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 absolute value than the corresponding LDA+U estimation. The LDA results do not give good agreement with experimental value of N β for (ZnMn)Se. They, however, remind us the experimental situation for (GaMn)As where some of the estimated values of N β are larger than in the case of (ZnMn)Se and there is a strong variation of N β from estimation to estimation. To understand the origin of the difference between LDA and LDA+U results we compare in Fig. 10 the DOS of (ZnMn)Se calculated within both approaches. We obtain very strong difference between two DOS: Whereas the LDA DOS possesses substantial Mn 3d contribution at the top of the valence

DOS(st/Ry)

(ZnMn)Se 12.5% LDA 12.5% LDA+U total

20

total

ZnSe

0 -20

DOS(st/Ry)

-0.4 -0.2 0

0.2

E(Ry)

20 0 -20

Mn3d -0.4 -0.2 0

E(Ry)

Mn3d 0.2

-0.4 -0.2 0

0.2

E(Ry)

Fig. 10. The total and partial Mn 3d DOS of Zn0.875 Mn0.125 Se. For comparison the DOS of ZnSe is presented. In LDA+U calculation U = 0.3 Ry

First-Principles Study of the Magnetism of DMS

125

band the account for Hubbard U results in a strong decrease of the Mn 3d contribution in this energy region. The failure of the mean-field treatment to describe the LDA splittings is directly related to the strong admixture of the Mn 3d states. Since the atomic Mn moment in (ZnMn)Se does not depend substantially on the Mn concentration the exchange splitting of the Mn 3d states cannot be described within a mean-field approximation. Summarizing we formulate some conclusions: First, in the case of substantial admixture of the impurity 3d states to the states at the top of the valence band the mean-field approach to the characterization of the p–d interaction does not apply. Second, in the case of (ZnMn)Se the application of the LDA+U scheme leads to the shift of the Mn 3d spin-up states from the top of the valence band to lower energies making the mean-field approximation applicable. Third, for (ZnMn)Se the LDA+U gives better overall agreement with experiment than LDA [29].

7 Comparative Study of (GaMn)As and (GaMn)N In this section we compare the magnetism of two prototype systems, (GaMn)As and (GaMn)N, focusing again on the properties of the holes. To circumvent the difficulty of the estimation of the N β parameter we will characterize the strength of the p–d interaction by the value of the admixture of the 3d states to the hole. This quantity is difficult to measure but it is easily accessible theoretically. This section is based on the paper [30]. In Figs. 11,12, we present the DOS of (GaMn)As and (GaMn)N for x = 12.5%. The main features of the DOS discussed below are valid also for other Mn concentrations. We found that (GaMn)As and (GaMn)N differ strongly in both LDA and LDA+U calculations. In LDA, the spin-up impurity band of (GaMn)As merges with the valence band. On the other hand, in (GaMn)N the impurity band lies in the semiconducting gap. (We performed calculations for both zinc-blende and wurzite crystal structures of (GaMn)N. The main part of the results is qualitatively similar for both crystal structures.) For both systems the LDA calculations give large Mn 3d contribution into impurity band (Fig. 12). In the LDA+U calculations, the impurity band of (GaMn)As disappears from the energy region at the top of the valence band. In contrast, (GaMn)N still possesses impurity band which lies now closer to the valence band. The Mn 3d contribution to the impurity band decreases compared with the LDADOS but is still large (Fig. 12). Now we turn to the discussion of the properties of the holes. We again present the results for one Mn concentration. From Fig. 13 and Table 1 we see that for the calculations performed within the same calculational scheme the holes in (GaMn)N are much stronger localized about the Mn atom than

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LDA

total-DOS LDA+U 12.5%

U=0.3Ry

(GaMn)As

10

10

DOS(states/Ry)

0

0

-10

-10

(GaMn)N-ZB

10

10

0

0

-10

-10 -0.4 -0.2

0

0.2 -0.4 -0.2

0

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E(Ry) Fig. 11. The total DOS of Ga0.875 Mn0.125 As and Ga0.875 Mn0.125 N calculated within LDA and LDA+U approaches. The circles highlight the part of the DOS that is most important for the discussion

LDA

Mn3d-DOS LDA+U 12.5%

U=0.3Ry

(GaMn)As

20

20

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DOS(st/Ry)

0

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-10

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0

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-10

-10

-20

-20

-0.4 -0.2

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0

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E(Ry) Fig. 12. The Mn 3d DOS of Ga0.875 Mn0.125 As and Ga0.875 Mn0.125 N

in (GaMn)As. The Mn 3d contribution into the hole, and therefore the p– d interaction, is also stronger in (GaMn)N. Thus there are two competing trends in the properties of the holes. To predict the relation between the Curie temperatures of both systems a direct calculation of the Mn-Mn exchange

First-Principles Study of the Magnetism of DMS U=0.3Ry 12.5%

Mn

(GaMn)As

LD

A

0.2

As (4)

As

LDA+U

Ga

(4)

Ga

Ga

0 0.4

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+U

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0 0

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U A+ LD

LD A

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A LD

hole distribution

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N (4)

Ga

Ga

1

0.5

distance from Mn (a)

Mn3d

Fig. 13. Distribution of the hole in LDA and LDA+U calculations. Numbers in parentheses give the number of atoms in the coordination sphere. The hole part is given for one atom. The distance from Mn is given in units of the lattice parameter. The thick bars to the left and right of the ordinate axis present the Mn 3d contribution into the hole for LDA and LDA+U calculations Table 1. Hole distribution and Mn 3d contribution (in percent). As-1, As-2, N-1 and N-2 denote the first and second coordination spheres of As and N (GaMn)As Mn(Mn3d) As-1 As-2 LDA LDA+U

29(16) 9(4)

35 45

14 23

(GaMn)N Mn(Mn3d) N-1 N-2 50(43) 25(19)

33 5 48 12

interactions with account for the complexity of the electron structure of the systems is necessary. Another important conclusion from the analysis of Fig. 13 and Table 1 concerns the influence of the Hubbard U on the hole localization and the p–d interaction. For both systems, the account for the on-site Coulomb interaction beyond the LDA leads to a strongly increased hole delocalization and, simultaneously, to decreased p–d interaction. In (GaMn)As, the Mn 3d contribution to the hole drops from 16% to a small value of 4%. For (GaMn)N we get 43% and 19% and the p–d interaction is still strong. Again the resulting influence of U on the Curie temperature cannot be predicted without direct calculation. In Fig. 14, we present the calculated Curie temperature as a function of the band occupation, n, for one Mn concentration. It is useful to present the Curie temperature in the form TC (n) = (TC (n)−TC (1)) + TC (1) where (TC (n)−TC (1)) gives the hole contribution to the Curie temperature and TC (1) corresponds to the completely occupied bands and can be treated as

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Curie temperature (K)

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3.125% (GaMn)As

(GaMn)N

200

0 LDA LDA+U

-2

-1

0

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-1

0

variation of band occupation, n

1

2

Fig. 14. TC as a function of the electron number for x = 3.125%. n = 0 corresponds to the nominal electron number (one hole per Mn atom)

the contribution of the Anderson’s superexchange. Both contributions play an important role (Fig. 14). Thus the final value of the Curie temperature can be interpreted as a result of the competition between antiferromagnetic superexchange through the filled bands and ferromagnetic kinetic exchange mediated by the holes. It is remarkable that Hubbard U produces opposite trends in the variation of TC in (GaMn)As and (GaMn)N (Fig. 14). Taking the nominal number of electrons (n = 0) we get a strong decrease of the Curie temperature in (GaMn)As and a substantial increase in (GaMn)N. Therefore in (GaMn)As the strong drop of the p–d interaction prevails over the delocalization of the holes. On the other hand, in (GaMn)N the increased delocalization of the holes prevails over decreased p–d interaction. The comparison of the role of U for different Mn concentrations (Fig. 15) shows that in (GaMn)N the same trend to increasing Curie temperature is obtained for the whole interval of concentrations studied. In (GaMn)As

(GaMn)As

Curie temperature (K)

600

300

TC, LDA

0

TC, L+U

(GaMn)N

600

300

0

0

0.1

x

0.2

Fig. 15. TC as a function of the Mn concentration for nominal electron number

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there is strong decrease of TC for low Mn concentrations. For larger x there is a crossover to increasing Curie temperature. These results are in good agreement with the coherent-potential-approximation (CPA) calculations [30] for both systems. The calculations have demonstrated the importance of both strong p– d interaction and the hole delocalization for the formation of the high-TC ferromagnetism. These conclusions are in good correlations with a recent model-Hamiltonian study by Bouzerar et al. [31].

8 Conclusions We have reported 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 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.

Acknowledgements The support from Bundesministerium f¨ ur Bildung und Forschung is acknowledged.

References 1. H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, S. Kutsumoto, Y. Iye: Appl. Phys. Lett. 69, 363 (1996) 113 2. T. Dietl, H. Ohno, F. Matsukura, J. Cibert, D. Ferrand: Science 287, 1019 (2000) 113 3. N. Theodoropoulou, A.F. Hebard, M.E. Overberg, C.R. Abernathy, S.J. Pearton, S.N.G. Chu, R.G. Wilson: Appl. Phys. Lett. 78 3475 (2001) 113 4. M.L. Reed, N.A. El-Masry, H.H. Stadelmaier, M.K. Ritums, M.J. Reed, C.A. Parker, J.C. Roberts, S.M. Bedair: Appl. Phys. Lett. 79, 3473 (2001) 113 5. S. Sonoda, S. Shimizu, T. Sasaki, Y. Yamamoto, H. Hori: J. Cryst. Growth 237, 1358 ((2002) 113

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6. K. Ando: Appl. Phys. Lett. 82, 100 (2003) 114 7. R. Giraud, S. Kuroda, S. Marcet, E. Bellet-Amalric, X. Biquard, B. Barbara, D. Fruchart, D. Ferrand, J. Cibert, H. Mariette: cond-mat/0307395 114 8. P. Kacman: Semicond. Sci. Technol. 16, R25 (2001) 114, 124 9. T. Dietl: Semicond. Sci. Technol. 17, 377 (2002) 114 10. J. K¨ onig, J. Schliemann, T. Jungwirth, A.H. MacDonald: cond-mat/0111314. 114 11. K. Sato, and H. Katayama-Yosida: Semicond. Sci. Technol. 17, 367 (2002) 114 12. S. Sanvito, G.J. Theurich, N.A. Hill: J. Superconductivity 15, 85 (2002) 114 13. L.M. Sandratskii, P. Bruno: Phys. Rev. B 66, 134435 (2002) 114, 115 14. L.M. Sandratskii, P. Bruno: Phys. Rev. B 67, 214402 (2003) 114, 115, 117 15. G. Bouzerar, J. Kudrnovsk´ y, L. Bergqvist, P. Bruno: Phys. Rev. B 68, 081203 (2003) 114 16. L. Bergqvist, P.A. Korzhavyi, B. Sanyal, S. Mirbt, I.A. Abrikosov, L. Nordstr¨ om, E.A. Smirnova, P. Mohn, P. Svedlindh, O. Eriksson: Phys. Rev. B 67, 205201 (2003) 114 17. K. Sato, P.H. Dederics, H. Katayama-Yoshida: Europhys. Lett. 61 403 (2003) 114 18. In the nonrelativistic systems with collinear magnetic structure the exchange interaction takes place between the states with the same spin projection. 114 19. V.I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein: J. Phys.: Condens. Matter 9, 767 (1997) 115 20. L.M. Sandratskii: Phys. Rev. B 68, 224432 (2003) 115, 130 21. L.M. Sandratskii: Advances in Physics 47, 91 (1998) 115 22. J. Okabayashi, A. Kimura, O. Rader, T. Mizokawa, A. Fujimori, T. Hayashi, M. Tanaka: Phys. Rev. B 58, 4211 (1998) 116, 123 23. K.C. Ku, S.J. Potashnik, R.F. Wang, S.H. Chun, P. Schiffer, N. Samarth, M.J. Seong, A. Mascarenhas, E. Johnston-Halperin, R.C. Myers, A.C. Gossard, D.D. Awschalom: Appl. Phys. Lett. 82, 2302 (2003) 119 24. F. Matsukura, H. Ohno, A. Shen, Y. Sugawara: Phys. Rev. B 57, 2037 (1998) 120, 123 25. H. Saito, V. Zayets, S. Yanagata, K. Ando: Phys. Rev. Lett. 90 207202 (2003) 121 26. L.M. Sandratskii, P. Bruno: J. Phys.: Cond. Matt. 15, L585 (2003) 121 27. J.K. Furdyna: J. Appl. Phys. 64, R29 (1988) 123 28. J. Szczytko, W. Mac, A. Stachow, A. Twardowski, P. Becla, J. Tworzydlo: Solid State Commun. 99, 927 (1996) 123 29. (ZnMn)Se is not ferromagnetic. Instead for low Mn concentration it shows an interesting combination of giant Zeeman splitting with the absence of any magnetic ordering up to very low temperatures. The reader is referred to [20] for detailed discussion of this system. 125 30. L.M. Sandratskii, P. Bruno, J. Kudrnovsk´ y: Phys. Rev. B 69 (2004) 125, 129 31. G. Bouzerar, J. Kudrnovsk´ y, P. Bruno: Phys. Rev. B 68, 205311 (2003) 129

Exchange Interactions and Magnetic Percolation in Diluted Magnetic Semiconductors J. Kudrnovsk´ y1 , L. Bergqvist2 , O. Eriksson2 , V. Drchal1 , I. Turek3,4 and G. 5 Bouzerar 1

2 3

4

5

Institute of Physics Academy of Science of the Czech Republic, Prague, Czech Republic Department of Physics, Uppsala University, Uppsala, Sweden Institute of Physics of Materials Academy of Science of the Czech Republic, Brno, Czech Republic Department of Electronic Structures, Charles University, Prague, Czech Republic Institut Laue – Langevin, Grenoble, France

Abstract. We propose a theory that combines first-principles evaluations of interatomic exchange interactions with a classical Heisenberg model and Monte Carlo simulations. Exchange interactions are determined using the magnetic force theorem and the one-electron Green functions. The magnetic properties of diluted magnetic semiconductors are dominated by short ranged interatomic exchange interactions that have a strong directional dependence. We show that critical temperatures of a broad range of diluted magnetic semiconductors, involving Mn-doped GaAs and GaN as well as Cr-doped ZnTe, are reproduced with a good accuracy only when the magnetic atoms are randomly positioned on the Ga (Zn) sites, whereas an ordered structure of the magnetic atoms results in critical temperatures that are too high. This suggests that the ordering of diluted magnetic semiconductors is heavily influenced by magnetic percolation, and that the measured critical temperatures should be very sensitive to details of the sample preparation, in agreement with observations.

1 Introduction The quantitative description of ground-state and finite-temperature properties of metallic systems represents a long-term challenge for solid state theory. Practical implementation of density functional theory (DFT) has led to an excellent parameter-free description of ground-state properties of metallic magnets, including traditional bulk metals and ordered alloys as well as systems without the perfect three-dimensional periodicity, like, e.g., disordered alloys. On the other hand, an accurate quantitative treatment of excited states and finite-temperature properties of these systems remains an unsolved problem for the first-principles theory despite the formal extension of the DFT to time-dependent phenomena and finite temperatures. The usual local spindensity approximation fails to capture important features of excited states, J. Kudrnovsk´ y et al.: Exchange Interactions and Magnetic Percolation in Diluted Magnetic Semiconductors, Lect. Notes Phys. 678, 131–145 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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in particular the magnetic excitations responsible for the decrease of the magnetization with temperature and for the magnetic phase transition. In developing a practical parameter-free scheme for the finite-temperature magnetism, one has to rely on additional assumptions and approximations the validity of which has to be chosen on the basis of physical arguments. Magnetic excitations in itinerant ferromagnets are basically of two different types: (i) Stoner excitations associated with longitudinal fluctuations of the magnetization; and (ii) the spin-waves or magnons, which correspond to collective transverse fluctuations of the direction of the magnetization. Near the bottom of the excitation spectrum, the density of states of magnons is considerably larger than that of corresponding Stoner excitations, so that the thermodynamics in the low-temperature regime is completely dominated by magnons and Stoner excitations can be neglected. It seems reasonable to extend this approximation up to the Curie temperature, and to estimate the latter by neglecting Stoner excitations. With thermodynamic properties in mind, we are primarily interested in the long-wavelength magnons with the lowest energy. We adopt the adiabatic approximation in which the precession of the magnetization due to a spinwave is neglected when calculating the associated change of electronic energy. The condition of validity of this approximation is that the precession time of the magnetization should be large as compared to characteristic times of electronic motion, i.e., the hopping time of an electron from a given site to a neighboring one and the precession time of the spin of an electron subject to the exchange field. In other words, the spin-wave energies should be small as compared to the band width and to the exchange splitting, a condition which is justified for magnetic atoms with large exchange splittings, like, e.g., Fe-, Co-, and f-metal ferromagnets as well as the diluted magnetic semiconductors containing transition metal impurities (Fe, Co, Mn, Cr). For more details concerning the validity of the adiabatic approximation see [1, 2]. This procedure corresponds to a mapping of the total energy of an itinerant electron system onto an effective Heisenberg Hamiltonian with classical spins  Jij ei · ej , (1) Heff = − i=j

where Jij is the exchange interaction energy between two particular sites (i, j), and ei , ej are unit vectors pointing in the direction of local magnetic moments at sites (i, j), respectively. In the present formulation the values and signs of the magnetic moments are already absorbed in the definition of the Jij ’s so that positive (negative) Jij ’s correspond to ferromagnetic (antiferromagnetic) couplings. Note that intracell non-collinearity of the spinpolarization is neglected since in this approach we are primarily interested in low-energy excitations due to intercell non-collinearity. There are basically two approaches to calculate the exchange parameters. The first one which we adopt here, referred to as the real-space approach,

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consists in calculating directly Jij by employing the change of energy associated with a constrained rotation of the spin-polarization axes in cells i and j [1, 2]. In the framework of the so-called magnetic force-theorem [1, 3] the change of the total energy of the system can be approximated by the corresponding change of one-particle energies which significantly simplifies calculations. In the second approach, referred to as the frozen magnon approach or the reciprocal-space approach [4], one chooses the constrained spinpolarization configuration to be the one of a spin-wave with the wave vector q and determines the energy E(q) by employing the generalized Bloch theorem for a spin-spiral configuration [5]. The exchange parameters Jij are then obtained by using the Bloch Fourier transform. Both approaches can be applied to crystals as well as to random alloys (the reciprocal-space in the framework of supercells [6]) but for random alloys the real-space approach is quite a natural choice [7]. Once the effective Heisenberg Hamiltonian (1) is constructed, statistical mechanical methods are used in the second step to estimate the corresponding Curie temperature. Recently, this two-step procedure was successfully applied to evaluate Curie temperatures and magnon spectra for bulk transition metals [8], rare-earth metals [9] as well as for low-dimensional systems such as ultrathin films [10] which were in a good agreement with available experimental data. Here we extend this approach to the random DMS alloys, namely Mn-doped GaAs and GaN and Cr-doped ZnTe.

2 Formalism A brief summary of the computational details and the direct evaluation of the effective pair exchange interactions is given here. 2.1 Electronic Structure We have determined the electronic structure of the DMS in the framework of the first principles all-electron tight-binding linear muffin-tin orbital (TBLMTO) method in the atomic-sphere approximation using empty spheres in the interstitial tetrahedral positions of the zinc-blende lattice needed for a good space filling. We used equal Wigner–Seitz radii for all sites. The valence basis consists of s-, p-, and d-orbitals, we included scalar-relativistic corrections but neglected spin-orbit effects. Substitutional disorder due to various impurities, both magnetic and non-magnetic, is included by means of the coherent potential approximation (CPA). Charge selfconsistency is treated within the framework of the local spin density approximation using the Vosko–Wilk–Nusair parametrization for the exchange-correlation potential. The experimental lattice constants of the host semiconductor are used also for the DMS alloys but we have verified that we can neglect a weak

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dependence of the sample volume on defect concentrations. Further details of the method can be found in [11]. 2.2 Effective Pair Exchange Interactions For applications to the DMS, we have to generalize the theoretical approach developed in [1, 8] to disordered systems. The Heisenberg parameters are obtained in terms of the magnetic force theorem [1] by (i) directly evaluating the change in energy associated with a small rotation of the spin-polarization axes in atomic cells i and j, and (ii) using the vertex-cancellation theorem (VCT) [12]. The VCT justifies the neglect of disorder-induced vertex corrections in (2) below and greatly simplifies calculations. The VCT was derived in [12] under rather general conditions and facilitates an efficient evaluation of exchange interactions, exchange stiffnesses, spin-wave energies, etc. The derivation of configurationally averaged effective pair exchange interactions M,M
J¯ij between two magnetic atoms M, M  located at sites i and j follows therefore closely the case without randomness [1, 8]:


1 M,M
M,M
↑ M
,M ↓ Im J¯ij = trL δiM (z) g¯ij (z) δjM (z) g¯ji (z) dz . (2) 4π C Here, trL denotes the trace over angular momenta L = (m), the energy integration is performed in the upper half of the complex energy plane along a contour C starting below the bottom of the valence band and ending at the Fermi energy, and δiM (z) = PiM,↑ (z) − PiM,↓ (z), where the PiM,σ (z) are the L-diagonal matrices of potential functions of the TB-LMTO method for σ =↑, ↓ corresponding to a particular magnetic atom M . The matrix δiM (z) reflects the exchange splitting of atom M . The quanM,M
↑ M
,M ↓ tities g¯ij (z) and g¯ji (z) refer to site off-diagonal blocks of the conditionally averaged Green function [11], namely, the average of the Green function over all configurations with atoms of the types M and M  fixed at sites i and j, respectively, determined in the framework of the CPA [11]. The exchange interactions between magnetic moments induced on non-magnetic atoms are negligible as compared to the exchange interactions between magnetic atoms, e.g., M =M  =Mn in (Ga,Mn)As alloys. The main advantage of M,M which can be evaluthe present approach is an explicit expression for J¯ij ated straightforwardly even for large distances d = |Ri − Rj | between sites i and j and thus allows to study their asymptotic behavior as a function of the distance d. The effect of impurities on the host bandstructure, which need not be a small perturbation, is usually neglected in model theories [14], where the unperturbed host Green function appears in (2) rather than its M,M
σ (z). conditionally averaged counterpart g¯ij In the low-concentration limit (2) reproduces correctly the RudermanKittel-Kasua-Yosida (RKKY)-type expression for exchange interactions between two magnetic impurities in a non-magnetic host [13]

Exchange Interactions and Magnetic Percolation in DMS RKKY Jij =

1 Im 4π

C

  imp,↑ imp,↓ trL δiimp (z) gij (z) δjimp (z) gji (z) dz .

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(3)

imp,σ (z) is the Green function of two Mn impurities embedded in Here, gij the semiconductor host, evaluated at sites i, j where impurities are located. A conventional RKKY expression is obtained by replacing the spinimp,σ (z) by the non-magnetic Green dependent impurity Green function gij host function gij (z) of the ideal semiconductor host crystal. The neglect of the renormalization of the host Green function by scatterings on impurities is two-fold: it introduces a phase factor and modifies the amplitude of the oscillations as compared to the conventional RKKY formula [13]. It should be noted that the numerical evaluation of expressions (2) and (3), in particular for large distances, requires careful energy and Brillouinzone (BZ) integrations (typically, 105 − 106 of k-points in the full BZ are needed for large distances such as 10 lattice constants and more, and for energy points on the contour C close to the Fermi energy).

