Local-Moment Ferromagnets

Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Los Angeles, CA, USA S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz, Köln, Germany

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
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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-

VI

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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

XV

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


2

M. Donath, W. Nolting

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.

4

M. Donath, W. Nolting

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-

6

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

2.2 Initial Susceptibility Initial susceptibility, defined as χ0 = lim [∂M/∂H]T , diverges at TC as H→0

(1)

Critical Behaviour of Heisenberg Ferromagnets −

χ0 (T ) = Γ − (−ε)−γ , χ0 (T ) = Γ + ε−γ , +

ε<0

13

(2)

ε>0

(3)

2.3 Critical Isotherm At T =TC , magnetization M varies with field H as M (TC , H) = A0 H 1/δ

or

H = DM δ

ε=0

(4)

2.4 Specific Heat 
The zero-field specific heat, defined as CH=0 = −T ∂ 2 G(T, H)/∂T 2 H=0 , where G is the free energy, diverges at TC as CH=0 (T ) =

− A− [(−ε)−α − 1] + B − , − α

ε<0

(5)

A+ −α+ [ε − 1] + B + , ε>0 (6) α+ Specific heat exhibits a cusp at TC when α < 0 whereas for α = 0 the singularity is logarithmic. B + and B − represent the non-singular background for 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)

14

S.N. Kaul

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 and the other 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| << 1, the function f0 (z) can be expanded β/∆ in a Taylor series around z = 0 with the result M (ε, h) = |h| *    1/∆ β/∆  f0 (0) + · · · . At |ε| = 0, M (0, h) = f0 (0) |h| f0 (0) + |ε| / |h| or equivalently, (27) M (0, H) = f0 (0)H 1/δ Comparing (27) with (4) gives β/∆ = 1/δ or βδ = ∆. We have shown above that β + γ = ∆. Thus, β + γ = βδ (which is the Widom scaling equality). The foregoing calculations thus serve to illustrate how the renormalization group approach provides a theoretical basis for the universality hypothesis. Having defined a renormalization group (RG), the RG transformation is iterated through the repeated application of the RG operator, R, i.e., H = R[H], H = R[H ], . . . At the critical point, the Hamiltonian maps onto itself under the RG transformation and such a Hamiltonian is called a fixed point Hamiltonian, H∗ . The fixed point Hamiltonian is defined by its invariance under RG transformation, i.e., R[H∗ ] = H∗ . A visualization of the approach to criticality is facilitated when the renormalization group operator is linearized about H∗ , e.g., H = R [H] = R [H∗ + µQ] = H∗ + µLQ + ϑ(µ2 )

(28)

where L is a linear operator defined by the eigenvalue equation LQi = eyi l Qi

(29)

This equation also defines eigenoperators (or eigenperturbations) Qi , eigenvalues eyi l and scaling exponents yi . Assuming that the eigenoperators form a complete set of operators, any Hamiltonian H0 can be expanded in terms of  this complete set, i.e., H0 = H∗ + µi Qi . In this expression, the Hamiltonian i

is parameterized by the scaling fields µi . Thus,  µi eyi l Qi Hl = R [H0 ] = H∗ + i

(30)

Critical Behaviour of Heisenberg Ferromagnets

19

For instance, we saw earlier that when H0 = H∗ + µε Qε + µh Qh with µε = |ε| and µh = h, the singular part of the free energy density is given 2−α ∆ Y± (±1, h/ |ε| ). If one starts with the Hamiltonian by gsing (ε, h) = |ε| H0 = H∗ + µε Qε + µh Qh + µi Qi , generalization of this procedure yields, within the linear approximation, the Hamiltonian Hl = H∗ + µε eyε l Qε + µh eyh l Qh + µi eyi l Qi and the singular part of the free energy density  gsing (ε, h, µi ) = |ε|

2−α

Y±

±1,

h |ε|

∆

,

µi

(31)



φi

|ε|

(32)

Depending upon the sign of φi , three different cases may arise. φi negative. As the critical point is approached, (|ε| → 0), the reduced φ fields µi = µi / |ε| i tend to zero. The leading singularity in the asymptotic behaviour is the same as if all the scaling fields µi were zero. This implies that the perturbations leave the pure (unperturbed) fixed point unaltered. Hence the operators Qi are irrelevant and µi are called irrelevant scaling fields. However, a Taylor expansion of the scaling function Y± (x) about x = 0 corrects the dominant singular terms by additive terms (called the “correction-to-scaling” terms) proportional 2−α+|φi | to |ε| . (II) φi positive. As |ε| → 0, three distinct temperature regions can be idenφ φ φ tified, |ε| i  µi , |ε| i ≈ µi and |ε| i  µi . In the first region, far from the critical point, µi are extremely small and the behaviour is as if µi are zero (i.e., the critical behaviour resembles that of the pure system in which the perturbing interactions are absent). In the second region, 1/φ centred at the crossover temperature |ε∗ | ≈ µi i , the perturbation begins to make its presence felt. Finally in the third region, closest to the critical point, the reduced fields µi become very large and grow rapidly as |ε| → 0 with the result that the perturbative treatment breaks down. Either there is no critical behaviour at all, as for instance, in the presence of a magnetic field, which suppresses the transition (∆ = βδ is always positive and hence can be thought of as a crossover exponent φi ), or else the actual critical behaviour is quite different from that corresponding to µi = 0 and depends on the nature of the operators Qi (i.e., the perturbation). Such crossover phenomena from one kind of critical behaviour to another reflect competition between two critical regimes; the µi = 0 critical behaviour yields progressively to the µi = 0 critical behaviour as the critical point is approached such that the latter critical behaviour takes over at ε = 0. This also implies that the presence of the relevant operator Qi and relevant scaling fields µi leads to a crossover from one critical regime to the other and causes a shift in the critical temperature from TC (µi = 0) to TC (µi = 0). (I)

20

S.N. Kaul

(III) φi zero. In this case, the operators Qi and scaling fields µi are marginal and they can have various consequences, such as logarithmic corrections to the power laws, or critical exponents varying continuously with the strength of the scaling fields µi .

5 The Gadolinium Case Gadolinium metal is made up of spherically symmetric 8 S7/2 Gd3+ ions and hence the magnetocrystalline anisotropy due to crystal-field interactions should be negligibly small or even zero. Moreover, the isotropic RudermanKittel-Kasuya-Yosida (RKKY) interactions between localized 4f magnetic moments give rise to ferromagnetism in this metal. Gadolinium is thus expected to behave as an isotropic three-dimensional Heisenberg ferromagnet in the critical region and should belong to the d = 3, n = 3 universality class. Contrary to this expectation, overwhelming experimental evidence [12, 13, 14] exits in favour of a small uniaxial anisotropy, which ensures that the c-axis of the hexagonal-close-packed (hcp) lattice is the preferred orientation of magnetization in gadolinium at temperatures above the spin reorientation temperature, TSR = 230 K. This asserts that the critical behaviour of gadolinium is that of a three-dimensional Ising ferromagnet upon approaching TC = 293 K. Thus, gadolinium (Gd) should fall within the d = 3, n = 1 universality class. The conventional theories of magnetocrystalline anisotropy hold crystalfield interactions solely responsible for the existence of magnetocrystalline anisotropy. Since Gd3+ is a S-state ion, crystal-field interactions are virtually absent in Gd metal. The conventional theories thus fail to account for the uniaxial magnetic ordering in Gd at temperatures in the vicinity of Curie point TC . According to a recent theory due to Fujiki, De’Bell, and Geldart [15], uniaxial anisotropy in Gd originates from the long-range dipole-dipole interactions operating between magnetic moments localized at the sites of the hcp lattice. These interactions favour [15] the c-axis as the easy direction of magnetization for temperatures above the spin reorientation temperature, TSR , where the lattice parameter ratio c/a of the hcp unit-cell falls below its ideal value of c/a ≈ 1.63. In Gd, the ratio c/a ≈ 1.59 for temperatures in the close proximity to TC . This viewpoint is further strengthened by the fact that the characteristic temperature scale for uniaxial anisotropy ∆Taniso , estimated from magnetic susceptibility data [16, 17] taken along the c-axis and in the basal plane on a single crystal of Gd in the critical region (see, Fig. 1 and its caption for details), is completely accounted for [16, 18] by the dipole-dipole interactions. As already stated in the introduction, the renormalization group (RG) calculations [19, 20] show that the crossover scenario in the critical region gets quite complicated when both dipolar interaction and magnetic anisotropy are present in an otherwise isotropic short-range (d = 3, n = 3) Heisenberg spin

Critical Behaviour of Heisenberg Ferromagnets 0.30

TSR = 230 K

[0001]

0.25

21

TC=292.77 K

[1010]

χ'

ext

0.20 [0001]

0.15

c a3

0.10 0.05 a 1 0.00 0.015

a2

[1010]

TSR

χ''

ext

**

T

0.010

130 K

0.005 0.000

100

150

[0001]

TC

*

T

[1010]

180 K

200

250

300

T (K)

Fig. 1. Temperature dependence of the real, χ
ext , and imaginary, χ

ext , components of susceptibility for Gd single crystal, when an ac field of rms amplitude Hac = 10 mOe and frequency 87 Hz is applied in the [0001] (open circles) and [10¯ 10] (open triangles) crystallographic directions [17]. In Gd, [0001] and [10¯ 10] are the “easy” and “hard ” directions of magnetization for T ≥ TSR . The inset displays the hexagonal close packed structure of Gd and indicates the crystallographic directions along which χ
ext (T ) and χ

ext (T ) were measured. Horizontal dashed lines indicate the demagnetisation limited values (= 1/(4πNd )). A comparison between the temperature variations of χ
ext along the easy and hard directions of magnetization demonstrates the following. In the hard direction, (i) uniaxial anisotropy progressively depresses the value of χ
ext with respect to the demagnetization-limited value as the temperature increases above TSR so much so that the maximum depression occurs at T = TC where the uniaxial anisotropy energy as a function of temperature goes through a broad peak [12, 13], and (ii) χ
−1 ext (T ) extrapolates to zero at a temperature which is below TC = 292.77 K (the Curie temperature in the easy direction of magnetization) by ∆Taniso = 0.58(2) K. The theory due to Fujiki et al. [15] offers a quantitative explanation for both the above observations (i) and (ii) in terms of the uniaxial anisotropy of dipolar origin

system. This is so because the dipolar interaction and magnetic anisotropy act as relevant scaling fields in the RG sense (Sect. 4) and make the Heisenberg fixed point unstable. In this case, “zero-field” susceptibility at temperatures above TC takes the scaling form   gD gA −γH , (33) χ(εH , gD , gA ) ∝ εH X εφHD εφHA

22

S.N. Kaul

where εH = [T − TC (0)]/TC (0), TC (0) = TC (gD = gA = 0) and γH , respectively, are the reduced temperature, transition temperature and susceptibility critical exponent of pure (gD = gA = 0) isotropic short-range Heisenberg (d = 3, n = 3) spin system. The crossover exponents φD and φA are positive while gD (gA ) is the dimensionless ratio of dipolar energy (anisotropy energy) and isotropic short-range exchange energy. Alternatively, gD and gA are a direct measure of the dipolar and anisotropy (relevant) perturbations. 1/φ 1/φ For sufficiently high temperatures, i.e., εH  gD D , gA A , the critical behaviour is that of an isotropic Heisenberg ferromagnet. As the temperature is lowered towards the critical point, a series of crossovers occur depending on the initial values of gA and gD and their relative strengths. In the tempera1/φ 1/φ 1/φ 1/φ ture ranges gD D  εH  gA A and gA A  εH  gD D , the spin system exhibits anisotropic short-range (e.g., d = 3, n = 1) and isotropic dipolar critical behaviour, respectively. The behaviour of the system in the asymptotic 1/φ 1/φ critical region, i.e., at temperatures εH  gD D , gA A or equivalently, in the limit ε → 0, is determined by both anisotropy and dipolar interactions; the reduced temperature ε measures the temperature deviation from the critical temperature TC (gD = 0, gA = 0) of the anisotropic dipolar fixed point. The RG calculations [19, 20] have addressed three distinctly different cases. Based on the calculated temperature dependence of the effective critical exponent for susceptibility, γef f (ε) = dlnχ−1 (ε)/dlnε, at temperatures spanning the asymptotic critical region and crossover regimes, these RG theories predict the following sequences of crossovers as the temperature is lowered from high temperatures to the critical point, TC . Case I: When both gD and uniaxial anisotropy (gU ) are extremely large [19], Gaussian regime → short-range Ising (I) → uniaxial dipolar (UD) fixed point (characterized by mean-field critical exponents with logarithmic corrections [19, 20, 21, 22]). Case II: When gU  gD [20], Gaussian → Isotropic short-range Heisenberg (IH) → I → UD. Case III: when gU  gD [20], Gaussian → IH → isotropic long-range dipolar (ID) → UD. Theoretical investigations of the cases I, II, and III were basically motivated by the expectation that the materials such as LiTbF4 or GdCl3 , Fe14 Nd2 B or Fe14 Y2 B, and Gd could be their respective experimental realizations. While the existence of logarithmic corrections to the mean-field power laws, characteristic of the uniaxial dipolar fixed point, in uniaxial ferromagnets GdCl3 , LiTbF4 and TbF3 [23, 24, 25, 26] as well as a crossover [19] from Ising to asymptotic uniaxial dipolar critical behaviour in LiTbF4 was experimentally confirmed [25], experimental observations fell short of theoretical expectations [20] in the case of hard magnets Fe14 Nd2 B and Fe14 Y2 B. The critical behaviour of Gd remained a highly controversial issue for nearly four decades. From a compilation (Table 1) of the most reliable values of the critical exponents α± , β, γ + and δ published prior to 1994, it is clear that, on the one hand, electrical-resistivity (ρ) [31] and specific-heat (Cp ) [32] data strongly indicated that the asymptotic critical behaviour of Gd could be that of a uniaxial dipolar ferromagnet since these data are consistent with

Critical Behaviour of Heisenberg Ferromagnets

23

Table 1. Critical exponent values for gadolinium reported till 1994. Abbreviations: ACS – ac susceptibility; BM – Bulk Magnetization; BN – Barkhausen Noise; Cp – Specific heat; PAC – Perturbed Angular Correlation; PC – Polycrystalline; ρ – Electrical resistivity; SC – Single Crystal; SES – Scaling Equation of State; TE – Thermal Expansion; TF – Thin Film Critical

Exponent

Exponent

Value

TC (K)

|εmin | − |εmax |

Type

Ref.

Method

α+

–0.09(5)

291.21

1.6 × 10−3 − 1.0 × 10−1

SC

[27]

Cp

α−

–0.32(5)

291.21

2.0 × 10−3 − 1.0 × 10−1

SC

[27]

Cp

10−2

α−

Sample

10−4

–0.20(2)

291.16

2.0 ×

SC

[28]

ρ, Cp

α+

–0.25

293.62

7.0 × 10−4 − 4.0 × 10−2

SC

[29]

TE

α+ = α−

–0.121(2)

293.425

1.3 × 10−3 − 6.6 × 10−2

SC

[30]

TE

293.471

4.5 × 10−5 − 1.0 × 10−3

SC

[31]

ρ

10−3

SC

[32]

Cp

SC

[33]

BM, SES

10−2

α+

=

α+ = α−

0

10−4

− 1.0 ×

α+

0

294.6

3.2 ×

β

0.37(1)

292.5

2.0 × 10−3 − 2.0 × 10−1 10−3

− 1.7 × − 8.0 ×

β

0.381(15)

293.3(1)

4.0 ×

SC

[34]

BM, SES

β

0.390(5)

291.9

2.7 × 10−2 − 1.9 × 10−1

PC

[35]

BM

β

0.385

293.59

1.7 × 10−3 − 8.5 × 10−2

PC

[36]

BN

β

0.399(16)

291.85

1.1 × 10−3 − 1.0 × 10−1

SC

[37]

PAC

10−3

SC

[37]

PAC

SC

[38]

BM

10−1

SC

[39]

BM

SC

[33]

BM, SES

10−2

10−4

β

0.362(8)

291.75

1.0 ×

β

0.375(5)

293.370

6.0 × 10−4 − 5.1 × 10−2 10−3

− 1.0 ×

γ+

1.3(1)

292.5(5)

2.2 ×

γ+

1.25(10)

292.5(1)

3.3 × 10−3 − 2.0 × 10−1 10−3

− 2.0 ×

γ+

1.196(3)

293.3(1)

4.5 ×

SC

[34]

BM, SES

γ+

1.24(3)

291.1(1)

9.9 × 10−3 − 3.7 × 10−2

PC

[40]

ACS

γ+

1.22(2)

293.51(3)

1.2 × 10−3 − 1.5 × 10−1

SC

[41]

ACS

γ+

1.39(2)

293.370

2.0 × 10−3 − 3.7 × 10−2

SC

[38]

BM

292.1(5)

2.0 ×

10−2

TF

[42]

ACS

293.57(2)

4.0 × 10−4 − 1.3 × 10−2

SC

[16]

ACS

1.327(2)

293.57(2)

1.0 × 10−3 − 1.0 × 10−2

SC

[18]

ACS

δ

4.39(10)

292.5

SC

[33]

BM, SES

δ

3.615(15)

293.3(1)

SC

[34]

BM, SES

δ

4.8(1)

293.370

SC

[38]

BM

γ+

1.235(25)

γ+

1.23(2)

γ+

10−3

− 5.8 ×

− 8.0 ×

24

S.N. Kaul

the mean-field behaviour (i.e., dρ(T )/dT , Cp (T ) ≈ |ε|−α with the specificheat critical exponent α = 0, where ε = (T − TC )/TC ) with logarithmic corrections in the asymptotic critical region. On the other hand, contrary to the above theoretical expectations, measurements of the bulk magnetization [33, 34, 35, 38, 39], perturbed angular correlation [37] and magnetic susceptibility [40, 41, 42]) yielded values for static critical exponent β (γ) of spontaneous magnetization (susceptibility) close to that predicted by d = 3 Heisenberg (d = 3 Ising) model. To make matters worse, the exponent δ had a value [33, 34] that is in complete disagreement with the value predicted by either the d = 3 Heisenberg model or the d = 3 Ising model or even the mean-field model. The reported values for the critical exponents α, β, γ and δ thus led to a serious violation of the scaling relations α + 2β + γ = 2 and β + γ = βδ. Furthermore, critical spin dynamics in gadolinium is inconsistent [43, 44] with the predictions of both d = 3 Heisenberg and Ising models. A major breakthrough in understanding the complex critical behaviour of Gd was achieved recently when an elaborate analysis of extensive highresolution ac susceptibility [45] and bulk magnetization [46] data taken along the c-axis (easy direction of magnetization) of a high-purity Gd single crystal over four decades in the reduced temperature ε = (T − TC )/TC clearly demonstrated the following. The single power laws, i.e., (1)–(4), alone neither adequately describe the observed temperature variations of spontaneous magnetization, M (ε, 0), and initial susceptibility, χ(ε), in the asymptotic critical region |ε| ≤ 2 x 10−3 , nor reproduce the observed field dependence of magnetization at TC , M (ε = 0, H), but do so only when the multiplicative logarithmic corrections to these power laws, predicted by the RG calculations for a d = 3 uniaxial dipolar ferromagnet, are taken into account, i.e., only when the RG expressions [21, 22] β  |ln |ε|| M (ε, 0) = B(−ε)

χ−1 (ε) = Γ−1 εγ |ln ε|

x−

−x+ −

 δ |ln |M ||−3x H = DM −

ε<0 ε>0 ε=0 +

(34) (35) (36)

with β = 0.5, γ = 1, δ = 3, x = 3/(n + 8), x = (n + 2)/(n + 8) and n = 1, are used for the data analyses. Considering that the experimental values for the asymptotic critical exponents β, γ, δ, the logarithmic correction B  δ−1 Γ match exponents x− , x+ , and the universal amplitude ratio Rχ = D only those predicted by the RG calculations [21, 22] for a d = 3, n = 1 dipolar ferromagnet (see, Table 2), these results unambiguously establish that the asymptotic critical behaviour of Gd is that of a d = 3 uniaxial dipolar (UD) ferromagnet. Moreover, these values [45, 46] of the asymptotic critical exponents β, γ and δ together with the previously reported [31, 32] value α± = 0 get rid of the earlier violation of the scaling relations completely. In addition, these results provide a conclusive experimental evidence for the existence of crossovers: UD → isotropic dipolar (ID) crossover at

Critical Behaviour of Heisenberg Ferromagnets UD

T

0.50

C

1.6

= 292.78(1) K

UD

T

C

25

= 292.77(1) K

1.4 βeff

γeff

0.49 UD εCO

0.48

IH

1.2

*

1.0

T

0.47

0.001

0.010 UD

0.0001

UD

ε = (T - TC ) / TC

χac(T) χ(T)

UD

ε CO

0.0010

ID

0.0100 UD

0.1000 UD

ε = (T - TC ) / TC

Fig. 2. Temperature variations of the effective critical exponents βef f and γef f [46]. The downward and upward arrows in the above figures indicate the Uniaxial Dipolar – to – Isotropic Heisenberg and Uniaxial Dipolar – to – Isotropic Dipolar crossover temperatures while T ∗ is the onset temperature of the peak, where a transition from the linear domain wall to Bloch domain wall occurs [36, 46] Table 2. Comparison between experiment and theory

Method

Gd

x−

γ

[Ref.]

β

M (T, H)

0.5002(6) 0.330(2)

x+

δ

1.0008(5) 0.329(1)

Rχ = B  δ−1 Γ D

3.005(5) 0.58(12)

[46] Gd

1.0003(3) 0.329(9)

χac [45]

d = 3, n = 3 RG Heisenberg

d = 3, n = 3 RG Dipolar

1.386(4)

4.80(4)

1.33(1)

0.381

1.372

4.45(4)

0.325(2)

1.241(2)

4.82(3)

1.6(1)

3.0

0.5

3.0

1.0

[2, 22]

d = 3, n = 1 RG Ising

0.365(3)

[2]

[2, 6]

d = 3, n = 1 RG Dipolar

[6, 21, 22]

Mean-field

[2]

0.5

3 n+8

=

1 3

1.0

for n = 1 0.5

n+2 n+8

=

1 3

for n = 1 1.0

D→ID εU = 2.05(10) x 10−3 followed at higher temperatures by a sluggish CO ID → Gaussian crossover (the latter crossover proceeds without the intervening, theoretically predicted [19, 20], ID → isotropic short-range Heisenberg (IH) crossover) as the temperature is raised from TC (critical temperature corresponding to the UD fixed point) to T  TC , and UD → IH crossover as the temperature is lowered from TC to T  TC . The UD → IH crossover is accompanied by a transition from linear (uniaxial dipolar/ Ising) domain wall

26

S.N. Kaul

1.8

Hac = 100 mOe; ν = 87 Hz Hac = 10 mOe; ν = 87 Hz

γeff

1.6 1.4 1.2

Hac = 10 mOe ν = 187 Hz Hac = 10 mOe ν = 187 Hz Hdc = 14.4 mOe

1.0

0.01 ID

0.1 ID

ID

ε = (T - TC )/TC

Fig. 3. γef f as a function of reduced temperature εID = (T − TCID )/TCID [45]. The ordinate scale for the data with symbols ×,
, ◦ and • should read as the specified scale minus 0.0, 0.1, 0.2, and 0.3, respectively. The Isotropic Dipolar (ID) – to – Gaussian regime crossover is extremely slow and begins at a reduced temperature close to the one at which a peak in the γef f (εID ) curve occurs [45] Note that γef f should approach unity in the Gaussian regime

to Bloch (Heisenberg) domain wall and no theoretical predictions exist for this crossover. The temperature variations of the effective critical exponents βef f and γef f for spontaneous magnetization and “zero-field” susceptibility, displayed in Figs. 2 and 3, highlight the crossovers UD → IH for T < TC and UD → ID → Gaussian for T > TC . Furthermore, the observation that the uniaxial anisotropy takes over exactly in the same temperature range |ε| ≤ 2 x 10−3 as that predicted by the theory [15] for the uniaxial anisotropy of dipolar origin asserts that the dipolar interactions are solely responsible for the uniaxial magnetic ordering in Gd for temperatures in the vicinity of TC . The other relevant details of the Gd case are given in references [17, 45, 46, 47]. The ID → IH crossover (not observed in Gd) has recently been observed [48] in amorphous re-entrant ferromagnets where long-range dipolar interactions play a decisive role in establishing long-range ferromagnetic order.

6 Summary and Future Scope A detailed comparison between the theoretical predictions and recent experimental observations concerning the critical behaviour of Gadolinium (Gd) permits us to draw the following conclusions. (i) The asymptotic critical be-

Critical Behaviour of Heisenberg Ferromagnets

27

haviour of Gd is that of a d = 3 uniaxial dipolar (UD) ferromagnet. (ii) For temperatures above TCU D , the uniaxial dipolar (UD) to isotropic dipolar (ID) D→ID = 2.05(10)×10−3 crossover occurs at a sharply defined temperature εU CO and this crossover is followed at higher temperatures by isotropic dipolar to Gaussian crossover. The latter observation is at variance with the theoretical prediction that the UD → ID crossover should be followed by ID → isotropic short-range Heisenberg crossover. (iii) For temperatures below TCU D , a single crossover from uniaxial dipolar to isotropic short-range Heisenberg fixed point D→IH = −2.08(5) × 10−3 and this crossover is accompanied is observed at εU CO by a transition from linear (uniaxial dipolar/Ising) domain wall to Bloch (Heisenberg) domain wall. No theoretical prediction exists for T < TCU D . (iv) Uniaxial anisotropy in Gd solely originates from the dipole-dipole interactions. The above conclusions (i), (ii) and (iv) have also been drawn recently from a quantitative interpretation of the critical spin dynamics data on Gd (obtained from M¨ ossbauer spectroscopy [37], Perturbed Angular Correlation [44] and Muon Spin Rotation [49] experiments) in terms of the modecoupling theory [50]. The crossover scenario turns out to be more complicated in Gd than in other uniaxial dipolar ferromagnets such as GdCl3 , LiTbF4 , TbF3 , Fe14 Nd2 B and Fe14 Y2 B since the relevant perturbations gU and gD in Gadolinium are not independent. This is so because in Gd, the dipolar interactions that give rise to gD are also responsible for gU . With these recent developments, Gd joins the select group of uniaxial dipolar ferromagnets such as GdCl3 , LiTbF4 and TbF3 [23, 24, 25, 26] in which the crossover phenomena are relatively better understood. The recent experimental results [17, 45, 46] thus assert that any rigorous theoretical treatment of the ground state and finite-temperature properties of Gd cannot afford to ignore the role of anisotropy and hence of dipolar interactions. Most of the renormalization group (RG) studies of the type and sequence of crossovers in uniaxial dipolar ferromagnets deal with the calculation of the effective susceptibility exponent γef f in the paramagnetic (PM) phase. However, for a better understanding of the crossover phenomena such theoretical calculations of the effective spontaneous magnetization exponent βef f (effective specific heat exponent αef f ) in the ferromagnetic (FM) phase (FM and PM phases) are needed particularly when the experimental results on βef f (ε) in Gd [46] and Fe14 Nd2 B [51] (αef f (ε) in Gd [32]) are already available for comparison. RG calculations of αef f (ε) in dipolar ferromagnets cover only the isotropic dipolar [52] and the Ising [19] cases. On the experimental front, γef f (ε) data in the asymptotic critical region are needed for the uniaxial dipolar ferromagnets Fe14 Nd2 B and Fe14 Y2 B to compare with the existing theoretical predictions [19, 20] concerning the crossover scenario for temperatures above TCU D .

