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The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research to serve the following purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic; • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany
[email protected] Markus Donath Wolfgang Nolting (Eds.)
Local-Moment Ferromagnets Unique Properties for Modern Applications
ABC
Editors Professor Markus Donath Westfälische-Wilhelms Universität Münster Physikalisches Institut Wilhelm-Klemm-Str. 10 48149 Münster, Germany
[email protected] Professor Wolfgang Nolting Humboldt-Universität zu Berlin Institut für Physik Newtonstr. 15 12489 Berlin, Germany
[email protected] Markus Donath and Wolfgang Nolting, Local-Moment Ferromagnets, Lect. Notes Phys. 678 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b135699
Library of Congress Control Number: 2005930333 ISSN 0075-8450 ISBN-10 3-540-27286-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-27286-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Printed on acid-free paper
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Preface
For an understanding of the fascinating phenomenon of ferromagnetism, one needs a description of the mechanism that underlies the coupling of the magnetic moments. In some materials, the magnetic moments are caused by itinerant electrons of partially filled conduction bands: the band ferromagnets. In others, they are due to localized electrons of a partially filled atomic shell: the local-moment ferromagnets. The latter class comprises the classical localmoment systems like some rare-earth elements and compounds but also more complex materials like diluted magnetic semiconductors and half-metallic ferromagnets. These materials are a hot topic of current scientific research for two reasons. On the one hand, the exchange interaction between the localized magnetic moments and the quasi-free charge carriers in these materials is far from being fully understood. On the other hand, some of these materials are promising candidates for modern applications in magnetoelectronic as well as spintronic devices because of their unique magnetic properties. The present book provides a status report on our current knowledge about these interesting materials gained from experimental investigations as well as theoretical descriptions. The various chapters in this book “Local-Moment Ferromagnets: Unique Properties for Modern Applications” are written in tutorial style by experts in the field. They were invited to an international specialists’ conference held under the same title in Wandlitz near Berlin (Germany) from 15 to 18 March 2004. It was the third seminar of this type in Wandlitz. The first seminar in 1998 dealt with magnetism and electronic correlations in classical local-moment systems: Magnetism and Electronic Correlations in LocalMoment Systems: Rare-Earth Elements and Compounds, ed. by M. Donath, P.A. Dowben, W. Nolting (World Scientific Publishing, Singapore, 1998). The second seminar in 2000 was dedicated to the microscopic understanding of band-ferromagnetism as an electron correlation effect: Band-Ferromagnetism: Ground-State and Finite-Temperature Phenomena, ed. by K. Baberschke, M. Donath, W. Nolting, Lecture Notes in Physics 580 (Springer, Berlin, 2001). The III. Wandlitz Days on Magnetism in 2004 came back to the phenomenon of local-moment ferromagnetism but with a special focus on particular materials with unique properties as described above. The presentations of twenty-seven invited speakers from thirteen different countries initiated in-
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tense and fruitful discussions between the sixty participants of the conference. More results were presented in form of posters during the three days of the seminar. The organizers hope that the lively discussions in Wandlitz support actual and future collaborations between the various specialists in the field of local-moment ferromagnets. Of course, this book cannot give a complete account of these fascinating subjects, given the tremendous worldwide activity, but rather focuses on the authoritative work of the contributors to the conference. Generous financial support by the Deutsche Forschungsgemeinschaft for this conference made it possible to bring together experimentalists and theoreticians, senior researchers and graduate students, to discuss the present state of affairs, to learn from each other, and to define joint projects for the future. Sincere thanks are due to the staff and associates of the Lehrstuhl Festk¨ orperphysik of the Institute of Physics at the Humboldt-Universit¨ at zu Berlin for doing an excellent job with the organization of the seminar. We wish to thank Prof. Dr. J¨ urgen Braun for his time-consuming work in collecting and composing the contributions to this book. We enjoyed the always effective collaboration with the Springer Verlag.
M¨ unster, Berlin August 2005
M. Donath W. Nolting
Contents
Introduction M. Donath, W. Nolting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Part I Concentrated Local-Moment Systems Critical Behaviour of Heisenberg Ferromagnets with Dipolar Interactions and Uniaxial Anisotropy S.N. Kaul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Critical Exponents and Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Scaling and Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Renormalization Group and Crossover Phenomena . . . . . . . . . . . . . . . 5 The gadolinium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary and Future Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 12 14 15 20 26 28
Aspects of the FM Kondo Model: From Unbiased MC Simulations to Back-of-an-Envelope Explanations Maria Daghofer, Winfried Koller, Alexander Pr¨ ull, Hans Gerd Evertz, Wolfgang von der Linden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Monte Carlo Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31 31 32 35 36 44 44
Carrier Induced Ferromagnetism in Concentrated and Diluted Local-Moment Systems Wolfgang Nolting, Tilmann Hickel, Carlos Santos . . . . . . . . . . . . . . . . . . . 1 Local Moment Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kondo-Lattice (s-f) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Electronic Selfenergy of “Concentrated” Local-Moment Systems . . . . . . . . . . . . . . . . . . . . . .