2.3 Asymptotic Behavior of Exchange Interactions M,M We will first discuss qualitatively the dependence of J¯ij on the distance Rij = |Ri − Rj |. In the limit of large values of Rij the expression (2) can be evaluated analytically by means of the stationary phase approximation [8]. By generalizing the approach developed in [8], we obtain

sin(k↑F · Rij + Φ↑ + Φ↓ ) M,M ∝ exp(− λ↑ij · Rij ) exp(− κ↓F · Rij ) . J¯ij 3 Rij

(4)

The quantity kσF , which characterizes the period of oscillations, is the Fermi ¯ σ (kF ) wave vector in a direction such that the associated group velocity ∇k E σ ¯ is parallel to Rij . Because of disorder, the band energy E (k) is modified by the real part of the spin-dependent selfenergy at the Fermi energy (determined within the CPA) while its imaginary part characterizes the damping λσij . The damping due to disorder is anisotropic inside the BZ [11], the factor λ↑ij therefore depends on the direction of Rij . Because of the half-metallic character of the DMS alloys (a fully occupied minority band), the corresponding critical Fermi wave vector k↓F of the minority states is complex, κ↓F = Imk↓F . This situation is similar to that studied in [8] for strong ferromagnets with a fully occupied majority band. Finally, Φ↑ and Φ↓ denote the phase factors. For the alloy case we thus find an exponential damping of J¯M,M with the distance, because of: (i) the damping due to substitutional disorder which was predicted by de Gennes [15], and (ii) additional damping due to the half-metallic character of the DMS alloys. It should be noted that damping due to disorder refers to configurationally averaged exchange interactions whereby exchange interactions in each alloy configuration can exhibit a slower decay with distance [16]. The averaged exchange interactions are

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sampled directly or indirectly in experiments when measuring, e.g., Curie temperatures and spin-wave spectra. 2.4 Exchange Interactions in Selected DMS’s Exchange interactions for (Ga0.95−y Mn0.05 Asy )As without (y = 0) and with (y = 0.01) As-antisites are illustrated in Figs. 1–4. The As-antisites, i.e., As atoms on the Ga-sublattice in (Ga,Mn)As alloys, acting as double donors which add two electrons into the valence band, are now considered to be the most probable compensating defects (together with interstitial Mn-atoms). We find that leading exchange interactions for y = 0 (Fig. 1) are essentially ferromagnetic, and, because of the damping, the antiferromagnetic values are very small due to a large oscillation period. It should be noted that next interactions between more distant pairs are important and contribute significantly e.g. to the mean-field value of Tc (see (6)): the first two (four) nearest neighbor shells contribute about 50 % (75 %) of its value. Exchange interactions are anisotropic for different directions. This is illustrated in Fig. 3, where we Mn,Mn (multiplied by the RKKY prefactor d3 ) show the dependence of the J¯ij with respect to the distance d along the directions [100], [110], and [111] in the Ga-sublattice of (Ga0.95 Mn0.05 )As alloy. The dominating character of the

1st-NN JMn-Mn (mRy)

2.5

J

Mn,Mn

(mRy)

2 1.5 1

4 3 2 1 0 -1 -2 -3 -4

y=0.0 y=0.01 0

0.5

0.05

0.1

0.15

Mn-concentration x

0 -0.5 0.5

1

1.5

2

2.5

3

3.5

(d/a) Fig. 1. Exchange interactions in ferromagnetic (Ga0.95−y Mn0.05 Asy )As for two different concentrations y of As-antisites plotted as a function of the distance d between two Mn atoms (in units of lattice constant a). Inset: the first nearestneighbor exchange interactions shown as a function of the Mn concentration x

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ln | (d/a)3 JMn,Mn | (mRy)

2.5 0 -2.5 y=0.0 -5

y=0.01

-7.5

(Ga0.95-y Mn0.05 Asy)As

-10 1

2

3

4

5

6

7

8

9

10 11

(d/a) along [110]-direction Fig. 2. Exchange interactions in ferromagnetic (Ga0.95−y Mn0.05 Asy )As for two different concentrations y of As-antisites as a function of the distance d (in units of the lattice constant a) between two Mn atoms plotted along the nearest-neighbor direction [110]. The form ln |(d/a)3 J¯Mn,Mn | is used to demonstrate the exponential damping of interactions with the distance

2.5 (Ga0.95 Mn0.05)As (d/a)3 JMn,Mn (mRy)

[100] [110] [111]

0

1

2

3

4

5

6

7

8

9

10 11

(d/a) along different directions Fig. 3. Exchange interactions (d/a)3 J¯Mn,Mn in ferromagnetic (Ga0.95 Mn0.05 )As calculated as a function of the distance d (in units of lattice constant a) between two Mn atoms along [100], [110], and [111] directions

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3.5

(d/a)3 JMn,Mn (mRy)

3

x=0.005

2.5

x=0.02

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x=0.05

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x=0.08

1 0.5 0 -0.5 1

2

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9

10 11

(d/a) along [110]-direction Fig. 4. Exchange interactions (d/a)3 J¯Mn,Mn in ferromagnetic (Ga1−x Mnx )As calculated as a function of the distance d (in units of lattice constant a) between two Mn atoms along the nearest-neighbor direction [110] for a set of Mn concentrations

exchange interactions along the [110]-direction can clearly be seen as well as the smallness of the interactions along the [100]-direction. The Fermi surface anisotropy along different directions in the BZ whose spanning vectors determine the periods of the damped oscillations, is also obvious. The expected decay of the exchange interactions with distance is clearly seen, for a quantitative analysis, however, we need a different presentation of results. A suitable way, e.g., is to plot ln|(d/a)3 J¯Mn,Mn (d)| as a function of the distance d along the dominating [110]-direction (Fig. 2). We indeed observe exponentially damped oscillations with a period of about 5.5 a, where a is the lattice constant. This period is larger than the average distance d¯Mn,Mn between two Mn nearest neighbors at a given concentration x. A simple estimate for d¯M,M in the zincblende lattice is  3 3 Mn,Mn ¯ a. (5) =2 d 16πx For (Ga0.95 Mn0.05 )As this estimate yields d¯Mn,Mn = 2.12 a. This finding supports the basic assumption made in model studies: the coupling between two Mn atoms is ferromagnetic and oscillatory with a period that exceeds the typical interatomic distance between magnetic atoms because of the small size of the corresponding hole Fermi surface. The latter feature is further illustrated in Fig. 4 where we plot (d/a)3 J¯Mn,Mn (d) as a function of the distance between two Mn atoms for a set of Mn concentration x: the smaller

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the concentration x, i.e., the smaller the size of the corresponding hole Fermi surface, the larger the period of the damped oscillations. The effect of As-antisites on the Ga-sublattice on exchange interactions is illustrated in Figs. 1 and 2, where we show results for (Ga0.94 Mn0.05 As0.01 )As. The presence of As-antisites, which reduces the number of holes in the valence band, leads in turn to an enlarged period of damped oscillations for y = 0.01. The nearest-neighbor interactions are reduced by nearly an order of magnitude as compared to the case without antisites while the reduction of other interactions is less pronounced (see Fig. 1). It should be noted that also the oscillations for y = 0.01 are exponentially damped, their damping is slightly larger as compared to the case without As-antisites (Fig. 2). The dependence of the exchange interactions between the nearest neighbors as a function of the Mn concentration with and without the doping by As-antisites is shown in the inset of Fig. 1. In general, we observe a decrease of the leading J¯1Mn,Mn with increasing concentrations of both Mn atoms and As defects, the decrease with increasing concentration y of As-antisites being weaker. The J¯1Mn,Mn are ferromagnetic in alloys without donors, but may change their sign for highly compensated systems, i.e., for smaller Mn concentrations. This indicates the instability of the ferromagnetic state with increasing Asantisite concentration. The kink in the dependence of J¯1Mn,Mn for x = 0.02 and y = 0.01 marks the change of p-type doping to n-type doping. Finally, in Fig. 5 we compare calculated values of the exchange interactions for Mn-doped GaAs and GaN as well as for Cr-doped ZnTe. It should be noted that the calculations were made for the zincblende structure, which is the correct structure for GaAs and ZnTe, whereas GaN crystallizes in the wurtzite structure. We do not expect this approximation for GaN to be of importance for the calculated critical temperatures below. We observe that the exchange interactions are strongly material dependent: the behavior of Mn in GaAs is quite different from that of Mn in GaN or Cr in ZnTe. The exchange interactions are of shortest range for Mn in GaN and longest range for Mn in GaAs, while Cr in ZnTe is found to lie between these limits. A pronounced directional anisotropy of exchange interactions, presented in Fig. 3 for the case of Mn-doped GaAs, is even stronger for the case of Mn in GaN as illustrated in the inset of Fig. 5. These results demonstrate that one needs to employ material-dependent exchange interactions rather than ad hoc chosen functional forms or empirical parameterizations in order to calculate critical temperatures in the DMS alloys.

3 Curie Temperatures The evaluation of the Curie temperature Tc in the framework of the two-step model reduces to the estimation of the critical temperature corresponding to the random Heisenberg model (1) with classical spins. Despite of an enormous simplification of the original problem of finding Tc in the framework of the

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2.5 2.5

x=0.05

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(d/a) J(mRy)

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(d/a) along different directions

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(Ga, Mn)N

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5

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(d/a) along (110)-direction Fig. 5. Exchange interactions in ferromagnetic Ga0.95 Mn0.05 As, Ga0.95 Mn0.05 N, and Zn0.95 Cr0.05 Te multiplied by the RKKY factor (d/a)3 (a is the lattice constant) and plotted as a function of the distance d between magnetic impurities. Inset: the same for Mn in GaN but calculated along different crystallographic directions

density functional theory, the problem is still quite difficult due to the randomness of exchange interactions in (1). The values of exchange interactions between different moments are not independent but they obey condition that the energy corresponding to the rotation of all magnetic sites by the same angle is zero (if the spin-orbit coupling is neglected), i.e., the magnon spectrum starts from zero energy. It can be shown [1] that this condition is automatically fulfilled for ordered crystals or for each random configuration in alloy. It is, however, difficult to obey this condition in the analytical alloy theory, like the CPA, because it requires to include the local environment effects. This problem was solved for the quantum Heisenberg model under certain plausible approximations only very recently [17]. Its implementation in the framework of the realistic, first-principles approach, remains a challenge. One way to get around this problem is the application of the virtualcrystal approximation (VCA) to the Hamiltonian (1) which restores the translational symmetry, and a consequent application of conventional approaches known from ordered systems [1, 8], like the mean-field approximation (MFA) or the random-phase approximation (RPA-VCA) [18]. The corresponding expression for the MFA value of Tc is

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kB TcMFA =

2x  ¯M,M J0i , 3

141

(6)

i=0

while the RPA-VCA result is 

kB TcRPA

−1

=

−1 3 1 
¯ ¯ J(0) − J(q) . 2x N q

(7)

M,M ¯ Here, J(q) is the lattice Fourier transform of exchange interactions J¯ij and x is the concentration of magnetic atoms. In addition, there have been attempts to solve the thermodynamical part of the problem by Monte Carlo (MC) simulations performed also in the framework of the VCA (MC-VCA) [18, 19], i.e., on an ordered network of sites. All these calculations seem always to overestimate the critical temperature in comparison with experiment. Experimental studies also indicate that the magnetic properties like, e.g., the ordering temperature and the magnetic moment, depend critically on the details of the sample preparation: the Mn concentration, possible clustering of Mn atoms as well as the concentration of non-magnetic defects. The corresponding critical temperature can vary over a large range, sometimes reaching room temperature. One may speculate that at least part of discrepancies between experiment and estimates based on realistic models is due to the compensating defects, like As-antisites [18, 7] or Mn-interstitials. However, the role of randomness in distribution of magnetic atoms, neglected in above studies, needs clarification. This seems to be particularly important for DMS alloys with a low concentration of magnetic atoms and the anisotropic and localized (exponentially damped) exchange interactions as discussed in Sect. 2.3. In such case the percolation effect can play an important role. An extreme example is the fcc-lattice with the nearest-neighbor interactions: the critical temperature is zero for concentrations smaller than about 20 per cent [20]. On the contrary, the estimates based on the ordered lattice (MFA, RPA-VCA, etc.) give the concentration dependence of Tc which starts from zero concentration. Although the real systems, even Mn-doped GaN, do not represent such extreme case, a strong modification of results due to possible magnetic percolation has to be expected.

3.1 Monte Carlo Simulations on Random Magnetic Sublattice M,M In principle, exchange interactions Jij between sites i and j have to be determined for each geometry of the Monte Carlo simulation cell (typically 50000 atoms) using electronic structure calculations. Since this is computationally impossible, in the present paper we calculate them approximately using the CPA and the local spin density approximation method as described in Sects. 2.1 and 2.2. This approach neglects possible local environment effects in the system because the electronic structure is determined within the single-site CPA, but it is nevertheless a good approximation since: (i) the CPA

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properly describes concentration variations of alloy properties, in particular the carrier concentration and electronic properties of various defects that are of key importance for properties of DMS; (ii) exchange splittings are robust with respect to local environment effects for atoms with large local magnetic moments, like e.g. Mn or Cr atoms; and (iii) the local environment effects can influence the electronic structure of the system, but numerical studies indicate that the CPA gives an accurate description of disordered states if the Fermi energy is located inside the valence band, i.e., in the metallic regime which is of interest here. Their influence is further reduced by energy integration present in (2). The local environment effects can be important for the case of compensated alloys with the Fermi level in quasi-localized states at band edges, and in particular, for the transport properties in this regime [21]. It should be noted that a similar approach was successfully employed in a related problem of transport in disordered multilayers, where randomness in lateral two-dimensional supercells was also approximated by the CPA [22]. However, spin-fluctuations which are relevant for a reliable determination of the Curie temperature are fully taken into account in the present MC simulations as well as the random distribution of magnetic impurities on the cation sublattice. The MC treatment of the Heisenberg Hamiltonian (1) employed the Metropolis algorithm. Magnetic atoms were distributed completely randomly on the Ga (Zn) positions of the zincblende lattice and the number of Mn (Cr) atoms was varied between 3000-13000, in order to use finite size scaling. All thermodynamic observables were averaged over 5 different disorder configurations and the critical temperatures were determined using the cumulant crossing method [19, 23]. Exchange interactions up to 17 shells of neighboring spins in the Heisenberg model were used to achieve convergence with respect to the shell number (this is important in particular for Mn-doped GaAs with the largest spatial extent of exchange interactions, Fig. 5). For more details we refer the reader to [24]. The results for Mn-doped GaAs and GaN and Cr-doped ZnTe at concentrations similar to those reported in experiments are summarized in Table 1. The large differences in experimentally determined Curie temperatures refer to samples with a different amount of compensating defects, like e.g., Mninterstitials and As-antisites in GaMnAs alloys. Annealing procedures usually reduce the amount of compensating defects, e.g., Mn-interstitials in GaMnAs samples. The effect of As-antisites in Mn-doped GaAs is also considered as a prototype of native defects [25]. The most important finding from Table 1 is that only the MC simulations that assume a realistic, random distribution of magnetic atoms give ordering temperatures of Cr-doped ZnTe and Mn-doped GaAs and GaN that are either in a good agreement with experiment or that lie within the range of experimentally observed ordering temperatures. It should be noted that experimental values for Mn-doped GaAs, and in particular for Mn-doped GaN, exhibit a broad range of ordering temperatures,

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Table 1. Critical temperatures of Mn-doped GaAs without and with As-antisites, Mn-doped GaN, and Cr-doped ZnTe in K. MFA denotes the mean field result, RPA holds for RPA-VCA [18], MC-O denotes the result from MC simulations on an ordered network of magnetic atoms (MC-VCA), and MC-R the main result from MC simulations on a disordered network of magnetic atoms. Expt denotes experimental values from [25, 26, 27] (Ga0.95−y Asy Mn0.05 )As (Ga1−x Mnx )N (Zn1−z Crz )Te y = 0.0 y = 0.005 y = 0.01 x = 0.03 x = 0.05 x = 0.08 z = 0.05 z = 0.10 z = 0.20 Expt. 45–140 MFA 287 RPA 275 MC-O 272 MC-R 162

45–140 228 212 215 150

45–140 139 122 137 111

0–370 342 293 305 55

0–370 376 327 330 77

0–370 376 323 334 96

– 356 324 315 120

– 451 400 410 215

300 ± 10 557 477 491 330

in some cases even the absence of ordering was reported. Calculations show that critical temperatures can indeed vary over a large interval, depending on structural arrangement and concentration of defects, e.g., for Mn-doped GaAs and typical As concentration of ∼0.5% is the critical temperature at the upper end of the interval of ordering temperatures reported experimentally. The MFA values overestimate the critical temperatures somewhat as compared to the RPA-VCA (related to a non-random lattice) and MC simulations on an ordered lattice which, in turn, agree well with each other [18]. It can also be inferred from Table 1 that the differences between MC results for ordered and random lattices become smaller with increasing concentration of the magnetic atoms, whereas the differences between the MFA, MC (ordered), and RPA-VCA results become larger for systems with more localized exchange interactions.

4 Conclusions We have reproduced the ordering temperatures of several typical group III-V and II-VI diluted magnetic semiconductors. We have demonstrated, by combining first-principles theory with Monte Carlo simulations, that a proper theoretical treatment of magnetic atoms in a diluted magnetic semiconductor, in particular their random distribution on the host lattice, is essential in order to reproduce observed ordering temperatures. The reason for this is the short-ranged and anisotropic behavior of the exchange interactions in these materials. The magnetic ordering in these systems thus displays features that are analogous to percolation phenomena. In fact a geometrical percolation that assumes nearest neighbor interactions only predicts the absence of magnetic ordering for a concentration of magnetic atoms lower than 20 % on the fcc-lattice. The fact that the exchange interactions are somewhat

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more extended in space as demonstrated in Sect. 2.4 leads to the magnetic ordering for lower concentrations of magnetic atoms as it is also observed. The present results also demonstrate that percolation becomes more important for lower concentrations of magnetic impurities and for systems with exchange interactions strongly localized in the real space. Our analysis most likely explains the large range of experimentally reported ordering temperatures, since the distribution of magnetic atoms on the semiconductor lattice depends critically on the sample preparation, resulting in a large range of ordering temperatures. It is also clear that if magnetic atoms form an ordered lattice, and if only few native defects are present, one may find critical temperatures of Mn-doped GaAs, Mn-doped GaN and Cr-doped ZnTe that are at, or above, room temperature. Finally we remark that the present formalism represents a general scheme that can be applied to a number of other problems, not addressed in this work. Examples of these are the effect of clustering of the magnetic atoms, or the occurrence of magnetic atoms and defects on several sublattices (e.g. GaAs with both substitutional and native interstitial Mn atoms or Mn-doped Ge).

Acknowledgements We acknowledge support from the RTN-Network of the EC Computational Magnetoelectronics (HPRN-CT-2000-00143). L.B. and O.E. acknowledge support from the Swedish Natural Science Foundation (VR), the Swedish Foundation for Strategic Research (SSF), Seagate Inc., the Center for Dynamical Processes Uppsala University, the G¨oran Gustafsson foundation, and the National Supercomputer Centre (NSC) and High Performance Computing Center North (HPC2N). J.K., V.D., and I.T. acknowledge support from the project AVOZ1-010-914 of the Academy of Sciences of the Czech Republic, the Grant Agency of the Academy of Sciences of the Czech Republic (A1010203, S2041105), and the Czech Science Foundation (202/04/0583).

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The Role of Interstitial Mn in GaAs-Based Dilute Magnetic Semiconductors Perla Kacman1,2 and Izabela Kuryliszyn-Kudelska1 1 2

Institute of Physics , Polish Academy of Sciences, Warsaw, Poland ERATO Semiconductor Spintronics Project

Abstract. Ferromagnetic (Ga,Mn)As continues to be the subject of intense research due to the interesting physical properties and possible spin-electronics applications. For the latter, however, the temperature of the transition to the ferromagnetic phase has to be considerable increased. In the reported studies it was shown that the limit for such an increase is imposed by the presence of Mn ions in the interstitial positions (MnI ). However, low-temperature annealing can remove a significant fraction of highly unstable MnI ions - a proper choice of the annealing conditions resulted in an outstanding increase of the Curie temperature of (Ga,Mn)As epilayers. Theoretical calculations have shown that the Mn ions in the interstitial position do not contribute to ferromagnetism mediated by free holes. Moreover, MnI double donors may form pairs with the nearest substitutional (MnGa ) acceptors - the calculations evidence that the spins in such pairs are antiferromagnetically coupled by superexchange. This provides a clear explanation of the effects observed after annealing, i.e., the increase of the hole concentration, of the Curie temperature, and of the saturation magnetization.

1 Introduction The ferromagnetic dilute magnetic semiconductors (DMS) were discovered 12 years ago [1]. Since then the main effort in the studies of these materials has been directed towards increasing their critical temperature. This is a fundamental condition for a technological breakthrough in the field of semiconductor spintronics, i.e., in the attempts to integrate, for applicational purposes, the spin degrees of freedom with the semiconducting properties in the same material. The low-temperature (LT), MBE-grown GaMnAs has become a favorite material for spintronics when it was shown [2] that the substitution of Mn for Ga in GaAs leads to ferromagnetism at temperatures reaching 110 K, for a long time the highest Curie temperature (TC ) observed in DMS [3, 4, 5, 6]. In the last two years, however, higher and higher values of TC in GaMnAs thin films, even exceeding 160 K, have been reported [7, 8, 9, 10, 11]. This progress has been achieved basically though optimization of the post-growth annealing time and temperature. At first, the “optimal” annealing temperatures slightly exceeded the LT MBE growth temperature but recently very high TC was obtained by annealing at temperatures considerably lower than the growth temperature [10, 11]. It was also shown that P. and I. Kuryliszyn-Kudelska: The Role of Interstitial Mn in GaAs-Based Dilute Magnetic Semiconductors, Lect. Notes Phys. 678, 147–161 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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d = 50 nm 140

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xMn [ % ]

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As grown

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Fig. 1. The effect of annealing. Curie temperature of GaMnAs: (a) as a function of Mn content x, with fixed epilayer thickness d (after [11]). (b) as a function of d for fixed x (after [10])

a proper choice of the Mn content [11] and of the thickness of the GaMnAs epilayer [10] enables a further increase of TC upon annealing, as presented in Fig. 1. In the theoretical models describing the ferromagnetism in DMS (e.g., in [12, 13, 14]) TC is expected to increase with both the magnetic ion and hole concentrations. In LT MBE-grown Ga1−x Mnx As this was indeed experimentally established for Mn concentrations up to about x = 0.06 [3]. In the as-grown samples a further increase of the Mn concentration to about x = 0.09 does not lead to higher TC , instead it results in lowering of the TC [4, 15]. Preparation of samples with still higher Mn content has been proven to be very difficult [16]. The Mn ion in the substitutional, cation position in the GaAs lattice (MnGa ) acts as an acceptor, but in all Ga1−x Mnx As samples the hole concentration is substantially lower than the Mn content. This has been ascribed to the presence of compensating donors, in particular to the formation of arsenic antisites (AsGa ) during the epitaxial growth of Ga1−x Mnx As at As overpressure [16, 17]. In [4, 6] and [18] the observed annealing-induced changes of the hole concentration and, therefore, TC were attributed to the decrease of the concentration of arsenic antisites. These antisites, however, are relatively stable defects – it was shown that to remove AsGa from LT MBE grown GaAs annealing temperatures above 450 C are needed [19]. In 2002 simultaneous channeling Rutherford backscattering (c-RBS) and particle-induced X-ray emission (c-PIXE) experiments [15] shed new light on this problem. Namely, they have revealed that in LT MBE-grown ferromagnetic Ga1−x Mnx As with high x a significant fraction of incorporated Mn atoms occupies well defined, commensurate with the GaAs lattice, interstitial

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Fig. 2. The effect of annealing. Percentage of Mn ions in various sites of the GaAs lattice as a function of: (a) the annealing temperature (after [15]). (b) the concentration of Be co-doping [23])

positions. Moreover, it has been shown that the LT post-growth annealing leads to a rearrangement of the Mn ions in the host lattice, in particular to the removal of Mn from the interstitial sites, as shown in Fig. 2(a ). On the other hand, theoretical ab initio studies on the properties of Mn in the GaMnAs ternary compound [20, 21] also predicted the formation of Mn interstitials (MnI ). According to the calculations, MnI ions, like AsGa , serve as double donors, decreasing the hole concentration and hence TC . The results of channeling experiments, presented in Fig. 2(a), directly showed that in the process of LT annealing the MnI ions are moved to random, incommensurate with the GaAs lattice positions (e.g., MnAs clusters), where the Mn ions are electrically inactive, which increases both the hole concentration and TC . According to the ab initio calculations in [10], this is due to out diffusion of Mn interstitials towards the surface – the binding energy for MnI in GaMnAs allows for significant diffusion of this defect at temperatures above ∼150C. The role of interstitial Mn ions occurred to be even more pronounced in Ga1−x−y Mnx Bey As samples, grown at Notre Dame with the hope to increase the hole concentration by introducing another acceptor [22, 23]. Instead, it was shown, again by the channeling experiments, that adding Be to Ga1−x Mnx As increases the concentration of MnI at the expense of MnGa (Fig. 2(b)). At the same time, although the hole concentration in the Ga1−x−y Mnx Bey As samples does not change significantly, the TC drops dramatically with the increase of y, as shown in Fig. 3. This was observed in the as-grown as well as in the annealed Ga1−x−y Mnx Bey As samples. We note that annealing diminishes the number of MnI ions but does not affect the number of substitutional MnGa ions, reduced by Be co-doping (compare Fig. 2(b)).

