the parameters η and ξ , defined by Eqs. (2.16), (2.20), a third parameter ζ needs to be introduced, to describe the relative intensity of the (axial) sixth-order CF: ζ =

B60 . ex

(2.22)

The ηξ ζ parameter space is divided by a number of planes into many domains, according to the order of the eigenvalues EM . Within the central domain (containing the origin) the eigenvalue sequence is monotonic: the ground state has the maximum (or negative maximum) possible M, the M of the first excited level differs from the latter by 1, etc. The eigenvalue sequences in the other domains correspond to permutations of the monotonic one. The boundaries of the central polyhedron are obtained from cross-over conditions for the levels with M = ±1. The respective gaps can be presented as ex + i , where ex is the exchange contribution (2.17), common for all pairs of adjacent levels, i are the CF contributions (numbered from bottom to top, Kuz’min, 1995): 3 5 1 = – (2J – 1)B20 – (2J – 1)(2J – 2)(2J – 3)B40 2 4 21 – (2J – 1)(2J – 2)(2J – 3)(2J – 4)(2J – 5)B60 , 32 8 9

According to the adopted convention, the positive z direction is parallel to M 3d and antiparallel to B ex . The following values were used: Bex = 295 T, B20 = –2 meV, B40 = 0 (Kuz’min et al., 2002).


3 5 2 = – (2J – 3)B20 – (2J – 10)(2J – 2)(2J – 3)B40 2 4 21 – (2J – 21)(2J – 2)(2J – 3)(2J – 4)(2J – 5)B60 , 32 etc.

175

(2.23)

Equating ex + i to zero, one obtains upon dividing by ex the equations of the boundaries of the central domain, e.g. 5 3 (2J – 1)η + (2J – 1)(2J – 2)(2J – 3)ξ 2 4 21 + (2J – 1)(2J – 2)(2J – 3)(2J – 4)(2J – 5)ζ = 1 (2.24) 32 and so on. In the special case of J = 5/2 Eq. (2.24) turns, as expected, into Eq. (2.21). A RE system with an axial CF corresponds to a point in the ηξ ζ space. One can demand that this point be close to the origin, the proximity being related to the dimensions of the central polyhedron. The same condition can be expressed as a strong inequality, |i | ex .

(2.25)

A system satisfying (2.25) will be called exchange-dominated in the strict sense. Alternatively, the locus must lie inside the central polyhedron, not too close to its boundaries: |i | < ex (1 – δ),

0 < δ 1.

(2.26)

This is a broadly exchange-dominated RE system. Clearly, any strictly exchangedominated system is also broadly exchange-dominated. The converse is not true. In order to ensure the free-ion value of the low-temperature magnetic moment, gJ J μB , it suffices for an axially-symmetric RE system to be broadly exchangedominated. All the above-said can be applied to real materials—which do not possess the axial symmetry—provided that Bnm with m = 0 are not too large as compared to Bn0 (which appears to be fulfilled in most cases). Then, to first order in Bnm /ex , the presence of non-axial CF terms has no effect on the low-temperature magnetic moment of the RE.10 Summarising, we have formulated two different concepts of an exchangedominated RE system. When the first of them applies, it is strictly justified to use the first-order perturbation theory in Bnm /ex (see subsection 2.511 ). Most real RE-based magnets, however, fit the second (broad) but not the first definition. For 10 Indeed, reduction of the ground-state magnetic moment is a time-even effect. The corresponding expression, μCF /μfree-ion = 1 – const1 (Bnm /ex ) – const2 (Bnm /ex )2 – . . . , may not contain odd powers of ex ∝ Bex , therefore const1 = 0. 11 This does not include Sm-based magnets, on account of the failure of the single-multiplet approximation. Luckily, the magnetic moment of Sm in intermetallics is so small—even its sign varies from compound to compound (Givord et al., 1980)—that it can be safely neglected altogether.

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such materials the equations of subsection 2.5 are still qualitatively correct, despite some lack of rigorous foundation. These equations are strictly inapplicable when neither of the above two definitions is satisfied.

2.3 The single-ion model for 3d-4f intermetallics Our goal in this subsection is to formulate a general recipe for computing the RE contribution to the magnetic anisotropy energy Ea proceeding from the parameters entering in the single-ion RE Hamiltonian Hˆ 4f . The latter will be treated in the single-multiplet approximation, the only exception being Section 2.9, dedicated specifically to J -mixing. As stated in Section 1.1, the single-ion approach to describing the properties of the RE subsystem in 3d-4f intermetallics relies on the peculiar hierarchy of exchange interactions in these compounds. Thanks to it, the 3d subsystem can be regarded as something external, whose action on the RE is described by means of an exchange field B ex . This enables one to treat the RE subsystem as an ensemble of non-interacting ‘ions’, each one of which is described by the following Hamiltonian:

Bnm Cm(n) (Jˆ). Hˆ 4f = 2(gJ – 1)μB Bex · Jˆ + gJ μB B · Jˆ + (2.27) n,m

As pointed out in Section 2.2, B ex is antiparallel to the 3d magnetisation M 3d . If Hˆ 4f is related to the crystallographic coordinate axes, the dependence on the orientation of M 3d enters into the first term of Eq. (2.27). The angles θ and φ defining the orientation of M 3d are external thermodynamic parameters in relation to the RE subsystem. The latter is described by the usual equilibrium canonical distribution, F4f = –kT ln Z4f (θ , φ)

(2.28)

where the RE partition function is

Hˆ 4f . Z4f (θ , φ) = tr exp – kT

(2.29)

With respect to the 3d subsystem (and therefore to the combined 3d-4f system) θ and φ are internal parameters, i.e. the former is described by means of a nonequilibrium thermodynamic potential 3d (θ , φ). The equilibrium values of θ and φ are determined through minimisation of the combined thermodynamic potential, min 3d (θ , φ) + F4f (θ , φ) . (2.30) θ,φ

At this stage we do not specify the form of 3d (θ , φ). Suffice it to say that, like Eq. (2.27), 3d contains anisotropy energy and a Zeeman term. Despite the fact that it only takes a few equations (2.27)–(2.30) to formulate the single-ion model, it proves impossible to obtain a general explicit expression for F4f (θ , φ). The main difficulty is taking the trace of the matrix exponential in Eq. (2.29). One exception is the special case of J = 1, when such an expression does exist (Kuz’min, 1995). This was used to demonstrate that F4f (θ , φ) cannot in


177

general be presented as a truncated expansion of type (2.8)–(2.13). Such a presentation requires that the anisotropy constants Ki or κnm contain incrementing powers of a small parameter, which is only true in some special cases when additional approximations can be made. If valid, the same approximations enable one to evaluate the anisotropy constants. It is interesting to note that validity of the very same approximations allows to settle the often raised question about the effect of non-collinearity of the 3d and 4f sublattices. The point is that the above-defined angles θ and φ determine the orientation of just the 3d sublattice magnetisation M 3d . As regards the RE moment μR , its orientation is described by the angles θR and φR , which are generally speaking distinct from θ and φ. The RE magnetisation is an internal thermodynamic parameter, so it does not figure explicitly in the above algorithm (2.8)–(2.13). For any given θ and φ, the angles θR = arccos(μRz /|μR |) and φR = arctan(μRy /μRx ) can be found from the relation ˆ ∂F4f gJ μ B H4f μR = – (2.31) = –gJ μB Jˆ = – tr Jˆ exp – . ∂B Z4f (θ ,φ) kT The problem posed by the non-collinearity is that the angles obtained through the minimisation of the total thermodynamic potential of the system (2.30) do not correspond to the orientation of the system’s total magnetisation. This difficulty will be shown to disappear as soon as there is a valid reason to truncate the series for Ea . To summarise, there are two possibilities. Either, one of the above-mentioned approximations is valid and then one can (i) truncate the expansion for Ea , (ii) obtain explicit expressions for the anisotropy constants entering in the truncated expansion, and (iii) neglect the non-collinearity of the 3d and 4f sublattices. Or, there is no such a valid approximation; then the standard description by means of anisotropy constants is impossible, as none of the above three preconditions can be secured. Of course, the general equations (2.27)–(2.31) are then still valid and can be used to compute M(B) (not the anisotropy constants) numerically. Consequently, we shall no longer dwell on the sterile general formalism, but rather proceed to the most important special cases.

2.4 The high-temperature approximation Let us recast Eqs. (2.28), (2.29) in a different form, J ex Hˆ 4f F4f (θ , φ) = – ln tr exp –x x J ex

(2.32)

where x is a quantity equivalent to Langevin’s magneto-thermal ratio, x= and ex is the exchange splitting (2.17).

J ex kT

(2.33)

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Assuming that x is a small parameter, we can expand F4f in powers of x in the spirit of Kramers-Opechowski: trHˆ 4f2 J ex ln(2J + 1) – x x 2J (2J + 1)ex trHˆ 4f3 + 2 (2.34) x2 + · · · 6J (2J + 1)2ex It has been taken into account that tr Hˆ 4f = 0. The first term in (2.34) is obviously isotropic, i.e. it does not depend on either θ or φ. Let us demonstrate that the second term is isotropic, too. To this end it is convenient to rewrite the Hamiltonian (2.27) as follows: F4f (θ , φ) = –

Hˆ 4f = sign(1 – gJ )ex n · Jˆ + Hˆ CF . (2.35) Here n = M 3d /|M 3d | is a unit vector in the direction of the 3d magnetisation, M 3d ↑↓ B ex , and it has been assumed that B Bex , in order to ensure that μR remains essentially independent of B. We shall make use of a helpful orthogonality relation for the operators Cm(n) (Jˆ) (Kuz’min, 1995), which readily follows from the well-known orthogonality of the CGC: –2n (2J + n + 1)! (n ) ˆ m 2 δnn δmm tr Cm(n) (Jˆ)C–m (2.36) (J ) = (–1) 2n + 1 (2J – n)! where the trace is taken over the states of any J multiplet, such that 2J ≥ n. Directing the z axis along n and noting that Jˆz ≡ C0(1) (Jˆ), we write 2 . (2.37) trHˆ 4f2 = 2ex tr(Jˆz2 ) + 2sign(1 – gJ )ex tr Hˆ CF C0(1) (Jˆ) + trHˆ CF The first term is just 13 J (J + 1)(2J + 1)2ex . The second term vanishes by virtue of the orthogonality relation (2.36), as Hˆ CF contains only Cm(n) (Jˆ) with n even. Finally, the third term of Eq. (2.37) can be considered in the crystallographic coordinates, then its independence of the orientation of M 3d or B ex becomes obvious. Going back to Eq. (2.34), the first non-vanishing contribution to the anisotropy energy comes from the term in x 2 , which contains trHˆ 4f3 . We thus proceed to its evaluation. We shall use the presentation of Hˆ 4f as a binomial (2.35), just as we did when computing trHˆ 4f2 . The invariance of the free energy with respect to time inversion means that 3 obviously does not depend on the all terms odd in Jˆ must vanish, while trHˆ CF orientation of M 3d . The only source of anisotropy in trHˆ 4f3 is the mixed product 32ex tr[(n · Jˆ)2 Hˆ CF ]. Let us write out this expression in the crystallographic coordinates, limiting ourselves to tetragonal, trigonal and hexagonal point groups as most relevant to permanent magnets: 2 32ex tr Jˆx sin θ cos φ + Jˆy sin θ sin φ + Jˆz cos θ × B20 C0(2) (Jˆ) + 4th - and 6th -order terms . (2.38)


179

Note that the square of the parenthesis in this equation contains products of the Cartesian components of the total angular momentum, which are linear combinations of Cm(n) (Jˆ) with n = 2 and 0, C0(0) ≡ 1: 12 1 (2) ˆ 1 1 (2) ˆ (2) ˆ 2 ˆ Jx = J (J + 1) – C0 (J ) + (J ) C2 (J ) + C–2 3 3 6 12 1 (2) ˆ 1 1 (2) ˆ (2) ˆ 2 ˆ Jy = J (J + 1) – C0 (J ) – C2 (J ) + C–2 (J ) 3 3 6

(2.39)

2 1 Jˆz2 = J (J + 1) + C0(2) (Jˆ) 3 3 .................................................... This is just a transformation inverse to Eqs. (1.17). There is no need to write out the mixed products, Jˆx Jˆy etc., since they do not contain C0(2) (Jˆ). Due to the orthogonality relation (2.36), only the terms in C0(2) (Jˆ) survive in Eq. (2.38). Since C0(2) (Jˆ) enters into Jˆx2 and Jˆy2 with the same coefficient – 13 , the terms in sin2 θ cos2 φ and in sin2 θ sin2 φ will enter in the final expression for F4f (θ , φ) also with the same factor. Therefore, in this approximation F4f does not depend on the azimuthal angle φ. Carrying out the calculations, we present the RE free energy as follows: F4f (θ , φ) = F0 + Ea , where F0 is the isotropic part and Ea is the anisotropy energy, Ea = K1 sin2 θ + K2 sin4 θ + · · · K1 = –

(J + 1)(2J – 1)(2J + 3) B20 x 2 + O(x 3 ). 40J

(2.40) (2.41)

Thus, the leading term of the high-temperature expansion of the first anisotropy constant is proportional to the second-order CF parameter B20 and to x 2 . Its independence of the higher-order CF parameters arises from the orthogonality relation (2.36). Quite similarly one arrives at the conclusion that K2 = const. × B40 x 4 + O(x 5 ) (Kuz’min, 1995). In general, the high-temperature series for an anisotropy constant multiplying sinn θ begins with a term in x n , whose coefficient is a linear combination of nth -order CF parameters Bnm .12 This fact ensures the convergence of the expansion (2.40) at small x (high T ). 12

In most high-symmetry cases these combinations contain a single CF parameter. For example, for the anisotropy constants in Eq. (2.13) one gets K1 ∝ x 2 B20 , K2 ∝ x 4 B40 , K2 ∝ x4 ReB44 , K2 ∝ x 4 ImB44 , K3 ∝ x 6 B60 , K3 ∝ x 6 ReB64 , K3 ∝ x 6 ImB64 .

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Figure 3.5 The unit cell of RE2 Fe14 B. The triangles indicate the positions of the 4f sites, occupied by the RE. The other RE sites (4g, not shown) are situated on the vacant diagonals.

Thus, to terms in x 2 , the anisotropy energy is simply K1 sin2 θ, where 2 1 μB Bex 2 . K1 = – J (J + 1)(2J – 1)(2J + 3)(gJ – 1) B20 10 kT

(2.42)

Note the very special role of the second-order CF parameter B20 . It and it alone can guarantee the persistence of the anisotropy (and therefore of the coercivity) of a permanent magnet material to high temperature, which is of vital importance for most industrial applications. Even more important is to have a large exchange field Bex , since K1 is propor2 . The value of Bex depends on temperature. To minimize its reduction tional to Bex at elevated temperatures, one should not just seek to increase the TC , but also to reduce the parameter s describing the shape of the dependence M3d (T ) (Kuz’min, 2005), 32 52 13 T T – (1 – s) . Bex ∝ M3d ∝ 1 – s (2.43) TC TC So far in this Section it has been assumed for simplicity that the local symmetry of the RE site and the symmetry of the crystal as a whole are described by point groups allowing just one second-order CF parameter B20 and accordingly, a single secondorder anisotropy term, K1 sin2 θ . However, we have already (Section 1.3) seen an example of permanent magnet materials, RE2 Fe14 B, whose crystallographic class is tetragonal, D4h , while the local symmetry is orthorhombic, C2v . The latter admits an extra second-order CF parameter B22 (purely real in Wybourne’s notation). Consequently, an extra term in sin2 θ should appear in Eq. (2.40), that describing the anisotropy in the basal plane, K1 sin2 θ cos 2φ, with K1 = const.×B22 x 2 +O(x 3 ). It turns out upon a closer look at the structure (Fig. 3.5) that four equivalent RE sites split into two pairs with different orientations of the local symmetry axes. (Recall that the simplest form of the CF—with two second-order CF parameters—refers


181

to the local axes). Before the macroscopic anisotropy energy can be obtained by summing up the single-ion contributions, these have to be related to the same coordinates. The simplest way to do it is to rotate the local axes of one-half of the RE sites (e.g. of those situated at z = 1/2, see Fig. 3.5) through 90° about [001]. Such a rotation is equivalent to a change of sign of B22 for those sites, while their B20 remains unchanged. As the summation over all RE sites is performed, the terms ∝ B22 cos 2φ, incompatible with the crystallographic class, simply cancel out. Thus, the presence of a nonzero B22 has no bearing on Eqs. (2.40)–(2.42) or on the exceptional role of B20 . This applies to most permanent magnet materials, insofar as they belong to one of the medium-symmetry crystal systems: tetragonal, trigonal or hexagonal. To conclude the discussion of RE2 Fe14 B, we note that there are two nonequivalent RE sites, 4f and 4g. For simplicity, only the former are shown in Fig. 3.5. The 4g sites are situated on the vacant diagonals and possess similar symmetry properties, to the extent that the above-mentioned cancellation of the terms in B22 takes place for both kinds of RE sites independently. The final expression for K1 should be a sum of two terms identical to (2.42) but with different B20 and Bex . There is, however, direct experimental evidence that in Gd2 Fe14 B the exchange fields on the two Gd sites are equal to within a few percent (Loewenhaupt et al., 1996). This fact enables us to still use the simple expression (2.42) for the RE2 Fe14 B compounds, 4g 4f provided that B20 is understood as 12 (B20 + B20 ). The averaging here is justified 4g 4f 4g 4f = Bex and does not require that B20 ≈ B20 . A recent X-ray by the fact that Bex 4g 4f diffraction experiment has revealed that B20 and B20 are essentially different (Haskel et al., 2005). In order to set the lower bound to the domain of validity of the hightemperature approximation, the expansion in powers of x should be continued. It was established (Kuz’min, 1995) that the main contribution to K1 beyond x 2 comes from the term ∝ B20 x 4 (even though nonzero contributions from other CF parameters may be present as well—not necessarily linear). In this approximation K1 = –

(J + 1)(2J – 1)(2J + 3) B20 x 2 (1 – dJ x 2 ) 40J

(2.44)

where dJ =

8J 2 + 8J + 5 . 84J 2

(2.45)

The quantity dJ is practically independent of J , varying between 0.11 for J = 8 and 0.14 for J = 5/2 (Kuz’min, 1995, Table I). For estimations one can take the fractional error of the high-temperature approximation to be just 0.12x 2 for all RE’s. This translates to 7% at T = 300 K for Tm2 Fe14 B. Though it may not always be sufficient for accurate calculations in the room-temperature range, the simplicity of Eq. (2.42) makes it nevertheless useful for analysing the behaviour of permanent magnet materials at T > 300 K.

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In the same approximation, to T –2 , Eq. (2.31) yields for the RE magnetic moment (Boutron, 1973): 1 B20 C μRx,y = Bx,y (2.46) 1 + (2J – 1)(2J + 3) + ··· T 20 kT 1 B20 C μRz = Bz (2.47) 1 – (2J – 1)(2J + 3) + ··· T 10 kT where B = 2(1 – gJ–1 )Bex + B is the effective magnetic field on the RE, C = J (J +1)gJ2 μ2B /3k is the Curie constant. Within the same accuracy, the susceptibility can be recast in the Curie–Weiss form, C χ,⊥ = (2.48) T – θ,⊥ where 1 kθ = (2J – 1)(2J + 3)B20 (2.49) 10 1 kθ⊥ = – (2J – 1)(2J + 3)B20 . (2.50) 20 Note the validity of the Elliott formula (Elliott, 1965): 3 (2J – 1)(2J + 3)B20 . (2.51) 20 The first anisotropy constant K1 can be presented as χ – χ⊥ 2 (B ) K1 = (2.52) 2 which in the considered approximation is equivalent to the earlier obtained result (2.42), provided that B Bex . In other words, the RE subsystem behaves in this approximation as an anisotropic paramagnet in an effective magnetic field. For this reason at high T the susceptibility anisotropy, χ – χ⊥ , just like K1 , depends on a single CF parameter B20 . To finalise this section, let us consider the effect of non-collinearity of the sublattices and the possibility to allow for it by introducing two sets of orientation angles and anisotropy constants, one for each sublattice. (Up no now we had to do with a single set of angles, θ and φ, corresponding to the orientation of M3d , while the respective anisotropy constants were mere sums, K3d + K4f ). Examination of Eqs. (2.46), (2.47) reveals two sources of non-collinearity: (i) possible violation of the condition B Bex while BBex , and (ii) the terms in B20 in square brackets—a purely CF effect. As a result, M3d Bex B μR . We exclude from the outset the possibility that the strong inequality B Bex may fail—the concept of anisotropy constants formulated in Section 2.1 requires that |μR | be independent of B.13 Our consideration in this section will therefore be limited to the CF-induced non-collinearity (ii). k(θ – θ⊥ ) =

13 A situation when B ∼ B is neither unattainable experimentally (see e.g. Kostyuchenko et al., 2003) nor intractable ex theoretically—the formalism of Section 2.3 still applies.


183

Let us rewrite the square bracket of Eq. (2.47) as follows, (2J – 1)(2J + 3) ηx (2.53) 10J where η is defined by Eq. (2.16). Assume that the system under consideration is at least broadly exchange-dominated. Then its parameter η must not exceed the η-axis intercept of the plane (2.24) delimiting the central polyhedron, 1–

2 (2.54) . 3(2J – 1) In a strictly exchange-dominated case this should become a strong inequality, cf. (2.25). By virtue of (2.54), the square bracket of Eq. (2.47) can be recast as |η|
0, 0 < n ≤ 2J ,

dBJ(n) (x) > 0. (2.74) dx The proof for arbitrary n is rather complicated and is not reproduced here. For n = 1 it follows from the fact that the square bracket in Eq. (2.80) is the dispersion of Jˆz , a positive-definite quantity. 2.6.1.5 The limit x → 0 (T → ∞)

BJ(n) (0)

=

1, if n = 0 0, if n > 0.

(2.75)

Taking the average in Eq. (2.63) is straightforward in this limit, since all Boltzmann’s exponentials equal unity and therefore BJ(n) (0) ∝ (2J + 1)–1 trC0(n) (Jˆ).


189

2.6.1.6 Asymptotic behaviour at low temperatures (x J ) n(n + 1) x (2J )! 1 (n) 1– exp – + ··· . BJ (x) ≈ (2.76) (2J )n (2J – n)! 2J J The prefactor is obtained in the limit T = 0 by averaging over the ground state (M = J ) using Eq. (1.24) and the explicit expression for the CGC CJJJJn0 (Varshalovich et al., 1988). Allowing for the (exponentially small) population of the first excited level yields the correction term of Eq. (2.76). The special case of n = 1 describes the magnetic moment: μR ∝ BJ (x) ≈ 1 – J –1 e–x/J .

Hence follows the approximate relation: BJ(n) (x) ∝ (μR )n(n+1)/2 (2.77) known as Zener’s n(n + 1)/2 power law (Zener, 1954; Callen and Callen, 1966; Goodings and Southern, 1971). 2.6.2 Differential relations and properties thence deduced 2.6.2.1 First derivative The first derivative of a GBF is given by dBJ(n) (x) n + 1 (n+1) = BJ (x) – BJ (x)BJ(n) (x) dx 2n + 1 n (2J + n + 1)(2J – n + 1) (n–1) + (2.78) BJ (x). 2n + 1 4J 2 To prove this relation, differentiate the identity (2.62) with respect to x one more time (the discrete parameters J and n = k being fixed): dJˆzk sign(1 – gJ ) ˆ ˆk =– cov Jz , Jz dx J sign(1 – gJ ) ˆk+1 ˆ ˆk =– (2.79) – Jz Jz . Jz J Multiplying this relation by an appropriate coefficient and summing up over k of the same parity so as to assemble the expression for C0(n) (Jˆ), one arrives at sign(1 – gJ ) ˆ (n) ˆ ˆ (n) ˆ dC0(n) (Jˆ) =– Jz C0 (J ) – Jz C0 (J ) (2.80) dx J or, on foot of the definition (2.63), [–sign(1 – gJ )]n+1 ˆ (n) ˆ dBJ(n) (x) = Jz C0 (J ) – BJ (x)BJ(n) (x). (2.81) n+1 dx J Transforming the first term by means of the identity n + 1 (n+1) ˆ Jˆz C0(n) (Jˆ) = C (J ) 2n + 1 0 n (2J + n + 1)(2J – n + 1) (n–1) ˆ C0 (J ) + (2.82) 2n + 1 4 one finally obtains Eq. (2.78).

190


It remains to prove Eq. (2.82). It is apparently an expansion of the product C0(1) C0(n) in irreducible tensor operators of appropriate parity. The triangle rule admits only terms in C0(n±1) . The prefactor of the term in C0(n+1) is readily obtained by equating the coefficients of Jˆzn+1 on both sides of Eq. (2.82). Note that the factor of Jˆzn in C0(n) (Jˆ) is the same as that of the leading term of the corresponding Legendre polynomial, i.e. 2–n (2n)!/(n!)2 . Thus, we get n + 1 (n+1) ˆ C Jˆz C0(n) (Jˆ) = (J ) + f (n, J )C0(n–1) (Jˆ). 2n + 1 0

(2.83)

The remaining unknown factor f (n, J ) is evaluated by substituting Eq. (2.83) into (2.81) and letting x go to infinity, whereas BJ(n) (x) → (2J )–n (2J )!/(2J – n)! and dBJ(n) (x)/dx → 0, cf. Eq. (2.76). This completes the proof of Eq. (2.82) and consequently of Eq. (2.78). 2.6.2.2 Power series expansion presented as follows:

BJ(n) (x) =

At small x (high temperatures) the GBF can be

1 (2J + n + 1)! 2n (2n + 1)!!(2J ) (2J + 1)(2J – n)! J (J + 1) + 18 (n + 3) n+2 n+4 x + O(x ) . × xn – n 3J 2 (2n + 3)

(2.84)

The leading coefficient of this expansion is readily obtained by applying the differential relation (2.78) k times, in order to compute the k th derivative of the GBF. Note that only the lowest-order GBF needs to be followed. Thus we write: dBJ(n) (x) n (2J + n + 1)(2J – n + 1) (n–1) BJ (x) + higher-order GBF = dx 2n + 1 (2J )2 d 2 BJ(n) (x) n(n – 1) = dx 2 (2n + 1)(2n – 1) ×

(2J + n + 1)(2J + n)(2J – n + 1)(2J – n + 2) (n–2) BJ (x) (2J )4

+ higher-order GBF ........................................................................... n!/(n – k)! d k BJ(n) (x) = dx k (2n + 1)!!/(2n – 2k + 1)!! ×

(2J + n + 1)!(2J – n + k)! B (n–k) (x) + higher-order GBF. (2J )2k (2J + 1 + n – k)!(2J – n)! J

191


By virtue of Eq. (2.75), one gets at x = 0:

d

k

BJ(n) (x) dx k

=

⎧ ⎨0,

if k < n

(2.85) n! (2J + n + 1)! , if k = n. 2n (2n + 1)!(2J ) (2J + 1)(2J – n)! Hence follows the leading coefficient of the expansion (2.84). The coefficient of the term in x n+2 is derived along the same lines, but this requires more cumbersome algebra and will not be reproduced here. x=0

⎩

2.6.2.3 Explicit expressions

Equation (2.78) can be recast as a recurrence formula 2n + 1 dBJ(n) (x) (n+1) (n) + BJ (x)BJ (x) BJ (x) = n+1 dx n (2J + n + 1)(2J – n + 1) (n–1) BJ (x). – (2.86) n+1 4J 2 Starting from the already known GBF of zeroth and first orders (2.70), (2.71) and working up, one can obtain in explicit form the GBF of an arbitrarily high order. Note the cancellation of the terms in coth2 [x(2J + 1)/2J ] arising from the derivative and from the product of two GBF in the square bracket of Eq. (2.86). As a consequence, GBF of any order n > 1 are linear in coth[x(2J +1)/2J ]. Conversely, the highest power of coth[x /2J ] increments by one each time Eq. (2.86) is used. Therefore, the GBF can be presented as follows (Kuz’min, 1992): 2J + 1 (n) x BJ (x) = Pn (ξ , η) – Qn (ξ , η) coth 2J 1 x 1 ξ= (2.87) coth η= . 2J 2J 2J The polynomials Pn and Qn obey the following recurrence relations: 2n + 1 2 2 ∂Pn (η – ξ ) – (1 + η)Qn – ξ Pn Pn+1 = n+1 ∂ξ n – 1 + 2η + (1 – n2 )η2 Pn–1 n+1 (2.88) 2n + 1 2 2 ∂Qn (η – ξ ) – (1 + η)Pn – ξ Qn Qn+1 = n+1 ∂ξ n – 1 + 2η + (1 – n2 )η2 Qn–1 . n+1 Table 3.3 contains explicit expressions for Pn (ξ , η) and Qn (ξ , η) with n ≤ 7 obtained by means of Eqs. (2.88). Note that for n odd these functions differ in sign from the analogous expressions of Magnani et al. (2003).14 The convention adopted herein ensures that the plots of all GBF at x > 0 lie entirely in the first quadrant. 14 We also note a misprint in their A (ξ , η): the inner-most parenthesis (–22 + 3η) should be multiplied by –η rather 5 than by +η.

192


Table 3.3 Polynomials Pn (ξ , η) and Qn (ξ , η) entering in the explicit expression (2.87) for the generalised Brillouin functions

P1 = –ξ Q1 = –(1 + η) P2 = 3ξ 2 + 1 + 2η Q2 = 3ξ(1 + η) P3 = –3ξ(5ξ 2 + 2 + 4η – η2 ) Q3 = –(1 + η)(15ξ 2 + 1 + 2η – 3η2 ) P4 = 105ξ 4 + 45ξ 2 (1 + 2η – η2 ) + 1 + 4η – 4η2 – 16η3 Q4 = 5ξ(1 + η)(21ξ 2 + 2 + 4η – 9η2 ) P5 = –15ξ [63ξ 4 + 14ξ 2 (2 + 4η – 3η2 ) + 1 + 4η – 7η2 – 22η3 + 3η4 ] Q5 = –(1 + η)[945ξ 4 + 105ξ 2 (1 + 2η – 6η2 ) + 1 + 4η – 14η2 – 36η3 + 45η4 ] P6 = 10395ξ 6 + 4725ξ 4 (1 + 2η – 2η2 ) + 105ξ 2 (2 + 8η – 20η2 – 56η3 + 15η4 ) + 1 + 6η – 20η2 – 120η3 + 64η4 + 384η5 Q6 = 21ξ(1 + η)[495ξ 4 + 30ξ 2 (2 + 4η – 15η2 ) + 1 + 4η – 19η2 – 46η3 + 75η4 ] P7 = –7ξ [19305ξ 6 + 4455ξ 4 (2 + 4η – 5η2 ) + 225ξ 2 (2 + 8η – 26η2 – 68η3 + 27η4 ) + 4 + 24η – 110η2 – 600η3 + 556η4 + 2376η5 – 225η6 ] Q7 = –(1 + η)[135135ξ 6 + 17325ξ 4 (1 + 2η – 9η2 ) + 189ξ 2 (2 + 8η – 48η2 – 112η3 + 225η4 ) + 1 + 6η – 41η2 – 204η3 + 463η4 + 1350η5 – 1575η6 ]

The explicit expressions for the GBF were first obtained by Brillouin (1927, n = 1), Yoshida (1951, n = 2), Kazakov and Andreeva (1970, n = 4, 6), Kuz’min (2002, n = 3), Magnani et al. (2003, n = 5, 7). 2.6.2.4 The quasi-classical limit, J → ∞ In the limit of very large J the GBF turn into the so-called reduced modified Bessel functions:

lim BJ(n) (x) = Iˆn+ 1 (x).

J →∞

2

(2.89)

The latter were introduced to the theory of magnetic anisotropy by Keffer (1955) and are defined as follows: Iˆn+ 1 (x) = In+ 1 (x)/I 1 (x) 2

2

2

(2.90)

where Iν (x) are modified spherical Bessel functions of the first kind (Abramowitz and Stegun, 1972, Chapter 10). To prove (2.89), note that by definition Iˆ1 (x) ≡ 1 = BJ(0) (x), while 2

Iˆ3/2 (x) = coth x – 1/x = L(x) = lim BJ (x) J →∞

(2.91)


193

that is the well-known Langevin function. Finally, the recurrence formula (2.86) goes over to 2n + 1 d Iˆn+1/2 (x) n ˆ ˆ ˆ ˆ + I3/2 (x)In+1/2 (x) – In–1/2 (x) (2.92) In+3/2 (x) = n+1 dx n+1 whose validity can be easily verified using the recurrence relation for the modified spherical Bessel functions, Eq. (10.2.19) of Abramowitz and Stegun (1972). Thus, by induction, for any n > 0 the function Iˆn+3/2 (x) obtained by means of Eq. (2.92) is the quasi-classical limit of the GBF BJ(n+1) (x), q.e.d. 2.6.3 Selected properties of the functions DJ(n) (x) Most of these readily follow from the definition of DJ(n) (x), Eq. (2.69), and the corresponding properties of the GBF. For example, the parity: DJ(n) (–x) = (–1)n–1 DJ(n) (x).

(2.93)

It follows from the monotonicity of the GBF that DJ(n) (x) > 0,

if x > 0, n > 0.

(2.94)

The asymptotic behaviour at large x(x J ) is described by DJ(n) (x) ≈

n(n + 1) (2J )! x e–x/J . 2n+1 J 3 (2J – n)!

(2.95)

Note that DJ(n) (x) → 0 as x → ∞. Alternatively, for x small, one has DJ(n) (x) =

3n (2J + n + 1)! x n–1 + O(x n+1 ). 2n n (2n + 1)!!2 J (J + 1)(2J + 1)(2J – n)!

(2.96)

Obviously, DJ(n) (0) = 0 for all n > 1. Vanishing at both ends of the semi-infinite interval 0 < x < ∞, positive and continuous everywhere within, the functions DJ(n) (x), n > 1, must have at least one maximum at a certain point xmax > 0. In fact, there is exactly one maximum on the positive semi-axis, see Fig. 3.6. For larger n the maxima are situated farther to the right, their height scaling roughly as J n–1 /n.

2.7 The linear-in-CF approximation (continued) Let us now return to the discussion of the main results of the linear theory, Eqs. (2.64), (2.65), (2.68). We shall rely on our newly acquired knowledge of the properties of the GBF. It is convenient to plot the GBF BJ(n) (x), n = 2, 4, 6, against inverse Langevin’s ratio 1/x, which is approximately proportional to absolute temperature T , Fig. 3.7. A feature that immediately draws attention in this graph is the presence of plateaus in the low-temperature region. On account of the exponentially rapid approach to saturation characteristic of the GBF, cf. Eq. (2.76), the plateaus in Fig. 3.7 have fairly sharply defined widths ∼1/5J .

194


(n)

Figure 3.6 Graphs of the functions DJ (x) (rescaled) with J = 6 and n = 2, 4, 6 (a); rescaled (n) positions (b) and rescaled heights (c) of the maxima of the functions DJ (x), plotted vs J .


Figure 3.7

195

Generalised Brillouin functions for J = 8, plotted against 1/x.

The plateaus disappear completely in the limit J → ∞, as the GBF go over to the reduced Bessel functions (2.89), employed in the quasi-classical theory of magnetic anisotropy (Keffer, 1955; Callen and Callen, 1966). The awkward fact that this constitutes a violation of the third law of thermodynamics was circumvented in the classical theory by putting to the fore the dependence of the anisotropy constants on magnetisation, rather than on temperature. This maneuvering has become obsolete after the introduction of the GBF. Let us reiterate: below a certain point all anisotropy constants become independent of temperature, or saturated. One of the consequences of this is that spontaneous spin reorientation transitions (SRT) can take place only above a certain temperature. Quantitatively this temperature is determined by the exchange field on the RE. For example, for HoFe2 , where the relation between x and T is given by x ≈ 750/T (Kuz’min, 2001), the functions B8(n) (x) are saturated below 1/x ≈ 0.02 (Fig. 3.7), or T ≈ 15 K. The height of a low-temperature plateau is determined by the numerical factor in front of the square bracket in Eq. (2.76); it tends to unity only when J → ∞. For example, BJ(2) (∞) = 1 – 1/2J . Accordingly, the low-temperature values of the anisotropy constants are given by (Goodings and Southern, 1971): (2J )! (2.97) Bnm . – n)! Here Bnm are the CF parameters normalised according to Wybourne. They include the Stevens factors, therefore, their magnitudes decrease as n increases. As against that, the coefficients of Bnm in Eq. (2.97) grow with n. As a result, at low temperatures the anisotropy constants κ2m , κ4m and κ6m are of the same order of magnitude. Similarly, there is no reason to presume that either K3 or K2 in Eq. (2.12) could be neglected at low temperatures. (Terms of order higher than six in sin θ do vanish though.) κnm |T =0 =

2n (2J

196

Figure 3.8


Three possible forms of temperature dependence of second anisotropy constant.

The situation is quite different in the high-temperature case. According to Eqs. (2.65), (2.84), Ki fall off with temperature as T –2i . Therefore, at around room temperature and above it one can neglect BJ(4) (x) and BJ(6) (x). In this approximation K2 and K3 vanish, while K1 is given by 3 K1 = – B20 J 2 BJ(2) (x). (2.98) 2 The quality of this approximation can be judged by the linearity of the magnetisation curves along the hard direction. Thus, the room-temperature magnetisation curves of RE2 Fe14 B with the heaviest RE are practically linear (Yamada et al., 1988). However, in the case of Nd2 Fe14 B one can still see some residual curvature at T = 290 K, which disappears at higher temperatures. In such a situation it may be sensible to leave K1 and K2 and to neglect K3 . From the fact that the GBF with n > 0 vanish at x = 0 and grow at any x > 0 it follows that these GBF are positive within the physically meaningful interval of values of x, 0 < x < ∞. Therefore, the signs of the anisotropy constants κnm entering in Eq. (2.10) are determined by the signs of the respective CF parameters Bnm (these in turn depending on the signs of the Stevens factors) and cannot change as temperature varies. The same is true in relation to the anisotropy constants K3 and K3 , cf. Eqs. (2.65). The latter quantity determines the orientation of the easy magnetisation direction in the basal plane for most hexagonal crystals (point groups D6 , C6v , D3h and D6h ). Equations (2.65) predict for the dependence K2 (T ) three possible shapes. These are sketched in Fig. 3.8. Apparently, K2 can change sign no more than once. We


197

shall come back to this argument in Section 3, in connection with the SRT in TbCo5 . As regards K1 , it can in principle change sign twice, and even more times when the 3d contribution is taken into account. Let us now consider the connection between the linear theory and the hightemperature expansion of Section 2.4. That these two approximations are closely related should not come as a surprise—we have already noted that the leading term in the high-temperature expansion of anisotropy constants is always linear in CF. For example, for the axial anisotropy constants one can write in general (Kuz’min, 1995): Ki = const. × B2i,0 x 2i + O(x 2i+1 ).