28

S.N. Kaul

Acknowledgements The author thanks all his collaborators, especially S. Srinath for his major contribution to the experimental work on gadolinium, and H. Kronm¨ uller and M. F¨ ahnle for fruitful discussions. The financial support from the Ministry of Education, Spain, and the warm hospitality of the University of Cantabria, Santander, is gratefully acknowledged. The author thanks J. I. Espeso for rendering assistance in the preparation of this manuscript.

References 1. H.E. Stanley: Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, London, 1971) 12, 14 2. S.N. Kaul: J. Magn. Magn. Mater. 53, 5 (1985) and references cited therein 12, 14, 15, 25 3. E.K. Riedel, F.J. Wegner: Phys. Rev. B 9, 294 (1974) 14 4. M. Seeger et al: Phys. Rev. B 51, 12 585 (1995); S.N. Kaul, M. Sambasiva Rao: J. Phys.: Condens. Matter 6, 7403 (1994) 14, 15 5. P.D. Babu, S.N. Kaul: J. Phys.: Condens. Matter 9, 7189 (1997); S.F. Fischer, S.N. Kaul, H. Kronm¨ uller: Phys. Rev. B 65, 064443-1 (2002) 14, 15 6. V. Privman, P.C. Hohenberg, A. Aharony: Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1991) 15, 25 7. N. Goldenfeld: Renormalization Group in Critical Phenomena (AddisonWesley, New York, 1994) 15 8. J.L. Cardy: Scaling and Renormalization in Statistical Physics (Cambridge University Press, London, 1996) 15 9. Review Articles: in Fractals and Disordered Systems, edited by A. Bunde and S. Havlin (Springer, Berlin, 1996) 15 10. C. Domb: The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena (Taylor & Francis, London, 1996) 15 11. S.N. Kaul, Phase Transitions 47, 23 (1994); S.K. Ma: Modern Theory of Critical Phenomena (W. A. Benjamin, Massachusetts, 1976) 16, 17, 18 12. W.D. Corner, W.C. Roe, K.N.R. Taylor: Proc. Phys. Soc. London 80, 927 (1962); W.D. Corner, B.K. Tanner: J. Phys. C 9, 627 (1976) 20, 21 13. C.D. Graham, Jr.: J. Appl. Phys. 34, 1341 (1963) 20, 21 14. G. Will, R. Nathans, H.A. Alperin: J. Appl. Phys. 35, 1045 (1964); J. W. Cable and E. O. Wollan, Phys. Rev. 165, 733 (1968) 20 15. N.M. Fujiki, K. De’Bell, D.J.W. Geldart: Phys. Rev. B 36, 8512 (1987) 20, 21, 26 16. D.J.W. Geldart et al: Phys. Rev. Lett. 62, 2728 (1989) 20, 23 17. S.N. Kaul, S. Srinath: Phys. Rev. B 62, 1114 (2000) 20, 21, 26, 27 18. R.A. Dunlap et al: J. Appl. Phys. 76, 6338 (1994) 20, 23 19. E. Frey, F. Schwabl: Phys. Rev. B 42, 8261 (1990) 20, 22, 25, 27 20. K. Ried, Y. Millev, M. F¨ ahnle, H. Kronm¨ uller: Phys. Rev. B 51, 15229 (1995) 20, 22, 25, 27 21. A.I. Larkin, D.E. Khmel’nitskii: Sov. Phys. JETP 29, 1123 (1969); F.J. Wegner, E.K. Riedel: Phys. Rev. B 7, 248 (1973); E. Bre´zin, J. Zinn-Justin: Phys. Rev. B 13, 251 (1976) 22, 24, 25 22. E. Bre´zin: J. Phys. (Paris) 36, L-51 (1975); A. Aharony, P.C. Hohenberg: Phys. Rev. B 13, 3081 (1976); A.D. Bruce, A. Aharony: Phys. Rev. B 10, 2078 (1974) 22, 24, 25

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23. J. K¨ otzler, W. Scheithe: Solid State Commun. 12, 643 (1973) ; J. K¨ otzler, G. Eiselt: Phys. Lett. 58A, 69 (1976) 22, 27 24. G. Ahlers et al: Phys. Rev. Lett. 34, 1227 (1975); J. Als-Nielsen, ibid. 37, 1161 (1976); J.A. Griffin, J.C. Litster, A. Linz, ibid. 38, 251 (1977); P. Bean-villain et al: J. Phys. C 13, 1481 (1980) 22, 27 25. R. Frowein et al: Z. Phys. B 25, 279 (1976); R. Frowein, J. K¨ otzler, W. Assmus: Phys. Rev. Lett. 42, 739 (1979); R. Frowein et al: Phys. Rev. B 25, 4905 (1982) 22, 27 26. J. Brinkmann et al: Phys. Rev. Lett. 40, 1286 (1978) 22, 27 27. E.A.S. Lewis: Phys. Rev. B 1, 4368 (1970) 23 28. D.S. Simons, M.B. Salamon: Phys. Rev. B 10, 4680 (1974) 23 29. K. Robinson, P.C. Lanchester: Phys. Lett. A 64, 467 (1978) 23 30. D.A. Dolejsi, C.A. Swenson: Phys. Rev. B 24, 6326 (1981) 23 31. D.J.W. Geldart et al: Phys. Rev. B 35, 8876 (1987) 22, 23, 24 32. G. Bednarz, D.J.W. Geldart, M.A. White: Phys. Rev. B 47, 14 247 (1993) 22, 23, 24, 27 33. M. Vincentini-Missoni et al: Phys. Rev. B 1, 2312 (1970) 23, 24 34. M.N. Deschizeaux, G. Develey: J. Phys. (Paris) 32, 319 (1971) 23, 24 35. A.G.A.M. Saleh, N.H. Saunders: J. Magn. Magn. Mater. 29, 197 (1982) 23, 24 36. P. Molho, J.L. Porteseil: J. Magn. Magn. Mater. 31–34, 1023 (1983) 23, 25 37. A.R. Chowdhury et al: Phys. Rev. B 33, 6231 (1986) 23, 24, 27 38. Kh.K. Aliev, I.K. Kamilov, A.M. Omarov: Sov. Phys. JETP 67, 2262 (1988) 23, 24 39. C.D. Graham, Jr.: Jpn. J. Appl. Phys. 36, 1135 (1965) 23, 24 40. G.H.J. Wantenaar et al: Phys. Rev. B 29, 1419 (1984) 23, 24 41. P. Hargraves et al: Phys. Rev. B 38, 2862 (1988) 23, 24 42. U. Stetter et al: Phys. Rev. B 45, 503 (1992) 23, 24 43. A.R. Chowdhury, G.S. Collins, C. Hohenemser: Phys. Rev. B 30, 6277 (1984) 24 44. G.S. Collins, A.R. Chowdhury, C. Hohenemser: Phys. Rev. B 33, 4747 (1986); A.R. Chowdhury, G.S. Collins, C. Hohenemser: ibid. 33, 5070 (1986) 24, 27 45. S. Srinath, S.N. Kaul, H. Kronm¨ uller: Phys. Rev. B 59, 1145 (1999) 24, 25, 26, 27 46. S. Srinath, S.N. Kaul: Phys. Rev. B 60,12 166 (1999) 24, 25, 26, 27 47. S.N. Kaul: Z. Metallkd. 93, 1024 (2002) 26 48. S. Srinath, S.N. Kaul, M. –K. Sostarich: Phys. Rev. B 62, 11 649 (2000); S. Srinath, S.N. Kaul: Europhys. Lett. 51, 441 (2000) 26 49. E. Frey, F. Schwabl, S. Henneberger, O. Hartmann, R. W¨ appling, A. Kratzer, G.M. Kalvius: Phys. Rev. Lett. 79, 5142 (1997) 27 50. S. Henneberger, E. Fray, P.G. Maier, F. Schwabl, G.M. Kalvius: Phys. Rev. B 60, 9630 (1999) 27 51. K. Ried, D. K¨ ohler, H. Kronm¨ uller: J. Magn. Magn. Mater. 116, 259 (1992) 27 52. A.D. Bruce, J.M. Kosterlitz, D.R. Nelson: J. Phys. C 9, 825, (1976) 27

Aspects of the FM Kondo Model: From Unbiased MC Simulations to Back-of-an-Envelope Explanations Maria Daghofer1 , Winfried Koller2 , Alexander Pr¨ ull1 , Hans Gerd Evertz1 1 and Wolfgang von der Linden 1

2

Institute for Theoretical and Computational Physics Graz University of Technology [email protected] Department of Mathematics, Imperial College

Abstract. Effective models are derived from the ferromagnetic Kondo lattice model with classical corespins, which greatly reduce the numerical effort. Results for these models are presented. They indicate that double exchange gives the correct order of magnitude and the correct doping dependence of the Curie temperature. Furthermore, we find that the jump in the particle density previously interpreted as phase separation is rather explained by ferromagnetic polarons.

1 Introduction Manganites [1] are often described by the ferromagnetic Kondo lattice model, which is considered to explain some of their features, e.g., the transition from antiferromagnetic to ferromagnetic order with doping [2]. The application of the model is motivated by the fact, that crystal field splitting divides the five d-orbitals into two eg and three t2g orbitals, where the latter are energetically favored in the case of manganites. All three t2g orbitals are singly occupied and rather localized. Due to a strong Hund’s rule coupling, these electrons are aligned in parallel and form a core spin with length S = 3/2. The filling of the eg orbitals is determined by doping and these electrons can hop from one Mn ion to the next via the intermediate oxygen. Hund’s rule coupling leads to a ferromagnetic interaction between the itinerant eg electrons and the t2g core spin. The core spins interact through super exchange leading to a weak antiferromagnetic coupling between them. In this chapter, we derive effective models for the ferromagnetic Kondo lattice model and introduce suitable Markov chain Monte Carlo (MC) algorithms. The presented results, were not obtainable by simple analytic considerations, are partly found by this MC method and partly by use of the Wang-Landau algorithm [3].

M. Daghofer et al.: Aspects of the FM Kondo Model: From Unbiased MC Simulations to Backof-an-Envelope Explanations, Lect. Notes Phys. 678, 31–45 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


32

Daghofer et al.

2 Model Hamiltonian The spin and charge degrees of freedom in manganites can be described by the ferromagnetic Kondo lattice model with two orbitals (x2 − y 2 and 3z 2 − r2 ):    ˆ =− H tiα,jβ c†iασ cjβσ − J˜H σ iα · S i + J  Si · Sj , (1) i,j,α,β,σ

iα



where c†iασ (ciασ ) creates (annihilates) an electron with spin σ in orbital α at site i, σ iα denotes electron spin in orbital α and S i the core spin. The first term of the Hamiltonian describes the hopping between the nearest neighbor sites; the hopping strength tiα,jβ depends on the involved orbitals and the direction. As matrices in the orbital indices α, β = 1(2), corresponding to the x2 − y 2 (3z 2 − r2 ) orbitals (see e.g. [2]), this hopping reads √   00 3/4 ∓ 3/4 √ , ti,i+ˆx/ˆy = t ti,i+ˆz = t . (2) 01 ∓ 3/4 1/4 The overall hopping strength is t, which will be used as unit of energy, by setting t = 1. The second term contains the ferromagnetic interaction between the electrons and the core spins and the third term is the AFM superexchange of the core spins. The S = 3/2 core spin can approximately be treated as a classical spin, which corresponds to the limit S → ∞ [4, 5]. It is then replaced by a vector of unit length and the factor 3/2 is incorporated into J˜H . This approximation simplifies calculations enormously and should not lead to much difference from the quantum case except possibly for very low temperatures T ≈ 0 [6, 7, 5, 8, 9]. For large J˜H , the electronic density of states of the Hamiltonian (1) is split into the lower and upper Kondo band, separated by approximately J˜H . The eg electrons move mostly parallel to the core spins in the lower band, and anti-parallel in the upper band. In order to derive effective low-energy models for J˜H  t, J  , we change the quantization axis for the electron spin from the global quantization axis (e.g. the z-direction) to a local quantization axis, namely the direction of the local t2g core spin. Spin up (down) then means that the eg electron spin is parallel (antiparallel) to the core spin. The Hund’s rule term J˜H σ iα · S i becomes JH (ˆ niα↓ − n ˆ iα↑ ), with the factor 1/2 coming from the electron spin also absorbed into JH . While the eg -spin is preserved in global quantization, this is no longer the case in the local quantization. An up electron at site i can therefore become a down electron at site j, which is denoted by the superscript for the hopping strength. Furthermore, the hopping now depends on the core spins:
σ,σ
(3) tσ,σ iα,jβ = tiα,jβ uij The first factor is the orbital-dependent hopping strength (2) and the second factor contains the relative orientation of the core spins:

Aspects of the FM Kondo Model

uσ,σ i,j = σ,−σ ui,j

ci cj + si sj eiσ(φj −φi )

33

= cos(ϑij /2) eiψij

(4)

= σ(ci sj e−iσφj − cj si e−iσφi ) = sin(ϑij /2) eiχij

(5)

with the abbreviations cj = cos(θj /2) and sj = sin(θj /2) and the restriction 0 ≤ θj ≤ π, where θi , φi are the polar coordinates for core spin S i . These factors depend on the relative angle ϑij of the core spins S i and S j and on some complex phases ψij and χij . With a shift of the chemical potential µ → µ − JH , the Hamiltonian (1) in local spin-quantization reads:   
†  ˆ =− H tσ,σ c c + 2J n ˆ + J Si · Sj . (6)
H iα↓ iα,jβ iασ jβσ i,j,α,β,σ,σ


iα



This is still the same Hamiltonian as (1) without any approximation besides the use of classical core spins. 2.1 Effective Spinless Fermions Most of the experimental results on manganites and all of the theoretical work presented here concerns electron densities 0 ≤ nel ≤ 1, i. e. predominantly the lower Kondo band. As JH is much larger than the hopping t and the AFM superexchange J  , one can simplify the model by a separation of energy scales [10]. As a first approximation, one can take JH → ∞ and thereby leave out the configurations with eg electrons antiparallel to the core spins completely. This approximation is widely used [11, 12], but misses some important effects discussed in Sect. 4.1. However, if one treats these configurations in second order perturbation theory [13, 14], almost perfect agreement to the original Kondo lattice model is obtained without any additional numerical effort [15]. This approach is similar to the derivation of the t−J model from the Hubbard model, while the JH → ∞ method corresponds to U → ∞ for the Hubbard model. In this effective model, the dynamical degrees of freedom are the low energy states with the eg -spins parallel (i. e. up in local quantization) to the t2g -spins. The virtual excitations meditated by the hopping matrix are configurations where one eg electron is antiparallel (down): (iα ↑) → (jβ ↓) → (i α ↑). As the low energy states contain only up electrons, this lead to an effective spinless fermion Hamiltonian:   ↑↑   t↑↓ t↓↑ † iα
,jβ jβ,iα ˆ HESF = − tiα,jβ ciα cjβ − c†iα
ciα 2JH
i,α,α i,j,α,β j,β  (7)    t↑↓ t↓↑ † i
α
,jβ jβ,iα  − c + J S · S . c i j i
α
iα 2JH [i=i
],α,α
j,β



The first term of this Hamiltonian contains the kinetic energy of the electrons moving in the lower Kondo band. As t↑↑ iα,jβ is largest for parallel core spins, this term favors ferromagnetism. The second term describes electrons that

34

Daghofer et al.

get excited into the upper Kondo band and then hop back to the original site. It yields a density dependent antiferromagnetic interaction between the core spins. The third term is a ‘three-site-term’ of minor influence and will be neglected [15]. On the other hand, its inclusion does not increase the numerical effort. The reduction of the Hilbert space achieved by this effective model is the same as for the JH → ∞ limit, and finite JH can thus be treated with the same numerical effort. 2.2 Uniform Hopping Approach A significant further simplification is the uniform hopping approximation proposed by van den Brink and Khomskii [16]. This approximation replaces the different angles of neighboring core spins by a mean value. In order to treat anisotropies, two different angles are chosen, θz = S i · S i±z in z-direction and θxy = S i ·S i±x = S i ·S i±y within the xy-plane. These should not be confused with the polar angle of an individual core spin θi . It is assumed that the relative orientation is the same between all nearest neighbor pairs. The hopping matrix therefore becomes translationally invariant. Spin configurations that are still treated exactly include, among others, ferro- and antiferromagnetism and spin canted states. The impact of the core spins on the hopping simplifies to = cos( uσ,σ z

θz ) = uz 2

,

uzσ,−σ = sin(

 θz ) = 1 − u2z 2

(8)

in z-direction and analogously in x/y-direction. Likewise, the inner product of the t2g spins entering the superexchange term can be expressed by Si · Si+ˆz = cos θz = 2u2z − 1 .

(9)

The energy of this model can easily be evaluated, especially in the thermodynamic limit and the ground state can be obtained by minimizing the energy with respect to θz and θxy . For a one orbital model in one dimension with periodic boundary conditions, the Hamiltonian simplifies to ˆ = −uz H

 ij

c†i cj −

  1 − u2z  † ci ci + J  L 2u2z − 1 , JH i

(10)

This Hamiltonian yields a shifted tight-binding band structure k = −2u cos(k) − (1 − u2z )/JH

(11)

with a band width of 4 uz , which has its maximum for ferromagnetic core spins and vanishes for antiferromagnetic order.

Aspects of the FM Kondo Model

35

Similar calculations can be done for three dimensions with both orbitals and a ground state phase diagram can thus be obtained, see [16, 15]. This UHA approach was extended to finite temperatures in [17]. To introduce this method, we proceed as follows: For a given core spin configuration S, characterized by the set of angles {θi , φi }, we define the average u-value u(S) =

1  ↑↑ uij (S) . Np

(12)

ij

Here Np is the number of n.n. pairs ij. In the ESF Hamiltonian (7), u↑↑ ij σ,−σ 2 is then replaced by u(S). Besides u↑↑ the Hamiltonian depends on |u | ij ij and on Si · Sj , which correspond to sin2 (ϑij /2) and cos ϑij , respectively. As a further approximation, these terms are replaced by 1 − u2 (S) and 2 u2 (S) − 1 respectively, which leads to the one-orbital UHA Hamiltonian ˆ = −u H

 ij

c†i cj −

  1 − u2  † zi ci ci + J  Np 2u2 − 1 , 2JH i

(13)

with zi being the number of nearest neighbors for site i. This Hamiltonian defines the Boltzmann factor for the spin configuration S. In order to calculate thermodynamical expectation values, one still needs to calculate the density of states Γ (u), i.e. the number of spin configurations with the same average value for u. It can be calculated exactly in one dimension and by use of the Wang-Landau algorithm [3] in higher dimensions. Once the density of states Γ (u) has been obtained, observables can be obtained for any temperature, much larger lattices can be treated and a 3D phase diagram for finite temperatures can be obtained. The numerical effort is reduced from an integration over the L-dimensional space of the core spin configurations to an integral over the one-dimensional unit interval for u.

3 Monte Carlo Algorithm The algorithm used to simulate the Kondo lattice model and the effective spinless fermion model in the grand canonical ensemble is the one proposed in [18]. For each core spin configuration, the resulting Hamiltonian for the eg electrons is a one-particle problem. The statistical weight for the core-spin configuration S in the grand canonical ensemble is the starting point of grand canonical Markov chain Monte Carlo simulations: ˆ

w(S|µ) =

ˆ

trc e−β(H(S)−µN ) . Z(µ)

(14)

ˆ It is calculated from the eigenvalues of H(S) by use of free fermion formulae, which is denoted by the trace over the fermionic degrees of freedom trc .

36

Daghofer et al.

As some particle numbers are not stable in the grand canonical ensemble, we developed a canonical algorithm. An exact approach would mean calculating the Boltzmann weight for every possible distribution of Nel electrons on L eigenvalues and summing over these contributions. This is numerically too demanding. But for low temperatures, only very few of these distributions actually contribute to the partition function. They can be obtained by filling N0 < Nel electrons into the N0 lowest eigenenergies and distributing only the remaining Nel − N0 ≈ 5 electrons on the states around the Fermi-energy. The canonical weight then becomes: 

ˆ

˜ P

w(S|Nel ) =

˜

e−β H(S,P(Nel )) , Z(Nel )

(15)

where P˜ denotes these restricted permutations. In order to decrease autocorrelations, particle fluctuations within a set of 3 to 5 densities were allowed. The amount of core-spin rotations was small for most updates in order to ensure high acceptance, but occasionally a complete spin flip was proposed. In one dimension, whole sections of the chain were rotated at once. 50 to several hundreds of sweeps were skipped between measurements. This ensured statistical independence for the 1D calculations, in 2D, remaining autocorrelations were treated by autocorrelation analysis. While observables depending only on Z are independent of the spin quantization (global/local), care must be taken when evaluating e.g. the one particle Greens function, which in global quantization can be written as

 † S  aiσ ; a†jσ ω = D[S] w(S|µ)u↑↑ (16) ji (S)  ci ; cj ω , σ

where  ci ; c†j Sω is the Green’s function in local spin quantization. It can be expressed in terms of the one-particle eigenvalues (λ) and the corresponding ˆ eigenvectors ψ (λ) of the Hamiltonian H(S):  ci ; c†j Sω =

 λ

ψ (λ) (i) ψ ∗(λ) (j) ω − ((λ) − µ) + i0+

(17)

It should be pointed out that the one-particle density of states (DOS) is identical in global and local quantization; for details see [17].

4 Results In the first subsection, we will demonstrate the validity of the simplified effective models. We will then use the uniform hopping approach (UHA) in Sect. 4.2 to determine the Curie temperature of the one-orbital model in three dimensions. In Sect. 4.3, we will present results for the one- and

Aspects of the FM Kondo Model

37

two-dimensional model and we will show that they do not indicate phase separation, but rather ferromagnetic polarons. A phase diagram for the 2D model is given in Sect. 4.4. 4.1 Validity of the ESF Model and the UHA in one Dimension In this section, we will present results from unbiased Monte Carlo Simulations for the ESF model in one dimension with one orbital and open boundary conditions and we will compare them to results for the full ferromagnetic Kondo lattice model and the JH → ∞ approximation. Simulations were done for L = 20 sites, β = 50, J  = 0.02 and JH varying from 4 to 10. Figure 1 shows the core-spin structure factor for the three models. For electron density n ≈ 0.75 (inset), the t2g correlations are ferromagnetic, driven by the kinetic energy of the eg electrons. For n = 1, the lower Kondo band is completely filled, no hopping is possible and the kinetic energy therefore vanishes. Excitations into the upper Kondo band (virtual for the ESF), which are favored by antiferromagnetism, then dominate the energy. For both densities and even for moderate JH = 4, the ESF model (7) and the original Kondo Model (6) produce virtually identical results. The JH → ∞ model on the other hand does not reproduce the antiferromagnetic correlations for the completely filled lower Kondo band correctly, because the virtual excitations are missing from this model, and it also overestimates the ferromagnetic correlations at n ≈ 0.75. The AFM effect coming from the virtual excitations can be described by a density dependent effective parameter Jeff = J  + 1/(2JH ) at n = 1 and it is generally much stronger than the small superexchange J  also favoring AFM.

Fig. 1. Spin structure factor for the t2g spins at n = 1 (inset: n ≈ 0.75) for β = 50, J
= 0.02, L = 20 and different values of JH . Circles: spinless fermion model (7); crosses: DE model (6). In the limit JH → ∞ (dashed line) the intensity of the AFM peak is considerably smaller than for finite JH . From [15]

38

Daghofer et al. 1 0.8

JH=4 J =6 H J =10 H

0.6 n 0.4 0.2 0 −12

−10

−8 µ−J

−6

−4

−2

H

Fig. 2. Electron density versus chemical potential for JH = 4, 6, and 10 (right to left), and J
= 0.02. MC results at β = 50, L = 20 for the spinless fermion model HESF (circles) are compared with those for the DE model H (crosses). Error bars of the MC data are smaller than the symbols. The lines correspond to groundstate UHA. From [15]

Figure 2 shows the electron density versus the chemical potential µ for the Kondo model, the effective spinless fermions and groundstate UHA for JH = 4, 6, 10. All three models give almost identical results. For very small µ, the band is empty and J  leads to antiferromagnetism. At a critical µc1 depending on Jeff = J  , the filling jumps to n ≈ 0.2 and the correlations become ferromagnetic. At a second critical µc2 , depending on Jeff = J  + 1/(2JH ), it becomes antiferromagnetic again and the density jumps from n ≈ 0.7 to n ≈ 1. These discontinuities mean that intermediate particle numbers are not stable in the grand canonical ensemble. They have been interpreted as phase separation. However, we will show in Sect. 4.3 that their cause lies rather in small ferromagnetic polarons. 4.2 Finite Temperature UHA and Curie Temperature Although the uniform hopping approach replaces the fluctuating core spins by an average u, it reproduces not only the expectation value of the energy, but also its width with astonishing accuracy, even for higher temperatures. The results of UHA also remain valid upon the inclusion of n.n Coulomb repulsion [17]. All the while, the numerical effort is reduced from sampling over hundreds of thousands of core-spin configurations to scanning the single parameter u within the unit interval. The fact that the results remain valid with inclusion of the Coulomb repulsion indicates that UHA is a reliable approximative method that can safely be extended to more complicated situations.