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Fig. 3. Hole concentration and TC in Ga1−x−y Mnx By As sample as a function of y

2 High Resolution X-ray Diffraction (HRXRD) Measurements The presence of interstitial Mn defects in GaMnAs was proven also by X-ray measurements. It is well known that the lattice constant of Ga1−x Mnx As layers increases with the increase of Mn content [3] – the lattice constant measurements were even used as a method of determining the Mn concentration x. While there clearly exists a phenomenological correlation between the experimentally established lattice parameter and x, it has not been really understood. On the one hand, it is known that the presence of arsenic antisites leads to an increase of lattice constant. On the other hand, in a recent paper [25] Maˇ sek et al. suggested that the Mn interstitials are responsible for the observed expansion of the lattice constant. The results of these first-principles calculations indicate that the relaxed lattice parameter of Ga1−x Mnx As has the following form: a = a0 + 0.02 × xsub + 1.05 × xint + 0.69 × y

(1)

where a0 is the lattice parameter of GaAs, xsub and xint are the concentrations of substitutional and interstital Mn, and y is the concentration of As antisites AsGa . Structural investigations of the as-grown and annealed GaMnAs samples have been undertaken [26] in order to verify the theoretical predictions. High resolution X-ray diffraction (HRXRD) measurements for a wide range of Mn concentrations (0.027 ≤ x ≤ 0.086) were performed in the Applied Crystalography Laboratory in the Institute of Physics of Polish Academy of Sciences. Measurements of the (004) Bragg reflections allowed to calculate the lattice parameters perpendicular to the layer plane (a⊥ ) for all samples studied. The

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combination of this and the measurements of the asymmetric (224) Bragg reflections were then used to determine the in-plane lattice parameter (a ), and subsequently the values of relaxed layer lattice parameter arelax . The primary result of the HRXRD measurements is the observation that the Ga1−x Mnx As lattice parameter decreases when the epilayers are annealed at the optimal conditions, i.e., when the interstitial Mn atoms are removed from the alloy. Using the values of a before and after annealing obtained from HRXRD studies and assuming xsub and y to remain constant one gets from (1) a change of the interstitial concentration smaller, but of the same order of magnitude, as the change in xint estimated from the RBS/PIXE experiments [26].

3 Channeling Experiments (c-RBS and c-PIXE) All the channeling experiments cited above, i.e., channeling Rutherford backscattering (c-RBS) and channeling particle-induced X-ray emission (cPIXE) measurements, were performed in Material Science Division Lawrence Berkeley National Laboratory. Channeling is guiding a beam of energetic ions into the open spaces (channels) between close-packed rows or planes of atoms in a crystal, as shown schematically in Fig. 4. The channeling is the result of a correlated series of small-angle, screened Coulomb collisions between the ion and the atoms bordering the channel. Thus, the channeled ions do not penetrate closer than the screening distance of the vibrating atomic nuclei and the probability of large-angle Rutherford collisions (back-scattering), nuclear reactions, or inner-shell X-ray excitation is greatly reduced as compared to the probability of such interactions from a non-channeled (random) beam of ions. Hence, for the ions incident at small angles, Ψ , to a close-packed direction a large reduction in back-scattering yield is observed (see Fig. 5). The normalized yield χh from host atoms is defined by the ratio of the yield for ions incident at an angle Ψ to the yield for a “randomly” directed beam.

Surface

Host atoms

Ψ

Solute atoms

Detector Fig. 4. Schematic of the channeling experiment

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Fig. 5. The c-PIXE spectra of the as-grown Ga0.908 Mn0.092 As sample. After [15]

The position of solute atoms in a crystal lattice can be determined from channeling experiments by measuring the normalized yield χh , from host atoms, and χs , from solute atoms, for the same depth increment in the crystal [27]. If solute atoms are in the same lattice sites as the host atoms, then χs ∼ = χh for any angle Ψ . If, however, the solute atoms project into the channel, the channeled ions interact with them causing an increased yield χs . A peak in χs near the perfect alignment (Ψ = 0) is an unambiguous evidence that the solute atoms lie near the center of the channel. By comparing yields for different channels, the position of solute atoms can be determined rather accurately by a triangulation procedure. In the diamond cubic crystal lattice there are three possible interstitial positions, two tetrahedral sites and one hexagonal, in which the atoms are shadowed along 100 and 111 direction and exposed in the 110 axial channel, as seen in the experiment (Fig. 5). The increased yield from Mn in the 110 channel has revealed that in the as-grown ferromagnetic Ga1−x Mnx As samples a fraction of incorporated Mn atoms occupies interstitial positions in the lattice, the more significant the higher the Mn content (e.g., ∼ 15% for the epilayer with x = 0.092 presented in Fig. 5, whereas 5% for x = 0.02). Recently, “perfect” samples with a low Mn concentration were grown, in which indeed the hole concentration p is only slightly less than the total Mn content and p as well as TC only slightly increase upon annealing [11]. In [23] thermodynamical arguments were used to explain why the amount of interstitials in

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Fig. 6. (a) MnGa −MnI pair in the GaAs structure. (b) The nearest four cation and six anion neighbors for an ion in the tetrahedral interstitial position in the zinc-blende lattice

the sample, and subsequently the sample’s reaction to the annealing, depend on the Mn content: as the Fermi level is pushed deeper towards or into the valence band by the increasing number of Mn substitutional acceptors, the formation of interstitials becomes energetically favorable. The tetraheadral and hexagonal interstitial sites can be distinguished by studying angular scans around the 110 axial direction [27]. From the scans presented in [15] the authors concluded that the MnI ions observed in Ga1−x Mnx As occupy tetrahedral positions, in which the interstitial is surrounded by four nearest neighbors. These can be cations or anions. It was first suggested in [15] that the electrostatic attraction between positively charged MnI donors and negative MnGa acceptors stabilizes the otherwise highly mobile MnI in the interstitial sites adjacent to MnGa , i.e., that the tetrahedral position between four cations – as presented in Fig. 6 – is preferred. Recently, this was confirmed by a calculation of the total energy of Mn ions located at different interstitial sites [24], which has shown that the tetrahedral interstitial position between cations is energetically favorable in strongly p-type material. It should be mentioned, however, that the authors of another ab initio study [10] claim that according to their calculation, in a p-type sample the tetraheadral interstitial site between the four As anions should be favored. In contrast, for GaMnAs co-doped with beryllium the angular scans around the 110 axial direction seem to suggest that the Mn interstitial donor is located not at the tetrahedral but at the hexagonal, ( 38 58 38 ), position [23]. In the hexagonal site √ the MnI has three anion and three cation nearest neighbors at a distance a 11/8, as shown in Fig. 7. According to first principle calculations, such a position should be highly unstable in the GaMnAs lattice, but it was shown [24] that the binding energy of Mn at various interstitial sites may be sensitive to the type and the degree of co-doping. The RBS/PIXE studies revealed, as shown in Fig. 2(b), that for higher Be concentrations the

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Fig. 7. (a) An Mn ion in the hexagonal interstitial position in zinc-blende GaAs lattice. (b) The six (three cation and three anion) nearest neighbors for MnI hexagonal

fraction of Mn in the interstitial position increases enormously, even to 50% of x. Thus, one can imagine the following scenario for the process of codoping GaMnAs with Be: after a certain hole concentration is reached, it is energetically more favorable that Mn rather than Be ions occupy interstitial positions and that it is the BeGa acceptor which stabilizes the MnI atom at the hexagonal site. For a constant Mn concentration, the increase of Be content results in more Mn atoms being displaced to interstitial positions. The most important result of the reported studies on Be co-doping of GaMnAs [22, 23] is the observed dramatic drop of TC in samples with higher Be concentration, even though the hole concentration remains nearly constant (see Fig. 3). This means that Mn interstitials act against the ferromagnetism in Ga1−x Mnx As, not only due to the compensating character of this defect, but also because of their distinct magnetic behavior.

4 SQUID Measurements In the annealed GaMnAs samples, in which the concentration of the compensating MnI donors decreases considerably, the hole concentration increases, the observed Curie temperature is much higher and the lattice constant decreases. Furthermore, it was demonstrated that appropriate annealing increases the saturation magnetization. Thus, it was concluded [15, 7, 30] that the presence of MnI reduces the net magnetic moment. The magnetic properties of Ga1−x Mnx As thin films were studied by direct magnetization measurements as well as magnetooptical (Magnetooptical Kerr Effect – MOKE) and magnetotransport investigations [7, 10, 11], all showing many interesting features. Here we present only the magnetic field dependence

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Fig. 8. SQUID measurements results: (a) Magnetization vs. magnetic field for the as-grown and annealed Ga0.938 Mn0.062 As sample. (b) Magnetic moment per Mn for as-grown and annealed sample obtained from saturation magnetization under a 3 kG magnetic field (after [11])

of the magnetization of as-grown and annealed GaMnAs epilayers, measured by a SQUID (superconducting quantum interference device) magnetometer. As an example, in Fig. 8(a), the hysteresis loops for as-grown and annealed (at 280◦ C and a higher temperature of 350◦ C) are presented. An increase of the saturation magnetization as well as a decrease of the coercive field is observed after heat treatment at the optimal conditions. In contrast, thermal annealing at a higher temperature leads to an increase of the coercive field and a decrease of the saturation magnetization. Such behavior was seen in every

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of the wide range of studied samples. While it is easy to imagine that proper annealing diminishes the coercive field, because it removes defects which can affect the magnetic domains movements, the answer to the question why the removal of interstitials increases the saturation magnetization is not so obvious. Moreover, the recent discovery of strong magnetic anisotropy fields in GaMnAs (up to 3 kG) [28], with the easy directions depending on the Mn as well as hole concentrations, thus on annealing [29], called in question the correctness of determining the saturation magnetization from the hysteresis loops taken in the range of magnetic fields ±500 G. Importantly, the increase of saturation magnetization upon annealing was confirmed by the SQUID measurements under an external magnetic field well exceeding the anisotropy field. The increase seems, however, not to be so pronounced as expected before (see Fig. 8(b)).

5 Exchange Interactions of Mn Interstitials In order to provide theoretical basis for the understanding of the experimental findings, which can not all be explained by the compensating properties of the interstitials (i.e., the increase of magnetization in the annealed Ga1−x Mnx As and the decrease of TC in the Ga1−x−y Mnx Bey As samples), studies of spin interactions for MnI were performed [30, 31, 24]. In [30, 31] the hybridization of the d-orbitals of magnetic ions in various lattice positions with the valence band p-states was analyzed. This effect is essential for both the superexchange and the RKKY-type ion-ion interactions in DMS. In the Zener model [12], the transition temperature to the hole-induced ferromagnetic phase is proportional to the square of the kinetic p-d exchange constant β, i.e., to the fourth power of the hybridization constant V at the center of the Brillouin zone. At the Γ point the valence band states in GaAs are built primarily (ca 80%, [32]) from the anion p-orbitals. Thus, in the simple tight-binding model presented in [31] the p-d hybridization for a given magnetic ion was assumed to be determined by the positions of its nearest-neighbor anions. In zincblende lattice, the Mn ion in the cation √ substitutional position has four anion nearest neighbors at distances a 3/4 (where a is the lattice constant) along the [1, 1, 1], [1,–1,–1], [–1, 1,–1] and [-1,-1, 1] directions. For these positions the inter-atomic matrix elements, Ex,xy , Ex,yz , Ex,zx , etc., expressed in terms of the Harrison parameters Vpdσ and Vpdπ [33], add up constructively to the hybridization constant V, at the center of the Brillouin zone equal to √ 4(Vpdσ − 2/ 3Vpdπ ). In contrast, the ion in the tetrahedral interstitial position, e.g., ( 14 14 34 ), has 6 anion neighbors along the [0, 0, ± 1], [0, ± 1, 0] and [±1, 0, 0] directions, at distances a/2 (see Fig. 6). In this case at the center of the Brillouin zone the inter-atomic matrix elements mutually cancel and the hybridization constant V and the kinetic exchange constant βIt are equal to 0. Thus, in [31] it was concluded that the MnI d-orbitals do not

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hybridize with the p-states of the holes at the top of the valence band and that the tetrahedral interstitials do not contribute to the hole-induced ferromagnetism. Similarly, it was shown that for Mn at a hexagonal interstitial site a part of the inter-atomic matrix elements in the hybridization with the three anion nearest neighbors also mutually cancel at the centre of the Brillouin zone, which should lead to a kinetic exchange constant βIh considerably smaller than that for the substitutional Mn ion (β/βIh ≈ 5). As the Curie temperature in the Zener model depends on β 2 , the contribution to the holeinduced ferromagnetism from the Mn ions occupying hexagonal interstitial sites should be very much suppressed as well. In [31] it was, therefore, suggested that the decrease of TC in Ga1−x−y Mnx Bey As can be explained by the decrease of the number of Mn ions participating in the Zener-type ferromagnetism, due to an increased fraction of Mn interstitials observed upon adding Be to GaMnAs. Contrary to the results presented [31], the density functional calculations [24] indicate that the exchange interactions between Mn ions and the holes do not depend on the position of the magnetic ion in the GaMnAs lattice. The authors of [24] argued that this discrepancy is related to the fact that in [31] the hybridization with the p-orbitals of the cation nearest neighbors was neglected. It is obvious that the kinetic exchange for interstitial Mn in GaMnAs derived in [31] should be treated as an approximation – the contribution mentioned in [24], as well as many other neglected contributions, e.g., the direct s-d exchange constant α, could affect the exchange interactions between the MnI ion and the valence band holes. Still, the model describes the dominant part of the p-d interaction and all the neglected effects lead to a considerably lower value of the exchange constant – the hybridization of MnI with the four nearest Ga ions results in βIt approximately three times smaller than β. If we take into account also the fact that one of the Ga ions in the vicinity of MnI is probably replaced by MnGa (see Fig. 6), we will obtain a four times smaller βIt . Such a value leads to a TC sixteen times lower, thus, it does not change the conclusion that the hole-induced ferromagnetism is extremely suppressed for the interstitial Mn in GaMnAs. The point raised in [24], that at the top of the GaAs valence band the wavefunctions have a considerable admixture of cation p-orbitals, appears however, to be important for the conclusions about p-d interaction for the Mn ion in the hexagonal interstitial position. Namely, it supports the result that p-d exchange is very much suppressed for the ion in this site. Including the hybridization of d-orbitals of the MnI in the hexagonal site with p-orbitals of the three cation nearest neighbors (compare Fig. 7(b)) leads, by symmetry, to further cancellation of the inter-atomic matrix elements for all the six nearest neighbors, thus to a further decrease of the value of βIh obtained in [31]. Neither the hexagonal interstitial position nor the binding energy of Mn ions at different sites of the GaAs lattice in the presence of Be acceptors were considered in [24]. The authors also do not offer an explanation for the

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experimentally observed behavior of TC in Ga1−x−y Mnx Bey As, if the p-d exchange does not depend on the position in the lattice, as obtained in their study. To explain why the removal of interstitials leads to an increase of magnetization in the GaMnAs samples in [31] it was noticed that the ionic spins in the MnGa -MnI pair shown in Fig. 6 can be coupled by superexchange mechanism. The spins of the two ions, S 1 and S 2 , are correlated due to the spin-dependent p-d exchange interaction between each of the ions and the valence band electrons in the entire Brillouin zone. The superexchange Hamiltonian: ˆ1 · S ˆ2 ˆ superexchange = −2J S H (2) can be obtained by a proper selection of spin-dependent terms in the matrix of the fourth order perturbation with respect to the hybridization for a system of two ions in the crystal [34]: −

 f | H ˆ h | l l | H ˆ h | l l | H ˆ h | ll | H ˆ h | i (El

− E0 )(El
− E0 )(El − E0 )




(3)

l,l ,l

Using the virtual transition picture, one can say that the superexchange is a result of four virtual transitions of an electron – from the band onto the d-shell of the ion and from the ionic d-shell to the band, in different sequences [35]. The quantitative determination of the superexchange constant J requires the knowledge of the energies of these virtual transitions, which are represented by the energy differences between the intermediate and initial states of the system of two ions and the completely filled valence bands, in the denominator of (3). In [31] a very simplified model, in which the dispersion of the valence bands was completely neglected, but which accounted for the wave-vector dependence of the hybridization matrix elements, allowed to determine the sign of this superexchange interaction and to estimate roughly its exchange constant J. The resulting formula for the exchange constant J reads:    1 1 1 1 1 1 + + + 2 J =− 25 Ea1 Ea2 Ea1 Ea2 Ea1 (Ea1 + Ed2 ) Ea22 (Ea2 + Ed1 )  × Vν∗1 ,k1 ,m (2)Vν2 ,k2 ,m (2)Vν∗2 ,k2 ,n (1)Vν1 ,k1 ,n (1) (4) ν1 ,ν2 ,k1 ,k2 ,m,n

In (4) the summation runs over the valence band indices ν1 and ν2 , the wave-vectors k1 , k2 from the entire Brillouin zone, and over the Mn d-orbitals m, n. The energies Eai and Edi (i = 1,2) are the transfer energies for the electron from the valence band onto the ion i and from the ion i to the valence band, respectively. It should be noted that these energies for the interstitial Mn ion are completely unknown. Still, since all these energies as well as the sum calculated numerically are positive, it was concluded that

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the spins in the MnGa -MnI pair are antiferromagnetically coupled. A crude estimate of the strength of this coupling gave J ≈ 71 K and an about 60% smaller value of the superexchange constant for a MnGa -MnGa closest pair. These values, although much larger than the ones observed in canonical II-VI DMS, seem to be reasonable in view of the extremely strong dependence of the hybridization mediated exchange interactions on the size of the lattice cell of the material [33]. Recently, it was shown also by ab initio calculations that superexchange is the mechanism responsible for the spin pairing in GaMnAs [24]. The energy of the antiferromagnetic coupling, 0.3 eV, obtained by this calculation, fits well the values estimated within the simple tight-binding model. The antiferromagnetic superexchange coupling explains why MnGa MnI pairs do not contribute to the low-field magnetization. Annealing breaks these pairs, removes the interstitial Mn and releases the spins of MnGa – this should lead to the observed increase of the saturation magnetization. In conclusion, during the recent years it was shown that the formation of interstitial Mn ions plays a crucial role in controlling the ferromagnetic transition in GaAs-based DMS. First of all, the compensating properties of the interstitial magnetic ions impose a limit to the Curie temperature in ferromagnetic Ga1−x Mnx As and Ga1−x−y Mnx Bey As samples. Also the magnetic properties of the Mn ion in both (tetrahedral and hexagonal) interstitial sites, i.e., the negligible kinetic exchange constant and strong antiferromagnetic superexchange with the adjacent substitutional Mn ion, act towards diminishing the transition temperature. Recognition of these defects resulted in a considerable increase of the TC in GaMnAs epilayers. The authors are very much obliged to all the collaborators and other colleagues whose results are presented here, especially to M. Dobrowolska, W. Dobrowolski, K. Edmonds, M. Sawicki, T. Wojtowicz, W. Walukiewicz and K. M. Yu, for their versatile help. Special thanks should be given to Prof. J. K. Furdyna, who pioneered these studies and involved both of us in the subject. Support of the FENIKS project (EC:G5RD-CT-2001-00535) and of the Polish State Committee for Scientific Research grant PBZ-KBN044/P03/2001 is also gratefully acknowledged.

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Magnetic Interactions in Granular Paramagnetic-Ferromagnetic GaAs:Mn/MnAs Hybrids Wolfram Heimbrodt and Peter J. Klar Dept. Physics and Material Sciences Center, Philipps-Universit¨ at Marburg, Renthof 5, D-35032 Marburg, Germany Abstract. The paramagnetic-ferromagnetic hybrid structure with a Curie temperature above room temperature formed by metallic MnAs clusters embedded in a semiconducting GaAs:Mn host matrix is a promising system for spintronic applications. In this chapter we present the results of detailed studies of the magnetooptical as well as the galvano-magnetic properties of the paramagnetic matrix, of the ferromagnetic clusters and of their interplay in the hybrid system.

1 Introduction Currently, the most extensively studied diluted magnetic semiconductor (DMS) is Ga1−x Mnx As. The RKKY-interaction is commonly accepted as the underlying mechanism yielding ferromagnetism. There exists, however, a discrepancy between experimental feasible Curie temperatures and theoretical expected values TC > 300 K [1]. For (Ga,Mn)As the highest Curie temperature reported so far is TC = 180 K based on the RKKY interaction mediated by free holes. In this chapter we present another phase of this material, a semiconductor ferromagnetic hybrid structure formed by MnAs clusters embedded in a paramagnetic GaAs:Mn host matrix. This hybrid system allows n-type doping, which may tremendously enhance the spin-dephasing time, by avoiding the strong spin-orbit coupling of the valence band. The system exhibits furthermore ferromagnetism above room temperature due to the MnAs clusters. An important aspect in the design of spintronic and magneto-electronic devices is to maximize magnetoresistance (MR) effects. To date, there are only a few reports on the MR effects in granular paramagnetic-ferromagnetic hydrids such as GaAs:Mn/MnAs [2, 3, 4], GaAs:Er/ErAs [5], GaAs:Mn/MnSb [6], GaAs:Mn/MnSb [7] or Ge1−y Mny /Mn11 Ge8 [8]. Negative and positive MR effects were observed in such granular paramagnetic-ferromagnetic hybrids [2, 3, 4, 8] and the underlying physical effects of the observed MR effects are still not fully clarified. All relevant interactions for GaAs:Mn/MnAs hybrids are illustrated in Fig. 1. An optimization of the MR effects can be achieved on a microscopic level by controlling the interactions between the ferromagnetic material and the free carriers in the matrix as well as the interactions in the paramagnetic material itself between the localized magnetic W. Heimbrodt and P.J. Klar: Magnetic Interactions in Granular Paramagnetic-Ferromagnetic GaAs:Mn/MnAs Hybrids, Lect. Notes Phys. 678, 163–184 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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Mn-subsystem in paramagnetic (Ga,Mn)As matrix

ferromagnetic MnAs cluster TC > 300K

doping

free carriers in paramagnetic (Ga,Mn)As matrix

Fig. 1. Schematic representation of the internal and external interactions in the granular paramagnetic-ferromagnetic hybrid system GaAs:Mn/MnAs

moments and the free carriers. But it can also be done on a macroscopic level by optimizing the current path through the device (e.g. by artificial structuring, by choosing an appropriate magnetic field geometry etc.). A thorough knowledge about the various interactions in the hybrids is a prerequisite for optimizing the complex system for future applications. The results of detailed studies will be discussed in this contribution.