(2.99)

2i+1

and of the subsequent terms may be non-linear in CF The coefficient of x parameters. (In fact, terms even in x must be odd in Bnm while those odd in x must be even in Bnm ). The first term of Eq. (2.99) may be regarded as the leading term of an expansion of Ki in powers of several variables: x, B20 , B40 etc. (or x and the dimensionless quantities introduced in Section 2.2: η, ξ etc.). The same series can be obtained in a different way: first Ki is expanded in powers of Bnm , then the coefficients of the obtained expansion (first of all, those of the terms linear in Bnm ) are further expanded in powers of x. In other words, Eqs. (2.65) of the linear theory are expanded in a series of powers of x using Eq. (2.84): (2J + n + 1)! (–1)n/2 Bn0 x n + O(x n+2 ) 2n n 2 n!(2n + 1)J (2J + 1)(2J – n)! n = 2, 4, 6.

Kn/2 =

(2.100)

Comparing this with the general expansion (2.99), one immediately gets an expression for the unknown constant therein. Equation (2.44) is not but a special case of (2.100) with n = 2, where the expansion is taken to the next, quartic in x term. Omitted from the approximate relation (2.44) are the term in x 3 (whose coefficient is a homogeneous quadratic form in Bnm ) and parts of the term in x 4 , namely, one which is linear in B40 and the other one cubic in Bnm . The omitted contributions were evaluated for TbCo5 and found small (Kuz’min, 1995). Thus, the linear theory becomes asymptotically exact at high temperatures (x small) because the leading terms of both expansions, (2.99) and (2.100) coincide. In other words, non-linear terms die out more rapidly with temperature. For typical permanent magnet materials the linear in Bnm contribution is also dominant at moderately high temperatures, where terms in x 4 are no longer negligible, the term in B20 x 4 prevailing over the one in B40 x 4 . Hence Eq. (2.44). One concludes that the high-temperature version of the linear theory (2.98) should be more accurate than Eq. (2.41) for exchange-dominated systems at intermediate temperatures, because the former includes—albeit approximately—the higher-order terms: in x 4 , x 6 etc. An added advantage of Eq. (2.98) is its sensible behaviour in the limit of very low temperatures, where it is generally speaking invalid. Thanks to the presence of an unexpanded GBF, Eq. (2.98) inoffensively tends to a finite limit as x → ∞, whereas (2.41) diverges.

198


As against that, Eq. (2.41) has the advantage of being very simple and is ideally suited for discussing the high-temperature behaviour of permanent magnets. It can also be easily solved for x. Before closing the subsection, let us touch upon the question of the influence of CF on the RE magnetic moment, described by Eq. (2.68). According to the abovestated properties of the functions DJ(n) (x), namely Eq. (2.95), this effect should be negligibly small at the temperature of liquid He, that is exactly where experimentalists usually try to detect it. The reason why they prefer to compare the measured RE moment with the free-ion expression (2.59) just at low temperatures is rather mundane. The Brillouin function is saturated there, so one does not need to worry about its unknown argument. Unfortunately, the saturation also kills off the sought CF effect. This effect reaches its maximum at a finite temperature. For example, in Nd2 Fe14 B the six-order CF contribution to μNd peaks at T ≈ 80 K (x ≈ 9), where it amounts to –2% of the total Nd moment (at the same temperature).15 At room temperature the sixth-order effect is a factor of 20 weaker, –0.1%, and may be safely neglected. The fourth-order contribution is maximum at T = 120 K, or x ≈ 6.2, where it reaches –1.6%. This reduces to about one-half of a per cent at ambient temperature, negligible in most cases. Finally, the second-order CF effect is maximum at room temperature, T = 290 K (x ≈ 2.3), where the function 2 (x) equals approximately 2.6. Accordingly, its relative contribution to μNd is D9/2 +7.7%. One should bear in mind however, that less than one-seventh of the total magnetisation of Nd2 Fe14 B at ambient temperature comes from the Nd sublattice. When related to the total magnetisation, the second-order CF effect reduces to a mere 1%. Thus, the CF contribution to magnetisation hardly needs to be taken into account in technical calculations of permanent-magnet devices. In any case, at or above room temperature only second-order CF matters. The influence of the CF on the magnetisation is most noticeable near SRT,16 where rotation of the easy magnetisation direction leads to a rapid change of the primed CF parameters (2.61) with temperature. Obviously, this effect is more pronounced in ferrimagnetic intermetallic compounds with the heavy RE and at first-order SRT. Further consideration of this phenomenon will be deferred till Section 3. At very high temperatures (x 1) the influence of the CF on μR decreases. The least rapidly falls off the second-order CF effect. According to Eq. (2.96), at small x (2J – 1)(2J + 3) x. DJ(2) (x) ≈ (2.101) 10J Putting this into Eq. (2.68), we arrive at Eqs. (2.46), (2.47). Note that by virtue of Eq. (2.61), B20 = B20 when B z (θ = 0), and B20 = – 12 B20 when B ⊥z 15 Our estimates are based on the CF parameters of Cadogan et al. (1988), converted to the Wybourne normalisation and averaged over the two Nd sites: B20 = –4.4 K, B40 = 0.092 K, B60 = 0.02 K. For simplicity, no distinction was made between primed and non-primed CF parameters below the SRT point, TSR = 135 K. The exchange splitting, ex = 168 K at T = 0, comes from the same source. At finite temperatures ex was scaled down in proportion to the iron sublattice magnetisation. We used the scaling factors 0.987, 0.976 and 0.891 for T = 80, 120 and 290 K. These were computed using Eq. (2.43) with s = 0.7 and TC = 592 K. 16 This effect is not attributable to the RE sublattice alone.


199

(θ = π /2). This manifests once again the intimate connection between the linearin-CF and the high-temperature approximations.

2.8 The low-temperature approximation Following the introduction of the general single-ion model in Section 2.3, we have considered two closely inter-related approximations: the high-temperature one (Section 2.4) and the one linear in CF (Sections 2.5–2.7). These were ‘good’ approximations, in the sense that their validity in any particular case was easy to verify, at least a posteriori. In this subsection we shall formulate an approximation that unfortunately lacks the same quality. It has been already said that the standard theory of magnetic anisotropy is generally inapplicable to RE-transition metal magnets at low temperatures—the expansion of the free energy in powers of sin θ cannot be truncated and the very description in terms of a single angle θ is no longer meaningful on account of significant non-collinearity of the sublattices. When an analytical description of some sort at low T is possible, it should be necessarily limited to a special case, i.e. it should involve extra assumptions apart from the smallness of T . As such an additional condition, we shall now assume that the CF has a predominantly axial character, or that Bnm with m = 0 are small in comparison with Bn0 . A rigorous verification of this hypothesis would require the knowledge of all CF parameters, including those with m = 0, which in this approach we do not presume to have. In that sense, this is an uncontrolled approximation. So let the CF be approximately axial and let the system under consideration be broadly exchange-dominated. This implies that the ground state is |J M, with M = – sign(1 – gJ )J , the first excited state has M = – sign(1 – gJ )(J – 1), etc. The positive z direction is, as usual, that of the 3d magnetisation vector. The latter coincides with the high-symmetry crystallographic axis [001], or at the most makes with it an infinitesimal angle θ . Such a situation is characteristic of permanent magnets—otherwise the material would simply lose its hard magnetic properties, first of all the coercivity. The reason why it makes sense to limit the consideration to the vicinity of the point θ = 0 is explained in Fig. 3.9. At T = 0 the free energy is the ground state energy, shown as a solid line. It may not always be a smooth function of the angle θ, because the RE energy levels may cross over at some finite values of θ . Just such a case is shown in Fig. 3.9a. A more realistic example is REFe11 Ti (Hu et al., 1990, Fig. 8 therein). It is clear that the displayed dependence F (θ) cannot be approximated with a smooth function like K1 sin2 θ or K1 sin2 θ + K2 sin4 θ across the entire interval from 0 to 90°. For θ small, such a presentation—in the spirit of Landau’s theory (Landau and Lifshitz, 1958)—is still possible and even useful. Indeed, when a second-order SRT occurs (Fig. 3.9b), the main events take place near the point θ = 0 (or perhaps near θ = π /2, in which case the small angle π /2– θ should be regarded as the order parameter). Of course, the benefit of presenting the anisotropy energy as K1 sin2 θ in close vicinity to θ = 0 is not limited to secondorder SRT. A number of other phenomena—nucleation, transverse alternatingcurrent susceptibility, magnetic resonance etc.—require such a presentation.

200


Figure 3.9 Angular dependence of the free energy at T = 0 (solid line) when a level cross-over takes place. The presence of a cusp in the solid curve does not interfere with the spin reorientation transition—the displacement of the minimum from the origin to a nonzero angle.

Thus, under the above assumptions, the RE Hamiltonian (2.35) falls into two parts:

Hˆ 4f = sign(1 – gJ )ex cos θ Jˆz + Bn0 C0(n) (Jˆ) n=2,4,6

+ sign(1 – gJ )ex sin θ Jˆx .

(2.102)

The first one of them is diagonal in the J M representation, while the second one can be treated as a perturbation since it contains an infinitesimal quantity sin θ. We restrict ourselves to the region of low temperatures, kT 2ex + 1 + 2 , where we can neglect thermal population of all but the lowest two levels of the RE. It will be recalled that ex + 1 is a gap separating the ground and the first excited states when θ = 0. Its two parts, due to the exchange and the CF, are defined by Eqs. (2.17) and (2.23), respectively. Similarly, ex + 2 stands for the gap between the first and the second excited levels. Expanding the centre of gravity of the lowest two levels and the gap between them in powers of the small parameter sin2 θ, Ec.g. (θ ) = Ec.g. (0) + V sin2 θ + · · · (θ ) = ex + 1 + 2W sin2 θ + · · · we get

ex + 1 . (2.103) 2kT The quantities V and W are evaluated using the standard second-order perturbation theory: 2J – 1 2J – 1 2ex sin2 θ Ec.g. = ECF + ex cos θ – 2 4 ex cos θ + 2 (2.104) 2 ex sin2 θ 2J – 1 2ex sin2 θ (θ ) = 1 + ex cos θ + J – . ex cos θ + 1 2 ex cos θ + 2 K1 = V – W tanh


201

Here ECF is the CF contribution to Ec.g. , independent of θ. Corrections of order higher than 2 in perturbation theory contain only terms of order higher than 2 in sin θ and therefore need not be considered. Presenting cos θ as 1 – 12 sin2 θ + · · · and collecting terms in sin2 θ in Eqs. (2.104), we arrive at 2J – 1 ex 2 (2.105) 4 ex + 2 J ex 1 W =V – . (2.106) 2 ex + 1 The set of Eqs. (2.103), (2.105) and (2.106) gives the coefficient K1 of the Landau expansion in powers of sin θ at low temperatures. The interplay of the CF and the exchange interactions can be best seen at T = 0, as the hyperbolic tangent in Eq. (2.103) becomes unity and K1 reduces to V =

J ex 1 (2.107) . 2 ex + 1 Insofar as their influence on K1 is concerned, the exchange and the CF act like two resistors connected in parallel—additive are the reciprocal splittings rather than the splittings themselves. In the extreme case of exchange domination, 1 /ex → 0, Eq. (2.107) turns into K1 =

J (2.108) 1 . 2 Here one can recognise the first one of the equations (2.65) of the linear theory, taken at T = 0, or x = ∞. Indeed,

1

n(n + 1) (2J )! J K1 = – n(n + 1)J n BJ(n) (∞)Bn0 = – Bn0 = 1 n+1 2 2 (2J – n)! 2 n=2,4,6 n=2,4,6 K1 =

where Eqs. (2.76) and (2.23) have been used. When 1 is small (as compared with ex ) but finite, Eq. (2.107) can be expanded in powers of the ratio 1 /ex : 1 J + ··· . K1 = 1 1 – (2.109) 2 ex Thus, an added advantage of the low-temperature approximation is that it provides a quantitative criterion of the performance of the linear theory at T = 0. Indeed, according to Eq. (2.109), the fractional error of the ‘linear’ equation (2.108) can be judged by the smallness of the ratio 1 /ex , or 2K1 /J ex . The latter combination contains quantities more readily accessible to experiment. (We would like to remind that K1 stands here for the RE contribution to the first anisotropy constant at T = 0.) For example, at T = 4.2 K TbCo5 has K1Tb = –99 K (Ermolenko, 1980) and Bex ≈ 220 T (Ballou et al., 1989), or ex ≈ 150 K. Hence 2K1Tb /6ex = –0.22, i.e. the linear approximation is accurate within 22%. This estimate, referred to T = 0, is the upper bound of the inaccuracy of the linear-in-CF approximation.

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As stated in the previous subsection, the linear theory performs better at higher temperatures and becomes exact in the limit T → ∞. Let us now demonstrate how the low-temperature approximation to K1 can be used to evaluate the temperature of a second-order SRT of the type easy axis—easy cone. Such transitions are not uncommon in hard magnetic materials, Nd2 Fe14 B being the best-known example with TSR = 135 K (Deryagin et al., 1984; Givord et al., 1984). A necessary condition for such a transition is that K1 = 0 at T = TSR , or17 K3d + V – W tanh

ex + 1 =0 2kTSR

whence kTSR =

ex + 1 . V + K3d ln 1 + 2 W – V – K3d

(2.110)

Taking for Nd2 Fe14 B the exchange and CF parameters of Cadogan et al. (1988) averaged over the two Nd sites, one finds from Eqs. (2.23) 1 = –74.7 K and 2 = 178 K, as well as ex = 167 K. Hence V = 172 K and W = 476 K, by way of Eqs. (2.105) and (2.106). The anisotropy constant of the iron sublattice K3d is taken equal to that of Y2 Fe14 B, which at low temperatures was found to be 6.0 K/Y atom (Givord et al., 1984). Then Eq. (2.110) yields TSR = 117 K. This compares rather well with TSR = 122 K, obtained numerically by Piqué et al. (1996) using the full algorithm of the single-ion model (Section 2.3) and the same parameters as above. The discrepancy between the experimental transition point, TSR = 135 K, and the calculated ones is inherent in the exchange and CF parameters of Cadogan et al. (1988) rather than being a consequence of the approximations introduced in this subsection—see the discussion by Piqué et al. (1996). For Ho2 Fe14 B the situation is similar. The transition point found from Eq. (2.110), TSR = 57 K (Kuz’min, 1995), agrees well with that obtained numerically using the same parameters, TSR = 56 K (Piqué et al., 1996), both being somewhat lower than the experimental value, TSR = 63 ± 2 K (Piqué et al., 1996). Let us recapitulate: at low temperatures the free energy of a RE-based hard magnetic material generally cannot be presented as a truncated expansion in powers of sin θ , Eq. (2.12) or similar. Such a presentation is however possible in a certain neighbourhood of the point θ = 0, where Eq. (2.12) has the meaning of Landau’s expansion, its convergence ensured by the smallness of sin θ . Then, upon some additional assumptions, a useful analytical expression (2.103) can be obtained for K1 (and in principle also for K2 etc.). 17

Here K1 is of course the total anisotropy constant, including the contributions from the 3d and the 4f subsystems.

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2.9 J -mixing made simple18 In this last part of Section 2 we shall finally get down to elucidating the importance of the so far neglected effect of J -mixing on the single-ion magnetic anisotropy. What most workers in the field know about J -mixing can be summarised as the following ‘Three Myths’. 1. J -mixing is a complex phenomenon, not readily amenable to quantitative treatment. Allowing for it necessitates large-scale computer calculations. Too many factors play a role, so little can be demonstrated conclusively. 2. Quantitatively, J -mixing leads to serious consequences only in samarium compounds. For the other REs this effect is not very important, if not exactly unnoticeable. Given the formidable difficulties of its description (Myth 1), it is better to neglect it. The neglect is certainly justified for the heavy REs. 3. When it comes to allowing for J -mixing, the effect can be best visualised at low temperatures. There, both the calculations are more transparent and the anomalies are sharper. When proved unimportant at low T , J -mixing may be neglected at any temperature. The above three statements contain only one grain of truth: that the J -mixing is most distinctly manifest in samarium compounds (and also in those of trivalent europium, not often come across among hard magnetic materials). The rest is misconceptions. In this subsection, according as the truth will gradually unfold, we shall be coming back to the ‘Three Myths’ to point out the falsity of this or that constituent statement. Since J -mixing in the light and the heavy REs is described by slightly different equations, we shall first consider in some detail only the former. The main results for the heavy RE case will be stated briefly towards the end of the subsection. We begin with writing down a model Hamiltonian for a single RE ion in the absence of applied magnetic field. The Hamiltonian is defined on the ground LS term, treated in the Russell–Saunders approximation, and contains terms describing spin-orbit coupling, exchange interaction and the CF:

ˆ (2.111) Anm Cm(n) (L). Hˆ 4f = Hˆ so + Hˆ ex + Hˆ CF = λLˆ · Sˆ – 2μB Bex Sˆz + n,m

Here the z axis has been chosen to be parallel to the 3d sublattice magnetisation M 3d (therefore antiparallel to the exchange field B ex ), which does not necessarily coincide with any of the high-symmetry crystallographic directions. Accordingly, the CF parameters in Eq. (2.111) are primed, to distinguish them from the usual, non-primed CF parameters defined in the crystallographic coordinate system. We aim at describing the lower part of the energy spectrum of the RE, involved in forming the thermodynamic properties of the solid. To this end we construct an effective Hamiltonian H˜ defined on the ground J manifold of the light RE, J = L – S,

Bnm Cm(n) (Jˆ) + δ V˜ . H˜ = ex Jˆz + (2.112) n,m 18

This subsection follows the work of Kuz’min (2002).

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Here Bnm are CF parameters in the J representation, incorporating the Stevens factors. They are related to the quantities Anm in Eq. (2.111) through known ra tional factors. E.g. for n = 2 this relation is given by Eq. (1.23): Bnm /Anm = (L + 1)(2L + 3)/(J + 1)(2J + 3). Similar expressions can also be written for n = 4 and 6. The effective Hamiltonian H˜ (2.112) differs from the usual single-multiplet Hamiltonian (2.14) in a very important way: it incorporates an operator δ V˜ containing second-order corrections bilinear in Hˆ ex and Hˆ CF , the latter two regarded as perturbations with respect to Hˆ so . Although it operates within the ground multiplet, δ V˜ contains inter-multiplet matrix elements of Hˆ ex and Hˆ CF . Subsequently we intend treating the last two terms of Eq. (2.112) as perturbations with respect to the first one, limiting ourselves to first-order corrections. Therefore, all terms with m = 0 may be omitted from the sum in Eq. (2.112). As regards the operator δ V˜ , we only need to compute its matrix elements diagonal in M: 2 δ VˆMM = – (2.113) J + 1, M|Hˆ ex |J MJ + 1, M|Hˆ CF |J M so where so = λ(J + 1) is the spin-orbit splitting between the centres of gravity of the ground (J ) and the first excited (J + 1) multiplets. The inter-multiplet matrix element of Hˆ ex = –2μB Bex Sˆz , required for Eq. (2.113), is given by (L + 1)(J + 1)(2J + 1) J +1,M J + 1, M|Hˆ ex |J M = – CJ M10 ex . (2.114) S(2J + 3)

This expression has been obtained from the well-known formula (88) of Van Vleck (1932) by setting J = L – S, as appropriate for the ground multiplet of a light RE, and factoring out the CGC: (J + 1)2 – M 2 +1,M . = CJJM10 (2J + 1)(J + 1) One of the advantages of Eq. (2.114) is that it depends on the exchange field Bex through the quantity ex , which is the exchange splitting of the ground multiplet defined by Eq. (2.17). This will enable us to reduce the effect of J -mixing to a renormalisation of the standard (without J -mixing) expression for the anisotropy constants (2.64), which depends on the characteristics of the ground multiplet only. For the inter-multiplet matrix element of the CF one can write J + 1, M|Hˆ CF |J M L S J

J n Bn0 = L S n=2,4,6 J n

+1 L 1 (2J + n + 1)! C J +1,M . n 2 (2J + 1)(2J – n)! J Mn0 J L

(2.115)

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This is a generalisation of the intra-multiplet form (1.24) of the Wigner-Eckart theorem. The change from Eq. (1.24) to Eq. (2.115) consists, apart from the obvious modification of the CGC, in adding the ratio of the two 6j symbols, as follows from Eqs. (6-4) and (6-5) of Wybourne (1965). The CF parameters Bn0 employed in Eq. (2.115) include the Stevens factors. This is in line with our strategy of describing the J -mixing in terms of the characteristics of the ground multiplet. As we further restrict ourselves to the Hund ground state of the light RE, Eq. (2.115) simplifies significantly,19 to become J + 1, M|Hˆ CF |J M

1 n(n + 1)S(2J + n + 1)! +1,M =– Bn0 . (2.116) CJJMn0 n 2 (L + 1)(2J + n + 2)(2J – n + 1)! n=2,4,6 Putting Eqs. (2.114) and (2.116) into Eq. (2.113) results in

1 n(n + 1)(J + 1)(2J + 1)(2J + n + 1)! ex Bn0 δ V˜MM = – n–1 so n=2,4,6 2 (2J + 3)(2J + n + 2)(2J – n + 1)! +1,M J +1,M CJ Mn0 . × CJJM10

(2.117)

Note the cancellation of explicit dependence on the quantum numbers L and S. Of course, Eq. (2.117) still depends on L and S implicitly, through the relation J = L – S, used in the derivation. Replacing the product of two CGCs in Eq. (2.117) by a linear combination of two CGCs through the following identity [a particular case of Eq. (8.7.37) of Varshalovich et al. (1988)], n(n + 1)(2J + 3) 1 J +1,M J +1,M CJ M10 CJ Mn0 = 2(2J + 1)(2n + 1) J +1 M × (2J + n + 1)(2J + n + 2)CJJM,n–1,0 JM – (2J – n)(2J – n + 1)CJ M,n+1,0 and comparing the result with Eq. (1.24), one concludes that the effective Hamil˜ or rather its part diagonal in M, can be presented as follows: tonian H,

ex n(n + 1) ˆ ˜ H = ex Jz + Bn0 C0(n) (Jˆ) – so 2n + 1 n=2,4,6 2 2J + n + 1 (n–1) ˆ (n+1) ˆ C0 (J ) – C (J ) . × 2 2J + n + 2 0 (2.118) 19 To obtain Eq. (2.116) one should use the recurrence relation (9.6.5) of Varshalovich et al. (1988), with a = f = L = J + S, b = S, c = d = J , and note that the last 6j symbol therein vanishes because its three upper indices do not satisfy the triangle rule. One can then isolate the ratio of the remaining two 6j symbols and substitute it into Eq. (2.115).

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It is now apparent that the effect of J -mixing is a renormalisation of the CF, which vanishes in the limit ex /so → 0. The remaining steps are quite similar to what was done in Section 2.5. The sum in Eq. (2.118) is treated as a perturbation with respect to the first term, ex Jˆz . The primed CF parameters are transformed to the crystallographic coordinates using Eq. (2.61). The thermal averages of the irreducible tensor operators are replaced by GBFs according to Eq. (2.63), recalling that sign(1 – gJ ) = 1 for the light REs. The energy corrections of first order in Bnm take the form of the anisotropy energy (2.10), in which ex n(n + 1) n κnm = Bnm J BJ(n) (x) + so 2n + 1 2J + n + 1 (n–1) 2J (n+1) BJ (x) – B × (2.119) (x) 2J 2J + n + 2 J with x = J ex /kT . Thus, the effect of J -mixing on the nth -order anisotropy constant κnm consists in renormalising its temperature dependence, which in the absence of the J -mixing is described by a single GBF of the same order n, cf. Eq. (2.64). Now Eq. (2.119) contains two extra terms, with BJ(n±1) (x). As one would expect, these corrections vanish when ex /so → 0. Equation (2.119) enables us to reach a definite conclusion about the sense of the effect. Let us consider the square bracket of Eq. (2.119) in the limit T → 0, or x → ∞. Making use of Eq. (2.76), we get 2J 2J + n + 1 (n–1) (n+1) (∞) BJ (∞) – B 2J 2J + n + 2 J 2J + n + 1 2J – n (n) = BJ (∞) (2.120) – > 0. 2J – n + 1 2J + n + 2 Obviously, the numerator of the first fraction in the parenthesis is greater than, and its denominator is less than their respective counterparts in the second fraction. It is easy to see that the square bracket of Eq. (2.119) will remain positive at any temperature, as BJ(n+1) (x) decays with temperature more rapidly than BJ(n–1) (x). Therefore, J -mixing always enhances the intra-multiplet anisotropy, irrespective of the sign of the latter. It follows from Eq. (2.120) that in the classical limit, J → ∞, the contribution to κnm from the J -mixing in the light REs vanishes at T = 0—a first indication that low temperatures may not be the best choice for appreciating the size of the effect, contrary to the generally accepted view (Myth 3). Equation (2.119) can be put to a further good use: setting to zero all GBFs of order higher than two, one arrives at a simple high-temperature version of the formalism. Let us additionally limit ourselves to tetra-, hexa- or trigonal crystals. Then the anisotropy energy is just K1 sin2 θ, where 3 3 6 2J + 3 ex (2) 2 BJ (x) . (2.121) K1 = – κ20 = – B20 J BJ (x) + 2 2 5 2J so


207

Figure 3.10 Quantities relevant to Eq. (2.119) for RE = Nd (J = 9/2, n = 2), plotted against (3) 1/x: — y = 1.6B9/2 (x), - - - y = 54 65 B9/2 (x), · · · the difference of the previous two.

Here the second term in the square brackets, representing the correction for J mixing, depends on temperature through the familiar Brillouin function—a sign of simplicity yet to come (contrary to Myth 1). The correction term should be small due to the ratio ex /so . However, at elevated temperatures it tends to zero more slowly (as 1/T ) than the first, intra-multiplet term, BJ(2) (J ex /kT ) ∝ 1/T 2 . Therefore, the relative importance of the J -mixing effect must increase with temperature. This constitutes a final departure from Myth 3. As has been demonstrated in Section 2.4, the leading term in the rigorous hightemperature expansion for K1 is linear in the CF parameter B20 . In other words, all other terms, including those nonlinear in Bnm , die out more rapidly as T → ∞. Therefore, Eq. (2.121) becomes asymptotically accurate at elevated temperatures. This happens irrespective of the strength of the CF in relation to the exchange, as both become weak in comparison with the thermal energy kT . In practice, Eq. (2.121) applies to 3d-4f compounds upward of room temperature. Thus, according to Fig. 3.10, in Nd-based magnets the approximation breaks down (the third-order term is no longer negligible) at x ∼ 3. This corresponds to T ≈ 220 K, assuming for the exchange field on Nd a value typical for hard magnetic materials, Bex = 400 T. The above argument is unaffected by the slow temperature variation of the exchange splitting ex . Implicitly, we assume that, while in the high-temperature regime, the system is still not close to the Curie point (the TC of a good permanent magnet should exceed ambient temperature by a factor of at least 2). As a measure of relative importance of the J -mixing in the room-temperature range one can use the ratio of the second term in the square brackets of Eq. (2.121)

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to the first one, ε=

6 2J + 3 ex BJ (J ex /kT ) . 5 2J so BJ(2) (J ex /kT )

(2.122)

Obviously, this quantity does not depend on the CF. In the high-temperature regime, which is primarily of interest to us, ε is also independent of the exchange field Bex , or ex . Indeed, expanding the Brillouin functions in Eq. (2.122) by means of Eq. (2.84) and keeping just the leading terms, one gets 12 kT . (2.123) 2J – 1 so Thus, for a given RE, the fractional contribution of J -mixing to second-order anisotropy is determined by temperature alone and does not depend on characteristics of the solid, such as CF or exchange field. This conclusion is valid in the room-temperature range as well as at higher temperatures. The importance of J mixing grows in direct proportion to absolute temperature. Calculations similar to the above can also be carried out for the second half of the RE series, where the ground multiplets have J = L + S. Omitting the details, we only state the result. The heavy-RE counterparts of the general expression (2.119), the high-temperature approximation (2.121) and the fractional contribution estimate (2.123) are, respectively, the following relations: ex n(n + 1) 2J – n + 1 (n–1) 2J κnm = Bnm J n BJ(n) (x) + BJ (x) – BJ(n+1) (x) so 2n + 1 2J 2J – n (2.124) 2J – 1 3 6 ex K1 = – B20 J 2 BJ(2) (x) + (2.125) BJ (x) 2 5 2J so 12 kT . ε= (2.126) 2J + 3 so One peculiar feature of the J -mixing in the heavy REs is that the effect is strictly nil at T = 0 [for verification put BJ(n) (∞) = (2J )–n (2J )!/(2J – n)! into Eq. (2.124)]. The physical reason is that the ground state of an exchange-dominated heavy RE does not take part in the J -mixing (see Fig. 3.11) since it cannot find itself a partner with the same magnetic quantum number, M = J , among the states of the first excited multiplet, whose M do not exceed J = J – 1. To summarise, contrary to the common perception (Myth 3), the influence of J -mixing on thermodynamic properties of RE magnets grows with temperature. In the limit T → 0 the effect either vanishes completely (light RE with J → ∞, heavy RE with arbitrary J ) or is very small. Its smallness at low temperatures is no indication that it may be neglected in the room-temperature range. The insuperable complexity of the J -mixing has proved to be a yet another myth (Myth 1). Where it matters most—at ambient temperature and above—this effect can be accounted for by means of back-of-the-envelope calculations using Eqs. (2.123), (2.126). ε=

209


Figure 3.11 Comparison of J -mixing in a light and a heavy RE. In both cases the mixing states have the same M and different J . The ground state of a heavy RE is not involved in the J -mixing. Table 3.4 Fractional contribution of J -mixing to the second-order anisotropy constant K1 , as given by Eq. (2.123) for the light REs (upper part of the table) and by Eq. (2.126) for the heavy REs (lower part)

RE

J

so (K)*

ε (T = 300 K)

ε (T = 400 K)

Pr Nd Sm Tb Dy Ho

4 9/2 5/2 6 15/2 8

3100 2740 1440 2880 4740 7480

0.166 0.164 0.625 0.083 0.042 0.025

0.221 0.219 0.833 0.111 0.056 0.034

* Taken from Elliott (1972).

Finally, the insignificance of the J -mixing in REs other than Sm (Myth 2) is disproved by the data presented in Table 3.4. At T = 400 K even terbium—a heavy RE—is subject to an 11% correction, while in Pr and Nd it is as high as 22%. In heavier REs the effect is noticeably smaller, for two reasons. Firstly, the larger denominator in the prefactor of Eq. (2.126) as compared with Eq. (2.123).

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This alone makes the size of the effect in Ho a factor of 0.3 smaller than in Pr. A reduction by a further factor 0.4 comes from the larger spin-orbit splitting so , especially high towards the end of the RE series. Sm-based magnets should be considered separately. There, the second-order J -mixing correction to K1 , ∝ 1/so , is the size of the main, intra-multiplet contribution. It is therefore likely that corrections of higher orders in 1/so , neglected in Eqs. (2.119)–(2.126), are not small. The contribution of J -mixing to sixth-order anisotropy constants (responsible e.g. for the anisotropy in the basal plane in the hexagonal ferromagnets Sm2 Fe17 and Sm2 Co17 ) can hardly be called a correction, since the intra-multiplet effect is strictly nil. Equation (2.115) for the matrix element of Hˆ CF is invalid, because the 6j symbol in the denominator equals zero (the numerator is also nil since B60 ∝ γJ = 0). Calculations in that case should be performed using an alternative approach developed by Magnani et al. (2003) specially for the purpose. The role of the Stevens coefficient γJ is then played by another quantity, δ6 , which is negative for Sm. When the sign of δ6 is taken into consideration, the anisotropic properties of Sm compounds—first of all the SRTs—are no longer incomprehensible. Our review of the theoretical apparatus for the description of single-ion magnetocrystalline anisotropy has reached its close. All along we tried to illustrate the relevance of the various approximations by performing simple calculations for wellknown hard magnetic materials. Our intention was to encourage experimentalists to use, where possible, the approximate equations for do-it-yourself calculations. Perhaps the best demonstration of the advantages of the single-ion model cast in analytical form is still to come. We are just turning to the phenomena where magnetic anisotropy manifests itself most vividly—spin reorientation transitions (SRT).

3. Spin Reorientation Transitions 3.1 General remarks The third and last section of this Chapter is dedicated to the phenomenon of spin reorientation transitions (SRT). Of interest to us here are not SRTs as such—a vast subject covered in excellent reviews and monographs (Belov et al., 1976, 1979)— but only some peculiar features of the SRTs viewed from the standpoint of the single-ion anisotropy model. Our main goal is to demonstrate that the single-ion model is more than an ad hoc theory explaining already known experimental facts. Rather, it possesses a certain power of prediction. Where the underlying approximations are valid, the strength of the model is such that all experimental findings not fitting in its framework eventually prove wrong. This point will be illustrated with a number of examples. A spin reorientation transition (SRT) is a phase transition consisting in a change of orientation of ordered magnetic moments—which can be distributed among several sublattices—with respect to crystallographic axes. Obviously, such magnetic transitions (called order-order transitions) are essentially distinct from the usual magnetic ordering of e.g. a ferromagnet at the Curie point. Even among order-order


211

transitions SRTs can be singled out in a separate class, on the grounds that they do not involve any change of the mutual orientation of the sublattice moments, only their orientation in relation to the crystal axes changes. Following this definition, metamagnetic transitions or field-induced transitions of ferrimagnets into a noncollinear state, are excluded from the scope of this section. Metamagnetism was reviewed by Levitin and Markosyan (1988) and by Goto et al. (2001). A chapter about the field-induced transitions in ferrimagnets appeared in Volume 9 of this Handbook (Zvezdin, 1995). Just like phase transitions in general, SRTs can be of first or of second order. In the former case the orientation angle experiences a discontinuity at the transition point, whereas in the latter case the angle itself varies continuously, but its first derivative is discontinuous. For all that, the change of symmetry, essential in second-order SRTs, is always abrupt. Therefore, no matter if an SRT is of first or of second order—it takes place at a point, rather than within an interval. For instance, the process of spontaneous20 spin reorientation shown in Fig. 3.12a comprises two second-order SRTs as well as continuous rotation of the magnetisation vector in the interval between the two transition points. This rotation is of course not a phase transition. Likewise, a mere change of slope of the θ (T ) dependence (Fig. 3.12b) does not amount to an SRT. The crucial difference is that at a transition point the angle θ takes a special high-symmetry value, 0 or π /2. When this is the case, the derivative of the orientation angle diverges on approach to the transition point from the lower-symmetry phase (Landau and Lifshitz, 1958). In a first-order SRT there is no restriction on the critical values of θ : in general both are distinct from 0 or π /2 (Fig. 3.12c), but either one of them or both may also take higher-symmetry values. Second-order SRTs need not always come in pairs as shown in Fig. 3.12a. It is not inconceivable that the process of spin reorientation starting at the point T2 may not reach completion before the temperature reaches 0 K. Well-known examples of single second-order SRTs are those taking place in Gd metal (Corner et al., 1962) and in Nd2 Fe14 B (Deryagin et al., 1984; Givord et al., 1984). All the above applies, practically without change, also to magnetic field-induced transitions. Interestingly, an infinitesimal magnetic field applied at an angle to the easy magnetisation direction is sufficient to provoke a second-order SRT. Saturation in a finite field, characteristic of magnetisation along a hard direction, is also a second-order SRT (by contrast, under general orientation of the field the approach to saturation is asymptotic, without an SRT). Apart from such ubiquitous and trivial second-order SRTs, there are also field-induced SRTs of first order. In this case the upper critical value of the angle θ may correspond either to low or to high symmetry (Figs. 3.13d, 3.13f), however, the lower critical value is always low-symmetry. The reason is the afore-mentioned transition to a low-symmetry phase induced by an infinitesimal magnetic field. Hereafter we shall concentrate on spontaneous SRTs. Magnetic field-induced SRTs of first order (FOMPs) were described in detail in Volume 5 of this Handbook (Asti, 1990). 20 According to the generally accepted definition, spontaneous SRTs take place as a result of temperature change, at zero magnetic field and ambient pressure.

212


Figure 3.12 Temperature dependence of orientation angle θ: (a) 2 second-order SRTs, (b) no SRT, (c) a first-order SRT.

It is an interesting peculiarity of second-order SRTs that Landau’s theory of second-order phase transitions (Landau and Lifshitz, 1958) applies to them practically without restrictions. In this sense SRTs differ significantly from order-disorder transitions. Estimations show that the interval where Landau’s theory fails due to critical fluctuations is very narrow in the case of SRTs, 10–7 . . . 10–4 K (Belov et al., 1976). Physically, this is because the fluctuations arising near an SRT have a very large correlation length.


Figure 3.13 Examples of magnetic field-induced SRTs: (a, b) 2 second-order SRTs, (c, d) 2 second-order SRTs and 1 first-order SRT (type II FOMP), (e, f) 1 second-order SRT and 1 first-order SRT (type I FOMP). Note the ubiquity of the trivial second-order SRT at B → 0.

213

214


3.2 SRTs in uniaxial magnets 3.2.1 Graphic representation In this subsection we shall introduce the concept of phase diagrams and relate the diagrams of different levels within the hierarchy of approximations to the anisotropy energy. We begin with the simplest expression, Ea = K1 sin2 θ.