47 47 49 52
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4
Magnetic Properties of “Concentrated” Local-Moment Systems . . . . . . . . . . . . . . . . . . . . . . 5 “Diluted” Local-Moment Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57 62 66 68
An Origin of CMR: Competing Phases and Disorder-Induced Insulator-to-Metal Transition in Manganites Yukitoshi Motome, Nobuo Furukawa, Naoto Nagaosa . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71 71 74 75 80 84 85
A Neutron Scattering Investigation of MnAs K.U. Neumann, S. Dann, K. Fr¨ ohlich, A. Murani, B. Ouladdiaf, K.R.A. Ziebeck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structural Aspects of MnAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Magnetic Properties of MnAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Paramagnetic Neutron Scattering Investigation . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87 87 89 91 93 95 96
Epitaxial MnAs Films Studied by Ferromagnetic and Spin Wave Resonance T. Toli´ nski, K. Lenz, J. Lindner, K. Baberschke, A. Ney, T. Hesjedal, C. Pampuch, L. D¨ aweritz, R. Koch, K.H. Ploog . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic FMR/SWR Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Phase Transitions in MnAs Studied by FMR . . . . . . . . . . . . . . . . . . . . 4 Magnetic Anisotropy in MnAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Inter- and Intra-Stripe Coupling in the MnAs Films . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99 99 100 102 105 106 109 110
Part II Diluted Magnetic Semiconductors First-Principles Study of the Magnetism of Diluted Magnetic Semiconductors L.M. Sandratskii, P. Bruno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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2 Calculational Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Single Band in the Frozen-Magnon Field . . . . . . . . . . . . . . . . . . . . . . . 4 Results for (GaMn)As,(GaCr)As,(GaFe)As . . . . . . . . . . . . . . . . . . . . . 5 (ZnCr)Te . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Properties of the Holes and Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . 7 Comparative Study of (GaMn)As and (GaMn)N . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 118 120 123 124 127 131 131
Exchange Interactions and Magnetic Percolation in Diluted Magnetic Semiconductors J. Kudrnovsk´y, L. Bergqvist, O. Eriksson, V. Drchal, I. Turek, G. Bouzerar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Curie Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133 133 135 141 145 146
The Role of Interstitial Mn in GaAs-Based Dilute Magnetic Semiconductors Perla Kacman, Izabela Kuryliszyn-Kudelska . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 High Resolution X-ray Diffraction (HRXRD) Measurements . . . . . . 3 Channeling Experiments (c-RBS and c-PIXE) . . . . . . . . . . . . . . . . . . . 4 SQUID Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Exchange Interactions of Mn Interstitials . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149 149 152 153 156 158 161
Magnetic Interactions in Granular ParamagneticFerromagnetic GaAs: Mn/MnAs Hybrids Wolfram Heimbrodt, Peter J. Klar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Growth and Preparation of Hybridstructures . . . . . . . . . . . . . . . . . . . . 3 Magneto-Optical Properties of the GaAs:Mn Matrix . . . . . . . . . . . . . 4 Galvano-Magnetic Properties of Paramagnetic GaMn:As Epitaxial Layers . . . . . . . . . . . . . . . . . . . . . 5 Ferromagnetic Properties of MnAs Clusters in GaAs:Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Galvano-Magnetic Properties of Hybrid structures . . . . . . . . . . . . . . . 7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165 165 166 168 171 174 176 183 183
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Dilute Ferromagnetic Oxides J.M.D. Coey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187 187 190 194 198 199
Part III Half-Metallic Ferromagnets Half-Metals: Challenges in Spintronics and Routes Toward Solutions J.J. Attema, L. Chioncel, C.M. Fang, G.A. de Wijs, R.A. de Groot . . . 1 Half-Metals with a Covalent Band-Gap . . . . . . . . . . . . . . . . . . . . . . . . . 2 Half-Metals with a Charge-Transfer Band-Gap . . . . . . . . . . . . . . . . . . 3 Half-Metals with a d − d Band-Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Experiments at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Finite Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Modifications in the Magnetic Anisotropy . . . . . . . . . . . . . . . . . . . . . . 8 Nano-Sized Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonquasiparticle States in Half-Metallic Ferromagnets V.Yu. Irkhin, M.I. Katsnelson, A.I. Lichtenstein . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Origin of Nonquasiparticle States and Electron Spin Polarization in the Gap . . . . . . . . . . . . . . . . . . . . . . 3 First-Principle Calculations of Nonquasiparticle States: a Dynamical Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 X-ray Absorption and Emission Spectra Resonant x-ray Scattering 5 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy S.J. Jenkins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Theory of Spintronic Materials: a Surface Science Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Surface Stoichiometries in a Supercell Approach . . . . . . . . . . . . . . . . . 3 Stoichiometry and Spintronic Structure . . . . . . . . . . . . . . . . . . . . . . . . 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
203 204 206 207 209 211 213 214 216 217
219 219 220 227 232 237 241 242
247 247 249 259 260 261
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Magnetization, Spin Polarization, and Electronic Structure of NiMnSb Surfaces Markus Donath, Georgi Rangelov, J¨ urgen Braun, Wolfgang Grentz . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Sample Preparation and Characterization . . . . . . . . . . . . . . . . . . . . . . 3 Spin-Resolved Appearance Potential Spectroscopy . . . . . . . . . . . . . . . 4 Spin-Resolved Inverse Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
263 263 265 268 271 273 274
Spin Injection Experiments from Half-Metallic Ferromagnets into Semiconductors: The Case of NiMnSb and (Ga,Mn)As Willem Van Roy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 NiMnSb-Based Spin Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ga1−x Mnx As-Based Spin Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
277 277 278 285 287 288
Growth and Room Temperature Spin Polarization of Half-metallic Epitaxial CrO2 and Fe3 O4 Thin Films M. Fonin, Yu. S. Dedkov, U. R¨ udiger, G. G¨ untherodt . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Half-Metallic Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Magnetite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chromium Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
291 291 291 293 300 308