2 Growth and Preparation of Hybridstructures A segregation of ferromagnetic MnAs clusters can occur under certain conditions in the MBE [2, 9, 10, 11, 12, 13] and MOVPE growth [14] of (Ga,Mn)As epitaxial layers. In the case of MBE, the segregation of the MnAs clusters is usually induced by thermal annealing after the growth, whereas in the MOVPE-case the cluster formation takes place during the growth process. An alternative approach to obtain MnAs clusters in GaAs is Mn+ ion implantation and subsequent annealing [15, 16]. Ferromagnetic MnSb clusters also occur in the MBE growth of (Ga,Mn)Sb [7]. Theodoropoulou et al. studied ferromagnetic (Ga,Mn)N platelets in GaN:Mn [17]. The segregation of binary ferromagnetic Mn-V can be considered a general feature of all Mn-containing III-V. Figure 2 depicts schematically the phases of the (Ga,Mn)As system formed at different MOVPE growth conditions. For substrate temperatures below 450 ◦ C and a Mn/Ga ratio in the gas phase of less than 10% a whisker growth is observed. For higher temperatures up to about 600 ◦ C and Mn/Ga ratios below about 8%, Ga1−x Mnx As alloy layers of high structural quality can be grown, but the incorporated Mn-content x is limited to less than about 0.5%. With increasing partial pressure of the Mn-precursor the formation of MnAs clusters takes place. The clusters are embedded in the (Ga,Mn)As matrix, almost lattice matched and without any indication of dislocations. In addition, their crystallographic orientation is well defined, i.e. the c-axis

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Tg [oC]

600 cluster growth

Ga1-xMnx As alloy

500

400

whisker growth

0.1

1

10

Mn/Ga ratio (gas phase)

Fig. 2. Sketch of the phases of the (Ga,Mn)As system formed in MOVPE growth. The growth parameters varied are the growth temperature Tg and the Mn/Ga partial pressure in the gas phase

of the hexagonal MnAs is basically parallel to the (111) directions of the zincblende Ga1−x Mnx As host. Figure 3 exemplarily shows TEM images of GaAs:Mn with MnAs clusters grown by MOVPE at 500◦ C with layer thicknesses of 150 nm (#13080), 500 nm (#13076) and 1000 nm, (#13072) respectively. The layers were grown on GaAs (001) substrates at a growth rate of 0.5 µm/h, and a nominal Mn/Ga ratio of 24% in the gas phase. In the thin layer of 150 nm thickness (left-hand image of Fig. 3), only one type of clusters (type A) is detected. These clusters possess a lens-like shape and are all situated close to the sample surface. The layer of 500 nm thickness (center image of Fig. 3) contains, in addition to type A clusters, two other cluster types. These are a type B of lens-shaped clusters with a bigger d /d⊥ ratio than type A, and a type C with a spherical or stout cylindrical shape. d⊥ denotes the extension of the clusters in the layer plane and d that along the growth direction. All three types of clusters are situated close to the surface for the 500 nm layer. However, the clusters of type C show a tendency to be overgrown. This is confirmed by the TEM image of the 1000 nm layer (right-hand image of Fig. 3). With increasing layer thickness the shape of the dominant clusters changes from lens-shaped clusters of type A to cylindrical or spherical clusters of type C. There are efforts by various groups to control the formation (e.g. size, position in the structure, etc.) of such ferromagnetic clusters in semiconductor structures grown by MBE or MOVPE. De Boeck et al. [18, 19] as well as Shimizu et al. [20, 21] demonstrated that size control and positioning of MnAs clusters is possible. Lampalzer et al. showed that in MOVPE growth the position of the clusters within a structure can be controlled by overgrowing them with AlAs [14]. Finally it should be mentioned that it is possible to change the majority carrier type from holes to electrons in the GaAs:Mn matrix by Te co-doping,

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A A

A

B B C C

100nm

#13080

100nm

#13076

200nm

#13072

Fig. 3. TEM images of MnAs clusters formed in GaAs:Mn layers of different thickness in the MOVPE growth of (Ga,Mn)As at high Mn/Ga ratios

which does not affect the magnetic properties of the clusters. Electronic and magnetic properties of these hybrids can be adjusted independently.

3 Magneto-Optical Properties of the GaAs:Mn Matrix Besides the usual Zeeman splitting and the Landau-level splitting in an external magnetic field, the exchange splitting needs to be considered for DMS. Excitonic effects are primarily devoted to the bottom of the conduction band and to the top of the valence band at k = 0. Furthermore, the band gap as well as the spin-orbit splitting are much bigger than the magnetic-field induced splittings, so that it is sufficient to treat the Γ6 conduction band (CB) and the Γ8 valence band (VB) separately. On the basis of the mean field and the virtual crystal approximation the splitting is commonly described by the parameters N0 α and N0 β which account for the exchange interaction between the localized magnetic moments and the extended CB and VB states, respectively. The shift and splitting of the CB states and VB states are: EcΓ6 = −xN0 α
Sz mj 1 EvΓ8 = − xN0 β
Sz mj 3

with with

1 2

(1)

1 3 mj = ± , ± . 2 2

(2)

mj = ±

For (II,Mn)VI DMS the exchange parameters are well known. The situation is somewhat more involved in (III,Mn)V alloys. This is illustrated by the values determined for N0 β in (Ga,Mn)As and (In,Mn)As. For very small Mn-concentrations (≈1017 to 1019 cm−3 ) a ferromagnetic p-d coupling was found by Heimbrodt et al. [22]. This is in agreement with earlier results by Liu et al. and Szczytko et al. [23] on Mn-doped GaAs [24] and Szczytko et al. [25] on Ga1−x Mnx As alloys with x ≈ 0.04. Only recently Hartmann et al. found that the sign of the exchange integral N0 β in GaAs:Mn is correlated with the type of conductivity as can be seen in Fig. 4. In n-type GaAs:Mn,Te

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(where all the Mn-acceptors were compensated by co-doping with Te; samples C and D), N0 β < 0 was observed, in contrast to the p-type samples without Te co-doping (A and B) where N0 β > 0 [26]. Sample C is at an intermediate stage, it is co-doped with Te, but still p-type. A negative N0 β is also reported for n-type In1−x Mnx As with x ranging from 0% to 12% [27, 28]. The correlation of the sign of N0 β with the local electronic configuration of the Mn-ion can qualitatively be understood by considering virtual jumps of the p-electrons to the d-states and back. In the case of half-filled d-shells (Mn2+ ), or even in the case of Fe2+ (3d6 ) or Co2+ (3d7 ), the spin has to be aligned antiferromagnetically, as jumps are possible only into the unoccupied spin 3d-states. For ions with less than half-filled d-shells, like Cr2+ (3d4 ) in IIVI, a ferromagnetic coupling is observed, as the spin of the hopping electron should be aligned parallel to the spins already located in the d-shell according to Hund’s rule. A Mn 3d5 + h center in GaAs behaves somewhat similar to the 3d4 state of Cr in II-VI semiconductors[29]. The hole with a polarized spin can accommodate preferably valence band electrons with a spin parallel to the spins in the 3d-shell, i.e. the p-d coupling is ferromagnetic.

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Fig. 5. Left-hand graph: Spin-flip Raman spectra of p-type GaAs:Mn. The observed g-values of 2.77 and 5.54 reflect that all Mn-ions are incorporated as A0 . Top righthand : Schematic representation of the mechanism leading to N0 β > 0 and N0 β < 0. Bottom right-hand: EPR spectrum of n-type GaAs:Mn,Te. The observed g-value of 2.0 confirms that all Mn-ions are in the A− configuration

These results are corroborated by studies of the local electronic structure of the Mn-ion. Schneider et al. demonstrated that the different Mn-centers in GaAs, i.e. A− with a 3d5 configuration and A0 with a 3d5 + h configuration, can be distinguished in EPR experiments due to different g-values [30]. Figure 5 shows spin-flip Raman spectra of p-type GaAs:Mn (sample A of Fig. 4) taken with the laser excitation close to the band gap. The observed signals are paramagnetic resonance signals arising from multiple spin-flips within the Zeeman-split ground state of the Mn 3d configuration. The signals have g-factors of 2.77 and 5.54 (two times 2.77), thus they originate from A0 centers. There is no evidence for signals with g = 2.0 and therefore it can be assumed that all Mn-ions are incorporated as A0 . Similarly, it can be anticipated that all Mn-acceptor in fully compensated n-type GaAs:Mn:Te samples are in the A− state. This is confirmed by the bottom graph on the right-hand of Fig. 5. It shows the EPR spectrum of sample E. For this n-type sample only a Mn-related signal with g = 2.0 was observed. There are also results for N0 β in the literature obtained from ferromagnetic, MBE-grown Ga1−x Mnx As samples with x of several percent. Matsukura et al. [31] determined absolute values |N0 β| ≈ 3.5 eV from transport measurements assuming that the negative MR curves above TC arises solely from spin-disorder scattering between the hole spins and the Mn-spins. Satoh et al. [32] determined values for |N0 β| between 3.0 eV and 3.5 eV from the

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dependence of TC on hole concentration. Core-level photoemission experiments of Ga0.926 Mn0.074 As gave a value of N0 β = −1.2 eV [33] assuming a A− -center for the Mn-ion. In the latter case, however, the authors declare that their results are consistent with both A− and A0 configuration of the Mn-ion and no definite prediction of the sign of N0 β can be given. In general, it appears to be rather difficult to extract reliable values for N0 β from ferromagnetic samples as direct measurements are hardly feasible. By now it is well established experimentally [34, 35] as well as theoretically [36, 37, 38] that Mn in ferromagnetic Ga1−x Mnx As samples is not solely incorporated on Ga-sites, but also incorporation as an interstitial acting as a double donor is common. This means that Mn-ions on Ga-sites acting as acceptors are to a large extent compensated. Consequently, A− and A0 -centers are often observed simultaneously in EPR experiments on MBE-grown ferromagnetic Ga1−x Mnx As [39, 40, 41]. Taking into account the strong correlation of N0 β and the type of Mn-center, this implies that N0 β is usually ill defined in Ga1−x Mnx As with x of several percent and a high degree of compensation. The N0 β-values determined on those samples have to be considered an average determined by the ratio of A− to A0 . In various theoretical and experimental papers N0 β is assumed to be antiferromagnetic and of a constant value [1, 42, 43, 44]. A TC variation is then solely determined by the Mn and free carrier concentration. Taking into account the strong correlation of N0 β and the type of Mn-center, N0 β is usually not a constant. It will be an average value determined by the ratio of A− to A0 . It seems to be essential to incorporate the dependence of N0 β on the local Mn-configuration in current theories.

4 Galvano-Magnetic Properties of Paramagnetic GaMn:As Epitaxial Layers It is essential to note that already the paramagnetic DMS alloy alone (i.e. without clusters) exhibits positive as well as negative MR effects [45, 46, 47, 48, 49]. However, these are different from those in the corresponding hybrids[4], as will be discussed in Sect. 6. The unusual MR effects of the paramagnetic DMS are explained by the interplay of band filling, magneticfield induced tuning of the band structure, carrier-carrier interactions, spinscattering and quantum corrections [46, 50, 51, 52, 53, 54]. An aspect whose influence on the galvano-magnetic properties of DMS was included so far only in the magnetic polaron picture [55], is the magnetic-field induced tuning of the alloy disorder in these materials. It arises due to fluctuations in the Mnconcentration which, in an applied magnetic field, lead to local fluctuations of the Mn-induced band splitting. Magnetic-field tuning of alloy disorder is a well known feature of DMS [56, 57, 58, 59]. On the other hand, it is well established that disorder in crystalline semiconductor alloys has a considerable impact on the transport properties [60].

W. Heimbrodt and P.J. Klar 40

8 MR Ga1-xMnxAs x < 0.005 T=1.6K

MR Ga0.999Mn0.001As 1.6K 10K 50K 100K 150K 280K

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Fig. 6. Left graph: MR effect at 1.6 K of samples with different Mn-content up to 0.5%, the arrow indicates increasing Mn-content. Right graph: Magnetoresistance of p-type paramagnetic Ga1−x Mnx As with x ≈ 0.001 at various temperatures

As an example, Fig. 6 shows the MR curves at T = 1.6 K of five paramagnetic Ga1−x Mnx As alloy samples with x up to about 0.5%. By increasing the Mn-concentration from doping levels to about 0.5%, the MR changes from a positive MR effect to a large negative MR effect. For the sample with the highest manganese concentration (x ≈ 0.005), the resistance decreases rapidly at low magnetic fields by about 30% up to 0.5 T. At higher fields the resistance decreases further but at a smaller rate of about −3% per Tesla for H > 5 T over the whole accessible field range. At 10 T the MR value is about −60%. The MR behavior in the intermediate regime of Mn-concentrations is rather complicated as demonstrated in the right graph of Figure 6 by the temperature dependence of the sample with x ≈ 0.001. At 1.6 K, the MR effect first increases at low magnetic fields, then goes through a maximum and finally decreases slowly. With increasing temperature the whole MR curve flattens (T = 10 K to T = 50 K) and eventually increases again exhibiting a positive parabolic MR effect above about T = 50 K. At low temperatures effects due to the giant Zeeman-splitting dominate which scale with B/T . The temperature dependence in this range below 50 K is very similar to the MR effects observed in diluted magnetic n-(Cd,Mn)Se and n-(Cd,Mn)Te [45, 47]. The usual MR-effect (proportional to H 2 ) dominates at higher temperatures above ≈ 50 K. It is caused by the cyclotron movement of the carriers and is typical for non-magnetic semiconductors. The prefactor of the parabola depends on the dominant scattering mechanism which varies with temperature. The behavior at low temperatures can be phenomenologically understood considering the interplay of Zeeman-splitting and occupation effects. For

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small Mn concentrations the Fermi-level is close to the VB edge. The reduced effective density of states caused by the Zeeman splitting leads to a positive MR which follows almost the Brillouin-function of the Zeeman-splitting. For higher Mn- and respective hole concentrations the Fermi-level shifts below the VB edge and the reduced spin scattering with the Mn ions aligned in the external field now dominates the transport. Michel et al. have compared two models (referred to as mobility model and network model) for describing theoretically the magneto-transport in p-type wide-gap dilute magnetic semiconductors in the paramagnetic phase [63]. In both models, band filling effects, magnetic-field splitting of the band states due to the p-d exchange interaction as well as effects of magnetic-field independent disorder are included whereas carrier-carrier interactions other than those responsible for the local magnetism of the Mn-ions are neglected. Therefore, the calculated MR curves in both models arise solely from the interplay of magnetic-field induced disorder effects and occupation effects. Two types of occupation effects occur due to the Giant Zeeman splitting: (i) the magnetic-field induced band shifts of the individual bands change the number of holes within each band; and (ii) the magnetic-field induced band splitting of the four hole bands changes the total density of states of the valence band as a function of energy. These magnetic-field dependent occupation effects lead to a negative MR contribution if the Fermi-level is above the VB-edge. Positive MR effects might arise in this models due to the interplay of local band gap and local Zeeman splitting. To illustrate this, consider a situation with alloy disorder where the local band gap at zero-field depends on (xloc −x). Due to the local Zeeman splitting (which scales linearly with xloc ) it might occur that the overall disorder increases for the hole bands approaching the acceptor and decreases for the hole bands moving away from the acceptor. This means qualitatively that the mobility decreases with increasing field for the bands whose occupation increases with the field, i.e. a positive MR might arise. Exemplarily, Fig. 7 depicts MR curves calculated for the above scenario. The network model was used for p-type DMS of different x. The parameters used in the calculation do not correspond to those of the (Ga,Mn)As samples discussed in the left-hand graph of Fig. 6. Nevertheless, similar trends are observed, i.e. the MR changes from positive to negative with increasing Mn-concentration. Further experimental as well as theoretical studies are necessary to reveal the actually underlying mechanisms.

5 Ferromagnetic Properties of MnAs Clusters in GaAs:Mn The properties of the hybrid systems will be strongly determined by the ferromagnetism of the clusters and their interaction with the electronic states of the host matrix. Therefore, it is essential to investigate the magnetic behavior

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Magnetoresistance

0.4

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-0.4 0.005 Mn 0.01 Mn 0.02 Mn 0.03 Mn

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Fig. 7. Magnetoresistance of a Mn-containing DMS for various Mn-contents x calculated using the network model and taking alloy disorder into account. T = 30 K, EA = 60 meV, |N0 β| = 2.5 eV

of the clusters themselves. To date there are only very few detailed studies of the ferromagnetism of such MnAs clusters [9, 64]. The FMR measurements were performed at temperatures below the Curie temperature TC = 330 K of the MnAs clusters. Figure 8 depicts a typical ESR spectrum obtained on the GaAs:Mn/MnAs hybrid layer (#13080) with

ESR signal [arb. units]

1

GaAs:Mn/MnAs #13080

0 Lorentzian -1

T = 150 K = 9.35 GHz = 155° 0

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10

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Fig. 8. ESR spectrum for sample #13080. The solid line indicates a fit with two Lorentz curves on a linear background

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Fig. 9. Upper frame: angular dependence of the FMR spectra for sample #13080 at T = 150 K with rotation around GaAs [–110] axis; lower frame: simulation for lens-shaped clusters with their hexagonal axes oriented along GaAs [111] and [11–1] directions

clusters of type A. The spectrum consists of two ferromagnetic resonance lines with a weak substructure. The solid line represents a fit by the field derivative of two Lorentz curves on a linear background. The asymmetric deviation of the experimental line shape from the ideal Lorentz curve indicates a distribution of resonance fields, which can be ascribed to minor variations of the MnAs-cluster shape and orientation. Performing angle-dependent FMR measurements allows one to analyze the anisotropic magnetic behavior of the MnAs clusters. Such measurements were performed for samples with different clusters (types A to C) and for different sample orientations with respect to the external magnetic field H. As an example, the upper frame of Fig. 9 shows a two-dimensional map in grey scale of the ESR signal of the GaAs:Mn/MnAs hybrid #13080 in dependence on magnetic field and rotation angle. The crystallographic GaAs [–110] direction served as rotation axis. The FMR pattern consists of two coexisting 180◦ -periodic curves with their minimum resonance field near 4 kOe. The two arrows in the figure indicate the positions where H is perpendicular to the sample plane (i.e. H  [001]) and where H lies in the sample plane (i.e. H  [1–10]). The angle and the field dependence of the observed ferromagnetic resonance curves of the MnAs clusters can be described theoretically using the classical equation of motion for the magnetization and accounting for Zeeman, magneto-crystalline anisotropy and demagnetization effects. The classical equation of motion is given by [65] ∂M = −γM × Heff + damping ∂t

(3)

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with the gyromagnetic ratio γ = gµB /
. The effective magnetic field includes the magneto-crystalline anisotropy field Han and the demagnetization field Hdem due to the shape of the ferromagnetic sample Heff = Hext + Han + Hdem

(4)

Neglecting the damping, 3 describes the precession of the magnetization with the Larmor frequency ω = γHeff (5) which is driven on resonance by a transverse magnetic microwave field h cos(ωt) ⊥ Hext . In general the directions of the three contributions to the effective magnetic field are not parallel to each other. For the calculation we used an uniaxial magneto-crystalline anisotropy, which is a sufficient approximation as the c-axis of the cluster is a six-fold rotational axis and the corresponding hexagonal plane are fairly isotropic. Using this model, the resonance behavior observed for GaAs:Mn/MnAs hybrids with different types of MnAs clusters can be entirely described by the variation of demagnetization factors due to different dominant cluster shapes. Assuming an ellipsoidal shape of the clusters allows one to fit the resonance curves by varying the height-to-width ratio of the axes of the ellipsoid. The lower frame of Fig. 9 depicts calculated curves for sample #13080 assuming a width to height ratio d : d⊥ of 0.4, in good agreement with the TEM investigations which yielded average dimensions of d = 13 nm and d⊥ = 35 nm for the MnAs clusters of this hybrid. Figure 10 summarizes the angular dependence of the resonance fields of the strongest lines obtained for the three layers of 150 nm, 300 nm and 500 nm thickness with the rotation axis along the GaAs [–110] direction together with the theoretical curves calculated for the three different cluster shapes. One can clearly recognize the systematic evolution from dominating thin lenses (type A) in the 150 nm layer via lensshaped clusters with increasing height to width ratio (type B) in the 300 nm layer to finally sphere-like clusters (type C) in the 500 nm layer.

6 Galvano-Magnetic Properties of Hybrid structures Now we want to discuss the MR effect in the GaAs:Mn/MnAs. MR measurements were performed between 15 K and 300 K and in magnetic fields H up to 10 T. The orientation of H with respect to the sample axis and the current direction I was varied. H was applied either perpendicular (along GaAs [001]) or parallel (along [110]) to the sample plane. In the latter case, the MR was measured with I (along [110])  H or I (along [1–10]) ⊥ H. The angular dependence of the FMR signals in the GaAs:Mn/MnAs hybrid with the smallest clusters in Fig. 9 shows that the behavior of the total magnetization of the MnAs clusters is comparable for the two field geometries where H  [001] and H  [1–10].