(3.1)

According to Section 2.4, this formula is relevant to a uniaxial magnet at high temperatures (T 300 K). Obviously, a system described by Eq. (3.1) may have just two stable states: 0, if K1 > 0 θ= (3.2) π /2, if K1 < 0. These are traditionally called ‘easy axis’ and ‘easy plane’. A first-order phase transition takes place at the point K1 = 0. This information is summarised in Fig. 3.14. Let us now turn to a more complicated example. Consider a system whose anisotropy energy is given by Ea = K1 sin2 θ + K2 sin4 θ.

(3.3)

Minimisation with respect to θ yields the following equilibrium phases (Casimir et al., 1959): ⎧ if K1 > max(0, –K2 ) ⎨0, θ = arcsin –K1 /2K2 , if – 2K2 < K1 < 0 (3.4) ⎩ π /2, if K1 < min(–2K2 , –K2 ). This rather complex combination of if-statements can be visualised with the aid of a simple diagram in the K1 –K2 plane, Fig. 3.15. Note the presence of a new phase ‘easy cone’, with intermediate values of θ between 0 and π /2. The bold lines separating the domains of different phases are phase transition lines of first (dashed) or second (solid) order. For simplicity we shall not go into the difference between the existence of a phase and its stability. Introducing a dimensionless ratio K1 /|K2 |, one can display the same information in quasi-one-dimensional diagrams, Fig. 3.16. Obviously, two such graphs are needed to show the qualitatively distinct cases of K2 > 0 and K2 < 0, Figs. 3.16a and 3.16b, respectively. What happens if K2 = 0? In other words, how can one graphically go over to the limit K2 → 0? The above-considered Eq. (3.1) is not but a particular case of the more general Eq. (3.3). Therefore, there must be a way to obtain Fig. 3.14 from Figs. 3.15 and/or 3.16. The graphic operation turning Fig. 3.16 into Fig. 3.14 is zooming out. Indeed, letting K2 go to zero means scaling Fig. 3.16 down, reducing it. Looking at Fig. 3.16 on an ever decreasing scale, one gradually ceases to distinguish the details. On a very small scale the easy-cone domain shrinks to non-existence and the transition occurs


215

Figure 3.14 Phase diagram of the simplest uniaxial magnet.

Figure 3.15 Phase diagram of a uniaxial magnet with two anisotropy constants (Casimir et al., 1959).

Figure 3.16 The information of Fig. 3.15 presented as two one-dimensional diagrams: (a) K2 > 0, (b) K2 < 0.

at the origin. Figures 3.16a and 3.16b are no longer different from each other, both becoming identical to Fig. 3.14. Moving in the opposite direction, one can regard Fig. 3.16 as a refinement of Fig. 3.14. According as one zooms in, it becomes apparent that the transition occurs not quite at the origin and that the sign of K2 does matter.

216


Let us apply these ideas to the analysis of a more complete expression for the anisotropy energy: Ea = K1 sin2 θ + K2 sin4 θ + K3 sin6 θ.

(3.5) 21

This expression is relevant to exchange-dominated RE magnets at arbitrary temperature. The properties of such a system cannot be concisely formulated either as inequalities of type (3.2) or (3.4) (the latter is already too cumbersome and far too involved), nor as a phase diagram similar to Fig. 3.15 in the K1 K2 K3 parameter space—comprehensibility does not belong to the virtues of 3-dimensional drawings. We are left with the only acceptable choice—a quasi-2-dimensional diagram in reduced coordinates. (We say ‘quasi’, because one always needs two drawings to show all possible cases, cf. Figs. 3.16a and 3.16b.) Choosing the reduced variables, we follow certain guidelines. The diagrams should be easy to relate to those of the cruder approximations (3.1) and (3.3). Quantities prone to changing sign are unsuitable candidates for the denominators. For instance, Asti’s choice of K2 /K1 and K3 /K1 (Asti, 1990) is rather inconvenient: every time K1 (T ) changes sign, the locus of the system goes to infinity, to reappear on the other sheet of the diagram. Clearly, K1 /K3 and K2 /K3 would be a better choice because, as stated in Section 2.7, K3 as a function of temperature is always sign-definite. We find the variables K1 /|K3 | and K2 /|K3 | even more suitable, in accordance with the requirement that the shape of the diagram should depend possibly little on the sign of K3 and that |K3 | should act as a scaling factor when going over to the limit K3 → 0. The phase diagram in such coordinates is displayed in Figs. 3.17a (K3 > 0) and 3.17b (K3 < 0). The notation for the phases is the same as in Figs. 3.15 and 3.16, however, now the angle θ in the easy-cone phase is determined from the condition (Asti, 1990): K22 – 3K1 K3 – K2 2 sin θ = (3.6) . 3K3 SRTs of five different kinds are possible: there can be a first- and a second-order SRT between every two phases out of the three present in Fig. 3.17, except the pair easy axis—easy plane, where only a first-order transition can take place. A secondorder transition easy axis—easy plane is impossible in principle, because none of the two phases is more symmetric than the other (in other words, neither of the two symmetry groups is a subgroup of the other one). The domain boundaries in Fig. 3.17 are mainly straight lines, the curved portions AO and BC being parabolic arcs. The equations describing these boundaries— the necessary conditions of the SRTs—are collected in Table 3.5. A general necessary condition of a second-order SRT is that the second derivative of the anisotropy energy with respect to the angle must vanish at the point corresponding to the higher-symmetry phase. E.g. for the transition easy axis—easy cone this condition is ∂ 2 Ea /∂θ 2 |θ=0 = 0, whence K1 = 0. In the case of first-order SRTs, a general 21 In real crystals the anisotropy energy may also depend on the angle φ, cf. Eqs. (2.12), (2.13). Our simplified analysis makes use of the well-known fact that in the vast majority of SRTs only the angle θ changes, while φ remains constant, fixed by the symmetry. For instance, in the case of the hexagonal crystallographic classes D6 , C6v , D3h and D6h this means φ = 0 or π /6, which reduces Eq. (2.12) to (3.5) with K3 → K3 ± K3 .

217


Figure 3.17 Phase diagram of a uniaxial magnet with three anisotropy constants: (a) K3 > 0, (b) K3 < 0.

Table 3.5

Necessary conditions of the SRTs in Fig. 3.17

Phases involved Easy axis–easy cone Easy plane–easy cone Easy axis–easy plane

1st -order SRT √ K2 = –2 K1 K 3 K2 /K3 = 1 – 2 1 + K1 /K3 K1 + K2 + K3 = 0

2nd -order SRT K1 = 0 K1 + 2K2 + 3K3 = 0 no SRT

218


necessary condition is that the function Ea (θ ) must take equal values at the points corresponding to the two phases in question. Thus, for the first-order transition easy axis—easy cone this condition is as follows: Ea (θcone ) = Ea (0) = 0.√Substituting Eq. (3.6) into Eq. (3.5) and equating the latter to zero yields K2 = –2 K1 K3 . Finally, neglecting graphically the third anisotropy constant (K3 → 0) consists in zooming out of Fig. 3.17. Then the small details are gradually lost, the points A, B and C merge with the origin and both sheets of the phase diagram turn into Fig. 3.15. Thus, Fig. 3.15 can be regarded as a cruder, lower-resolution version of Fig. 3.17, and vice-versa Fig. 3.17 is a more precisely defined version of Fig. 3.15, taking account of the sign of K3 . 3.2.2 Peculiarities following from the single-ion model In this subsection some specific predictions of the single-ion model regarding spontaneous SRTs in uniaxial magnets will be considered. Admittedly, these predictions can only be formulated as a number of separate statements, or ‘rules’, unlike in the case of cubic magnets, where a fully-fledged coherent theory can be developed (see Section 3.3 below). Nevertheless, these ‘rules’ deserve some respect. Experience shows that ignoring them may lead to easily avoidable mistakes. In the crudest approximation, zooming out of all phase diagrams, an SRT takes place in a uniaxial magnet when its first anisotropy constant changes sign. This statement is certainly true for the room-temperature range and might be somewhat qualified for low temperatures. Staying for the moment near room temperature, there is one natural reason for K1 to change sign in a 3d-4f intermetallic compound—the competition of the 3d and the 4f contributions. Indeed, since the high-temperature approximation applies to the RE, 3 K1 = K3d – αJ B20 J 2 BJ(2) (x). (3.7) 2 Here αJ is the first Stevens factor and B20 is the leading CF parameter in the coordinate representation, Eq. (1.4). Note that the product αJ B20 in Eq. (3.7) equals the quantity B20 in Eq. (2.98). On account of the known properties of the function BJ(2) (x), Section 2.6, the second term in Eq. (3.7) is sign-definite and falls off monotonically with temperature. The same is true in respect of K3d ;22 it is smaller in magnitude but also decreases more slowly with temperature than the RE contribution. Therefore, in order for K1 to become zero near ambient temperature, the two terms in Eq. (3.7) must have opposite signs. Applied to a particular family of RE-iron or RE-cobalt compounds, where K3d and B20 are both sign-definite, this means that, depending on the combination of signs of K3d and B20 , spontaneous SRTs will occur either in the compounds of the REs with αJ positive, or on the contrary, only in those where αJ is negative. This ‘Stevens αJ rule’ is followed by the vast majority of uniaxial RE-iron and RE-cobalt intermetallics (Buschow, 1988, 1991; Kirchmayr and Burzo, 1990; Li and Coey, 1991; Franse and Radwański, 1993). Exceptions do happen, however, for the obvious reason that higher-order CF terms may interfere in the anisotropy energy balance, as the high-T approximation gradually breaks down below room temperature. 22

A rare exception is the compound Y2 Fe14 B, where K1 (T ) is non-monotonic (Bartashevich et al., 1990).


219

We wish to emphasise that the single-ion theory is not concerned with the task of evaluating ab initio the CF parameters Bnm , nor with predicting their signs. Rather, it regards them as phenomenological parameters. It should not be confounded with the point-charge model. The well-known fact that the latter generally fails in metallic systems leaves the single-ion theory undefeated. Let us refine our analysis and take into consideration the second anisotropy constant K2 [in the case of tetragonal magnets, Eq. (2.13), this should be understood as the combination K2 – (K2 )2 + (K2 )2 ]. As we still remain in the room-temperature range, we may neglect all anisotropy constants but K1 and K2 , taking for the latter just the fourth-order CF term in Eq. (2.65): 35 (3.8) βJ B40 J 4 BJ(4) (x). 8 Here βJ is the second Stevens factor and B40 is the CF parameter in the coordinate representation, Eq. (1.4). The product βJ B40 in Eq. (3.8) is equivalent to the quantity B40 in Eq. (2.65). In tetragonal magnets the role of B40 in Eq. (3.8) is played by the combination B40 – ( 352 )1/2 sign βJ |B44 |. The occurrence of a spontaneous spin reorientation is still subject to the ‘Stevens αJ rule’. However, the presence of a nonzero K2 brings about some variety: the reorientation may proceed as a single first-order SRT or by way of two secondorder SRTs (Fig. 3.12a displays the latter possibility). Which of the two scenarios will take place is decided by the sign of K2 (Horner and Varma, 1968), which is in turn determined by the sign of the second Stevens factor βJ (the 3d contribution to K2 is negligible). As an illustration, let us consider the well-studied archetypal permanent magnet materials RECo5 . In accordance with the ‘Stevens αJ rule’, spontaneous SRTs are observed in all RECo5 where αJ < 0, that is with RE = Pr (Yermolenko, 1983), Nd, Tb (Lemaire, 1966), Dy (Ohkoshi et al., 1977) and Ho (Lemaire, 1966; Chuev et al., 1981a). Moreover, for RE = Pr, Nd, Dy and Ho the SRTs are distinctly of second order. This is in perfect agreement with the ‘Stevens βJ rule’, because all four above-mentioned REs have βJ < 0 (Table 3.1). In contrast, Tb has βJ > 0, therefore, the SRT in TbCo5 must be of first order. This definite prediction of the single-ion theory is apparently at variance with experiment, which interprets the reorientation process in TbCo5 as two closely situated second-order SRTs. The situation is aggravated further by the fact that it is not just from bulk magnetic measurements (Ermolenko, 1980) that this conclusion was made. Also neutron diffraction experiments (Lemaire and Schweizer, 1967; Kelarev et al., 1980) reportedly detected in TbCo5 , in a narrow interval just above 400 K, the presence of an easy-cone phase with intermediate values of θ between 0 and π /2. It should be noted that these data are open to another interpretation. Namely, that two phases—easy axis and easy plane—coexist in the vicinity of the SRT (which is of first order, as predicted by the single-ion model). The relative content of the two phases varies gradually with temperature, from the pure easy plane below ∼400 K to the pure easy axis above ∼425 K. This interpretation is corroborated by the peculiar shape of the temperature dependence of the angle θ observed K2 =

220


Figure 3.18 Orientation angle of the easy magnetisation direction versus temperature. Two paradigms of continuous spin reorientation in a uniaxial magnet: (i) two second-order SRTs at T1 and T2 , solid line, (ii) one first-order SRT broadened by normally distributed inhomogeneity, dashed line (after Kuz’min, 2000).

in TbCo5 . It resembles the dashed curve in Fig. 3.18. Such a shape of the dependence θ (T ), described by the complementary error function, is characteristic of ‘smeared-out’ first-order SRTs (Kuz’min, 2000). It is clearly distinct from the arcsine-type dependence (solid line) with two sharp second-order transition points T1 and T2 . The inhomogeneity of composition, necessary for smearing out of a first-order SRT, is present in TbCo5 , whose real stoichiometry is TbCo5+δ , with δ ≈ 0.1. According as the experimental arguments in favour of two second-order SRTs in TbCo5 become less unambiguous, the single-ion theory, on the contrary, strengthens its insistence on a single first-order SRT. Beyond all doubt is the general analysis of Horner and Varma (1968) demonstrating that an SRT must be of first order if K2 < 0 and of second order if K2 > 0. On the other hand, the shape of the dependence θ (T ) in the other RECo5 undergoing spontaneous spin reorientation (RE = Pr, Nd, Dy, Ho) bears close resemblance to the continuous curve of Fig. 3.18, with two square-root-type anomalies characteristic of Landau’s theory.23 The SRTs are clearly of second order, and therefore K2 > 0, in the aforementioned RECo5 with βJ < 0. By virtue of Eq. (3.8), K2 must be negative in TbCo5 , where βJ > 0. The only (unlikely) loophole in our logic might be that T = 400 K is not high enough a temperature and that K2 in TbCo5 , negative at the highest temperatures as it should be, becomes positive somewhere above T = 425 K due to an extraordinarily large sixth-order CF term, cf. the second one of Eqs. (2.65). This remote possibility can be ruled out completely. Indeed, as stated in Section 2.7, K2 cannot change sign more than once. And in TbCo5 K2 < 0 at T = 4.2 K, as found experi23 In the case of PrCo , where the reorientation process is incomplete, only the higher-temperature anomaly is observed, 5 T1 being effectively negative.


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mentally by Ermolenko (1980). Therefore, K2 is negative at any temperature. Thus, the single-ion theory still insists that the SRT in TbCo5 must be of first order. The true situation in TbCo5 has been finally established in recent scanning differential calorimetry experiments (Tereshina et al., 2007), which revealed a nonzero latent heat of the SRT—an ultimate proof that it is of first order. The single-ion model made no mistake. Let us formulate a yet another specific prediction of the model. When the easy direction lies in the basal plane, its orientation within the plane for a certain family of tetragonal compounds is determined by the sign of the second Stevens coefficient βJ , whereas in hexagonal compounds it follows the third Stevens factor γJ . Moreover, in the latter case the orientation within the basal plane cannot change as temperature is lowered. In tetragonal compounds such a reorientation may take place no more than once, but in reality SRTs of this kind are extremely rare. For definiteness, we limit ourselves to the hexagonal crystallographic classes D6 , C6v , D3h and D6h , where the anisotropy energy is presentable as Eq. (2.12) with the anisotropy constants given by Eqs. (2.65). By assumption, θ ≡ π /2. Therefore, the equilibrium value of the angle φ, either 0 or π /6, is determined by the sign of the quantity K3 , √ 231 γJ B66 J 6 BJ(6) (x). K3 = (3.9) 16 Again, we have factored out the Stevens coefficient γJ , so that the product γJ B66 in Eq. (3.9) is equivalent to the quantity B66 in the last one of Eqs. (2.65). Thus, if the product γJ B66 is negative, the easy magnetisation direction within the basal plane is the a axis, or [100], corresponding to φ = 0. If γJ B66 > 0, then the easy axis is b, or [120], φ = π /6. For a given family of RE-iron or RE-cobalt compounds, all having B66 of the same sign, the easy direction is determined by the sign of the third Stevens factor γJ . This ‘Stevens γJ rule’ can be best illustrated by our favourite example—the RECo5 compounds—whose symmetry is described by the holohedral hexagonal group D6h . Back in the 1960’s it was found from neutron diffraction on NdCo5 and TbCo5 (Lemaire, 1966; Lemaire and Schweizer, 1967) and also from magnetic measurements on a single crystal of HoCo5 (Katsuraki and Yoshii, 1968) that the easy direction within the basal plane in all those compounds is the a axis. This is not illogical, since Nd, Tb and Ho have γJ < 0, see Table 3.1. Following the same logic, the easy direction in PrCo5 and in DyCo5 must be the b axis, because γJ > 0 for both Pr and Dy. Indeed, magnetisation data obtained on a PrCo5 single crystal (Yermolenko, 1983) confirmed that the easy direction there rotates from the c towards the b axis (even though the reorientation process is not completed down to T = 0). Unexpectedly, DyCo5 falls out of line. A magnetisation study of a single crystal (Ohkoshi et al., 1977) concluded that the easy direction at and below room temperature is the a rather than the b axis. This statement was reiterated in the work of Berezin et al. (1980) and even in the neutron diffraction paper of Chuyev et al. (1981b). The single-ion model seems to have received a fatal blow and will never recover.

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However, a more careful perusal of the above articles reveals a number of discrepancies. Thus, according to Ohkoshi et al. (1977), the a axis is the same as the [110] direction and the b axis is [100], which is rather unusual. A clue to their unconventional notation may be found in their previous paper (Ohkoshi et al., 1976), stating that in NdCo5 the easy direction below T = 245 K is the b axis, or [100] (while in reality it is the a axis, or [100]). Apparently, the authors use a nonstandard set of Bravais vectors where two of them, lying in the basal plane, make an angle of 60°, rather than 120° as in the standard hexagonal set. Moreover, the direction of those Bravais vectors is called the b axis, while their bisector is called the a axis. In short, Ohkoshi et al. (1976, 1977) seem to have swapped the a and b axes, as compared with the standard notation. Of course, this is not but a plausible conjecture. Furthermore, the text of the paper of Berezin et al. (1980) speaks of an easy axis a and a hard axis b. However, from the experimental magnetisation curves in Fig. 1 thereof one concludes that the opposite is true. Namely, that the easy magnetisation direction is the b axis. Finally, in their neutron diffraction experiments the unsuspecting Chuyev et al. (1981b) did not at all pose the problem of checking the orientation of the easy direction within the basal plane of DyCo5 , having taken for granted that is was along the a axis. At our instance, Skokov (2007) have recently conducted a series of purposeful tests on a single crystal of DyCo5 , which have established that the easy magnetisation direction at and below room temperature lies along the crystallographic axis b, i.e. [120], exactly as predicted by the single-ion model, or by the ‘Stevens γJ rule’. The above examples demonstrate—quite convincingly in our view—that the single-ion theory of magnetocrystalline anisotropy has the power of prediction. To the extent that it enables an armchair theoretician to find mistakes in experimental papers. In this connection, the recent attempts to question the validity of the singleion approach and even to supplant it with a new mechanism (Irkhin, 2002) can only arouse bewilderment. We reiterate, however: in order for the strength of the singleion model to be fully appreciated, it has to be kept strictly apart from the task of computing the CF parameters ab initio. Certain progress has been achieved in the latter field, too (Hummler and Fähnle, 1996; Novák, 1996), based on the density functional theory rather than on the naive point-charge model.

3.3 Spontaneous SRTs in cubic magnets In the crudest approximation, the anisotropy energy of any cubic crystal is given by Ea = K1 n2x n2y + n2y n2z + n2z n2x . (3.10) The equilibrium phases are the 6-fold degenerate [100] and 8-fold degenerate [111]. The former is energetically favourable at K1 > 0, the latter at K1 < 0. A first-order SRT takes place at the point K1 = 0. This information is presented graphically in a one-dimensional phase diagram, Fig. 3.19.


223

Figure 3.19 Phase diagram of a cubic magnet described by Eq. (3.10). The positive semi-axis is the domain of the [100] phase, the negative semi-axis (hatched) is the domain of the [111] phase, the origin being the transition point.

Let us consider a more realistic expression for the anisotropy energy (2.8), valid for the cubic crystallographic classes possessing 4-fold symmetry axes,24 O, Td and Oh : Ea = K1 n2x n2y + n2y n2z + n2z n2x + K2 n2x n2y n2z . (3.11) This expression is relevant to cubic exchange-dominated 3d-4f compounds, such as e.g. Laves phases REFe2 and RECo2 . In that case the omission of terms of order higher than six is a valid approximation. It is also justified to neglect the 3d contribution to the fourth- and sixth-order anisotropy constants. This contribution, originating from the relatively weak spin-orbit coupling, decreases rapidly as the order of the anisotropy constants n increases (∝ λn ). In practice, it can play a role only in second-order anisotropy constants, here forbidden by the symmetry. Thus, not unreasonably we expect that the linear theory of Section 2.5 should apply to the systems under consideration to full extent. Proceeding from the CF Hamiltonian (1.26), where Onm are meant to be the usual Stevens operators in the J representation, and repeating the manipulations of Section 2.5, we get for the anisotropy constants the following expressions: K1 = –40J 4 b4 BJ(4) (x) – 168J 6 b6 BJ(6) (x) K2 = 1848J 6 b6 BJ(6) (x).

(3.12)

Here BJ(4,6) (x) are the fourth- and sixth-order generalised Brillouin functions (GBF, Section 2.6) and x is the magneto-thermal ratio (2.33). Figure 3.20 displays the phase diagram of a cubic magnet described by Eq. (3.11) (Smit and Wijn, 1959). We observe the presence of an additional phase—the 12fold degenerate [110]. In the considered approximation all the transitions are of first order. It is convenient to present the information contained in the two-dimensional phase diagram (Fig. 3.20) as quasi-one-dimensional diagrams, Fig. 3.21. There, the role of the coordinate is played by the ratio K1 /|K2 |. The need to distinguish two essentially distinct cases according to the sign of K2 brings about the two sheets of the diagram, Figs. 3.21a and 3.21b. This insignificant complication is outweighed by the advantages of Fig. 3.21. The latter is related in a rather straightforward way to Fig. 3.19, this relation being a graphic realisation of neglecting K2 , or taking the limit K2 → 0. Zooming out of Fig. 3.21, one gradually loses out of sight the details like the intermediate [110] domain or the deviation of the transition point 24 It will be recalled that in the case of the cubic classes T and T an extra six-order term (2.9) must be taken into h account.

224


Figure 3.20 Phase diagram of a cubic magnet with two anisotropy constants (Smit and Wijn, 1959). The oblique phase boundaries are 9K1 + 4K2 = 0 (second quadrant) and 9K1 + K2 = 0 (fourth quadrant).

Figure 3.21 A quasi-one-dimensional presentation of the phase diagram of Fig. 3.19: (a) K2 > 0, (b) K2 < 0.

from the origin. What eventually remains of either part of Fig. 3.21 is two semiaxes, the blank positive [100] and the hatched negative [111], that is just Fig. 3.19. The more important advantage of Fig. 3.21 is its one-dimensionality. Thanks to it, the SRT conditions can be presented simply as taking on of certain universal values, –4/9, 0 and 1/9, by the variable K1 /|K2 |. By means of Eqs. (3.12) these conditions can be readily expressed in terms of the dimensionless CF ratio b4 /|b6 |. It is still necessary to distinguish two particular cases according to the sign of b6 . Namely: A. b6 > 0 (K2 > 0). There are two transitions: 1. [111]–[110] at K1 /|K2 | = –4/9. By virtue of Eqs. (3.12), this is equivalent to 49 B (6) (x) b4 = J 2 J(4) . |b6 | 3 BJ (x)

(3.13)


225

Figure 3.22 Phase diagrams of a cubic RE magnet in the dimensionless coordinates ‘temperature–crystal field’.

2. [110]–[100] at K1 /|K2 | = 0, whence b4 21 B (6) (x) = – J 2 J(4) . |b6 | 5 BJ (x)

(3.14)

B. b6 < 0 (K2 < 0). One further transition is possible: 3. [111]–[100] at K1 /|K2 | = 1/9, which yields 14 B (6) (x) b4 = – J 2 J(4) . |b6 | 15 BJ (x)

(3.15)

It is time to take advantage of the one-dimensionality of the chosen representation. Having saved a dimension in Fig. 3.21, we now have the option of adding a new second dimension to our diagrams. We choose the quantity 1/x for this role, which will enable us to include temperature evolution of the system into the picture. The variable 1/x is more convenient for the purpose than x, because at low temperatures 1/x is directly proportional to T ; in any case its dependence on T is monotonic. Now Eqs. (3.13)–(3.15) describe curves in the plane 1/x – b4 /|b6 |, Fig. 3.22. It is interesting to note that one and the same special function is involved in all three cases—the ratio of the sixth- to the fourth-order GBF—only the prefactors differ. The advantage of the coordinates employed in Fig. 3.22 is that the phase boundaries therein are universal (apart from their dependence on the quantum number J , Fig. 3.22 corresponds to J = 8). Anyhow, the topology of the phase diagrams does not depend on J , while the ordinates of the points A, B and C are given by (J – 2)(2J – 5) and – 157 (J – 2)(2J – 5), resimple formulae: 496 (J – 2)(2J – 5), – 21 10 spectively. Therefore, for any other J diagrams similar to Fig. 3.22 can be sketched rather straightforwardly. Accurate drawings should present no major difficulties either, since all GBF have been tabulated (Kuz’min, 1992). Temperature evolution of a specific compound can be depicted in Fig. 3.22 by a horizontal line, because the quantity b4 /|b6 | proper of the system remains constant

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as ‘temperature’ 1/x varies. If this horizontal line crosses one of the curves, a spontaneous SRT takes place at the temperature T corresponding to the abscissa of the crossing-point. Thus, spontaneous SRTs follow a scenario fully determined by the ratio b4 /|b6 | and independent of the strength of the exchange interaction (as long as the system is exchange-dominated, see Section 2.2). Knowledge of the exchange field on the RE and of its temperature dependence is only needed for establishing the quantitative relation between x and T . The sign of b6 is very important, since it decides to which sheet of the phase diagram, Figs. 3.22a or 3.22b, the system belongs. To determine sign(b6 ), it suffices to find sign(K2 ) at any temperature. A number of more specific conclusions can be drawn. 1. No more than one spontaneous SRT can take place in any one system. 2. If the low-temperature phase is [100], a spontaneous SRT is in principle impossible. 3. If the low-temperature phase is [110], a spontaneous SRT will take place inevitably. 4. If the low-temperature phase is [111] and b4 > 0, no spontaneous SRT is possible. 5. If the low-temperature phase is [111] and b4 < 0, an SRT is inevitable, [100] being the high-temperature phase. Table 3.6 summarises the predictions of the single-ion theory and the experimental information on the easy magnetisation directions and spontaneous SRTs in the cubic Laves phases REFe2 and RECo2 . Full-potential density-functional calculations (Diviš et al., 1995) yielded a positive sign for the fourth-order CF parameter in the coordinate representation (i.e. for the quantity b4 /βJ ) and a negative sign for b6 /γJ . Accordingly, in Table 3.6, sign(b4 ) = sign(βJ ) and sign(b6 ) = – sign(γJ ), cf. Table 3.1. For RE = Sm γJ is undefined, its role being played by the quantity δ6 < 0 (Magnani et al., 2003). Therefore, b6 > 0 for SmFe2 and SmCo2 . Examining Table 3.6 one observes that the single-ion model agrees with experiment in all cases without exception. It should be noted that in each case the model makes a binding prediction of the high-T phase as well as predicting an optional low-T phase. In order for the low-T option to be realised, i.e. in order for the spontaneous SRT to actually happen, the ratio b4 /|b6 | must be within certain bounds, also predicted by the theory. In some exceptional cases, e.g. in ErFe2 and in ErCo2 , the model can rule out the possibility of an SRT altogether. However, in general the single-ion theory is not concerned with calculating CF parameters ab initio, therefore it cannot be held responsible for wrongly predicted SRTs in specific compounds. Allegations of failure of the single-ion model sometimes found in the literature are in fact reports of failures of various modifications of the point-charge model (confused with the single-ion one). In those cases when the CF parameters are considered known, their noncompliance with the single-ion model is a sure sign of mistake. Thus, according to Gignoux et al. (1975, last line of Table I) b4 /|b6 | in HoCo2 is about –204. However, the fact that HoCo2 undergoes a spontaneous SRT [110]–[100] means that this ratio must be between 0 and –138.6, cf. Fig. 3.22a. Further checks unearth an apparent inconsistency between the values b6 = 2.3 × 10–5 K and K2 = 109 erg/cm3 ,

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Table 3.6 Easy directions of magnetisation in the cubic Laves phases REFe2 and RECo2 . In each case the model makes a binding prediction of the high-temperature orientation (righthand symbol), the predicted low-temperature phase being optional

RE

Pr Nd Sm Tb Dy Ho Er Tm Yb

Single-ion model Sign b4 Sign b6 –1 –1 +1 +1 –1 –1 +1 +1 –1

–1 +1 +1 +1 –1 +1 –1 +1 –1

Easy direction [111]/[100] [110]/[100] [110]/[111] [110]/[111] [111]/[100] [110]/[100] –/[111] [110]/[111] [111]/[100]

Experiment REFe2 –/[100]a [110]/[100]a [110]/[111]d –/[111]f –/[100]f –/[100]f –/[111]f –/[111]f –/[100]a

RECo2 –/[100]b [110]/[100]c –/[111]e –/[111]b –/[100]b [110]/[100]g –/[111]b –/[111]h paramagnetism

a Meyer et al. (1981). b Atzmony and Dublon (1977). c Atzmony et al. (1976). d Van Diepen et al. (1973). e Gratz et al. (1994). f Taylor (1971). g Gignoux et al. (1975). h Déportes et al. (1974).

reported by Gignoux et al. (1975). According to the second one of Eqs. (3.12), at T = 0 and J = 8, K2 must equal 166,486,320 b6 = 3.8 × 103 K/f.u., or 1.15 × 1010 erg/cm3 , which is an order of magnitude too high. Given that K2 was determined experimentally, one is left to conclude that the reported value of b6 (and most likely of b4 as well) is mistaken. There are also examples of false SRTs in the literature, later proved to be artefacts. For instance, an impossible transition [100]–[111] ‘discovered’ by Shimotomai et al. (1980) in PrFe2 (a quick glance at Fig. 3.22 is sufficient to conclude that [100] cannot be the low-T phase in a spontaneous SRT). Or, e.g. an incomplete transition [100]–[110] in YbFe2 (Meyer et al., 1981), where the easy direction ‘slightly deviates’ from [100] above T ≈ 50 K. Apart from the afore-mentioned fact that [110] can only be the low-T phase in a spontaneous SRT and [100] can only be the high-T one, the general theory (Section 3.1) states that the orientation angle always changes sharply near an SRT, even if the latter is of second order (Fig. 3.12a). It is like an airplane, which cannot take off or land ‘slightly’. Most tortuous was the way to the truth in the case of HoFe2 . An early review by Taylor (1971, Table 7 thereof) gave the correct easy magnetisation direction, [100], without any SRT. This view was soon reiterated by Atzmony et al. (1972), who interpreted the Mössbauer spectrum of HoFe2 at T = 4.2 K as being characteristic of the [100] phase. Had the authors seen the above Conclusion 2, they would have put a full stop at this point. The [100] phase has the simplest, most reliably identified Mössbauer spectrum, our Conclusion 2 is therefore quite useful. Anyhow, this was not to be, and during the decade of the 1970’s articles of the same authors were coming out (Atzmony and Dublon, 1977 and references therein), claiming having observed an SRT in HoFe2 at around 14 K. The disproval finally came in the form of a direct specific heat measurement (Germano et al.,

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1979), which found in that temperature range no anomaly characteristic of a phase transition. To be fair to Atzmony et al., their version of the events was not impossible, but rather improbable. Between T = 0 and 14 K the ratio B8(6) (x)/B8(4) (x) in Eq. (3.14) changes by as little as a quarter of a percent [for HoFe2 , x ≈ 750 K/T (Kuz’min, 2001)]. In order for the SRT at about 14 K to become reality, nature would have to set the ratio b4 /|b6 | within a very narrow interval immediately above –138.6. In reality, b4 /|b6 | ≈ –184 (Germano et al., 1979). The controversy around the SRT in HoFe2 is typical of the state of the theory in the 1970’s. The basics of setting and diagonalising the RE Hamiltonian were well known by then, whereas the approximations introduced in this Chapter were not. Thus, it was unknown that the sequence of phases is determined by a sole quantity—the ratio b4 /|b6 |—and does not depend on the exchange field at all. That is, out of the three disposable model parameters only one is relevant to deciding if a spontaneous SRT is to take place. The presence of irrelevant parameters in the early calculations could not bring about but confusion. Also unsound were the attempts to ‘explain’ the complex Mössbauer spectra observed in some REFe2 by way of intermediate low-symmetry orientations of the easy magnetisation direction (Atzmony and Dariel, 1974). Such an explanation involves necessarily a large anisotropy constant of eighth order. The linear theory— whereby this and all higher-order anisotropy constants are strictly nil—is admittedly an approximation. It is, however, a well-founded approximation, so what is forbidden by it can only be small. A rather more plausible but prosaic explanation could be that the samples investigated by Atzmony and Dariel (1974) were not single-phase. The theory developed in this subsection and expressed graphically in Fig. 3.22 is not limited to the cubic Laves phases. Without major modifications it applies e.g. to the RE6 Fe23 compounds. One subtlety needs to be taken into consideration, however: the local symmetry of the 24e sites occupied by the RE in RE6 Fe23 is not cubic, but rather tetragonal, C4v . Therefore, five nonzero CF parameters are allowed, cf. Eq. (1.32). Equations (3.12) are still valid, provided that linear combinations of 4th- and 6th-order CF parameters, 121 (7B40 + B44 ) and 241 (3B60 – B64 ), are substituted for b4 and b6 , respectively. These combinations arise in the process of averaging over the 24e sites with differently oriented local 4-fold axes (parallel to [100], [010], and [001]). The second-order CF parameter B20 is averaged out completely. Figure 3.22 is then valid too, provided the ordinate is defined as (14B40 + 2B44 )/|3B60 – B64 |.

4. Conclusion We are about to close this Chapter about crystal-field effects in 3d-4f intermetallics. From the subject of CF on REs we moved on to magnetocrystalline anisotropy and further on to SRTs. En route we touched upon the influence of the CF on the magnetic moment of the RE. Of course, the narrow path we took does


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not cover the whole area of CF-related phenomena. For example, magnetostriction can be described—similarly to anisotropy constants—by expressions involving generalised Brillouin functions (Kuz’min, 1992). However, in this Chapter we deliberately did not enter into the topic of magnetostriction. It will be exhaustively covered in an extensive monograph by Professor A. del Moral, due to appear shortly. Likewise, the route towards SRTs is not but one of many ways to proceed from the subject of magnetic anisotropy. Other possible connections include micromagnetism, ac susceptibility, coercivity, magnetic resonance etc. This Chapter is addressed primarily to experimentalists. At the intuitive level, most of them would be well familiar with the physics of the phenomena discussed above. They might be less forthcoming when it comes to committing themselves to a quantitative estimate, for the obvious reason that a computer program able to ‘take account of everything’ is hard to come by. A message we tried to get across is that taking everything into account is not always necessary. When valid, a suitable approximation may offer the invaluable advantages of a concise analytical expression—greater transparency and simplicity of calculations. This turned out particularly well in the case of the J -mixing effect, Section 2.9, where expressions suitable for back-of-the-envelope calculations (2.123, 2.126) were obtained. Another example worthy of mention is the newly developed in Section 3.3 theory of spontaneous SRTs in exchange-dominated cubic magnets. Its main statement is ultimately simple: as temperature varies, the system goes through a sequence of (at the most two) phases which is unambiguously determined by a single quantity—a ratio of fourth- and sixth-order CF parameters. As regards uniaxial magnets, there the main results can be formulated as the ‘Stevens αJ , βJ and γJ rules’. Even though they do not constitute an accomplished theory, these rules are nonetheless binding necessary conditions, to the extent that their violation is a nearly certain sign of a mistake. It was also graphically shown how the classical phase diagram of a uniaxial magnet (Fig. 3.15) is modified when a third anisotropy constant is allowed for, establishing a simple visual relation between Figs. 3.15 and 3.17. Last but not least, the interplay of the 3d-4f exchange and the CF in the expression for the leading anisotropy constant K1 , was shown to take a particularly transparent form at high T (2.42) and also when T → 0 (2.107). Our goal was to bring all these simple findings to the notice of workers in the field of magnetic materials.

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CHAPTER

FOUR

Magnetocaloric Refrigeration at Ambient Temperature Ekkes Brück *

Contents List of Symbols and Abbreviations 1. Brief Review of Current Refrigeration Technology 2. Introduction to Magnetic Refrigeration 3. Thermodynamics 4. Materials 4.1 Gd metal and alloys 4.2 Gd5 (Ge,Si)4 and related compounds 4.3 La(Fe,Si)13 and related compounds 4.4 MnAs based compounds 4.5 Heusler alloys 4.6 Fe2 P based compounds 4.7 Other Mn intermetallic compounds 4.8 Amorphous materials 4.9 Manganites 5. Comparison of Different Materials and Miscellaneous Measurements 6. Demonstrators and Prototypes 7. Outlook Acknowledgements References

237 237 239 241 247 247 248 253 256 259 262 265 267 269 270 274 280 281 281

Abstract Modern society relies on readily available refrigeration. Magnetic refrigeration has three prominent advantages compared to the most commonly used compressor-based refrigeration. First there are no harmful gasses involved, second it may be built more compact as the working material is a solid and third magnetic refrigerators generate much less noise. Recently a new class of magnetic refrigerant-materials for room-temperature applications was discovered. These new materials have important advantages over existing magnetic coolants: They exhibit a large magnetocaloric effect (MCE) in conjunction with a magnetic phase-transition of first order. This MCE is, larger than that of Gd metal, which *

Department of Mechanical Engineering, Federal University of Santa Catarina, Florianopolis SC, Brazil and Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands E-mail: [email protected]



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E. Brück

is used in the demonstration refrigerators built to explore the potential of this evolving technology. In the present review we compare different materials, however, concentrating on transition metal containing compounds, as we expect that the limited availability of Rare-earth elements will hamper the industrial applicability. Recently also more and more demonstrators and prototypes are being developed. We compare the different concepts and discuss some important design issues.