Aspects of the FM Kondo Model

400

39

PM

paramagnetic

300 Tc

PI 200 ferromagnetic 100 SCI 0

0

FI 0.1

FM 0.2

0.3

0.4

0.5

x Fig. 3. Curie temperature (dashed line) of the one-orbital DE model for a 163 cluster and t = 0.2 eV calculated in UHA. Circles and phases PM (paramagnetic metal), PI (paramagnetic insulator), FM (ferromagnetic metal), FI (ferromagnetic insulator), and SCI (spin canted insulator) are experimental results for La1−x Srx MnO3 [19], UHA results [17]

As observables can be evaluated for all temperatures once the density of states Γ (u) is known, it can be used to determine the Curie temperature for the three dimensional one-orbital model with JH = ∞ and J  = 0. If one sets the only free parameter, namely the hopping strength t, which was also used as unit of energy, to t = 0.2eV , in accordance with experiments, one obtains the Curie temperature in reasonable agreement with experiment, see Fig. 3. In order to obtain the different low temperature phases besides FM and PM observed in experiments for low carrier concentrations, finite JH and J  would be needed as well as two orbitals with Coulomb repulsion. 4.3 Phase Separation versus Ferromagnetic Polarons The discontinuity of the filling as a function of the chemical potential, see Fig. 2, is usually interpreted as phase separation [18], i.e. the system is expected to split into antiferromagnetic domains with low and ferromagnetic domains with higher carrier concentration. Taking Coulomb interactions into account, PS has been argued to lead to either small [20] or large [21] (nano-scale) clusters, which have been the basis for a possible though controversial [22] explanation of CMR [2, 23]. More thorough evaluation of the MC data for the transition near the filled lower Kondo band reveals however, that single hole ferromagnetic polarons are stabilized instead, even without any Coulomb repulsion. All results presented in this section are for the effective spinless fermion model (7). A first indication for ferromagnetic polarons is the behavior of the electron density nS with the MC time near the critical chemical potential as depicted in Fig. 4, where nS is the thermodynamical expectation value for

40

Daghofer et al. 168

164

167.5

162

167

160

131 130.5 130 158

166

129.5

156

Ne

Ne

Ne

166.5

154

129

128.5

165.5 152

128

150

127.5

165 164.5

(a) 164

50

148 100 150 200 250 300

MC time

127 (c)

(b) 50

100 150 200 250 300

MC time

50

100 150 200 250 300

MC time

Fig. 4. Mean particle numbers Ne in a grand canonical MC simulation in 2 dimensions (L = 14 × 12, JH = 6, J
= 0.02, β = 50) as a function of MC time. One time step corresponds to 200 sweeps of the lattice. (a) µ = 1.26 > µ∗ : almost filled, (b) µ = 1.19  µ∗ : polaron regime, (c) µ = 1.12 < µ∗ : FM regime. For visibility, only the first 350 time steps are shown

the filling given the core-spin configuration S sampled by the MC run. In the FM regime [Fig. 4(c)], the density fluctuates slightly with the fluctuating core-spin configurations and takes non-integer values in accordance with standard results for free electrons. For the almost filled band [Fig. 4(a)], the density is nS = 1 (0 holes) for most spin configurations and occasionally, configurations occur, which contain exactly one, exactly two or exactly three holes. In between [Fig. 4(b)], the particle number fluctuates strongly, but as for nS ≈ 1, almost only integer fillings occur. While these integer fillings can hardly be understood in a PS scenario which is supposed to be a mixture of the low- and the high-density phases, it can easily be explained by independent polarons containing one hole each. As they are independent from each other, all have the same energy exactly balanced by the critical chemical potential µ∗ and their number therefore fluctuates strongly. To measure the size of the ferromagnetic domains, we use the dressed core-spin correlation function Sh (l) =

L−l 1  h n S i · S i+l L − l i=1 i

(18)

that measures the correlations around a hole. The hole- density operator nhi is related to the electron density via nhi = 1 − ni . Equation (18) holds for open boundary conditions (employed in 1D), the formula for periodic boundary conditions used in 2D is analogous. Figure 5 shows this dressed core-spin correlation. The ferromagnetic regions around the holes are small and their size does not grow with doping neither for the one- nor for the the two-dimensional case. This indicates that

Aspects of the FM Kondo Model 1

1

1

1

0.5

0.5 0.8

0

−0.5

0.4

−1 0.2

0

5

10

S(r)

0.6

Sh(r) / Sh(0)

S(l)

Sn(l) / Sn(0)

0.8

41

0.6

0

−0.5

0.4

−1 0

0.2

10

r2

20

0

0 0

2

4

6

8

−0.2 0

10

5

10

15

20

25

r2

l

Fig. 5. Dressed spin-spin correlation function (18) from unbiased MC for β = 50, J
= 0.02, and JH = 6. Left panel: 1 dimension: L = 50-site chain containing one (×), two (), three (◦), four (), and five ( ) holes. The dashed line is calculated within the simple polaron picture, while the solid line represents the generalized UHA result for a single polaron, see [24]. Right panel: 2 dimensions: 12 × 14 lattice with 1 (◦), 6 (×) and 20 () holes. Continuous lines are data for the simple Polaron model (see Fig. 6, right panel): 1 Polaron (dotted), 6 (solid) and 20 Polarons (dashed). L−lThe insets show the conventional spin-spin correlation function 1 S(l) = L−l i=1 S i · S i+l . Left from [24], right from [25]

0.4 density

nh

0.2

0 1 2 3 4 5

1

2

3

4

5

Fig. 6. Toy model for FM polarons in one (left) and two (right) dimensions. Height represents hole density. In 1 D, a FM domain of Lf = 4 lattice sites is embedded in an AFM background, in 2D one spin is flipped from the perfect AFM. A single hole is localized in the FM domain giving rise to the depicted hole density (different from the schematic shape in Fig. 4 of [26]). Left from [24], right from [25]

the introduction of more holes leads to more small FM polarons rather than to a growth of the existing ones. These observations lead to the development of a simple toy model for the FM polarons. Each polaron consist of a small (3-4 sites in 1D, 5 sites in 2D) FM well, in which the hole can delocalize. These wells are embedded into an AFM background. For a schematic representation, see Fig. 6. The value of the critical chemical potential can be obtained from the toy model by simple

42

Daghofer et al.

0 −3

0

−2

2

−1

ω

0

1

2

−2

0 −3

0

−2

2

−1

ω

0

1

2

dos

π wave number

−2

dos

π wave number

wave number

π

dos

energy considerations. It is simply the difference between the energy gained by the delocalized hole and the energy payed for the breaking of AFM bonds. In this simplest model, the hopping strength is given by uf = 1 for the FM regions and ua = 0 for AFM bonds. The impact of thermal fluctuations of the core spins can be modeled by a generalization of UHA to inhomogeneous structures, where Γ (uf , ua ) has to be determined. This was done for the one-dimensional model, for results and details of the algorithm see [24]. In order to compare the toy model to the MC data in 2 dimensions, random deviations were added to the core spins, see [25]. The principal effect of the fluctuations is a finite bandwidth for the AFM at half filling, their amount was therefore fitted to yield the same width for the AFM band as the MC data, see Fig. 8. Because the FM wells in which the hole can move are so small, they give rise to only a few well separated signals in the spectral density. Figure 7 shows the spectral density and the density of states for one, two and three holes in one dimension. On sees a broad band in the center which comes from holes moving in the imperfect AFM background. Separated from this central band by a (mirror) pseudogap are dispersionless states from the FM polarons at ω  ±1.5. The weights of these signals increase upon the introduction of more holes.

−2

0 −3

0

−2

2

−1

0

1

2

ω

Fig. 7. Spectral density for Nh = 1 hole, Nh = 2 holes Nh = 3 holes, corresponding to 1, 2 and 3 polarons in one dimension. Dashed lines: MC data, solid: generalized UHA. Parameters as in Fig. 4. From [24]

Figure 8 shows the spectral density for the two dimensional model with 6 and 20 holes and compares it to the data for the toy model with added random fluctuations. There is again a broad central band from the AFM featuring a mirror band due to the doubling of the unit cell and again the polaronic states separated by a pseudogap. There is strong correspondence of the unbiased MC data to the simplified model for both the one- and the two dimensional system.

−2

0

2

(π,π)

(0,π)

−2

(0,0)

wave number

wave number

(0,0)

(0,0) −3

43

dos

dos

Aspects of the FM Kondo Model

0

2

(π,π)

(0,π)

−2

−1

0 ω

1

2

(0,0)

−2

0

2

ω

Fig. 8. Spectral density for J
= 0.02, β = 50, JH = 6 on a 12 × 14 lattice: left: 6 holes (x ≈ 0.035); right: 20 holes (x ≈ 0.12). Solid lines are the unbiased MC data, dashed lines the simplified polaron model

The pseudogap, which is also observed in experiments [27, 28, 29, 30], can easily be explained by the few well separated eigenenergies of the holes trapped in the small polarons. In a phase separation scenario with larger FM regions, additional states would fill the gap between the AFM band and the polaron states, in contrast to experimental results and the MC data. 4.4 Phase diagram in 2D Although 0 < J  < 0.1 is by far the smallest parameter of the Hamiltonian, it has a considerable effect; especially at low carrier density, when the kinetic energy is small. While we observed nearly independent polarons for β = 50 and J  = 0.02, they tend to form larger clusters and eventually phase separation for decreasing J  and avoid each other for larger J  . The reason for this effect is the stabilization of the AFM background by J  . In order to determine the phase boundary between the polaronic regime and phase separation, we chose the filling, at which the nearest AFM signal (at the distance r2 = 5) in the dressed core-spin correlation (18) became ferromagnetic. This criterion is somewhat arbitrary and the transition is not sharp, polarons rather coexisting with larger clusters. For large doping, J  = 0.05 destroys ferromagnetism and leads to the so called flux phase around half filling [11, 31, 12, 32]. In the phase diagram, one sees a small window in parameter space for phase separation, but for realistic parameters J  > 0.01, polarons dominate, especially at small doping.

44

Daghofer et al.

AFM 160

N

e

140

Pol

0%

Pol./PS

10 %

PM

120 100

20 %

FM

80

Flux

1.5 1

40 % 50 %

0.04

0.5

µ

x

30 %

0.02

0 −0.5 0

J′

Fig. 9. Electron number Ne as a function of µ and J
, and phase diagram for −0.5 < µ < 1.6, 0 ≤ J
≤ 0.05, 0 ≤ x ≤ 0.6 (i. e. filled to 40% filled lower Kondo band),JH = 6, β = 50 on a 14 × 12-lattice. “Pol.”: polaronic regime, “Pol./PS”: mixture of both polarons and larger ferromagnetic clusters, “FM”: ferromagnet, “AFM”: antiferromagnet, “PM”: regime without magnetic structure, “Flux”: Flux phase

5 Summary An effective spinless fermion model (ESF) was derived from the FM Kondo lattice model (1) with classical core spins and for JH  t, leading to the ESF Hamiltonian (7). It allows the treatment of finite JH with the same numerical effort as the JH → ∞ approximation, but gives better correspondence to the original model. A further simplification was achieved by treating the fluctuating core spins by a uniform hopping strength (UHA). With this much simpler model (13), we obtained the Curie temperature for the 3D model with one itinerant orbital in accordance with experimental values. By unbiased MC simulations for the ESF model (7) with non-degenerate conduction band in one and two dimensions, we found that ferromagnetic polarons are the reason for features previously attributed to phase separation. This polaronic behavior is enhanced by larger J  > 0.02. A phase diagram was obtained for the 2 dimensional case.

References 1. T.A. Kaplan, S.D. Mahanti: editors. Physics of Manganites. Kluwer Academic/ Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, 1. edition, 1998. 31 2. E. Dagotto, T. Hotta, A. Moreo: Phys. Rep. 344, 1 (2001) 31, 32, 39 3. F. Wang, D.P. Landau: Phys. Rev. E 64, 5 (2001) 31, 35 4. N. Furukawa: in Physics of manganites, Kluwer Academic Publisher, New York, 1. edition (1998) 32

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5. E. Dagotto, S. Yunoki, A.L. Malvezzi, A. Moreo, J. Hu, S. Capponi, D. Poilblanc, N. Furukawa: Phys. Rev. B 58, 6414 (1998) 32 6. D. Meyer, C. Santos, W. Nolting: J. Phys. Condens. Matter, 13, 2531 (2001) 32 7. W. M¨ uller, W. Nolting: Phys. Rev. B 66, 085205 (2002) 32 8. D.J. Garcia, K. Hallberg, C.D. Batista, S. Capponi, D. Poilblanc, M. Avignon, B. Alascio: Phys. Rev. B 65, 134444 (2002) 32 9. D.R. Neuber, M. Daghofer, R.M. Noack, H.G. Evertz, W. von der Linden: cond-mat/0501251, (2005) 32 10. A. Auerbach: Interacting Electrons and Quantum Magnetism, Springer-Verlag, New York, Berlin, Heidelberg, 1. edition (1994) 33 11. H. Aliaga, B. Normand, K. Hallberg, M. Avignon, B. Alascio: Phys. Rev. B 64, 024422 (2001) 33, 43 12. M. Yamanaka, W. Koshibae, S. Maekawa: Phys. Rev. Lett. 81, 5604 (1998) 33, 43 13. S. Yarlagadda, C. S. Ting: Int. J. Mod. Phys. B 15, 2719 (2001) 33 14. S.Q. Shen, Z.D. Wang: Phys. Rev. B 61, 9532 (2000) 33 15. W. Koller, A. Pr¨ ull, H.G. Evertz, W. von der Linden: Phys. Rev. B 66, 144425 (2002) 33, 34, 35, 37, 38 16. J. van den Brink, D. Khomskii: Phys. Rev. Lett. 82, 1016 (1999) 34, 35 17. W. Koller, A. Pr¨ ull, H.G. Evertz, W. von der Linden: Phys. Rev. B 67, 104432 (2003) 35, 36, 38, 39 18. S. Yunoki, J. Hu, A.L. Malvezzi, A. Moreo, N. Furukawa, E. Dagotto: Phys. Rev. Lett. 80, 845 (1998) 35, 39 19. A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Kido, Y. Tokura: Phys. Rev. B 51, 14103 (1995) 39 20. A.O. Sboychakov, A.L. Rakhmanov, K.I. Kugel, M.Yu. Kagan, I.V. Brodsky: J. Exp. Theor. Phys. 95, 753 (2002) 39 21. A. Moreo, S. Yunoki, E. Dagotto: Science 283, 2034 (1999) 39 22. D.M. Edwards: Adv. Phys. 51, 1259 (2002) 39 23. T. Wu, S.B. Ogale, J.E. Garrison, B. Nagaraj, A. Biswas, Z. Chen, R.L. Greene, R. Ramesh, T. Venkatesan, A.J. Millis: Phys. Rev. Lett. 86, 5998 (2001) 39 24. W. Koller, A. Pr¨ ull, H.G. Evertz, W. von der Linden: Phys. Rev. B 67, 174418 (2003) 41, 42 25. M. Daghofer, W. Koller, H.G. Evertz, W. von der Linden: J. Phys. Condens. Matter, 16, 5469 (2004) 41, 42 26. A. Moreo, S. Yunoki, E. Dagotto: Phys. Rev. Lett. 83, 2773 (1999) 41 27. D.S. Dessau, T. Saitoh, C.H. Park, Z.X. Shen, P. Villella, N. Hamada, Y. Moritomo, Y. Tokura: Phys. Rev. Lett. 81, 192 (1998) 43 28. T. Saitoh, D.S. Dessau, Y. Moritomo, T. Kimura, Y. Tokura, N. Hamada: Phys. Rev. B 62, 1039 (2000) 43 29. Y.D. Chuang, A.D. Gromko, D.S. Dessau, T. Kimura, Y. Tokura: Science 292, 1509 (2001) 43 30. J.H. Park, C.T. Chen, S.W. Cheong, W. Bao, G. Meigs, V. Chakarian, Y.U. Idzerda: Phys. Rev. Lett 76, 4215 (1996) 43 31. D.F. Agterberg, S. Yunoki: Phys. Rev. B 62, 13816 (2000) 43 32. M. Daghofer, W. Koller, W. von der Linden, H.G. Evertz: Physica B 359-361, 804 (2005) 43

Carrier Induced Ferromagnetism in Concentrated and Diluted Local-Moment Systems Wolfgang Nolting, Tilmann Hickel and Carlos Santos Institut f¨ ur Physik, Humboldt-Universit¨ at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany Abstract. For modeling the electronic and magnetic properties of concentrated and diluted magnetic semiconductors we use the s-f model (ferromagnetic Kondolattice model), which traces back the characteristic properties of such materials to an interband exchange coupling between itinerant conduction electrons and localized magnetic moments. For the electronic part an interpolation scheme between a maximum number of rigorously tractable limiting cases is desirable, since the parameter dependence of the selfenergy cannot be calculated exactly. Two possible implementations, which allow the evaluation of temperature-dependent quasiparticle densities of states and spectral densities in different parameter regimes, are presented. For constructing the magnetic phase diagram again two approaches are proposed. A modified RKKY theory maps the interband exchange to an effective Heisenberg model, the exchange integrals of which turn out to be functionals of the electronic selfenergy acquiring therewith a distinct temperature and carrier concentration dependence. A second method derives the magnetic phase diagram from the paramagnetic susceptibility of the itinerant conduction electrons avoiding therewith a mapping of the s-f exchange onto an effective Heisenberg model. The latter procedure allows to inspect the influence of the moment disorder in diluted systems on magnetic stability by adding a CPA-type concept to the theory. For almost all moment concentrations x ferromagnetism is possible, however, only for carrier concentrations n distinctly smaller than x. This can be understood by inspecting the respective quasiparticle density of states. The charge carrier compensation in real magnetic semiconductors such as Ga1−x Mnx As seems to be a necessary condition for getting carrier induced ferromagnetism.

1 Local Moment Magnetism In contrast to the band ferromagnets (e.g. Fe, Co, Ni) the magnetic and the conductivity properties of local-moment systems are due to two different electron groups. The one consists of strictly localized electrons from an inner only partially filled shell giving rise to a permanent magnetic moment (4f electrons in the prototypical rare earth compounds such as EuO, Gd, ...). The other group is built up by itinerant charge carriers of an only partially filled valence or conduction band. Mutual influence (interband exchange coupling) of these two subsystems leads to some extraordinary physical properties, W. Nolting et al.: Carrier Induced Ferromagnetism in Concentrated and Diluted Local-Moment Systems, Lect. Notes Phys. 678, 47–69 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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which are interesting from a basic point of view, and also because of the related technological potential. Prototypical local moment systems are the classical magnetic semiconductors or insulators, today better classified as “concentrated” magnetic semiconductors. In first place there are the europium chalcogenides [1], which take their magnetism from the half-filled 4f-shell of the europium-ion. As to their purely magnetic properties the EuX are considered as rather well understood representing almost ideal realizations of the Heisenberg model. More striking is therefore the temperature reaction of the unoccupied band states on the magnetic order of the moments, manifesting itself, e.g., in a red shift of the optical absorption edge upon cooling below TC [2]. That shift must be due to a respective shift of the lower conduction band edge. This assignment has led to the simple picture of a spin-polarized splitting of the conduction band in the ferromagnetic phase. It, indeed, qualitatively explains a series of interesting experimental observations as the spin filter properties in field emission experiments [3] and above all the spectacular insulator-metal transition in Eu-rich EuO [4, 5]. To our information this transition exhibits the biggest resistivity jump ever observed in nature, up to 14 orders of magnitude. In any case one has to conclude that the reason for the remarkable temperature dependence of the unoccupied band states must be due to an interband exchange coupling between the localized magnetic 4f states and the extended band states. The metallic counterpart of EuO is the ferromagnetic element Gd, again with localized magnetic moments due to the half-filled 4f shell while the conductivity properties are determined by itinerant (5d/6s) electrons [6]. An interesting question concerns the carrier-induced ferromagnetic ordering and the applicability of the conventional RKKY theory. In a systematic manner this could be inspected by the alloy Eux Gd1−x S. Via x a controlled populating of the conduction band is possible without diluting the 4f spin system resulting in a change from the ferromagnetic insulator EuS to the antiferromagnetic metal GdS. An important topic discussed controversially is the temperature behaviour of the induced exchange splitting of the Gd-(5d/6s) conduction bands [7, 8]. In any case, the same interband exchange as for the above-mentioned semiconductors is certainly responsible for a great part of the Gd or Eu1−x Gdx S properties, too. The exciting research field of “spintronics” refers to new phenomena of electronic transport, for which the electron spin plays a decisive role, in contrast to conventional electronics for which the electron spin is practically irrelevant. For a full exploitation of spintronics (or spin-electronics) one should have materials that are simultaneously semiconducting and ferromagnetic. That is the reason for the intensive effort that has been focused on the search for magnetic semiconductors with high Curie temperatures. It is to the merit of Ohno and coworkers to reach a TC of up to 160 K in Ga1−x Mnx As [9, 10] and to prove the electric control of TC by means of a gate voltage [11]. Intense

Carrier Induced Ferromagnetism in Local-Moment Systems

49

experimental as well as theoretical research on the outstanding phenomena associated with the interplay between ferromagnetic cooperative features and semiconducting properties is currently going on [12]. It is an important challenge of materials science to understand the ferromagnetism in compounds such as Ga1−x Mnx As, and to find out the conditions for Curie temperatures TC exceeding sufficiently room temperature. The implentation of Mn2+ ions into the prototype semiconductor GaAs provides local moments (S = 52 ) which decisively influence the electronic GaAs band states giving them, e.g., a distinct temperature dependence. Furthermore, each divalent Mn ion creates in principle one valence band hole. There is no doubt that an interband exchange between localized 3d-Mn2+ moments and the itinerant valence-band holes creates the ferromagnetic ordering of Ga1−x Mnx As. The nature of this exchange interaction is the same as that in EuO or Gd. An important question is whether and how the disorder of the localized magnetic (Mn2+ ) moments influences the magnetic stability. Does the disorder weaken or even strengthen ferromagnetic coupling? With respect to the main goal, namely reaching room temperature ferromagnetism, the disorder aspect has to be considered a central point to clarify. Local moment systems certainly represent interesting and fundamental physics, which is far from being fully understood. In the following we will first concentrate on a proper modeling of these systems. It is commonly accepted [12] that the (ferromagnetic) Kondo-lattice model (KLM) represents a good starting point for the description. The next section is therefore devoted to a detailed inspection of this model. Sections 3 and 4 will describe theoretical approaches to the, respectively, electronic and magnetic part of the not exactly solvable many-body problem of the KLM. Typical model results for the concentrated local-moment systems are presented and discussed. In the last part (Sect. 5) we will discuss how to implement the disorder into the Kondo-lattice model and to inspect its influence on magnetic stability.

2 Kondo-Lattice (s-f ) Model 2.1 Hamiltonian The Kondo-lattice model (KLM) [13, 14, 15, 16] regards the interband exchange between localized magnetic moments and itinerant conduction electrons as an intra-atomic exchange, i.e. a local interaction between the localized spin S i and itinerant electron spin σ i ,   Tij c†iσ cjσ − J S i · σi . (1) H = H0 + Hex = ij

i

c†iσ and ciσ are, respectively, creation and annihilation operator of an electron with spin σ at site Ri . Tij are the hopping integrals. J is the exchange

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coupling constant. In case of a ferromagnetic exchange (J > 0) the notation “Kondo-lattice model” is certainly somewhat misleading, better are the older denotations “s-f, s-d model” or in the strong coupling region “double-exchange model”. Using second quantization the interaction can be decomposed into three terms, an Ising-like interaction between the z-components of the spin operators and two terms which express spin exchange processes:   Siz (ni↑ − ni↓ ) + Si+ c†i↓ ci↑ + Si− c†i↑ ci↓ . (2) Hex = −J i

niσ is the occupation number operator. Spin exchange may happen by, respectively, magnon emission by a down electron and magnon absorption by an up electron. In both cases the electron reverses its spin. Since the magnon may carry any wave-vector from the first Brillouin zone a rather diffuse scattering spectrum is to be expected. However, there is a third possibility, the formation of a magnetic polaron by spin exchange of the electron with its immediate spin-neighbourhood. It can be described as a propagation of the electron accompanied by a virtual cloud of repeatedly emitted and reabsorbed magnons. The many-body problem of the KLM is solved as soon as we have found the retarded single-electron Green function +∞

  i † = −i dt e
Et [ciσ (t), c†jσ (0)]+  . Gijσ (E) = ciσ ; cjσ E

(3)

0

[· · · , · · · ]+ means the anticommutator, and · · ·  is the thermodynamic average. More convenient for the following is the wave-vector dependent Fourier transform i 1  Gijσ e
k·(Ri −Rj ) . (4) Gkσ (E) = N ij All other terms are transformed in the same manner. The introduction of the selfenergy Mkσ (E) via     = Mkσ (E) ckσ ; c†kσ , (5) [ckσ , Hex ]− ; c†kσ E

E

where [. . . , . . .]− means the commutator, formally solves the Green-function equation of motion Gkσ (E) =


. E − ε(k) − Mkσ (E) + i0+

(6)

ε(k) is the Fourier transform of the hopping integral Tij . We will present our results mainly in terms of the spectral density Skσ (E) (SD) and the quasiparticle density of states ρσ (E) (QDOS):

Carrier Induced Ferromagnetism in Local-Moment Systems

ρσ (E) =

1  −1  Skσ (E − µ) = ImGkσ (E − µ) . N
N
π k

51

(7)

k

µ is the chemical potential. QDOS as well as SD will in general be temperature, carrier concentration, and lattice structure dependent. Most important for ferromagnetic systems is of course the spin dependence. 2.2 Exactly Solvable Limiting Case Fortunately, the very involved many-body problem of the Kondo-lattice model can be solved exactly for a non-trivial and very illustrative limiting case, by which we can inspect in detail the spin exchange processes. It is the situation of a single electron in an otherwise empty conduction band interacting with a ferromagnetically saturated spin system (think of EuO at T = 0 K). Because of the empty band and the totally aligned spin system the hierarchy of equations of motion of the Green function decouples exactly [17, 18, 19], because one can exploit relationships of the following kind: · · · ciσ  = c†iσ · · ·  = · · · Si+  = Si− · · ·  = 0 ,

(8)

· · · Siz  = Siz · · ·  =
S· · ·  .