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The left-hand graph of Fig. 11 shows a comparison of the temperature dependence of the resistivity of GaAs:Mn with and without MnAs clusters. The temperature dependence is very similar in the range from about 100 K 1e+5 #13080 (type A) #13077 (type B) #13076 (type C) #13072 (type C) GaAs:Mn reference

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Fig. 11. Comparison of transport results of GaAs:Mn/MnAs hybrids with different cluster types (given in brackets) and a GaAs:Mn reference sample without MnAs clusters. Left-hand graph: Temperature dependence of the resistivity. Right-hand graph: Temperature dependence of the MR value at H = 10 T in the geometry where H (along [001]) ⊥ I

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Fig. 12. Comparison of transport results of the GaAs:Mn/MnAs hybrid (#13080) for different transport geometries. Left-hand graph: Magnetoresistance curves at various temperatures. Right-hand graph: Temperature dependence of the MR values of sample #13080 and the reference sample at H = 10 T (and H = 3 T; bottom). The three geometries are indicated on the left-hand

to 300 K, but start to strongly deviate at lower temperatures. At low temperatures the resistivity of the cluster samples is by several orders of magnitude bigger. The temperature T0 where the deviation sets in seems to shift slightly to higher temperatures with increasing cluster size. It has been confirmed by Hall measurement that the hole concentrations at room temperature are comparable for the samples with clusters and without clusters. The strong increase of the resistivity below T0 in the cluster samples is correlated with a strong decrease of the hole concentration p. As will be discussed in what follows the holes get trapped at the interface between the MnAs clusters and the GaAs:Mn matrix. The left-hand graph of Fig. 12 depicts the temperature-dependence of the MR effect in the GaAs:Mn/MnAs hybrid with the smallest clusters for three different transport geometries. The temperature dependent MR in the

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geometry where H (along [001]) ⊥ I is depicted at the top. At low temperatures, a strong negative MR is observed. A MR value of –30% is achieved at H = 10 T. With increasing temperature, the negative MR effect is suppressed and then changes to a positive MR at a temperature T1 = 40 K. The magnitude of the positive MR effect increases further with increasing temperature and reaches a maximum at a temperature T2 = 60 K. It always increases with increasing H and does not reach a saturation up to H = 10 T. At T2 a MR value of 160% for H = 10 T is observed. Above T2 , the MR effect becomes smaller again, but remains positive up to 300 K. In the middle, the temperature dependence of the MR in the geometry where H (along [1–10]) ⊥ I is depicted. In this geometry, the MR behavior as a function of field and temperature is similar, but the magnitude of the effect is reduced. The maximum positive MR effect is only about 50% at H = 10 T. It is worth mentioning, that the MR effects in the GaAs:Mn/MnAs hybrids for the two geometries (top and middle) are more than an order of magnitude bigger than in the GaAs:Mn samples without clusters. When H (along [1–10])  I, as depicted at the bottom, the MR effects are smallest. However, at low fields H < 5 T, the MR behavior is similar to that of the other two geometries. The MR changes very slowly with increasing H; it changes from negative to positive with increasing temperature; and the MR effect is largest at T2 = 60 K. At high fields H > 5 T, the MR is always negative and keeps decreasing monotonically without saturation up to H = 10 T. The negative MR effect at 10 T is again biggest at T2 . The magnitude of MR effect in this orientation is comparable to that in the paramagnetic GaAs:Mn reference sample without MnAs nanoclusters. The right-hand graph of Fig. 12 summarizes the typical temperature dependence of the MR effects for the three transport geometries. The MR values at H = 10 T are depicted as a function of temperature for the sample with the smallest clusters and for the GaAs:Mn reference sample without clusters. The MR effect is biggest at the characteristic temperature T2 and approaches that of the sample without clusters at the highest temperatures. The MR effect due to the clusters is enhanced strongly in the hybrid compared with the reference sample, when I ⊥ H (top and middle). When I  H, the MR effects are comparable in magnitude in both samples indicating that the effect of the clusters is weaker in this geometry (bottom). The observed anisotropy of the MR for current direction I  H and I ⊥ H obviously reflects a difference in the current path through the sample. This leads obviously to a variation of the degree of interaction between the free carriers in the paramagnetic matrix and the ferromagnetic MnAs nanoclusters. For I ⊥ H the current path through the sample is extended due to the circular movement between two scattering events and the number of interactions between the free carriers and the ferromagnetic clusters is enhanced. Similar results for all three transport geometries were observed for samples with other cluster sizes. Exemplarily, the right-hand graph of Fig. 11 depicts

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the MR value at H = 10 T as a function of temperature for hybrid samples with different cluster sizes in the geometry where H (along [001]) ⊥ I. A negative MR nearly -30% independent of cluster size occurs at 15 K. With increasing temperature, the MR of all samples becomes positive at T1 , then increases further until it reaches a maximum at T2 . Afterwards it drops again, but remains positive up to 300 K. With increasing cluster size T1 and T2 shift to higher temperatures and the magnitude of the MR near T2 decreases. Wellmann et al. reported a similar behavior for GaAs:Mn/MnAs hybrids with smaller clusters (diameters < 50 nm) and observed the biggest MR effects for the 50 nm clusters [3]. This suggests that there is an optimum cluster size where the MR effects are maximum. This is further corroborated by results of Akinaga et al. who found a similar temperature dependence of a much smaller MR effect in GaAs:Mn/MnAs hybrids with cluster sizes below 10 nm [2]. The identification of the microscopic scattering mechanisms leading to the observed MR effects is rather difficult. The microscopic mechanism behind the solely negative MR effect in other granular alloys where ferromagnetic clusters are embedded in a diamagnetic (metallic or insulating) matrix [66, 67, 68, 69, 70, 71, 72, 73, 74, 75] is usually explained by spin-dependent tunnelling between clusters in insulating matrices [76] whereas in the case of metallic matrices the reason is spin-dependent scattering at the interface of the ferromagnetic cluster [67, 77]. It is assumed, in the model, that no spinflip scattering occurs in the diamagnetic matrix. Therefore, it is possible to employ the ‘two-channel model’ postulated by Mott in 1936 [78]. The spinup and spin-down carriers can be considered as two independent families of charge carriers, each with its own distinct properties. This allows one to model the current flow through the structure by two resistor networks for spin-up and for spin-down connected in parallel. As a consequence, the resistivity drops when an external magnetic field aligns the magnetization of the ferromagnetic clusters as one obtains a ‘short-circuit’ in one of the spin channels. The assumption that no spin-flip scattering occurs in the matrix does not hold for at least two reasons in our hybrid structures. Firstly, it is well known that scattering by magnetic impurities may cause a spin-flip and secondly the semiconducting matrix is p-type and due to the spin-orbit coupling the hole spin-orientation relaxes much faster than an electron spin in an s-like conduction band. Another possibility is the positive extraordinary MR (EMR) effect, which dominates the MR behavior in high-mobility diamagnetic semiconductors with diamagnetic metal inclusions [79, 80, 81]. The basic idea of this effect is shown on the left of Fig. 13. If the conductivity contrast between the semiconductor and the metal is big (i.e. σS  σM ), the local electric field E loc at the interface will always be perpendicular to the cluster surface. In zero magnetic field (Hext = 0), the current flowing through the material is

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Fig. 13. Left: Schematic representation of the extraordinary MR effect (positive MR). The arrows indicate the current paths. Middle: Schematic representation of a possible positive MR effect. Right: Schematic representation of a possible negative MR effect. See text for explanations

focused into metallic regions with the metal clusters acting as a short circuit; the current density j is parallel to the local electric field E loc . At finite magnetic field, the current deflection due to the Lorentz-force results in a directional difference between j and E loc , the angle between them being the Hall angle θH given by: θH = arctan (µH Hext ) = arctan

RH Hext ρ0

(6)

where µH is the Hall mobility, RH the Hall constant and ρ0 the resistivity at Hext = 0. For sufficiently high fields the Hall angle approaches 90◦ in which case j is parallel to the cluster interface and the current is deflected around the metal clusters. This effect gives rise to the EMR effect. As a rule of thumb, the bigger the mobility µH (larger Hall angles) and the lower the carrier density in the semiconductor (large conductivity contrast) the larger is the EMR effect. The GaAs:Mn reference sample has a mobility of about 80 Vs/cm2 which corresponds to a Hall angle (at Hext = 10 T) of about 5◦ only. Therefore, it is not very likely that the EMR effect is the main cause of the observed positive MR effect in the GaAs:Mn/MnAs hybrids. The coexistence of negative and positive MR effects is a unique feature of paramagnetic-ferromagnetic granular hybrids [2, 3, 4, 8, 5, 6]. The paramagnetism of the matrix plays a major role in the underlying microscopic processes. In particular, the magnetic-field induced tuning of the band structure and the local disorder of GaAs:Mn due to the giant Zeeman splitting need to be accounted for. This is the major difference compared to granular diamagnetic-ferromagnetic hybrids where the magnetic-field dependence of diamagnetic matrix can be neglected. Two MR mechanisms based on the interplay of paramagnetic effects in the matrix and ferromagnetism of the clusters are illustrated in the middle

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and on the right of Fig. 13. On the right is a schematic representation of the mechanism yielding a negative MR. At Hext = 0 T, the magnetic field is not zero in the paramagnetic matrix due to the presence of the ferromagnetic clusters. An estimation of the dipolar field close to the surface of the clusters yields about 1 T (density of the Mn-ions of nMn ≈ 3 × 1028 m−3 and a magnetic moment per Mn-ion of 3.4µB [2, 82]). At low temperatures, this yields a splitting in the paramagnetic matrix close to the surface and gives trap depth of about 15 meV. This means that the holes are trapped at the cluster interface at low temperatures due to the local giant Zeeman splitting and form a bound magnetic polaron (BMP). In an applied external field Hext the giant Zeeman splitting occurs throughout the entire GaAs:Mn matrix and releases the trapped holes leading to a negative MR effect. With increasing temperature, the Zeeman splitting and the trap depth will decrease following the Brillouin function. In addition, the thermal energy of the holes kB T will increase. Therefore, no BMP will be formed and the respective negative MR effect will disappear. The BMP formation at low temperatures is in concordance with the observed dramatic increase of the sample resistivity ρ0 with decreasing temperature in Fig. 11. The MR mechanism depicted in the middle of Fig. 13 yields a positive MR. At zero field Hext = 0 (top image), there is a preferential path for one spin orientation through the ferromagnetic clusters. Any spin information get lost immediately in the matrix due to scattering with the randomly oriented Mn S = 5/2 spins, as the spin-flip length is much smaller than the mean distance between the ferromagnetic clusters. In an applied magnetic field Hext = 0, the Mn S = 5/2 spins become aligned and via the p-d exchange interaction align in turn the hole spins. Furthermore, the magnetization of the clusters will be aligned along the field direction. This alignment of the cluster magnetization favors now the carrier spin-orientation, which is opposite to that of the majority spin carriers in the DMS matrix (bottom image). This yields a positive MR effect. The ability of the paramagnetic matrix to preferentially align the carrier spins decreases with increasing temperature, i.e. the thermal disorder destroys the preferential orientation of the Mn-ions. This is again in agreement with the experimental results, the positive MR in Fig. 11 decreases for higher temperatures and approaches that of the paramagnetic reference sample. The maximum is given by the competition between the positive and negative MR contributions. The positive MR effect is bigger for the smaller clusters as their surface to volume ratio is bigger. This results in more scattering events for the majority spin carrier. This qualitative discussion of the MR mechanisms highlights the importance of the paramagnetic matrix for the transport properties of the hybrids. A detailed theoretical modelling is required to analyze these effects in detail. For this purpose, it is possible to extent the network model described in [63].

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7 Concluding Remarks We presented a systematic study of the ferromagnetic-paramagnetic hybrid system GaAs:Mn/MnAs. The manifold interactions of the system have been addressed. It is important to note, that the relevant interactions are tunable, which makes these hybrid systems promising candidates for semiconductor based spintronic applications and devices. The advantage of these hybrids is their compatibility with the established device technology. The Curie temperature of MnAs TC = 330 K is still too low for most applications. The GaAs:Mn/MnAs hybrid system introduced in this chapter is, therefore, just a model system on the way to higher TC . The Curie temperature of the MnAs clusters inside the semiconducting matrix is somewhat higher compared to bulk MnAs. The reason is not yet clear. It could be caused by the strain or by the incorporation of Ga into the clusters, which may result in somewhat changed binding angles with a respective altering of the exchange interaction. This could be an important tool for raising the Curie temperature. Another approach is given by incorporating other clusters with higher Curie temperatures, as e.g. MnSb.

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Dilute Ferromagnetic Oxides J.M.D. Coey Physics Department, Trinity College, Dublin 2, Ireland

Abstract. Dilute ferromagnetic oxides can have Curie temperatures far in excess of 300 K and exceptionally large magnetic moments per transition-metal cation. It is suggested that the strong ferromagnetic exchange is mediated by electrons from donor states associated with lattice defects. These electrons form magnetic polarons in a spin-split impurity band in an appropriate concentration range. Minority-spin 3d states of Fe, Co or Ni dopants are pinned at the Fermi level. The Curie temper√ ature in the mean field approximation is proportional to (xδ) where x and δ are the concentrations of magnetic cations and donors, respectively. A magnetic phase diagram as a function of x and δ is presented, which includes regions of localised and metallic ferromagnetism below the cation percolation threshold.

1 Introduction The most interesting magnetic materials to emerge in the last few years are a new group of dilute ferromagnetic oxides [1-13] and nitrides [1,14-17] which are wide-gap semiconductors yet exhibit Curie temperatures TC well in excess of room temperature. Interest in these materials is focussed on understanding their magnetism, with a view to developing thin-film magneto-optic and spinelectronic devices that can operate in ambient conditions. A list of some of the oxides where high-temperature ferromagnetism has been reported is given in Table 1. First reports were for Co dilute in TiO2 (anatase) by Matsumoto et al. [3], in ZnO by Ueda et al. [6] and in SnO2 by Ogale et al. [10]. A syndrome, or set of common symptoms is associated with the ferromagnetism, which is usually associated with samples in thin film form. – All the oxides are n-type. They are often partially compensated. – The average moment per transition-metal cation mc approaches (or even exceeds) the spin-only moment at low concentrations x of magnetic cations. It falls progressively as x increases. – The ferromagnetism is present at concentrations that lie far below the percolation threshold associated with nearest-neighbour cation coupling. – The ferromagnets can be metallic or semiconducting. – The saturation magnetization may be quite different for different orientations of applied field relative to the film. – Properties vary greatly for samples of the same nominal composition prepared by different methods, or by different groups. The magnetization tends to decay with time on a timescale of months.

J.M.D. Coey: Dilute Ferromagnetic Oxides, Lect. Notes Phys. 678, 185–198 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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Table 1. Dilute ferromagnetic oxides and nitrides. Eg ; band gap. Tc ; Curie temperature Eg (eV) GaN AlN TiO2 SnO2

3.5 4.3 3.2 3.5

ZnO

3.3

Cu2 O In1.8 Sn0.2 O3

2.0 3.8

Doping x Mn – 9% Cr – 7% Co – 1–2% Fe – 5% Co – 5% V – 15% Fe – 5%, Cu1% Co – 10% Ni – 0.9% Co – 5%, A10.5% Mn – 5%

Moment (µB ) 1.2 1.8 7.5 0.5 0.75 2.0 0.06 0.45 0.8

Tc (K)

Reference

940 > 600 650–700 610 650 > 350 550 280–300 > 300 > 300 > 300

[12] [13] [3] [11] [11] [7] [8] [6] [9] [45] [46]

Data on the concentration-dependence of the ferromagnetic moment of Mn, Fe or Co in SnO2 in Fig. 1a illustrates points (ii) and (iii). The collection of data on Co in ZnO in Fig. 1b and TiO2 in Fig. 1c illustrate point (vi). Point (v) has been demonstrated particularly for 3d dopants in SnO2 [12] and ZnO [13]. A first reflex was to try to explain the results in terms of phase segregation. Ferromagnetic impurity phases and dopant clustering have indeed been identified in some cases, notably in Co-doped anatase and rutile [18, 19], where the solubility limit is low. These observations cast some doubt on the original claim of that Co-doped anatase was a dilute ferromagnetic insulator [3] but ferromagnetic phase segregation cannot be a general explanation [20]. The magnetic moments at low concentrations in Fig. 1 are unprecedented; they are comparable to the spin-only moments of the cations in their high-spin state (5µB for Mn2+ or Fe3+ ; 4µB for Mn3+ , Fe2+ or Co3+ ; 3µB for Mn4+ or Co2+ ). It would be an extraordinary coincidence for suitable ferromagnetic secondary phases to have occured in all three systems. Furthermore, the ordered moment per 3d cation at low concentrations exceeds that in practically any known oxide or alloy including the pure metal [21]. The moments per cation are simply too large to be explained in this way. These results challenge our understanding of magnetism in oxides. The common and well-understood superexchange mechanism [22], which is predominantly antiferromagnetic and short-ranged, cannot be invoked firstly because the magnetic order appears at concentrations of magnetic cations far below the percolation threshold xp , which is approximately 2/z, where z is the cation coordination number [8, 12] and secondly because the average net moment per magnetic cation in ferrimagnetically-ordered superexchange systems rarely exceeds about one Bohr magneton. The ferromagnetic double-exchange mechanism, originally described by Zener with reference to mixed-valence

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Fig. 1. Summary of literature results on the magnetic moment per transition-metal ion in (a) SnO2 doped with Mn, Fe or Co; (b) ZnO doped with V, Mn, Co, Fe or Ni; (c) TiO2 doped with Fe or Co

manganites can produce large moments, but it is also a nearest-neighbour interaction, and requires mixed cation valence so that 3dn ⇐⇒ 3dn+1 configuration fluctuations can take place. There is no general evidence for mixed valence in these dilute oxides.

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Longer-range ferromagnetic exchange can be mediated by carriers in a spin-polarized band. Well-studied semiconductor examples are: – n-type europium monochalcogenides such as EuS, where the carriers are ↑ electrons in the spin-split 5d/6s conduction band produced by doping with donors such as Gd3+ ; the Eu2+ 4f 7 magnetic core level lies in the gap [23]. – p-type III-V dilute magnetic semiconductors such as (Ga1−x Mnx )As where carriers are ↓ holes in the 4p valence band; the Mn2+ 3d5 core level lies below the top of this band [24]. The manganese solubility limit in GaAs is ∼7%. Curie temperatures in these semiconductor systems range up to about 170 K. A model by Dietl et al. based on exchange mediated by p-band holes has predicted higher Curie temperature of about 300 K for ZnO and GaN doped with 5% Mn [25]. In dilute metallic alloys, the Rudemann–Kittel–Kasuya–Yoshida (RKKY) exchange mechanism operates [26] whereby a magnetic impurity produces an oscillatory spin polarization in the conduction band, which normally leads to spin glass freezing at temperatures of order 10 K. The RKKY interaction is ferromagnetic at short distances and for small concentrations of conduction electrons. Here we discuss the strong ferromagnetic exchange coupling and the large magnetic moments in dilute n-type oxides [27]. We first develop a simple model based on the propensity of these oxides to form shallow donors, and then consider how the model has to be modified to account for the high TC .

2 Model The general formula of a dilute magnetic oxide is (A1−x Mx )On
δ

(1)

where A is a nonmagnetic cation, M is a magnetic cation and
represents a donor defect. Here n is usually 1 or 2. An electron associated with a particular defect will be confined in a hydogenic orbital of radius rH = ε(m/m∗ )ao , where ε is the relevant dielectric constant of the oxide, m∗ is the effective electron mass and a0 is the Bohr radius (53 pm). Weakly-bound donor electrons tend to form bound magnetic polarons, coupling the 3d moments of the ions within their orbits [28–31]. The basic idea is illustrated in Fig. 2. The binding energy of the donor electron in its ground-state 1s orbital is EB = (m∗ /mε2 )R, where R is the Rydberg energy (13.6 eV). The depth of the electron traps is therefore of order a few tenths of an electron volt if  ≈ 5 and m∗ /m < 1. As the concentration of va3 −1/2 ) exp (−r/rH ) cancies δ increases from zero, the 1s orbitals ψ(r) = (πrH begin to overlap, and form an impurity band. At first, the electrons remain localized because of the effects of correlations and potential fluctuations in a

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Fig. 2. Schematic representation of a magnetic polaron. A donor electron in its hydrogenic orbit couples with its spin antiparallel to magnetic dopants with a 3d shell which is half-full or more than half-full. The figure is for x = 0.1, γ = 12 (from [27])

narrow band, discussed by Mott [33], but there is a critical vacancy density beyond which the impurity band states become delocalized, and metalncrit
lic conduction sets in. The insulator-metal transition in the impurity band is an Anderson transition brought about by the random nature of the donor potential wells [33]. In many doped semiconductors, the condition is 1/3 rH ≈ 0.27 (ncrit
) crit

(2)

ncrit
/nO ,

The corresponding value of δ is δ = where the oxygen density nO is approximately 6.1028 m−3 for oxides with a close-packed oxygen lattice. Defining γ = ε(m/m∗ ) so that rH = γa0 , condition (2) becomes γ 3 δ crit ≈ 2.2

(3)

Values of δ crit are included in Table 2. Exchange coupling between electrons in hydrogenic orbitals is weakly antiferromagnetic, but no evidence has been found for magnetic ordering in simple impurity bands [31, 33]. Now we consider the interaction of the magnetic dopant cations M with the bound hydrogenic electrons. Overlap between the hydrogenic electron and the cations within its orbit leads to ferromagnetic exchange coupling between the cations, regardless of the sign of their coupling to the donor electron in its hydrogenic orbital. This interaction may be written in terms of the s-d

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Table 2. Parameters for the oxides; a, c – lattice parameters, Z – formula units per unit cell, nc – cation density, ε – high-frequency dielectric constant, rH = e(m/m*)a0 – donor centre radius, νc – number of cation sites within the donor vacancy concentration for insulator – metal transition, δ crit centre radius, ncrit
corresponding proportion of vacant oxygen sites, xp percolation concentration a

ZnO TiO2 TiO2 SnO2

wurtzite anatase rutile rutile

c

Z

pm

pm

335 379 459 474

522 951 296 319

nc

ε

m∗ /m rH

28

2 4 2 2

10 m−3 3.94 2.93 3.20 2.80

nm

4.0 9.0 7.4 3.9

0.28 1 0.3 0.24

νc

ncrit δ crit
24

10 m−3 0.76 64 45 0.48 14 4.3 1.31 150 8.8 0.86 75 31

xp

−3

10

1.14 3.4 0.14 0.55

0.5 0.25 0.25 0.25

exchange parameter Jsd as [33] ∆Eex = − Jsd S · s|Ψ (r)|2 Ω

(4)

where S is the spin of the 3d cation of volume Ω, and s is the donor electron spin. (This interaction is analogous to the Fermi contact interaction between the nuclear spin and the spin of atomic electrons). In Table 2, the number of cation sites νc within a sphere of radius rH is listed; rH is evaluated using the high-frequency dielectric constant, which is appropriate for a metallic impurity band. When the electrons are localized, a larger value of ε may be appropriate. This quantity νc ranges from a few tens to more than ten thousand, depending on γ, the ratio of dielectric constant to reduced effective mass. The number of these cation sites occupied by a magnetic cation is ν=xνc . In the case of a 3d5 ion such as Fe3+ or Mn2+ , only unoccupied ↓ orbitals are available. Hence the donor electron is ↓. The effective coupling between two impurities within the same orbital is therefore ferromagnetic. More generally, the coupling between the cation and the donor electron is ferromagnetic when the 3d shell is less than half full, and antiferromagnetic when the 3d shell is half-full or more. In either case, the resulting coupling between two similar impurity cations within the same donor orbital is always ferromagnetic. As the density of defects n
increases, the hydrogenic orbitals associated with the randomly-positioned defects begin to overlap. Taking them as randomly-packed spherical objects with radius rH , percolation occurs when 1/3 3 they occupy roughly 16% of space [34]; (4/3)πrH n
= 0.16 or n
rH ≈ 0.34. Provided γ > 4, this threshold condition for the appearance of long-range ferromagnetic order can also be expressed in terms of δ = n
/nO as γ 3 δp ≈ 4.3

(5)

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The average number N of donor electrons interacting with a particular 3 magnetic cation is (4/3)πrH n
. The polaron percolation threshold δp and the cation percolation threshold xp are two landmarks on the magnetic phase diagram. Provided the donor orbitals cover a sufficient number ν of magnetic cations, ferromagnetism can 3 xno /n. Antiferromagnetism or ferriappear above δp . ν is equal to (4/3)πrH magnetism can appear beyond xp where there are continuous paths throughout the crystal joining nearest-neighbour magnetic cations. When x < xp , antiferromagnetic superexchange cannot create long-range order; at most it will act within small groups of nearest-neighbour cations giving clusters of antiferromagnetically-coupled spins such as ↑↓, ↑↓↑, . . . . . . . . A comparison of conditions (3) and (5) suggests that the ferromagnetic phase should only form when the states in the impurity band are delocalized. However, the additional random potentials created by the cation cores and the spin splitting of the impurity band due to exchange interaction with the magnetic ions may enlarge the zone where the electrons are localized to the extent of forming a localized ferromagnetic region. Furthermore, the percolation threshold δp will be somewhat lower than that given by the hardsphere argument, (5). The exchange energy ∆Eex between a localized dopant core spin S and the donor electron of spin s is given by (4) where Ω = (4/3)πrc3 and rc is 3 ) is a the cation radius. For convenience, we suppose that |Ψ (r)|2 = 3/(4πrH constant within a sphere of radius rH , and zero elsewhere. Hence ∆Eex = −Jsd S · s/ρ3

(6)

where ρ = rH /rc . This interaction can be represented by a molecular field acting on the donor electron, HW = Jsd
Sz /2µ0 µB ρ3 . The electron will be almost completely spin polarized in the magnetically-ordered state if it interacts with many magnetic cations. The Curie temperature can be estimated in this approximation. The effective field acting on the cation spins S is HW ’ = N Jsd
sz /2µ0 µB ρ3 . Hence
Sz /S = BS (N Jsd S
sz /kB T ρ3 )

(7)


sz /s = tanh(νJsd s
Sz /kB T ρ3 )

(8)

and where BS (x) is the Brillouin function. Close to TC , the equations can be linearized using BS (x) ≈ [(S + 1)/3S]x and tanh(x) ≈ x , yielding TC = [S(S + 1)N ν/3]1/2 sJsd /(kB ρ3 ). This may be written in terms of x and δ 3 3 xnO /n, N = (4/3)πrH δnO , and s = 1/2 as using ν = (4/3)πrH TC = [S(S + 1)s2 xδn/3]1/2 Jsd ωc /kB

(9)

where ωc is the cation/anion volume ratio for the oxide (typical value 8%), multiplied by the oxygen packing fraction, 0.74, and n = 1 or 2, depending on

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the oxide composition. This expression is a little surprising because it shows that TC is independent of γ. It is sufficient for the donor orbital to extend over a sufficient number of cations, but there is nothing to be gained by any further increase of the dielectric constant. The interaction of the hydrogenic orbital with the magnetic cations is independent of the size of the orbital, −3 rH , when x is small, because the electron density |Ψ |2 in (4) varies as rH 3 whereas the interaction volume νΩ varies as rH . Exchange coupling with isolated impurities does not therefore alter the size of the orbit. However, overlap of the hydrogenic orbital with antiferromagnetically coupled spins tends to shrink the size of the orbital, and may help to localize the states in the impurity band [35].