Key Words: magnetic refrigeration, transition metal compounds, magnetic entropy, magnetocaloric effects

Magnetocaloric Refrigeration at Ambient Temperature

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List of Symbols and Abbreviations AMR Active magnetic regenerator A Helmholtz free energy α thermal expansion coefficient Brillouin function BJ maximal applied field Bmax C Curie constant effective Curie constant C* heat capacity at iso-magnetic intensity CH magnetic contribution to specific heat Cm COP coefficient of performance χ magnetic susceptibility DSC differential scanning calorimetry B magnetic field change from 0 to Bmax Sm magnetic entropy change Tad adiabatic temperature change g Lande’s-factor H magnetic intensity externally applied field H0 J quantum number of angular momentum k Boltzmann constant Bohr magneton μB M magnetization saturation magnetization Ms MCE Magnetocaloric effect N numbers of magnetic moments Weiss-field constant NW S entropy T absolute temperature Curie temperature TC effective Curie temperature T0* Curie temperature if the lattice was not compressible T0 U internal energy magnetic contribution to internal energy Um V volume volume in absence of exchange interaction V0 W magnetic work Z partition function

1. Brief Review of Current Refrigeration Technology The art and science of refrigeration is concerned with the cooling of matter to temperatures lower than those available in the surroundings at a particular time and place.

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Refrigeration can be achieved in many different ways. The principal methods of refrigeration, which have found practical use in industry and consumer products, are: (a) (b) (c) (d)

vapor compression, vapor absorption, air cycle, thermo-electric.

Vapor-compression refrigeration systems are the most common refrigeration systems in use today. Like in vapor absorption systems, the refrigerating effect is produced by making a volatile fluid boil at a suitably low temperature. The vast majority of cooling devices of all sizes from domestic refrigerators to large industrial systems use the vapor compression principle and, basically, this is owed to the better costto-benefit ratio attainable from a vapor compression installation in comparison with other systems. Absorption refrigeration systems are used in large chemical plants, in air-conditioning and in some domestic refrigerators. Because it needs an input of heat at a moderately high temperature to drive it, it finds applications where such heat is readily available or where mechanical power is not available. With the advent of cogeneration plants, absorption refrigeration is experiencing a renewal but limited to this field, for a review see (Srikhirin et al., 2001). Efficiency attainable from absorption systems is not competitive to that one attained from a vapor compression system. Air cycle refrigeration, in which the temperature of air is reduced by an expansion process in which work is done by the air, was used for many years as the principal method of refrigeration at sea, chiefly on account of the inherent safety of the method. The open cycle cold air machine or heat pump may seem very attractive by its simplicity and environmental advantage, and numerous attempts have been made over the years to revive the idea, eliminating some of its drawbacks by using turbo or other high speed rotary machinery, for a review see (Gigiel, 1996). The problem of excessive power consumption remains, however. It is clear that the open cold air system has little chance of gaining any importance for refrigeration or heat pumps in the normal temperature range unless a significant break-through should occur and this does not seem very likely at the present time. Thermo-electric refrigeration works on the principle of the Peltier effect; i.e. the cooling effect produced when an electric current is passed through a junction of two dissimilar metals. With the materials so far available its efficiency is rather low, but it has many uses in circumstances where efficiency does not matter much, as in very small refrigerators for specimens on microscope stages, instruments for measuring dew point, food and liquid storage for camping and a few others. Thermoelectrics, despite the billion dollars in research spent to date, have not cracked 10% in efficiency though they may achieve as high as 20% to 30% if recent reports are true, for a review see (Riffat and Ma, 2003). Still, the technology is far less efficient than vapor compression systems. The design of a refrigeration system is a type of problem of which the solution involves many considerations. Design invariably requires the critical evaluation of the solutions by considering factors such as economics, safety, reliability, and envi-


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ronmental impact, before choosing a course of action. The vapor compression cycle has dominated the refrigeration market to date because of its advantages: high efficiency, low toxicity, low cost, and simple mechanical embodiments. Perhaps this is because as much as 90% of the worlds heat pumping power; i.e. refrigeration, water chilling, air conditioning, various industrial heating and cooling processes among others, is based on the vapor compression cycle principle. In recent years environmental aspects have been becoming an increasingly important issue in the design and development of refrigeration systems. Especially in vapor compression systems, the banning of CFCs and HCFCs because of their environmental disadvantages has made way for other refrigeration technologies which until now have been largely ignored by the refrigeration market. As environmental concerns grow, alternative technologies which use either inert gasses or no fluid at all become attractive solutions to the environment problem. A significant part of the refrigeration industry R&D expenditures worldwide is now oriented towards the development of such alternative technologies to the replacement of vapor compression systems in a mid- to long-term perspective. One of these alternatives is magnetic refrigeration.

2. Introduction to Magnetic Refrigeration Magnetic refrigeration, based on the magnetocaloric effect (MCE), has recently received increased attention as an alternative to the well-established compression-evaporation cycle for room-temperature applications. Magnetic materials contain two energy reservoirs; the usual phonon excitations connected to lattice degrees of freedom and magnetic excitations connected to spin degrees of freedom. These two reservoirs are generally well coupled by the spin lattice coupling that ensures loss-free energy transfer within millisecond time scales. An externally applied magnetic field can strongly affect the spin degree of freedom that results in the MCE. In the magnetic refrigeration cycle, depicted in Fig. 4.1, initially randomly oriented magnetic moments are aligned by a magnetic field, resulting in heating of the magnetic material. This heat is removed from the material to the ambient by heat transfer. On removing the field, the magnetic moments randomise, which leads to cooling of the material below ambient temperature. Heat from the system to be cooled can then be extracted using a heat-transfer medium. Depending on the operating temperature, the heat-transfer medium may be water (with antifreeze) or air, and for very low temperatures helium. The cycle described here is very similar to the vapour compression refrigeration cycle: on compression the temperature of a gas increases, in the condenser this heat is expelled to the environment and on expansion the gas cools below ambient temperature and can take up heat from the environment. In contrast to a compression cycle the magnetic refrigeration cycle can be performed quasi static which results in the possibility to operate close to Carnot efficiency. Therefore, magnetic refrigeration is an environmentally friendly cooling technology. It does not use ozone depleting chemicals (CFCs), hazardous chemicals

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Figure 4.1 Schematic representation of a magnetic-refrigeration cycle, which transports heat from the heat load to the ambient. Left and right depict material in low and high magnetic field, respectively.

(NH3 ), or greenhouse gases (HCFCs and HFCs). The difference between vapourcycle refrigerators and magnetic refrigerators manifests itself also is the amount of energy loss incurred during the refrigeration cycle. From thermodynamics it appears feasible to construct magnetic refrigerators that have very high Carnot efficiency compared to conventional vapour pressure refrigerators (Brown, 1976; Steyert, 1978). This higher energy efficiency will also result in a reduced CO2 release. Current research aims at new magnetic materials displaying larger magnetocaloric effects, which then can be operated in fields of about 2 T or less, that can be generated by permanent magnets. The heating and cooling described above is proportional to the change of magnetization and the applied magnetic field. This is the reason that, until recently, research in magnetic refrigeration was almost exclusively conducted on superparamagnetic materials and on rare-earth compounds, see the earlier review in this Handbook (Tishin, 1999). For room-temperature applications like refrigerators and air-conditioners, compounds containing manganese or iron should be a good alternative. Manganese and iron are transition metals with high abundance. Also, there exist in contrast to rare-earth compounds, an almost unlimited number of manganese and iron compounds with critical temperatures near room temperature. However, the magnetic moment of manganese generally is only about half the size of heavy rare-earth elements and the magnetic moment of iron is even less. Enhancement of the caloric effects associated with magnetic moment alignment may be achieved through the induction of a first order phase-transition or better a very rapid change of magnetisation at the critical temperature, which will bring along a much higher efficiency of the magnetic refrigerator. In combination with currently available permanent magnets (Dai et al., 2000a) based on modern rare-earth transition-metal compounds (Kirchmayr, 1996), this opens the path to the development of small-scale magnetic refrigerators, which no more rely on rather costly and service-intensive superconducting magnets. Another prominent advantage of magnetocaloric refrigerators is that the cooling power can be varied by scaling from


241

milliwatt to a few hundred watts or even kilowatts. To increases the temperature span of the refrigerator, in comparison with the temperature change in a single cycle, all demonstrators or prototypes nowadays are based on the active magnetic regenerator design (Barclay and Steyert, 1981). In the first section we will discuss the thermodynamics of magnetic refrigeration and some numerical models that are developed to simulate the performance characteristics of magnetocaloric devices. Then we discuss recent developments in magnetocaloric materials and finally we shall discuss several recent demonstrators and prototypes.

3. Thermodynamics Recently a description of the thermodynamics of a refrigeration cycle was given starting with the first law of thermodynamics (Kitanovski and Egolf, 2006). Alternatively one may start from a microscopic view on the atomic magnetic moments in a solid in combination with textbook statistical mechanics e.g. (Pathria, 1972). With J the quantum number of angular momentum, a magnetic field will lift the (2J + 1) degeneracy of the eigenstates. At finite temperatures, thermal agitation prevents that only the eigenstates with lowest energy are occupied. For noninteracting magnetic moments like in a simple paramagnetic salt it has been shown that the partition function of the system is given by N 2J + 1 1 Z = sinh (1) x / sinh x , 2J 2J where x = gμB J H /(kT ), g is known as Lande’s-factor, μB is the Bohr magneton, k is the Boltzmann constant, H is the magnetic intensity, T is the absolute temperature, and N is the numbers of the magnetic moments. Using Eq. (1) and Helmholtz free energy A = –kT ln Z, we can calculate the entropy S, internal energy U , magnetization M and heat capacity CH at iso-magnetic intensity as, x ∂A 2J + 1 x – ln sinh – xBJ (x) S=– (2) = Nk ln sinh ∂T H 2J 2J U = A + T S = –NkT xBJ (x) (3) ∂A = NgμB J BJ (x) M=– (4) ∂H T and

CH =

∂U ∂T

= –Nkx

H

2

2J + 1 2J

2

2

csch

2 1 2J + 1 1 2 x – x csch 2J 2J 2J

(5)

242

where

E. Brück

2J + 1 1 1 2J + 1 coth x – coth x BJ (x) = 2J 2J 2J 2J

is the Brillouin function that varies between 0 and 1 for x = 0 and x = ∞, respectively. When x 1, Eq. (4) becomes the Curie law (Zemansky, 1968; Vonsovskii, 1974; Buschow and de Boer, 2003), CH (6) , T where C = Ng 2 μ2B J (J + 1)/3k is the Curie constant. This equation gives the well known inverse proportionality of the magnetic susceptibility χ = M /H . When the argument of the Brillouin function is very large, thus either at low temperatures and or high magnetic field, the magnetization will saturate to the maximal value MS M=

MS = NgμB J .

(7)

As initially stated the above is valid for noninteracting magnetic moments. When the distance between magnetic moments is small, the Pauli exclusion principle, which states that two identical fermions may not have the same quantum states, results in interaction between magnetic moments. Heisenberg introduced a model to describe this exchange interaction on microscopic scale. The Heisenberg exchange Hamiltonian may be written in the form

Hexch = – (8) 2Jij Si · Sj i<j

the summation extends over all magnetic moment pairs in the crystal lattice. For positive values of the exchange constant Jij one finds parallel alignment else antiparallel. Ferromagnetism is observed for positive exchange interactions below a critical temperature. The exchange interaction can be regarded as effective field acting on the moments. This field is produced by the surrounding magnetic moments and called molecular field. As the size of the surrounding moments is proportional to the magnetization, the molecular field Hm is written as Hm = NW M

(9)

with NW the Weiss-field constant, that was already introduced in the early 20th century long before the development of quantum physics. The total magnetic field experienced by a magnetic material is thus the sum of the externally applied field H0 and the internal field H = H0 + Hm

(10)

and Eq. (6) needs to be rewritten as M=

C (H0 + NW M) T

(11)

243


thus in the presence of ferromagnetic interaction much lower fields are sufficient to saturate the magnetization. The magnetic susceptibility is given by χ=

C C = T – NW C T – TC

(12)

this is the Curie–Weiss law, where TC is the Curie temperature. Below the Curie temperature spontaneous magnetization is observed. For most materials, the phase transition from the paramagnetic state to the ferromagnetic state is found to be of second order. This means that the temperature dependence of the first derivative of the free energy (S, M, V ) is continuous and the second derivative of the free energy (CH , χ , thermal expansion α) is discontinuous. Some materials like MnAs however, show a magnetic phase transition of first order, thus the volume changes abruptly at the critical temperature. To account for this, Bean and Rodbell (1962) assumed a strong distance dependence of the exchange energy. Within a molecular field model they introduced an additional parameter the volume dependent TC . TC = T0 1 + β(V – V0 )/V0 (13) here TC is the Curie temperature, T0 would be the Curie temperature if the lattice was not compressible, ß is the slope of the dependence of TC , V is the volume and V0 would be the volume in absence of exchange interaction. As pointed out by de Blois and Rodbell (1963), the Curie temperature will depend on the thermal expansion and the Curie–Weiss law will be modified χ=

C* C C = = T – TC T – T0 (1 + αβT ) T – T0*

(14)

where the effective Curie constant C * and the effective Curie temperatures T0* are given by C* =

C 1 – αβT0

and

T0* =

T . 1 – αβT0

(15)

Starting from experimentally determined parameters, the temperature and field dependence of the magnetization of MnAs were calculated using this model (Bean and Rodbell, 1962; de Blois and Rodbell, 1963). Another ansatz that accounts for magnetic ordering with a first order phase transition is the model of itinerant metamagnetism (Moriya and Usami, 1977). In this model spin fluctuations are treated in a Landau–Ginzburg type of approach, including higher order terms. For certain ranges of parameters a first order phase transition is observed. This situation may be considered as competing ferro- and antiferro-magnetic interactions. This model also produces a strong volume dependence of TC (Yamada et al., 2002a, 2002b). From Eq. (4) it is obvious, that the work necessary to magnetize a unit volume of a material is M W =– (16) μ0 H dM 0

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Figure 4.2 Schematic representation of the effect of a magnetic field on the entropy of a material. The response in a cooling cycle will lie in between the isothermal (A–C) and adiabatic (D–C) process.

in the same way one finds within the molecular field model for a ferromagnetic material a contribution to the internal energy per unit volume. M μ0 Hm dM Um = – (17) 0

with Hm = NW M the molecular field generated by the exchange interaction. The specific heat due to the spontaneous magnetization of a ferromagnet is thus dM 2 dUm = –1/2μ0 NW . (18) dT dT At low temperatures the magnetization M is usually almost temperature independent, therefore the magnetic contribution to the specific heat is very small. The largest magnetic contribution to the specific heat occurs close to TC where M strongly varies. Near TC also an applied magnetic field has an intense effect on M and on the temperature dependence of the magnetization in constant field. This then will change the magnetic contribution to the internal energy U . Under adiabatic conditions this will result in a change of temperature T (line D–C in Fig. 4.2). These changes in internal energy or temperature due to a magnetic field change are called magnetocaloric effect. The magnetocaloric effects are not restricted to the magnetically ordered state but also in the paramagnetic state they are observed. Equations (4) and (16) are valid in any magnetic state and thus magnetic work will change the total energy of the system. The temperature change due to the application of a magnetic field may be recorded directly. There exist nowadays a few experimental setups to do this direct measurement. Either an adiabatically mounted sample is exposed to a varying magnetic field or the Cm =


245

sample is quickly inserted into a high field region (Dankov et al., 1997; Giguere et al., 1999b; Tishin, 1999; Hu et al., 2003). Alternatively one may perform specific heat and magnetization measurements in different applied magnetic fields. For magnetization measurements made at discrete temperature intervals, Sm can be calculated by means of

Mi+1 (Ti+1 , B) – Mi (Ti , B) Sm (T , B) = (19) B, Ti+1 – Ti i where Mi+1 (Ti+1 , B) and Mi (Ti , B) represent the values of the magnetization in a magnetic field B at the temperatures Ti+1 and Ti , respectively. On the other hand, the field-induced magnetic entropy change can be obtained more directly from a calorimetric measurement of the field dependence of the heat capacity and subsequent integration: T C(T , B) – C(T , 0) Sm (T , B) = (20) dT , T 0 where C(T , B) and C(T , 0) are the values of the heat capacity measured in a field B and in zero field, respectively. It has been confirmed that the values of Sm (T , B) derived from the magnetization measurement coincide with the values from calorimetric measurement (Gschneidner et al., 1999; Tegus et al., 2002d). The adiabatic temperature change can be integrated numerically using the experimentally measured or theoretically predicted magnetization and heat capacity B ∂M T Tad (T , B) = – (21) dB . ) C(T , B ∂T 0 B In practice the adiabatic temperature change derived from direct measurements is generally somewhat smaller than the one derived from Eq. (21). The reason for this is probably because the specific heat near the phase transition is a function which is heavily dependent on temperature and field and some interpolation is inevitable but difficult. Recently, magnetic entropy changes are also calculated theoretically either based on local moment descriptions in the Bean Rodbell model or on itinerant electron descriptions (von Ranke et al., 2000, 2001, 2004; Yamada and Goto, 2003, 2004; de Oliveira and von Ranke, 2005). For the degree of agreement between experiment and calculations, the starting point does not matter too much. The mean field approaches all give reasonable agreement though the shape of the temperature dependence of the magnetic entropy change is generally not well reproduced. However, this shortcoming of the mean field approach is well known, as it does not account for so-called short-range ordering phenomena and local fluctuations. To account for the colossal magnetocaloric effect, von Ranke et al. added a lattice contribution to the magnetic entropy change (von Ranke et al., 2005, 2006). This contribution is found to be due to a change in Debye temperature when the crystal structure changes. The model put forward parameterizes this via the Grüneisen parameter that connects the thermal expansion and the lattice specific heat. Also for this model description, the magnitude of the effect is well reproduced

246

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Figure 4.3 T –S diagram representing the Brayton and Ericsson cycles (see text).

but the experimentally determined shape of the temperature dependence of the magnetic entropy change is quite different. Magnetic refrigeration near room temperature will always involve a cycle process in which the material will be magnetized and demagnetized. The cycles which are most discussed in literature either consist of two isomagnetic field and two isothermal processes (Ericsson) or two isomagnetic field and two adiabatic processes (Brayton) or two isothermal processes and two iso-magnetization processes (Stirling). The former two cycles are depicted in the T –S diagram see Fig. 4.3 where H1 < H2 . In the Brayton cycle the processes 5–1 and 3–4 are the isofield processes and 1–3 and 4–5 are the adiabatic processes. For the Ericsson cycle 6–2 and 3–4 are the isofield processes and 2–3 and 4–6 the isothermal processes. Regeneration takes place during the isofield processes. Today the literature theoretically discussing various aspects of these cycles is quite numerous see (Steyert, 1978; Huang and Teng, 2004; Lin et al., 2004; Allab et al., 2005; Kitanovski and Egolf, 2006; Xia et al., 2006; Yu et al., 2006; Zhang et al., 2006c). The performance of Gd in a Stirling cycle was discussed already 30 years ago by Steyert (Steyert, 1978). Huang et al. compare the Brayton and Ericson cycle in a numerical model based on the Brillouin function description of a magnetic material (Huang and Teng, 2004). Lin et al. consider the effect of irreversibility in the Stirling cycle with regeneration (Lin et al., 2004). Allab et al. implement the effect of finite heat transfer in their numerical model (Allab et al., 2005). Kitanovski et al. discuss various aspects of the thermodynamics involved in magnetic cooling (Kitanovski and Egolf, 2006). Xia et al. discuss irreversibility in the Ericsson cycle with regeneration (Xia et al., 2006). Yu et al. consider a paramagnetic material in an Ericsson cycle and study the effect of increasing the applied field on the temperature span and the performance of the refrigerator (Yu et al., 2006). Zhang et al. discuss the effect of irreversibility on a two stage Brayton cycle (Zhang et al., 2006c). Next to specific cycles also more general effects like demagnetizing effects (Peksoy and Rowe, 2005) and hysteresis (Basso et al., 2005, 2006) are considered theoretically.


247

4. Materials In the following section we shall discuss various materials that have been put forward to be used in magnetic refrigerators. From Eq. (21) it is obvious that a strong temperature dependence of the magnetization will result in a large magnetocaloric effect. This is true for a paramagnet near zero temperature and for a ferromagnet near the Curie temperature. However, we will see that there are still some other possibilities like a transition from antiferromagnetic to ferromagnetic. What was realized only quite recently, is that a material with a combined magnetic and structural transition, often exhibits a very large magnetocaloric effect. We will start with Gd as this is nowadays considered as a reference material, though it is rather unlikely that this will be employed in future refrigerators operating near room temperature.

4.1 Gd metal and alloys The only elemental ferromagnet with a Curie temperature near room temperature is Gd. The ordered magnetic moment of Gd at low temperatures is quite high 7.63 μB and magnetic ordering occurs below 293 K with a second order phase-transition. This metal is currently used as magneto refrigerant in various magnetocaloric refrigeration demonstrator prototypes (see section 6). As pointed out by Dan’kov et al. (Dan’kov et al., 1998), the Curie temperature and the magnetocaloric properties of Gd strongly depend on the impurity content. As the impurity content is generally quoted as weight percent, different batches with nominal the same impurity content may yet have quite different properties. It appears, that especially the interstitial impurities H, C, N, O and F, which are rather light and therefore hardly count in the weight percentage, weaken the exchange interaction, which leads to an reduction in Curie temperature and enhanced spin fluctuations below and above the Curie temperature. The importance of knowing the type of impurities is further demonstrated by the same authors when studying the time dependence of the magnetocaloric effect. For high purity samples they show that the adiabatic temperature change is the same when measured in quasi-static fields or in pulsed fields with full loop duration of 0.2 s, the latter simulates a refrigerator operating at 5 Hz. However, commercial Gd with a high C content exhibits in pulsed field measurement only about 60% of the maximal adiabatic temperature change compared to high purity materials. Additionally to the reduction in adiabatic temperature change the whole effect is shifted a few K to lower temperatures. On the other hand, pulsed field measurements, on a sample with a higher overall impurity content that contains mainly oxides, almost reproduce the results found for high purity materials. The thermal conductivity of Gd also depends on the impurity content (Glorieux et al., 1995). This should be especially taken into account when the performance of a refrigerator is simulated. It is not always possible to just plug in some literature data and start simulating. The high reactivity of Gd can be accounted for by adding some NaOH into the water in the heat exchanger (Zhang et al., 2005b).

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As the magnetocaloric effect of a material is large only in the vicinity of the Curie temperature, one needs more than one material if one wants to increase the temperature span of operation. Other rare-earth metals have lower ordering temperatures, however the magnetic ordering is often rather complicated and the associated magnetocaloric effects are often not very large. A way out from this is Gd that is doped with heavy rare-earth, these alloys show a lower ferromagnetic ordering-temperature with yet large ordered magnetic moments. A Gd alloy with 26% Tb orders at 280 K and shows a somewhat larger MCE compared to pure Gd (Gschneidner et al., 2000). A Gd alloy with 50% Dy orders at 230 K and the MCE reported for this alloy is about the same as for pure Gd (Dai et al., 2000b). Dai et al. also studied further addition of Nd to reach even lower temperatures, however the Nd doped alloys show strong time dependence of the magnetic response which may obstruct application. There also exist several compounds of Gd with nonmagnetic elements that have magnetic ordering temperatures above room temperature. The structural and magnetocaloric data of these are included in Table 4.1. In the case of Gd7 Pd3 also the eutectic composition Gd76 Pd24 has been proposed as composite magnetic refrigerant (Canepa et al., 2002, 2005).

4.2 Gd5 (Ge,Si)4 and related compounds The search for alternatives to rare-earth elements and their alloys lead to the discovery of a sub-room temperature giant-MCE in the ternary compound Gd5 (Ge1–x Six )4 (0.3 ≤ x ≤ 0.5) (Pecharsky and Gschneidner, 1997b). Since then there is a strongly increased interest from both fundamental and practical points of view to study the MCE in these materials (Choe et al., 2000; Morellon et al., 2000). The most prominent feature of these compounds is that they undergo a first-order structural and magnetic phase transition, which leads to a giant magnetic field-induced entropy change, across their ordering temperature. We here therefore will discuss to some extend the structural properties of these compounds. At low temperatures for all x Gd5 (Ge1–x Six )4 adopts an orthorhombic Gd5 Si4 type structure (Pnma) and the ground state is ferromagnetic (Pecharsky et al., 2002). However, at room temperature depending on x three different crystallographic phases are observed. For x > 0.55 the aforementioned Gd5 Si4 structure is stable, for x < 0.3 the materials adopt the Sm5 Ge4 -type structure with the same space group (Pnma) but a different atomic arrangement and a somewhat larger volume, finally in between these two structure types the monoclinic Gd5 Si2 Ge2 type with space group (P1121 /a) is formed, which has an intermediate volume. The latter structure type is stable only below about 570 K where again the orthorhombic Gd5 Si4 -type structure is formed in a first-order phase transition (Mozharivskyj et al., 2005). The thermal evolution of the structural change is depicted in Fig. 4.5. As one may guess, the three structure types are closely related (see Fig. 4.4) (Pecharsky and Gschneidner, 1997c); the unit cells contain four formula units and essentially only differ in the mutual arrangement of identical building blocks which are either connected by two, one or no covalent-like Si–Ge bonds, resulting in successively increasing unit-cell volumes. The giant magnetocaloric effect is observed for the

Structural, magnetic and magnetocaloric data of Gd based intermetallic compounds. |S| at B = 2 T if not indicated differently

Compound

Structure type

Space group

Tc (K)

Gd Gd0.74 Tb0.26

HCP HCP

P63/mmc P63/mmc

294 280

Gd0.5 Dy0.5 Gd7 Pd3 Gd4 Bi3 Gd5 Si4 Gd5 (Si0.5 Ge0.5 )4 Gd5 (Si0.5 Ge0.5 )4 Gd5 (Si0.5 Ge0.5 )4 Gd5 (Si0.425 Ge0.575 )4 Gd5 (Si0.425 Ge0.575 )4 Gd5 (Si0.425 Ge0.575 )4 Gd5 (Si0.5 Ge0.5 )4 Gd5 (Si0.45 Ge0.55 )4 Gd5 (Si0.365 Ge0.635 )4 Gd5 (Si0.3 Ge0.7 )4 Gd5 (Si0.25 Ge0.75 )4

HCP Th7 Fe3 Th3 P4 Gd5 Si4 Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 Si4 Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 ) Gd5 (Si2 Ge2 )

P63/mmc P63mc I43d Pnma P1121 /a P1121 /a Pnma P1121 /a P1121 /a P1121 /a P1121 /a P1121 /a P1121 /a P1121 /a P1121 /a

230 334 332 346 272 275 305 246 246 246 278 257 215 182 155

|S| (J/kg·K)

Comment

5 6 5 2.5 1.5 4.2 27 14 20(5T) 43 47 38 14 17 28 31 38

References Dan’kov et al. (1998) Gschneidner and Pecharsky (2000)

Optimal First Fe doped 2 T//a-axis 2 T//b-axis 2 T//c-axis

Dai et al. (2000b) Canepa et al. (2002) Niu et al. (2001) Spichkin et al. (2001) Pecharsky et al. (2003a) Pecharsky et al. (2003b) Provenzano et al. (2004) Tegus et al. (2002b) Tegus et al. (2002b) Tegus et al. (2002b) Casanova (2004) Casanova (2004) Casanova (2004) Casanova (2004) Casanova (2004)


Table 4.1

249

250

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Figure 4.4 Crystal structure adopted at room temperature in the pseudo-binary system Gd5 Si4 –Gd5 Ge4 (Pecharsky and Gschneidner, 1997c).

Figure 4.5 X-ray diffraction spectra of Gd5 Ge2.4 Si1.6 taken at different temperatures (Duijn, 2000).


251

Figure 4.6 Temperature dependence of the electrical resistivity of an annealed crystal of Gd5 Ge2.4 Si1.6 (Duijn, 2000).

compounds that exhibit a simultaneous paramagnetic to ferromagnetic and structural phase-transition that can be either induced by a change in temperature, applied magnetic field or applied pressure (Morellon et al., 1998a, 2004a). In contrast to most magnetic systems the ferromagnetic phase has a 0.4% smaller volume than the paramagnetic phase which results in an increase of TC on application of pressure with about 3 K/kbar. The structural change at the phase transition brings along also a very large magneto-elastic effect and the electrical resistivity behaves anomalous see Fig. 4.6. The strong coupling between lattice degrees of freedom and magnetic and electronic properties is rather unexpected, because the magnetic moment in Gd originates from spherical symmetric S-states that in contrast to other rare-earth elements hardly couple with the lattice. First principle electronic structure calculations in atomic sphere and local-density approximation with spin-orbit coupling added variationally, could reproduce some distinct features of the phase transition (Pecharsky et al., 2003b). Total energy calculations for the two phases show different temperature dependences and the structural change occurs at the temperature where the energies are equal. There appears a distinct difference in effective exchange-coupling parameter for the monoclinic and orthorhombic phase, respectively. This difference could directly be related to the change of the Fermi-level in the structural transition. Thus the fact that the structural and magnetic transitions are simultaneous is somewhat accidental as the exchange energy is of the same order of magnitude as

252

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the thermal energy at the structural phase-transition. The electrical resistivity and magneto resistance of Gd5 Ge2 Si2 also shows unusual behaviour, indicating a strong coupling between electronic structure and lattice. For several compounds of the series, next to a cusp like anomaly in the temperature dependence of the resistivity, a very large magnetoresistance effect is reported (Morellon et al., 1998b, 2001a; Levin et al., 2001; Tang et al., 2004). In view of building a refrigerator based on Gd5 (Ge1–x Six )4 , there are a few points to consider. The largest magnetocaloric effect is observed considerably below room temperature (see Table 4.1), while a real refrigerator should expel heat at least at about 320 K. Because the structural transition is connected with sliding of building blocks, impurities especially at the sliding interface can play an important role. The thermal hysteresis and the size of the magnetocaloric effect connected with the firstorder phase transition strongly depend on the quality of the starting materials and the sample preparation (Pecharsky et al., 2003a). For the compounds Gd5 (Ge1–x Six )4 with x around 0.5 small amounts of impurities like Al, Bi, C, Co, Cu, Ga, Mn or O may suppress the formation of the monoclinic structure near room temperature. These alloys then show only a phase transition of second order at somewhat higher temperature but with a lower magnetocaloric effect (Pecharsky and Gschneidner, 1997a; Provenzano et al., 2004; Mozharivskyj et al., 2005; Shull et al., 2006). The only impurity that appears to enhance the magnetocaloric effect and increases the magnetic ordering temperature are so far Pb and Sn (Li et al., 2006b; Zhuang et al., 2006). This sensitivity to impurities like carbon, oxygen and iron strongly influences the production costs of the materials which may hamper broadscale application. Next to the thermal and field hysteresis the magneto-structural transition in Gd5 (Ge1–x Six )4 appears to be rather sluggish (Giguere et al., 1999b; Gschneidner et al., 2000). This will also influence the optimal operation-frequency of a magnetic refrigerator and the efficiency. An aspect which is hardly ever taken into consideration is the availability of the components. For this compound Ge will be the limiting ingredient as the worldwide yearly production of Ge amounts to only about 90 metric tons. This Ge is mainly consumed for electronic and optical devices so about 10 t may be available for the production of magnetic refrigerants. Which, limits the yearly production of Gd5 (Ge1–x Six )4 to about 160 t (see Table 4.9). Other R5 (Ge,Si)4 compounds are also found to form in the monoclinic Gd5 Ge2 Si2 type structure and when the structural transformation coincides with the magnetic ordering transition a large magnetocaloric effect is observed (Table 4.2). This is most strikingly evidenced in the experiments of Morellon et al. (2004b) on Tb5 Ge2 Si2 where the two transitions were forced to coincide by application of hydrostatic pressure, which results in a strong enhancement of the magnetic entropy change at the ordering temperature. The magnetic ordering temperatures of other R5 (Ge,Si)4 compounds are all lower than for the Gd compound as expected. For cooling applications below liquid nitrogen temperatures some of these compounds may be interesting.

253


Table 4.2 Structural, magnetic and magnetocaloric data of several heavy rare-earth based intermetallic compounds. |S| at B = 2 T if not indicated differently

|S| (J/kg·K)

Compound

Structure type

Space group

TC (K)

Tb5 Si4

Gd5 Si4

Pnma

225

5.2

Tb5 Si4

Gd5 Si4

Pnma

223

4.5

Tb5 (Si3 Ge)

Gd5 (Si2 Ge2 )

P1121 /a

215

4.5(3T)

Tb5 (Si2 Ge2 )

Gd5 (Si2 Ge2 )

P1121 /a

110

Tb5 (Si2 Ge2 )

Gd5 (Si2 Ge2 )

P1121 /a

116

Tb5 (Si2 Ge2 )

Gd5 (Si2 Ge2 )

P1121 /a

76

Tb5 Ge4

Sm5 Ge4

Pnma

91a

DyTiGe

CeFeSi

P4/nmm

165

0.5

HoTiGe

CeFeSi

P4/nmm

90

1.0

HoTiGe

CeFeSi

P4/nmm

90

2.5//c

TmTiGe

CeFeSi

P4/nmm

15

4.2

10.4 7.8(3T) 12 1.0

References Morellon et al. (2001b) Spichkin et al. (2001) Thuy et al. (2002) Morellon et al. (2001b) Thuy et al. (2002) Tegus et al. (2002c) Morellon et al. (2001b) Tegus et al. (2002b) Tegus et al. (2001) Tegus et al. (2002b) Tegus et al. (2002b)

4.3 La(Fe,Si)13 and related compounds Another interesting type of materials are rare-earth–transition-metal compounds crystallizing in the cubic NaZn13 type of structure. LaCo13 is the only binary compound, from the 45 possible combinations of a rare-earth and iron, cobalt or nickel, that exists in this structure. It has been shown that with an addition of at least 10% Si or Al this structure can also be stabilized with iron and nickel (Kripyakevich et al., 1968). The NaZn13 structure contains two different Zn sites. The Na atoms at 8a and ZnI atoms at 8b form a simple CsCl type of structure. Each ZnI atom is surrounded by an icosahedron of 12 ZnII atoms at the 96i site. In La(Fe,Si)13 La goes on the 8a site, the 8b site is fully occupied by Fe and the 96i site is shared by Fe and Si. The iron rich compounds La(Fe,Si)13 show typical invar behavior, with magnetic ordering temperatures around 200 K that increase to 262 K with lower iron content (Palstra et al., 1983). Thus, though the magnetic moment is diluted and also decreases per Fe atom, the magnetic ordering temperature increases. Around 200 K the magnetic-ordering transition is found to be distinctly visible also in the electrical resistivity, where a chromium-like cusp in the temperature dependence is

254

E. Brück

observed. In contrast to Gd5 Ge2 Si2 this phase transition is not accompanied by a structural change, thus above and below TC the material is cubic. Recently, because of the extremely sharp magnetic ordering transition, the (La,Fe,Si,Al) system was reinvestigated by several research groups and a large magnetocaloric effect was reported (Hu et al., 2000, 2001b; Fujieda et al., 2002). The largest effects are observed for the compounds that show a field- or temperature-induced phase-transition of first order. Unfortunately, these large effects only occur up to about 210 K as the magnetic sublattice becomes more and more diluted. When using standard melting techniques, preparation of homogeneous single-phase samples appears to be rather difficult especially for alloys with high transition metal content. Almost single phase samples are reported when, instead of normal arc melting, rapid quenching by melt spinning and subsequent annealing is employed (Liu et al. 2004, 2005; Gutfleisch et al., 2005). Samples prepared in this way also show a very large magnetocaloric effect. To increase the total magnetic moment partial substitution of Ce for La has been successful (Fujieda et al., 2006a, 2006b). This substitution however, leads next to an enhanced magnetocaloric effect, to a lower magnetic ordering temperature and a broader thermal hysteresis. Small additions of Nd were also studied and were found to increase the ordering temperature, however at 50% Nd the phase transition becomes of second order and the entropy change steeply drops (Zhu et al., 2005). To increase the magnetic ordering temperature without loosing too much magnetic moment, one may replace some Fe by other magnetic transition-metals. Because the isostructural compound LaCo13 has a very high critical temperature substitution of Co for Fe is widely studied. The compounds La(Fe,Co)13–x Alx and La(Fe,Co)13–x Six with x ≈ 1.1 and thus a very high transition-metal content, show a considerable magnetocaloric effect near room temperature (Hu et al., 2001a, 2005; Shen et al., 2004; Proveti et al., 2005). This is achieved with only a few percent of Co and the Co content can easily be varied to tune the critical temperature to the desired value. It should be mentioned however that near room temperature the values for the entropy change steeply drop. The fact that the alloys with the highest Fe content have an antiferromagnetic ground-state indicates that antiferromagnetic direct exchange-interaction plays an important role in these compounds. Taking into account that this occurs at a very high Fe density, one may expect that expansion of the lattice will lead to an increase in ferromagnetic exchange. Hydrogen is the most promising interstitial element. In contrast to other interstitial atoms, interstitial hydrogen not only increases the critical temperature but also leads to an increase in magnetic moment (Irisawa et al., 2001; Fujieda et al., 2002, 2004a; Fujita et al., 2003; Nikitin et al., 2004; Mandal et al., 2005). The lattice expansion due to the addition of three hydrogen atoms per formula unit is about 4.5%. The critical temperature can be increased to up to 450 K, the average magnetic moment per Fe increases from 2.0 μB to up to 2.2 μB and the field- or temperature-induced phase-transition is found to be of first-order for all hydrogen concentrations. This all results for a certain Si percentage in an almost constant value of the magnetic entropy change per mass unit over a broad temperature span see Table 4.3 and Fig. 4.7. Heat capacity measurements on La(Fe,Si) and La(Fe,Si)H in applied magnetic field, confirm a large adiabatic temperature change (Podgornykh and Shcherbakova, 2006). The reduction of the

255


Table 4.3 Magnetic and magnetocaloric data of several NaZn13 type intermetallic compounds. |S| at B = 2 T if not indicated differently

Material La(Fe0.90 Si0.10 )13 La(Fe0.89 Si0.11 )13 La(Fe0.880 Si0.120 )13 La(Fe0.877 Si0.123 )13 LaFe11.8 Si1.2 La(Fe0.88 Si0.12 )13 H0.5 La(Fe0.88 Si0.12 )13 H1.0 LaFe11.7 Si1.3 H1.1 LaFe11.57 Si1.43 H1.3 La(Fe0.88 Si0.12 )H1.5 LaFe11.2 Co0.7 Si1.1 LaFe11.5 Al1.5 C0.1 LaFe11.5 Al1.5 C0.2 LaFe11.5 Al1.5 C0.4 LaFe11.5 Al1.5 C0.5 La(Fe0.94 Co0.06 )11.83 Al1.17 La(Fe0.92 Co0.08 )11.83 Al1.17

Remark

Melt spun

Tmax (K)

Smax (J/kg·K)

Ref.