On the Importance of Defects in Magnetic Tunnel Junctions P.A. Dowben, B. Doudin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chromium Oxide Interfaces and Surface Composition . . . . . . . . . . . . 3 Intermediate States in the Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Polarizable Defects in Cr2 O3 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Defect Mediated Coupling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion: Defects May Be Important . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
311 311 314 317 321 323 327 328
List of Contributors
S.N. Kaul School of Physics University of Hyderabad Hyderabad 500 046 India and CITIMAC, Facultad de Ciencias Universidad de Cantabria 39005 Santander, Spain
Maria Daghofer Institute for Theoretical and Computational Physics Graz University of Technology
[email protected] Winfried Koller Department of Mathematics Imperial College
Alexander Pr¨ ull Institute for Theoretical and Computational Physics Graz University of Technology
Hans Gerd Evertz Institute for Theoretical and Computational Physics Graz University of Technology
Wolfgang von der Linden Institute for Theoretical and Computational Physics Graz University of Technology
Wolfgang Nolting Institut f¨ ur Physik Humboldt-Universit¨ at zu Berlin, Newtonstr. 15 12489 Berlin, Germany
Tilmann Hickel Institut f¨ ur Physik Humboldt-Universit¨ at zu Berlin, Newtonstr. 15 12489 Berlin, Germany
Carlos Santos Institut f¨ ur Physik Humboldt-Universit¨ at zu Berlin, Newtonstr. 15 12489 Berlin, Germany
Yukitoshi Motome RIKEN (The Institute of Physical and Chemical Research) 2-1 Hirosawa, Saitama 351-0198 Japan
[email protected] XIV
List of Contributors
Nobuo Furukawa Department of Physics Aoyama Gakuin University 5-10-1 Fuchinobe, Sagamihara Kanagawa 229-8558, Japan
[email protected] Naoto Nagaosa CREST, Department of Applied Physics University of Tokyo 7-3-1 Hongo, Bunkyo-ku Tokyo 113-8656, Japan and Correlated Electron Research Center, AIST Tsukuba Central 4, 1-1-1 Higashi Tsukuba, Ibaraki 305-8562 Japan and Tokura Spin SuperStructure Project ERATO Japan Science and Technology Corporation, c/o AIST Tsukuba Central 4, 1-1-1 Higashi Tsukuba, Ibaraki 305-8562 Japan
[email protected] K.U. Neumann Department of Physics Loughborough University Loughborough LE11 3TU, UK S. Dann Department of Chemistry Loughborough University Loughborough LE11 3TU, UK K. Fr¨ ohlich Department of Physics Loughborough University Loughborough LE11 3TU, UK
A. Murani Institute Laue Langevin Rue Horowitz, 36048 Grenoble Cedex, France
B. Ouladdiaf Institute Laue Langevin Rue Horowitz, 36048 Grenoble Cedex, France
K.R.A. Ziebeck Department of Physics Loughborough University Loughborough LE11 3TU, UK
T. Toli´ nski Institut f¨ ur Experimentalphysik Freie Universit¨at Berlin Arnimallee 14 D-14195 Berlin, Germany and Institute of Molecular Physics, PAS, Smoluchowskiego 17 60-179 Pozna´ n, Poland
[email protected] K. Lenz Institut f¨ ur Experimentalphysik Freie Universit¨at Berlin Arnimallee 14 D-14195 Berlin, Germany
[email protected] J. Lindner Fachbereich Physik Experimentalphysik-AG Farle Universit¨at Duisburg-Essen Lotharstr. 1, D-47048 Duisburg Germany
[email protected] List of Contributors
K. Baberschke Institut f¨ ur Experimentalphysik Freie Universit¨at Berlin Arnimallee 14, D-14195 Berlin Germany
[email protected] Wolfram Heimbrodt Department of Physics and Material Sciences Center Philipps-Universit¨ at Marburg Renthof 5, D-35032 Marburg Germany
A. Ney Solid State and Photonics Lab Stanford University Stanford, CA 94305-4075, USA
Peter J. Klar Department of Physics and Material Sciences Center Philipps-Universit¨ at Marburg Renthof 5, D-35032 Marburg Germany
T. Hesjedal Paul-Drude-Institut f¨ ur Festk¨ orperelektronik Hausvogteiplatz 5-7, D-10117 Berlin, Germany C. Pampuch Specs GmbH, Voltastraße 5 13355 Berlin, Germany L. D¨ aweritz Paul-Drude-Institut f¨ ur Festk¨ orperelektronik Hausvogteiplatz 5-7, D-10117 Berlin, Germany R. Koch Paul-Drude-Institut f¨ ur Festk¨ orperelektronik Hausvogteiplatz 5-7, D-10117 Berlin, Germany K.H. Ploog Paul-Drude-Institut f¨ ur Festk¨ orperelektronik Hausvogteiplatz 5-7, D-10117 Berlin, Germany
Perla Kacman Institute of Physics Polish Academy of Sciences Warsaw, Poland
Izabela Kuryliszyn-Kudelska Institute of Physics Polish Academy of Sciences Warsaw, Poland
J. Kudrnovsk´ y Institute of Physics Academy of Science of the Czech Republic Prague, Czech Republic
L. Bergqvist Department of Physics Uppsala University Uppsala, Sweden
O. Eriksson Department of Physics Uppsala University Uppsala, Sweden
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XVI
List of Contributors
V. Drchal Institute of Physics Academy of Science of the Czech Republic Prague, Czech Republic
J.J. Attema Electronic Structure of Materials University of Nijmegen Toernooiveld 1, 6525 ED Nijmegen The Netherlands
I. Turek Institute of Physics of Materials Academy of Science of the Czech Republic, Brno Czech Republic and Department of Electronic Structures Charles University Prague, Czech Republic
L. Chioncel Electronic Structure of Materials University of Nijmegen Toernooiveld 1, 6525 ED Nijmegen The Netherlands
G. Bouzerar Institut Laue – Langevin Grenoble, France
L.M. Sandratskii Max-Planck Institut f¨ ur Mikrostrukturphysik Weinberg 2, D-06120 Halle, Germany
[email protected] P. Bruno Max-Planck Institut f¨ ur Mikrostrukturphysik Weinberg 2, D-06120 Halle Germany
[email protected] J.M.D. Coey Physics Department Trinity College Dublin 2, Ireland
C.M. Fang Electronic Structure of Materials University of Nijmegen Toernooiveld 1, 6525 ED Nijmegen The Netherlands G.A. de Wijs Electronic Structure of Materials University of Nijmegen Toernooiveld 1, 6525 ED Nijmegen The Netherlands R.A. de Groot Electronic Structure of Materials University of Nijmegen Toernooiveld 1, 6525 ED Nijmegen The Netherlands and Laboratory of Chemical Physics MSC, University of Groningen Nijenborgh 4 9747 AG Groningen The Netherlands V.Yu. Irkhin Institute of Metal Physics 620219, Ekaterinburg Russia
List of Contributors
M.I. Katsnelson Department of Physics Uppsala University Box 530, SE-751 21 Uppsala Sweden
A.I. Lichtenstein Institute of Theoretical Physics University of Hamburg Jungiusstrasse 9, 20355 Hamburg Germany
S.J. Jenkins Department of Chemistry University of Cambridge Lensfield Road Cambridge CB2 1EW United Kingdom
[email protected] Markus Donath Physikalisches Institut Westf¨alische Wilhelms-Universit¨at Wilhelm-Klemm-Str. 10 48149 M¨ unster Germany
Georgi Rangelov Physikalisches Institut Westf¨alische Wilhelms-Universit¨at Wilhelm-Klemm-Str. 10 48149 M¨ unster Germany
J¨ urgen Braun Physikalisches Institut Westf¨alische Wilhelms-Universit¨at Wilhelm-Klemm-Str. 10 48149 M¨ unster Germany
XVII
Wolfgang Grentz Kantonschule Z¨ urcher Oberland 8620 Wetzikon Switzerland Willem Van Roy IMEC, Kapeldreef 75 B-3001 Leuven, Belgium