3 Discussion The Curie temperature can be estimated in a typical case: (A0.90 M0.10 )O0.99
0.01

a monoxide with n = 1, nO = 6.1028 m−3 , x = 0.1, S = 5/2, rc = 0.06 nm and δ = 0.01. Derived values are rH = 0.8 nm, ν = 13, N = 1.3, and ρ = 13. The value of Jsd may be inferred for Co in ZnO from the red shift of the band gap with cobalt doping [36] to be 1.5 eV. The Curie temperature given by (8) is TC = 28 K. The exchange energy associated with the donor centre then is ν∆Eex(max) = 129 K. The positional disorder of the dopant ions may enhance TC somewhat and give rise to an unusual temperature-dependence of the magnetization [37], but the estimated Curie temperature is more than an order of magnitude too low, and in terms of our model there are few options for increasing it. We have assumed when writing (8), that the impurity band electrons are localized; if they are delocalized, TC is even lower [27]. It is unreasonable to expect the value of Jsd can be much greater than 1.5 eV. Nor can the donor concentration greatly exceed 1%; in any case TC varies only as the square root of δ. We need somehow to increase the donor electron density |Ψ (r)|2 in the vicinity of the cation, piling up more charge in the neighbourhood of the magnetic impurities. An extreme upper limit obtained by distributing the donor electron density entirely around the impurity cations is νΩ|Ψ |2 < 1. Hence Jsd /ρ3 in (6) is replaced by Jsd /ν, at most. In the example, ρ = ν = 13, so the upper limit on TC is 4700 K. To account for the observed values of TC , it is therefore necessary for about 10% of the donor charge density to be concentrated at the ν dopant sites rather than just ν(rc /rH )3 = 0.6% in the example where the charge density in the donor orbital is uniform. The exchange energy associated with the donor centre then becomes 2200 K. In order to achieve significant charge transfer from the donor states to the magnetic impurities, the 3d6 level in the case we are considering (3d5 impurity with S = 5/2) has to be pinned at the Fermi level, and hybridize with the ↓ impurity band states. Hybridization will redistribute the donor electron density over the impurity sites. Spin splitting of the impurity band is

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Fig. 3. Schematic band structure of an oxide with 3d dopants and a spin-split impurity band; the oxygen 2p band is full. The 4s band is practically empty, but the states are spin-split by interaction with 3d dopants. The impurity band is formed from donor states, which may be associated with lattice defects. It too may be spin-split by interaction with the 3d dopants. The left and right panels show the positions of the minority-spin 3d band for low and high Curie temperatures respectively

opposite in sign to that of the 4s conduction band, as shown in the schematic density of states diagrams in Fig. 3. Figure 2b shows the electronic structure necessary for a high Curie temperature when the 3d shell is at least half full. If the 3d shell is less than half full, there is also a possibility of hightemperature ferromagnetism when the 3d ↑ level lies near the top of the band gap, as shown in Fig. 3b. This may be the case for Ti or V in ZnO [7, 13] and for V in TiO2 [38]. As δ increases, there comes a point where the impurity-band model breaks down, and the donor states merge with the bottom of the conduction band. When the Fermi level lies above the mobility edge, the system is fully metallic. It is then appropriate to consider the RKKY exchange mechanism. This interaction may be operative even when the Fermi level lies below the mobility edge, provided the separation of magnetic cation neighbours is less than the localization length. The Fermi wavevector kF remains small at the bottom of the band, and the RKKY interaction is always ferromagnetic at low electron densities. At higher electron densities, the interaction will provide as many negative as positive exchange bonds, and the system then becomes a spin glass, with random spin freezing. The RKKY function F (ξ) = −[(ξcosξ − sinξ)/ξ 4 ] first changes sign for ξ = 2kF r = 4.49. Taking r = (xnO /n)−1/3 and kF = (3π 2 n
)1/3 , the condition for ferromagnetism becomes nn
/xnO < 0.4 or

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x > 2.6nδ

(10)

which is easily satisfied. Considering only the interaction with the closest neighbours, the Curie temperature in the molecular field approximation is given by TC = 2zJRKKY S(S + 1)/3kB where JRKKY is given by the expres2 m∗ kF4 n2 )/(32π 3
2 n2O )]F (ξ). The interaction is ferromagnetic sion [26] [(Jsd provided ξ is small, when F (ξ) ≈ 1/3ξ ≈ (1/6)(x/3π 2 nδ)1/3 . The expression for TC in the RKKY model is therefore 2 TC ≈ zm∗ δx1/3 nJsd S(S + 1)/48π
2 n1/3 n2/3 c kB

(11)

Here TC varies as δx1/3 . Taking Jsd = 1.5 eV, z = 12, S = 5/2, n = 1 and x = 0.1, δ = 0.01 as in the previous example, we find JRKKY = 0.53 K, and TC = 36 K. There is little that can be done to enhance the interaction because the polarization at the bottom of the 4s band is positive, and these states do not hybridize with the empty 3d↓ states at EF . The anticipated behaviour of the magnetism in dilute ferromagnetic oxides is summarized in Fig. 4. There is an extensive ferromagnetic range when γ 3 δ > 4 and x < xp , bounded only by the lines x = 2.6nδ and x = xp . The carriers in the antiferromagnetic state beyond but close to the per-

Fig. 4. The magnetic phase diagram for dilute ferromagnetic semiconductors (from [27])

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colation threshold will tend to produce a canted antiferromagnetic state. High-temperature ferromagnetism for the heavy 3d impurities depends on there being a significant 3d↓ density of states at EF . Figure 5 is a section at x = 0.1 as a function of δ, showing the schematic variation of magnetic ordering temperature.

Fig. 5. Schematic variation of Curie temperature with donor concentration

Thus far, the nature of the defects responsible for the shallow donors has been unspecified. What seems certain is that they provide n-type conduction. The presence of oxygen vacancies at the percent level is documented for the higher oxides SnO2 and TiO2 , [39] and also for ZnO [40], where Zn interstitials may also be found. The vacancies tend to trap one or two electrons, forming F+ or F0 centres, respectively [41]. The singly occupied vacancies are deep donors, with a binding energy in ZnO, for example, which is comparable to the band gap (3.2 eV). The doubly occupied F centre is in a 1s2 spin singlet state, and overlap of the orbitals will lead to a filled impurity band which cannot mediate direct exchange [42]. Overlap with the 4s band, or partial compensation would drain some electrons out of the donor impurity band leading to a ferromagnetic state. Otherwise, a 1s1 2p1 spin triplet excited state for the doubly-occupied F centre leaves the 2p electron in a shallow donor state, which might be stabilized by exchange with the dopants instead of the singlet. The charge density at the cations is somewhat increased if the donors are in a 2p instead of a 1s state, as charge density is removed from the centre of the orbital, and is piled up in a ring where it is more likely to encounter a cation. The energy needed to promote an electron to a 2p state is (3/4)(m∗ /mε2 )R, which is of order 0.1 eV. The exchange energy of the donor electron is about 0.2 eV. An

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intriguing consequence of 2p rather than 1s occupancy is the orbital moment of the donor. According to Hund’s rules, it is in a state with l = 1, s = 1/2 and j = 1/2 where the total moment is antiparallel to the spin, and is therefore parallel to the 3d moment when the 3d shell is half full or more. The Land´e g-factor is 2/3, and the magnetization is therefore enhanced by (1/3) µB per shallow donor. Hence a possibility of obtaining a moment per dopant cation which is a little greater than the spin-only value in the dilute limit. A contribution to the ferromagnetic moment also arises from the spin-splitting of the 4s band, but this will not exceed δµB . In any case, the total moment m per formula unit cannot exceed (x2SµB + δµB ). In the numerical example, this amounts to 5.1µB per cation. However, there is evidence of giant magnetic moments which greatly exceed the spin only moments of the magnetic ions in dilute ferromagnetic oxide [10, 12, 13], they cannot be explained in this way; nor can the enormous anisotropy of the magnetization of some thin films [13, 43]. Recent experiments on undoped HfO2 [43] and other oxides lightly doped with 3d transition metal ions (x  1%) [10, 12, 13, 44] suggest that the 3d dopants are not the only source of magnetism. The defects themselves seem to carry magnetic moments, which in some cases order spontaneously with a high Curie temperature. In that case the defects by themselves may form a spin-split impurity band, regardless of the presence of magnetic cations.

4 Conclusion In conclusion, a simple physical model of indirect exchange via shallow donors has been analysed in relation to dilute ferromagnetic oxides with high Curie temperatures. In order to account for the syndrome set out in the introduction, the Fermi level in the impurity band has to be pinned in the 3d↓ density of states for heavy 3d ions or in the 3d↑ density of states for light 3d ions. The model suggests how the interactions may be controlled by compensation, for example, and it points to how moments slightly larger than the spin-only value could arise. The model may be adapted to other classes of dilute magnetic semiconductors, such as the nitrides, where extended defects may play the role of donors in establishing a spin polarized impurity band. The model cannot account for moments per cation far in excess of the spin-only moment, which have been reported for some dilute oxide systems, nor can it explain the huge anistropy of the saturation magnetization. These observations require another, more unconventional source of magnetism for their explanation. Acknowledgement This work was supported by Science Foundation Ireland. The author is grateful to James Lunney, Sebastiaan van Dijken, Stefano Sanvito and Plamen

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Stamenov for their helpful comments, and to Ciara Fitzgerald for some of the figures, and the unpublished data on SnO2 in Fig. 1a.

References 1. S.J. Pearton et al: J. Appl. Phys 93 1 (2003) 2. W. Prellier, A. Fouchet, B. Mercey: J. Phys. Condens. Matter 15, R1583 (2003) 3. Y. Matsumoto, M. Murakami, T. Shono, T. Hasegawa et al: Science 292, 854 (2001) 185, 186 4. W.K. Park, R.J. Ortega-Hertogs, J. Moodera, A. Punnoose, M.S. Seehra: J. Appl. Phys. 91, 8093 (2002) 5. Z.J. Wang, J.K. Tang, L.D. Tung, W.L. Zhou, L Spinu: J. Appl. Phys. 93, 7870 (2003) 6. K. Ueda, H. Tabata, T Kawai: Appl. Phys. Lett. 79, 988 (2001) 185 7. H. Saeki, H. Tabata, T. Kawai: Solid State Commun. 120, 439 (2001) 193 8. S.J. Han, J.W. Song, C.H. Yang, S.H. Park et al: Appl. Phys. Lett. 81, 4212 (2002) 186 9. P.V. Radovanovic, D.R. Gamelin: Phys Rev. Lett. 91, 157202 (2003) 10. S.B. Ogale, R.J. Choudhart, J.P. Buban, S.E. Lofland et al: Phys. Rev. Lett. 91, 077205 (2003) 185, 196 11. J.M.D. Coey, A.P. Douvalis, C.B. Fitzgerald, M. Venkatesan: Appl. Phys. Lett. 84, 1332 (2004) 12. C.B. Fitzgerald, M. Venkatesan, J.M.D. Coey (unpublished) 186, 196 13. M. Venkatesan, C.B. Fitzgerald, J.G. Lunney, J.M.D. Coey: Phys. Rev. Lett. 93, 177206 (2004) 186, 193, 196 14. S. Sonoda, D.S. Shimizu, T. Sasaki: J. Cryst. Growth 237-239, 1358 (2002) 15. S.Y. Wu, H.X. Liu, L. Gu, R.K. Singh et al: Appl. Phys. Lett., 82, 3047 (2003) 16. D. Kumar, J. Antifakos, M.G. Blamire, Z.H. Barber: Appl. Phys. Lett. 84, 5004 (2004) 17. H.X. Liu, S.Y. Wu, R.K. Singh, D.J. Smith, N. Newman, N.R. Dilley, L. Montes, M.B. Simmonds: Appl. Phys. Lett. 85, 4076 (2004) 18. J.Y. Kim, J.H. Park, B.G. Park, H.J. Noh et al: Phys. Rev. Lett. 90, 017401 (2003) 186 19. A. Punnoose, M.S. Seehra, W.K. Park, J.S. Moodera, Appl. Phys. Lett. 93, 7867 (2003) 186 20. K. Rode, A. Anane, R. Mattana, J.P. Contuor et al: J. Appl. Phys. 93, 7676 (2003) 186 21. Two cases where the moment in an oxide ferromagnet comes close to the spinonly value are in double perovskites such as Sr2 FeMoO6 where the iron is coupled ferromagnetically by ↓ electrons in a 4d/3d conduction band, giving moments 4µB per iron and in mixed valence manganites such as (La0.7 Sr0.3 )MnO3 where the manganese is coupled ferromagnetically by double exchange. 186 22. J.B. Goodenough: Magnetism and the Chemical Bond, Interscience, New York (1963) 186 23. F. Holtzberg, S. von Molnar, J.M.D. Coey in Handbook on Semiconductors Vol 4, North Holland, p. 803, (1980) 188 24. T. Dietl: Semiconductor Science and Technology 17, 377 (2002) 188

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25. T. Dietl, H. Ohno, F. Matsukara, J. Cibert, D. Farrand: Science 287, 1019 (2000) 188 26. D. Matthis: Theory of Magnetism I, Springer, Berlin (1981) 188, 194 27. J.M.D. Coey, M. Venkatesan, C.B. Fitzgerald: Nature Materials, 4 (2005) 188, 192 28. T. Dietl, J. Spalek: Phys. Rev. B 28, 1548 (1983) 29. P.A. Wolff in Semiconductors and Semimetals. Vol 25 (J. R. Furdyna and J. Kossut, editors) Academic Press, San Diego 1988. 30. D.E. Angelescu, R.N. Bhatt: Phys. Rev. B 65, 075211 (2002) 31. A.C. Durst, R.N. Bhatt, P.A. Wolff: Phys. Rev. B 65, 235205 (2002) 189 32. S. Das Sarma, E.W. Hwang, A. Kaminski: Phys. Rev. B 67, 155201 (2003) 33. N.F. Mott: Conduction in noncrystalline materials, Oxford University Press (1987) 189, 190 34. R.M Zallen: Physics of Amorphous Solids, Wiley, New York (1983) 190 35. T. Kasuya: Solid State Commun. 8, 1635 (1970) 192 36. K.J. Kim, Y.R Park: Appl. Phys. Lett. 81, 1420 (2002). The red shift is proportional to x. Jsd is obtained by extrapolation to x = 1. 192 37. M. Berciu, R.N. Bhatt: Phys. Rev. Lett. 87, 107203 (2001) 192 38. N.Y. Hong, J. Sakai, A. Hassini: Appl. Phys. Lett. 84, 2602 (2004). 193 39. J. Yahia: Phys. Rev. 130, 1711 (1963) 195 40. A.F. Kohan, D. Ceder, D. Morgan, C. van de Walle: Phys. Rev. B 61, 15019 (2000) 195 41. B. Henderson and G. F. Imbusch, Optical spectroscopy of inorganic solids, Oxford University Press (1989) 195 42. The doubly-occupied 1s2 F centres will mediate a weak antiferromagnetic superexchange, analogous to the antiferromagnetic superexchange mediated by the 2p6 oxygen ion. 195 43. M. Venkatesan, C.B. Fitzgerald, J.M.D. Coey: Nature 430, 630 (2004) 196 44. J.M.D. Coey: J. Appl. Phys. (2005) 196 45. S.N. Kale, S.B. Ogale, S.R. Shinde, M. Sahasrabuddhe et al: Appl. Phys. Lett. 82, 2100 (2003) 46. J. Philip, N. Theodoropoulou, G. Berera, J.S. Moodera, B. Satpati: Appl. Phys. Lett. 85, 777 (2004)

Part I

Half-Metallic Ferromagnets

Half-Metals: Challenges in Spintronics and Routes Toward Solutions J.J. Attema1 , L. Chioncel1 , C.M. Fang1 , G.A. de Wijs1 and R.A. de Groot12 1

2

Electronic Structure of Materials, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands Laboratory of Chemical Physics, MSC, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

Abstract. The current status of half-metallicity is briefly reviewed. Three origins of half-metallicity are distinguished: In heusler C1b compounds covalent interactions similar as in III-V semiconductors are responsible. The effect is intimately related with the crystal structure, while the electron-electron interactions are weak. A second category is found in the limit of strong magnetism. Chromium-dioxide is an archetype here. In the third category, strong electron-electron interactions can be the origin of half-metallicity like in magnetite and the double perovskites. Experiments probing half-metallicity are reviewed with a distinction between the behaviour of genuine bulk, surfaces and interfaces. The effects of temperature and disorder on half-metallicity are discussed. In general increased temperature will reduce spin-polarisation as magnons are excited. Calculations show that suitably chosen impurities counteract this effect, and may actually improve the spin-polarization at finite temperature. Finally, spin injection from nano-scale structured half-metallic contacts is discussed.

The electronic structure of a magnetic solid is characterized by two bandstructures for the two spin-quantization directions[1][2]. A half-metal is defined as a material, where the band-structure for one spin-direction is metallic, while it is semi-conducting for the other spin-direction. (The description with two independent band-structures is possible as long as the electron-spin is a valid quantum number, i.e. when the spin-orbit interaction is small compared with the exchange interaction). There are several direct consequences of the half-metallic properties. For example, the electrical conduction takes place by charge carriers of one spindirection exclusively, which suggested possibilities of exploiting the spin degree of freedom in logical devices. Another consequence is that the magnetic moment per unit cell is necessarily an integral number. This follows from the integral total number of electrons as well as integral number of electrons for the semi-conducting spin-direction, since the number of electrons for the metallic spin-direction is simply the difference between these two quantities. A fundamental property of a half-metal is the band-gap. The origin of band-gaps can be quite distinct and consequently, the influence of atomic order, deviations from perfect stoichiometry, impurities, temperature, pressure etc. on the semi-conducting properties can be different. Since the half-metallic J.J. Attema et al.: Half-Metals: Challenges in Spintronics and Routes Toward Solutions, Lect. Notes Phys. 678, 201–216 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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properties depend sensitively on the band-gap, the origin of this band-gap forms a logical choice for the classification of half-metals[3]. In this classification we will distinguish three categories: half-metals with a covalent band-gap like in the group IV elemental and III-V semiconductors. The second category is formed by half-metals with a charge-transfer band-gap, where the wave-function character of the valence-band on one hand and the conduction band on the other hand are associated with different chemical species in an ionic compound. Finally we mention half-metals with a d − d gap, where both the valence band and the conduction-band show primarily d-wave-function character. The considerations in this chapter are mainly based on electronic structure calculations. No technical details are given here; we refer the interested reader to the original literature. Many of the concepts discussed here are worked out computationally in actual materials based on NiMnSb. This is because it could be considered the archetype of half-metallicity and consequently has been studied in more detail both experimentally as well as theoretically. Also, it is relatively simple computationally, it shows a high Curie temperature and shows strong similarities in crystal and electronic structure with zinc-blende semiconductors. There are no reasons, however, why similar concepts could not be implemented for other half-metals.

1 Half-Metals with a Covalent Band-Gap The archetype of half-metals in this category is NiMnSb. It crystallizes in the Heusler C1b structure, which is closely related to the diamond and zincblende structures. The latter ones are formed by an fcc cubic Bravais lattice with a basis of two atoms at 0,0,0 and 14 , 14 , 14 . The Heusler C1b structure differs only by the presence of a third species at 12 , 12 , 12 . The precise occupancy is of importance for the half-metallic properties: the nickel occupies the “middle” 14 , 14 , 14 position. NiMnSb shows a band-gap at the Fermi energy for the minority-spin electrons. For this spin-direction the compound is pseudo-iso-electronic with the zinc-blende semiconductors: the antimony is the pentavalent species while, since the manganese shows a magnetic moment of 4µb , its minority configuration is trivalent. The manganese d − t2g electron plays the role of the metal p electron in the zinc-blende semiconductors, which is possible because of the lack of inversion symmetry. It is vital that the trivalent and pentavalent atoms occupy the sites with tetrahedral coordination, as this is vital for the III-V semiconductors, while the role of the nickel is to mediate the manganese-antimony interactions and to supply the 10 electrons needed to complete the 18 electrons (9 per spin-direction) to qualify as an octet compound in period 4 of the periodic table. Because of the exchange splitting of the manganese 3d levels, the positions of the atomic energy levels are also quite comparable with the case of the zinc-blende semiconductors.

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For the majority spin-direction a different situation occurs, with the manganese d-levels much lower in energy now, roughly degenerate with the nickel d states and the antimony states. This leads to broad energy-bands characteristic for an alloy for the majority spin-direction. Three bands intersect the Fermi energy. As a consequence a fundamentally different situation exists in comparison with half-metals in other categories. As we will see, in the latter cases half-metallicity occurs because of a rigid-band like shift of the energybands for the two spin-directions with respect to each other caused by the exchange interaction, causing the Fermi-level to fall within the band-gap for one spin-direction but not for the other. In NiMnSb the band-structures for the two spin-directions are basically different with a band-gap for the minority spin-direction that simply does not exist for the majority spin. These findings are summarized in Fig. 1.

(A)

(B)

Fig. 1. Band structure of NiMnSb for the majority spin-direction (A) and the minority spin-direction (B)

The importance of the lack of inversion-symmetry in the Heusler C1b structure is best exemplified by considering the Heusler L21 structure. In this case the unoccupied ( 34 , 34 , 34 ), position in the C1b structure is occupied by a second Ni, leading to a compound of composition Ni2 MnSb. This introduces an inversion center with dramatic consequences for the band-gap in the minority spin-direction. The inversion center does discriminate d functions from p-functions, properly speaking in the language of group theory: the d functions transform as a different irreducible representation as p functions in the presence of an inversion center, consequently the interactions between the trivalent manganese d states and the pentavalent antimony states are no longer allowed. These bands now cross rather than repel each other and the band-gap has disappeared. The density of Ni2 MnSb is also different from that of NiMnSb. In order to separate the effect of the inversion center from that of the density, the band-structure of Ni2 MnSb with the density of NiMnSb is plotted in Fig. 2. In order to complete the comparison, the band-structure of NiMnSb with the density of Ni2 MnSb is plotted in Fig. 3. The higher den-

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(A)

(B)

Fig. 2. Band structure of Ni2 MnSb for the majority spin-direction (A) and the minority spin-direction (B) Note that the band crossings have eleminated the band gap

(A)

(B)

Fig. 3. Band structure of NiMnSb compressed to the density of Ni2 MnSb.(A): majority, spin, (B): minority spin

sity as compared with equilibrium NiMnSb has actually increased the size of the band-gap. An increase of the band-gap under compression is a rare but certainly not unique phenomenon: it occurs frequently in III-V semiconductors. Due to the increased kinetic energy in the phase under compression the Fermi-level is positioned in the bottom of the conduction-band now. There are several other Heusler C1b compounds, which show a bandgap for the minority spin-direction in the vicinity of the Fermi-level. PtMnSb is calculated to be a half-metal[4], but the inclusion of spin-orbit interaction is likely to produce an overlap between the top of the valance band and the Fermi-level. In PdMnSb, IrMnSb, RhMnSb, PtMnSn and OsMnSb[5] this overlap occurs already without the inclusion of spin-orbit interaction. It quite possible that these compounds could be turned into half-metals by suitable doping. CoMnSb[6] and FeMnSb[7] are calculated to be half-metals, but experimental reality is more complex here. CoMnSb actually forms a superstructure with a size of its magnetic moment that does not suggest half-metallic properties. FeMnSb is always deficient in iron, but the magnetic moment as a function

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of composition does extrapolate to the 2µb calculated for the stoichiometric compound. Consequently, the compound could very well be half-metallic. A more recent development is the growth of metastable half-metallic transitionmetal pnictides like CrAs[8] and CrSb[9] on top of II-V semiconductors.