(9)

The ↑ spectrum is then trivial because an ↑ electron cannot exchange its spin with the totally aligned localized spins. So only the Ising term works leading to a rigid shift of the total spectrum of − 12 JS. The ↓ spectrum, however, is non-trivial because magnon emission as well as the polaron formation is possible even at T = 0 K. After straightforward manipulations one finds the following exact expression for the electronic selfenergy: 1 2 1 J S G0 (E + 12 JS) 1 Mkσ (E) = − zσ JS + (1 − zσ ) 4 1
1 ≡ Mσ (E) , 2 1 − 2 J
G0 (E + 12 JS)

with

1  1 −1 G0 (E) = [E + µ − ε(k)] .
N

(10)

(11)

k

zσ = δσ↑ − δσ↓ is a sign factor. Figure 1 shows the exactly calculated quasiparticle density of states for the ferromagnetically saturated KLM in the case of a simple cubic (sc) lattice. The ↑-QDOS is undeformed, only rigidly shifted. Correlation effects are observable only in the ↓ spectrum. The low-energy tail consists of scattering states due to magnon emission by the ↓ electron. Since the process is accompanied by a simultaneous electron spinflip it can happen only if there are empty ↑ states within reach. That is the reason why this part of the ↓ spectrum occupies exactly the same energy region as ρ↑ (E). The upper part of the spectrum is built up by polaron states, which for the considered special case turn out to be bound states representing a quasiparticle with

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W. Nolting et al. 2.0 ↑ ↓

1.0

J = 0.05 eV

0.0

J = 0.10 eV

1.0

QDOS [1/eV]

0.0

J = 0.15 eV

1.0 0.0

J = 0.20 eV

1.0 0.0

J = 0.25 eV

1.0 0.0

J = 0.30 eV

1.0 0.0

-1.0

-0.5

0.0

0.5

1.0

E [eV] Fig. 1. Exact solution of the Kondo-lattice model for a ferromagnetically saturated semiconductor ( S z  = S, n = 0). Quasiparticle density of states as function of energy for various values of the interband exchange coupling J. Full line for spin ↑, dashed line for spin ↓. Parameters: sc lattice, bandwidth W = 1 eV, S = 72

infinite lifetime (δ-function in the spectral density!). Already for very moderate coupling strengths ρ↓ (E) splits into two subbands. The weight of the 2S 1 and that of the lower band to 2S+1 . upper subband amounts to about 2S+1 This exact result demonstrates the importance of the spinflip terms (2) in the Hamiltonian with respect to the characteristic properties of the KLM. They should be treated with special care when constructing the unavoidable approximations for the general case of finite temperatures and finite band occupations.

3 Electronic Selfenergy of “Concentrated” Local-Moment Systems The evaluation of the KLM requires the solutions of an electronic and a magnetic partial problem. Very often, only the electronic part is investigated while the local moment magnetization is phenomenologically simulated by a

Carrier Induced Ferromagnetism in Local-Moment Systems

53

Brillouin function [13, 20, 21]. Such a procedure presumes ferromagnetism without deriving it selfconsistently. We start with an investigation of the electronic selfenergy and complete the theory in Sect. 4 with an inspection of the magnetic terms which enter the selfenergy expression. We compare two approaches which are based on similar ideas, the “Interpolating Selfenergy Approach” (ISA) of [20] and the “Moment-Conserving Decoupling Approach” (MCDA) of [16]. 3.1 Interpolating Selfenergy Approach The “Interpolating Selfenergy Approach” (ISA) [20] fulfills a maximum number of exact limiting cases being, however, restricted to the low-density limit (n → 0), which is the relevant limit for magnetic semiconductors. The ISA consists of two partial steps: First we try to get the general structure of the selfenergy Mkσ (E) in such a way that by construction all known exact limiting cases of the KLM are correctly fulfilled. Relevant limiting cases are, e.g., the above-discussed T = 0 K-result (10) for the empty-band case, the zero-bandwidth limit [22] and second-order perturbation theory [21]. The following ansatz is able to reproduce all these limiting cases for n → 0: aσ G0 (E − 12 Jzσ S z ) 1 1 Mkσ (E) ≡ Mσ (E) = − Jzσ S z  + J 2 . 2 4 1 − bσ G0 (E − 12 Jzσ S z )

(12)

G0 (E) is the free propagator (11). S z  is the magnetization of the local moment system, which has to be determined separately. aσ and bσ are at first free parameters. They are fixed by exact high-energy expansions of the Green function Gkσ (E) =

+∞

∞ (n)  mkσ Skσ (E  ) dE  =
, E − E E n+1 n=0

(13)

−∞

for which we calculate the first four spectral moments rigorously according to their definition 

n ∂ (n) i
[ckσ (t), c†kσ (t )]+  . (14) mkσ = ∂t t=t
Equation (13) can be applied via Dyson equation to the selfenergy, too, resulting in ∞ m  Ckσ . (15) Mkσ (E) = Em m=0 m are linear combinations of the moments up to order m+1 The coefficients Ckσ explicitly given in [20]. Eventually one gets for the parameters aσ and bσ the following simple expressions:

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aσ = S(S + 1) − zσ S z (zσ S z  + 1) ,

(16)

J . (17) 2 We see that the resulting selfenergy (12) has the following general structure, which is found by other methods, too (see Sect. 3.2): bσ =

1 Mkσ (E) = − Jzσ S z  + J 2 Dkσ (E; J) . 2

(18)

The first term is linear in the coupling and proportional to the the magnetization S z . It is just the mean-field result being correct for sufficiently weak exchange coupling J. It stems from the Ising-part in the exchange Hamiltonian (2) and would give rise to a fully spin-polarized splitting of the conduction band if there were not the second term. Dkσ (E; J) incorporates all the consequences of the above-discussed spin exchange processes. Because of the assumed low-density limit the only local-moment correlation that enters Dkσ (E; J), is the magnetization S z , which introduces the temperature dependence to the electronic quasiparticle structure. Figure 2 shows the QDOS for various local-moment magnetizations. The latter works here only as a parameter, that brings into play the temperaturedependence. Strong deformations appear as a consequence of the exchange interaction. The chosen effective, rather moderate coupling JS W = 0.7 even provokes a temperature-driven bandsplitting. The upper part is predominantly due to polaron formation, while the lower part belongs to magnon 2.0 z

〈S 〉 = 3.5

1.0

-1

QDOS [eV ]

0.0 z

〈S 〉 = 2.5

1.0

0.0 z

〈S 〉 = 1.5

1.0

0.0 z

〈S 〉 = 0.0

1.0

0.0 -1.0

-0.5

0.0

0.5

1.0

E [eV]

Fig. 2. Quasiparticle density of states, calculated with ISA, as function of the energy for various values of the local-moment magnetization S z . Full line for spin ↑, dashed line for spin ↓. Parameters: J = 0.2 eV, S = 7/2, W = 1 eV, sc lattice

Carrier Induced Ferromagnetism in Local-Moment Systems

55

emission and absorption, respectively. The result for saturation (S z  = S, n = 0) is exact (see Fig. 1). Because of the assumed low-density restriction the ISA-QDOS does not show any n-dependence. That, however, must be considered a certain shortcoming of the ISA. 3.2 Moment Conserving Decoupling Approach The ISA is reliable only for very low charge carrier densities. For arbitrary band occupations we use the Green function theory of [16] which represents a “Moment Conserving Decoupling Approach” (MCDA). As for the ISA, the theory uses, at least partly, the concept of interpolating key-terms between exact limiting cases. Starting point is the equation of motion of the Green function (3)  m

1 ((E − µ)δim − Tim )Gmjσ =
δij − J(zσ Γii,jσ (E) + Fii,jσ (E)) . (19) 2

The higher Green functions     Fim,jσ (E) = Si−σ cm−σ ; c†jσ , Γim,jσ (E) = Siz cmσ ; c†jσ E

E

,

(20)

where Siσ = Six + izσ Siy , prevent an exact solution of (19). The equations of motion of the two Green functions (20) contain further higher Green func  † ±,z and [Si , Hex ]− Bm ; c†jσ , tions of the type Ai [cm±σ , Hex ]− ; cjσ E E where Ai and Bj are any combinations of local-spin and band operators. The decoupling procedure happens  in two steps [16]. We exemplify the procedure  † : The off-diagonal terms i = m are simplified in for Ai [cm±σ , Hex ]− ; cjσ E

analogy to the selfenergy (5) to      Ai [cm±σ , Hex ]− ; c†jσ → Mmrσ (E) Ai cr±σ ; c†jσ E

r

E

.

(21)

The right-hand side is a linear combination of simpler Green functions with the selfenergy-elements as coefficients. The spectral decompositions of the Green functions in (21) show that all poles on the left-hand side do appear also on the right-hand side. The respective spectral weights are approximated by the selfconsistent determination of the selfenergy-elements. To account for the strong local correlations the diagonal terms i = m are handled with special care:      → γn an ; c†jσ (22) Ai [ci±σ , Hex ]− ; c†jσ E

n

E

The right-hand side contains simpler Green functions, defined by the operator an and already involved in the hierarchy of equations of motion. The choice

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of these functions is done in such a way that all known exact limiting cases for the respective Green function (atomic limit, ferromagnetically saturated semiconductor, local spin S = 1/2, n = 0, n = 2, . . .) are reproducable. The at first unknown coefficients γn are then fixed, similar as in (13), by exact high-energy expansions (spectral moments) of the respective Green function.  ±,z † are approxThe Green functions of the type [Si , Hex ]− Bm ; cjσ E imated in the same manner, where, however, the exchange operator Hex is taken in the effective Heisenberg form (23) which is derived in the next section. Eventually we arrive at a selfenergy of the type (18) where the scattering part Dkσ (E; J) is a complicated expression [16], for conciseness not presented here. It is a functional of mixed spin correlations such as Siz niσ , Si+ c†i↓ ci↑ , . . .. Fortunately, all these correlations can be found by use of the spectral theorem in connection with a proper Green function already involved in the system of equations of motion. However, there appear also pure local-moment correlations like Siz , ± ∓ Si Si , (Siz )3 , . . . , which cannot be derived from electronic Green functions. For their determination we use the modified RKKY theory of [14, 16], which is explained in the next section. Figure 3 shows a typical model result for an sc lattice derived with MCDA for a moderate coupling J = 0.2 eV and a finite bandoccupation of n = 0.2. For this parameter constellation the modified RKKY of the next section yields selfconsistently a Curie temperature of TC = 238 K. Let us inspect here the reaction of the electronic spectrum on the magnetic state of the local-spin system. The results for the QDOS are qualitatively similar to those of the ISA (Fig. 2). At low temperatures we recognize that the main features of the exact (n = 0, T = 0 K) limiting case (Fig. 1) obviously remain valid for finite band occupations, too. The ↑ spectrum is more or less undeformed, only rigidly shifted to lower energies. Correlation effects appear only in the ↓ part. The interpretation is the same as for the exact limiting case. The upper part of the ↓-QDOS consists of polaron states while the low-energy tail is built up by scattering states due to magnon emission by the ↓ electron. With increasing temperature, i.e. decreasing magnetization and therewith increasing magnon density scattering states appear also in the ↑ spectrum. Magnonabsorption by a ↑ electron is of course equivalent to magnon-emission by a ↓ electron, with one exception. Magnon-absorption is possible only if there are any magnons in the system, which is indeed not the case in ferromagnetic saturation. That is the reason why the ↑ spectrum is so simple at T = 0 K. With increasing temperature, however, the two spin spectra become more and more similar until at TC the spin asymmetry is removed, but there remains in the bandstructure a splitting of the original dispersion into two quasiparticle branches, both obviously with comparable spectral weight. Note that this splitting does not appear if the spin-exchange terms in the interaction (2) are neglected as is done in mean-field treatments of the Kondo-lattice model.

Carrier Induced Ferromagnetism in Local-Moment Systems

57

Fig. 3. Spin dependent quasiparticle bandstructure (left column) as a function of wave vector and quasiparticle density of states (right column) as function of energy for four different temperatures, calculated by MCDA. Parameters: J = 0.2 eV, W = 1 eV, n = 0.2, S = 72 , sc lattice

The spinflip terms are also responsible for the fact that there opens a gap in the QDOS with increasing temperature.

4 Magnetic Properties of “Concentrated” Local-Moment Systems For constructing the magnetic phase diagram we use two different methods. The first will be called in the following “modified RKKY” (M-RKKY). It results from a mapping of the exchange term (2) onto an effective Heisenberg model as it is done in the conventional RKKY theory, too. This mapping is avoided by the second method, which derives the phase diagram from the paramagnetic susceptibility of the itinerant conduction electrons.

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4.1 Modified RKKY Theory The idea is to replace the exchange interaction (2) by an effective HeisenbergHamiltonian  Jˆij S i · S j Hf = − (23) i,j

by averaging out all conduction electron degrees of freedom, (. . .(c) ) [16, 14]   S i · σ i −→ −J S i · σ i (c) . (24) −J i

i

Averaging only in the band electron subspace means that the averaged interaction term retains an operator character in the subspace of the local spins. According to the alternative representation of the exchange interaction Hex = −J


  iqRi ˆ σσ
c†k+qσ ckσ
, e (S i · σ) N
iσσ

(25)

kq

ˆ is the vector of Pauli spin matrices, the mapping means to determine where σ the expectation value c†k+qσ ckσ
(c) performed in the subspace of conduction electrons, only: c†k+qσ ckσ
(c) =

  1 −βH
†
. Tr e c c kσ k+qσ Ξ

(26)

H  has exactly the same structure as H in (1), but for the averaging procedure the local spin operator has to be considered as a c number, therefore not affecting the trace. The expectation value does not vanish for q = 0 and σ = σ  as it would when averaging in the full Hilbert space of the KLM. To evaluate (26) we introduce a proper “restricted” Green function


σ (E) = Gσk,k+q

 (c) ckσ
; c†k+qσ , E

(27)

which has the normal definition of a retarded Green function, only the averages have to be done in the Hilbert space of H  . The equation of motion is readily been calculated leading to an exact series expansion which allows to calculate the “restricted” Green function (27) up to any desired accuracy. Truncating the series after the first non-trivial term (linear in J) leads to the conventional RKKY result [14], which can be equivalently derived by use of second-order perturbation theory starting from an unpolarized conduction electron gas. The “modified” RKKY [16, 14] takes into account to higher order the exchange-induced spin polarization of the conduction electrons by replacing in a proper manner “bare” Green functions by “dressed” ones. For special details the reader is referred to [14]. Exploiting the spectral theorem we get from the “restricted” Green function the expectation value

Carrier Induced Ferromagnetism in Local-Moment Systems

59

(26) and therewith eventually the Heisenberg-Hamiltonian (23) with effective exchange integrals of the following kind: Jˆij =

+∞

J 2  iq(Ri −Rj ) e dE f− (E) 4πN 2 kqσ

(28)

−∞

−1
. ×Im (E − ε(k) + i0+ )(E − ε(k + q) − Mk+qσ (E) + i0+ ) f− (E) denotes the Fermi function. The effective exchange integrals are decisively influenced by the conduction electron selfenergy Mkσ (E) which brings into play a distinct band occupation and temperature dependence of Jˆij . Neglecting the selfenergy Mk+qσ (E) in (28) leads to the conventional RKKY formula with Jˆij ∝ J 2 as a result of second order perturbation theory. Via Mkσ (E) higher order terms of the conduction electron spin polarization enter the M-RKKY being therefore not restricted to weak couplings, only. They have to be determined selfconsistently within our approximate procedure. To obtain from the effective operator (23) the magnetic properties of the KLM we apply an RPA method (“Tyablikow-approximation”) which is known to yield reasonable results in the low as well as in the high temperature region [23]. The approach concerns the spin Green function [24]   z (a) , (29) Pij (E) = Si+ ; eaSj Sj− E

where a is a real number, that helps to derive a differential equation to obtain the local moment magnetization S z  and other local correlation functions. (a) The RPA decoupling of the equation of motion of Pij (E) leads to quasiparticle poles at   ˆ = 0) − J(q) ˆ . (30) E(q) = 2
S z  J(q ˆ J(q) is the Fourier transform of the effective exchange integral (28) being therefore a functional of the electronic selfenergy. The average magnon number −1 1   βE(q) φ(S) = e −1 (31) N q determines the magnetization [24] S z  =


(1 + S + φ)φ2S+1 + (S − φ)(1 + φ)2S+1 (1 + φ)2S+1 − φ2S+1

(32)

and other local moment correlation functions such as S − S +  = 2
S z φ(S) .

(33)

The key quantity of ferromagnetism is the Curie temperature TC which can be derived from (32) performing the limiting process S z  → 0:

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W. Nolting et al.

2 kB TC =
2 S(S + 1) 3



−1 1  ˆ ˆ J(q = 0) − J(q) N q TC

−1 .

(34)

The effective exchange integrals are temperature-dependent and have to be used here for T → TC . If we choose a special expression for the electronic selfenergy, e.g. (18), then we eventually arrive at a closed system of equations which can be solved selfconsistently for all electronic and magnetic properties of the KLM. The temperature-dependent quasiparticle density of states and quasiparticlebandstructure, exhibited in Fig. 3, have been calculated by use of the MCDA (Sect. 3.2) together with the M-RKKY developed in this section. Spontaneous ferromagnetic ordering appears below the finite Curie temperature (T = 238 K for the parameter set chosen for Fig. 3) as a consequence of an indirect coupling between the local moments, mediated by a spin polarization of the conduction electrons and due to the interband exchange (2). Therefore, TC exhibits a strong carrier concentration dependence as shown in Fig. 4. Ferromagnetism appears for low electron (hole) concentrations, while being excluded around half-filling (n = 1). The ferromagnetic region increases with increasing J, but never reaches n = 1. A similar n-dependence of TC has been found in [25] by use of dynamical mean field theory. It is interesting to compare the results of the M-RKKY with those of the conventional RKKYtheory. For the special case of J = 0.2 eV a corresponding curve is inserted in Fig. 4. The maximum TC is higher, but ferromagnetism exists only in a very narrow region of low electron (hole) concentrations. This ferromagnetic n-region turns out to be independent of the coupling J [14].

Fig. 4. Curie temperature as function of the band occupation n for various exchange couplings J, calculated by use of MCDA and M-RKKY. Parameters: sc lattice, W = 1 eV, S = 72

Carrier Induced Ferromagnetism in Local-Moment Systems

61

Fig. 5. Curie temperature as function of the coupling J for various n, calculated by use of MCDA and M-RKKY. Parameters: sc lattice, W = 1 eV, S = 72

The J-dependence of the Curie temperature, plotted in Fig. 5, reveals two remarkable features. First, for those band occupations n for which the conventional RKKY does not allow a collective spin order there exists a critical J, which has to be exceeded to get ferromagnetism . In the example of Fig. 5 this is the case for n ≥ 0.13. This is in accordance with the fact that for J → 0 eV the modified RKKY reproduces the conventional RKKY. The second feature is that TC runs into saturation for strong couplings J, where the saturation value depends on the band occupation n. These results are by no means reproducable by the conventional RKKY. By combination with a “first principles” bandstructure calculation the MCDA together with the M-RKKY has been successfully applied to the electronic quasiparticle structure of the 4f ferromagnet Gadolinium [26] with an astonishingly accurate description of the experimental magnetic data. 4.2 Paramagnetic Susceptibility We present in this subsection an alternative approach to the magnetic phase diagram which avoids the mapping on an effective Heisenberg model. If we are only interested in the phase diagramm we can exploit the static susceptibility of the itinerant electron subsystem: B→0  ∂ nσ  . (35) χ(T ) = ∂B T ≥TC σ We investigate exclusively the possibility of ferromagnetism, the average occupation number niσ  ≡ nσ  is therefore site-independent. nσ  must be calculated in the presence of an external magnetic field. A corresponding

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W. Nolting et al.

Zeeman term has to be added to the model Hamiltonian (1). From the singleelectron Green function (3) and by use of the spectral theorem we then get straightforwardly the spin-dependent average occupation number. The Curie temperature TC = TC (n, J) and therewith the phase diagram is eventually read off from the susceptibility poles: χ−1 (T = TC ) = 0 .

(36)

This procedure has been performed in [27] by applying the ISA (Sect. 3.1). The key point is how to express the spontaneous magnetization S z  which cannot be derived from the single-electron Green function. One can exploit the fact that S z  and the spin polarization of the band electrons n↑  − n↓  are mutually conditional. They become critical for the same parameters, in particular at the same temperature. In the critical region we can therefore assume a proportionality B→0  ∂ z S  = η · χ(T ) . (37) ∂B T ≥TC The proportionality of the response functions can be traced back to a proportionality of the expectation values S z  and n↑ −n↓ , which is in the critical region according to a Taylor expansion certainly fulfilled. One might think of a simple ansatz for η that neglects a possible dependence on model parameters and temperature by assuming an equivalence of the reduced quantities n↑  − n↓  S z  ⇐⇒ . S n↑  + n↓ 

(38)

That results in η = Sn . This ansatz, plausible as it is, can probably been replaced by more profound theories in an improved approach. It has been used in [27] in the framework of the ISA to determine the Curie temperature of the KLM as function of the band occupation n and the coupling strength J. Qualitatively the results agree with those found by MCDA and M-RKKY (see Figs. 4 and 5). Therefore, we do not present details here, but rather concentrate ourselves in the next section on the phase diagram of the “diluted” Kondo-lattice model which we derive with the just-described susceptibility method.

5 “Diluted” Local-Moment Systems Let us now inspect the influence of a dilution of the local moment system on ferromagnetic stability in the KLM. That means that not all lattice sites are occupied by a magnetic moment and that the available localized moments are statistically distributed over the crystal with a concentration x < 1. This models the situation of a diluted ferromagnetic semiconductor such as

Carrier Induced Ferromagnetism in Local-Moment Systems

63

Ga1−x Mnx As. In this connection we assume that each moment can be accompanied at most by one free charge carrier, i.e. for the carrier concentration n holds n ≤ x. By this assumption we can simulate the effect of antisites or other mechanisms which can absorb free carriers. We are interested in the magnetic phase diagram as function of J, n, and x. To treat the statistical disorder of the moments by a CPA-type approach we introduce a fictitious binary alloy consisting of constituents α (concentration 1 − x) and β (concentration x). α stands for nonmagnetic sites (e.g. Ga3+ ), while β sites carry a magnetic moment (e.g. Mn2+ in Ga1−x Mnx As). The magnetic moment is exchange coupled via (2) to itinerant charge carriers. The atomic level of a nonmagnetic α site is in the presence of a homogeneous magnetic field B in z-direction ασ = T0 − zσ µB B .

(39)

On the magnetic β sites the local interband exchange Hex (2) acts on the itinerant charge carriers. That is accounted for by a “dynamic” atomic energy level which incorporates the electronic selfenergy Mσ (E) (Sect. 3): βσ (E) = T0 + Mσ (E) − zσ µB B .