184 188 195 208 195 233 274 287 291 323 274 185 210 238 255 273 303

28 24 20 14 25 20 19 28 24 19 12 5.5 5.7 5.5 5 4.8 4.5

Fujita et al. (2003) Fujita et al. (2003) Fujita et al. (2003) Fujita et al. (2003) Gutfleisch et al. (2005) Fujita et al. (2003) Fujita et al. (2003) Fujita et al. (2003) Fujita et al. (2003) Fujita et al. (2003) Hu et al. (2002) Wang et al. (2004) Wang et al. (2004) Wang et al. (2004) Wang et al. (2004) Hu et al. (2001a) Hu et al. (2001a)

Figure 4.7 Magnetic entropy change for different LaFe13 based alloys at a field change from 0 to 2 T (Fujita et al., 2003; Wang et al., 2004; Hu et al., 2005).

electronic contribution to the heat capacity observed for the hydrogenated sample however, is in conflict with the model of itinerant metamagnetism for these materials.

256

E. Brück

From the materials cost point of view the La(Fe,Si)13 type of alloys appear to be very attractive. La is the cheapest from the rare-earth series and both Fe and Si are available in large amounts (see Table 4.9). The processing will be a little more elaborate than for a simple metal alloy but this can be optimized. For the use in a magnetic refrigerator next to the magnetocaloric properties also mechanical properties and chemical stability may be of importance. The hydrogenation process of rare-earth transition-metal compounds produces always granular material due to the strong lattice expansion. In the case of the cubic NaZn13 type of structure this does not seem to be the case. At the phase transition in La(Fe,Si)13 type of alloys also a volume change of 1.5% is observed (Wang et al., 2003). If this volume change is performed very frequently the material will definitely become very brittle and probably break into even smaller grains. This can have distinct influence on the corrosion resistance of the material and thus on the lifetime of a refrigerator. The suitability of this material definitely needs to be tested.

4.4 MnAs based compounds MnAs exist similar to Gd5 Ge2 Si2 in two distinct crystallographic structures (Pytlik and Zieba, 1985). At low and high temperature the hexagonal NiAs structure is found and for a narrow temperature range 307 K to 393 K the orthorhombic MnP structure exists. The high temperature transition in the paramagnetic region is of second order. The low temperature transition is a combined structural and ferroparamagnetic transition of first order with large thermal hysteresis. The change in volume at this transition amounts to 2.2% (Fjellvag et al., 1984). The transition from paramagnetic to ferromagnetic occurs at 307 K, the reverse transition from ferromagnetic to paramagnetic occurs at 317 K. Very large magnetic entropy changes are observed in this transition (Kuhrt et al., 1985; Wada and Tanabe, 2001). Similar to the application of pressure (Menyuk et al., 1969; Yamada et al., 2002b) substitution of Sb for As leads to lowering of TC (Wada et al., 2002, 2003), 25% of Sb gives an transition temperature of 225 K (Table 4.4). However, the thermal hysteresis is affected quite differently by hydrostatic pressure or Sb substitution. In Mn(As,Sb) the hysteresis is strongly reduced and at 5% Sb it is reduced to about 1 K. In the concentration range 5 to 40% of Sb TC can be tuned between 220 and 320 K without loosing much of the magnetic entropy change (see Table 4.4) (Morikawa et al., 2004; Wada and Asano, 2005). Direct measurements of the temperature change confirm a T of 2 K/T (Wada et al., 2005b). On the other hand MnAs under pressure shows an extremely large magnetic entropy change (Gama et al., 2004) in conjunction with large hysteresis. A similar effect can be produced at ambient pressure when part of the Mn is substituted by Fe see Fig. 4.8 (De Campos et al., 2006). This effect is larger than what may be expected from aligning the Mn magnetic moments, to account for this excess magnetic entropy change von Ranke et al. introduced a model which involves the difference in phonon spectra for the different crystal structures (von Ranke et al., 2005, 2006). The materials costs of MnAs are quite low, processing of As containing alloys is however complicated due to the biological activity of As. In the MnAs alloy

|S| (J/kg·K)

Structure type

Space group

Tc (K)

M3d (μB /3d)

MnFeGe MnFe0.9 Co0.1 Ge MnFe0.8 Co0.2 Ge MnFe0.7 Co0.3 Ge MnFe0.6 Co0.4 Ge MnFe0.5 Co0.5 Ge MnFe0.4 Co0.6 Ge MnFe0.3 Co0.7 Ge MnFe0.2 Co0.8 Ge MnFe0.15 Co0.85 Ge MnFe0.1 Co0.9 Ge MnCoGe

Ni2 In Ni2 In Ni2 In Ni2 In Ni2 In Ni2 In Ni2 In Ni2 In Ni2 In TiNiSi TiNiSi TiNiSi

P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc Pnma Pnma Pnma

159 173 209 220 223 228 242 249 289 306 340 345

0.97 0.97 1.13 1.13 1.20 1.30 1.51 1.55 2.34 1.97 2.05 2.06

1.6a 1.8a 2.5a 2.9a 3.2a 3.5a 2.9a 4.0a 9.0a 5.3a 5.7a 6.1a

Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006) Lin et al. (2006)

Mn5 Ge2.5 Si0.5 Mn5 Ge2 Si Mn5 Ge1.5 Si1.5 Mn5 GeSi2

Mn5 Si3 Mn5 Si3 Mn5 Si3 Mn5 Si3

P63 /mcm P63 /mcm P63 /mcm P63 /mcm

299 283 258 198

2.56 2.52 2.46 2.36

7.8 7.6 6.9 6.8

Zhao et al. (2006) Zhao et al. (2006) Zhao et al. (2006) Zhao et al. (2006)

Mn5 Ge3 Mn5 Ge2.9 Sb0.1 Mn5 Ge2.8 Sb0.2 Mn5 Ge2.7 Sb0.3

Mn5 Si3 Mn5 Si3 Mn5 Si3 Mn5 Si3

P63 /mcm P63 /mcm P63 /mcm P63 /mcm

298 304 307 312

2.64 2.63 2.60 2.40

9.3 6.6 6.2 5.6

Songlin et al. (2002a) Songlin et al. (2002a) Songlin et al. (2002a) Songlin et al. (2002a)

LaMn1.9 Fe0.1 Ge2 LaMn1.85 Fe0.15 Ge2 LaMn1.8 Fe0.2 Ge2

ThCr2 Si2 ThCr2 Si2 ThCr2 Si2

I4/mmm I4/mmm I4/mmm

310 295 275

1.2 1.2 1.2

1.02b 0.93b 0.88b

Zhang et al. (2006b) Zhang et al. (2006b) Zhang et al. (2006b)

References

257

Compound


Table 4.4 Structural, magnetic and magnetocaloric data of Mn based intermetallic compounds. |S| at B = 2 T if not indicated differently. M3d is the magnetic moment at low temperature per 3d atom

258

Table 4.4

(Continued.)

Compound

Structure type

Space group

Tc (K)

M3d (μB /3d)

(Fe0.9 Mn0.1 )3 C (Fe0.8 Mn0.2 )3 C (Fe0.7 Mn0.3 )3 C

Fe3 C Fe3 C Fe3 C

Pnma Pnma Pnma

305 109 31

1.39 0.62 0.22

Mn3 GaC

CaTiO3

¯ Pm3m

160*

1.2/1.8

MnAs (Mn,Fe)As Mn1+δ As0.8 Sb0.2 MnAs0.75 Sb0.25 Mn1.1 As0.75 Sb0.25 Mn1.5 As0.75 Sb0.25

NiAs NiAs NiAs NiAs NiAs NiAs

P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc P63 /mmc

317 310 250 232 227 204

3.4 – 3.7 3.7 3.3 3.2

|S| (J/kg·K) 3.4 1.8 1.3 15 43 320 26 14c 17c 14c

References Brück et al. (2007) Brück et al. (2007) Brück et al. (2007) Tohei et al. (2003) Nascimento et al. (2006) De Campos et al. (2006) Wada and Asano (2005) Morikawa et al. (2004) Morikawa et al. (2004) Morikawa et al. (2004)

a B = 5 T; b B = 1.8 T; c B = 1.0 T; * T . t

E. Brück


259

Figure 4.8 Magnetic entropy changes of MnAs alloys at a field change from 0 to 5 T (Wada and Asano, 2005; De Campos et al., 2006; Nascimento et al., 2006).

the As is covalently bound to the Mn and would not be easily released into the environment. However, this should be experimentally verified, especially because in an alloy frequently second phases form that may be less stable. The change in volume in Mn(As,Sb) is still 0.7% which may result in aging after frequent cycling of the material.

4.5 Heusler alloys Heusler alloys frequently undergo a martensitic transition between the martensitic and the austenitic phase which is generally temperature induced and of first order. Ni2 MnGa orders ferromagnetic with a Curie temperature of 376 K, and a magnetic moment of 4.17 μB , which is largely confined to the Mn atoms and with a small moment of about 0.3 μB associated with the Ni atoms (Webster et al., 1984). As may be expected from its cubic structure, the parent phase has a low magneto-crystalline anisotropy energy (Ha = 0.15 T). However, in its martensitic phase the compound is exhibiting a much larger anisotropy (Ha = 0.8 T). The martensitic-transformation temperature is near 220 K. This martensitic transformation temperature can be easily varied to around room temperature by modifying the composition of the alloy from the stoichiometric one. The low-temperature phase evolves from the parent phase by a diffusionless, displacive transformation leading to a tetragonal structure, a = b = 5.90 Å, c = 5.44 Å. A martensitic phase generally accommodates the strain associated with the transformation (this is 6.56% along c for Ni2 MnGa) by the formation of twin variants. This means that a cubic crystallite splits up in two tetragonal crystallites sharing one contact plane. These twins pack together in compatible orientations to

260

E. Brück

minimize the strain energy (much the same as the magnetization of a ferromagnet may take on different orientations by breaking up into domains to minimize the magneto-static energy). Alignment of these twin variants by the motion of twin boundaries can result in large macroscopic strains. In the tetragonal phase with its much higher magnetic anisotropy, an applied magnetic field can induce a change in strain. This is the reason why these materials may be used as actuators. Next to this ferromagnetic shape memory effect, very close to the martensitic transition temperature, one observes a large change in magnetization for low applied magnetic fields. This change in magnetization is also related to the magnetocrystalline anisotropy. This change in magnetization is resulting in a moderate magnetic entropy change of a few J/mol · K, which is enhanced when measured on a single crystal (Hu et al., 2001c; Marcos et al., 2002). When the composition in this material is tuned in a way that the magnetic and structural transformation occurs at the same temperature, the largest magnetic entropy changes are observed (Kuo et al., 2005; Long et al., 2005; Zhou et al., 2005b). Recently in the Heusler alloy NiMnSn a large inverse magnetocaloric effect was reported (Krenke et al., 2005b), this effect is related to the increase of magnetization with increasing temperature over the martensitic transition temperature. Substitution of Co for Ni leads to an increase of the transition temperature close to room temperature (see Fig. 4.9). CoMnSb is a half Heusler alloy with a rather high ordering temperature when one considers magnetocaloric applications, the effect of Nb addition on the magnetic properties and magnetocaloric effect (MCE) of CoNbx Mn1–x Sb alloys was investigated recently (Li et al. 2006c, 2007). The Curie temperature of these compounds slightly decreases with Nb substitution. As seen in Table 4.5, Nb substitution strongly lowers the magnetic moment and the MCE of CoMnSb alloy. With increasing Nb content, the magnetic moment decreases linearly and the magnetic phase transition is smeared out over a larger temperature interval. These facts result in a reduction of the magnetic entropy change, but lead to a broader working temperature span. The Fe rich Heusler alloys generally show different behavior. A series of Fe2 MnSi1–x Gex compounds (x = 0–1) was prepared by Zhang et al. (2003) using a mechanically activated solid-state diffusion method. Both X-ray diffraction and differential scanning calorimetry evidenced the presence of an amorphous phase after 10 h of milling. The X-ray data reveal that in the high-temperature annealing the single D03 -type phase can be retained up to 50% substitution of Ge for Si in Fe2 MnSi. A metastable D03 phase is obtained after crystallization of the as-milled amorphous compounds with x > 0.5. High-temperature annealing transforms the low-temperature D03 phase into a single D019 phases (x = 1) or a mixture of D03 and D019 phase (x = 0.6 and 0.8). Low-field thermomagnetic measurements show a moderately sharp ferromagnetic-paramagnetic transition, which becomes enormously broad in higher magnetic fields. The Curie temperature is significantly enhanced when going from the D03 phase to the D019 phase. Neither a magneticfield-induced transition nor a reversible structural transition is observed throughout this compound series. The magnetocaloric effect associated with the magnetic transition is small. This may be illustrated for the compound Fe2 MnSi0.5 Ge0.5 listed in Table 4.5.

Structural, magnetic and magnetocaloric properties of Mn based Heusler alloys and intermetallic compounds with Fe2 P structure

Compound

Structure type

Space group

Tc (K)

Fe2 MnSi0.5 Ge0.5

BiF3

Fm3m

260

Ms (μB /3d) at 5 K 0.93 ∼1.3 ∼1.3 ∼1.3 ∼1.3 ∼1.3

|S| (J/kg·K)

Ref.

0.8

Zhang et al. (2003)

8.65T

Zhou et al. (2005a) Zhou et al. (2005a) Zhou et al. (2005a) Zhou et al. (2005a) Zhou et al. (2005a)

Ni52.9 Mn22.4 Ga24.7 Ni50.9 Mn24.7 Ga24.4 Ni55.2 Mn18.6 Ga26.2 Ni51.6 Mn24.7 Ga23.8 Ni52.7 Mn23.9 Ga23.4

BiF3 BiF3 BiF3 BiF3 BiF3

Fm3m Fm3m Fm3m Fm3m Fm3m

305 272 315 296 338

CoMnSb CoNb0.2 Mn0.8 Sb CoNb0.4 Mn0.6 Sb CoNb0.6 Mn0.4 Sb

MgAgAs MgAgAs MgAgAs MgAgAs

F43m F43m F43m F43m

472 470 465 463

Ni50 Mn35 Sn15 Ni50 Mn37 Sn13

Cu2 MnAl 10M

¯ Fm3m Pnnm

187 303

MnFeP0.45 As0.55 MnFeP0.47 As0.53 Mn1.1 Fe0.9 P0.47 As0.53 MnFeP0.89–x Six Ge0.11 x = 0.22 x = 0.26 x = 0.30 x = 0.33

Fe2 P Fe2 P Fe2 P

P62m P62m P62m

300 293 298

2.0 2.0 2.1

15 15 21

Brück et al. (2005) Brück et al. (2005) Brück et al. (2005)

Fe2 P Fe2 P Fe2 P Fe2 P

P62m P62m P62m P62m

270 292 288 260

2.1 2.1 2.1 2.1

38 41 39 36

Brück et al. (2007) Brück et al. (2007) Brück et al. (2007) Brück et al. (2007)

2.00 3.2 2.3 1.9

3.55T 20.45T 7.05T 15.65T 2.10.9T 1.40.9T 1.20.9T 0.60.9T

Li et al. (2007) Li et al. (2007) Li et al. (2007) Li et al. (2007)

5.8 6.8

Krenke et al. (2005b) Krenke et al. (2005a)


Table 4.5

261

262

E. Brück

Figure 4.9 Magnetic entropy change for a field change from 0 to 2 T for different Heusler alloys. Note that NiMnSn shows the inverse effect (Krenke et al., 2005b; Long et al., 2005).

For magnetocaloric applications the extremely large length changes in the martensitic transition will definitely result in aging effects. It is well known for the magnetic shape-memory alloys that only single crystals can be frequently cycled while polycrystalline materials spontaneously crack and convert to powder after several cycles. For the Ga containing alloys similar to Ge there is only a very limited supply of Ga metal as the worldwide production is of the order of 90 t. As most of the Ga is consumed for GaAs wafers and as dopand in semiconductor industries the yearly production of NiMnGa for transducer or magnetocaloric applications would be limited to about 140 t.

4.6 Fe2 P based compounds The magnetic phase diagram for the system MnFeP-MnFeAs (Beckman and Lundgren, 1991) shows a rich variety of crystallographic and magnetic phases. The most striking feature is the fact that for As concentrations between 30 and 65% the hexagonal Fe2 P type of structure is stable and the ferromagnetic order is accompanied by a discontinuous change of c/a ratio. While the total magnetic moment is not affected by changes of the composition, the Curie temperature increases from about 150 K to well above room temperature. We reinvestigated this part of the phase diagram (Tegus et al., 2002b; Brück et al., 2003) and investigated possibilities to partially replace the As (Tegus et al., 2003, 2005; Dagula et al., 2005, 2006; Zhang et al., 2005a; Ou et al., 2006; Thanh et al., 2006). Polycrystalline samples can be synthesised starting from the binary Fe2 P and FeAs2 compounds, Mn chips and P powder (red) mixed in the appropriate proportions by ball milling under a protective atmosphere. After this mechanical alloying


263

Figure 4.10 Temperature dependence of the magnetization of MnFeP0.45 As0.55 measured in an applied field of 50 mT. T1 and T2 indicate the onset and finalization of the phase transition. Th indicates the thermal hysteresis.

process one obtains amorphous powder. To obtain dense material of the crystalline phase, the powders are pressed to pellets wrapt in Mo foil and sealed in quartz tubes under an argon atmosphere. These are heated at 1273 K for 1 hour, followed by a homogenisation process at 923 K for 50 hours and finally by slow cooling to ambient conditions. The powder X-ray diffraction patterns show that the compound crystallises in the hexagonal Fe2 P type structure. In this structure the Mn atoms occupy the 3(g) sites, the Fe atoms occupy the 3(f) sites and the P and the As atoms occupy 2(c) and 1(b) sites statistically (Bacmann et al., 1994). From the broadening of the X-ray diffraction reflections, the average grain size is estimated to be about 100 nm (Tegus et al., 2002a). Figure 4.10 shows the temperature dependence of the magnetisation measured with increasing and decreasing temperature in an applied field of 50 mT. The thermal hysteresis is signature of a first-order phase transition. Because of the small size of the thermal hysteresis (less than 1 K), the magnetisation process can be considered as being reversible in temperature. From the magnetisation curve at 5 K, the saturation magnetisation was determined as 3.9 μB /f.u. This high magnetisation originates from the parallel alignment of the Mn and Fe moments, though the moments of Mn are much larger than those of Fe (Beckman and Lundgren, 1991). Variation of the Mn/Fe ratio may also be used to further improve the magnetocaloric effect. Recently, we have observed a surprisingly large magnetocaloric effect in the compound MnFeP0.5 As0.3 Si0.2 at room temperature (Dagula et al., 2006). After replacing all As a considerable large magnetocaloric effect is still observed for MnFe(P,Si,Ge) (Thanh et al., 2006). A Mössbauer study on the MnFe(P,As) series evidences the importance of competing AF and F interactions that depend on the local environment (Hermann et al., 2004).

264

E. Brück

Figure 4.11 Magnetic entropy change of various MnFe(P,As,Si,Ge) alloys for a field change from 0 to 2 T.

The magnetic-entropy change of different MnFe(P,As,Si,Ge) alloys is shown in Fig. 4.11. The origin of the large magnetic-entropy change should be attributed to the comparatively high 3d moments and the rapid change of the magnetisation in the field-induced magnetic phase transition. In rare-earth materials, the magnetic moment fully develops only at low temperatures and therefore the entropy change near room temperature is only a fraction of their potential. In 3d compounds, the strong magneto-crystalline coupling results in competing intra- and inter-atomic interactions and leads to a modification of metal-metal distances which may change the iron and manganese magnetic moment and favours the spin ordering. The large MCE observed in Fe2 P based compounds originates from a fieldinduced first-order magnetic phase transition. The magnetisation and structural change is reversible in temperature and in alternating magnetic field as was evidenced also in X-ray diffraction experiments in applied magnetic field (Koyama et al., 2005; Yabuta et al., 2006). The magnetic ordering temperature of these compounds is tuneable over a wide temperature interval (200 K to 450 K). The excellent magnetocaloric features of the compounds of the type MnFe(P,Si,Ge,As), rather simple sample preparation (Kim and Cho, 2005; Yan, 2006) in addition to the very low material costs, make it an attractive candidate material for a commercial magnetic refrigerator. However, similar to MnAs alloys, it should be verified that materials containing As do not release this to the environment. The fact that the magneto-elastic phase-transition is rather a change of c/a than a change of volume, makes it feasible that this alloy even in polycrystalline form will not experience severe aging effects after frequent magnetic cycling.


265

4.7 Other Mn intermetallic compounds MnFe1–x Cox Ge The intermetallic compound MnFeGe crystallizes in the hexagonal Ni2 In-type structure. In this structure Mn atoms occupy 2a sites with a moment of 2.3 μB /Mn, Fe atoms are at 2d sites with 1.1 μB /Fe, and Ge at 2c sites (Beckman and Lundgren, 1991). The Curie temperature of MnFeGe is 228 K. On the other hand, the compound MnCoGe crystallizes in the orthorhombic TiNiSi-type structure with a Curie temperature of 337 K. In this structure Mn has a moment of about 3 μB /Mn and Co has a moment of 0.78 μB /Co. When replacing Fe by Co, it is expected that both magnetic moment and Curie temperature should increase and a structural transformation from the hexagonal Ni2 In-type to the orthorhombic TiNiSi-type occurs. It turns out, that the samples have the Ni2 In-type structure (hexagonal, space group P63/mmc) for x < 0.8 and the TiNiSi-type structure (orthorhombic, space group Pnma) for x ≥ 0.85 (Lin et al., 2006). The MnFe0.2 Co0.8 Ge compound crystallizes mainly in Ni2 In-type, but a small amount of orthorhombic phase traces were present. The lattice parameter a decreases and c increases, but the unit cell volume becomes smaller with increasing of Co contents. Figure 4.10 depicts the concentration dependence of the structure and the Curie temperatures, which also are listed in Table 4.4. MnFeGe has a Curie temperature of 159 K, which is much lower than the value of 228 K reported earlier (Beckman and Lundgren, 1991). MnCoGe has a Curie temperature of 345 K, being close to the earlier reported values of 337 K. The data listed in the table show that the Curie temperature increases with increasing Co contents. The spontaneous moment of the MnFe1–x Cox Ge compounds at 5 K, derived from extrapolation to zero field of the high-field magnetization, has also been listed in Table 4.4. The magnetic moments increase with increasing Co content in the Ni2 In-type structure, reaching a maximum value of 2.34 μB /3d atom. for x = 0.8. In the NiTiSi-type structure, the magnetic moments almost saturate at a value of 2.06 μB /3d atom. When the symmetry changes from hexagonal to orthorhombic, TC and magnetic moment increase abruptly see Fig. 4.12. The magnetic-entropy change is derived from the magnetization data by using Eq. (19). Table 4.4 shows the magnetic-entropy change of MnFe1–x Cox Ge compounds in a field change from 0 to 2 and 5 T, respectively. The magnetic-entropy change in the compounds, which crystallized in the Ni2 In-type structure, increases with increasing Co content. A comparatively large magnetic-entropy change, which reaches 9 J/kg · K, is observed for x = 0.8 in a field change of 5 T. Mn5–x Fex Si3 Because field induced transitions can produce large magnetocaloric effects the material Mn5 Si3 crystallizing in the hexagonal Mn5 Si3 -type structure with space group P63 /mcm attracted some attention (Songlin et al., 2002b). The compound Mn5 Si3 is an antiferromagnet with a field-induced transition (Sm = 2.9 J/kg · K at 58 K and B = 5 T). On the other hand the isostructural compound Fe5 Si3 is a ferromagnet with a Curie temperature of 363 K. The magnetic phase transitions and the magnetocaloric properties have been investigated in the pseudo binary system

266

E. Brück

Figure 4.12 Magnetic and crystal structure of the MnFe1–x Cox Ge compounds.

Mn5–x Fex Si3 for x = 0, 1, 2, 3, 4, 5. With increasing Fe content, the antiferromagnetic ordering temperature shifts to higher temperatures. At 4.2 K, the Mn5–x Fex Si3 compounds with x = 1 and 2 display antiferromagnetic behavior up to 38 T. The compounds with x = 4 and 5 show ferromagnetic order. The largest value for the magnetic-entropy change is observed for the MnFe4 Si3 compound (SM = –4.0 J/kg · K at 310 K and B = 5 T). Mn5 Ge3–x Six One of the Mn rich alloys with group four elements that orders near room temperature is Mn5 Ge3 , the magnetocaloric effect of this alloy is fairly large but yet smaller than for Gd metal (Hashimotoa et al., 1981). The magnetic properties and the magnetocaloric effect of Mn5 Ge3–x Six alloys were investigated by (Zhao et al., 2006) for x = 0.1, 0.3, 0.5, 1.0, 1.5 and 2.0. All Mn5 Ge3–x Six compounds crystallize in the Mn5 Si3 -type hexagonal structure with space group P63 /mcm. The lattice parameters and the Curie temperature of Mn5 Ge3–x Six alloys decrease with increasing x. As can be seen in the table, a fairly large magnetic-entropy change has been observed in these alloys near room temperature. The average Mn magnetic moment decreases with increasing Si content. The substitution of Si in Mn5 Ge3 does not result in a change of the crystal structure. But the Si substitution has two kinds of effects on the magnetocaloric effects. One is that the magnetic-entropy change decreases with increasing Si content, the other one is that the magnetocaloric effect peak becomes broadened. Mn5 Ge3–x Sbx Compared to the Mn5 Ge3–x Six series, substitution of Ge by Sb should give the opposite effect on unit-cell volume and Curie temperature In this series the magnetic


267

and magnetocaloric properties were investigated by (Songlin et al., 2002a) for compounds with x = 0, 0.1, 0.2 and 0.3 (see Table 4.1) The compounds crystallize in the hexagonal Mn5 Si3 -type structure with space group P63 /mcm. The Sb substitution leads to slightly enhanced Curie temperatures but decreasing average magnetic Mn moments with increasing Sb content. The Sb substitution has two kinds of effects on the magnetocaloric effect (MCE) of Mn5 Ge3–x Sbx . One is the magnetic entropy change decreases with increasing Sb content, the other is that the MCE peak becomes broadened. LaMn2–x Fex Ge2 These compounds crystallize in the tetragonal ThCr2 Si2 -type structure and the Curie temperature gradually decreases with increasing Fe concentration from 310.7 K at x = 0.10 to 274.5 K at x = 0.20 (Zhang et al., 2003). As can be seen in Table 4.4, the magnetic entropy change in this series of compounds, measured with a field change of 1.8 T, also decreases with Fe content. (Fe1–x Mnx )3 C The (Fe1–x Mnx )3 C compounds crystallize in the orthorhombic Fe3 C structure. The Curie temperature can be adjusted very well from 31 to 483 K. However, there is a large loss of magnetization with the addition of manganese by changing the Fe/Mn ratio. The magnetocaloric effects remain relatively low see Table 4.4. Mn3–x Cox GaC The magnetocaloric effect in the Mn3–x Cox GaC compounds has been investigated by Tohei et al. (2003). Mn3 GaC shows a first-order antiferromagnetic to ferromagnetic transition at Tt = 160 K. Large magnetocaloric effects of Smag = 15 J/kg · K, were observed at this transition. The substitution of Co for Mn lowers Tt without significant loss of magnetocaloric effects (Table 4.4). It was indicated that the system could cover a wide temperature range of 50–160 K by combining the compounds with various compositions from x = 0 to 0.05.

4.8 Amorphous materials Super paramagnetic or very soft magnetic materials change their magnetization reversibly in very low magnetic fields. This is the motivation that a few amorphous materials are being studied for possible magnetocaloric applications. On the one hand the Fe rich alloys have good mechanical and corrosion properties but rather high magnetic ordering temperatures (Franco et al., 2006a, 2006b). Another problem with these materials is the rather low adiabatic temperature change, for effective heat transfer in a short time interval this temperature change should be a few degrees and not a few tenth of a degree. The flatter top of the temperature dependence of the magnetic entropy change, is another motivation for the use of amorphous materials in AMRs. It is expected that the performance of these materials is better as this reduces the cycle losses in e.g. an Ericsson cycle. With this motivation mainly rare-earth based materials are studied for rather low temperature applications (Foldeaki et al., 1997a, 1998;

268

E. Brück

Table 4.6 Magnetocaloric properties of a few amorphous or nanocrystalline transition metal based materials. |S| at B = 2 T if not indicated differently

Material

Remark

Fe90 Zr10 Fe82 Mn8 Zr10 Fe80 Mn10 Zr10 Co66 Nb9 Cu1 Si12 B12

Melt spun Melt spun Melt spun

Bmax (T)

Tmax (K)

Smax (J/kg·K)

Ref.

7 5 5 0.15

237 210 195 175

6.52 2.8 2.3 ?

Co66 Nb9 Cu1 Si12 B12

Partly recrystallize

0.15

120

?

Co66 Nb9 Cu1 Si12 B12

Partly recrystallize

0.15

80

?

Pd40 Ni22.5 Fe17.5 P20

Bulk amorphous Finemet

5

94

0.58

Maeda et al. (1983) Min et al. (2005) Min et al. (2005) Didukh and Slawska-Waniewska (2003) Didukh and Slawska-Waniewska (2003) Didukh and Slawska-Waniewska (2003) Shen et al. (2002)

480

1.1

Franco et al. (2006a)

FeMoSiBCuNb

1.5

Table 4.7 Magnetocaloric properties of a few amorphous or nanocrystalline rare-earth metal based materials

Material

Remark

Bmax (T)

Tmax (K)

Smax (J/kg·K)

Ref.

Gd70 Ni30 Gd70 Ni30 Gd70 Ni30 Gd70 Fe30 GdNiAl GdNiAl NdFe12 B6

Melt spun Melt spun Melt spun ground Melt spun Melt spun Ball milled Melt spun recrystallized Ball milled Poly

1 7 7 1 2 1 1

126 130 95 288 40 35 218

2.5 11.5 7.5 1.5 6.7 1.6 8.4

Foldeaki et al. (1997a) Foldeaki et al. (1997a) Foldeaki et al. (1998) Foldeaki et al. (1997a) Si et al. (2002) Chevalier et al. (2005) Zhang et al. (2006a)

9 9

130 50

2.2 4.9

GdMn2 GdMn2

Marcos et al. (2004) Marcos et al. (2004)

Giguere et al., 1999a, Si et al., 2001a, 2001b, 2002). The key data of a few amorphous or nanocrystalline alloys are summarized in Table 4.6 for the transition metal based materials and Table 4.7 for rare-earth based materials. In general the magnetic entropy change in amorphous materials is smeared out over a wider temperature interval than what is observed in crystalline materials. This behavior is beneficial for the efficiency of a regenerator. However, if the mag-


269

netocaloric effect results in less than 1 degree temperature change, the driving force for heat transfer becomes quite low and very low frequency operation will be required. Therefore evaluation of the cooling capacity of a material as proposed a few years ago (Wood and Potter, 1985) should not be done by just integrating over the width at half maximum. Instead, the region of the curve that results in a too low temperature change should be truncated. Some reports of very large cooling capacity of a certain material should therefore be taken with some caution as the temperature span used for the calculation is far too wide.

4.9 Manganites Field or temperature induced first order phase transitions are also a common feature of a large group of colossal magneto resistance CMR materials. In recent years a few of these rare-earth manganese oxide materials that crystallize in Perovskite type structure, were studied with respect to their magnetocaloric properties. Very recently a review on the magnetocaloric properties of these materials was published (Phan and Yu, 2007). Indeed a few materials produce decent magnetic entropy changes near or below room temperature see Fig. 4.13. Some key parameters of this type of materials are summarized in Table 4.8. Concerning the perspective of these materials for room temperature magnetic refrigeration, a few aspects need to be considered. The materials definitely have excellent corrosion stability and generally are quite cheap. However, though the magnetic entropy change is quite decent, the magnetic field induced change in temperature is often quite low in these materials. The reason for this is the low

Figure 4.13 Magnetic entropy change of several manganites (Zhong et al., 1999; Hueso et al., 2002; Phan et al., 2005).

270

E. Brück

Table 4.8 Magnetocaloric properties of a few perovskite type manganites. FO, SO stand for first and second order transitions, respectively

Material

Remark

Bmax (T)

Tmax (K)

Smax Ref. (J/kg·K)

La0.6 Ca0.4 MnO3 La0.67 Ca0.33 MnO3

SO? Sol gel process Sol gel process FO SO SO SO SO SO FO Sol gel process Hydrothermal

3 1

263 263

5 5

Bohigas et al. (1998) Hueso et al. (2002)

1.5

230

5.5

Guo et al. (1997)

1 3 3 1 1 1 1 1

216 90 103 334 220 283 345 312

6.3 2 1.3 2 1.5 1.1 3.1 2.1

Ulyanov et al. (2007) Bohigas et al. (1998) Bohigas et al. (1998) Zhong et al. (1998) Zhong et al. (1998) Zhong et al. (1998) Phan et al. (2005) Hou et al. (2006)

2

317

1.5

Li et al. (2006a)

La0.8 Ca0.2 MnO3 La0.7 Ca0.3 MnO3 La0.958 Li0.025 Ti0.1 Mn0.9 O3 La0.65 Ca0.35 Ti0.1 Mn0.9 O3 La0.799 Na0.199 MnO2.97 La0.88 Na0.099 Mn0.977 O3 La0.877 K0.096 Mn0.974 O3 La0.65 Sr0.35 Mn0.95 Cu0.05 O3 La0.7 Nd0.1 Na0.2 MnO3 La0.5 Ca0.3 Sr0.2 MnO3

number of magnetic ions per formula unit, thus the lattice specific heat is quite high compared to most other magnetocaloric materials. From Eq. (21) it is obvious that this will lead to a low T . The low electrical conductivity in these materials could be an advantage as the generation of eddy currents at high cycle frequencies is reduced. However this low electrical conductivity comes along with a low thermal conductivity which limits the cycle frequency.

5. Comparison of Different Materials and Miscellaneous Measurements The MCEs for field changes of 2 T (if available) are summarized in the tables. It is obvious that above room temperature a few transition-metal-based alloys perform the best. If one takes into account the fact that T also depends on the specific heat of the compound (Pecharsky and Gschneidner, 2001) these alloys are still favorable not only from the cost point of view. This makes them likely candidates for use as magnetic refrigerant materials above room temperature. However, below room temperature a number of rare-earth compounds perform better and for these materials a thorough cost vs performance analysis will be needed. The main parameters of the most important magnetocaloric materials are summarized in Table 4.9 which allows a fast comparison. Because of the limited availability, the Gd, Ge and Ga containing materials will be restricted to niche

Material

Limiting ingredient

Estimated availability

Total availability of MC material

Temperature range (°C)

Thermal hysteresis

Tmax for B 2 T

Gd metal Gadolinium silicon alloys Gd5 (Si1–x Gex )4 Manganese alloys Mn(As1–x Sbx ) MnFe(P1–x Asx )

Gd Ge

1000 WW prod. = 90 t, avail. 10 t?

1000 t 140 t

0–20 –100 to 0

+ –

6K 7K

None

None

No limitation for an industrial production

–50 to 50 –100 to 120

0 0

6K 8K

NaZn13 type alloys La(Fe13–x Mx )

La

4000

22000 t

–80 to 50

–

6K

Manganites LaMnO3 Heusler alloys Ni0.501 Mn0.227 Ga0.258

La

4000

7000 t

–100 to 50

–

3K

Ga

WW prod. = 90 t, avail. 10 t?

60 t?