2 Half-Metals with a Charge-Transfer Band-Gap There exists a direct relation between half-metallicity in the second category and strong magnetism. Strong magnetism is defined as a situation where an (hypothetical) increase of the strength of the exchange-interaction does not lead to an increase in magnetic moment. In practice this means that either the majority-spin d band is completely occupied or the minority spin d-band is empty. Thus, iron is a weak magnet, while nickel is a strong magnet. But in spite of its completely polarized d-band nickel is not a half-metal: its s and p electrons, which show hardly any spin-polarization whatsoever, determine the electrical conductivity. In the formation of ionic compounds it are exactly these itinerant electrons, which are preferentially transferred and thus localised on an anion, and consequently half-metallic properties can occur. In practice up till now only cases have been found, where the minority spin d-shell is empty, but there are no fundamental reasons why the occurrence of half-metallicity should be confined to these cases. The prime example of half-metals in this category is CrO2 [10]. The Cr4+ ion possesses 2 d-electrons, which occupy the majority d-band straddled by the Fermi-level. Due to the exchange splitting, the empty minority d-levels are higher in energy and form the conduction band above the band-gap. Many examples of half-metals in this category are found in the colossal magneto-resistance materials[1]. The occurrence of half-metallicity in this category is not critically dependent on the crystal structure and/or imperfections.

3 Half-Metals with a d − d Band-Gap In systems where, in the vicinity of the Fermi energy, bandwidths are small as compared with exchange splitting, ligand field splitting, etc. many band-gaps occur and consequently half-metals could be expected to be more the rule than the exception. However, most of these systems are actually Mott insulators rather than half-metals. Nevertheless, systems exist with bandwidths large enough to enable metallic conduction but narrow enough to limit this conductivity to one spin-direction exclusively by the scenario outlined above. The archetype in this category is magnetite (Fe3 O4 )[11]. It is an insulator at low temperatures and transforms at 120K into a half-metal. It crystallizes in the spinel structure, as do many transition-metal oxides. Among them, especially in the ternary ones, many half-metals could be expected. These compounds are usually Mott-insulators. An important recent development

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here is the occurrence of half-metallicity in the double perovskites. Where mixed oxides based on 3d-metals frequently show Mott-insulating behaviour, the insulating state can be suppressed by admixing 4d or 5d transition metals with more delocalised d-electrons[12]. These double-perovskites have high Curie temperatures as compared with the simple perovskites. The inclusion of the double-perovskites in category 3 is not uncontroversial. Since the dstates of the valence en conduction bands originate from different transition metals, inclusion in category 2 could be considered. Since a charge transfer gap is generally assumed to describe wave function characters of valence and conduction bands centered on oppositely charged ions, inclusion in category 3 is more appropriate. It is noteworthy to mention that several examples in this category have an energy-gap for the majority spin-direction. No specific sensitivity on the crystal structure is expected here, but the introduction of impurities in systems in this category can stabilize the competing Mott insulating state. We conclude the classification of half-metals with some remarks about the relation between strong magnetism and half-metals. Although the two phenomena may intuitively look related, actually only the half-metals in the second category are strong magnets. Half-metals of the third category are weak magnets by definition: a d−d gap for the semiconducting spin-direction implies both occupied and empty d-states. The half-metals in the first category are also weak magnets. The importance of the presence of minority-spin d electrons for the occurrence of a band-gap has been mentioned before and is exemplified in Fig. 4. It shows the bandstructure of NiMnSb where the compound is forced to be a strong magnet by a sufficiently strong increase in the exchange interaction. The half-metallic properties are lost, just as GaAs would be a metal if gallium were deprived of its p-states. Clearly halfmetallicity and strong magnetism are mutually exclusive in this category.

(A)

(B)

Fig. 4. Band structure of NiMnSb as a strong magnet: (A) majority spin, (B): minority spin

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4 Experiments at Low Temperatures Most considerations until now have been based on electronic structure calculations. This may come as a surprise, but as profound as a property as half-metallicity may be, there is no “smoking gun” experiment to prove or disprove it. The requirement for a half-metal to pocess an integer magnetic moment, as has been mentioned before, is a necessary but certainly not sufficient condition. Two different routes for the experimental verification of the half-metallic properties have been pursued. The first one is the study of the Fermi surface in order to establish that there exists one for only one spin-direction. The second strategy is based on measurements of the spin-polarization of electrons extracted from a half-metal with methods like spin-resolved photo-emission, Andreev reflection etc. The measurement of the spin-dependence of the Fermi surface seems to be the most direct proof, since it does not depend on surface reconstructions and segregations nor contaminations. (It should be realized that even an unreconstructed, stoichiometric surface of a half-metal is not necessarily half-metallic itself). On the other hand, more familiar methods of measuring the Fermi surface like the De Haas Van Alphen effect (DHVA) do not measure the spin-dependence of the Fermi surface directly. Of course one could construct experimental Fermi surfaces based on DHVA measurements and use a possible agreement with a half-metallic band-structure as evidence. However, DHVA measurements require high quality single crystals, high magnetic fields and low temperatures, even if these are available, small pockets of a Fermi surface can easily be overlooked leading to incorrect conclusions. A direct measurement of the spin-dependent Fermi surface is provided by positron annihilation. Also the sensitivity of this measurement does not depend on Dingle temperatures, so there are no problems associated with the sensitivity for small pockets or parts of the Fermi surface with higher effective masses of the charge carriers. Since these expensive and tedious measurements are not commonly described in textbooks on solid state physics, we will briefly describe them here. The positron is the anti-particle of the electron. When a positron-electron pair in the singlet state annihilate, two identical γ particles emerge in exactly opposite directions if both the electron and positron were at rest. Since a positron thermalizes instantaneously on entering a solid, deviations from 180 degree for the angles of emergence of γ quanta are a direct measure of the momentum of the annihilated electron in the crystal. So, annihilation of an electron at the Γ point of the Brillouin zone leads to angles of 180◦ , these angles will deviate from 180 degree progressively in following an energy-band towards the Brillouin-zone boundary. If an energy-band crosses the Fermi energy however, a cut-off in the detection of γ quanta as a function of the angle is observed, since no electrons are available above the Fermi energy at low temperatures. In other words, the measurements of the cut-offs (as a function of angle!) leads to a direct determination of the Fermi surface. The reason that this measurement is spin-selective orig-

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inates from the following. First, annihilation of an electron-positron pair in a triplet state leads to the emission of three γ quanta with very different energies (and angular distributions) as compared with the case of annihilation of a singlet pair. Thus these two processes can easily be discriminated. Second, a source of spin-polarized positrons is available because of the parity conservation in the β-decay of 22 N a. In this way two measurements using a single domain single-crystal with the direction of the magnetization parallel and anti-parallel to the direction of the positron beam allows the measurement of the Fermi surfaces for the two spin-directions directly. This way, the half-metallic properties of NiMnSb were established experimentally within the experimental accuracy of .01 electron[13]. Unfortunately, this is the only measurement reported on a half-metal. Partially this is because of the lack of single-crystals of half-metals. Unfortunately magnetite, which is omnipresent in single-crystalline form, transforms at 120 K to an insulating phase. Measurements of the spin-polarization of electrons extracted from halfmetals give a confusing picture. Spin-resolved photo-emission on the colossal magneto-resistance material La1−x Cax MnO3 showed half-metallic properties[14]. In NiMnSb, Andreev-reflection reported 58% polarization, spinpolarized tunnelling 28%[15], whereas inverse photo-emission resulted in close to 100%[16] under certain conditions. Probably the more consistent are obtained now for CrO2 , with, for example, Andreev reflection leading to polarizations in access of 90%[17]. It should be realized that many, if not all of these systems show surface segregations and results are very much dependent on experimental conditions. Only a few surfaces have been investigated with electronic structure calculations. The (001) surface of CrO2 has a stoichiometric composition[18] and consequently could be stable without reconstructions or segregations. It shows two two-dimensional surface states with primarily oxygen wavefunction character just at the top of the valence band for the minority spin direction. Nevertheless, the half-metallic properties are conserved because these surface states are still far away from the Fermi-level in energy. Remarkably the 001 faces do not occur on CrO2 crystals and data on more complex surfaces are lacking. For NiMnSb all possible (100), (110) and (111) surfaces were calculated including surface relaxations. Not one single surface was found that maintained the half-metallic properties of the bulk[19]. These negative findings raise doubt whether half-metals could ever play a role in spin-injection in for example semiconductors. But from the point of view of the half-metal, a III-V semiconductor at an interface constitutes a much smaller perturbation for the electronic structure as compared with the vacuum at a surface, at least for a half-metal of the first category. This expectation motivated computational studies of all possible low index interfaces between NiMnSb on one hand and CdS and InP on the other hand. These zinc-blende semiconductors were selected because they show a minimal mismatch with NiMnSb. Indeed an interface could be obtained that

Half-Metals: Challenges in Spintronics and Routes Toward Solutions Ni Ni5

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Fig. 5. The NiMnSb – CdS interface. Top: structure of the interface. Bottom: local density of states as a function of energy for majority and minority spin-direction (full lines). Dashed likes: density of states for bult NiMnSb. The numbers refer to the atoms in the top figure

showed genuine semi-conducting properties for the semiconductor, the interface as well as the half-metal for the minority spin-direction (Fig. 5). It is an interface along the (111) direction, where the half-metal is terminated by Sb, while the semiconductor is terminated by the anion[19]. The occurrence of these anion pairs may be somewhat unexpected, however they occur in minerals like gudmundite, costibite and paracostibite with also a metal coordination very similar to the interface. Clearly, much more work is needed, both experimentally as well as computationally to obtain a proper insight in the very complex issues of surfaces and interfaces.

5 Finite Temperatures All considerations until now did neglect the effect of finite temperatures. The thermal excitations of interest in half-metals at reasonable temperatures are the magnon excitations. They result in the occupation of spin-spiral states with wavelengths decreasing with increasing energy, so the long-wavelength

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magnons are thermally more easily accessible. The effect on a local scale is to generate a finite density of states in the gap of the half-metal, thus reducing the spin-polarization of the conduction electrons. This effect increases with temperature, till finally at the Curie temperature, no net spontaneous polarization persists. Because of this phenomenon, the application of half-metals as a source of spin-polarized charge carriers should be limited to temperatures well below the Curie temperature. Several half-metals with high Curie temperatures exist (NiMnSb 740K, Fe3 O4 868K, CrO2 397K). The magnon excitations are an intrinsic property of a magnetic solid and the effect given a compound and the temperature there is not too much to influence them except a rather weak influence of the sample shape (which is not a practical variable), an applied magnetic field and the influence of the particle size. From a materials point of view, there are several possible strategies to control to some extent the influence of temperature on the degree of spinpolarization of the conduction. For example, one could deliberately introduce disorder in the half-metal. This leads to the destruction of coherent magnon bands resulting in a discrete rather than continuous magnon spectrum. Depending on the details of these spectra, the influence of temperature will be modified, increasing the performance of the half-metal as source of spinpolarized particles at some temperatures, while degrading the performance at other temperatures. The drawback is of course that not only the coherence of the magnon states is affected, but the coherence of the electron states as well in general. The trick (or the art) is to design a type of disorder that has maximal impact on the coherence of the magnon states with minimal impact on the electronic states. Another route towards improvement is increasing the magnetic anisotropy. The magnetic anisotropy lifts the degeneracy in energy of the direction of the magnetization with respect to the crystal lattice. It is relativistic in origin and is dominated by the spin-orbit interaction, which directly couples the direction of the magnetization (spin) to that of the lattice (orbit). Usually it is weak for cubic systems, but much stronger for uniaxial systems. It is particularly weak in NiMnSb, which is cubic and shows a band-structure, which does not allow spin-orbit interaction to play a role of importance: The only intersections with the Fermi-level occur halfway the Brillouin zone by dispersive bands. The effect of the magnetic anisotropy is to introduce gaps in the magnon spectrum at the centre of the magnetic Brillouin zone, making magnon excitations thermally less accessible. Since the magnetic anisotropy operates exactly on those states, which are most easily influenced, a beneficial effect is to be expected at any temperature. The remainder of this chapter is devoted to materials aspects of these three strategies to optimise performances at finite temperatures: disorder, magnetic anisotropy and size effects (nano-structured contacts).

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6 Disorder At first glance the introduction of impurities in half-metals looks more of a curse than a blessing, especially in the first category. On closer examination it becomes clear that each imperfection should be investigated by itself and nothing is gained by a priori assumptions. In a computational study on the influence of 14 different kinds of intrinsic defects in NiMnSb 5 turned out to be detrimental for the half-metallic properties and these were energetically extremely unstable. These 5 cases are directly related to the destruction of the tetrahedral coordination of the manganese and antimony. So, both forms of manganese-nickel interchange (either as a neighbouring pair or as isolated entities) destroy half-metallic properties. In the case of nickel-antimony interchange, the half-metallic properties are only lost in the case of isolated atoms, as for nearest neighbour paired defects half-metallic properties are conserved. Contrary to this, the interchange of manganese and antimony destroys the half-metallic properties only as a paired defect, as isolated defects show a localized occupied impurity state for the minority spin, not corrupting the half-metallic properties. The same behaviour is found for an antimony atom displaced to an empty ( 34 , 34 , 34 ) position. Displacing manganese or nickel atoms to the empty position does not affect the half-metallic properties nor do reasonable deviations from the perfect nickel stoichiometry. There is no evidence, neither theoretically nor experimentally, for the occurrence of these defects and even if samples could be mistreated badly enough to show these forms of disorder, there is no indication why they should affect magnon-states more profoundly than electron-states. Consequently we will explore the possibilities of deliberate doping here. The primary concern in the minimization of the impact of the doping on electronic properties is clearly the preservation of the band-gap for the minority-spin direction. Magnonically, the influence is most directly realized by the random suppression of magnetic moments on the magnetic lattice, based at the manganese sites in NiMnSb. The band-gap is expected to remain unaffected, as long as we substitute manganese by any other trivalent transition metal. The suppression of the magnetic moment requires identical configurations for the two spin-directions for the substituting atom. In other words, a partial substitution of manganese by scandium fulfils all the requirements. There is a very good chance that this substitution could actually be realized in practice, since NiScSb does actually exist and even crystallizes in the Heusler C1b structure just as NiMnSb[20]. It is a non-magnetic semiconductor (A technicality here. Density functional theory seriously underestimates band-gaps. So the actual band gap in NiScSb is likely to be well in access of the 0.5 electron-volt calculated. Since the dielectric properties of a half-metal are those of a metal, density functional theory is believed to yield correct band-gaps in this case). Experimentally hardly anything more than the crystal structure is known about NiScSb. Figure 6 shows the density of states of NiMnSb with 3% of the manganese substituted by scandium. Clearly, the half-metallic properties

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are completely conserved, but magnetic moments are absent at the random Sc sites.

7 Modifications in the Magnetic Anisotropy The dominant mechanism behind the magnetic anisotropy is the spin-orbit interaction. Enhancing the spin-orbit interaction in a half-metal with spininjection as a goal is not without danger: the spin-orbit interaction of the charge-carriers leads to a mixing of the two spin-components already at zero temperature, with as a consequence a reduction of the spin-polarization of these charge carriers – exactly the quantity we tried to optimise at finite temperatures. Ideally, one would like to have a strong spin-orbit interaction to take place in some core level, safely locked away from the Fermi energy. This situation can actually be approached in the lanthanides, where the spin-orbit interaction of the localized 4f electrons, which are not involved in chemical bonding themselves, is responsible for the magnetic anisotropy. Several considerations are of importance in the selection of the lanthanide atom. First, since we are considering a partial substitution of manganese by a lanthanide and the half-metallic properties (band-gap) are expected to be conserved only for a trivalent atom, Europium (and Cerium) are to

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be avoided. Secondly, from the point of view of the spin-orbit interaction, one needs a spin-moment (optimally half-way the lanthanide series), but also an orbital moment (zero half-way the series, optimally one quarter and three quarter the series). Good candidates are elements like promethium or holmium. The likelihood that lanthanides can substitute for manganese in NiMnSb is indicated by the occurrence of stable phases of composition NiLnSb, which, for the heavier lanthanides, even crystallize in the Heusler C1b structure, identical to NiMnSb. The calculated density of states of NiMnSb with 17% of the manganese replaced by holmium is shown in Fig. 7 for the stable case of the holmium magnetic moment coupled anti-ferromagnetically with respect to the direction of the manganese sublattice magnetization. 40 30 20 10 0 −10 −20 −30 −40 −12

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Fig. 7. Density of states as a function of energy for holnium doped NiMnSb. Dashed line indicates the Fermi energy. Note the presentation of the band gap for the minority spin-direction. Horizontal scale: electron-volt. Vertical scale: states/f.u. eV

This introduces yet another consideration. So far, we have been considering only the frustration of magnon excitations by substitutions on the magnetic lattice by impurities, whose magnetic moment is so tightly bound to the lattice, that they will not participate in a spin-wave. If, however, the coupling of these impurities to the magnetic sublattice is weak, as compared with temperature, no beneficial effect is to be expected. Table 1 lists calculated coupling strengths between the magnetic moment of some rare-earth impurities and the manganese sublattice magnetization in degrees Kelvin, as well as the most stable magnetic configuration. Which rare earths should be selected depends on the goal. Promethium and uranium have been included for didactical purposes only: the first element does not occur in nature, consequently it is radio-active and expensive, the second should be avoided because of the depolarisation of the conduction electrons

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Coupling Strength (K) Coupling Type

NIMnSb: Pm NIMnSb: Ho NIMnSb: U

560 52 3000

AF AF F

to be expected from the spin-orbit interaction of the more diffuse 5f states. Holmium is the prime element to be considered at low temperatures, a light lanthanide could perform better at higher temperatures.

8 Nano-Sized Contacts The optimisation of spin-injection at finite temperature through the replacement of a continuous magnon-spectrum by a discrete one was discussed above with the partial substitution of manganese by scandium. Another way of realizing the same objective is to exploit the fact that finite size systems do not show a continuous excitation spectrum. In other words, rather than a normal interface we consider a contact comprising a multitude of nanoscopic halfmetallic particles on top of the material in which spin-polarized charge carriers are needed. At first instance this looks like a formidable task. The surface of a half-metal is seldomly halfmetallic and consequently, an isolated halfmetallic nanosized particle will not be halfmetallic for a substantial part of its volume. It does not suffice for a halfmetal to show one halfmetallic surface since nanoparticles usually show several surfaces. It is clear that the notion of freestanding nanosized particles has to be abandoned. It is worth to explore the possibility of embedding the nanocontacts in some suitable medium. The halfmetallic properties of NiMnSb, lost at the surface, could be restored at the interface with CdS (or InP). But this restoration only worked for one particular interface in one direction (111), while the embedding should restore the halfmetallic properties of the surface of NiMnSb in all possible directions. The choice of CdS and InP in the interface studies was motivated, however, by the general interest in zinc-blende semiconductors, the embedding phase for NiMnSb nanocontacts serves a much more restricted goal and subsequently other materials could be considered. The best candidate is NiScSb. The densities of states of the interfaces for the three principle stackings are given in Fig. 8. Genuine halfmetallic behaviour is found for all three directions, so there is good hope this will work. The size of the NiMnSb particles can only be determined after the temperature of optimal operation is specified. Another important parameter is the distance between particles. In the limit of small interparticle distance behaviour similar to a macroscopic contact will be obtained. In the limit of large interparticle distances the Curie temperature will drop till eventually the nanoscopic particles could even be-

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have as superparamagnetic entities, requiring an external magnetic field for alignement.

References 1. W.E. Pickett, J.S. Moodera: Phys. Today 54, 39-44 (2001) 201, 205 2. R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow: Phys. Rev. Lett. 50, 2024 (1983) 201 3. C.M. Fang, G.A. de Wijs, R.A. de Groot: Appl. Phys. 91, 8340-4 (2002) 202 4. R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow: J. Appl. Phys. 55, 2151-4 (1984) 204 5. R.A. de Groot, K.H.J. Buschow: J. Magn. Mater. 54-57, 1377-80 (1986) 204 6. J. K¨ ubler: Physica B 127, 257-63 (1984) 204 7. R.A. de Groot, A.M. van der Kraan, K.H.J. Buschow: J. Magn. Mater. 61, 330-6 (1986) 204 8. H. Akinaga, T. Manago, M. Shirai: Jpn. J. Appl Phys. 39, L 118-20 (2000) 205 9. J.H. Zhao, F. Matsukura, K. Tahamura, E. Abe, D. Chiba, H. Ohna: Appl. Phys. Lett. 79, 2776-8 (2001) 205 10. K. Schwartz: J. Phys. F 16, L 211-15 (1986) 205 11. A. Yanase, K. Siratori: J. Phys. Soc. Japan 53, 312-17 (1984) 205 12. H.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, Y. Tokura: Nature 395, 677-80 (1998) 206 13. K.E.H.M. Hanssen, P.E. Mijnarends, L.P.M. Rabou, K.J.H. Buschow: Phys. Rev. B 42, 1533-40 (1990) 208 14. J.H. Park, E. Vescova, H.J. Kim, C. Kwon, R. Ramesh, T. Venkatesan: Nature 392, 794-6 (1998) 208 15. C.T. Tanaka, J. Nowak, J.S. Moodera: J. Appl. Phys. 86, 6239-42 (1999) 208 16. D. Ristoiu, J.P. Nozieres, C.N. Borca, T. Komesu, H.K. Jeong, P.A. Dowben: Europhys. Lett. 49, 624-30 (2000) 208

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17. R.J. Soulen, J.M. Byers, M.S. Osofsky, B. Nadgarny, T. Ambrose, S.F. Cheng, P.R. Broussard, C.T. Tanaka, J.S. Moodera, A. Barry, J.M.D. Coey: Science 282, 85-8 (1998) 208 18. H. van Leuken and R.A. de Groot: Phys. Rev. B 51, 1171-3 (1995) 208 19. G.A. de Wijs, R.A. de Groot: Phys. Rev. B 64, 020402 (2001) 208, 209 20. V.K. Pecharskii, Y.V. Pankevich, O.I. Bodak: Kristallografiya 28, 173-4 (1983) 211 21. D.W. Ross: Lysosomes and storage diseases. MA

Nonquasiparticle States in Half-Metallic Ferromagnets V.Yu. Irkhin1 , M.I. Katsnelson2 and A.I. Lichtenstein3 1 2

3

Institute of Metal Physics, 620219, Ekaterinburg, Russia Department of Physics, Uppsala University, Box 530, SE-751 21 Uppsala, Sweden Institute of Theoretical Physics, University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany

Abstract. Anomalous magnetic and electronic properties of the half-metallic ferromagnets (HMF) have been discussed. The general conception of the HMF electronic structure which takes into account the most important correlation effects from electron-magnon interactions, in particular, the spin-polaron effects, is presented. Special attention is paid to the so called non-quasiparticle (NQP) or incoherent states which are present in the gap near the Fermi level and can give considerable contributions to thermodynamic and transport properties. Prospects of experimental observation of the NQP states in core-level spectroscopy is discussed. Special features of transport properties of the HMF which are connected with the absence of one-magnon spin-flip scattering processes are investigated. The temperature and magnetic field dependences of resistivity in various regimes are calculated. It is shown that the NQP states can give a dominate contribution to the temperature dependence of the impurity-induced resistivity and in the tunnel junction conductivity. First-principle calculations of the NQP states for the prototype half-metallic material NiMnSb within the local-density approximation plus dynamical mean field theory (LDA+DMFT) are presented.