[email protected] M. Fonin Fachbereich Physik Universit¨at Konstanz, 78457 Konstanz, Germany Yu. S. Dedkov Institut f¨ ur Festk¨ orperphysik Technische Universit¨at Dresden 01062 Dresden, Germany U. R¨ udiger Fachbereich Physik Universit¨at Konstanz, 78457 Konstanz, Germany G. G¨ untherodt II. Physikalisches Institut Rheinisch-Westf¨ alische Technische Hochschule Aachen 52056 Aachen, Germany P.A. Dowben Department of Physics and Astronomy and the Center for Materials Research and Analysis (CMRA) 116 Brace Laboratory of Physics University of Nebraska P.O. Box 880111 Lincoln, Nebraska USA 68588-0111
XVIII List of Contributors
B. Doudin Department of Physics and Astronomy and the Center for Materials Research and Analysis (CMRA) 116 Brace Laboratory of Physics
University of Nebraska P.O. Box 880111 Lincoln, Nebraska USA 68588-0111
Introduction M. Donath, W. Nolting
The phenomenon of spontaneous collective order of the magnetic moments in some solid materials (ferro-, ferri-, antiferromagnetism), still attracts the interest of researchers working in experiment and theory alike. Experimentalists carefully characterize the magnetic properties of these interesting materials as a function of the structure, morphology, composition, magnetic field, pressure, and temperature. The ultimate goal is to tailor the magnetic properties and optimize them for certain applications. The theoretical description is not a trivial task because collective magnetism is a many-body phenomenon of quantum-mechanical nature. So far, no complete theory is available which could describe all kinds of ferromagnetic materials. Two major classes of ferromagnets are distinguished according to the kind of electrons carrying the magnetic moments: itinerant or band ferromagnets on the one side and local-moment ferromagnets on the other side. In the latter case, the exhange interaction is not a direct interaction due to the localization of the electrons with no significant overlap of their wave functions from one atomic site to the next. An interaction between the localized magnetic moments and the itinerant charge carriers or interspaced anions is needed for a so-called indirect exchange interaction. The contributions of this book concentrate on three different subjects within the topic of local-moment ferromagnetism. The first part deals with concentrated local-moment systems comprising classical local-moment ferromagnets as well as manganites, which show the colossal magnetoresistance (CMR) effect. The second part covers a relatively new class of materials, the diluted magnetic semiconductors. The origin of ferromagnetic order in these materials is subject of an intense debate today. The third part focuses on half-metallic ferromagnets, an interesting class of materials, well-known for decades, but with new perspectives for applications in magnetoelectronic and spintronic devices.
Concentrated Local-Moment Systems The complex critical behaviour of Gd remains a highly controversial issue both from experimental and theoretical points of view, and that has been the case for nearly four decades. An elaborate analysis of high-resolution ac susceptibility and bulk magnetization data taken along the c-axis (easy axis M. Donath and W. Nolting: Introduction, Lect. Notes Phys. 678, 1–7 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com
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of magnetization) of a high-purity Gd single crystal made it possible to reveal several crossovers close to the Curie temperature TC : Gaussian regime, isotropic short-range Heisenberg, isotropic dipolar, uniaxial dipolar - as predicted by renormalization group calculations. The experimental investigations evidenced the decisive role played by dipolar interactions, despite their weak strength, in establishing uniaxial magnetic order in Gd for temperatures near TC . The ferromagnetic Kondo-lattice model is considered a candidate for describing CMR-manganites. Monte-Carlo simulations, assuming classical spins, reveal that the double-exchange mechanism does not lead to phase separation in the one-dimensional model but rather stabilizes inidvidual ferromagnetic polarons. The ferromagnetic polaron picture can explain the pseudogap in the one-particle spectral function. The physics of classical local-moment systems such as the “concentrated” ferromagnetic semiconductor EuS and the ferromagnetic 4f metal Gd is mainly due to the same interband-exchange interaction that also provides the carrier-induced ferromagnetism of the diluted magnetic semiconductors and, at least partly, the various magnetic phases of the manganites. The ferromagnetic Kondo-lattice model, better s-f (s-d) model, certainly covers the main aspects of the magnetic and quasiparticle features, however only if the model treatment goes beyond mean field. It can be shown that spin exchange processes, neglected by mean-field theories from the very beginning, are responsible for just the characteristic properties of such local-moment systems. It is demonstrated that a combination of a many-body evaluation of the Kondo-lattice model with a first-priniciples band structure calculation can reproduce almost quanitatively the temperature-dependent electronic and magnetic properties of Gd. The rich physics of the manganites La1−x Dx MnO3 (D=Ca2+ , Sr2+ ), which exhibit the CMR effect, appears to a large extent to be due to an exchange coupling between localized t2g electrons and itinerant eg electrons. The t2g particles form a localized S = 3/2 magnetic moment while the correlated eg band allows for a maximum filling n = 1 (Mott insulator for x = 0). Besides the complicated magnetic phase diagram, the convincing explanation of the metal-insulator transition, coinciding with the magnetic phase transition in the Ca-doped manganites, poses a sophisticated problem. It is commonly accepted that the exchange coupling of localized and itinerant particles is much bigger than the bandwidth, so that the double-exchange model, which is the strong coupling version of the ferromagnetic Kondolattice model, may represent a good frame for a description. However, there is evidence that the coupling of electrons to local phonon modes should be taken into consideration. The insulator-metal transition and the origin of the CMR has been investigated alternatively by using Monte-Carlo methods on finite-size clusters. Counter-intuitive observations are made with respect to the influence of randomness. The latter comes into play as charge randomness
Introduction
3
(valence mixing Mn3+ /Mn4+ ) or by lattice distortion (Jahn-Teller effect). Direct consequences are stabilization of short-range correlations of charge ordering, while long-range order is suppressed. A charge gap opens due to these correlations, and double-exchange ferromagnetism turns out to be robust against randomness. The ferromagnetic phase, therefore, delves into the charge order region what explains some pecularities in the temperature dependence of the resistivity. Most striking and really counter-intuitive is the finding that the insulator-to-metal transition may be due to randomness.