(40)

Later we will take the selfenergy from the ISA described in Sect. 3.1, which turns out to be a local quantity (12). We regard the charge carriers in the “diluted” Kondo lattice as a system of particles propagating in the above-defined fictitious binary αβ-alloy. We do not take into account for our model study a Coulomb disorder potential which might be important in some circumstances [28]. All what we need can be derived from the propagator +∞

dω Rσ (E) = −∞

ρ0 (ω) . E − ω − Σσ (E)

(41)

Σσ (E) is now the electronic selfenergy in the diluted system and ρ0 (ω) the Bloch-density of states of the non-interacting carriers. As soon as Σσ (E) is known we can derive from the propagator Rσ (E) the quasiparticle density of states of the interacting system (7), 1 ρσ (E) = − ImRσ (E) , π

(42)

and therewith the average spin-dependent occupation number which we need in (35) for the derivation of the susceptibility: +∞

dE f− (E)ρσ (E) . nσ  = −∞

(43)

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The decisive selfenergy Σσ (E) of the disordered moment system we get by applying a standard CPA formalism [29] to the above-defined fictitious binary alloy, i.e. we have to solve the following CPA equation: −zσ µB B − Σσ (E) 1 − Rσ (E)(−zσ µB B − Σσ (E)) Mσ (E) − zσ µB B − Σσ (E) +x . 1 − Rσ (E)(Mσ (E) − zσ µB B − Σσ (E))

0 = (1 − x)

(44)

This implicit equation for the selfenergy can be solved for any moment concentration x leading therewith, after some simple manipulations, to the magnetic susceptibility (35). As mentioned we use here for the electronic selfenergy the result (12) of the ISA, which is trustworthy in particular for low electron concentrations as it is the case in diluted magnetic semiconductors. From the poles of χ(T ) (36) we derive the Curie temperature as function of the coupling strength J, the concentration x and the bandoccupation n ≤ x. The actual evaluation has been done for an sc lattice with the width W of the non-degenerate Bloch band as energy unit. As already mentioned, we assume that each magnetic ion can in principle donate one electron to the conduction band. However, not all these charge carriers can be considered as really itinerant, they may be partly compensated. So it makes sense to inspect the situation of n ≤ x. Figure 6 shows the inverse paramagnetic susceptibility in the limit B → 0T as function of the temperature for various a) n = 0.1

-1

Inverse susceptibility χ [arb. units]

J = 0.2 eV J = 0.5 eV J = 1.0 eV

b) J = 0.5 eV n = 0.10 n = 0.15 n = 0.20

0

500

1000 Temperature [K]

1500

Fig. 6. Paramagnetic inverse susceptibility of the “diluted” Kondo lattice model as function of the temperature, in (a) for a fixed band occupation n = 0.1 and different exchange couplings, in (b) for a fixed exchange coupling J = 0.4 eV and different carrier concentrations n. Parameters: sc lattice, W = 1 eV, S = 52

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65

parameter constellations. The existence of zeros indicate instabilities of the paramagnetic system against ferromagnetic order. In some cases two zeros are found (not shown in the figure). The requirement that χ must be positive in the paramagnetic phase makes the choice of the physical solution unique. The resulting Curie temperatures for the special case x = 1 (“concentrated” local moment system) are qualitatively very similar to those in Fig. 4. Ferromagnetism is restricted to very low electron (hole) concentrations. Around half-filling (n = 1) ferromagnetism is excluded for all coupling strengths. Note that the KLM does not consider a direct exchange between the localized moments. So the collective ordering is exclusively due to the interband exchange and mediated by an induced spin polarization of the itinerant charge carriers. The resulting Curie temperatures for diluted systems are plotted in Fig. 7 as functions of the carrier concentration n for various moment concentrations 800

a) J = 0.1 eV 1.0

600

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5 0

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Fig. 7. Curie temperature as a function of the band occupation n for various concentrations x of magnetic moments in the “diluted”(x < 1) Kondo-lattice model. (a) J = 0.1 eV, (b) J = 0.4 eV, (c) J = 1.0 eV. Parameters: sc lattice, W = 1 eV, S = 52

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x. We find for practically all concentrations x ferromagnetism but, remarkably, only when the carrier concentration n is distinctly smaller than x. The compensation effects observed in diluted magnetic semiconductors (antisites, interstitials, ...) turn out to be a necessary precondition for ferromagnetism in the diluted system. This can be understood by the the quasiparticle density of states. For strong enough couplings J, the (paramagnetic) Q-DOS (Fig. 8) consists of three separated structures. The low-energy and the high-energy parts are built up by states from the correlated β sites. They are separated roughly by 12 J(2S + 1). The middle structure is due to uncorrelated α sites. Tendencies to magnetic order are exclusively due to the two correlated subbands. In case that the three structures are well separated, then the area under the two correlated ones amounts to x, while that of the uncorrelated middle one is 1 − x. Therefore, with increasing density of magnetic moments more and more spectral weight is shifted into the correlated subbands. First precondition for ferromagnetism is that the Fermi edge lies in one of the correlated subbands. The second precondition corresponds to what has been found for the “concentrated” (x = 1) KLM. According to Fig. 4 very low band occupations are sufficient and even favourable for carrier-induced ferromagnetism. That means in the diluted case (x < 1) that, roughly estimated, ferromagnetic ordering appears for band occupations 0 < n < nc (J) · x, where nc (J) denotes the critical band occupation for x = 1 at a given J. The situation, for which each magnetic moment creates exactly one free carrier, i.e. n = x, would result in a fully occupied low-energy structure corresponding to halffilling in the “concentrated” KLM. For this case ferromagnetism is excluded (Fig. 4). It is indeed observed for diluted ferromagnetic semiconductors that the number of itinerant carriers is substantially smaller than the number of local moments [12]. In Ga1−x Mnx As, e.g., each Mn2+ ion in principle provides one hole in the valence band. However, only a certain percentage of them are really itinerant, the others are compensated by antisites or interstitial Mn ions or anything else. Erwin and Petukhov [30] were the first to suggest that such compensation effects might be in favour of a collective order, in agreement with our model results.

6 Summary We have investigated the carrier induced ferromagnetism (RKKY) in concentrated and diluted local-moment systems within the framework of the ferromagnetic Kondo-lattice model. Candidates for this model are magnetic metals (Gd, Dy, Tb), (“concentrated”) magnetic semiconductors (EuO, EuS), diluted magnetic semiconductors (Ga1−x Mnx As, In1−x Mnx Sb) and CMR materials (La1−x Cax MnO3 ).

Carrier Induced Ferromagnetism in Local-Moment Systems

x = 0.2 x = 0.4 x = 0.6 x = 0.8 x = 1.0

b) J = 0.4 eV

x = 0.05 x = 0.1 x = 0.2 x = 0.3

c) J = 1.0 eV

x = 0.05 x = 0.1 x = 0.2 x = 0.3

-1

QDOS [eV ]

-1

QDOS [eV ]

-1

QDOS [eV ]

a) J = 0.1 eV

-1.5

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0 0.5 E [eV]

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Fig. 8. Paramagnetic quasiparticle density of states of the “diluted” Kondo-lattice model as function of energy for different values of the moment concentration x and three different exchange couplings J. Parameters: sc lattice, W = 1 eV, S = 52

Starting point was an instructive, exactly solvable limiting case (“ferromagnetically saturated semiconductor”) which helps to get an idea about decisive spin exchange processes between localized magnetic moments and itinerant band electrons. Besides that, this limiting case can serve as a testing criterion for unavoidable approximations of the general many-body problem. Two approaches to the electronic selfenergy have been proposed, which fulfill a maximum number of exact results of the KLM. A common feature of both is the distinct temperature dependence of the selfenergy mainly via interband exchange due to magnetic correlations in the local moment system. The similarity of the results, though the parameter regimes for a reliable application

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of both approaches differ, is an additional criterion for the quality of the performed approximations. For the magnetic part an extended RKKY mechanism has been worked out to derive the magnetic phase diagram. The method consists of a mapping of the interband exchange on an effective Heisenberg model. The resulting indirect exchange integrals are functionals of the electronic selfenergy, being therefore strongly temperature and carrier concentration dependent. For weak couplings J our theory reproduces the features of the conventional RKKY, which follows from second order perturbation theory with respect to J. However, already for very moderate J substantial deviations appear, because higher-order terms of the induced conduction spin polarization bring their influence to bear. Ferromagnetism turns out to be stable for low electron (hole) densities n, where the ferromagnetic n region increases with increasing J, however, never reaching half-filling n = 1. The main features of the phase diagram are confirmed by a second method which avoids the mapping on an effective Heisenberg model. Instead of this, the phase diagram is derived from the singularities of the paramagnetic susceptibility of the itinerant charge carrier subsystem. This method is suitable to inspect the influence of magnetic moment disorder on magnetic stability in a “diluted” KLM, an aspect of great importance for spintronics relevant materials such as Ga1−x Mnx As. By a CPA-type treatment of the disorder a qualitative explanation of the ferromagnetism in diluted magnetic semiconductors could be found. A main conclusion of our model study with respect to real diluted magnetic semiconductors is that a substantial compensation of the itinerant charge carriers (n < x) by antisites or other mechanisms appears to be a necessary condition for the existence of a ferromagnetic order. Acknowledgment This work benefitted from financial support by the “Volkswagenstiftung” as well as by the Sonderforschungsbereich 290 of the Deutsche Forschungsgemeinschaft.

References 1. P. Wachter: Handbook of the Physics and Chemistry of Rare Earth volume 1 chapter 19, Amsterdam North Holland (1979) 48 2. G. Busch, J. Junod, P. Wachter: Phys. Lett. 12, 11 (1965) 48 3. E. Kisker, G. Baum, A.H. Mahan, W. Raith, B. Reihl: Phys. Rev. B 18, 2256 (1978) 48 4. T. Penney, M.W. Shafer, J.B. Torrance: Phys. Rev. B 5, 3669 (1972) 48 5. P. Sinjukow, W. Nolting: Phys. Rev. B 68, 125107 (2003) 48 6. M. Donath, P.A. Dowben, W. Nolting: editors, Magnetism and Electronic Correlations in Local-Moment Systems:Rare-Earth Elements and Compounds Singapore (1998) World Scientific. 48

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7. D. Li, J. Zhang, P.A. Dowben, M. Onellion: Phys. Rev. B 45, 7272 (1992) 48 8. B. Kim, A.B. Andrews, J.L. Erskine, K.J. Kim, B.N. Harmon: Phys. Rev. Lett. 68, 1931 (1992) 48 9. H. Ohno: Science 281, 951 (1998) 48 10. F. Matsukara, H. Ohno, A. Shen, Y. Sugawara: Phys. Rev. B 57, R2037 (1998) 48 11. H. Ohno, D. Chiba, F. Matsukara, T. Omiya, E. Abe, T. Dietl, Y. Ohno, K. Ohtani: Nature 408, 944 (2000) 48 12. T. Dietl: Semicond. Sci. Technol. 17, 377 (2002) 49, 66 13. W. Nolting, G.G. Reddy, A. Ramakanth, D. Meyer, J. Kienert: Phys. Rev. B 67, 024426 (2003) 49, 53 14. C. Santos, W. Nolting: Phys. Rev. B 65, 144419 (2002) 49, 56, 58, 60 15. R. Schiller, W.M¨ uller, W. Nolting: Phys. Rev. B 64, 134409 (2001) 49 16. W. Nolting, S. Rex, S. Mathi Jaya: J. Phys.: Condens. Matter 9, 1301 (1997) 49, 53, 55, 56, 58 17. B.S. Shastry, D.C. Mattis: Phys. Rev. B 24, 5340 (1981) 51 18. S.R. Allan, D.M. Edwards: J. Phys. C 15, 2151 (1982) 51 19. W. Nolting, U. Dubil, M. Matlak: J. Phys.: Condens. Matter 18, 3687 (1985) 51 20. W. Nolting, G.G. Reddy, A. Ramakanth, D. Meyer: Phys. Rev. B 64, 155109 (2001) 53 21. T. Hickel, W. Nolting: Phys. Rev. B 69, 085110 (2004) 53 22. W. Nolting, U. Dubil: phys. stat. sol. (b) 130, 561 (1985) 53 23. R. Schiller, W. Nolting: Solid State Commun. 118, 173 (2001) 59 24. H.B. Callen: Phys. Rev. 130, 890 (1963) 59 25. A. Chattopadhyay, A.J. Millis: Phys. Rev. B 64, 024424 (2001) 60 26. C. Santos, W. Nolting, V. Eyert: Phys. Rev. B 69, 214412 (2004) 61 27. W. Nolting, T. Hickel, A. Ramakanth, G.G. Reddy, M. Lipowczan: Phys. Rev. B 70, 075207 (2004) 62 28. C. Timm: J. Phys.: Condens. Matter 15, R1865 (2003) 63 29. R.J. Elliott, J.A. Krumhansl, P.L. Leath: Rev. Mod. Phys. 46, 465 (1974) 64 30. S.C. Erwin and A.G. Petukhov: Phys. Rev. Lett. 89, 227201 (2002) 66

An Origin of CMR: Competing Phases and Disorder-Induced Insulator-to-Metal Transition in Manganites Yukitoshi Motome1 , Nobuo Furukawa2 and Naoto Nagaosa3,4,5 1

2

3

4

5

RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Saitama 351-0198, Japan [email protected] Department of Physics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan [email protected] CREST, Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan [email protected] Correlated Electron Research Center, AIST, Tsukuba Central 4, 1-1-1 Higashi, Tsukuba, Ibaraki 305-8562, Japan 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

Abstract. We theoretically explore the mechanism of the colossal magnetoresistance in manganese oxides by explicitly taking into account the phase competition between the double-exchange ferromagnetism and the charge-ordered insulator. We find that quenched disorder causes a drastic change of the multicritical phase diagram by destroying the charge-ordered state selectively. As a result, there appears a nontrivial phenomenon of the disorder-induced insulator-to-metal transition in the multicritical regime. On the contrary, the disorder induces a highly-insulating state above the transition temperature where charge-ordering fluctuations are much enhanced. The contrasting effects provide an understanding of the mechanism of the colossal magnetoresistance. The obtained scenario is discussed in comparison with other theoretical proposals such as the polaron theory, the Anderson localization, the multicritical-fluctuation scenario, and the percolation scenario.

1 Introduction Colossal magnetoresistance (CMR) in perovskite manganese oxides AMnO3 has attracted much attention in the physics of strongly-correlated electron systems [1, 2, 3]. In these materials, carrier doping by chemical substitutions of A-site ions, e.g., Ca substitutions into La-site in LaMnO3 , leads to a ferromagnetic metal (FM) at low temperatures. In this regime, the resistivity shows a rapid decrease by applying an external magnetic field near the critical point, which is called the CMR effect. The doped FM state and the negative magnetoresistance are basically understood by the Zener doubleexchange (DE) interaction [4, 5]. In the Zener scenario, the system consists of Y. Motome et al.: An Origin of CMR: Competing Phases and Disorder-Induced Insulator-toMetal Transition in Manganites, Lect. Notes Phys. 678, 71–86 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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conduction electrons and localized spins, and there is a strong ferromagnetic Hund’s coupling between them. Through this strong correlation, the external magnetic field which aligns the localized spins ferromagnetically increases the kinetic energy of electrons to induce a metallic state. Recent development in experiments, partly promoted by potential applications to electronic engineering such as spintronics devices, has achieved to enhance the CMR effect; the resistivity sharply decreases by a factor of 104 – 106 in the magnetic field of only a few Tesla. One of the important features of the enhanced CMR is a characteristic temperature dependence of the resistivity at zero magnetic field. The resistivity shows good metallic behavior below the ferromagnetic transition temperature TC while it shows insulating behavior above TC . That is, the resistivity shows a steep increase toward TC from above and suddenly drops near TC . The highly insulating state just above TC is very sensitive to the magnetic field, which leads to the enhanced CMR effect. Therefore, the origin of the highly insulating state above TC is a key to understand the mechanism of the enhanced CMR effect. Neither the highly-insulating nature nor the huge response to the external magnetic field can be explained by the simple DE theory. There have been many theoretical proposals which attempt to explain them. Several scenarios are based on the single-particle picture such as spin polaron [6] or Jahn-Teller (JT) polaron theory [7, 8] and Anderson localization scenario by quenched disorder [9, 10]. Recently, some attempts have also been made to understand the enhanced CMR as a cooperative phenomenon due to the many-body correlation, such as the multicritical-fluctuation scenario [11] and the percolation scenario [12, 13]. It is highly desired to clarify which is a suitable picture. Several experiments indicate the importance of a keen competition between different phases and quenched disorder for the CMR phenomena. One of the systematic investigations has been explored in a new class of materials A1/2 Ba1/2 MnO3 [14, 15, 16]. It is found that under a special condition of the synthesis, A ions and Ba ions constitute a periodic layered structure. In these A-site ordered materials, every Mn site has an equivalent environment of the surrounding ions, and therefore, there is no disorder from the alloying effect. In a usual synthesis process or by annealing the A-site ordered materials, one obtains the materials in which A and Ba ions distribute randomly, and there is structural and electrostatic disorder at Mn sites. Since the systematic change of the average radius of A-site ions is known to modify the electron bandwidth, it is now possible to control both electron correlation and quenched disorder in a systematic way in these new materials. Figure 1 shows the phase diagram of this class of materials obtained by Akahoshi et al. [16]. Open symbols connected by the solid line in the figure are the results for the A-site ordered materials. There, the system shows a typical multicritical phase diagram, that is, the FM transition temperature TC and the charge-ordered insulator (COI) transition temperature TCO meet at almost the same temperature (the multicritical point). In the disordered

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500

TCO

400 300

TC

COI

TN

T(K) 200 100

TN

TC

FM

TSG

0 1.20

1.25

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RA (A) Fig. 1. Experimental phase diagram for A1/2 Ba1/2 MnO3 obtained in [16]. The horizontal axis is the averaged ionic radius of the A-site ion. TC , TCO , TN , and TSG denote the transition temperatures for FM, COI, the antiferromagnetic state, and the glassy state, respectively. The hatched area is the disorder-induced COI-to-FM transition regime. The shaded area shows the region where COI is destroyed by introducing the A-site disorder. See the text and [16] for details

materials, the multicritical phase diagram shows a drastic change in an asymmetric manner as shown by the closed symbols connected by the dashed line in the figure. That is, the FM state is robust in spite of a suppression of TC , whereas the COI state completely disappears and is replaced by a glassy state at very low temperatures below TSG . The multicritical point is shifted to the left side of the phase diagram, and at the same time, is suppressed down to a lower temperature. Consequently, there appears a nontrivial regime in which the disorder induces the transition from COI to FM (the hatched area in Fig. 1). An important observation is that in this regime of the disorder-induced COI-to-FM transition, the resistivity shows the characteristic temperature dependence mentioned above, i.e., the highly-insulating state above TC followed by a sudden drop below TC . There, the typical enhanced CMR effect is obtained by applying a small external magnetic field [16]. Similar phenomena are also observed in A0.55 (Ca,Sr)0.45 MnO3 [17, 18] and in Cr doping in A1/2 Ca1/2 (Mn,Cr)O3 [19, 20, 21, 22]. Another crucial observation is that strong fluctuations of charge and lattice orderings are observed in the highlyinsulating state above TC , but they are largely suppressed below TC [18]. This reentrant behavior appears to correlate with the temperature dependence of the resistivity. These experimental results strongly suggest that the disorder plays

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a significant role in the multicritical phenomena to induce the enhanced CMR effect. The purpose of this Contribution is to investigate the phase competition in the multicritical regime and the disorder effect theoretically. We will explore a comprehensive understanding of the enhanced CMR effect on the basis of a cooperative-phenomenon picture rather than a single-particle one. In the next Sect. 2, we introduce an extended DE model to describe the phase competition, and briefly explain the numerical method. The results are presented in Sect. 3. We discuss the origin of CMR on the basis of our results in Sect. 4. We also compare our results with experimental data and other theoretical proposals. Section 5 is devoted to summary and concluding remarks.

2 Model and Method In the present study, we consider a minimal model which captures the competition between FM and COI. We take account of the conventional DE interaction as a stabilization mechanism of FM state. As for COI, we consider one of the simplest mechanisms, i.e., the electron-phonon coupling to the breathing-type distortions [23]. The explicit form of the Hamiltonian reads [24]  †  (ciσ cjσ + H.c.) − JH σiz Si H = −t ij σ

−g

 i

i

 1 2 λ ni Qi + Qi + Qi Qj + i ni , 2 i 2 i

(1)

ij

where the summation with ij is taken over the nearest-neighbor sites i and j, and the index σ denotes the spin of conduction electrons. The first line of (1) is for the DE part; the first term describes the kinetic energy of the single-band conduction electrons with the transfer integral t, and the second term denotes the ferromagnetic Hund’s coupling between the conduction electron spin σi and the localized spin Si . For simplicity, we consider the limit of JH → ∞ and the coupling of the Ising symmetry, i.e., Si = ±1, which retains the essential physics of the DE ferromagnetism [25]. The first term in the second line of (1) describes the electron-phonon coupling of the breathing  type where g is the coupling constant, ni = σ c†iσ ciσ is the local electron density, and Qi is the amplitude of the distortion at the site i. The next two terms denote the elastic energy of distortions. The latter term describes a cooperative aspect of the lattice distortion, where λ is taken to be positive because a shrinkage (expansion) of a MnO6 octahedron tends to expand (shrink) the neighboring MnO6 octahedra. Lattice distortions are treated as classical objects for simplicity. The last term in (1) is for the quenched disorder which couples to conduction electrons. In real materials, the alloying

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effect of A-site ions in AMnO3 as well as of the Cr substitution into the Mn sites causes structural and electrostatic disorder, which modifies the on-site potential through the Madelung energy. Here, we mimic this effect by the random on-site energy i . We consider the model (1) on the two-dimensional (2D) square lattice in the half-doped case, i.e., 0.5 electron per site on average. We set the halfbandwidth in the case of JH = g = i = 0 as an energy unit, i.e., W = 4t = 1. We take λ = 0.1 hereafter. We consider the binary-type distribution of the random potential, i.e., i = ±∆. When g is small (large bandwidth), the DE part is dominant and stabilizes FM at low temperatures. On the other hand, when g is large (small bandwidth), the electron-phonon coupling becomes dominant and induces the checkerboard-type charge order (CO) with the wave number (π, π). Hence, the competition between FM and COI is expected in our model (1) by controlling g/W . In the A-site ordered materials, the A-site substitution modifies mainly the bandwidth of conduction electrons through changes of the length and the angle of Mn-O-Mn bonds. The right (left) hand side of the phase diagram in Fig. 1 corresponds to larger (smaller) bandwidth regime where FM (COI) is stabilized at low temperatures. In the A-site disordered materials, the situation is more complicated; the A-site substitution may change the bandwidth as well as the strength of the disorder. In the following, we investigate the model (1) by changing both g and ∆ systematically to understand the physics of the competing phases observed in experiments. We study thermodynamic properties of the model (1) by employing the Monte Carlo (MC) simulation. In the MC sampling, physical quantities are averaged for configurations of localized spins {Si } and lattice distortions {Qi } which are randomly generated by using the importance sampling technique. The MC weight is calculated by the diagonalization of the electronic part for a given configuration of {Si } and {Qi }. In the presence of disorder, we take a quenched random average for different configurations of the on-site random potential {i }. For the details of MC calculations, readers are referred to [24]. In the present system, since the competition between different orders as well as the spatial inhomogeneity due to the disorder is important, it is crucial to distinguish short-range correlations and long-range orders. Therefore, although the calculations are performed on small planar systems, e.g., 8 × 8 lattices, we have carefully applied a systematic finite-size scaling analysis [26].

3 Results 3.1 Disorder Effect on Multicritical Phase Diagram In the pure case without disorder (∆ = 0), the model (1) shows the multicritical phase diagram as shown in Fig. 2 (a). There, we have four different

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FM

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g/W Fig. 2. (a)–(c) Phase diagrams of the model (1) for the change of the disorder strength ∆. Circles (squares) denote the transition temperatures for the ferromagnetism (the charge ordering) estimated by using the finite-size scaling up to 8 × 8 sites. The lines are guides for the eyes. In (b) and (c), the results for the pure case (a) are shown by gray symbols and lines for comparison. The change of the phase diagram is schematically summarized in (d)

phases; the high-temperature para phase, FM phase in the small g/W regime, COI phase in the large g/W regime, and the coexisting phase of ferromagnetism and charge ordering (F + COI) in between. Note that the coexisting F + COI state is uniform and not phase-separated. Thus, the phase diagram shows a tetracritical topology [27]. From the systematic study of the density of states (not shown here), we find a metal-insulator transition at the phase boundary between FM and F + COI phases.

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When we introduce the disorder to the system, the multicritical phase diagram shows the systematic and drastic change as shown in Figs. 2 (b) and (c). The FM phase is robust to the disorder showing a slight decrease of TC , whereas the COI phase is surprisingly unstable to the disorder. The critical temperature TCO rapidly decreases with increasing the strength of disorder ∆, and the COI phase finally collapses at ∆ ∼ 0.2W in the parameter range of the present MC calculations. The asymmetric change of the multicritical phase diagram for the disorder is schematically summarized in Fig. 2 (d). In the following, we will explore the mechanism of this change to clarify the origin of the enhanced CMR. 3.2 Fragility of Commensurate Insulator against Disorder The FM state is robust to the disorder because the DE ferromagnetism is stabilized by the kinetics of conduction electrons. TC is proportional to the kinetic energy which gradually decreases with the disorder [28]. On the contrary, we found that COI is surprisingly fragile against the disorder. The fragility is understood by the out-of-phase pinning phenomenon as follows. Figure 3 (a) shows the MC results of the density of states (DOS) in the COI regime. Even when the disorder destroys the long-range CO, the gap structure in DOS remains robust as shown in the figure. This means that short-range correlations survive and local lattice distortions persist to open the gap. Therefore, the fragility of the COI phase is not due to the rapid decrease of the amplitude of lattice distortions, but due to the disturbance of the phase of the commensurate ordering. The random potential to electrons is “a random field” to CO, and it easily pins the commensurate ordering pattern with introducing antiphase domain walls as schematically shown in Fig. 3 (b). 1.0 0.8

(b)

(a)

∆ 0.0 0.1 0.2

0.6

D(ω) 0.4 0.2 0.0 -3.0

-2.0

-1.0

0.0

ω

1.0

2.0

3.0

Fig. 3. (a) The density of states at g = 1.6 and T = 0.016 calculated for 8 × 8-sites systems. (b) A schematic picture of the out-of-phase pinning for the checkerboardtype charge ordering. White (black ) circles denote electrons (holes), and crosses are the random pinning centers. The dashed gray curve shows an antiphase domain wall

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3.3 Highly-Insulating State above TC The out-of-phase pinning picture suggests that there remain short-range charge correlations and fluctuations in the region where the long-range CO is destroyed by the disorder. This is indeed the case. Figure 4 shows the temperature dependences of the susceptibility of the staggered lattice distortion which is calculated by fluctuations of the COI order parameter. Although the long-range CO is destroyed by the disorder and the diverging behavior at TCO in the pure case is smeared out, fluctuations of charge and lattice orderings remain finite and are enhanced toward TC even at a finite ∆. This could be regarded as a reminiscence of the multicritical phenomenon in the pure case [11]. 25 20

χQ

15

∆ 0.0 0.1 0.2

TCO TC

10 5 0 0.00

0.02

0.04

0.06

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T Fig. 4. Temperature dependence of the staggered susceptibility of the lattice distortion calculated at g = 1.0 for 8 × 8-sites systems. The gray arrows indicate the values of TC in the presence of disorder

We note that the fluctuations are suppressed for large values of ∆. We suggest that for very strong disorder, the system may favor rather a static state with small COI clusters rather than the fluctuating state. This crossover will be discussed in Sect. 4.3. The remaining fluctuations of CO affect the electronic states in the hightemperature regime. Figures 5 show the MC results of (a) DOS and (b) the optical conductivity in this regime. As ∆ becomes larger, the dip in DOS at the Fermi energy becomes deeper, and the low-energy weight of σ(ω) becomes smaller to develop a quasi-gap feature. Thus, the disorder induces fluctuations of CO, and tends to make the system insulating in the hightemperature regime.