–50 to 50

–

3K


Table 4.9 Availability of different types of magnetocaloric materials, possible range of use, thermal hysteresis and temperature change in 2 T. The worldwide production is estimated from data of US geological service, hysteresis is strongly sample dependent, Gd has a second order transition, thus no hysteresis, below 2 K is 0, above 2 K is –

271

272

E. Brück

markets. At present it is not clear which material will really get to the stage of real life applications and one may expect that still other materials will be developed. Though it is already feasible that for applications with limited temperature span and a cooling power in the kW range like air conditioning, commercial competitive magnetic refrigerators are quite possible, it is not yet obvious, which of the above mentioned materials shall be employed. Currently most attention is paid to the pure magnetocaloric properties which are derived from magnetic measurements as described in section 3. In the last few years more specific experimental equipment has been developed to characterize magnetocaloric materials. At Moscow State University two setups exist to determine the adiabatic temperature change. One setup utilizes a rather simple electromagnet (B < 2 T) and a nitrogen cryostat. The other setup is a liquid nitrogen cooled pulse magnet in combination with a He flow cryostat. In the former equipment the variation of sample temperature during the sweeping of the field is monitored with a thermocouple. Sweeping to the maximum field takes approximately 3 s (Tishin, 1999). Data of the most important magnetocaloric materials exist from this equipment (Chernyshov et al., 2002; Hu et al., 2003, 2005; Brück et al., 2005; Ilyn et al., 2005; Wada et al., 2005b). The last few years however the maximal field employed is only 1.45 T. The reported temperature changes are 4.3 K at 292 K for Mn1.1 Fe0.9 P0.47 As0.53 , 4 K at 312 K for MnAs, 4 K at 188 K for LaFe11.7 Si1.3 , 3.2 K at 274 K for LaFe11.2 Co0.7 Si1.1 , and 3 K at 271 K for Gd5 Ge2.05 Si1.95 . The pulse field setup which provides up to 8 T fields with a pulse length of 0.2 s is up to now only used to study the classical materials (Dankov et al., 1997). At the University of Quebec a special sample holder for a Quantum Design PPMS system is constructed, which enables the insertion of the sample from a low field position into the 9 T maximum field within 1 s (Gopal et al., 1997). The sample temperature is monitored with a Cernox resistance thermometer and can be varied between 2 and 400 K. However, only the temperature at the high field position is controlled so that the measurements need to be performed rather fast to avoid excessive drift of the temperature. Eddy current heating of the sample may occur under these conditions. Next to classical magnetocaloric materials like (Gd,Y) alloys (Foldeaki et al., 1997b), also measurements on Gd5 Ge2 Si2 are reported, it should be noted that the latter results were quite controversial (Giguere et al., 1999b; Gschneidner et al., 2000; Sun et al., 2000). Direct measurements of the adiabatic temperature-change of La0.6 Ca0.4 MnO4 were reported by a Danish group. They utilize a nitrogen cryostat that can be inserted in the field of an electro magnet. For a field change of 0.7 T at 270 K a maximal adiabatic temperature change of 0.5 K is observed (Dinesen et al., 2002). Recently, the same group reports on the extension of the temperature range to well above room temperature for La0.67 Ca0.33–x Srx MnO3+δ (Dinesen et al., 2005). The largest effect measured directly in again 0.7 T is for the compound without Sr at 267 K a temperature change of 1.5 K. For 22% Sr at 344 K a temperature change of 0.5 K is observed. A pulsed magnet setup with rather long pulse duration of more than a second has been developed at Sichuan University (Tang et al., 2003) there are however little results published yet.


273

Figure 4.14 Photographs of the DSC in field insert open (left) and closed (right) the sample is mounted on the thermo batteries attached to the Cu block a reference sample is mounted on the backside (Marcos et al., 2003).

At Tohoku University also the mechanical insertion of the thermally insulated sample into a 2 T field region is employed for direct measurements (Fujieda et al., 2004b). The results on LaFe11.9 Si1.1 T = 5.9 K at 188 K are in good agreement with earlier published data (Hu et al., 2003), the temperature change observed for LaFe11.9 Si1.1 H1.6 T = 4.0 K at 319 K is rather unexpected. A group at the University of Genoa reports on a fast direct measurement device that employs an electromagnet with maximum field of 1 T and can be utilized from 100 K up to 340 K (Canepa et al., 2005). The adiabatic temperature change of several Gd eutectic alloys are reported, the authors also observe a rather high sensitivity of the MCE on impurities in Gd. A commercial adiabatic calorimeter from Thermis Ltd placed in a 6 T superconducting magnet is employed in Barcelona (Tocado et al., 2005). The calorimeter works in the temperature range 4–370 K and magnetic fields may be applied as long as the vacuum grease used to attach the sample to the sample holder keeps it at its place, the latter may become problematic near room temperature for magnetic samples. For the direct measurements performed in a field sweep to 5 T field a maximal temperature change T = 6.5 K at 109 K is reported for Tb5 Ge2 Si2 . With the aim to study the heat effects occurring in magnetostructural transitions, Marcos et al. developed a differential scanning calorimeter (DSC), which is depicted in Fig. 4.14, capable to work in magnetic fields up to 5 T (Marcos et al., 2003). In contrast to most other calorimeters a DSC can measure in a heating and a cooling mode so the thermal hysteresis is easily monitored. Also the entropy change in a first order phase-transition is normally difficult to determine as

274

E. Brück

Figure 4.15 Thermal conductivity of various magnetocaloric materials (Battabyal and Dey, 2004; Fujieda et al., 2004c; Fukamichi et al., 2006).

accurately as with a DSC. In combination with applied magnetic field the phase transition of giant magnetocaloric materials thus can be studied very accurately. Interesting cycle time dependent effects in the magnetic field induced entropy change in Gd5 Ge3.8 Si0.2 have been observed with this equipment (Casanova et al., 2005). Another important property of magnetic refrigerants is the thermal conductivity. At Tohoku University the temperature dependence of the thermal conductivity was studied for several magnetic refrigerants (Fujieda et al., 2004c; Fukamichi et al., 2006). At the Indian Institute for Technology, Karagpur the thermal conductivity of (La,Sr,Ag)MnO3 perovskites was studied (Battabyal and Dey, 2004). As one may expect the perovskites have a much lower thermal conductivity but astonishingly the conductivity of MnAs is not much higher and Gd5 Ge2 Si2 is found to be intermediate, the results are summarized in Fig. 4.15. In the near future also other properties like corrosion resistance, mechanical properties, heat conductivity, electrical resistivity and environmental impact should be addressed more.

6. Demonstrators and Prototypes The growing interest in magnetic refrigeration near room temperature is also reflected in a growing number of projects that do not study materials properties but


275

study the performance of certain refrigerator designs. Next to theoretical papers that discuss for example different thermodynamic cycles, in the last few years several demonstrators and prototypes were built. We shall here discuss a few of them in more or less chronological order, some key aspects of these prototypes are summarized in Table 4.10. A few years ago already reviews on magnetic refrigerators were published but the development is rather fast so it is worthwhile to discuss it here again (Yu et al., 2003; Gschneidner et al., 2005). Already more than 30 years ago, the advantages of a regenerator process for magnetic refrigeration near room temperature were pointed out in a paper that discusses Gd as a possible magnetic refrigerant near room temperature in combination with a 7 T superconducting magnet (Brown, 1976). The paper predicts near Carnot efficiency and a maximal temperature span of 46 K at a sink temperature of 340 K. Shortly after this a rotary design of the magnetic Stirling cycle was proposed (Steyert, 1978). Here also a 7 T magnetic field is used to magnetize a porous Gd disk that rotates in and out of the high field region and heats or cools a counterflowing heat-transfer fluid. This machine was predicted to reach a cooling power of 32 kW/l Gd at an operating frequency of 1 Hz. Note that the rule of thumb, low fluid thermal capacity is needed for high efficiency, which holds for a passive regenerator, does not hold in the case of an active regenerator. The first realization of a reciprocating magnetic refrigerator that very much resembles the design proposed by Brown was reported 20 years later (Zimm et al., 1998). This magnetic refrigeration demonstrator built in collaboration of Ames Laboratory and Astronautics in Madison WI, utilized a 5 T superconducting magnet and 3 kg of Gd spheres. Extremely good performance parameters were reported on this machine. A cooling power of 600 W at an operating frequency of 0.17 Hz and a temperature span of 10 K was realized. The COP reached a value of 10 or in other words a Carnot efficiency of 75% was reported, the authors mention that they neglected a few things, but in the light of more recent publications this high Carnot efficiency must be taken with caution. Other authors report for very similar heat-exchanger designs rather high losses due to the high flow resistance of the heat-exchanger (see below). Having in mind to design a very simple demonstrator a group in Barcelona utilizes the rotary device proposed by Steyert in combination with permanent magnets (Bohigas et al., 2000). A thin ribbon of Gd metal mounted on a plastic wheel rotates in and out of the field of 0.3 T. Commercial olive oil is used as heat transfer medium and regenerator. The maximal temperature difference achieved in steady state operation was 1.6 K which clearly demonstrates that regeneration worked even in this simple device. When the device is operated at 1/3 Hz steady state is reached after about 1000 s. The main losses in this device are probably due to flow of transfer medium between different sections. The authors do not comment on the efficiency of the device and the numbers quoted are not sufficient to determine it. To test the possibility of increasing the temperature span of an active magnetic regenerator a team at the University of Victoria BC designed a compact reciprocating device that accommodates AMR pucks of 2.5 cm diameter and 2.5 cm length (Richard et al., 2004). The field source for this device is a 2 T superconducting magnet and the operating frequency could be varied between 0.2 and 1 Hz. Two

276 Table 4.10 Magnetic refrigeration demonstrators, the magnetic field source is (S) superconducting magnet, (P) permanent magnet, (E) electromagnet. When authors quote the performance there exist COPT only taking into account the cooling power and the power dissipated at the hot heat exchanger and COPR cooling power divided by total electrical power input

AMR type

AMR material

Magnetic field (T)

Remarks

Ref.

Ames Laboratory/ Astronautics Barcelona University of Victoria

Reciprocating

Gd spheres

5 (S)

COPT 10

Zimm et al. (1998)

Rotary Reciprocating

Gd foil Gd, Gd0.74 Tb0.26

0.3 (P) 2 (S)

Bohigas et al. (2000) Richard et al. (2004)

Lab. Electric Grenoble Astronautics

Reciprocating

Gd foil

0.8 (P)

Olive oil Epoxy bonded pucks COPR 2.2

Clot et al. (2003)

Rotary

1.5 (P)

4 Hz

Zimm et al. (2006)

Tokyo Inst. Techno./Chubu Natl Inst. Appl. Sci./Cooltech Xian Jiaotong Univ.

Rotary

Gd, Gd-Er, spheres LaFeSiH particles Gd-Dy, Gd-Y spheres Gd plates

0.7 (P)

Okamura et al. (2006)

1 (P)

Torque 52 Nm COPR 0.2 Torque 10 Nm

Vasile and Muller (2006)

2.18 (E)

COPT 25

Gao et al. (2006)

University of Victoria

Reciprocating

2.0 (S)

T 50 K

Rowe and Tura (2006)

Rotary Reciprocating

Gd spheres; Gd5 (Si,Ge)4 pwdr. Gd, Gd0.74 Tb0.26 Gd0.85 Er0.15

E. Brück

Name


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AMR’s were alternating in and out of the high field region. The heat transfer fluid used in this experiment was He gas at up to 10 atm. pressure and a maximal mass flow of 0.4 g per half cycle. Two types of regenerators were tested one with two pucks of Gd with a total mass of 90 g, and one consisting of two materials with different TC , a puck of Gd at the hot side and a puck of Gd0.74 Tb0.26 at the cold side with a total mass of 85 g. The tests clearly show that for reaching larger temperature spans the multimaterial heat exchanger is more suited. Another interesting point is that the low heat capacity of the helium gas in combination with the low mass flow rate seemingly limited the performance of the device. The cooling power of this device is depending on the temperature span only a few W. This results in a very long startup period of about 1.5 h before steady state is reached. The authors do not quote a COP or other figures of performance. Just recently the same group reported on the use of three different materials, Gd, Gd0.74 Tb0.26 and Gd0.85 Er0.15 with a total mass of about 135 g for each AMR and a field of 2 T (Rowe and Tura, 2006). In this configuration a maximal no-load temperature span of 50 K was realized. The authors however comment that this span quickly decreases when the hot reservoir is at temperatures above 307 K. At the Laboratoir Electric de Grenoble a reciprocating magnetic refrigerator was designed that utilizes a Halbach magnet generating 0.8 T transverse field and Gd foil as magnetic refrigerant (Clot et al., 2003). The device is only poorly isolated which results in considerable thermal losses, therefore the authors only quote the COP for a temperature span of 4 K. For this condition the cooling power is found to be 8.8 W and the electrical power needed to operate the device is 4 W thus a COP of 2.2 is derived. This COP is a really measured value and not just estimated after neglecting parasitic influences. The second AMR constructed at Astronautics, Madison WI is a rotary device that utilizes a permanent magnet as field source (Zimm et al., 2006). In this machine the magnetic field is about 1.5 T and the regenerator rotates with up to 4 Hz in and out of the high field region while the heat-exchange fluid (water) is pumped in the opposite direction. This device has been tested with several different materials. Pure Gd, Gd0.94 Er0.06 , a combination of these two and the giant MCE material La(Fe0.88 Si0.12 )13 H. The performance of the different beds strongly depends on the temperature interval that is studied. The authors also mention that the performance of the device is strongly influenced by the fluid-flow resistance of the AMR matrices which depends on the material. The Gd and Gd-Er particles are spherical with a rather narrow size distribution but in the case of LaFeSiH the particles are with irregular shape and a large spread of sizes. The latter limited the operation frequency of the LaFeSiH AMR to 1 Hz. Interestingly, for low temperature spans the authors find the cooling power to be higher at lower cycle frequencies while the opposite is true for larger temperature spans. This effect is attributed to the dominance of valve friction and dynamic flow losses, which are lower at low frequency. This finding may be of interest for the optimization of startup procedures. Two other rotary devices have been presented recently which instead of moving the regenerator move the magnet (Okamura et al., 2006; Vasile and Muller, 2006). The obvious advantage of moving the magnet is that one avoids sliding seals that may deteriorate after extended period of use.

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The device developed at the Tokyo Institute of Technology and Chubu Electric power employs a 0.77 T permanent magnet and four AMRs that consist each of four different Gd alloys Gd0.92 Y0.08 , Gd0.84 Dy0.16 , Gd0.87 Dy0.13 and Gd0.89 Dy0.11 with TC ranging from about 5°C to 10°C (Okamura et al., 2006). The AMR’s of 1 kg each are packed beads of spheres with 0.6 mm diameter with a filling factor of 63%. In the cooling cycle the magnet is rotated in 90 degree steps switching from one AMR to the next. In the stopping period water is pumped through the AMR. The flow direction depends on the field if the field is high the water flows from the cold side to the hot side and vice versa. Actually the heat exchange period can extend into the rotating period. Typical rotation periods of 0.5 s and cycle times of 2.4 s were used. The authors realize a maximal cooling power of 60 W at 10°C with zero temperature span, however the coefficient of performance of this device is disappointing low (below 0.2). This is due to the high torque (52 Nm) needed to switch the magnetic field and the high flow resistance of the water in the AMR resulting in a very high power consumption. The authors conclude that new arrangement and configuration of the AMR beds are inevitable to improve the design. These two points are especially addressed in the design presented by the French National Institute of Applied Sciences and Cooltech Applications (Vasile and Muller, 2006). They employ a heat exchanger with micro channels and separated circuits for hot and cold flow. This heat exchanger consists of quadratic plates of pure Gd with a width of 45 mm and thickness 0.65 mm (see Fig. 4.16). The fluid channels are 0.2 mm wide. The filling factor is 77.7%. The magnets are rotating NdFeB based permanent magnets that generate a 1 T magnetic field. In the same publication a modified Hallbach magnet design is presented that generates 1.9 T. This magnet is open on one side and can thus be rotated over the heat exchangers. The regenerator with straight micro channels has a very low flow resistance and yet a good heat transfer rate. Dividing hot and cold flow in separate channels reduces the dead volume. Additionally the torque needed to rotate the magnet array is rather low as the AMRs are arranged almost continuous. This torque of 10 Nm is more than 5 times less compared to the Japanese design with yet a higher applied field (Muller, 2006). A near industrial design prototype is depicted in Fig. 4.17. At Xian Jiaotong University a study of different materials in a given magnetic refrigerator is performed (Gao et al., 2006). On the one hand Gd spheres with different diameters 0.3 and 0.55 mm and 0.3–0.75 mm particles of the alloy Gd5 Si2 Ge2 . An electromagnet with 160 mm diameter pole pieces and an air gap of 60 mm generates the magnetic field of 2.18 T. The AMR with dimension 140 × 76 × 36 mm3 is made of stainless steel and isolated by 2 mm thick layer of isolation material. The Gd particles are prepared by milling the Gd hydride as the pure Gd is too ductile to be easily milled. After the required size is achieved the material is dehydrided. However, the authors mention that the magnetocaloric effect of the powder is 10–25% lower than what was measured on the ingot which was used as starting material. The Gd5 Ge2 Si2 prepared from 2N5 commercial grade Gd which was not heat treated after preparation does not exhibit a first order phase transition but shows similar properties as the material reported by (Provenzano et al., 2004). The alloy is milled and particles with sizes between 0.15 and 0.3 mm


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Figure 4.16 Front view of a rotary prototype the black squares are the Gd plates of the micro-channel heat exchangers (Vasile and Muller, 2006).

are selected by sieving. The refrigeration cycle is of the type move, flush, move, flush where the periods and the flow rates can be varied and the flows and temperature variations are recorded automatically. Similar as Zimm et al. (1998) the power consumption of the magnet and the pumps are neglected for the calculation of the COP. Actually only the refrigeration capacity and the heat released at the hot reservoir are considered. Thus also the power consumption of the step motor driving the magnet is neglected. The results for the different materials are rather puzzling. 930 g of 0.3 mm diameter Gd produce a higher cooling power and a better performance than 1109 g of 0.55 mm diameter Gd. The worst performance comes from the 1213 g of 0.15–0.3 mm Gd5 Ge2 Si2 . As the external parameters like force and flow resistance are neglected, these differences must originate from heat transfer, regenerator and demagnetization parameters. The poor performance of Gd5 Ge2 Si2 may be explained by the rather low thermal conductivity and the fact that the very irregular shape of the particles can lead to enhanced demagnetization effects. On the other hand the higher specific heat of Gd5 Ge2 Si2 should improve the performance of the regenerator. All this can not hold for the two Gd batches as different cycle frequencies and fluid-flow rates were tested, the effect of differ-

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Figure 4.17

Photograph of most recent prototype of Cooltech (Muller, 2007).

ent sizes should have been compensated. This makes me suspect that parasitic heat sources were not detected in the system. One possible heat source is the pump in the primary circuit that probably will consume much more power when the packing factor and thus the pressure drop in the AMR is increased. Leakage of the heat released in the motor into the water circuit is quite feasible. The design of a permanent magnet field source is also an important issue for cost efficient magnetic refrigeration. The very simple bar magnets used in some prototypes are limiting the field to below 1 T. However, nowadays closed and open Halbach magnet arrays are built that can produce fields exceeding 2 T (Lee and Jiles, 2000; Lee et al., 2002a, 2002b; Xu et al., 2004, 2006a, 2006b). These arrays have in common that several magnetized bar magnets are combined and the magnetization is then concentrated in a soft magnetic pole piece. A soft magnetic shell that acts as a flux return path further enhances the performance. The art of getting maximal performance out of a minimal number of segments will determine the price of these advanced field sources (Russek and Zimm, 2006).

7. Outlook From the above it is obvious that the field of magnetic refrigeration is very fast developing, both in research on new materials, modeling and prototype design. The ideal magnetocaloric material is yet to be developed and the quest for higher efficiency refrigerators is becoming more and more important as the reduction of global CO2 production becomes a high priority issue. After the compressor being the technology of the 20th century, magnetic refrigeration has all potential to become the technology of the 21st century. However, to bring this into reality a broad research effort is needed on all three fields of research. The THERMAG conference series started in 2005 in Lausanne and continued in 2007 in Portoroz brings together researchers from industry and academia which is necessary to accelerate


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the development. Having the next conference held in Asia would also give credit to the vast amount of research performed on this subject on that continent.

ACKNOWLEDGEMENTS This work is supported by the CNPQ process number 304385/2006-9-PV and the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs. We want to especially thank A. Planes of University of Barcelona and C. Muller of Cooltech Applications for supplying photographs of the DSC in field and of prototypes, respectively.

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CHAPTER

FIVE

Magnetism of Hydrides Günter Wiesinger * and Gerfried Hilscher *

Contents 1. 2. 3. 4. 5.

Introduction Formation of Stable Hydrides Electronic Properties Basic Aspects of Magnetism Review of Experimental and Theoretical Results 5.1 Binary rare-earth hydrides 5.2 Binary actinide hydrides 5.3 Binary transition metal hydrides 5.4 Ternary rare-earth–transition-metal hydrides 5.5 Hydrides of amorphous alloys Acknowledgement References

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1. Introduction The present review is based upon our article (Wiesinger and Hilscher, 1991) which covered the literature until 1990. Despite the article of Vajda (1995a), dealing exclusively with binary RHx systems, no comprehensive review about metalhydrogen-systems was published since then. Thus, when studying the literature, it seems worthwhile to update the review to the articles published to date, particularly, since ten metal-hydrogen conferences took place in the last 15 years (MH 1988: Z. Phys. Chem. NF 163 (1989), MH 1990: J. Less-Comm. Met. 172–174 (1991), MH 1992: Z. Phys. Chem. NF 179 (1993), MH 1994: J. Alloys Comp. 231 (1995), MH 1996: J. Alloys Comp. 253–254, MH 1998: J. Alloys Comp. 293–295, MH 2000: J. Alloys Comp. 330–332 (2002), MH 2002: J. Alloys Comp. 356–357 (2003), MH 2004: J. Alloys Comp. 404–406 (2005), MH 2006: J. Alloys Comp. (2007). In order to avoid redundancy we have tried to condense the text of the previous article under the consideration of improvements in theory and experiment happening in *

Institute for Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria



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the last one and a half decades. This is visible particularly in the tables which have been comprehensively updated and completed. Intermetallic compounds of 3d metals (particularly Mn, Fe, Co and Ni) with rare earth elements exhibit a large variety of interesting physical properties. The magnetic properties of these intermetallics (for reviews see e.g. Wallace, 1973; Buschow, 1977a, 1980a and Kirchmayr and Poldy, 1979) are a matter of interest for two main reasons: Firstly their study helps to elucidate some of the fundamental principles of magnetism (RKKY interaction, crystal field effects, valence instabilities, magnetoelastic properties, coexistence of superconductivity and magnetic order). Secondly they are of technical interest, because several compounds (RCo5 , R2 Co17 , Nd2 Fe14 B, RFe11 T) were found to be a suitable basis for high performance permanent magnets. More recently the unique soft magnetic properties made amorphous metal-metalloid alloys to a further class of materials which has attained considerable importance with regard to industrial application. Since the discovery of LaNi5 as a hydrogen storage material roughly four decades ago, a vast number of intermetallic compounds and alloys has been involved in studies of the hydrogen induced changes of their physical properties. A large variety of techniques has been applied in order to elucidate the mechanism of hydrogen uptake which is particularly complex in intermetallic compounds. They can roughly be divided into surface sensitive methods (photo emission and related spectroscopies, X-ray absorption (XANES, EXAFS), X-ray magnetic circular dichroism (XMCD), transmission electron microscopy, conversion electron Mössbauer spectroscopy and to some extent susceptibility measurements, NMR and ESR) and surface insensitive experiments, where only the bulk properties can be studied (calorimetric and transport studies, magnetic measurements, neutron and X-ray diffraction, transmission Mössbauer spectroscopy). Despite the complex hydrogen absorption mechanism, some general statements concerning the influence of hydrogen upon the physical properties can be made. Hydrogen uptake commonly leads to a considerable lattice expansion. Although the absorption of hydrogen can lead to a volume increase of up to 30%, the overall crystal structure frequently is retained. The hydrogen induced rise in volume is to a large extent the essential reason for the altered magnetic properties in the hydrides. A larger volume implies narrower bands which, on the other hand, may reduce a hybridization having perhaps been present in the host compound. When a transition metal (TM) is alloyed to a rare earth or a related metal (R), the R-3d exchange interaction (3d-5d overlap) leads to a significant reduction of the TM moment. The strong hydrogen affinity of the R metals brings about a decrease of the 3d-5d overlap in the hydrides. Thus the absorption of hydrogen commonly cancels this moment depression to a certain degree. This partial restoration of the 3d moment is interpreted as an hydrogen induced screening effect. The predominant part of published results connected with magnetism considers binary R hydrides (R = rare earth element) and hydrides of binary compounds of the general formula Ry TMz , R being a rare earth element, which may be replaced by elements such as Sc,Y, Zr, Ti, and TM standing for a transition metal. Particularly Mn, Fe, Co and Ni compounds have been examined with regard to hydrogen absorption properties. Consequently, after some theoretical considerations the re-

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view will deal with the experimental results regarding binary rare earth hydrides, followed by a short treatment of transition metal hydrides. The main part covers ternary hydrides along the element order mentioned above, the final part containing hydrides of less common compounds and alloys (e.g. GdRh2 , oxygen stabilized TiTM compounds, several ternary R-compounds, amorphous alloys). Experimental data are only to some extent mentioned in the text. They have been summarized in several tables according to the transition element present in the compound. In order to limit the number of references to a reasonable number, initially attention was focused to the literature cited subsequently to 1980, except those papers which contain physical quantities given in the tables. For the remaining former literature the reader is referred to the comprehensive review articles of Buschow et al. (1982a), Buschow (1984a), Burger (1987) and Wiesinger and Hilscher (1988a). Since we suggest that a reference list, complete as much as possible, is desired by the reader, the articles published after 1991 are merely added to the initial reference list in Wiesinger and Hilscher (1991).

2. Formation of Stable Hydrides In order to predict the formation of metal hydrogen systems, the heat of formation has to be evaluated. Up to now, only a few first principle calculations have been performed. However, empirical and semi-empirical models have been proposed for the heat of formation and heat of solution of metal hydrides. For a recent review we refer to chapter 6 of Hydrogen in Intermetallic Compounds (Griessen and Riesterer, 1988). The cellular model of Miedema et al. (1976) and, more recently, the band structure model of Griessen and Driessen (Griessen and Driessen, 1984a, 1984b; Griessen et al., 1984) have successfully been applied in metal hydride research. While the former model is already known fairly well and thus needs not to be introduced separately, the latter one shall be described briefly, particularly because the electronic band structure is involved and thus the connection with magnetism is obvious. Empirical linear relations are proposed between the standard heat of formation H and characteristic band structure energy parameters of the parent elements in order to predict H of the ternary hydrides. In the case of binary metal hydrides the standard heat of formation is correlated with the difference between the Fermi energy and the energy of the centre of the lowest s-like conduction band of the host metal. In the case of ternary metal hydrides the energy difference for intermetallics of two d-band metals has been evaluated using the model of Cyrot and CyrotLackmann (1976). The exact density of states (DOS) of an alloy is approximated by a “simple” DOS, where the individual contributions of the elements are acting in an additive way (coherent potential approximation). There are various steps involved in the scaling of the DOS function of each metal. In the first step the widths of the d bands of both metals are set equal to their weighted average and the DOS curves are brought to a common width. In the second step the Fermi energies are equilibrated. The agreement of the calculated heat of formation values with the experiment was

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found to be remarkably good. In most of the cases the band structure model yields better results than the Miedema-model, which furthermore has the disadvantage of involving more fit parameters. The development of reasonable computational methods in the last decade led to a step forward compared to the semi-empirical models mentioned above. In particular, we want to mention the theoretical study of Gupta (1999) applying an ab initio self-consistent linear muffin-tin orbitals method to study the stability of several Zr- and La-3d ternary hydrides. As an example the results of ZrNiHx (x = 0, 3) are given. The total density of states (DOS) of both, parent compound and hydride is displayed in (Fig. 5.1). The comparison with the pure intermetallic reveals the several modifications: (i) On the low energy side a new structure associated with metal–hydrogen bonding and H–H interactions appears in the hydride between 4 and 13 eV below EF ; (ii) The energy separation between the Ni 3d and the Zr 4d main peaks decreases upon hydrogen uptake as a consequence of the lattice expansion, the Zr 4d DOS having been considerably modified by the presence of hydrogen; (iii) The Fermi energy of the hydride lies closer to the main Ni 3d peak. The observed reduction of EF is associated predominantly with the lattice expansion. Furthermore, the Zr–H interaction leads to a substantial lowering of the Zr 4d states located above the Fermi level in the parent intermetallic. This factor is of essential importance for the stability of the hydride. The contribution of the Ni 3d states to the metal–hydrogen bonding is sizable although it is lower than that of the Zr 4d states. This has to be attributed to the larger coordination number of H with Zr in both, the pyramidal (Zr3 Ni2 ) and the tetrahedral (Zr3 Ni) sites. The most important difference observed in the bonding of H with Ni and Zr lies in the fact that the Ni-d states are already occupied in the parent compound, while a majority of the Zr-d states are located above EF . This feature has important consequences on the position of the Fermi energy in the hydride. The effect of chemical substitution plays an important role in the reduced stability of a compound (e.g. LaNi4 M compared to LaNi5 ). The reason of the decreased stability of LaNi4 M compared to LaNi5 is found in the lattice expansion which accounts for about 50% of the decohesion of the compound. On the other hand, the Fermi energy is always found to rise upon hydrogenation, a factor which affects adversely the stability (Gupta, 2002).

3. Electronic Properties The knowledge of the electronic properties (band structure, DOS) considerably helps in understanding a material’s magnetic properties. In most of the stable metal hydrides, a simple interpretation of the band structure can be obtained, as

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(a)

(b) Figure 5.1 Total density of states of ZrNi (a) and ZrNiH3 (b) and number of electrons (dotted line). EF is chosen as origin of the energy axis (Gupta, 1999).

follows: since the hydrogen potential is more attractive than the metal atom potential, the lowest energy bands result from the hydrogen-metal bonding and the H–H antibonding interactions. The number of corresponding bands is usually equal to the number of hydrogen atoms in the unit cell (Gupta, 2002). Starting from the pioneering work of Switendick (1978, 1979), in the last years the number of papers dealing with band structure calculations has increased considerably (see e.g. Gupta, 1989, 1999; Singh and Papaconstantopoulos, 1994;

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Figure 5.2 Electronic density of states for majority spin and minority spin electrons for both (a) YCo3 H2 and (b) parent YCo3 . Zero of the energy axis is the Fermi energy (Cui et al., 2005).

Gupta and Rodriguez, 1995; Orgaz and Gupta, 1995, 2002; Elsässer et al., 1998a, 1998b, 1998c; Michalowicz et al., 2002; Crivello and Gupta, 2003; Wu et al., 2004; Jezierski et al., 2005; Orgaz et al., 2005). Moreover, the accuracy of the DOS and the Fermi energy calculations has grown substantially. Decomposition of the DOS into site and angular momentum components are now available for many metal hydrides. Even charge transfer calculations and hydrogen induced changes of the magnetic properties can now be explained in satisfactorily agreement with experimental data (Cr-H and YFe2 H4 , Crivello and Gupta, 2005). By using an ab initio density functional theory, Cui et al. (2005) succeeded in predicting the structure and the electronic structure of YCo3 H2 (Fig. 5.2). A somewhat different approach to study the electronic structure of LaNi5 Hx was performed by Monma et al. (2006) applying the DV-Xα method. The kind of the electronic charge transfer upon hydrogenation is an essential point for interpreting the hydrogen-induced change of the magnetic properties. In


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order to explain the magnetization data of rare-earth-transition-metal hydrides, a few earlier works favored a hydrogen-transition-metal charge transfer in connection with the rigid-band model (see e.g. Wallace, 1978, 1982). Later on, a similar interpretation has been given with regard to Mn, Fe and Ni hydrides (see e.g. Antonov et al., 1989). However, theory (energy band and DOS calculations, see e.g. Vargas and Christensen, 1987; Gupta, 1982, 1987, 1989, 1999) and experiment (Mössbauer studies performed on R nuclei and X(U)PS investigations, see e.g. Cohen et al., 1980; Schlapbach, 1982; Schlapbach et al., 1984; Höchst et al., 1985; Osterwalder et al., 1985) proved the indefensibility of this position. Details will be found below. There are a number of experimental methods in order to compare theory and experiment in the field of the electronic properties. The Pauli contribution of the magnetic susceptibility and the electronic specific heat coefficient γ are proportional to N(EF ). Resistivity measurements yield valuable results for binary hydrides (see section 5.1), for hydrides of intermetallic compounds this method is rarely applied because of experimental difficulties (contacting brittle samples or disintegration of the specimens into powder). Spectroscopic techniques such as electron and X-ray photo emission belong to the most powerful methods to study the electronic structure. A valence-band photoelectron spectrum resembles a one-electron DOS curve. Within some approximations, photoelectron spectra yield directly position and width of the occupied bands, charge transfer is indicated by XPS core level and Mössbauer isomer shifts. In valence fluctuation systems X-ray absorption experiments are particularly valuable. The X-ray absorption near-edge structure (XANES) contains information about the partial DOS and has become an increasingly important technique. Compton scattering, in particular, when associated with band structure calculations, was proved to be a powerful tool in studying the electronic structure of metal hydrogen systems (Mizusaki et al., 2003, 2005; Yamaguchi et al., 2007). Even complicated ternary hydrides are subject to theoretical studies nowadays. As an example, hydrides of the iron rich rare earth intermetallics, R2 Fe17 , are given, where a significant enhancement of the Curie temperature TC is observed which was claimed to correlate with the rate of the increase in the lattice constant a upon the introduction of hydrogen (Fujii et al., 1995). This effect is further discussed on the basis of the electronic band structure. According to the spin fluctuation theory of Moriya (1987), Lonzarich (1987), Mohn and Wohlfarth (1987), TC is proportional to the inverse of the density of states in the spin-up and spin-down bands at the Fermi level, N↑ (EF ) and N↓ (EF ). Several experimental results carried out on R2 Fe17 compounds suggest that the a axis expansion due to the introduction of interstitial hydrogen into the 9e sites within the dense (001) plane mainly brings about the reduction of the hybridization between Fe 3d and R 5d states, leading to a decrease in both N↑ (EF ) and N↓ (EF ). This hybridization reduction might play an important role in the suppression of the spin fluctuations yielding a rise in TC . Later on, Beurle and Fähnle (1992) presented a study on hexagonal Y2 Fe17 H3 (Th2 Ni17 type of structure) containing calculations within the framework of the local-spindensity approximation (LSDA) and the linear-muffin-tin-orbital (LMTO) method in atomic sphere approximation (ASA). The authors were able to show that there are two counteracting effects of the interstitial hydrogen: a geometrical effect (vol-

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ume expansion and local relaxation), increasing Fe moment and hyperfine field and a hybridization effect of the hydrogen atom with the neighboring Fe atoms, reducing these values. Concerning the magnetic moments, the results were found to be in good agreement with self-consistent augmented spherical wave (ASW) calculations for the rhombohedral Th2 Zn17 phase (Coehoorn and Daalderop, 1992). Band structure calculations with the LAPW method and the LMTO-ASA method were used to study the magnetic properties of YFe2 and its hydrides (Singh and Gupta, 2004 and Crivello and Gupta, 2005, respectively). The magnetic properties were explained by the two competing effects, mentioned above. In the hydride, the majority spin states of Fe were found to be fully occupied, the Fermi energy falling in a peak of the minority spin density of states. The increase in magnetization, observed for limited hydrogen concentration is attributed almost entirely to the lattice expansion. For a comprehensive description of the electronic properties of metal–hydrogen systems the reader is referred to the reviews of Switendick (1978), Gupta and Schlapbach (1988), Gupta (2002) and, published quite recently, to chapter 7 in Fukai’s book on basic bulk properties of metal–hydrogen systems (Fukai, 2005).