Towards Diluted Magnetic Semiconductors Currently, experimentalists worldwide are highly active in preparing and characterizing diluted magnetic semiconductors and related systems, e.g., MnAs as bulk samples, MnAs thin films deposited on GaAs, Ga1−x Mnx As thin films, MnAs clusters embedded in paramagnetic GaMnAs. From an applications point of view, high Curie temperatures are highly desirable. Therefore, the conditions for high transition temperatures have to be explored. For MnAs, a non-typical first-order transition from a hexagonal lowtemperature ferromagnetic phase to an orthorhombic high-temperature paramagnetic phase has stimulated intense research activity. Spin-polarized neutron scattering provides insight into magnetic correlations in MnAs, where magnetism is related to a structural instability. Neutron scattering sees magnetic correlations to be ferromagnetic with essentially no temperature dependence. This is in contrast to magnetization measurements which indicate an unusual temperature dependence in the orthorhombic phase. Epitaxial MnAs films on GaAs were characterized by ferromagnetic and spin wave resonance aiming at anisotropy and intrinsic exchange interaction. The first order phase transition described above manifests itself in the resonance spectra as a jump of both the resonance field and the resonance line width and turns out to be dominated by a coexistence of phases (stripe pattern). A granular hybrid structure formed by ferromagnetic MnAs clusters embedded in paramagnetic GaMnAs exhibits ferromagnetism above room temperature (TC = 330 K) due to the MnAs clusters. By co-doping with Te, the majority carrier type of the matrix can be changed from holes to electrons. The magnetoresistance of p-type and n-type samples differs considerably because of different s-d and p-d exchange integrals. The experimental data can qualitatively be understood as a result of the interplay between Zeeman splitting (field-induced tuning of band states), band filling and disorder. In Ga1−x Mnx As, Mn atoms substituting Ga promote ferromagnetism by exchange interaction with GaAs holes. The highest Curie temperature reported so far is 172 K. Interstitial Mn ions are thought to counteract this tendency via antiferromagnetic superexchange interaction with neighbouring substitutional Mn ions. An increase of the Curie temperature was observed for epitaxial GaMnAs layers after low-temperature post-growth annealing.
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The interstitial Mn segregates from the bulk to the surface during annealing, giving rise to a further enhancement of the bulk magnetic transition temperature. Experimental evidence of Mn interstitial enrichment at the surface comes from x-ray absorption spectroscopy and x-ray resonant magnetic scattering. It is widely accepted that Mn interstitials are mainly affected by annealing. In addition, measurements have shown that co-doping with Be ions increases the number of Mn interstitials at the expense of substitutional Mn resulting in a strong decrease of TC without an appreciable change of the free hole concentration. Theoretical studies have been performed to understand the magnetic properties of Mn ions in interstitial positions. One finds that the p-d exchange interaction matrix element is strongly reduced for interstitial Mn ions. The transfer of Mn ions from substitutional to interstitial positions (tetrahedral sites) diminishes the number of magnetic ions contributing to the carrier-induced ferromagnetism. That explains the experiments on GaMnBeAs. Furthermore, interstitial Mn acts as double donor, thus reducing the hole concentration with a respective influence on the ferromagnetism of the diluted magnetic semiconductor. The interaction between neighbouring interstitial and substitutional Mn ions could theoretically be identified as antiferromagnetic superexchange. - Since some difficult technological issues connected with the growth and lithography of magnetic semiconductors are now solved, it has become possible to explore the physics of nanostructures for promising spintronic applications. A model study in the framework of the ferromagnetic Kondo-lattice model was used to explore the influence of magnetic moment disorder in diluted magnetic semiconductors. The carrier-induced ferromagnetism exhibits a strong band occupation dependence. In “concentrated” local-moment systems rather low electron (hole) concentrations favour ferromagnetic ordering. By a CPA-type evaluation of the static susceptibility it was shown that this effect transfers to the diluted systems, i.e. for ferromagnetism the number of free carriers must be substantially smaller than the number of magnetic ions. Compensation effects (antisites) appear to be a necessary precondition for ferromagnetism. With a combination of first-principles calculations of interatomic exchange integrals for a classical Heisenberg model and Monte Carlo simulations, the observed Curie temperatures of a series of diluted magnetic semiconductors (Mn-doped GaAs and GaN, Cr-doped ZnTe) could be reproduced with good accuracy. However, a random moment distribution appears to be necessary to explain the measured TC values. An ordered structure of the magnetic moments leads to transition temperatures that are by far too high. The actual exchange interaction seems to be exponentially damped by disorder. Magnetic percolation plays an important role for the magnetic properties of diluted magnetic semiconductors. Furthermore, the role of the holes for the carrier-mediated exchange interaction has been reexamined in a parameterfree theoretical scheme. Holes must be delocalized from the magnetic ion, but
Introduction
5
simultaneously must experience a strong (local) exchange interaction with the magnetic impurity, which seems somewhat conflicting. By inspecting the resulting correlation energy, the different magnetic behaviour of Ga1−x Mnx As and Ga1−x Mnx N can be understood. Collective spin excitations in diluted magnetic semiconductors have been studied in the frame of the p-d (s-f, Kondo-lattice) model. It turns out that a proper modeling of the band structure by a six-band Kohn-Luttinger ansatz is important. The multiplet of spin wave dispersions (one optical and several acoustic modes) exhibits a strong band-occupation dependence reflecting to a certain degree the carrier-concentration dependence of the Curie temperature. Highly promising new magnetic materials might be a group of diluted ferromagnetic oxides such as (Co,Fe,Mn)-doped ZnO, TiO2 , and SnO2 . These wide-gap semiconductors exhibit, surprisingly, Curie temperatures well in excess of room temperature. They could fulfill the fundamental criteria for spin electronics: Long diffusion lengths realized in a semiconductor or semimetal and a Curie temperature above 500 K. There are doubts, however, that this can be fulfilled by Ga1−x Mnx As. All these oxides are n-type, often partially compensated. The average moment per transition-metal ion is higher than the spin-only moment of the magnetic ion, maybe because of the spin-split 4s band. Ferromagnetism is already present for very low magnetic impurity concentrations, far below the percolation threshold for nearest-neighbour coupling. The materials can be metallic or semiconducting. Unfortunately, so far, the properties depend critically on the preparation method. Model calculations show that the minority-spin 3d level must be pinned at the Fermi energy in order to get high Curie temperatures.