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3.4 Disorder-Induced Insulator-to-Metal Transition In contrast to the enhanced insulating nature at high temperatures, we find a remarkable phenomenon at low temperatures in the multicritical regime, that is, the disorder-induced insulator-to-metal transition. Figures 6 show (a) DOS and (b) the optical conductivity close to the metal-insulator phase boundary. In the pure case (∆ = 0), DOS shows a finite gap and σ(ω) has a gap structure due to the long-range CO. When we introduce the disorder, the gap is suddenly filled and there appears a finite DOS at the Fermi energy. At the same time, the low-energy spectral weight of σ(ω) rapidly increases. These results strongly indicate that the disorder induces the transition from the commensurate insulator to a metallic state. Such a phenomenon caused by the multicritical phase competitions is surprising and counter-intuitive because in general the disorder tends to localize electrons and to make the system more insulating. We note that the optical conductivity in the multicritical regime in Fig. 6 (b) as well as in Fig. 5 (b) shows drastic reconstruction in the wide energy 0.25

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range up to ω ∼ W by a small disorder of only the order of 0.1W . This is a consequence of the cooperative phenomena in the phase-competing regime. 3.5 Brief Summary of Results Here, we give a short summary of our results. MC simulation revealed that our model (1) exhibits the multicritical phenomenon due to the competition between FM and COI states, and the competing phases show the asymmetric response to the disorder. The COI phase is very unstable to the disorder while the FM phase is rather robust, which leads to the disorder-induced insulator-to-metal transition at low temperatures. In spite of the fact that the long-range CO is destroyed at low temperatures, the short-range correlations of CO are enhanced by the disorder at high temperatures, which tends to make the system more insulating. Thus, we obtain reentrant behavior, i.e., charge and lattice correlations are enhanced with decreasing temperature, but they are suppressed below TC . This is considered to be entropy-driven reentrant behavior [29]; the short-range CO state has rather high entropy related to configurations of the antiphase domain walls, and is favored at high temperatures due to the high entropy. We consider that this reentrant behavior is an origin of CMR phenomena as discussed in Sect. 4.1. Our results elucidate an important role of the contrasting nature of the competing phases. The FM phase is stabilized by the kinetics of conduction electrons, and has a long-range ordering with the wave number (0, 0). On the other hand, the COI phase is stabilized by a commensurate lattice ordering with the wave number (π, π). The former is robust but the latter is fragile against the out-of-phase pinning by disorder. The competition between the uniform metal and the commensurate insulator plays a central role.

4 Discussions 4.1 Origin of CMR In this section, we discuss the origin of CMR suggested by our results. We found in Sect. 3 the contrasting effects of disorder on the electronic states in the phase competing regime, i.e., the insulator-to-metal transition at low temperatures and the enhanced insulating nature at high temperatures. The former occurs in the hatched area in the schematic phase diagram shown in Fig. 7 (a). The latter is conspicuous in the shaded area in Fig. 7 (a) where we found the enhanced fluctuations of charge and lattice orderings. The contrasting effects of disorder and the resultant reentrant behavior of charge and lattice correlations provide a key to understand the enhanced CMR effect observed in experiments. Schematic temperature dependences of the resistivity along the downward arrow in Fig. 7 (a) are shown in Fig. 7 (b), suggested by our numerical results of σ(ω) in Sect. 3. In the pure case

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Fig. 7. (a) Schematic phase diagram of the model (1). Dashed curves denote TC and TCO in the absence of disorder, and the solid curve denotes TC in the presence of disorder. See Fig. 2 (d). The hatched area shows the region where the disorderinduced insulator-to-metal transition takes place. The shaded area denotes the relevant regime to the huge CMR effect where COI fluctuations are substantial and the highly-insulating state is realized. (b) Temperature dependences of the resistivity along the downward arrow in (a) suggested by the present study. Dashed and solid curves correspond to the cases without and with disorder, respectively

with ∆ = 0, the resistivity sharply increases below the COI transition temperature TCO as shown by the dashed curve in Fig. 7 (b). When the disorder is introduced into the system, the high-temperature resistivity increases because the disorder enhances the insulating nature there. On the contrary, at a low temperature, we have the insulator-to-metal transition induced by the disorder, where the resistivity should show a sudden drop. Hence, we obtain the characteristic temperature dependence for ∆ = 0 as shown by the solid curve in Fig. 7 (b). Note that the temperature dependence is in accord with that of charge and lattice fluctuations found in Fig. 4, which illuminates the close relation between the resistivity and these fluctuations. The enhanced insulating state just above TC is known to show a huge response to a small external magnetic field, i.e., the enhanced CMR as mentioned in Sect. 1. Such a huge response is expected in the shaded region in Fig. 7 (a) where fluctuations of charge and lattice orderings are enhanced by the disorder. Thus, the enhanced CMR phenomena can be understood by the contrasting effects of the disorder in the phase competition between the uniform metal and the commensurate insulator. 4.2 Comparison with Experimental Results Our results reproduce well the experimental results in CMR manganites of the A-site ordered/disordered materials [15, 16], A0.55 (Ca,Sr)0.45 MnO3

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[17, 18], and Cr-doped materials A1/2 Ca1/2 (Mn,Cr)O3 [19, 20, 21, 22] in the following aspects; – multicritical phase diagram in clean cases, – asymmetric change of the multicritical phase diagram by introducing disorder, in particular, the fragility of COI phase, – disorder-induced insulator-to-metal transition in the multicritical regime, – highly-insulating state above TC in the presence of disorder, – remaining short-range correlations or fluctuations of charge and lattice orderings in the insulating state above TC observed in Raman scattering and diffuse X-ray scattering experiments, – reentrant temperature dependence commonly observed in the resistivity and the charge and lattice fluctuations. The agreement strongly indicates that our model (1) captures the essential physics of phase competing phenomena and disorder effects on them in CMR materials. We also note that our results give an understanding of the close relation between the charge fluctuations and CMR found in La1−x Cax MnO3 at x ∼ 1/3 [30, 31]. 4.3 Comparison with Other Theoretical Scenarios Finally, we compare our scenario with other theoretical proposals for the mechanism of CMR. As discussed in Sect. 1, CMR is a kind of localizationdelocalization phenomenon at the ferromagnetic transition temperature TC . The low-temperature metallic phase is basically understood by the Zener DE mechanism, and hence, the problem is how to understand the hightemperature insulating state above TC . Several proposals are based on the single-particle picture. One is the polaron scenario including both spin polaron [6] and JT polaron theories [7, 8]. In this scenario, the high-temperature insulating state is described by the self trapping of electrons to form small polarons. Another scenario is the Anderson localization due to the quenched disorder [9, 10]. For both scenarios, it is difficult to answer the following questions on the general aspects of the enhanced CMR comprehensively: – Why does the localized state at high temperatures become unstable at lower temperatures? – Why does the localization-delocalization transition always coincide with the ferromagnetic transition without fine tuning of parameters? – Why does the disorder cause the insulator-to-metal transition as observed in the A-site ordered manganites? – Why is CMR observed in wide region of the phase diagram, or why does the localization occur even at highly carrier-doped region? Hence, the single-particle pictures appear to be insufficient to explain the enhanced CMR. In particular, they cannot explain the experimental fact that

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the “clean” system shows the multicritical phase diagram while the disorder induces the CMR phenomenon in the multicritical regime. On the contrary, our results based on the cooperative-phenomenon picture give a comprehensive understanding of CMR as a localization-delocalization phenomenon. Essential ingredients are the phase competition and the disorder, which are found in general in manganites and do not need fine tuning of parameters. Our theory answers why the delocalization occurs at TC as well as why the disorder induces the insulator-to-metal transition. We note that there have been proposed other cooperative-phenomenon scenarios. One is the multicritical-fluctuation scenario [11] in which enhanced fluctuations near the multicritical point play an important role in the insulating state above TC . The scaling law for the magnetization curve (the so-called Arrott plot) shows evidence for this scenario. Although the scaling holds for a class of materials with weak disorder in which the multicritical point is at rather high temperature, it does not show good agreement with the others where the multicritical point is suppressed down to low temperatures by the disorder and the typical CMR is observed. Another proposal is the percolation scenario which explicitly takes account of the disorder. In this scenario, it is supposed that the disorder induces the percolated mixtures of FM and COI islands below T ∗ which is the transition temperature TC or TCO in the absence of the disorder [12, 13]. The resistivity is determined by the percolating path, and is sensitive to the pattern of the coexisting FM and COI regions which is easily changed by an external magnetic field. However, the theory is limited to a phenomenological level and does not give a microscopic explanation for the disorder-induced insulator-to-metal transition as well as the reentrant behavior of CO fluctuations. Moreover, this rather static picture appears to be incompatible with recent experimental results which indicate a dynamically fluctuating state just above TC [32]. Compared with above two scenarios based on the cooperative-phenomenon picture, our theory is considered to interpolate these two limiting pictures, i.e., it elucidates what happens in weak or moderate disorder regime in the phase-competing system. There, short-range correlations or dynamical fluctuations of charge and orbital orderings remain substantial and are relevant to the localization at high temperatures. In our numerical results in Sect. 3, we do not find any clear indication of the percolated cluster formation below T ∗ [24], and such static phase separation may be relevant in rather strong disorder regime where CO fluctuations are largely suppressed as implied by Fig. 4. Our theory, which describes the crossover from the clean to dirty limits, strongly suggests that the enhanced CMR occurs in the weak or moderate disorder regime with large fluctuations. Note that the unbiased numerical calculations play key roles to reveal the highly nontrivial properties in the weak or moderate disorder regime.

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5 Summary and Concluding Remarks In this Contribution, we discussed the phase competition between ferromagnetic metal and charge-ordered insulator and the role of the disorder by applying the Monte Carlo calculations to an extended double-exchange model. Highly nontrivial phenomena are revealed such as the disorderinduced insulator-to-metal transition as well as the entropy-driven reentrant behavior of charge-ordering fluctuations. Our results show good agreement with recent experimental results in CMR manganites, and give a comprehensive understanding of the origin of the enhanced CMR effect. There still remain many open problems. One is the role of other interactions which are neglected in our simplified model. One of the neglected elements is the orbital degree of freedom in the twofold eg orbitals which strongly couples with the JT lattice distortion. This is necessary, at least, to describe a complicated orbital ordering in the COI phase [33], and possibly introduces orbital fluctuations and enhances the insulating nature in the CMR regime as the fluctuations of charge and lattice orderings. Another omitted element is the AF superexchange interaction between localized spins. This is important to reproduce the complicated magnetic structure in the socalled CE phase as well as the A-type AF metallic phase, and possibly plays a substantial role in the glassy state at low temperatures. Recent theoretical study including these elements reported similar results to ours [34] although the numerical analysis is laborious and very limited for such a complicated model thus far. Another issue is the calculation in realistic 3D systems. The present calculations have been performed in 2D. The pinning effect due to the disorder is sensitive to the dimensionality, which should be carefully examined further. Moreover, in real materials, there may be some spatial correlation between the disorders, in other words, a long-range nature of the influence of disorder through the lattice strain effect or the cooperative effects of lattice distortions [35]. There, the orbital degree of freedom should play an important role through the Jahn-Teller coupling. It is highly desired to examine such effects for realistic electronic models in the higher dimension. The direct calculation of the resistivity also remains as a hard task for numerical studies. There are several reports on the resistivity in small-size clusters which strongly indicate a huge response to the magnetic field [23, 36]. To explain the enhanced CMR effect more quantitatively, further investigations are necessary including the development of theoretical tools to calculate the resistivity directly in larger-scale systems.

Acknowledgment The authors acknowledge Y. Tokura, Y. Tomioka, and E. Dagotto for fruitful discussions. This work is supported by Grants-in-Aid for Scientific Research

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and NAREGI Nanoscience Project from the Ministry of Education, Culture, Sports, Science, and Technology.

References 1. A.P. Ramirez: J. Phys. Cond. Matt. 9, 8171 (1997), and references therein 71 2. Y. Tokura, N. Nagaosa: Science 288, 462 (2000), and references therein 71 3. E. Dagotto, T. Hotta, A. Moreo: Phys. Rep. 344, 1 (2001), and references therein 71 4. C. Zener: Phys. Rev. 82, 403 (1951) 71 5. N. Furukawa: in Physics of Manganites, edited by T. Kaplan and S. Mahanti (Plenum, New York, 1999), and references therein 71 6. C.M. Varma: Phys. Rev. B 54, 7328 (1996) 72, 82 7. A.J. Millis, P.B. Littlewood, B.I. Shraiman: Phys. Rev. Lett. 74, 5144 (1995); A.J. Millis, R. Mueller, B.I. Shraiman: Phys. Rev. B 54, 5405 (1996) 72, 82 8. H. R¨ oder, J. Zang, A.R. Bishop: Phys. Rev. Lett. 76, 1356 (1996); J. Zang, A.R. Bishop, H. R¨ oder: Phys. Rev. B 53, R8840 (1996) 72, 82 9. B.M. Letfulov, J.K. Freericks: Phys. Rev. B 64, 174409 (2001) 72, 82 10. E.E. Narimanov, C.M. Varma: Phys. Rev. B 65, 024429 (2002) 72, 82 11. S. Murakami, N. Nagaosa: Phys. Rev. Lett. 90, 197201 (2003) 72, 78, 83 12. A. Moreo, S. Yunoki, E. Dagotto: Science 283, 2034 (1999) 72, 83 13. J. Burgy et al: Phys. Rev. Lett. 87, 277202 (2001) 72, 83 14. F. Millange et al:, Chem. Mater. 10, 1974 (1998) 72 15. T. Nakajima et al:, J. Phys. Soc. Jpn. 71, 2843 (2002) 72, 81 16. D. Akahoshi et al: Phys. Rev. Lett. 90, 177203 (2003) 72, 73, 81 17. Y. Tomioka, Y. Tokura: Phys. Rev. B 66, 104416 (2002) 73, 82 18. Y. Tomioka et al: Phys. Rev. B 68, 094417 (2003) 73, 82 19. A. Barnab´e et al., Appl. Phys. Lett. 71, 3907 (1997); B. Raveau, A. Maignan, C. Martin: J. Solid State Chem. 130, 162 (1997) 73, 82 20. T. Kimura et al: Phys. Rev. Lett. 83, 3940 (1999) 73, 82 21. Y. Moritomo et al: Phys. Rev. B 60, 9220 (1999) 73, 82 22. T. Katsufuji et al: J. Phys. Soc. Jpn. 68, 1090 (1999) 73, 82 23. J.A. Verg´es, V. Mart´ın-Mayor, L. Brey: Phys. Rev. Lett. 88, 136401 (2002) 74, 84 24. Y. Motome, N. Furukawa, N. Nagaosa: Phys. Rev. Lett. 91, 167204 (2003) 74, 75, 83 25. Y. Motome, N. Furukawa: J. Phys. Soc. Jpn. 70, 1487 (2001); 70, 2802 (2001). The Ising symmetry enables us to study finite-temperature phase transitions in 2D systems 74 26. Y. Motome, N. Furukawa, N. Nagaosa: J. Mag. Mag. Mat., 272-276, 1805 (2004) 75, 85 27. The tetracritical phase diagram can be turned into the bicritical one by explicitly including a competing term between FM and CO, which makes the coexisting F + COI state unstable [26]. The effects of disorder are qualitatively common to both tetracritical and bicritical cases, and therefore we focus on the tetracritical case in this Contribution 76 28. Y. Motome, N. Furukawa: Phys. Rev. B 68, 144432 (2003) 77 29. N. Furukawa, Y. Motome, N. Nagaosa: to be published in Physica B, condmat/0406504 80

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Y. Motome et al. P. Dai et al: Phys. Rev. Lett. 85, 2553 (2000) 82 C.P. Adams et al: Phys. Rev. Lett. 85, 3954 (2000) 82 S. Mori, private communications 83 S. Yunoki, A. Moreo, E. Dagotto: Phys. Rev. Lett. 81, 5612 (1998) 84 H. Aliaga et al: Phys. Rev. B 68, 104405 (2003) 84 J. Burgy, A. Moreo, E. Dagotto: Phys. Rev. Lett. 92, 097202 (2004) 84 C. Sen, G. Alvarez, E. Dagotto: preprint (cond-mat/0401619) 84

A Neutron Scattering Investigation of MnAs K.U. Neumann1 , S. Dann2 , K. Fr¨ ohlich1 , A. Murani3 and B. Ouladdiaf3 1 and K.R.A. Ziebeck 1 2 3

Dept. of Physics, Loughborough University, Loughborough LE11 3TU, UK Dept. of Chemistry, Loughborough University, Loughborough LE11 3TU, UK Institute Laue Langevin, Rue Horowitz, 36048 Grenoble Cedex, France

Abstract. MnAs is a ferromagnet which undergoes a transition to the ferromagnetically ordered state via a 1st order phase transition. Here some of the crystallographic and magnetic properties are presented and discussed. In particular the question of the magnetic moment stability and its coupling to the lattice is addressed by a combination of different neutron scattering and magnetisation experiments. MnAs exhibits a strong magnetoelastic coupling. An external magnetic field is able to induce a structural transformation between orthorhombic and hexagonal modifications of MnAs. In zero field this transformation is accompanied by a change in atomic volume of ∼2%. The magnitude of the volume anomaly decreases with the application of an external magnetic field. The magnetic response in the orthorhombic phase is found to be strongly non-linear. Neutron measurements show that the magnitude of the Mn moment in the paramagnetic phase is similar to the ground state value and that both the temperature response of the β and γ phases is characterised by ferromagnetic correlations.

1 Introduction The compound MnAs has been intensively studied for more than 40 years. At low temperatures MnAs is ferromagnetically ordered. At a temperature of approximately 40◦ C a first order phase transition is observed to a paramagnetic state. This phase transition is accompanied by a structural change from the low-temperature hexagonal phase with ferromagnetic order, α-MnAs, to a paramagnetic orthorhombic phase, β-MnAs. At higher temperatures, ∼120o C, the structure reverts back to the hexagonal phase (γ-MnAs) in a second-order phase transition. Both high-temperature modifications of MnAs are paramagnetic. The phase diagram is indicated below.

α-MnAs NiAs-structure ferromagnetic

318K

β-MnAs MnP-structure paramagnetic

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K.U. Neumann et al.: A Neutron Scattering Investigation of MnAs, Lect. Notes Phys. 678, 87–96 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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The first-order phase transition around 40◦ C has both a lattice and a magnetic contribution. Just below the phase transition a ferromagnetically ordered moment is observed of a magnitude amounting to ∼70% (2.4 µB ) of the ordered ground state moment (∼3.45 µB ). The transition can be influenced by the application of an external magnetic field. Indeed in the paramagnetic region an applied field can induce a metamagnetic phase transition. These characteristics have led to the investigation of the magnetocaloric effect in MnAs as a potential candidate for magnetic refrigeration. In experiments MnAs exhibits an entropy change of ∆Smag = - 32 J/(kg·K) at the first order phase transition [1, 2, 3]. The hexagonal-to-orthorhombic phase transition around 40◦ C is accompanied by a contraction of the lattice. On going from the ferromagnetic to the paramagnetic state the unit cell volume decreases by ∼2% [4]. This has led some authors [5, 6] to propose a change in magnitude of the magnetic moment with a significant rearrangement of electrons on the Mn-atoms. This high spin – low spin transition model, however, is difficult to reconcile with the very similar magnitudes of atomic magnetic moments in the ferromagnetically ordered state and the paramagnetic moment in the orthorhombic and the high-temperature hexagonal phases as obtained from magnetisation and neutron measurements. At high temperature the magnetic susceptibility of the γ-MnAs phase follows a Curie Weiss law. From the Curie Weiss constant the effective paramagnetic moment and hence the moment (gS) per Mn atom, can be determined. Despite some temperature and applied-magnetic-field hysteresis the α-βMnAs phase transition is fully reversible. The change of volume at the phase transition opens up the possibility of using this material in applications akin to shape memory materials [7]. Although solid lumps of the material have the tendency to develop cracks on traversing the 1st order phase transition some applications may exploit the change in volume in bulk samples, or alternatively, take advantage of MnAs in the form of thin films. The magnetic field dependence of the phase transition, and the possibility to induce a metamagnetic phase transition in the paramagnetic state, are reflected in the physical properties of MnAs in the β-phase. Mira et al. [6] have measured the magnetoresistance just above the α-β-MnAs phase transition and reported a “colossal-like” magnetoresistive response of 17% in a field of 5 Tesla. However, the coupling of magnetic degrees of freedom to the lattice has to be taken into account for a full description of the magnetic properties as an external magnetic field may induce a crystallographic phase transformation. This is supported by observations of a significant magnetoelastic response by Chernenko et al. [8]. In order to experimentally identify and separate the magnetic and structural contributions characterising the magnetic properties of MnAs, neutron scattering measurements have been undertaken on a powder sample. Some structural aspects such as unit cell volume, atomic positions and size of

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magnetic moments, can be investigated using neutron powder diffraction in both zero field as well as in applied magnetic fields. The magnetic properties of the paramagnetic phases have also been investigated using spin polarised neutrons. These measurements are supported by magnetisation measurements up to 350 K.

2 Structural Aspects of MnAs

M [J/(T*kg)]

Neutron powder diffraction experiments have been carried out on MnAs using the high resolution diffractometer D2b at the ILL in Grenoble, France [9]. The diffraction pattern at 100 K in zero applied field is shown in Fig. 1. At this temperature the MnAs adopts the hexagonal α-MnAs structure. The refinements of the pattern in zero external field and in a field of 5 Tesla indicate that the crystallographic structure does not change with the application of the magnetic field. The moment refined on the Mn atoms, increases as a result of the alignment of the magnetic moment perpendicular to the scattering vector.

150

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Fig. 1. Neutron diffraction pattern of MnAs in the ferromagnetic phase at 100 K. The external magnetic field is zero. The dots indicate the measurement and the line is the fit. The difference between observation and calculation is indicated by the line at the bottom of the figure. Vertical bars indicate the positions of Bragg peaks for the various phases (from top to bottom): nuclear MnAs structure, magnetic contribution of MnAs, MnO impurity phase (volume fraction of ∼0.06%). The insert shows a magnetisation measurement at T = 100 K in the form of an isotherm

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Similar measurements at a temperature of 320 K are shown in Figs. 2 and 3 for zero field and in a field of 5 Tesla, respectively. This temperature falls within the paramagnetic regime of MnAs but close to the α-β phase transition. The pattern in zero field indicates the coexistence of the α- and β-MnAs phases with a volume proportion of 10% and 90%, respectively. The application of an external magnetic field oriented approximately perpendicular to the scattering plane reverses this ratio to ∼88% in volume for the ferromagnetic α-MnAs phase and ∼12% for the orthorhombic β-MnAs phase. The small contribution of an MnO impurity phase, well below the 1% contribution, remains unaltered. The observed coexistence of the α- and β-MnAs phases at 320 K enables a precise determination of the volume anomaly at this temperature. On going from the ferromagnetic to the paramagnetic state the volume contracts by 2.3% in zero magnetic field. The application of an external magnetic field of 5 Tesla increases the unit cell volume of the α-MnAs phase by ∼0.2%, while the unit cell volume of the β-MnAs increases by almost 1%. Thus at 320 K the application of a field reduces the volume difference between these phases from ∼2.3% at zero external field to 1.6% for a field of 5 Tesla. The major part of the volume variation arises due to the length change of the c-axis in the orthorhombic cell. This axis is obtained from the hexagonal cell by cortho = cos(30o ) · 2 · ahex . With the application of a field of 5 Tesla this axis increases by 1% while all the other axes change by less than 0.1%.

3 Magnetic Properties of MnAs The applied magnetic field dependence of the crystallographic structure is also observed in magnetisation measurements as shown in Fig. 4. For fixed temperatures of 310 K, 325 K and 330 K the magnetisation as a function of increasing or decreasing field shows a distinct step due to the large difference in magnetisation of the α- and β-MnAs phases. With the creation of the ferromagnetic α-MnAs phase the magnetisation is increased signalling the field-induced structural transformation. The magnetisation measurements and the structural investigations confirm that the transition temperature depends on the external magnetic field. Such a conclusion is consistent with structural investigation in an applied magnetic field using X-rays by Ishikawa et al. [10, 11] or neutron measurements by Mira et al. [6]. The stability range of the orthorhombic phase as a function of temperature and applied magnetic field has been given by Zieba [12]. However, the transition is not purely magnetic but involves the lattice. This coupling has to be taken into account if both the crystallographic field dependence and the magnetic properties of the orthorhombic β-MnAs phase are to be understood. The complex nature of M(T,H) is illustrated in Fig. 5. Magnetisation measurements presented in the form of Arrott plots as a function of H/M versus

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Fig. 2. Neutron diffraction pattern of MnAs at T = 320 K with no external magnetic field applied. The symbols have the same meaning as in Fig. 1 with the Bragg positions being indicated (from top to bottom) for the hexagonal α-MnAs phase (nuclear and magnetic), the MnO impurity phase and the orthorhombic βMnAs phase

Fig. 3. Neutron diffraction pattern of MnAs at T = 320 K with external magnetic field of B = 5 Tesla applied perpendicular to the scattering plane. The symbols have the same meaning as in Fig. 2

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M [J/(kg*T)]

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B [Tesla] Fig. 4. Magnetisation as a function of increasing and decreasing field for various temperatures in the orthorhombic phase of MnAs. The metamagnetic phase transition is related to the transformation between the paramagnetic orthorhombic β-MnAs and the ferromagnetic α-MnAs phases

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H/M [kg*T /J] Fig. 5. Magnetisation measurements presented in the form of Arrott plots in the paramagnetic β-MnAs phase. The curvature of the isotherms indicate the non-linear dependence of the magnetisation on the external magnetic field. The metamagnetic transition is seen in the Arrott plots as a sudden increase of the M2 values by a factor of 103 . The inset shows the Arrott plots at higher temperatures. For 310 K the measurements have been plotted for increasing (up) and decreasing (down) applied field (compare to magnetisation measurement at 310 K in Fig. 4)

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M2 are expected to be straight lines for a homogeneous ferromagnet. As seen in Fig. 5 the magnetisation is strongly non-linear also in the orthorhombic phase before the onset of the field induced β-MnAs to α-MnAs transformation. This transformation is revealed in the Arrott plots by a strong increase of the M2 values by a factor of 103 . As a consequence of the non-linearity of the magnetisation a measurement of the magnetic moment in a constant applied magnetic field and as a function of temperature has resulted in inverse magnetic susceptibilities being obtained which increase with decreasing temperature [14]. Such an interpretation, however, assumes a linear dependence of the magnetisation on the applied field. This assumption is not valid for MnAs in the orthorhombic phase. The change in magnetisation, as a function of temperature and field, has to take into account the lattice. It has been suggested (Goodenough and Kafalas [13]) that a magnetic moment instability is responsible for the anomalous magnetic and lattice response of MnAs. The moment instability arises from a rearrangement of electrons on the Mn atom resulting in a high spin – low spin transition. However, analysis of the magnetisation results of Ido et al. [14] yield an ordered ground state moment of magnitude 3.45 µB /Mn-atom and a magnetic moment µp at high temperatures of 3.56 or 3.82 µB /Mn-atom. µp is related to the effective magnetic moment µeff by µ2eff = µp (µp + 2) with µeff being obtained from the slope of the Curie-Weiss susceptibility. As may be seen from the data of Ido et al. [14] shown in Fig. 6 the size of the magnetic moment is not appreciably altered between the lowand high-temperature hexagonal phases of MnAs. Within the paramagnetic orthorhombic phase, the susceptibility is field dependent and hence non Curie Weiss like. Due to this strongly non-linear response of the system, it is difficult to extract the size of the local Mn moment from magnetisation measurements. Thus an alternative means, such as neutron scattering, has to be employed in order to determine the magnetic moment in the orthorhombic β-MnAs phase. With the magnetic moments of the ferromagnetic ground state and the hexagonal γ-MnAs phases being similar it is difficult to reconcile this observation with the model of a high spin – low spin transition between the α- and γ-MnAs phases. Paramagnetic neutron scattering experiments are able to adress this question.