4. Basic Aspects of Magnetism Metallic magnetism covers a wide range of phenomena, which are intimately correlated with both the electronic structure and the metallurgy of a given metal or compound. Particularly the latter appears to be an important factor when considering the formation and properties of intermetallic compounds and binary (ternary) hydrides. Frequently the studies of hydrides of intermetallic compounds have led to a deeper insight into the fundamental properties of the parent system. For quite a long time 3d magnetism has been a controversial topic Wohlfarth, 1980, where still some problems are not completely settled. The reason for this controversy is the absence of a general agreement upon the microscopic nature of the magnetic state above and below the Curie temperature. Two opposite standpoints have so far been used to explain the magnetic order as a function of temperature. In the Heisenberg model magnetism is described in terms of localized moments and the magnetization vanishes at TC because of disorder in the local moments due to thermal fluctuations. Nevertheless, their absolute value remains almost independent of temperature. In the Stoner–Wohlfarth itinerant-electron model the magnetic moment is determined by the number of unpaired electrons in the exchange-split spin-up and spin-down bands. Within this model the thermal excitations of electron-hole pairs (single particle excitations of the Fermi–Dirac distribution) reduce the exchange splitting and thus favor the paramagnetic state. Consequently, the magnetization disappears only if the absolute value of the magnetic moment goes to zero, which only happens if the exchange splitting is zero, too. This model sufficiently describes magnetism in metals at 0 K, provided that the band structure and density of states is known to a sufficient accuracy and electron correlations are not too strong as in


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heavy fermion materials. For an analysis of experimental data obtained from ternary hydrides in terms of the Stoner–Wohlfarth theory, see section 5.4.2.4 (Fruchart et al., 1995). Parallel to the development of band structure theory in terms of the density functional formalism and the local spin density approximation (LSDA) there was a search for “simple toy-models” as e.g. the Hubbard model to reproduce solid state magnetism. Thus calculations of the ground state properties with high accuracy and reliability are now available. Accordingly, the understanding of complex mechanisms in the solid improved significantly. Among the remaining problems the temperature dependence of the magnetization was one of the most crucial ones. While for the localized moment models finite temperature effects became reasonably clear at a rather early stage this development, however, took some time for itinerant electrons in most solids since the calculated exchange splitting of the spin up and down bands was too large. Thus, the corresponding Curie temperatures were also too large by a factor of 4–8 and the inverse susceptibility is expected to show a T 2 rather than a linear temperature dependence as frequently observed. It become clear that the single particle excitations in the Stoner model are not—or only to a small extent— responsible for the finite temperature behavior of metallic magnetism. These results suggest that one has to consider two extreme limits: (i) the localized limit for which the magnetic moments and their fluctuations are localized in real space (delocalized in reciprocal space), with their amplitudes being large and fixed; (ii) the itinerant limit for which the moments and their fluctuations are localized in reciprocal space (delocalized in real space), with their amplitudes being temperature dependent. A Curie–Weiss law is observed in both cases, however, its physical origin and the corresponding Curie constant are different. To solve these inconsistencies Moriya (1987) included thermally induced collective excitations of the spin system—as they were already known for localized spins—in order to formulate an unified picture of magnetism. A similar approach was introduced by Murata and Doniach (1972) who introduced local and random classical fluctuations of the spin density (spin fluctuations) which should be excited thermally. The latter two models become equivalent at high temperatures and lead to a Curie–Weiss law. Thus, about hundred years after Langevin, there exists a fairly good knowledge about the basic mechanisms of localized and itinerant electron magnetism but many open questions still remain and a practical unified picture of magnetism is still not at hand. Magnetism of strongly correlated electron systems (in particular with unstable 4f and 5f moments as e.g. in Ce-, Yb- and U-intermetallics) is still not well understood but an actual research topic, see e.g. the conference series on Strongly Correlated Electron Systems (SCES) Physica B 378–380 (2006): One of the goals of modern condensed matter research is to couple magnetic and electronic properties to develop new classes of material behavior, such as high temperature superconductivity or colossal magneto-resistance, spintronics and the newly discovered multi-ferroic materials. Strong correlations between electrons lead to a renormalization of the electron mass by a factor of 1000 or more, which are therefore called heavy Fermions and are usually well described in terms of the Fermi liquid theory. Heavy electron materials lie frequently at the verge of a magnetic instability,

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in a regime where quantum fluctuations of the magnetic and electronic degrees are strongly coupled and significant deviations from Fermi liquid behavior may occur. Thus, these materials appear to be an important test-bed for development of the understanding about the interactions between magnetic and electronic quantum fluctuations, see e.g.: Coleman (2006) and v. Löhneysen et al. (2006). Contrary to the magnetism of the 3d-metals, the magnetic properties of the “stable” trivalent rare earth (R) elements are unambiguously described in terms of the RKKY theory; because of the localized nature of the 4f electrons no overlap exists between 4f wave functions on different lattice sites. Thus, the magnetic coupling can only proceed indirectly via the spatially non-uniform polarization of the conduction electrons. The pure 4f -4f interaction and its behavior upon the absorption of hydrogen can be studied directly not only in binary rare-earth hydrides, but also in ternary hydrides with a zero transition metal moment. As a first approximation one would expect that hydrogen induced changes in the magnetic properties of the latter can be explained in analogy with the binary hydrides, i.e. in terms of the anionic model. There, the conduction electron concentration is lowered after hydrogen uptake which in turn reduces the RKKY interaction. The rare earths form binary hydrides with the stoichiometries x = 2 and x = 3. In the case of x approaching 3, metallic conductivity disappears, which has been attributed by Switendick (1978) to the formation of a low-lying s-band with the capacity to hold six valence electrons. This in fact equals the number of electrons supplied to the conduction band by one R and three H atoms. Since this low lying bonding band is completely filled up with electrons, in the RH3 conduction electrons are no longer present, prohibiting the transmission of the RKKY interaction. This accounts for the suppression of the magnetic interactions, which indeed is generally observed experimentally. However, as will be seen later, details the physical properties of the rare-earth hydrides in the α-phase and of the broad homogeneity range of the R-dihydrides are only partly solved and several exceptions from the simple approach can be found. In R-3d intermetallics and their hydrides, where both the R and the 3d element carry a magnetic moment we can distinguish between three main types of magnetic interactions which are quite different in nature: that (i) between the localized 4f moments; (ii) between the more itinerant 3d moments; and (iii) between 3d and 4f moments. Generally, it is observed that these interactions decrease in the following sequence: 3d-3d > 4f -3d > 4f -4f . Actually, it is the combined effect of itinerant 3d electrons (providing a large Curie temperature), and localized 4f states (providing the magnetocrystalline anisotropy), which frequently make these compounds suitable for permanent magnet application. In contrast to the binary 4f hydrides, for ternary R-3d hydrides no similar straightforward arguments can be used about the hydrogen induced change of the magnetic order. The only statement being generally valid is that upon hydrogen absorption the magnetic order of Co and Ni compounds is considerably weakened which is not observed in the case of Fe compounds. As will be described in detail below, hydrogen absorption usually weakens the magnetic coupling between the 4f and the 3d moments and can lead to substantial


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changes of the 3d transition-metal moment in either way. As mentioned earlier, hydrogen in the lattice reduces the 4f -3d exchange interaction. This is explained by a reduced overlap of the 3d-electron wave functions with the 5d-like ones due the narrower bands as a consequence of the hydrogen induced increase in volume. Furthermore concentration fluctuations of H atoms over a few atomic distances may frequently occur, leading to a difference in electron concentration between one site and an other and, therefore, to a varying coupling strength. Additionally, a disturbance of the lattice periodicity takes place in the hydrides, reducing the mean free path of the conduction electrons (see section 5.1). This leads to a damping of the RKKY conduction electron polarization which in turn decreases the magnetic coupling strength. If the magnetic order in R-intermetallics is dominated by the 4f moments, the concept of an R-H charge transfer in analogy with the binary rare earth hydrides has proved to be a reasonable explanation for the hydrogen induced changes in magnetism (see isomer shift data obtained from Mössbauer studies on rare earth nuclei). In the case where 3d magnetism is dominant in the R-3d compounds, no general rule can be given. Commonly, hydrogen absorption leads to a loss in the 3d moment in Ni- and Co-based intermetallics, but to an enhancement of the Fe moment. As an example, the hydrogen induced change of the magnetic moment in GdCo2 H4 has been computed by using self-consistent electronic structure calculations, where further the significant drop in TC found experimentally could be confirmed (Severin et al., 1993). For Mn-intermetallics both changes from para- to ferromagnetism and vice versa are obtained. In Fe-containing intermetallic hydrides the 3d states are localized to a greater extent compared to the parent compound. This leads to an enhancement of the molecular field which, on the other hand, is opposed by the influence of the grown Fe–Fe distance, tending to reduce it. As it is observed experimentally, the former is apparently the dominating one, yielding an increased or at least an unchanged molecular field constant nRFe upon hydrogenation. When discussing the hydrogen induced change of the magnetic properties one is, among other things, faced with the problem of finding confidential moment data. Frequently, one has to rely on magnetization measurements, which may lead to wrong results in those cases, where from experimental reasons (lack of a high field facility) only incomplete saturation has been achieved. As will be seen below, particularly in the case of ternary hydrides, magnetic saturation is difficult to obtain. The situation, however, has been improved in the last decade, since in rare cases, ultra high magnetic fields, exceeding 100 T are available. An alternative way is offered by Mössbauer measurements carried out in zero applied field. However, the problem of correlating the hyperfine field unambiguously with the magnetic moment (particularly in the case of the hydrides) still remains. Fortunately, however, an increasing number of neutron diffraction results have been achieved more recently, showing that the assumption of a unique hyperfine coupling constant is an oversimplification. For a detailed summary of neutron diffraction data we refer to chapter 4 in Hydrogen in Intermetallic Compounds I (Yvon and Fischer, 1988) and the proceedings of the more recent metal hydrogen conferences (see section 1).

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5. Review of Experimental and Theoretical Results 5.1 Binary rare-earth hydrides 5.1.1 α-, α * -RHx solid solutions Hydrogen is readily absorbed by the rare earths (R) and forms solid solutions (αphases) at high temperatures. The solubility limits at a certain temperature generally increase with the atomic number. An extensive review of the situation has been given by Vajda (1995a). Thus, here we shall only recall the most essential facts with emphasis put on the magnetic properties. The rare-earth–hydrogen (R-H) phase diagram as presented in Fig. 5.3 is generally valid for the hcp heavy lanthanides Gd through Lu (with some limitations for Yb) and for Y. It is characterized by relatively broad existence ranges around the stoichiometric compositions, both in the α-phase solid solutions and for the cubic β-phase dihydrides and h.c.p. γ -phase trihydrides. As concerns the magnetic properties of the rare earths with incomplete 4f shells, their interaction with hydrogen is favored by the stability of the single-phase regions displayed in Fig. 5.3 at low temperatures, where these metals are magnetically ordered. In certain cases a metastable low-T α * -phase occurs (see below). The (upper) phase boundaries lie in the range between x = 0.03 (Ho) up to x = 0.35 (Sc), those for the β-phase RH(D)2+x between x = 0.03 (Lu) and x = 0.3 (Gd) (Vajda, 1995a, 2005; Udovic et al., 1999). The lower limits of the β-phase are purity dependent and reach ideally values close to 2.00; the width of the γ -phase has not been established definitely in most cases but is of the order of 0.1 H atoms/R. An important fact following from the particular shape of the phase diagrams is the tendency of the excess hydrogens, x, in each phase to form sub-lattices at higher x- and lower T -values, with a strong influence upon the electronic and thus, upon the magnetic properties. As mentioned above, the unusual fact of an existing H solid solution phase down to the lowest temperatures, without precipitation of the dihydride, has permitted to study its interaction with magnetism in Ho, Er, and Tm. Thus, it was found (for details and references, see Vajda, 1995a) that the transition temperatures to sinusoidal or helical antiferromagnetism (AFM), TN or TH , decreased in all three metals as well as the TC towards ferrimagnetism in Tm, while TC towards conical ferromagnetism (FM) in Er increased strongly upon hydrogenation. At the same time, a kind of magnetic hardening took place manifesting itself e.g. by a decrease of the critical field needed for the ferri-to-ferromagnetic spin-flip transition in Tm (Fig. 5.4) or by the increasing spiral period of the helical phase in Ho (Fig. 5.5). The observations were explained by the competition of several processes: on the one hand, a diminishing role of the RKKY exchange mechanism due to a decrease of the carrier density with increasing H concentration and, on the other, by a simultaneously growing influence of magneto-elastic and anisotropy effects. It should be pointed out that the special configuration of the α * -phase, in fact, is not a real solid solution but-as determined by neutron scattering-consists of H-R-H pairs on second-neighbor T-sites aligned in modulated quasi-unidimensional chains along the c-axis (Vajda, 1995a).


305

Figure 5.3 Generic phase diagram for R-H solubility, valid for bulk specimens, the hydrogen solubility is usually larger for thin films (Vajda, 2005).

Figure 5.4 1995a).

Magnetization as a function of applied field for several α * -Tm hydrides (Vajda,

This reminds one of the equally modulated AFM of those rare earths, where the α * -phase is present. In Fig. 5.5 this parallelism between the spin-density waves (SDW) in the concerned metals and the charge-density wave (CDW) formed by the zig-zagging structure of the α * -phase is demonstrated. The occurrence of the CDW of the α * -phase is related to an electronic topological transition on the Fermi surface, in particular to its webbing features. The α * -phase was found to form in systems where a situation is present similar to a Peierls transition. The three metals

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Figure 5.5 c-axis modulated magnetic configurations (SDW) in Ho, Er and Tm (to the left) and the modulated H-R-H chain structure (CDW) of the α * -phase (to the right) (Vajda, 2005).

with modulated configurations exhibiting a suitable turning angle ωi are just those forming an α * -phase, while the equally modulated AFM of Dy and Tb with a lower ωi do not (Vajda, 2005 and references therein). Theoretical efforts to study the eventual evolution of the Fermi surface upon introduction of hydrogen have been undertaken (to begin with) on the non-magnetic YHx system (Garcés et al., 2005). In the heaviest lanthanides (and Sc and Y) which retain hydrogen in solution (metastable α * phase) down to 0 K no evidence is found of an α–β phase transition, however, a resistivity anomaly in the range between 150 and 170 K is observed. This anomaly was attributed to short range ordering of the interstitial H atoms. In the case of α-LuHx it has been identified by neutron scattering as creation of linear chains of H–H pairs on tetrahedral sites along the 3c-axis surrounding a metal atom (Blaschko et al., 1985). Contrary to the heaviest rare earths, hydrogen in solution appears to be unstable in the lighter R elements below a certain temperature (decreasing from ≈700 K to ≈400 K for La to Dy, respectively) and precipitates into the β-phase (dihydride). α-HoHx , α-ErHx and α-TmHx yield a hydrogen interstitial solubility limit of 3, 7 and 11 at.%, respectively (Daou and Vajda, 1988), which was previously believed to be lower. Hydrogen in solution reduces the antiferromagnetic ordering temperature TN of Ho (133 K) at a rate of about 2 K/at.% H(D) which is in agreement with the results obtained for the two other magnetic R-hydrogen systems, α-ErHx and αTmHx (Daou et al., 1987). Details of this phenomenon have been described in the review of Vajda (1995a). These results were confirmed by resonant magnetic X-ray scattering at the Ho LIII -edge on hydrogenated thin Ho films (Sutter et al., 2001), where furthermore a hydrogen induced increase of the length of the magnetic spiral could be observed.


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The effect of hydrogen absorption upon the magnetic properties in α-ErHx has been studied by resistivity (Daou et al., 1980; Vajda et al., 1987b; Daou and Vajda, 1992), magnetic (Vajda et al., 1983; Ito et al., 1984; Vajda and Daou, 1984; Burger et al., 1986b; Boukraa et al., 1993a, 1993b) and specific heat measurements (Schmitzer et al., 1987). Er is well known to exhibit three different magnetic structures: below the Néel temperature TN of 85 K there is a sinusoidally modulated magnetization along the c axis, while the basal plane magnetization remains zero. The basal plane component starts to order in a helicoidal structure at TH = 51 K and spiral (conical) ferromagnetism is stable below TC = 19.5 K (for a review see Coqblin, 1977). The above mentioned measurements show that TN and TH decrease, while TC and TC1 (at which a transition to an incommensurate structure of 15 layers occurs) increase with H or D content (see Fig. 5.6). The rise of TC is interpreted by Burger et al. (1986b, 1987) in terms of a hydrogen induced dilatation of the c axis as a consequence of the interplay between the uniaxial anisotropy and the magnetoelastic energy and the coupling between the axial and basal plane magnetization. The increase in the electronic specific heat with rising H content (up to 1.5 at.%) indicates a growing density of states at EF for Er in the α-phase (at least at low concentrations), which leads to the suggestion that the decrease of TN , TH and the spin disorder resistivity ρspd is due to a reduction in the exchange interaction in agreement with magnetic measurements. The thermal variation of the resistivity and the specific heat in the ferromagnetic cone structure regime (below TC ) gives evidence for the existence of a gap like behavior in the spin wave spectrum, which is reduced with the addition of hydrogen in solution. From this variation of the exchange anisotropy gap and the nuclear specific heat Schmitzer et al. (1987) and Vajda et al. (1987b) draw the same conclusion as above, namely that the exchange field is reduced with rising hydrogen content in α-ErHx . Contrary to α-ErHx , for α-TmHx both TN (57.5 K) and the order-order transition at TC (39.5 K) decrease with rising H content down to 45.5 K and 29 K respectively for x = 0.1 (Daou et al., 1981; Vajda and Daou, 1984; Vajda et al., 1989c). This different behavior is suggested to arise from the specific magnetic structures, which is in the case of Tm sinusoidally modulated antiferromagnetic below TN and gradually squares up to yield below TC = 39 K an antiphase ferromagnet with three spins up and four spins down and has therefore no basal plane component in contrast to Er below TC . The fall of TN and TC in α-TmHx is governed also by the reduction of the indirect exchange interaction. In Tm, there is no interaction present between the short range ordered H–H pairs and the uniaxially aligned moments. On the other hand, in α-ErHx a strong interaction between the H–H pairs and the conical structure appears to change the magnetoelastic energy giving rise to a TC enhancement in Er with H in solution. In Fig. 5.7 a global view of the temperature dependence of the electrical resistivity of an α-TmHx single crystal parallel to the b and c axis is presented, showing a significantly different ρ(T ) behavior between the two crystal orientations with x. While the b axis oriented crystal exhibits a nearly linear increase of the residual resistivity ρrb with x (Fig. 5.7a), the apparent increase of ρrc (ρ parallel to the c axis) is much larger and nonlinear. In fact the resistivity decrease due to ferrimagnetic ordering is strongly suppressed

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Figure 5.6 (a) Variation of the magnetization with temperature for pure Er (x = 0) and ErH0.035 . A field of H = 0.02 T is applied parallel to the c axis [(o) x = 0, (+) x = 0.035] (Burger et al., 1986b). (b) Heat capacity of α-ErHx with various hydrogen concentrations: (!) x = 0; (E) x = 0.01; (1) x = 0.03 and (2) ErD0.03 (Schmitzer et al., 1987).


309

Figure 5.7 Temperature dependence of the resistivity parallel to the b axis for various α-TmHx crystals with the x values labeled; the insert shows the magnified low-temperature region. The arrows indicate the high-temperature anomaly above 150 K as well as the variation of the Néel temperature TN . (b) The same as (a) for single crystals with a c axis orientation. Both figures after Daou et al. (1988b).

by hydrogen in solution, disappearing completely for x > 0.05, which is attributed to the evolution of magnetic superzones, a phenomenon already observed to a smaller degree in single crystals of α-ErHx (Vajda et al., 1987b). The insert of Fig. 5.7a shows a substantial rise of the magnetic contribution to the resistivity

310


ρmag (T ) with x. The analysis of these data in terms of a sum of a power function and an exponential expression for the anisotropy gap b ρmag (T ) = AT n + BT 2 exp(–/kT)

shows that both n and decrease with rising x. Specific heat measurements show a similar trend (Daou et al., 1988b), namely that an anisotropy gap occurs in the spin wave spectrum which is reduced with growing amount of hydrogen and goes hand in hand with the development of a complex magnetic structure below 4 K. Of the magnetically disordered elements Sc, Y and Lu, the H-system of the latter has been investigated thoroughly by specific heat and susceptibility measurements (Stierman and Gschneidner, 1984). These authors state that Lu is a spin fluctuation system, where the fluctuations are quickly degraded by impurities and by hydrogen in solid solution. Both the susceptibility χ and the electronic specific heat coefficient γ show a similar variation as a function of H composition yielding a peak at 3 at.% and 1.5 at.% H, respectively. This difference is attributed to hydrogen tunneling giving rise to a linear contribution to the heat capacity. Correcting for this brings into good agreement the concentration dependence of both γ and χ with a peak at about 3% H. In this context it is worth to note that also in α-ErHx and presumably in α-TmHx an increase of the electronic specific heat with x is observed. However, in α-TmHx the increase of γ with growing x is not established for x > 0.02, since in this regime the magnetic contribution to the heat capacity could not unambiguously be resolved, a not yet determined magnetic structure occurring below 4 K (Daou et al., 1990). Susceptibility measurements of α-ScHx by Volkenshtein et al. (1983) indicate that the spin paramagnetism is reduced by a factor of two for x = 0.36. This can be associated (neglecting a possible change of the Stoner enhancement factor) with the decrease of the density of states at EF , whereby EF passes through a maximum of the N(E) curve down to lower energies. This trend of the α-phase is also observed in the dihydride. According to band structure calculations, susceptibility and spinlattice relaxation time measurements the DOS at EF in comparison with that of parent Sc is reduced by a factor of 3.5, 3 and 4.5, respectively. 5.1.2 Rare-earth dihydrides The rare earths commonly form dihydrides and trihydrides. The dihydrides exhibit a broad homogeneity range and crystallize except for Eu and Yb in the CaF2 structure. The CaF2 structure forms a f.c.c. unit cell, where in the ideal case all tetrahedral (T) sites are occupied. On increasing the hydrogen content, the octahedral (O) sites become gradually filled up with hydrogen atoms to form the BiF3 structure. Dihydrides of Eu and Yb are of the orthorhombic (Pnma) structure. In reality, the “pure” dihydride is frequently substoichiometric. The occupation of the octahedral (O) sites already starts sometimes for x = 1.8, depending on the material purity. In particular the oxygen content is of importance and also the sample shape (foils, powder etc.) used for hydrogen loading. It seems that the larger the purity of the parent material, the closer is the approach to the ideal stoichiometry of the dihydride. In the case of heavy rare earth with a purity of


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99.9% and 99.99%, the stoichiometry of the dihydride is usually 1.90 < x < 1.95 and 1.96 < x < 2.0 (Vajda, 2000, 2004), respectively. The absorption of hydrogen affects the magnetic properties of the rare earths indirectly via a reduction of the number of conduction electrons and a volume expansion. Both effects lead to a drastic decrease of the RKKY indirect exchange interaction between the localized 4f electrons mediated by the conduction electrons. Consequently, the magnetic ordering temperatures are much lower than in the parent metals (e.g. TC = 291 K for Gd, TN = 20 K for GdH1.93 ). On raising the hydrogen content above the pure dihydride, it is obvious that a random and/or ordered occupation of the octahedral sites with hydrogen significantly influences the crystalline electric field (CEF). This plays an important role in the change of the magnetic properties of those compounds located in the intermediate range between the di- and trihydrides. The rare earth dihydride acts as a monovalent metal (one conduction electron per atom), the trihydride as an insulator or a semiconductor. Whereas the electronic structure and the magnetic properties of stoichiometric dihydrides and trihydrides seem to be rather well understood, considerable confusion exists in the intermediate composition range (–0.15 to –0.05 < x < 1 of REH2+x ). To elucidate the transition between these two extreme situations this regime became therefore of growing interest about three decades ago, the research activities still ongoing. Simplifying the matter, one may expect a continuous decrease of the conduction electron density with rising x, implying that each H atom depopulates the conduction band by one electron through the formation of a low-energy metal–H band. However, this simple model is complicated by several structural transitions: attractive H–H interactions lead to a phase segregation with the formation of a dilute metallic phase (x = 0.1–0.2) and a concentrated nearly insulating phase or γ phase (x = 0.8–0.9). This seems to be the case in the heavy rare earths (R = Gd to Lu), where no homogeneous dihydride exists for x > 0.3. In the lightest rare earths (La, Ce and Pr) such segregations are not observed and mainly order-disorder transitions occur within the H-sublattice, presumably due to more repulsive H–H interactions (Burger et al., 1988). From electronic band-structure calculations, it is known that a charge transfer occurs from the metal atoms to the hydrogen atoms in the tetrahedral sites, whereas the hydrogen atoms at the octahedral sites can be considered as essentially neutral (Fujimori et al., 1980; Misemer and Harmon, 1982). Although the theoretical values of the charges transferred are still a matter of debate, the occurrence of charge transfer is supported by experimental XPS data on the metal core-level shifts obtained on hydrogenation (Schlapbach, 1982; Osterwalder, 1985; Gupta and Schlapbach, 1988). Negative charges at the tetrahedral sites yield crystalfield ground states, that have been confirmed experimentally by neutron scattering, Mössbauer spectroscopy, susceptibility and specific heat measurements. This favors the so-called anionic or hydridic model for the formation of binary rare-earth hydrides which, however, has to be regarded with some caution. The limits of this model have been assessed by Gupta and Burger (1980) by means of a site and angular momentum analysis of the DOS. These authors were able to show that there exists a considerable hybridization of the low-lying hydrogen-metal bands. For fur-

312


ther discussions of band structure calculations on RH2 and RH3 we refer to Gupta and Schlapbach (1988). Their results support a metal to semiconductor transition, whereby the explanation concerning the origin and nature of the gap and its opening particularly in the intermediate concentration range was still not resolved at that time. Later on, by considering the strong “breathing” of the hydrogen ion (i.e. a large change in the H 1s orbital radius upon orbital occupation); Eder et al. (1997) where able to interpret the opening of a substantial gap in YH3 . Their conclusion was that the ground state of YH3 corresponds closely to that of a Kondo insulator, with H binding two electrons in a singlet state. More recently, the gap in YH3 was confirmed by first-principle calculations, independent of the crystal structure which still is an open issue (Wolf and Herzig, 2002, 2003). Sufficient agreement with available experimental data were obtained. For a compilation of electronic structure problems in metal hydrides we refer to the review of Gupta (2002). The rare earth dihydrides order antiferromagnetically with ordering temperatures below 20 K except CeH2+x , NdH2+x and EuH2 which also exhibit ferromagnetic order. In view of their orthorhombic structure (Pnma), dihydrides of Eu and Yb are also an exception among their cubic neighbors. The magnetic structure of the RD1.95 for the light rare earth (R = Ce, Pr, Sm) and GdD1.95 has been resolved by neutron diffraction and appears to be rather similar: for Sm and Gd dideuterides ferromagnetic coupling occurs within the (111)-plane which couple antiferromagnetically with the adjacent sheets (MnO-type structure), while for Ce and Pr an additional modulation within the (111) plane is ob¯ and [112] ¯ direction respectively (Arons and Schweizer, 1982; served into the [110] Arons and Cable, 1985). Thus NdH1.95 is the only ferromagnet in the antiferromagnetic series of the pure (x = 0) R deuterides of the cubic CaF2 or BiF3 structure and seems therefore to be an unresolved exception. The RD2 compounds in which R is a heavy rare earth element (R = Tb, Dy, Ho ) exhibit √modulated magnetic structures (see Fig. 5.8) where the modulation period 4a0 11 along [113] is commensurate with the crystallographic lattice (Shaked et al., 1984). This corresponds to a ferromagnetic coupling of the magnetic moments within the (113) planes and an antiferromagnetic alignment between ¯ for those planes. The direction of the spin axis is [001] for Tb and Dy but [863] HoD2 . The magnetic structure of ErD2 contains both a commensurate component (belonging to the magnetic lattice (4a0 ) and an additional incommensurate component, details of which are treated in the experimental section below. Furthermore, it should be pointed out, that for the heavy rare-earth dihydrides (R = Tb to Dy) generally magnetic short-range order is present (see experimental section). The ordering temperatures and the magnetic structure of the single-phase dihydrides with the respective x values according to various authors are collected in Table 5.1. The orthorhombic dihydrides of Eu and Yb are not included. From this Table 5.1 it is obvious that those elements situated on the boundaries of the 4f series exhibit the more complicated antiferromagnetic structure, while Nd with ferromagnetic order is rather exceptional in this type of compounds. The transition temperatures reported for the pure dihydrides agree fairly well with each other, although the hydrogen content given for the pure dihydride varies significantly in

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Figure 5.8 Table 5.1

R

Magnetic structure of TbD2 and HoD2 according to Shaked et al. (1984).

Magnetic properties of cubic single-phase dihydrides RH2+x

Transition temperature (K)

Type of magnetic order

x

Ref. a

Ce

6.2 6.9

AF AF

¯ axis MnO type modulated along the [110]

–0.05 0

[1] [2]

Pr

3.3 3.5

AF AF

¯ axis MnO type modulated along the [112]

–0.05 –0.03

[3, 4] [5]

Nd

6.8

F

Sm

9.6 9.6

AF AF

Gd

21

0.0

[6]

MnO type

–0.15 –0.12

[7] [8]

AF

MnO type

–0.7 ∼ =0.0 –0.5 –0.08 ∼ =0.0 ∼ =0.0 ∼ =0.0 ∼ =0.0 ∼ =0.0 ∼ =0.0 ∼ =0.0 ∼ =0.0 –0.1 ∼ =0.0 ∼ =0.0 0.0

[7]

Tb

17.2 18.5 18.0 19.0

AF AF AF AF

¯ Commensurate modulated along [113], spin axis along [001] k1 = 1/4[113], k2 ∼ = 1/8[116]

Dy

3.5 5.0 5.2

AF AF AF

Ho

4.5 6.5 6.3

AF AF AF

¯ Commensurate modulated along [113], spin axis along [001] k1 ∼ = 1/4[113], k2 ∼ = 1/40[11, 11, 30] ¯ Commensurate modulated along [113], ¯ spin axis along [863] k1 ∼ = 1/4[113], k2 ∼ = 1/40[11, 11, 30]

Er

2.15 2.13 2.23

AF AF AF

Commensurate and incommensurate components k1 =?, k2 ∼ = 1/40[11, 11, 30] ∼ = 1/8[116]

Tm

No magnetic order down to 2 K

[9] [10] [5] [11] [9] [12] [11] [9] [12] [11] [9] [13] [11] [14] [15]

a References: [1] Arons et al. (1987c), [2] Vajda et al. (1990), [3] Arons et al. (1987a), [4] Arons and Cable (1985), [5] Vajda et al. (1989a), [6] Senoussi et al. (1987), [7] Arons and Schweizer (1982), [8] Vajda et al. (1989b), [9] Shaked et al. (1984), [10] Arons et al. (1982), [11] Vajda (2005), [12] Daou et al. (1988a), [13] Opyrchał and Biegański (1976), [14] Kubota and Wallace (1963b), [15] Burger et al. (1986a).

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Figure 5.9

Magnetic phase diagram of CeH2+x (Vajda, 1995a).

particular cases. This indicates as discussed above that the purity of the starting material is of crucial importance since impurities presumably occupy the tetrahedral lattice sites which prevents the formation of the strictly stoichiometric dihydride RH2 . This suggestion deduced from the comparison of the transition temperatures and magnetic phase diagrams (Table 5.1, see also Fig. 5.9) seems to be in contradiction with the statement of Arons et al. (1987c). From entropic arguments that a stoichiometric dihydride with all tetrahedral sites occupied by H atoms does not exist, the occupation of the octahedral sites starts already at RH1.95 with 2.5% vacancies (Schlapbach et al., 1987) on the tetragonal lattice sites. Already a slight increase of absorbed hydrogen leads to a loss of long-range magnetic order and shows sometimes spin-glass-like behavior at low temperatures. In this context it should be noted that the nature of the magnetic transition of hyperstoichiometric dihydrides at low temperatures depends sensitively on the cooling rate. From a resistivity anomaly at about 150 K which is different for quenched and slowly cooled samples (10 K/min and 0.3 K/min) Vajda et al. (1985, 1989a, 1993) deduced that in the latter case short range ordering occurs within the octahedral H-sublattice. This significantly affects the magnetic transition, presumably as a result of a modified crystal field scheme due to local symmetry distortions. The influence of the amount of absorbed hydrogen on the magnetic ordering temperature of several heavy rare-earth β-dihydrides was found to be by no means straightforward (Boukraa et al., 1993a, 1993b). Valuable information about the behavior of hydrogen in binary hydrides has first been obtained from the analysis of the susceptibility by Wallace and Mader (1968) and from the Schottky anomaly in the low-temperature specific heat (Biegański and Staliński, 1970, 1979). Further information is found in references given by Arons (1982). The energy level scheme which has been derived clearly favors the anionic model. Inelastic neutron scattering and Mössbauer spectroscopy are further techniques which have been applied in order to determine the crystal-field level scheme in binary RH2 hydrides (Shenoy et al., 1976; Knorr and Fender, 1977; Knorr et al.,


315

1978; Friedt et al., 1979a, 1979b; Arons, 1982; Arons et al., 1986, 1987b). From these results the anionic state of the hydrogen ions has been corroborated, too. In the heavy rare earth dihydrides the analysis of the paramagnetic spin-disorder resistivity in terms of crystalline-field effects gives furthermore reasonable agreement with the generally considered anionic H model (Daou et al., 1988a, 1988b). In the following we present experimental results predominantly obtained since 1980, in particular concerning the intermediate range between the di- and trihydrides. For previous data of the magnetic properties of those hydrides we refer to the comprehensive compilation by Arons (1982) and the reviews by Libowitz and Maeland (1979) and Wallace (1978, 1979). Details concerning more recent investigations can be found in the comprehensive review of Vajda (1995a). CeH2+x Cerium and its compounds exhibit an exceptional behavior in the series of rare earths. The ambivalent character of the one 4f electron, behaving either atomic like as in γ -Ce or less localized and stronger hybridized as in α-Ce, gives rise to fascinating magnetic and electronic properties as e.g. mixed valency, Kondo and heavy fermion behavior (Fisk et al., 1988). Later on, in monophase β-CeH2+x , order-disorder transformations, exhibiting features of a first-order phase transition were found for x < 0.35 and 0.65 < x < 0.75, whereas the features of a continuous second-order phase transition in the interval 0.35 < x < 0.65 were obtained applying the static-concentration-waves theory (Ratishvili et al., 1993, 1994). The pure dihydride (x = 0) behaves like a monovalent metal, with one conduction electron of d-character per Ce atom. CeH2 is antiferromagnetic below 7 K. Figure 5.9 shows that the magnetic order changes from antiferromagnetism in the slightly hydrogen deficient dihydride CeD1.95 via no magnetic order at about x = 0.05 (for T > 1.3 K) to ferromagnetism for 0.1 < x < 0.75 and again to antiferromagnetism at x > 0.8 (Arons et al., 1987a). Additional magnetic transitions have been observed in CeH1.95 by heat capacity measurements (Abeln, 1987) and by resistivity measurements at various x values 0 < x < 0.4 (Vajda et al., 1990). These additional transitions, whose nature is not yet resolved, are also presented in the phase diagram proposed by Abeln (1987) and Arons et al. (1987a). Good agreement between the two data sets of Abeln (1987) and Vajda et al. (1990) is obtained, if the hydrogen stoichiometry of the respective samples is shifted by 0.05 at.% relative to each other. This means that the substoichiometric dihydride CeH1.95 of Abeln (1987) corresponds to CeH2.00 of Vajda et al. (1990). A narrow paramagnetic interval separates the intermediate ferromagnetic and the antiferromagnetic range for x > 0.8. In this context it is worth noticing that Kaldis et al. (1987) reported on a miscibility gap for 0.56 < x < 0.64 between the tetragonally distorted and the cubic structure at room temperature. However, for CeH2.8 , a two phase region (cubic and tetragonal) occurs below 238 K while above this temperature only the cubic phase appears to be stable. The antiferromagnetic structure of CeD1.95 and CeD2.91 is presented in Fig. 5.10. The obvious difference between these magnetic structures is the additional antiferromagnetic modulation along the [110] direction with a period of 5 times a0 for CeD1.95 . The magnetic moment determined by neutron diffraction (Schefer et al., 1984) in the ferromagnetic range (1.1 μB at 1.3 K for x = 0.29) as well as in the antiferromagnetic CeD2.96 (0.61 μB at 1.3 K) is by far smaller than that expected for

316


Figure 5.10 Magnetic structure of (a) CeD1.95 and (b) CeD2.91 (after Abeln, 1987). The antiferromagnetic coupling between (111) planes is additionally modulated along the [1-11] direction for CeD1.95 .

a free Ce3+ ion (2.14 μB ). These moments lie just between the calculated moments of 1.56 μB and 0.71 μB corresponding to a 8 and 7 ground state, respectively. From high field measurements up to 30 T on a single crystal also a rather low saturation moment of 0.9 μB has been derived which hardly changes with the hydrogen content (Arons et al., 1984). Except for the ferromagnetic CeD2.46 compound the moment attains 1.03 μB . The reduced moment may be attributed to crystal field effects since the overall crystal-field splitting was determined from susceptibility measurements by Osterwalder et al. (1983) to be 285 K (assuming the 8 quartet to be the ground state). This finding is corroborated by inelastic neutron scattering and susceptibility measurements, according to which the 7 doublet is situated 20 meV above the 8 ground state (Abeln, 1987). The fourfold degenerate 8 ground state of CeH1.95 will be split into two doublets separated by 12 K as the hydrogen content is increased up to CeH2 but remains almost unchanged for higher hydrogen contents. Abeln (1987) deduced this from inelastic neutron scattering and from a pronounced Schottky anomaly occurring in the heat capacity of CeH2 at about 5 K. Later on, Vajda et al. (1990) interpreted their electrical resistivity measurements on some CeH2+x specimens to reflect incoherent and coherent Kondo lattice behavior above and below T ≈ 20 K, respectively. The incoherent term was found to exhibit a log T behavior, showing the influence of the crystal field degeneracies. While the incoherent Kondo properties are only weakly x-dependent, the magnetic properties are extremely sensitive to them. For the low-temperature resistivity the typical T 2 -dependence of a Kondo lattice was obtained. The electronic specific heat coefficient γ of CeH2+x is rather large (≈100 mJ/mol K2 ) (Schlapbach et al., 1987). With the value of A ≈ 0.2 μ cm/K2 , the ratio A/γ 2 fits nicely into the Kadowaki-Woods plot. This and the comparatively large γ -value indicate the tendency of Ce hydrides towards heavy Fermion behavior. Eventually, low-temperature anomalies were observed due to magnetic transitions (see Table 5.1) which tend to disappear when upon quenching (cooling rate ≈103 K/min) atomic disorder is introduced into the sublattice of the supplementary x H-atoms (Vajda et al., 1990).