Half-Metallic Ferromagnets Half-metallicity means in the ideal case that only electrons with one and the same spin direction will contribute to the electric current, i.e. the Fermi edge lies in a gap of one spin part of the density of states. Heusler alloys such as (Fe,Co,Ni,Cr,Pt)MnSb are promising candidates. Actually, NiMnSb with a TC of 728 K is in the center of intensive investigation. The origin of the band gap is equally diverse as the origin of half-metallicity. Therefore, the origin of the bandgap is chosen as a criterion for the classification of half-metals: (1) weak ferromagnets with a covalent band gap, where structure and symmetry matters (NiMnSb), (2) strongly ferromagnetic ionic systems with a chargetransfer band gap (manganese perowskites La1−x (Ca,Pb,Sr)x MnO3 , CrO2 ), (3) narrow-band ferromagnets with a d-d band gap like Fe3 O4 , which is nearly a Mott insulator. The preservation of the band gap, however, is intimately related to the surface/interface structure, imperfections and temperature. Incoherent non-quasiparticle states in the band gap near the Fermi edge are theoretically predicted and may give considerable contributions to ther-
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M. Donath, W. Nolting
modynamic and transport properties. LDA+DMFT calculations for NiMnSb give evidence for the existence of these non-quasiparticle states which have not been observed experimentally yet. Core-level spectroscopy has been proposed as possible tool for this purpose. These states should also influence the temperature dependence of impurity scattering in a system like CrO2 tunneling junction. An important question is whether it is possible to get 100% spin polarization at the Fermi level, at least at T = 0, as first-principles bandstructure calculations predict. And if so, is this also true for surfaces and interfaces with their broken symmetry? Theoretically, insight into the stability of different NiMnSb surface terminations can be gained by density-functional calculations and by studying MnSb, NiSb, and NiMn surfaces. Furthermore, the influence of surface and interface electron states within the gap can be examined for uncovered surfaces and selected interfaces. What happens with the gap at finite temperatures? Magnon and phonon effects may lead to a depolarization, so that strict half-metallicity appears to be limited to T = 0. Proper doping and certain geometrical structures may lead to an optimization of spin polarization at the Fermi energy. It was shown that embedding NiMnSb in a NiScSb matrix, i.e. alloying NiMn1−x Scx Sb, the system changes for x = 1 to x = 0 from a nonmagnetic semiconductor via a diluted and even quasi-concentrated magnetic semiconductor to a genuine half metal. Surface sensitive experiments on NiMnSb so far failed to detect 100% spin polarization at the Fermi level. Spin-resolved (inverse) photoemission as well as spin-resolved appearance potential spectroscopy give smaller spin polarization values by at least a factor of two. This is true not only at the Fermi level but also for the minority density of Mn states above the Fermi level. Besides the problem of surface/interface states destroying the complete surface spin polariztion at the Fermi level, the surface magnetization appears to be reduced. The reason for that is not clear at present. Spin-resolved photoemission on epitaxial Fe3 O4 (111) films grown on W and Al2 O3 exhibit a spin polarization of about -80% at room temperature. For epitaxial CrO2 (100) films deposited on TiO2 (100) substrates, a “record” value of 95% spin polarization at the Fermi level was found at room temperature, yet with a relatively small density of states. Half-metallic ferromagnets are particularly attractive for spin injection. In this respect, NiMnSb turns out to be advantageous compared with oxides because no barriers are needed to protect the semiconductor from oxidation. Experiments suggest that it may be feasible to fabricate half-metallic NiMnSb/GaAs contacts. However, it is not straightforward to combine a low chemical disorder with a stoichiometrically and structurally controlled interface to suppress the formation of metallic interface states. Better results have been obtained by introducing a tunnel barrier, which results in about 6% spin injection at 80 K from polycrystalline NiMnSb films across an amorphous AlOx barrier. Better results are achieved with the diluted magnetic
Introduction
7
semiconductor Ga1−x Mnx As, which, to a certain degree, can also be considered as a half-metal and can be combined with III-V semiconductors. In combination with a Zener tunnel junction to convert holes into polarized electrons, more than 80% spin injection was reported at 4.2 K. It is known that defects influence the electronic properties, electric transport, and magnetic coupling, which is also true for the insulating oxides used as tunnel junctions. Therefore, the importance of defects in magnetic tunnel junctions has to be considered in detail in the future.
Part I
Concentrated Local-Moment Systems
Critical Behaviour of Heisenberg Ferromagnets with Dipolar Interactions and Uniaxial Anisotropy S.N. Kaul School of Physics, University of Hyderabad, Hyderabad 500 046, India CITIMAC, Facultad de Ciencias, Universidad de Cantabria, 39005 Santander, Spain Abstract. In any real magnetic system, weak anisotropic long-range dipole-dipole interactions are invariably present besides crystal-field interactions and dominant isotropic short-range Heisenberg interactions (in insulating systems) or isotropic long-range Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions (in metallic systems) that couple the localized magnetic moments. In many magnetic systems, the dominant interactions normally sustain long-range magnetic order and govern the ground state and finite-temperature magnetic properties. Crystal-field interactions lead to magnetocrystalline anisotropy, which constrains the domain magnetizations to lie along the “easy directions”. Even in such systems, the magnetic behaviour in the critical region is significantly altered by the dipolar interactions so much so that the interplay between crystal-field, dipolar and Heisenberg or RKKY interactions gives rise to a complex scenario of crossovers between different critical regimes. A thorough study of critical behaviour of these systems has yielded rich dividends in that their magnetic properties are now much better understood. We will make only a few passing remarks about such systems. However, there are certain exceptional cases where dipolar interactions, despite their weak strength, play a very important role in deciding the nature of magnetic order. That gadolinium metal belongs to this rare category of magnetic systems is demonstrated by the latest advances in understanding its complex magnetic behaviour in the critical region. We present recent experimental results on the critical behaviour of gadolinium and the relevant theoretical background so as to bring out these latest developments clearly.
1 Introduction In insulating magnetic systems, the localized magnetic moments interact with one another not only through Heisenberg exchange interactions but also via relatively weak dipole-dipole interactions. Compared to isotropic short-range (Heisenberg) exchange interactions, magnetic dipole-dipole interactions have both a long range and a reduced symmetry. These attributes of dipolar interactions result in important modifications to the critical behaviour of a pure Heisenberg ferromagnet. The presence of significant crystal–field interactions (which are responsible for magneto-crystalline anisotropy in non-S-state magnetic ions), besides the isotropic short-range Heisenberg and long-range dipolar interactions, in a magnetic system leads to a variety of interesting but S.N. Kaul: Critical Behaviour of Heisenberg Ferromagnets with Dipolar Interactions and Uniaxial Anisotropy, Lect. Notes Phys. 678, 11–29 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com
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complicated physical (crossover) phenomena in the critical region because of the interplay between different interactions. Experimental investigations in the critical region thus provide a unique and direct means of probing the type of interactions present and the interplay between them that finally decides the nature of magnetic order prevailing in the systems under study. Phase transitions and critical phenomena belong to the rare category of research fields that permit one to draw unambiguous conclusions from a quantitative comparison between theory and experiment. With a view to “set the stage” for such a comparison, we define the asymptotic critical exponents and amplitudes, which quantify the critical behaviour near the magnetic orderdisorder phase transition, in Sect. 2. The concepts crucial to understanding the theoretical aspects of critical phenomena such as the scaling hypothesis, universality, renormalization group approach and crossover between different critical regimes, are introduced in the Sects. 3 and 4. In Sect. 5, we set out to make a detailed comparison between the theoretical expectations and the experimental reality with specific reference to the local-moment metallic ferromagnet gadolinium (Gd) in which the electrons responsible for magnetism do not contribute to electrical conductivity. Such a comparison between theory and experiment brings out clearly the role of dipolar interactions in giving rise to a complex magnetic behaviour of Gd within and outside the critical region.