4 Paramagnetic Neutron Scattering Investigation Using spin-polarised neutrons and polarisation analysis it is possible to separate the magnetic scattering contribution from all other sources of scattering (Squires [15], Lovesey [16]). If after the scattering event the neutron is counted irrespective of its energy, an energy integrated neutron scattering cross section is determined providing information on the size of the magnetic

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M0 = 3.45 µB

µp = 3.82 µB

1.0

χ-1 [emu/g]

M(T)/M0

2*104

390K

0.5 Θpara = 288K Θpara = 260K 100

200

µp = 3.56 µB 318K

0.0 0

1*104

300

400

500

600 700 T [K]

800

Fig. 6. Magnetisation and magnetic susceptibility of MnAs as a function of temperature for MnAs according to Ido [14]. The atomic magnetic moments are indicated as either Mo for the ground state moment of the ferromagnetically ordered phase, or as µp for the paramagnetic phase. µp is obtained from the effective magnetic moment µeff the inverse square of which determines the slope of the reciprocal magnetic susceptibility

moment and the nature of the magnetic correlations, i.e. whether the magnetic moments are correlated ferro- or antiferromagnetically, or disordered as expected for Curie Weiss behaviour of local moments. The paramagnetic scattering of MnAs has been determined, as a function of temperature, in the orthorhombic as well as in the high-temperature hexagonal phases. Results obtained in the two phases are shown in Fig. 7. The magnitude of the paramagnetic scattering is found to be essentially independent of temperature. This suggests that the size of the magnetic moment giving rise to the scattering does not change between the β-MnAs and α-MnAs phases. For both temperatures the scattering is enhanced in the forward direction. This is an indication of ferromagnetic correlations. A combination of polarised and thermal energy neutron scattering experiments has enabled the spatial correlation and total paramagnetic scattering to be determined as a function of temperature. The results obtained by using spin polarised neutrons on the instrument D7 are supported by using unpolarised neutrons in a time of flight experiment on IN4 enabling integration over the entire magnetic response.

Neutron Scattering on MnAs

95

( dσ/dΩ )mag [arb. units]

250

MnAs

200

T = 550K T = 325K

150

100

50

0 0

1

2 -1

q [A ] Fig. 7. Paramagnetic neutron scattering of MnAs in the orthorhombic β-MnAs phase (at T = 325 K) and the high temperature hexagonal γ-MnAs phase at 550 K [17]

Apart from a neutron guide field of several 100 Gauss the spin-polarised neutron scattering measurement was essentially carried out in zero magnetic field. The results indicate that the magnetic moment on the Mn atom is stable and does not appreciably vary between the α, β and γ phases of MnAs. Furthermore the magnetic correlations are predominantly ferromagnetic. Within the temperature interval investigated the results show that models invoking a change in the magnetisation of Mn moment or the presence of antiferromagnetic correlations are inappropriate.

5 Conclusions The MnAs system exhibits some intriguing physical properties. Firstly with increasing temperature a first-order phase transition from a ferromagnetically ordered state to a paramagnetic one is observed with a lattice contraction of ∼2% and a change of structure. Secondly MnAs displays a high sensitivity of magnetic and structural characteristics on an external magnetic field in the orthorhombic phase. Thirdly a transition occurs at high temperatures back to the low-temperature hexagonal phase. The sequence of phase transitions, and the dependence of the orthorhombic structure on external fields, suggests that MnAs is a system for which several states are very close in energy to the ground state. Thus modest magnetic fields, pressure or alloying will be

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able to change the balance and result in phase transitions. Here these changes have been illustrated with neutron and magnetisation measurements. Despite the strong response of the system in terms of structural changes or magnetic response, a zero external field measurement of the paramagnetic scattering of MnAs in the high temperature phases indicates that the magnetic moment is stable, and that magnetic correlations are strongly ferromagnetic and independent of temperature. Measurements of structural properties in external magnetic fields have revealed that the transformation of α-MnAs and β-MnAs can be induced by an external field. There is a region in which orthorhombic and hexagonal phases coexist, with a temperature and magnetic field hysteretic region in the phase diagram. The strong response of MnAs to temperature changes and applied field make this compound a candidate for magnetocaloric applications such as magnetic cooling or, due to the large volume change as a function of temperature, as an actuator.

References 1. H. Wada, Y. Tanabe: J. Appl. Phys. Lett. 79, 3302 (2001) 88 2. S. Fujieda, Y. Hasegawa, A. Fujita, K. Fukamichi: J. Appl. Phys. 95, 2429 (2004) 88 3. R. Locke: Final Year Research Project, Department of Physics, Loughborough University (2002) 88 4. R.H. Wilson, J.S. Kasper: Acta Cryst. 17, 95 (1964) 88 5. J.B. Goodenough, J.A. Kafalas: Phys. Rev. 157, 389 (1967) 88 6. J. Mira, F. Rivadulla, J. Rivas, A. Fondado, T. Guidi, R. Caciuffo, F. Carsughi, R.P. Radaelli, J.B. Goodenough: Phys. Rev. Lett. 90, 097203 (2003) 88, 90 7. K. Otsuka, C.M. Wayman: Shape Memory Materials, Cambridge University Press, (1998) 88 8. V.A. Chernenko, L. Wee, P.G. McCormick, R. Street: J. Appl. Phys. 85, 7833 (1999) 88 9. Details of instruments and the experimental set-up can be obtained from www.ill.fr/YellowBook. 89 10. F. Ishikawa, K. Koyama, K. Watanabe, H. Wada: Jpn. J. Appl. Phys. 42, L918 (2003) 90 11. F. Ishikawa, K. Koyama, K. Watanabe, H. Wada: Physica B346-347, 408 (2004) 90 12. A. Zieba, Y. Shapria, S. Foner: Phys. Lett. 91A, 243 (1982) 90 13. J.B. Goodenough, J.A. Kafalas: Phys. Rev. 157, 389 (1967) 93 14. H. Ido: J Appl. Phys. 57, 3247 (1985) 93, 94 15. G.L. Squires: Introduction to the Theory of Thermal Neutron Scattering, Dover Publication (1996) 93 16. S.W. Lovesey: Theory of Neutron Scattering from condensed Matter, Clarendon Press, Oxford (1984) 93 17. K.-U. Neumann, K.R.A. Ziebeck, F. Jewiss, L. D¨ aweritz, K.H. Ploog, A. Murani: Physica B335, 34 (2003) 95

Epitaxial MnAs Films Studied by Ferromagnetic and Spin Wave Resonance T. Toli´ nski1,3 , K. Lenz1 , J. Lindner1,4 , K. Baberschke1 , A. Ney2,5 , T. aweritz2 , R. Koch2 and K.H. Ploog2 Hesjedal2 , C. Pampuch2,6 , L. D¨ 1

2

3

4

5

6

Institut f¨ ur Experimentalphysik, Freie Universit¨ at Berlin, Arnimallee 14, D-14195 Berlin, Germany [email protected] Paul-Drude-Institut f¨ ur Festk¨ orperelektronik, Hausvogteiplatz 5-7, D-10117 Berlin, Germany permanent address: Institute of Molecular Physics, PAS, Smoluchowskiego 17, 60-179 Pozna´ n, Poland new address: Fachbereich Physik, Experimentalphysik-AG Farle, Universit¨ at Duisburg-Essen, Lotharstr. 1, D-47048 Duisburg, Germany new address: Solid State and Photonics Lab, Stanford University, Stanford, CA 94305-4075, USA new address: Specs GmbH, Voltastraße 5, 13355 Berlin, Germany

Abstract. We investigated the anisotropy and intrinsic exchange interaction within MnAs films using ferromagnetic resonance (FMR) and spin wave resonance (SWR), respectively. Apart from the dominating in-plane easy axis a presence of an independent contribution (independent FMR mode) characterized by an out-of-plane easy axis is found in agreement with our previous magnetometric studies. The temperature sweep of the resonance spectra shows a jump both for the resonance field and the resonance linewidth at a temperature of 10◦ C, i.e., at the transition from the hexagonal (ferromagnetic) α-phase to the region of the coexisting α- and orthorhombic (paramagnetic) β-phase. In the coexistence region the main easy axis lies in-plane and perpendicular to the stripe direction being the direction of the c axis. In the SWR measurements with magnetic field applied close to the normal of the film a set of lines resulting from the excitation of spin waves is observed. The extracted exchange constant is as small as A = 17.7 ×10−10 erg/cm. Moreover, the temperature dependence of the spin wave stiffness constant D = 2A/M has been determined within the coexistence region.

1 Introduction Most of the devices used in or being designed for the contemporary data storage, magnetic random access memory or reading heads consist of two different films with alternating stacking sequence. The possible combinations are systems like ferromagnet/nonmagnet, metal/insulator(semiconductor) and others. Such an alternating structure can be created not only in the direction normal to the film plane but also in the plane of the sample. However, a preparation of magnetically, structurally as well as electrically different planar objects requires complicated lithographic operations. T. Toli´ nski et al.: Epitaxial MnAs Films Studied by Ferromagnetic and Spin Wave Resonance, Lect. Notes Phys. 678, 97–109 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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An excellent alternative is provided by MnAs films grown epitaxially on GaAs(001) substrates. In the temperature range of 10-40◦ C these films spontaneously create self-organized stripe domains [1, 2]. Each hexagonal ferromagnetic metallic stripe (α-phase) is adjacent to orthorhombic and paramagnetic neighbors (β-phase). In the case of bulk MnAs there exists only a direct α-β-phase transition at 40◦ C [3] followed by a β-γ-phase transition at 125◦ C, where the γ denotes a paramagnetic hexagonal MnAs phase. In contrast to the situation within bulk MnAs for thin films of MnAs on GaAs(001) substrates the structural α-β-phase transition proceeds through the intermediate coexistence region (Fig. 1), which is favored for the accommodation of the stress between the film and the substrate [4]. The elastic domains, the structural and magnetic topography, and static magnetization have already been widely studied [1, 2, 4, 5, 6, 7, 8, 9]. A powerful tool for investigations of the magnetic anisotropy, magnetic interactions as well as for identification of divers phases is ferromagnetic resonance (FMR) and its generalization for non-uniform excitations known as spin-wave resonance (SWR). In this paper we will use the advantages of the FMR and SWR technique for the studies of MnAs/GaAs(001) films. 10°C

40°C

FM  phase

PM  phase

bulk MnAs

MnAs film on GaAs

striped  Fig. 1. Illustration of the α-β-phase transition at 40◦ C for bulk MnAs (top) and for a thin film (bottom)

2 Basic FMR/SWR Formalism In this section we introduce the basic theoretical formalism of FMR and SWR methods, which will be employed for the analysis of the experimental results. The standard Landau-Lifshitz equation of motion for a magnetization vector precessing around the direction of the magnetic field is given by dM = −γM × Heff dt

(1)

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where γ = gµB /
. If only the resonance frequency is of interest (1) can be equivalently written within the free energy formalism by [10].    2 2 Fθθ Fϕϕ − Fθϕ ω = (2) γ M 2 sin2 θ The free energy F is expressed in the spherical coordinate system and for the case of our films becomes [11] F = − HM (sin θ sin θH cos (ϕ − ϕH ) + cos θ cos θH )   Nb 2 Na 2 2 − K2⊥ − M cos θ − K2 − M sin2 θ cos2 ϕ 2 2

(3)

The respective angles are marked on the sketch in Fig. 2. Na and Nb are the demagnetizing factors along the a and b axis. Inserting (3) into (2) yields the resonance equation for the polar angular dependency  2    ω  ⊥ = H cos (θ − θH ) − Meff − Meff cos 2θ γ      ⊥ × H cos (θ − θH ) − Meff − Meff cos2 θ − Meff

(4)

assuming ϕH = 0◦ . Simultaneously, the equilibrium condition Fθ = 0 (free energy derivation) has to be fulfilled, i.e. H sin (θ − θH ) =

 1 ⊥  Meff − Meff sin 2θ 2

(5)

Analogously, for the in-plane angular dependency (θH = 90◦ ) one gets: [0001]

[1120]

b

[1100]





 

c







a



Fig. 2. Sketch of the stripe pattern and the coordinate system

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 2  ω  = H cos (ϕ − ϕH ) − Meff cos 2ϕ γ    M ⊥ − eff (1 + cos 2ϕ) × H cos (ϕ − ϕH ) + Meff 2

(6)

with the equilibrium condition Fϕ = 0 in the form 

H sin (ϕ − ϕH ) =

Meff sin 2ϕ 2

(7)



⊥ where Meff = Nb M − 2K2⊥ /M and Meff = Na M − 2K2 /M . If the thickness of a thin film is of the order of the spin wave length and pinning occurs at the film surface then instead of the uniform mode a series of lines can be observed in the resonance spectrum [12, 13]. These lines correspond to standing spin waves of wave number k = nπ/L. Provided that a uniform microwave field is used to excite the spin waves a net absorption and thus a resonance signal occurs only for odd n. The spin wave modes are observed close to the normal direction of a film, for θ < θcrit . The general (2) has to be rebuilt by adding the exchange energy term [14]:

 2    2 1 ω 1 1 2 2 F F F = + Dk + Dk (8) × − ϕϕ θθ θϕ γ M sin2 θ M M sinθ which leads to the modified (4)  2    ω  ⊥ = H cos (θ − θH ) − Meff − Meff cos 2θ + Dk 2 γ      ⊥ × H cos (θ − θH ) − Meff − Meff cos2 θ − Meff + Dk 2

(9)

where D is the spin wave stiffness constant connected with the exchange constant by D = 2A/M .

3 Phase Transitions in MnAs Studied by FMR The transition between the α- and the coexisting α-β-phase at 10◦ C entails a transformation from a pseudo-two dimensional thin film to pseudo-one dimensional stripes. As the magnetic resonance field depends strongly on the shape of the magnetic object, it is clear that this geometrical collapse should be easily detectable by FMR experiments. In Fig. 3 exemplary resonance spectra collected close to the lower phase transition and a temperature dependence of the resonance field Hres in the range from −40◦ C to 40◦ C are displayed. The measurements are carried out for θ = 0, i.e., at the normal direction, at a microwave frequency of 9 GHz. It is at once visible that at the

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Fig. 3. FMR measurements: (a) Spectra close to the transition between the αphase and the striped α-β-phase. (b) Temperature dependence of the resonance field measured in the direction normal to the film (hard direction)

lower-temperature phase transition a sharp decrease of Hres occurs. It can be quantitatively explained by considering the correct demagnetizing factors just before and after the drop [15]. Aharoni et al. [16] derived the formulas for the demagnetizing factors of a prism characterized by a width a, length c and height b. For θ = 0 (4) simplifies to  2   
 ω  ⊥ ⊥ = H − Meff − Meff · H − Meff γ

(10)

Below the transition at 10◦ C, Na = Nc = 0 and Nb = 4π, which is typical for thin films. Above this transition a finite stripe width of about 300 nm was measured by magnetic force microscopy (MFM). The respective demagnetizing factors are Na = 0.16·4π, Nc = 0, and Nb = 0.84·4π [16, 11]. Substituting these values into (10) provides the correct upper and lower resonance field at the vicinity of 10◦ C. It is visible from Fig. 3a and the spectra in Fig. 3b that between 12◦ C and 15◦ C two resonance lines are present. The lower field line is ascribed to the striped α-phase and the upper one is connected with a remaining contribution of the primary non-striped α-MnAs film. For higher temperatures only one line remains with its resonance position approaching the paramagnetic limit (the value for which all effective fields disappear and the resonance

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field is determined only by the frequency and the gyromagnetic ratio γ, i.e. ω/γ = Hres with temperature approaching Tc = 40◦ C), where the α-phase diminishes. In general the resonance field of a magnetic specimen is determined not only by the employed frequency and γ but also by the inherent anisotropies present within the sample. In the case of strong anisotropy fields it may easily happen that the resulting Hres value is not accessible with the electromagnet used and thus the signal of the magnetic sample or a magnetic phase is not detectable. Therefore, a good solution is to test various microwave frequencies. Apart from the measurements at 9 GHz shown in Fig. 3 the MnAs films have been also studied at 35 GHz. The higher frequency implies that in the normal direction the FMR line observed at 9 GHz shifts to higher magnetic fields and is above the accessible field of the magnet below room temperature. However, a different line appears at low fields. In the inset of Fig. 4a a spectrum obtained at 23◦ C is displayed showing the low-field line disappearing at H = 0 and the high-field line appearing with increasing temperature in the accessible range of the magnetic field. Figure 4a presents the temperature dependence of the low-field line. Like at 9 GHz a well pronounced drop in

Fig. 4. FMR line of grains with easy direction of magnetization pointing out-ofplane. (a) Temperature dependence of the resonance field measured in the direction normal to the film. Inset: An example of a spectrum showing the disappearing low resonance line (due to the easy out-of-plane inclusions direction) and the main resonance appearing at high fields (hard out-of-plane direction) (b) Temperature dependence of the FMR linewidth ∆Hpp

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the Hres (T ) dependence is noticeable at the phase transition at 10◦ C. The paramagnetic limit for 35 GHz is 12.5 kOe. Since the line is below it, this implies that the low-field FMR signal corresponds to the easy axis of the involved phase. As the external field is oriented normal to the film plane within the measurement displayed in Fig. 4a, the behavior of the low-field signal is opposite to the one of the main magnetic phase of the sample exhibiting an in-plane easy direction of magnetization (see the angular dependence of the high-field line analyzed in the next section). In turn this observation implies that there is a secondary phase with an easy axis oriented out-of-plane. A previous observation of the presence of this out-of-plane component of magnetization has already been reported by Ney et al. [17] using magnetometric investigations. It is probably connected with Mn inclusions within the film. It was also proposed to employ these features of the MnAs films in magnetologic devices [18]. A surprising behavior in the Hres (T ) dependence is shown in Fig. 4a, namely, the resonance field of the low-field mode is decreasing with temperature. It means that the anisotropy is increasing with temperature. This atypical behavior can be explained by two effects. First of all, this secondary phase can be coupled to the main in-plane oriented phase and, hence, is not perfectly aligned in the normal direction. With temperature the coupling is decreasing, therefore, the tilted out-of-plane magnetization turns into the normal orientation resulting in the observed decrease of the resonance field. Secondly, the grains characterized by their out-of-plane easy axis may become thinner with temperature in the planar dimensions leading to the increased value of the anisotropy due to shape effects. It is visible in Fig. 4b that not only the resonance field but also the linewidth ∆Hpp is sensitive to the phase transition. However, after the drop at 10◦ C ∆Hpp grows with increasing temperature in a typical way by the reason of spin fluctuations.

4 Magnetic Anisotropy in MnAs In the previous section we showed how the magnetic phase transitions and the various secondary phases can be investigated with ferromagnetic resonance. However, a quantitative analysis, e. g. based on (10), requires the values of the involved anisotropies as input parameters. They can be determined very precisely from the measurements of the angular dependence of the ferromagnetic resonance. In Fig. 5 such an angular dependence for the main resonance mode (main magnetic phase) is presented for the polar and in-plane measurements. The solid lines represent a fit according to (4) and (6), which provide  ⊥ = −14 kOe and Meff = −20 kOe. the values of the effective fields Meff Using the saturation magnetization value from SQUID measurements M = 680 emu/cm3 and Na = 0.16 · 4π, Nc = 0, Nb = 0.84 · 4π the anisotropy constants K2⊥ = 7.2 × 106 erg/cm3 and K2 = 7.3 × 106 erg/cm3 are obtained [11]. These constants are close to the value for bulk MnAs crystals

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Fig. 5. Angular dependences of the main resonance line measured at a frequency of 35 GHz for 57 nm thick MnAs films at the ambient temperature, i.e., at the coexistence region of the α- and β-stripes

[19]. Again, the choice of the frequency, being 35 GHz in the case of Fig. 5, was a crucial point because a different frequency band would not provide the nearly full angular dependence (from the easy to the hard magnetization direction).

5 Inter- and Intra-Stripe Coupling in the MnAs Films So far we have shown how to use FMR for identification of the magnetic phase transition and extracting the values of the inherent anisotropies of the system. However, there are even more fundamental properties which can be investigated with the FMR technique. First, the question arises what the driving force behind the ferromagnetism of the α-stripes is. Second, it is expected that there should be an interaction between the ferromagnetic αstripes, especially close to the transition temperature where the paramagnetic β-phase already starts to form although the distance between the α-stripes is still small. It is also of interest what the result of the competition between the intrinsic intra-stripe and the additional inter-stripe coupling is. Concerning the interaction between the stripes a useful analogy can be borrowed from the multilayer systems. It is well established [20] that in alternatively stacked ferromagnetic and nonmagnetic layers an interlayer exchange interaction is present, which can be determined by FMR. Due to the interlayer coupling a secondary resonance line is excited in the FMR experiment, which is characterized by an out-of-phase precession of the magnetizations

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(optical mode) in the ferromagnetic films in contrast to the main resonance line with the in-phase precession of magnetizations (acoustical mode). The separation between the resonance modes is directly related to the strength of the coupling. In the current MnAs/GaAs(001) films a similar but planar stacking occurs. For the striped α-phase in the in-plane easy direction of magnetization it is possible to observe a small secondary mode (Fig. 6a). The separation of the resonance fields for the two modes is decreasing with temperature (Fig. 6b) supporting the possibility of the existence of the inter-stripe coupling which ought to decrease with temperature because of the increasing distance between the ferromagnetic α-stripes (increasing stripe-width of the paramagnetic β-phase). Furthermore, it is found that the difference between the resonance fields of the modes is smaller for thicker MnAs films exhibiting a larger separation of the stripes than for the thinner samples. In the case of the multilayer systems the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction causes the interlayer coupling via the non-magnetic spacer and disappears at about 5 nm of spacer thickness. In the striped MnAs the coupling is observed even for more than 100 nm-wide β-stripes. Therefore, a dipolar coupling is the most plausible source of the inter-stripe correlations. A magnetically similar system, i.e., stripes with magnetic moments ordered

Fig. 6. (a) In-plane FMR measurements showing two modes observed in the easy direction (b) Temperature dependence of the resonance fields of the low-field (squares) and high-field (circles) mode. The decreasing separation (decreasing interstripe coupling) between the modes is visible due to the increasing fraction of the paramagnetic β-phase

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in-plane but perpendicularly to the stripe direction, was studied by Pratzer and Elmers [21]. They considered monolayer-thick Fe stripes on a tungsten substrate and estimated the strength of the dipolar field assuming that the field created by one infinite line of atoms is µ/(2πar2 ) and the sum over W lines (indices i, j) and neighboring stripes (index n) can be expressed by Hdip =

∞ W W 1 1 µ 1  π a3 W n=1 i=1 j=1 (nW0 + j − i)2

(11)

where W0 −W is the number of atomic lines between the ferromagnetic stripes and a is the lattice constant. The application of this approximation to our system leads to Hdip = 2.4 Oe but owing to the large thickness of the MnAs stripes it is expected that the real value most likely is much higher. Now we proceed to the question concerning the nature of the intra-stripe coupling responsible for the interaction between the spins within the αstripes. The answer is provided by the SWR experiment already described in Sect. 2. Figure 7 shows the spin wave spectra collected at θ = 10◦ , i.e. close to the normal direction, at various temperatures. The excitation of the standing spin waves requires a pinning at the surface of the film. In the studied MnAs films the strong pinning of the surface spins is ensured by the natural oxidation of the surface. From the fit of the resonance fields of the spin wave modes Hn (Fig. 8a) to (9) anisotropy constants are obtained which

Fig. 7. Spin wave spectra collected at several temperatures close to the normal direction

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are in good agreement with those extracted from the angular dependencies analyzed in Sect. 4. However, the key parameter derivable from Fig. 8 is the stiffness constant D = 2A/M with A = 17.7 × 10−10 erg/cm [22]. In addition, we have measured the temperature dependence of D, which is plotted in Fig. 8b.

Fig. 8. (a) Dependence of Hn on the square of the spin wave number n2 [22]. The fit is performed with (9). (b) Temperature dependence of the stiffness constant D

The field equivalence of A is 2Aπ/M L2 = 24 Oe showing that the coupling within the stripes is rather weak. Therefore, a double exchange mechanism can be responsible for the ferromagnetism of the MnAs films. From comparison with the estimation Hdip  2.4 Oe one deduces that these two interactions can compete depending on the temperature and the thickness of the MnAs films.