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Compton scattering was applied by Yamaguchi et al. (2007) to study the electronic structure of CeH2.1 . In this experiment the change in photon energy is related to the electron momentum just before the scattering. The full three dimensional electron momentum density can be reconstructed from directional Compton profiles. Thus, this technique is suitable to elucidate important features of the Fermi surface. PrH2+x Below TN = 3.3 K PrD1.95 orders antiferromagnetically (MnO type, see above) with an ordered moment of 1.5 μB /Pr atom (Arons et al., 1987c), while PrH2.25 is a weak Van Vleck paramagnet down to 2 K (Wallace and Mader, 1968, and references given by Arons, 1982). From the analysis of susceptibility measurements of PrH2x in terms of crystal field effects Wallace and Mader (1968) proposed the anionic model for hydrogen in these types of compounds which has later been supported by specific heat measurements (Biegański and Staliński, 1970; Biegański, 1972, 1973). Both experimental results are satisfactorily described by the 5 ground state caused by the crystal field splitting of the degenerate 34 H ground state of Pr3+ due to the surrounding of negatively charged hydrogen ions. This assumption is furthermore confirmed by inelastic and polarized neutron scattering (Knorr and Fender, 1977; Knorr et al., 1978; Arons et al., 1987a). The antiferromagnetic transition and the resistivity minimum at about 28 K, which has been attributed by Vajda et al. (1989a), Burger et al. (1990) and (Vajda, 1995a) to crystal field effects, if the first excited state 1 (non magnetic) is close to the ground state. Thus, the contribution of spin-disorder scattering to the resistivity first is reduced upon rising temperature, before increasing again due to the taking over by phonon scattering. For samples with x > 0.2 neither susceptibility (Wallace and Mader, 1968) nor resistivity measurements down to 1.5 K manifest magnetic order. According to the specific heat measurements (Drulis and Biegański, 1979) the ground state for PrH2.57 is a nonmagnetic singlet which is in line with the nearly temperature independent Van Vleck susceptibility at low temperatures. The transition from antiferromagnetism to Van Vleck paramagnetism was explained by Arons et al. (1987b) in terms of a degeneracy of the magnetic 5 and the singlet 1 states. However, this needs a fairly large change of the cubic crystal field parameter x (in the notation of Lea et al., 1962) from x > 0.54 to x < 0.54. This seems to be questionable, because the additional hydrogen atoms on octahedral sites should be considered as essentially neutral. In view of the significant resistance anomaly at 150 K it seems likely that non cubic ordering of the octahedral hydrogens modifies the local symmetry of the crystal field. While for PrD2 the crystal field experienced by the majority of Pr ions is cubic, Knorr et al. (1978) demonstrated by a careful analysis of their neutron data that in PrD2.5 the distribution of the octahedral hydrogen interstitials occurs not at random but rather in a mer-XA3 configuration leading to an orthorhombic crystal field at the Pr site. With this orthorhombic crystal field symmetry, they could explain their neutron and susceptibility data of PrD2.5 satisfactorily. Furthermore, it is stated that Pr does not undergo a valence change from PrH2 to PrH2.5 . Besides the pronounced resistivity anomaly at 150 K commonly found in these superstoichiometric samples a further anomaly occurs in the hydrogen richest com-

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pound x = 0.76 at about 220–250 K with indication for a first order transition (similar to Ce and La) (Burger et al., 1988; Vajda et al., 1989a). The strongly x-dependent structural transformations affects the magnetic transition at low temperatures via a modification of the local crystal field symmetry. For high x-values the analysis of the phonon and the residual resistivity by Burger et al. (1988) implies that the carrier density decreases strongly, thus the system approaches the metalinsulator transition. NdH2+x NdH2 has been reported by Kubota and Wallace (1963a) to order ferromagnetically at 9.5 K with a moment of 1.36 μB , while Carlin et al. (1982) found TC = 5.6 K and a saturation moment of 1.97 μB . The latter ordering temperature is in good agreement with specific heat measurements of Biegański et al. (1975c), NMR investigations (Kopp and Schreiber, 1967) and resistivity studies (Daou et al., 1992), indicating magnetic ordering at 6.2 K. Senoussi et al. (1987) performed systematic hysteresis measurements on NdH2+x up to x = 0.7 which indicate for x = 0 ferromagnetic behavior below 6.8 K with a coercivity of 150 Oe and a spontaneous moment of 1.06 μB . For increasing H-content the spontaneous moment is drastically reduced. Moreover thermomagnetic irreversibilities observed by zero field cooled (ZFC) and field cooled (FC) M versus T measurements point to freezing effects and spin glass behavior. A clear cut spinglass behavior, however, is rather unlikely for the stoichiometric dihydride since in their specific heat measurements Biegański et al. (1975c) obtained a pronounced sharp peak at 6.2 K. Both the considerably reduced moment (relative to the free Nd3+ value 3.37 μB ) and the freezing phenomena (growing with rising hydrogen content) may arise from a complex interplay between the RKKY interaction and the magnetic anisotropy. In particular the random uniaxial anisotropy, which could be induced by the crystal field and the local fluctuations of the hydrogen concentration on the octahedral interstitials, together with the reduced conduction electron concentration are suggested to suppress long range magnetic order for x > 0. SmH2+x For the antiferromagnetic transitions in SmH2+x good agreement is obtained from susceptibility and resistivity measurements (Arons and Schweizer, 1982; Vajda et al., 1989a). However, the H-stoichiometries of the corresponding samples differ significantly: the pure dihydride is referred to as SmH1.85 and to SmH1.98 by the above authors, respectively. With rising H-content TN is shifted from 9.6 K to lower temperatures (to 5.5 K for x = 0.08, Arons, 1982 and to 8 K for x = 0.16, Vajda et al., 1989a). In the resistivity curve the antiferromagnetic transition is observed as a sharp transition for x < 0.1 which becomes smoother at higher x values and disappears at x = 0.26. Furthermore, a resistivity minimum occurs just above TN remaining observable up to x = 0.26. The ρ(T ) minimum was tentatively attributed to incommensurate magnetic order or to corresponding critical fluctuations (Vajda et al., 1989b). The magnetic structure is of the MnO type and changes to an incommensurate structure, as found for GdH2+x (Arons et al., 1987a). Vajda et al. (1989a) stated that not only the concentration of octahedral H-atoms but also their configuration is of importance for the shape of the transition. The reason for this is that quenching from room temperature introduces local disorder due to


319

Figure 5.11 Resistivity of GdH(D)2+x for various x-values. Note the magnetic transitions below 100 K, the structural anomalies and the metal–semiconductor transition (for x = 0.305) at 260 K (Vajda and Daou, 1993; Vajda, 1995b).

the presence of isolated H-atoms on octahedral sites which can be recovered above about 150 K where a resistivity anomaly occurs. EuH2+x EuH2 is a ferromagnetic semiconductor with TC = 18.3 K, θ = 23.13 K, Ms = 7.1 μB and μeff = 7.94 μB (Bischof et al., 1983). The semiconducting behavior of the divalent dihydride is in accord with the anionic model. These susceptibility studies were corroborated by specific heat measurements on dihydrides and dideuterides, small deviations being probably due to uncertainties in the hydrogen concentration (Drulis and Staliński, 1989; Drulis, 1993). GdH2+x The antiferromagnetic MnO type structure in GdD1.93 (with TN = 20 K) changes with the introduction of octahedral x-atoms into an incommensurate helical structure with the axis along [111] and TN = 15.5 K (Arons, 1982; Arons and Schweizer, 1982). The excess hydrogen (x) located on octahedral interstitials has a tendency to order below room temperature in short-range and long-range ordered structures which influences the type of magnetic order. Based on resistivity measurements, Vajda et al. (1991) proposed a structural and magnetic phase diagram for 0 ≤ x ≤ 0.25 with MnO-type antiferromagnetic, helical and three incommensurate antiferromagnetic structures. In a further work, Vajda and Daou (1993) succeeded in preparing hydrides with x-values up to the β-phase limit. The richness of the magnetic manifestations in the resistivity of the system GdH2+x is evident from Fig. 5.11. Later on, the results obtained by Vajda’s group were corroborated by investigations on single crystalline films (Hémon et al., 2000). TbH2+x In the dihydride TbH2+x the Néel temperature (TN = 18.5 K for x = –0.05; TN = 16.11 K for x = –0.07, Drulis et al., 1984b; Biegański et al., 1975a) rises strongly with x up to TN = 40 K for x = 0.12 (Arons et al., 1982). For

320


Figure 5.12 Magnetic phase diagram of TbH(D)2+x , exhibiting regions of two commensurate phases (a) and (b) and an incommensurate phase (c). (×) neutron data, o resistivity data, (1) susceptibility data, (F) specific heat data (Vajda et al., 1993).

x = –0.05 a commensurate antiferromagnetic structure appears below 15.8 K. For larger hydrogen concentrations different incommensurate structures with an axial sinusoidal modulation are observed which are stable from TN down to liquid helium temperature. The transition from the former type of structure into the latter has been ascribed to the random occupation of some octahedral sites in the f.c.c. Tb lattice. By performing cold-neutron-diffraction experiments Vajda et al. (1993) were able to determine the magnetic structure more precisely. Two magnetic phases were found to be present at low temperatures, one commensurate antiferromagnetic below TN , the other incommensurate antiferromagnetic phase between TI (I for intermediate) and TN = 16 K. For the pure dihydride, TbH2 , the propagation vector of the former phase is k = 1/4(113); for TN < T ≤ TI = 19–21 K the latter incommensurate phase with k ≈ (0.123, 0.137, 0.754) was obtained. Both, resistivity and neutron diffraction measurements enabled Vajda and Daou (1993) and Vajda et al. (1993) to establish a tentative magnetic phase diagram of β-TbH2+x (Fig. 5.12). In stoichiometric TbH2 (TN = 17.2 K, x ≈ 0.0), the saturation moment equals μ = 7.2 μB which is reduced by the crystal field as compared to the free ion value of 9 μB . Shaked et al. (1984) reported on a modulation of the magnetic √ structure (period 4a0 11) that is commensurate with the crystal lattice. For higher x values the magnetic contribution to the resistivity changes drastically (Vajda et al., 1987a): two peaks occur below 38 K for x = 0.16 (Fig. 5.13). Their height and absolute value depend furthermore upon the cooling rate. This phenomenon was attributed by André et al. (1992) to a DO22 tetragonal distortion of the f.c.c. lattice of the superstoichiometric compound which, later on, could be confirmed by the theoretical treatment of Ratishvili and Vajda (1993). The importance of axial symmetry, but without taking an ordering process into account, was stressed by Drulis et al. (1984b) to explain the specific heat data of


321

Figure 5.13 Resistivity as a function of temperature for several β-TbH2+x compounds (Vajda and Daou, 1993).

TbH2.06 : hydrogen in the octahedral positions generate a crystal field with axial symmetry for 30% of the Tb3+ ions with a doublet as the ground state, while the remaining 70% experience the cubic crystal field potential with a singlet ground state. In this view it seems rather plausible that the configuration of octahedral H interstitials determines the magnetic properties in the range 0 < x < 1, while a simple reduction of the conduction electrons with rising x would mainly lower the Néel temperature. √ DyH2+x The antiferromagnetic structure is modulated with a period of 4a0 11 along [113] with an ordered moment of about 3 μB –4 μB below TN = 3.3–3.5 K (Biegański et al., 1975b; Shaked et al., 1984). Carlin and Krause (1981) derived from the broad maximum of the magnetic susceptibility a Néel temperature of 3.2 K. Considering specific heat measurements (Biegański et al., 1975b; Biegański and Staliński, 1976) and 161 Dy Mössbauer (Eγ = 26 keV) spectroscopy studies (Friedt et al., 1979a) the crystal field-level scheme was established by Daou et al. (1988a) to consist of a 7 doublet ground state as expected for the anionic model. Furthermore, 161 Dy Mössbauer spectroscopy showed a significant increase of the line width already below about 7 K. Between 5 K and 3.3 K unresolved magnetic hyperfine patterns indicated a distribution of magnetic hyperfine fields (Friedt et al., 1979a) with a gradual increase of the average hyperfine field on lowering the temperature. Below 3.3 K a sharp magnetic pattern characteristic of a single magnetic field is observed (Fig. 5.14). Moreover, the large drop of the spin-disorder resistivity in the range from 140 K to TN is in good agreement with the crystal field ground state configuration (Daou

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Figure 5.14 161 Dy Mössbauer spectra recorded from DyD2 at temperatures close to the onset of magnetic ordering (Friedt et al., 1979a).

et al., 1988a). By performing resistivity measurements Vajda and Daou (1992) were able to show the close interaction between magnetic and structural ordering. Later on, by cold-neutron diffraction in β-DyD2 below T2 = 5 K Vajda et al. (1997) observed two sinusoidally modulated magnetic configurations, one for T ≤ T1 ≈ 2.5–3.5 K, the other for T1 ≤ T ≤ T2 . The magnetic structure below T1 was found to be nearly commensurate, that between T1 and T2 being more incommensurate. The presence of a spontaneous precession signal in muon spin rotation (μ+ SR) experiments (Gygax et al., 2000, 2002) revealed the development of a commensurate short-range ordered magnetic structure below 10 K in DyD2.135 (Fig. 5.15) which is compatible with the neutron diffraction data of Vajda et al. (1997) on the same sample. This finding has been attributed to a possible octahedral-hydrogen lattice ordering in this compound. Nothing of that kind could be observed in β-DyD2 , even at temperatures down to 1.9 K. HoH2+x The magnetic structure is similar to those of the Tb and Dy dihydrides. ¯ with an ordered moment of 6.4 ± 0.4 μB However, the spin axis is along [863] below TN = 4.5 K. The small moment, as compared to the free-ion moment, is due to moment reduction by the crystal field (Shaked et al., 1984). By means of cold-neutron-diffraction experiments, Vajda et al. (1998) succeeded in characterizing all existing magnetic configurations in the pure Ho dideuteride, HoD2 . Below T2 = 6.3 K an incommensurate modulated structure with a propagation vector k 2 ≈ (0.273, 0.273, 0.748) could be established. Between T1 ≈ 3 K up to 4 K a


323

Figure 5.15 Zero-field muon spin rotation spectrum taken from DyD2.13 at T = 4 K (Gygax et al., 2002).

Figure 5.16 Electrical resistivity of β-HoH2+x as a function of temperature; open symbols: x = 0, full symbols: x = 0.12. In the inset an enlarged view is displayed of the low-T resistivity observed for the parent sample (x = 0) and of the derivative, –ρ/T , for the superstoichiometric sample (x = 0.12) (Vajda et al., 1998).

commensurate AF configuration occurred simultaneously with a propagation vector k 1 = 1/4(113). The two structures coexist down to 1.4 K. In the temperature region T2 < T 45 K anisotropic short-range-order magnetism is observed. For HoD2.12 , magnetic short range order is measured up to 7 K at the same position as for x = 0; an analysis shows correlation with the h.c.p. γ phase indicating martensitic memory effects and electronic phase separation. The transitions in both samples are evident from the resistivity behavior upon temperature (Fig. 5.16).

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Figure 5.17 Resistivity as a function of temperature for β-ErH2+x in slowly cooled (relaxed, R, !) and quenched (Q, ") states. Note the hysteresis between cooling and heating (Vajda, 1995b).

Mössbauer spectra were taken from the dideuteride doped with Er using the Er Mössbauer effect (Eγ = 80.6 keV) (Friedt et al., 1979b). A 7 ground state was proposed for the Er impurity in HoD2 , surprisingly unlike that in ErD2 , since both ErD2 and HoD2 have nearly the same lattice parameter and should have similar electronic structures. By the μ+ SR experiments on β-HoD2.0 and β-HoD2.12 , Gygax et al. (2002) were able to show a critical behavior around about 6 K, the magnetic short range order probably being present well above this temperature, as seen by neutron diffraction (Vajda et al., 1998). 166

ErH2+x The magnetic structure below TN = 2.15 K contains both a commensurate and an incommensurate component whereby the former belongs to a cubic magnetic lattice with a lattice parameter of 4a0 (Shaked et al., 1984). 166 Er (80.6 keV) Mössbauer spectra (Shenoy et al., 1976; Friedt et al., 1979b) clearly identify a 6 ground state in this compound which is corroborated by susceptibility and specific heat measurements interpreted in terms of crystal field effects (Arons, 1982). We want to point out, that the magnetic ordering temperatures of the three latter compounds determined from the disappearance of the Zeeman splitting in the Mössbauer spectra well agree with those values obtained from the anomaly in the specific heat measurements. As it is obvious from inspection of Fig. 5.17, around 250 K a semiconductor– metal transition develops for x > 0.07. The low-temperature minimum at 15–20 K,


325

Figure 5.18 Susceptibility of YbH2.41 as a function of temperature (Drulis et al., 1988).

for x = 0.07 and 0.088, having being supposed to be of magnetic origin, changes its shape drastically and shifts to 120 K in the case of x = 0.091, evidently signifying a semiconductor-metal transition. Thus, the authors came to the conclusion that the metallic behavior of the latter sample is limited to the interval between the two transitions, i.e. from 120 K to 240 K. Recent neutron diffraction studies (Vajda et al., 2005) revealed below TN = 2.23 K two coexisting sinusoidally modulated antiferromagnetic configurations, however, no commensurate antiferromagnetic configuration could be detected down to 120 mK. On the other hand, similar to the observations on TbD2 and HoD2 (Vajda et al., 1993, 1998), magnetic short range order shows up near 1.5 K in ErD2 and remains up to 10 K. TmH2+x No magnetic order was detected by susceptibility measurements (Kubota and Wallace, 1963b) and by resistivity measurements (Burger et al., 1986a) down to 2 K. The spin-disorder resistivity is almost constant above 150 K and decreases rapidly at low temperatures with a tendency to vanish for T = 0 K. This result is interpreted by Burger et al. (1986a, 1987) on the basis of a nonmagnetic ground state separated from the first excited state by a gap of 150 K. YbH2+x The orthorhombic dihydrides (YbH2+x , –0.2 < x < 0.0) are semiconducting and divalent according to optical reemission measurements (Gupta and Schlapbach, 1988; Büchler et al., 1989). Semiconductivity of the dihydride appears to be a consequence of the divalency, as in EuH2 . Substoichiometric trihydrides with x > 0.2 are metallic and are reported to be of the f.c.c. structure (Drulis et al., 1988, 1999; Büchler et al., 1989). Susceptibility measurements of YbH2.41 down to 2 K (Fig. 5.18) indicated together with specific heat measurements the presence of a Kondo scattering mechanism and/or an intermediate valence behavior. The susceptibility was found to deviate from Curie–Weiss behavior, leveling off below 4 K.

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The heat capacity showed a pronounced upturn at low temperatures with a high C /T value of 589 mJ/mol K2 at 2.48 K (Drulis et al., 1988). Finally, the photoelectron spectra according to Büchler et al. (1989) clearly show a valence transition from the divalent YbH2 with a 4f 14 configuration to a mixed valent behavior in YbH2.6 with a 4f 13 /4f 14 configuration. These studies were completed by further specific heat measurements (1.75 K, Iwasieczko et al., 1997, 1999, 2001; Drulis et al., 1999), the average valency of Yb having been derived to lie in the range 2.6–2.7. The effective moment was found to be slightly increasing with the amount of hydrogen in the sample, its maximum value of 3.85 μB /Yb being significantly smaller than expected for a Yb3+ ion (4.54 μB ) (Iwasieczko et al., 2001). Finally, it could be demonstrated that in the metastable β phase YbH2.25 Yb is in the trivalent state, exhibiting a magnetic behavior characteristic for normal light lanthanide trivalent elements (Iwasieczko et al., 2001). 5.1.3 Rare-earth trihydrides Trihydrides exhibit a hexagonal closed packed (h.c.p.) metal atom arrangement, except the lighter rare earth La, Ce and Pr (Renaudin et al., 2002) which form trihydrides where the cubic closed packed (ccp) metal atom arrangement is sustained at least under ordinary conditions. Complete structure data for RH3 have been reported for R = Y (Udovic et al., 1996), Pr (Renaudin et al., 2002), Nd (Renaudin et al., 2000), Sm (Kohlmann et al., 2007), Ho (Mansmann and Wallace, 1964) and Dy (Udovic et al., 1999). The crystal structure of the trihydrides with the hcp metal atom arrangement is a trigonal LaF3 (tysonite) type structure in which hydrogen or deuterium fills one tetrahedral and two trigonal metal interstices. While the trihydrides are ionic semiconductors, the dihydrides are metallic with low hydrogen content and semiconductors with high concentration of hydrogen (Libowitz, 1972). However, by applying sufficiently high pressures (≈15 GPa) upon some RH3 compounds (R = Gd, Ho, Er, Lu) a reversible structural transition to a cubic phase could be achieved which is supposed to be metallic (Palasyuk and Tkacz, 2004, 2005; Palasyuk et al., 2005). The rare-earth trihydrides (except CeH2.753 ) were believed to show no magnetic ordering, down to liquid helium temperature (Kubota and Wallace, 1963b; Wallace, 1978; Birrer et al., 1989). This was claimed to be consistent with the assumption of anionic hydrogen and a completely depopulated conduction band, giving rise to the semiconducting or insulating behavior of those hydrides. In the Ce-H system, the trihydride (though not really a γ -phase, since cubic) exhibits the same AFMcoupled (111) planes as the dihydride but without its modulation (Vajda, 1995a). In the trihydrides the magnetic interaction is mostly dipole-dipole, with large crystal-field effects and RKKY exchange is not possible because of the absence of carriers in these insulating materials. Daou et al. (1992) observed in the case of NdH2+x that the occurrence of ferromagnetic ordering (for x ≤ 0.32) changes into incommensurate antiferromagnetic ordering with the opening of a gap for x ≥ 0.6. Alternatively, this has been attributed by the authors to the localization of the conduction electrons. The large resistivity over the whole temperature range is attributed to the almost insulating behavior of the γ -phase (trihydride).


327

Figure 5.19 Intensity of the magnetic peak (0, 0, 1/2) obtained from a neutron diffraction experiment on DyD3 (left); magnetic structure of DyD3 (right) (Udovic et al., 1999).

SmD3 is an insulating paramagnet down to 2 K with an effective moment of 0.36 μB and crystallizes with the tysonite type structure with three independent ordered deuterium atom sites having trigonal planar, trigonal pyramidal and tetrahedral metal environments (Kohlmann et al., 2007). The reduced effective moment with respect to that of the free Sm3+ state (0.845 μB ) and the significant curvature of the inverse susceptibility is typical for Sm compounds and can be attributed to crystal field effects. GdH3 is a semiconducting antiferromagnet with a low Néel temperature (1.8 K, Carlin et al., 1980). Miniotas et al. (2002) discovered in GdH3–x thin films a large negative magnetoresistance and an anomalous temperature and field dependence of the magnetization. These observations were interpreted by the authors in terms of an electronic phase separation, promoted by substoichiometric regions of the hydride phase. From an early 161 Dy Mössbauer study, Friedt et al. (1979a) concluded DyH3 to be a paramagnet down to 1.6 K. More recently, however, Udovic et al. (1999) had determined the magnetic structure of DyD3 by neutron scattering as AFM-stacked FM planes along the c-axis of the type aabb (Fig. 5.19), confirming the transition temperature of TN = 3.3 K which was earlier observed by Carlin and Krause (1981). Thus, the absence of any magnetic hyperfine interaction in the Mössbauer experiment has to be obviously attributed to the occurrence of fast spin fluctuations. Neutron diffraction experiments on γ -ErD3 (Vajda et al., 2005) indicated an asymmetric magnetic short range order structure which was found to disappear at TN = 590 mK. This result is in fine agreement with TN ≈ 0.6 K, obtained from magnetization measurements (Flood, 1978). A proton NMR study by Weizenecker (2001) showed that γ -TmH2.73 remains a van Vleck paramagnet down to liquid helium temperature, just like the cubic βTmH2 , but the splitting between the non-magnetic ground state, 2 , and the first excited magnetic state, 5(2) , has decreased by more than 1/3: from 174 to 111 K.

328


(a)

(b) Figure 5.20 Magnetic susceptibility data and main susceptibility contribution as a function of temperature for Yb2.67 (a); Specific heat as a function of temperature of YD2.46 and YbD2.71 (b) (Drulis M. and Drulis H., 2004).

Due to experimental limitations, in the early study by Wakamori et al. (1986) no magnetic order could be determined in almost stoichiometric Yb trihydride (YbH2.96 ). The magnetic moment was found to be μeff = 4.37 μB /Yb. Later on, for non-stoichiometric Yb trihydrides Iwasieczko et al. (1997) proposed an antiferromagnetic transition just below 4.2 K. Since this result was in contradiction to neutron diffraction results, the system was re-investigated by means of specific heat and susceptibility measurements (Drulis, 1999; Iwasieczko et al., 1999). The lowtemperature maximum observed in χ (T ) (Fig. 5.20) was interpreted to reflect an


329

intermediate valence state of the Yb ions in this material. The specific heat studies were extended over a larger temperature range by Drulis et al. (2002) and by M. Drulis and H. Drulis (2004). Besides the anomaly at 4 K, in the case of YbH2.71 another one could be observed at 230 K (Fig. 5.20), the latter being attributed to a metal-insulator transition. Nothing of that kind was obtained for YbH2.46 . The low temperature specific heat anomaly is explained as a feature characteristic of a Kondo semiconductor with a small hybridization gap suppressed by the divalent Yb ions, disrupting the Yb3+ lattice periodicity. The magnetic moment per Yb atom, μeff = 3.85 ± 0.05 μB was found to be almost independent of the hydrogen concentration. This value is significantly smaller than that predicted by Hund´s rule for an Yb3+ ion (μeff = 4.54 μB ), confirming that Yb stays in an intermediate valence state.

5.2 Binary actinide hydrides The early actinide hydrides exhibit fascinating properties. In particular the structural properties may be classified as being unique in the periodic table. Complex phases form for the ThHx and UHx systems that are not observed for other metals. In the PaHx system, simple b.c.c. cubic C15 Laves and A15 phases occur depending on temperature and composition. Rare earth like hydrides with the CaF2 structure are found beyond uranium for the NpHx and PuHx systems with a trivalent metallic state. For a general review on the properties of actinide hydrogen systems we refer to Ward (1985a). The magnetic and electronic properties of the actinides and their intermetallics are largely determined by the partly filled 5f shell (for details and references we refer to the review by Sechovský and Havela, 1988). Concerning the localization of the 5f electrons the actinides may be placed between the d transition metals and the rare earth elements. The 5f electrons in the actinides are less localized than the 4f electrons in the corresponding rare earth series, but the 5f -5f overlap decreases on going from the early to the late actinides. The fact that the degree of 5f localization is determined by the 5f -5f overlap is documented in the well known Hill plot which correlates the most simplest ground states (superconductivity, paramagnetism or magnetic order) with the actinide–actinide distances. Superconductivity occurs in the early actinides (Th, Pa and U) while spin fluctuation effects are found in Np and Pu. For the transplutonium elements the 5f electrons become more localized and thus, starting from Am, the series becomes rare earth like. As the hydrogen absorption generally expands the lattice and reduces the 5f -5f overlap, a more localized behavior is expected and indeed observed in the hydrides than in the parent metals. Th4 H15 is a superconductor with a rather high transition temperature (8 K). Magnetism occurs in the U and Pa hydrides, but disappears in the PuHx system and reappears in the Pu hydrides. The 5f electrons finally become fully localized for the transplutonium elements and the heat of formation approaches that of typical rare earths hydrides (Ward, 1985b). The following later actinide hydrogen systems are expected to exhibit properties similar to those of the rare earth. Unfortunately, very few experiments have been performed because of the intense radioactivity of the transplutonium elements.

330


5.2.1 ThHx ThH2 with the face centered tetragonal structure is isostructural with the dihydrides of Ti, Zr and Hf but exhibits an appreciably larger lattice constant. The higher hydride Th4 H15 is superconducting below 8 K and crystallizes in a complex b.c.c.structure containing 16 atoms per unit cell (Satterthwaite and Toepke, 1970; Ward, 1985a, 1985b). No evidence for superconductivity could be found for ThH2 down to 1 K, although the parent metal is superconducting below 1.37 K. The reappearance of superconductivity in Th4 H15 initiated band-structure calculations, inelastic neutron scattering experiments and heat capacity measurements (Miller et al., 1976; Winter and Ries, 1976; Dietrich et al., 1977). From specific heat measurements with and without external field Miller et al. (1976) concluded that Th4 H15 is a bulk type II superconductor whose properties are in fair agreement with the BCS theory. The electronic specific heat coefficient (γ = 8.07 mJ/mol K2 ), the Debye temperature (θ = 211 K) and the electron-phonon enhancement factor (λ = 0.84) of Th4 H15 is by 87%, 29.5% and by 58%, larger than in the parent metal. According to valence band spectra (Weaver et al., 1977) the increase of the γ value is not caused by an enhanced density of states at the Fermi energy. No significant increase of the phonon enhancement factor is derived from band-structure calculations by Winter and Ries (1976). According to their calculation, they predicted a TC -enhancement if Th is substituted by elements with a lower valency. This, however, is not in agreement with the experimental results which show a depression of TC (Oesterreicher and Bittner, 1977). 5.2.2 PaHx No magnetic order was detected by susceptibility measurements above 4 K in the C15 Laves phase and the A15 phase. The effective paramagnetic moment is 0.84 μB and 0.98 μB , respectively (Ward et al., 1984). 5.2.3 UHx Usually the β-UH3 phase occurs which crystallizes in the A15 structure, while the α-UH3 phase is difficult to prepare and contains frequently a mixture of αand β-phases (Ward, 1985b). Both crystal structures belong to the Pm3n group. There are many magnetic and NMR measurements of the β-hydride and few of the α-hydride which are reviewed by Ward (1985a). Both order ferromagnetically. The paramagnetic Curie temperature of α-UH3 is between 174 K and 178 K. According to specific heat, neutron diffraction and magnetic measurements TC of β-UH3 is in the range between 170 K and 181 K. Due to the lack of saturation the data of the spontaneous moment exhibit a considerable scatter (0.87– 1.18 μB ), while the neutron diffraction result of Shull and Wilkinson (1955) gives a moment of 1.39 μB . This is obviously a consequence of a rather high magnetocrystalline anisotropy, which is also reflected in the heat capacity: Fernandes et al. (1985) analyzed the specific heat data of Flotow and Osborne (1967) in terms of spin wave contributions and found good agreement with the experimental data if an energy gap of about 80 K is taken into account. The appearance of an energy gap in the ferromagnetic spinwave spectrum is a strong indication for a high magnetocrystalline anisotropy. The electronic specific heat coefficient of β-UH3


331

Figure 5.21 High-field magnetization curves (in decreasing field) of UH3 at different temperatures at ambient pressure (left); Arrott plots taken at ambient pressure (right) (Andreev et al., 1998).

(γ = 28.7 mJ/mol K2 ) is nearly by a factor of 3 larger than that of U metal. By analogy with Ce hydrides this γ -enhancement may presumably arise from a f –d correlation effect as in heavy fermion systems rather than from a simple increase of the density of states at the Fermi energy. Concluding, uranium hydride β-UH3 holds an outstanding position in actinide magnetism, being the first 5f electron ferromagnet experimentally determined. Andreev et al. (1998) found the following magnetic properties: high spontaneous volume magnetostriction, large high-field magnetic susceptibility, reduction of magnetic moment and TC upon external pressure, low magnetic anisotropy and low anisotropic magnetostriction, all features pointing to an itinerant character of magnetism and, thus, to a significant 5f -delocalization (Figs. 5.21, 5.22). 5.2.4 NpHx Neptunium forms analogous to the rare earth hydrogen systems a cubic dihydride (CaF2 -structure) and a hexagonal trihydride. The susceptibility of NpHx (x = 2.04, 2.67, 3.0) exhibits only a weak temperature dependence which is nearly constant below 200 K (Aldred et al., 1979). A crystal field calculation based on the Np3+ (5f 4 ) ground state yields good agreement with the experimental data. 5.2.5 PuHx Magnetic order occurs in the PuHx system for all x values (1.99 < x < 3.0) and changes from antiferromagnetic order in PuH1.99 (TN = 30 K) to ferromagnetism (Aldred et al., 1979; Ward, 1985a). Instead of antiferromagnetism in the powdered dihydride ferromagnetic order was reported for a bulk sample with x = 1.93 (TC = 45 K; Willis et al., 1985). The Curie temperatures increase with the hydrogen content up to 101 K for the hexagonal trihydride, while the spontaneous moments

332


Figure 5.22 Magnetic moment as a function of applied pressure at 4.2 K (left); Curie temperature as a function of applied pressure (right) (Andreev et al., 1998).

decrease from 0.57 μB for x = 1.93 to 0.353 for x = 3.0. By analogy to the Np-system Aldred et al. (1979) suggested from susceptibility measurements a Pu3+ (5f 5 ) ground state. With the same crystal field parameters as for NpH2 the magnetic ground state consists mainly of the J = 5/2 manifold with 3% admixture of J = 7/2. Thus the expected ordered moment (1.0 μB ) is significantly higher than the experimental value ( 5000 with modulus of elasticity of 95 MPa). A schematic cross section of this device at different stages of the fabrication process is shown in Fig. 6.29. It was fabricated using a mixture of bulk and surface micromachining. The final device had a full deflection of the 500 µm and 1000 µm length beams with coil currents/powers of 80 mA/320 mW and 24 mA/19 mW respectively. Deflection forces up to 200 µN were demonstrated. A range of different approaches have been taken to fabricate a suitably compact and low power device. Fullin et al. (1998) have further developed the idea of a Permalloy cored device by using a soft magnetic substrate of FeSi in place of the thick film NiFe substrate used by Wright et al. (1997). A simplified diagram of the magnetic circuit used by Fullin is shown in Fig. 6.30. For this circuit an analytical expression for the force can be obtained and is given by equation (18). Ftot =

μ0 Ag (NI )2 4(lg +

Ag l c 2 ) 2Ac μrc

– kt

(19)

Magnetic Microelectromechanical Systems: MagMEMS

513

Figure 6.29 Schematic cross section of a relay employing a high permeability plated Permalloy core with a copper coil and gold contacts. (a) After bulk machining of vias in silicon. (b) After deposition of contact pads, plating of coil and deposition of insulating oxide layer. (c) After plating of Permalloy beam over sacrificial layer of hard baked resist. (d) After plating of Permalloy on back of wafer and stripping of sacrificial resist layer.

Figure 6.30 Equivalent magnetic circuit given by Fullin et al. (1998), for the MEMS relay using a magnetic substrate.

where Ag and lg are the area and length of the two air gaps, N /2 is the number of turns on each coil and I the current flowing through both, Ac and lc are the cross sectional area and length of the keeper, μrc is the relative permeability of the circuit. Figure 6.31 compares the results of this analytical calculation with those obtained by finite element analysis. The agreement is sufficiently good for the analytical expression to be used in device design. To fabricate a practical device a two chip construction, as illustrated in Fig. 6.32, was used. The upper quartz chip carries the NiFe moving keeper which is supported on two NiFe cantilever beams. The two chips are bonded using a conductive paste to give a rest air gap of 20 µm. When the coil on the lower substrate is energised

514

M.R.J. Gibbs et al.

Figure 6.31 Force as a function of pole gap for data taken from Fullin et al. (1998), Cugat et al. (2003). Analytical calculation (eq. (18)) is the solid curve and data points are from FEA simulation.

Figure 6.32

Schematic construction of relay after Fullin et al. (1998)

the keeper is pulled down onto the pole pieces and closes the contact. The device can be switched up to 500 Hz with a time to open of 0.2 msec and a time to close of 0.3 msec. The open resistance is greater than 10 G at 100 V. The device was modified by Tilmans et al. (1999) to provide a fully packaged microrelay in an SOIC-16 package. In order to reduce total power dissipation, which would be critical for RF relays in mobile equipment, latching devices are required. Two main types of magnetic


Figure 6.33

515

Functional diagram of hybrid switch after Borwick et al. (2003).

device have been fabricated. The first uses a hybrid technique where the actuation is provided by the long-range magnetic force and the holding, or latching force, is provided by the shorter-range electrostatic interaction. The second uses magnetic bi-stability to provide the latching action. A schematic illustrating the operation of a hybrid device produced by Borwick et al. (2003) is shown in Fig. 6.33. Here the suspension carries the activation current. A magnetic field is produced by external permanent magnets and provides the Lorentz force for actuation. The capacitors provide the electrostatic holding force. The structure is of a lateral type where the motion is in the plane of the substrate. Several structures were fabricated with differing spring constants for the suspension and various types of contact. The softest structure, having a resonant frequency of 750 Hz, required only 0.9 mA to actuate and 0.9 V to hold the switch closed. Although the force generated with the current is 0.9 µN it is sufficient to provide enough displacement for the electrostatic actuator to take over and hold the switch closed. This device had very low power but suffered from being very slow and with a tendency for the suspension to buckle when opening a closed switch with more than a few micro-Newtons of force. A stiffer device was also fabricated which had a resonant frequency of 4.5 kHz, but required 20 mA to actuate and 6 V to hold closed. Guan et al. (2005) have also produced a hybrid device, but in their case the motion is normal to the plane of the substrate. A schematic cross section of this device is shown in Fig. 6.34. It is fabricated by surface micromachining using an electroplated Cu sacrificial layer to produce the working gap in the relay. The magnetic field (B) is generated by permanent magnets that form part of the package in a similar way to that for the lateral device previously described. The current passing through the upper electrode and its suspension provides the Lorentz force that causes the actuation of the relay. The electrostatic force between the upper

516

M.R.J. Gibbs et al.

Figure 6.34

Schematic cross section of device after Guan et al. (2005).

electrode and the lower holding electrodes then provides the latching action. The crossbar and the lower contacts form the signal path for the relay. Devices fabricated in this way required a holding voltage of between 6 V and 12 V with an average value of 8 V. The contact resistance was in the range 150 m to 340 m with an average value of 170 m . Electrostatic breakdown was between 106 V and 225 V with an average value of 161 V and typical current carrying capacity was 1 A. The RF isolation for this device was approximately 70 dB at 20 MHz. A device utilising the bi-stability of a magnetic dipole in an applied field to provide a latching action was developed by Ruan et al. (2000) and was subsequently commercialised as the Maglatch™. The principle of operation is illustrated in Fig. 6.35, which shows the two stable states for the magnetic armature. Making the armature out of a soft magnetic material (Permalloy) with significant shape anisotropy to ensure uniform magnetisation along the armature ensures that it can easily be magnetised along its length and the direction of magnetisation can be readily switched in a relatively small external applied field. Thus the system has two stable states where the external bias field, produced by a permanent magnet, holds the magnetised armature either in the position illustrated by the solid line drawing or the dashed line drawing in Fig. 6.35. To switch from one state to the other a field orthogonal to the bias field is applied from the integral coil to magnetise the armature in the opposite sense. The armature then rotates about the pivot to the opposite stable state. The pivot point is deliberately made off centre along the armature’s length to increase the force on the contacts and reduce the on resistance of the relay. Finally, Table 6.9 summarises the characteristics of some different devices fabricated to date. It is clear that non use a permanent magnet as the armature, indicating the lack of good thin film permanent magnet materials for MEMS applications. 5.2.2 Motors The drive towards developing MEMS based magnetic motors is provided by such application areas as Robotics (Hayashi and Iwatuki, 1998), Micro-Surgery (Polla, 2001) and Optical Positioning (Collins et al., 1999). A range of magnetic actuators for various special applications have been demonstrated using electroplated soft magnetic materials such as Permalloy (Liu, 1998), commercial permanent mag-

517


Figure 6.35 Schematic diagram illustration the principle of operation of a bi-stable latching relay using Permalloy strips to provide uniform magnetization in the armature. The two stable states for the armature are illustrated. The flux from the coil is used to switch from one state to the other. (After Ruan et al., 2000). Table 6.9

A summary of the characteristics of some devices fabricated to date

Ref.

Contact Activation resistance power (m ) (mW)

Wright et al. – (1997) Fullin et al.

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