2 Critical Exponents and Amplitudes In the asymptotic critical region, the behaviour of a magnetic system is characterised by a set of critical exponents and amplitudes [1, 2]. Critical exponents are the exponents in the power laws that define the asymptotic behaviour of various thermodynamic quantities near the critical point TC and the corresponding amplitudes are the prefactors in these power laws. The asymptotic critical exponents and amplitudes for the second-order ferromagnetic (FM) to paramagnetic (PM) phase transition are defined as follows. 2.1 Spontaneous Magnetization In the asymptotic critical region, the spontaneous magnetization, MS , i. e. the order parameter for the FM-PM phase transition, varies with the reduced temperature ε = (T − TC )/TC as MS (T ) = lim M (T, H) = B(−ε)β , H→0
ε 0 and ε < 0, respectively. CH=0 (T ) =
2.5 Spin-Spin Correlation Function At TC , the correlation function for the spin fluctuations at the points 0 and r in space, G(r) ≡ [s(r) − s][s(0) − s], decays with distance, r, as G(|r|) = N |r|
−(d−2+η)
[large |r| , ε = H = 0]
(7)
where d is the dimensionality of the lattice and η is a measure of deviation from the mean-field behaviour. 2.6 Spin-Spin Correlation Length The correlation length, ξ, is the distance over which the order parameter fluctuations are correlated and is defined through the relation G(|r|) = e−|r|/ξ(T ) / |r|, for d = 3 and |r| → ∞. In the critical region, ξ depends on temperature as ξ(T ) = ξ0− (−ε)−ν
−
ε < 0, H = 0
(8)
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ξ(T ) = ξ0+ ε−ν
+
ε > 0, H = 0
(9)
In (1)–(9), β, γ − , γ + , δ, α− , α+ , η, ν − and ν + are the critical exponents and B , Γ − , Γ + , A0 or D, A− , A+ , N , ξ0− and ξ0+ are the corresponding amplitudes. There are nine exponents in total but only two of them are independent. This is a consequence of the scaling relations [1, 2] between them, e.g., α+ = α− , γ + = γ − , ν + = ν − , βδ = β + γ (Widom equality), α + 2β + γ = 2 (Rushbrooke equality), α + β(δ + 1) = 2 (Griffiths equality), (2 − η)ν = γ (Fisher equality) and dν = 2 − α (Josephson equality), to name a few. Strictly speaking, the single power laws are valid only in the asymptotic limit T → TC . In practice, however, the power laws are fitted to the data over a finite temperature range. Consequently, such an approach yields only average exponent values since, in general, the amplitudes as well as the exponents are temperature-dependent and they assume temperature-independent values only in the asymptotic critical region [2]. In order to tackle this problem effectively, the concept of effective critical exponent was introduced by Riedel and Wegner [3]. The effective critical exponents provide a local measure for the degree of singularity of physical quantities in the critical region. The effective critical exponent of a function f(ε) is defined by the logarithmic derivative λef f (ε) = d ln f (ε)/d ln ε. In the limit ε → 0, λef f (ε) coincides with the asymptotic critical exponent λ.
3 Scaling and Universality In one of its forms [1, 2], the scaling hypothesis asserts that in the asymptotic critical region, the singular part of the Gibbs free energy, Gs (ε, H), is a generalized homogeneous function of its arguments ε and H . Scaling makes two specific predictions, both of which have been vindicated by experiments [1, 2, 4, 5]. First, it relates various critical exponents through the scaling equalities. Second, it predicts that all the magnetization, M (ε, H), curves (either magnetization M (H) isotherms at different temperatures or M (ε) at different fields) taken in the critical region collapse onto two universal curves, β one for ε 0, if the scaled magnetization, M/ |ε| , is ∆ plotted against the scaled field, H/ |ε| , where ∆=βδ is the gap exponent. The renormalization group approach (Sect. 4) puts these predictions on a sound theoretical footing. A related concept is the universality, which basically amounts to cataloging, under a single category (class), all types of systems that possess the same values for critical exponents and critical amplitude ratios and for which the equation of state and the correlation functions become identical near criticality provided the order parameter, the ordering field and the correlation length (time) are scaled properly by material-dependent factors. Thus, the critical exponents and the ratios between critical amplitudes (but not the
Critical Behaviour of Heisenberg Ferromagnets
15
amplitudes themselves) are universal [2, 4, 5, 6] in the sense that they possesses exactly the same numerical values for a number of widely different systems belonging to the same universality class. The universality class, in turn, is determined by (i) the spatial dimensionality “d ”, (ii) the number of order parameter components or equivalently, the order parameter dimensionality “n”, (iii) the symmetry of the Hamiltonian, and (iv) the range of interactions. For instance, d = 3, n = 1 corresponds to a three-dimensional Ising system in which the spins on a three-dimensional lattice are constrained to point either in the +z (up) or –z (down) directions. In this example, the range of interactions is too short compared to the spin-spin correlation length and the symmetry of the Hamiltonian is reflected through the extremely large uniaxial anisotropy which constrains the spins to point up or down. The basic physical idea behind universality is that as ξ becomes very large as T → TC microscopic details loose their importance for the critical behaviour.