6 Conclusions Employing FMR we have investigated the magnetic phase transitions at 10◦ C and 40◦ C in MnAs films grown on GaAs(001). Apart from the main phase with the in-plane easy axis we identify a secondary phase with out-of-plane easy direction of magnetization.

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The magnetic anisotropy can be precisely determined by FMR. The effect of the shape anisotropy of the stripes was taken into consideration providing Na = 0.16 · 4π, Nc = 0, Nb = 0.84 · 4π for the demagnetizing factors and K2⊥ = 7.2 × 106 erg/cm3 , K2 = 7.3 × 106 erg/cm3 for the anisotropy constants. Inter-stripe coupling has been observed with FMR and ascribed to the ferromagnetic dipolar interaction between stripes. The intra-stripe exchange constant extracted from spin wave resonance measurements was small, thus supporting that a double exchange mechanism can be responsible for the ferromagnetism of the α-phase.

Acknowledgment The work was supported by the DFG (Sfb 290, TP A2, A5).

References 1. V.M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L.D¨ aweritz, K.H. Ploog: Phys. Rev. Lett. 85, 341 (2000) 98 2. T. Plake, M. Ramsteiner, V.M. Kaganer, B. Jenichen, M. K¨ astner, L.D¨ aweritz, K.H. Ploog: Appl. Phys. Lett. 80, 2523 (2002) 98 3. C. Guillaud: J. Phys. Radium 12, 223 (1951) 98 4. A.K. Das, C. Pampuch, A. Ney, T. Hesjedal, L. D¨ aweritz, R. Koch, K. H. Ploog: Phys. Rev. Lett. 91, 087203 (2003) 98 5. F. Schippan, G. Behme, L. D¨ aweritz, K.H. Ploog, B. Dennis, K.-U. Neumann, K.R.A. Ziebeck: J. Appl. Phys. 88, 2766 (2000) 98 6. V.M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L. D¨ aweritz, K.H. Ploog: Phys. Rev. Lett. 85, 341 (2000) 98 7. T. Plake, T. Hesjedal, J. Mohanty, M. K¨ astner, L. D¨ aweritz, K.H. Ploog: Appl. Phys. Lett. 82, 2308 (2003) 98 8. M. Tanaka, J.P. Harbison, M.C. Park, Y.S. Park, T. Shin, G.M. Rothberg: J. Appl. Phys. 76, 6278 (1994) 98 9. F. Schippan, A. Trampert, L. D¨ aweritz, K.H. Ploog: J. Vac. Sci. Technol. B 17, 1716 (1999) 98 10. J. Smit, H.G. Beljers: Phillips Res. Rep. 10, 113 (1955) 99 11. J. Lindner, T. Toli´ nski, K. Lenz, E. Kosubek, H. Wende, K. Baberschke, A. Ney, T. Hesjedal, C. Pampuch, R. Koch, L. D¨ aweritz, K.H. Ploog: J. Magn. Magn. Mater. 277, 159 (2004) 99, 101, 103 12. H. Puszkarski: Acta Phys. Polon. A 38, 217 (1970) 100 13. H. Puszkarski: Acta Phys. Polon. A 38, 899 (1970) 100 14. L.J. Maksymowicz, H. Jankowski: J. Magn. Magn. Mater. 109, 341 (1992) 100 15. A. Ney, T. Hesjedal, C. Pampuch, A.K. Das, L. D¨ aweritz, R. Koch, K.H. Ploog, T. Toli´ nski, J. Lindner, K. Lenz, K. Baberschke: Phys. Rev. B 69, 081306(R) (2004) 101 16. A. Aharoni: J. Appl. Phys. 83, 3432 (1998) 101

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17. A. Ney, T. Hesjedal, C. Pampuch, J. Mohanty, A.K. Das, L. D¨ aweritz, R. Koch, K.H. Ploog: Appl. Phys. Lett. 83, 2850 (2003) 103 18. C. Pampuch, A.K. Das, A. Ney, L. D¨ aweritz, R. Koch, K.H. Ploog: Phys. Rev. Lett. 91, 147203 (2003) 103 19. R.W. De. Blois, D.S., Rodbell: Phys. Rev. 130, 1347 (1963) 104 20. J. Lindner, K. Baberschke: J. Phys. Condens. Matter 15, S465 (2003); J. Phys. Condens. Matter. 15, R193 (2003) 104 21. M. Pratzer, H.J. Elmers: Phys. Rev. B 66, 033402 (2002) 106 22. T. Toli´ nski, J. Lindner, K. Lenz, K. Baberschke, A. Ney, T. Hesjedal, C. Pampuch, L. D¨ aweritz, R. Koch, K.H. Ploog: Europhys. Lett., 68, 726 (2004) 107

Part II

Diluted Magnetic Semiconductors

First-Principles Study of the Magnetism of Diluted Magnetic Semiconductors L.M. Sandratskii and P. Bruno Max-Planck Institut f¨ ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany, [email protected] [email protected] Abstract. We report the density-functional-theory calculations of the exchange interactions and Curie temperature for a series of III-V and II-VI diluted magnetic semiconductors. We focus the discussion on the role of the holes in the establishing the ferromagnetic order in various systems. We suggest a method of the quantitative characterization of the properties of the holes. It is shown that there are two conflicting properties of the holes – delocalization from impurity and p–d interaction – whose combination determines the hole influence on the Curie temperature. We demonstrate that Hubbard U increases delocalization of the holes and decreases the strength of p-d interaction. Depending on the system these competing trends can lead to both increase and decrease of the Curie temperature. We show that high value of the Curie temperature for low impurity concentration is possible only in the case of substantial admixture of the impurity 3d states to hole states.

1 Introduction An important current problem towards practical use of the spin-transport in semiconductor devices is the design of the materials that make possible the injection of spin-polarized electrons into a semiconductor at room temperature. One of the promising classes of materials are diluted magnetic semiconductors (DMS). A strong interest to these systems was attracted by the observation [1] in Ga0.947 Mn0.053 As of the ferromagnetism with the Curie temperature (TC ) as high as 110 K. To design materials with Curie temperature higher than room temperature the knowledge of physical mechanisms governing the exchange interactions in DMS is of primary importance. Despite much work devoted to the study of DMS the situation is still controversial from both experimental and theoretical points of view. The complexity of the experimental situation is well illustrated by the case of (GaMn)N. After the theoretical prediction of high-temperature ferromagnetism in (GaMn)N by Dietl et al. [2] this system was produced and investigated in many experimental groups. The spectrum of the magnetic states reported ranges from paramagnetic and spin-glass states to the ferromagnetism with extremely high TC of 940 K [3, 4, 5]. A further proof of the complexity of the magnetism of (GaMn)N was given by recent magnetic circular dichroism

L.M. Sandratskii and P. Bruno: First-Principles Study of the Magnetism of Diluted Magnetic Semiconductors, Lect. Notes Phys. 678, 113–130 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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measurements by Ando [6]. The measurements were performed on a highTC sample and led Ando to the conclusion that the (GaMn)N phase in this sample is paramagnetic. The ferromagnetism of the sample was claimed to come from an unidentified phase. On the other hand, in a recent preprint by Giraud et al. [7] a high temperature ferromagnetism was detected in samples with a low Mn concentration of about 2%. The authors rule out the presence of precipitates in the system and argue that the room temperature ferromagnetism is an intrinsic property of (GaMn)N. Likewise, the theoretical situation is not uniform. The theoretical works on the ferromagnetism in the DMS systems can be separated into two groups. The first group models the problem with an effective Hamiltonian containing experimentally determined parameters. This part of the studies was recently reviewed in [8, 9, 10] Different assumptions concerning the energy position and the role of the 3d sates of the magnetic impurities are formulated. Different types of the exchange interactions – potential exchange, kinetic exchange, superexchange, double exchange, RKKY exchange, Blombergen-Rowland exchange and others – are invoked to describe the properties of the system. Most of the model-Hamiltonian studies agree with each other in that the presence of holes plays important role in ferromagnetism of DMS. The second group of theoretical studies is based on the parameter-free calculations within the density functional theory (DFT) (see, e.g., reviews [11, 12] and more recent work [13, 14, 15, 16, 17]). The DFT calculations intend to treat realistic systems in their detailed complexity. In particular, every electron state is involved in effective exchange interaction with all other electron states [18]. Therefore different types of exchange interactions intermix and influence each other. The complexity of the DFT picture reflects the complexity of the real systems. To gain deeper qualitative understanding of the magnetism of the system it is, however, very useful to relate the DFT results to the results of the model-Hamiltonian treatments attempting to single out the leading exchange mechanisms responsible for the establishing of the long range magnetic order. In this paper we report DFT calculations for a number of III-V and IIVI DMS. To develop a general platform for discussion of various systems we focus on the role of the holes in the magnetism of DMS. First, we reexamine the importance of the holes for establishing the long-range ferromagnetic order. Then we discuss the properties of the holes that are relevant for the ferromagnetic ordering and consider quantitative characterization of these properties. We investigate the influence of the on-site Coulomb interaction on the properties of the holes. We study the connection between the combination of the hole features, on the one hand, and the effective exchange interactions between the magnetic impurities and the Curie temperature, on the other hand. The main body of calculations is performed within the local density approximation (LDA) to the DFT. An important issue in the physics of DMS

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is the role of the Coulomb correlations of the 3d electrons. The so-called LDA+U method introduces explicitly the on-site Coulomb interaction (Hubbard U ) and leads to a better description of many systems with strong Coulomb correlations [19]. However, the superiority of the LDA+U method is not universal. For numerous systems with 3d atoms the LDA gives better agreement with experiment. In the paper, we report both LDA and LDA+U calculations for a number of DMS. The influence of Hubbard U on various electron properties of the systems is analyzed. Where possible the results of both calculations are compared with experiment to establish which of two approaches provides better description of the DMS studied.

2 Calculational Technique The calculational scheme is discussed in [13, 14, 20] to which the reader is referred for more details. The scheme is based on DFT calculations for supercells of semiconductor crystals with one cation atom replaced by a Mn atom. The size of the supercell determines the Mn concentration. Most of the calculations are performed for the zinc-blende crystal structure of the semiconductor matrix. To calculate the interatomic exchange interactions we use the frozenmagnon technique and map the results of calculation of the total energy of the helical magnetic configurations en = (cos(q · Rn ) sin θ, sin(q · Rn ) sin θ, cos θ) onto a classical Heisenberg Hamiltonian  Jij ei · ej Hef f = −

(1)

(2)

i=j

where Jij is an exchange interaction between two Mn sites (i, j) and ei is the unit vector pointing in the direction of the magnetic moment at site i, Rn are the lattice vectors, q is the wave vector of the helix, polar angle θ gives the deviation of the moments from the z axis. A deviation of the atomic moments from the parallel directions causes a change of the atomic exchange-correlation potentials [21] which leads to a perturbation of the electron states. The value of the perturbation of a given state depends on other states of the system, both occupied and empty, since these states enter the secular matrix of the problem. Within the Heisenberg model (2) the energy of frozen-magnon configurations can be represented in the form E(θ, q) = E0 (θ) − sin2 θJ(q)

(3)

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where E0 does not depend on q and J(q) is the Fourier transform of the parameters of the exchange interaction between pairs of Mn atoms:  J0j exp(iq · R0j ) . (4) J(q) = j=0

Performing back Fourier transformation we obtain the parameters of the exchange interaction between Mn atoms: J0j =

1  exp(−iq · R0j )J(q) . N q

(5)

The Curie temperature is estimated in the mean-field (MF) approximation kB TCM F =

2 J0j 3

(6)

j=0

We use a rigid band approach to calculate the variation of exchange parameters and Curie temperature with respect to electron occupation. We assume that the electron structure calculated for a DMS with a given concentration of the 3d impurity is basically preserved in the presence of defects. The main difference is in the occupation of the bands and, respectively, in the position of the Fermi level. In the LDA+U calculations we use U = 0.3 Ry that corresponds to the value determined experimentally [22].

3 Single Band in the Frozen-Magnon Field Pursuing the aim to place the role of the holes in the focus on the discussion we begin with the consideration of a simple model of one spin-degenerate band in a helical exchange field. Such a field corresponds to the exchange field originating from the frozen magnon. We discuss the role of electrons in almost empty band and holes in almost filled band in establishing the ferromagnetic order. The frozen magnons are defined by (1). The exchange field acting on the band is taken in the form bn = ∆ en where ∆ is the amplitude of the field. The secular matrix of a tight-binding model takes in this case the form  2 θ cos ( 2 )H− + sin2 ( θ2 )H+ − ∆ − 12 sin θ(H− − H+ ) 2 (7) − 12 sin θ(H− − H+ ) sin2 ( θ2 )H− + cos2 ( θ2 )H+ + ∆ 2 where H− = H(k − 12 q), H+ = H(k + 12 q), and H(k) describes spindegenerate bands of a non-magnetic crystal.

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The eigenvalues of the matrix (7) have the form 1 (H− + H+ ) 2 1 1 1 1 ±( (H− − H+ )2 − ∆ cos θ(H− − H+ ) + ∆2 ) 2 4 2 4

ε± (k) =

(8)

and are illustrated in Fig. 1.

q

energy (abit. units)

4

3

ferro magnon 2

1

0

-0.4

-0.2

0

0.2

0.4

k/2π

Fig. 1. The energy bands (thick solid line) of the spiral structure with q = 0.8π, ∆ = 2, and θ = 45◦ . H(k) = 1 − cos(k). The lattice parameter is assumed to be unity. The thin solid lines show the bands of a ferromagnetic configuration. The broken lines give the ferromagnetic bands shifted in the reciprocal space according to the given q

For completely filled bands the total energy

1 2 dk[(ε− (k) + ε+ (k)] = dkH(k) Eb = ΩBZ BZ ΩBZ BZ

(9)

does not depend on magnetic configuration. [ΩBZ is the volume of the Brillouin zone (BZ)]. This means that the kinetic exchange taken into account by (7) does not contribute into effective intersite exchange interaction in the case of completely filled bands. For almost empty and almost filled bands the minimum of the total energy always corresponds to the ferromagnetic configuration. For a detailed proof of this statement the reader is referred to [14]. Figure 1 illustrates that under the influence of the frozen magnon the bands become narrower: The minimal electron energy of the ferromagnet is lower than the minimal electron energy of the spiral and the maximal electron energy of the ferromagnet is higher than the maximal energy of the spiral. In combination with (9) this is sufficient to prove that small number of electrons or small number of holes lead to the ferromagnetic ground state.

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Thus within the model of a single spin-degenerated band in a frozenmagnon exchange field the presence of holes makes the ferromagnetic structure favorable. In the following sections we show that the importance of the holes for establishing ferromagnetic order is preserved in the full–scale DFT calculations.

4 Results for (GaMn)As,(GaCr)As,(GaFe)As Next we consider calculations for three III-V DMS: (GaMn)As, (GaMn)As and (GaFe)As. Figure 2 presents the DOS of the systems for the impurity concentration of 3.125%. For comparison the GaAs DOS is shown. In the case of (GaMn)As the replacement of one Ga atom in the supercell of GaAs by a Mn atom does not change the number of spin-down states in the valence band. In the spin-up chanel there are, however, five additional energy bands which are related to the Mn 3d states. Since there are five extra energy bands and only four extra valence electrons (the atomic configurations of Ga and Mn are 4s2 4p1 and 3d5 4s2 ) the valence band is not filled and there appear unoccupied (hole) states at the top of the valence band. There is exactly one hole per Mn atom. In (GaCr)As, the Cr 3d states assume a higher energy position relative to the semiconductor-matrix states than the Mn 3d states in (GaMn)As. As a result, the spin-up impurity band is separated from the valence band and lies in the semiconducting gap of GaAs. This impurity band contains three energy bands. Since Cr has one electron less than Mn, only one third of the impurity band is occupied: the occupied part contains one electron per Cr atom and there is place for further two electrons.

DOS(states/Ry)

1e/Cr 1h/Mn (GaFe)As (GaCr)As (GaMn)As

DOS 3.125% GaAs

20 0 -20 -0.4 -0.2 0

-0.4 -0.2 0

-0.4 -0.2 0

-0.4 -0.2 0

E(Ry) Fig. 2. The spin-resolved DOS for (GaMn)As, (GaCr)As, (GaFe)As. For comparison the DOS of GaAs is shown. The concentration of Mn is 3.125%. Upper part of the graph shows spin-up DOS. The inserts zoom the important energy regions about the Fermi level. In (GaMn)As there is one hole per Mn atom. In (GaCr)As there is one electron per Cr atom in the impurity band

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MF-Curie temperature (K)

In (GaFe)As the number of valence electrons is by one larger than in (GaMn)As. Simultaneously, the Fe 3d states assume lower energy position with respect to the semiconductor matrix than Mn 3d states. As a result, the spin-down Fe 3d states become partly occupied. There are carriers in both spin-channels. In Fig. 3 we show calculated mean-field Curie temperatures. For (GaMn)As and (GaCr)As we obtained rather similar results at least for realistic impurity concentrations below 15%. In the case of (GaFe)As there is a clear trend to antiferromagnetism.

GaCrAs

400

GaMnAs

200 0 -200

GaFeAs -400 0

0.1

x

0.2

Fig. 3. The mean-field Curie temperature. The negative values of the Curie temperature reveal the prevailing antiferromagnetic exchange interactions and instability of the ferromagnetic state

(GaCr)As and (GaFe)As are not yet sufficiently studied experimentally. For (GaMn)As our result of high Curie temperature for low Mn concentrations is in good correlation with experiment. Our estimation of the Curie temperature is substantially higher than the early experimental value of 110 K for Ga0.947 Mn0.053 As. However, later measurements [23] on samples with less uncontrolled defects gave higher Curie temperature of 150 K that is closer to our estimation. An important issue in DMS on the GaAs basis is the presence of donor defects, in particular AsGa antisites, compensating part of the holes. Thus the number of holes per Mn atom is usually smaller than one. To study this effect we performed calculations for varied band occupation. An increase of band occupation leads to a decrease of the number of holes and vice versa. The results are shown in Fig. 4. In the case that all energy bands are either completely filled or empty [these are the point of n = 1 for (GaMn)As and the points n = −1 and n = 2 for (GaCr)As (Fig. 4)] the antiferromagnetic interatomic exchange interactions always prevail. This agrees with the picture of Anderson’s antiferromagnetic superexchange. In (GaMn)As there is a clear trend to decreasing TC for hole number less than the nominal value of one.

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(GaMn)As

Curie Temperature (K)

400

12.5%

(GaFe)As

(GaCr)As

12.5%

12.5%

200

6.25% 3.125%

6.25% 6.25%

0

3.125%

3.125%

-200 -400 -1

0

1

-1

0

1

electron number, n

-1

0

1

Fig. 4. The Curie temperature as a function of the electron number. n is the number of the excess electrons (or missing electrons for negative values) per magnetic impurity atom. The nominal electron number corresponds to n = 0. In all cases of completely filled electron bands (kinks on the curves) the prevailing exchange interactions are antiferromagnetic 500

Curie Temperature (K)

GaMnAs

n=-0.4

400

n=0.0

300

n=0.2

200

* experiment

n=0.4

100

n=0.6

0

n=0.8 -100 0

0.1

0.2

Mn concentration Fig. 5. The Curie temperature of (GaMn)As as a function of Mn concentration for different numbers of holes. The experimental values are taken from [24]

In Fig. 5 we show the Curie temperature as a function of Mn concentration for different numbers of holes per Mn atom. For the case of n = 0.6 (that is 0.4 hole per Mn atom) the calculated Curie temperatures are close to the experimental values. In (GaCr)As the situation is different. Because of a decreased number of valence electrons the impurity band can accept two further electrons per Cr atom and is far from the full occupation. As a result the donor defects do not

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exert in this case the strong influence obtained for (GaMn)As: One compensating electron makes (GaMn)As antiferromagnetic whereas the Curie temperature of (GaCr)As changes only weakly. On the other hand, the acceptor defects influence strongly the Curie temperature of (GaCr)As: a decreasing number of valence electrons below the nominal value marked by n = 0 in Fig. 4 leads to a fast decrease of the interatomic exchange parameters resulting in the change of the sign of the parameters and in the transition from ferromagnetic to antiferromagnetic exchange interactions. At n = −1 the impurity band becomes empty that corresponds to the strongest antiferromagnetic interactions (minimum of TC in Fig. 4). The result for (GaFe)As differ drastically from the results for both (GaMn)As and (GaCr)As. Here, already for nominal electron number (n = 0) the antiferromagnetic interactions become stronger than ferromagnetic. The reasons for this are a small number of the holes in the spin-up chanel and a very narrow impurity band in the spin-down chanel. Summarizing, our calculations show that the partial occupation of the energy bands is essential for the formation of ferromagnetic order.

5 (ZnCr)Te The calculational technique discussed above is universal and can be applied to both III-V and II-VI DMS. In this Section we discuss the study of the II-VI DMS (ZnCr)Te. A strong interest to this system is attracted by the recent experiment by Saito et al [25] who detected the ferromagnetism of Zn0.8 Cr0.2 Te with the Curie temperature of 300 K, together with magnetooptical evidence in favor of an intrinsic (i.e., hole mediated) mechanism of ferromagnetism. The cations in II-VI semiconductors have one additional valence electron compared to the cations in III-V systems. Therefore the replacement of a cation atom by a Mn atom is not expected to result in partially filled bands. The situation changes in the case of Cr doping since one Cr atom has one valence electron less than one Mn atom. We performed calculations for Zn0.75 Cr0.25 Te [26] and found a number of features similar to the properties of (GaMn)As. The calculated DOS is shown in Fig. 6. The system is half-metallic. There is exactly one hole per Cr atom in the spin-up valence band. The calculated Curie temperature is shown in Fig. 7. The decreasing number of holes leads to a transition from ferromagnetic to antiferromagnetic exchange interactions. As expected, the doping with Mn does not result in partially filled bands. Correspondingly (ZnMn)Te is antiferromagnetic for nominal number of valence electrons. Remarkable that TC (n) curves for (ZnCr)Te and (ZnMn)Te are similar with the major difference consisting in the shift by one along the axis of band occupation. This result shows the validity of the rigid band model for qualitative considerations.

L.M. Sandratskii and P. Bruno

DOS(st/Ry)

122

20

0

-20

(ZnMn)Te

(ZnCr)Te -0.4

-0.2

0

0.2 -0.4

-0.2

ZnTe 0

0.2 -0.4

-0.2

0

0.2

E(Ry)

Fig. 6. The DOS of Zn0.75 Cr0.25 Te and Zn0.75 Mn0.25 Te. For comparison the DOS of ZnTe is shown 600

TC

TC(K)

300

(ZnCr)Te

0

-300

-2

(ZnMn)Te

-1

0

1

2

n Fig. 7. The Curie temperature of (ZnCr)Te and (ZnMn)Te as a function of the electron number

6 Properties of the Holes and Magnetism In previous sections we have shown the crucial role of the partially filled bands in the ferromagnetism of III-V and II-VI DMS. To gain a deeper insight in the mechanisms of the formation of ferromagnetism we take a closer look at the properties of the holes relevant to the establishing the long-range order. Two features of the holes are important: the delocalization from the impurity, on the one hand, and the interaction between the hole and the magnetic impurity (p–d interaction), on the other hand. Figure 8 illustrates schematically the necessity of both components. For comparison of different systems it is important to characterize the properties of the holes quantitatively. The numerical characterization of the hole delocalization is rather straightforward: The spatial distribution of the hole can easily be obtained within DFT calculations on the basis of the known wave functions of the electron states. A quantitative characterization of the p–d interaction is less straightforward. In the mean-field approximation, the parameter N β describes the spin-splitting of the semiconductor valence-band states

First-Principles Study of the Magnetism of DMS Delocalization Polarization

123

Ordering

−

+

−

+

−

−

+

+

+

Fig. 8. Both delocalization of the holes and the exchange interaction hole-impurity are crucial for establishing ferromagnetic order. The upper line shows schematically the holes localized about the impurity atoms. The holes are strongly spin-polarized but do not mediate the exchange interaction between impurities. In the second line, the holes are not disturbed by the impurities and are completely delocalized. They, however, are not spin-polarized by the impurities and also cannot mediate the exchange interaction. The third line presents an intermediate situation: The hole states are disturbed by the impurities. They are, however, not completely localized about the impurity. Combination of the delocalization and spin-polarization allows to mediate exchange interaction between Mn atoms

ε↓ − ε↑ = N β Sx

(10)

which appears as a consequence of the interaction of these states with Mn 3d electrons. In (10), S is the average atomic spin moments of the magnetic impurities, x is the impurity concentration. However, the applicability of the mean-field approach is not self-evident and needs to be investigated. The analysis of the experimental data shows that for II-VI DMS the value of N β is well established [27]. For III-V systems the situation is different. For instance, the experimental estimations of N β made for (GaMn)As on the basis of different experimental techniques vary strongly from large values of |N β| = 3.3 eV [24] and N β = 2.5 eV [28] to a much smaller value of N β = −1.2 eV [22]. ( Negative N β corresponds to antiparallel directions of the spins of the d and p states.) In Fig. 9 we present the calculated exchange splitting at the top of the valence band in (ZnMn)Se. For low Mn concentrations this system is paramagnetic up to very low temperatures. It shows, however, a giant Zeeman splitting in external magnetic field. The giant Zeeman splitting reveals strong p–d exchange interaction. Two DFT approaches are used: LDA and LDA+U (Fig. 9). The LDA+U exchange splittings are well described by the mean-field formula (10) that assumes proportionality between the splitting and the Mn concentration. The coefficient proportionality gives the value of N β = −1.3 eV in very good agreement with experiment. On the other hand, the LDA results deviate strongly from the mean-field behavior. The LDA calculations does not give the proportionality between splitting and concentration. Correspondingly, the ratio between the splitting and the concentration varies with the

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splitting at Γ (Ry)

0

ZnMnSe

-0.05

LDA LDA+U, U=0.3Ry EXP Nβ=−1.3eV

-0.1 0

0.05

0.1

0.15

0.2

Mn concentrartion

0.25

Fig. 9. The exchange splitting at at the top of the valence band (Γ point). Calculations are performed within LDA and LDA+U. The solid line corresponds to the experimental value of the N β parameter [8]

concentration and has larger