4 Renormalization Group and Crossover Phenomena Wilson’s renormalization group scheme has provided a firm theoretical basis for understanding scaling and universality, and offered a powerful tool to calculate critical exponents, scaling functions and correlation functions. The renormalization group approach has, therefore, formed the subject of many authoritative reviews (for recent reviews, see [7, 8, 9, 10]). In what follows, we give the main essence of one of the variants of renormalization group and its practical implications. As an illustrative example, we begin with the Hamiltonian for N Ising spins on a simple cubic lattice of volume V in the presence of an ordering (magnetic) field H , i.e., H=−
1 J (r − r ) S (r) S (r ) − H S (r) 2 r r
(10)
r
In terms of the Fourier components Sq = v S(r) e−iq·r
(11)
r
where v is the volume per lattice site r, the above Hamiltonian can be expressed as HS0 1 (12) Jq Sq S−q − H=− 2N q v with Jq = v −2 J(r)e−iq·r and Sq=0 = S0 = v S(r). For short-range r
r
interactions, Jq can be expanded with the result Jq = J0 − J2 q 2 + · · ·
(13)
16
S.N. Kaul
where Jq=0 = J0 = v −2
r
J(r) and J2 =
1 6v 2
J(r)r2 . Substituting (13) for
r
Jq in (12), we get H=−
1 HS0 1 J0 S02 + J2 q 2 Sq S−q − 2N 2N v q
(14)
The free energy is defined by F = E − T S, where E and S are the energy and entropy as functions of S0 . If the mean magnetization per spin is m, S0 = N vm and (14) becomes [11] E=−
N v2 N v2 J0 m2 + J2 (∇m)2 − N mH 2 2
(15)
The entropy S = kB lnW with W = N↑N!N! ↓ ! where W is the number of configurations for a given number N↑ and N↓ of spins pointing up and down, respectively, on a lattice of N sites. Using Stirling’s formula and recalling and N , = N 1−m that m = (N↑ − N↓ )/N , or alternatively, N↑ = N 1+m ↓ 2 2 the entropy is given by
1+m 1−m S = −N kB − ln 2 + ln(1 + m) + ln(1 − m) (16) 2 2 Since m 1, ln(1 + m) and ln(1 − m) can be expanded in powers of m 2 4 with the result S = −N kB − ln 2 + m2 + m + · · · . Dividing the lattice 12 into cells of sufficiently large size (the cell volume being still smaller than the macroscopic volume V so that the mean magnetization of the cells can be described by the continuous variable m(r)), the free energy assumes the form
1 1 F = dd r a0 (T ) + a(T )m2 (r) + b(T )m4 (r) (17) 2 4 1 H 2 + c(T ) {∇m(r)} − m(r) 2 v where a0 (T ) = −kB T ln2/v, a(T ) = (kB T −J0 v 2 )/v, b(T ) = kB T /3v, c(T ) = J2 v and d is the lattice dimensionality. From (17), the free energy density can be defined as (18) g(T, H) = g0 (T ) − gsing (T, H) with g0 (T ) = a0 (T ) and gsing (T, H) = (F/V )−g0 (T ). Equation (17) suggests that the effective cell Hamiltonian has the Ginzburg-Landau form. The same result as (17) would have been obtained if, instead of an Ising spin system, we had considered a three-dimensional Heisenberg spin system. Starting with an effective cell Hamiltonian, the renormalization group (RG) transformation proceeds in two steps [11]. First, the cell size in each direction is increased by a factor b and the bigger cell Hamiltonian is constructed out of the smaller cell Hamiltonian. Second, a scale transformation, in which the length scale
Critical Behaviour of Heisenberg Ferromagnets
17
changes by a factor b = el in all linear dimensions, is performed such that the bigger-cell volume shrinks back to the original smaller-cell volume, i.e., V (l) = e−dl V (0). As a consequence, the free energy is left unaltered but the free energy density transforms according to gl=0 = e−dl gl while (as we shall see in the later part of this section) the Hamiltonian H0 = H{µi } transforms into Hl = H{µi eyi l }, where µi are the scaling fields and yi are the scaling exponents. RG transformation thus requires that (19) g {µi } = g(µ0 ) − e−dl gsing µi eyi l We now identify the scaling fields µi with the relevant fields µε = |ε| and µh = H = h. Since the parameter l is arbitrary, it can be chosen such that |ε|eyε l = 1 or e−yε l = |ε|. Consequently, e−dl = (e−yε l )d/yε = |ε| and
d/yε
2−α
= |ε|
eyh l = (e−yε l )−yh /yε = |ε|
(20)
−∆
(21)
where 2−α = d/yε and ∆ = yh /yε . Combining (19), (20) and (21), we obtain 2−α
g(T, H) = g0 (T ) − |ε|
∆
Y± (±1, h/ |ε| )
(22)
For convenience, we set the macroscopic volume of the system equal to unity ( V = 1). Thus, the first- and second-order derivatives of g with respect to H yield the magnetization M (T, H) and the “in-field’ susceptibility χ(T, H), respectively, whereas the second-order derivative of g with respect to temperature yields the specific heat C(T, H). Therefore [11], ∆ ∂g(T, H) ∂Y± (±1, h/ |ε| ) 2−α−∆ = |ε| or M (T, H) = − ∂H ∂h T T
β
∆
M (ε, h) = |ε| f± (h/ |ε| ) (23) χ(T, H) =
C(T, 0) = −T
2
∂M (T, H) ∂H
∂ g ∂T 2
∆ ∂f± (h/ |ε| ) = |ε| or ∂h T ∆ ∂f± (h/ |ε| ) −γ χ(ε, h) = |ε| ∂h
2−α−2∆
T
(24)
T
−α
= C(ε, 0) = (1 − α)(2 − α)TC−1 Y± (0) |ε|
(1 + ε)
H=0
(25) In (22)–(25), Y± (±1, h/ |ε| ), f± (h/ |ε| ), (∂f± (h/ |ε| )/∂h)T and Y± (0) are the scaling functions, which in the asymptotic limit assume constant values and + and – signs denote ε > 0 and ε < 0, respectively. A comparison of ∆
∆
∆
18
S.N. Kaul
(23), (24) and (25) with the definitions (1), (2), (3), (5) and (6) reveals that these constant limiting values are nothing but the asymptotic critical amplitudes and that β = 2 − α − ∆ and γ = −2 + α + 2∆. From these relations, it immediately follows that β + γ = ∆ and α + 2β + γ = 2 (which is the Rushbrooke scaling equality). Furthermore, (23) is the magnetic equation of ∆ state (MES) or just the scaling equation of state. As ε → 0, |ε| /h → 0 and the MES can be cast into an alternative form [11] |ε| β/∆ f0 (26) M (ε, h) = |h| 1/∆ |h| 1/∆
In the limit |ε| / |h|