1 Introduction Half-metallic ferromagnets (HMF) [1, 2, 3] have attracted recently scientific and industrial attention due to their importance for spin-dependent electronics or “spintronics” [4]. The HMF have metallic electronic structure for one spin projection (majority- or minority-spin states), but for the opposite spin direction the Fermi level lies in the energy gap [1]. Therefore the spin-up and spin-down contributions to electronic transport properties have different orders of magnitude, which can result in a huge magnetoresistance for heterostructures containing the HMF [2]. At the same time, the HMF are very interesting conceptually as a class of materials which may be suitable for investigation of the essentially manybody physics “beyond standard band theory”. In most cases many-body effects lead only to renormalization of the quasiparticle parameters in the sense of Landau’s Fermi liquid theory, the electronic liquid being qualitatively similar to the electron gas (see, e.g., [5, 6]). On the other hand, due to specific V.Yu. Irkhin et al.: Nonquasiparticle States in Half-Metallic Ferromagnets, Lect. Notes Phys. 678, 217–243 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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band structure of the HMF, an important role belongs here to incoherent (nonquasiparticle, NQP) states which occur near the Fermi level because of correlation effects [2]. The appearance of NQP states in the energy gap near the Fermi level is one of the most interesting correlation effects typical for the HMF. The origin of these states is connected with “spin-polaron” processes: the spin-down low-energy electron excitations, which are forbidden for the HMF in the standard one-particle scheme, turn out to be possible as superpositions of spin-up electron excitations and virtual magnons. The density of these nonquasiparticle states vanishes at the Fermi level, but increases drastically at the energy scale of the order of a characteristic magnon frequency ω. The NQP states were first considered theoretically by Edwards and Hertz [7] in the framework of a broad-band Hubbard model for itinerant electron ferromagnets. Later it was demonstrated [8] that for a narrow-band (infiniteU ) Hubbard model the whole spectral weight for one spin projection belongs to the NQP states which is of crucial importance for the problem of stability of Nagaoka’s ferromagnetism [9] and for an adequate description of the corresponding excitation spectrum. The NQP states in the s − d exchange model of magnetic semiconductors have been considered in [10]. It was shown that depending on the sign of the s−d exchange integral, the NQP states can form either only below the Fermi energy EF or only above it. Later it was realized that the HMF are natural substances for theoretical and experimental investigating of the NQP effects [11]. A variety of these effects in the electronic and magnetic properties has been considered (for a review of the earlier works see [2]) and some recent developments will be discussed in the present paper. As an example of the highly unusual properties of the NQP states, we note that they can contribute to the T -linear term in the electron heat capacity [11, 12], despite of the fact that their density at EF vanishes at temperature T = 0. Existence of the NQP states at the HMF surface has been predicted in [13] and may be important for their detection by surface-sensitive methods such as ARPES [14] or by spin-polarized scanning tunneling microscopy [15]. Recently the density of NQP states has been calculated from first principles for a prototype HMF, NiMnSb [16]. Some effects of the NQP states on physical properties of the HMF will be considered below. Because of the length restrictions we will concentrate on several examples skipping the temperature dependence of nuclear magnetic relaxation rate [17] and many others.

2 Origin of Nonquasiparticle States and Electron Spin Polarization in the Gap From the theoretical point of view, the HMF are characterized by the absence of magnon decay into the Stoner excitations (electron-hole pairs with opposite spins). Therefore spin waves are well defined in the whole Brillouin zone, similar to Heisenberg ferromagnets and degenerate ferromagnetic semiconductors. Thus, unlike for the usual itinerant ferromagnets, effects of

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electron-magnon interactions (so-called spin-polaron effects) are not masked by the Stoner excitations in the HMF and may be studied in a “pure” form. As we will see below, the electron-magnon scattering results in the occurrence of NQP states. We start our consideration of the interaction of charge carriers with local moments in the standard s-d exchange model [18]. The s-d exchange Hamiltonian reads


†
tk ckσ ckσ − Ik,k+q Sq c†kα σ αβ ck−qβ − Jq Sq S−q (1) H= kσ

qk

αβ

q

where c†kσ , ckσ and Sq are operators for conduction electrons and localized spins in the quasimomentum representation, the electron spectrum tk is referred to the Fermi level EF , Ik,k+q is the s-d exchange parameter, σ are the Pauli matrices. We include in the Hamiltonian explicitly the “direct” d-d exchange interaction (last term in (1)) to construct a perturbation theory in a convenient form. In real materials, this interaction may have a superexchange nature or result from the indirect exchange via conduction electrons (in the HMF situation, this is not reduced to the RKKY interaction). In the latter case, the d-d exchange interaction comes from the same s-d interaction and cannot be considered as an independent parameter. However, as demonstrated by direct calculations (see e.g. [10, 19]), the corresponding terms with magnon frequencies occur in higher order of the I perturbations, for the case where the bare d-d exchange interaction is absent. The s-d exchange model does not describe properly the electronic structure for such HMF as the Heusler alloys or CrO2 , where there is no domination of the sp-electrons in electronic transport, and a separation of electrons into a localized d-like and a delocalized s-like group is questionable. In such a case, the Hubbard model which describes the Coulomb correlations in a d-band is more appropriate. However, qualitative effects of electron-magnon interaction do not depend on the microscopic model. The calculations of the electron and magnon Green’s functions in the non-degenerate Hubbard model were performed in [7, 11] and gave practically the same result as the s-d exchange model with simple replacement of I by the Hubbard parameter U . As demonstrated by analysis of the electron-spin coupling, the NQP picture turns out to be different for two possible signs of the s − d exchange parameter I. For the case I < 0, the spin-up NQP states appear below the Fermi level as an isolated region in the energy diagram (Fig. 1). The occupied states with the total spin S − 1 are a superposition of the states |S| ↓ and |S − 1| ↑. The entanglement of the states of electron and spin subsystems which is necessary to form the NQP states is a purely quantum effect formally disappearing at S → ∞. For qualitative understanding why the NQP states are formed only below the EF in this case, we consider a limit I → −∞ . Then the charge carrier is really a many-body state of the occupied site with total spin S − 1/2, which propagates in the ferromagnetic medium with spin

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Fig. 1. Density of states in a half-metallic ferromagnet with I < 0 (schematically). Non-quasiparticle states with σ =↑ occur below the Fermi level. ∆ denots the spin splitting.

S at any other site. The fractions of the states |S| ↓ and |S − 1| ↑ in the charge mobile carrier state are 1/(2S + 1) and 2S/(2S + 1), respectively, so that the first number is just a spectral weight of occupied spin-up electron NQP states. At the same time, the density of empty states is measured by the number of electrons with a given spin projection which can be added to the system. It is obvious that one cannot put any spin-up electrons in the spin-up site with I = −∞. Therefore the density of NQP states should vanish above EF . On contrary, for the I > 0 case, the spin-down NQP scattering states form a “tail” of the upper spin-down band, which starts from EF (Fig. 2) since the Pauli principle prevents electron scattering into occupied states. A similar analysis of the limit I → +∞ helps to understand the situation qualitatively. It is worthwhile to note that in most known HMF an energy gap exists for minority-spin states [2] which is similar to the case I > 0, therefore the NQP states should arise above the Fermi energy. For exceptional cases with the majority-spin gap such as a double perovskite Sr2 FeMoO6 [20] one should expect the NQP states below the Fermi energy. This would be very interesting since in the latter case the NQP states can be probed by spin-polarized photoemission which is technically much simpler than spin-polarized BIS spectra [21] needed to probe the empty NQP states. Let us consider now the density of states (DOS) scheme for the HMF within the s − d exchange model more quantitatively [2, 10]. Neglecting the k dependence of s − d exchange interaction, the electron Green’s function has the following form Gσk (E) = [E − tkσ − Σkσ (E)]

−1

(2)

Nonquasiparticle States in Half-Metallic Ferromagnets

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Fig. 2. Density of states in a half-metallic ferromagnet with I > 0 (schematically). Non-quasiparticle states with σ =↓ occur above the Fermi level

where tkσ = tk − σI
S z  is the mean-field electron spectrum and Σkσ (E) is the self-energy which describe the electron-magnon interactions. Within the second order approximation in I one has Σkσ (E) = 2I 2 SQσk (E) with Q↑k (E) =




Nq + n↓k+q

q

E − tk+q↓ + ωq

, Q↓k (E) =


1 + Nq − n↑k−q q

E − tk−q↑ − ωq

(3)

Below we will present more accurate results for the Green’s functions (see (29)) but here the lowest-order perturbation expression (3) will be sufficient. Using an expansion of the Dyson  equation (2) we obtain a simple expression for the electron DOS (− π1 Im k Gkσ (E)) Nσ (E) =




δ(E − tkσ ) −

k


k

δ  (E − tkσ )ReΣkσ (E) −

1
ImΣkσ (E) (4) π (E − tkσ )2 k

The second term on the right-hand side of (4) describes the renormalization of quasiparticle energies. The third term, which arises from the branch cut of the self-energy Σkσ (E), describes the incoherent (nonquasiparticle) contribution owing to scattering by magnons. One can see that the NQP does not vanish in the energy region, corresponding to the “alien” spin subband with the opposite projection −σ. Substituting (3) into (4) and neglecting the quasiparticle shift we obtain for the case of HMF with I > 0 
 2I 2 SNq N↑ (E) = 1− δ(E − tk↑ ) (tk+q↓ − tk↑ )2 kq

N↓ (E) = 2I 2 S


kq

1 + Nq − nk↑ δ(E − tk↑ − ωq ) (tk+q↓ − tk↑ − ωq )2

(5)

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The DOS for the case of an the empty conduction band is shown in Fig. 3. The T 3/2 -dependence of the magnon contribution to the residue of the Green’s function (2), which follows from (3), i.e. of the effective electron mass in the lower spin subband, and an increase with temperature of the incoherent tail from the upper spin subband, together lead to a strong temperature dependence of partial densities of states Nσ (E), the corrections being of opposite sign.

Fig. 3. Density of states in the s-d model in the case of an empty conduction band (I > 0). At T = 0 (solid line) the spin-polaron tail of spin-down states reaches the band bottom. The dashed line corresponds to finite temperatures

The behaviour of N (E) near the Fermi level in the HMF (or degenerate ferromagnetic semiconductor) turns out to be non-trivial as well (Figs. 2,3). If we neglect magnon frequencies in the denominators of (5), the partial density of incoherent states should occur as a jump above or below the Fermi energy EF for the case of I > 0 and I < 0 respectively owing to the Fermi distribution functions. An account of finite magnon frequencies ωq = Dq 2 (D is the spin stiffness constant) leads to the smearing of these singularities on the energy interval ω  EF , with the N (EF ) being equal to zero. For |E − EF |  ω we obtain  3/2 1  E − EF  N−α (E) =  θ(α(E − EF )), Nα (E) 2S  ω

(6)

where α = sign(I) = ±1 is the spin projections ↑, ↓ of corresponding NQPstates. With increasing |E − EF |, N−α /Nα tends to a constant value which is of order of I 2 within the perturbation theory. In the strong coupling limit where |I| → ∞ we have for the |E − EF |  ω N−α (E) 1 = θ(α(E − EF )) Nα (E) 2S

(7)

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In a simple s-d model case, qualitative considerations [24], as well the Green’s functions calculations [10, 25], give a spin polarization of conduction electrons in the spin-wave region proportional to the magnetization: P ≡

N↑ − N↓ = 2P0
S z  N↑ + N ↓

(8)

A weak ground-state depolarization 1 − P0 occurs in the case of I > 0. The behavior P (T ) 
S z  is qualitatively confirmed by experimental data on field emission for ferromagnetic semiconductors [22] and transport properties for the half-metallic Heusler alloys [23]. Note that equation 8 is valid for a hole spin-wave region only for the narrow-band case (large I), whereas for the case of a small I it described spin-polarization only for very low temperatures. An attempt to generalize the result (8) to the HMF case has been made on the basis of qualitative arguments for the atomic limit [26]. We will demonstrate that the situation for the HMF is more complicated. Let us focus on the magnon contribution to the DOS (5) and calculate a following function: Φ=


kq

2I 2 SNq δ(EF − tk↑ ) (tk+q↓ − tk↑ − ωq )2

(9)

Using the parabolic electron spectrum tk↑ = k 2 /2m and averaging over angles of the vector k, we obtain Φ=

Nq 2I 2 Sm2
ρ , 2 2 ∗ 2 kF q (q ) − q

(10)

where ρ = N↑ (EF , T = 0). We have used the condition q  kF , q ∗ = m∆/kF = ∆/vF , where ∆ = 2 |I| S is the spin splitting. The corresponding 2 crossover energy scale is equal to T ∗ = D (q ∗ ) ∼ (∆/vF )2 TC . Finally, we have the following expression for Φ  ∞ 1/2 1 x dx I 2 S m2 , (11) ρ P Φ= 2 ∗ /T ) − x kF 2π 2 exp x − 1 (T 0 where P is the symbol for the principal value. At the very low temperatures T < T ∗ , this result is in agreement with the qualitative considerations presented above:  3/2 T S −
S z  ρ∝ ρ (12) Φ= 2S TC Nevertheless, for T > T ∗ we have absolutely different temperature dependence of the spin polarization: (q ∗ ) Φ = 1.29 4Sπ 2 3



T T∗

1/2 ρ

(13)

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This conclusion is rather important since the crossover temperature T ∗ can be small and a simple estimation (8) may be valid only for very low temperatures. Moreover, it turns out that the temperature dependence of the polarization at T > T ∗ is not universal at all. Note that the model of rigid spin splitting used above is not applicable for real HMF where the gap hybridization originates from [1, 2], in contrast to the case of degenerate ferromagnetic semiconductors. The simplest model for the HMF consists of a “normal” metallic spectrum for the majority electrons and a hybridization gap for the minority ones (ξk ≡ tk↑ − EF )   k 2 − kF2 1 tk↑ − EF = , tk↓ − EF = ξk + sgn (ξk ) ξk2 + ∆2 , (14) 2m 2 where we assume for simplicity that the Fermi energy lies exactly in the middle of the hybridization gap. Otherwise one needs to shift ξk → ξk + E0 − EF in the last equation, E0 being the middle of the gap. Further, in the expression for tk+q↓ one can replace ξk+q by vk q, vk = k/m, and use the fact that ξk = 0 owing to the delta-function in the definition of Φ. Since a small q give the main contribution to the estimated integral, we can assume Ik,k+q  Ik,k . Then one has the following expression


 1 ∂ 2  Φ = 2S Ik,k Nq δ(ξk )Λkq , Λkq = − (15) ∂ωq tk+q↓ − tk↑ − ωq ωq =0 kq

where the angular brackets means the avere over angles of the vector k. Simple calculations gives the final result:  

2 8 X 3 − (X 2 + 1)3/2 + 1 + X , (16) Λkq = vF q∆ 3 where X = kF q/m∆ ≡ q/q ∗ (q ∗ is linear in ∆). At X  1 corresponding to T  T ∗ = Dq ∗2 , one has, instead of (13), the following estimation Φ=


kq

2I 2 SNq δ(ξk )


Nq T ∗1/2 16 T ∝ q∗ ∝ 1/2 T ln ∗ 3vF q∆ q T TC q

(17)

At X  1 (T  T ∗ ) we get an universal T 3/2 behavior Φ=ρ


q

Nq ∝

T 3/2 1/2

(18)

TC

The density of NQP states is zero at the Fermi energy only for T = 0, while for finite temperatures it is proportional to the following integral  ∞ K(ω) , (19) dω N (EF ) ∝ sinh(ω/T ) 0

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where K(ω) is a spectral density of the spin fluctuations [24, 10, 12]. Generally speaking, for temperatures which are comparable with the Curie temperature TC there are no essential differences between the half-metallic and “ordinary” ferromagnets since a gap of the HMF is filled. A corresponding analysis for a model of conduction electrons interacting with “pseudospin” excitations in “ferroelectric” semiconductors is performed in [12]. The symmetrical part of N (E) with respect to EF in the gap can be attributed to the smearing of electronic states by the electron-magnon scattering, while the asymmetric part is the density of NQP states due to the Fermi distribution function. Note that this filling of the gap is very important for possible applications of the HMF in spintronics: they really have some advantages only in the region of T  TC . Since a single-particle Stoner-like theory leads to a much less restrictive, but unfortunately completely wrong condition T  ∆, a manybody treatment of the HMF problem is inevitable.

3 First-Principle Calculations of Nonquasiparticle States: A Dynamical Mean Field Theory A history of the HMF starts from the band-structure of semi Heusler alloy NiMnSb [1]. Later numerous first-principle electronic structure investigations of the HMF have been carried out (see, e.g., recent papers [27, 28, 29, 30] and a review of early works in [2]). All of them are based on a standard local density approximation (LDA) or generalized gradient approximation (GGA) to the density functional theory, and, sometimes, on the LDA+U approximation (see [31] for CrO2 ). Of course, essential correlation effects such as NQP states cannot be considered in these techniques. Recently, a successful approach has been proposed [32, 33] to include correlation effects into the first-principle electronic structure calculations by combining the LDA scheme with the dynamical mean-field theory (DMFT). The DMFT maps a lattice many-body system onto quantum impurity models subject to a self-consistent condition (for a review, see [34]). In this way, the complex lattice many-body problem splits into simple one-body crystal problem with a local self-energy and the effective many-body impurity problem. In a sense, the approach is complementary to the local density approximation [35, 36, 37] where the many-body problem splits into one-body problem for a crystal and many-body problem for homogeneous electron gas. Naively speaking, the LDA+DMFT method [32, 33] treats localized d- and f -electrons in spirit of the DMFT and delocalized s, p-electrons in spirit of the LDA. Due to numerical and analytical techniques developed for solution the effective quantum-impurity problem [34], the DMFT become a very efficient and extensively used approximation for local energy dependent self-energy Σ(ω). The accurate LDA+DMFT scheme can be used for calculating a large number of systems with different strength of electron correlations (for detailed

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description of the method and computational results, see [38, 39, 40]). Following the recent work [16] we present here first LDA+DMFT results for the electronic structure calculations of a “prototype” half-metallic ferromagnet NiMnSb. Before considering the real HMF case, it is worthwhile to check the applicability of DMFT scheme for quantitative description of the NQP states. The DMFT is considered as an optimal local approximation which means that the self-energy depends only on the energy and not on the quasimomentum [34]. At the same time, the NQP states are connected with the self-energy (3) which is almost local. It will be exactly local if we neglect magnon energies in comparison with the electron bandwidth, which is rather accurate approximation for realistic parameters. The local approximation means formally that we replace the q-dependent magnon spectral density by the average one, as in the (19). Such a procedure has been analyzed and justified in the [41]. It should be stressed that an accurate description of the magnon spectrum is not important for existence of the NQP states as well as for proper estimation of their spectral weight, but can be important for an explicit shape of the DOS tail in the vicinity of the Fermi level (see (6)). Let us start from the DMFT calculations for the one-band Hubbard Hamiltonian

tij (c†iσ cjσ + c†jσ ciσ ) + U ni↑ ni↓ , (20) H=− i,j,σ

i

on the Bethe lattice with coordination z → ∞ and nearest-neighbor hoping √ tij = t/ z (in this limit the DMFT is formally exact [34]). In this case the DOS have a semicircular form: 1  2 4t − 2 (21) N () = 2πt2 In order to stabilize the HMF state in our toy model, we have added an external magnetic spin splitting ∆, which mimics the local Hund polarization from other electrons in the real NiMnSb compound. This HMF state corresponds to a mean-filed (Hartree-Fock) solution with a LSDA-like DOS (Fig. 4). We can study an average magnon spectrum in this model through the two-particle correlation function. The local spin-flip susceptibility † † + − χ+− loc (τ ) =
S (τ )S (0) =
c↑ (τ )c↓ (τ )c↓ (0)c↑ (0) ,

(22)

represents the response function required. We have calculated this function using the numerically exact QMC procedure [42]. The model DMFT results are presented in Fig. 4. In comparison with a simple Hartree-Fock solution (dashed line) one can see an additional wellpronounced states appearing in the spin-down gap region, just above the Fermi level. This new many-body feature corresponds to the NQP states. In addition to these states visible in both spin channels of the DOS around 0.5 eV, a many-body satellite appears at the energy of 3.5 eV.

Nonquasiparticle States in Half-Metallic Ferromagnets 0.8

Im Σ (eV)

0.0

DMFT 0.4

Σ

EF Σ

-0.1

EF

-1

DOS(eV )

227

-2

-1

0

E(eV)

1

2

0.0

+-

-0.4

Im χloc(eV-1)

0.3

0.0

-0.8

-2

0

HF

1 E(eV) 2

-1

0

1 E(eV)

2

3

4

Fig. 4. Density of states for HMF in the Hartree-Fock (HF) approximation (dashed line) and the QMC solution of DMFT problem for semi-circular model (solid line) with the band-width W = 2 eV, Coulomb interaction U = 2 eV, spin-splitting ∆ = 0.5 eV, chemical potential µ = −1.5 eV and temperature T = 0.25 eV. Insets: imaginary part of the local spin-flip susceptibility (left) and the spin-rezolved selfenergy (right)

The left-hand inset in the Fig. 4 represents the imaginary part of local spin-flip susceptibility. One can see a well pronounced shoulder (around 0.5 eV), which is connected with an average magnon DOS. In addition there is a broad maximum (at 1eV) corresponding to the Stoner excitation energy. The right-hand inset in the Fig. 4 represents the imaginary part of self-energy calculated from our “toy model”. The spin up channel can be described by a Fermi-liquid type behavior with a parabolic energy dependence −ImΣ ↑ ∝ (E − EF )2 , whereas in the spin down channel the imaginary part −ImΣ ↓ shows the 0.5 eV nonquasiparticle shoulder. Due to the relatively high temperature of our QMC calculation (an exact enumeration, technique with the number of time-slices equal to L = 24) the NQP tail goes a bit below the Fermi level, in agreement with (19); at temperature T = 0 the NQP tail should ends exactly at the Fermi level. Let us move to the calculations for a real material – NiMnSb. The details of computational scheme have been described in [16], and only the key points will be mentioned here. In order to integrate the DMFT approach into the band structure calculation the so called exact muffin-tin orbital method (EMTO) [43, 44] was used. In the EMTO approach, the effective one-electron potential is represented by the optimized overlapping muffin-tin potential, which is the best possible spherical approximation to the full oneelectron potential. The implementation of the DMFT scheme in the EMTO method is described in detail in [45]. One should note that in addition to the usual self-consistency of the many-body problem (self-consistency of the self-energy), a charge self-consistency has been achieved [40].

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For the interaction Hamiltonian, a most general rotationally invariant form of the generalized Hubbard Hamiltonian has been used [33]. The effective many-body impurity problem is solved using the spin polarized T -matrix plus fluctuation-exchange appriximation (a so-called SPTF)scheme proposed in the [46], which is a development of the earlier approach [33]. The SPTF approximation is a multiband spin-polarized generalization of a well-known fluctuation exchange (FLEX) approximation [47], but with a different treatment of the particle-hole (PH) and particle-particle (PP) channels. The particleparticle (PP) channel is described by a T -matrix approach [48] yielding a renormalization of the effective interaction. The static part of this effective interaction is used explicitly in the particle-hole channel. There are various methods to estimate the required values of the on-site Coulomb repulsion energy U and exchange interaction energy J for realistic materials. The constrained LDA calculation [49] estimates an average Coulomb interaction between the Mn d electrons as U = 4.8 eV with an exchange interaction energy of J = 0.9 eV. However, this method is adequate for a typical insulating screening and in general is not accurate for a metallic kind of screening. The latter will lead to a smaller value of U . Unfortunately, there are no reliable schemes to calculate U for metals, therefore the results for different values of U in the energy interval from 0.5 eV to the constrained LDA value U = 4.8 eV have been tested. At the same time, the results of constrained LDA calculations for the Hund exchange parameter J do not depends on metallic screening and should be reliable enough. It appeared that the LDA+DMFT results are not very sensitive to the value of U , due to the T -matrix renormalization. Figure 5 represents the results for DOS using LSDA and LDA+DMFT (with U = 3 eV and J = 0.9 eV) approaches. It is important to mention that the magnetic moment per formula unit is not sensitive to the U values and is equal exactly µ = 4 µB , which suggests that the half-metallic state is stable with respect to the introduction of the correlation effects. In addition, the DMFT gap in the spin down channel, defined as the distance between the occupied part and the starting point of nonquasiparticle state’s “tail”, is also not very sensitive to the U values. For different U ’s a slope of the “tail” is slightly changed, but the total DOS is weakly U -dependent due to the same T -matrix renormalization effects. Thus the correlation effects do not effect too strongly a general feature of the electron energy spectrum (except for smearing of DOS which is due to the finite temperature T = 300 K in our calculations). The only qualitatively new effect is the appearance of a “tail” of the NQP states in the energy gap above the Fermi energy. Their spectral weight for realistic values of the parameters is not very small, which means that the NQP should be well pronounced in the corresponding experimental data. A relatively weak dependence of the NQP spectral weight on the U value (Fig. 6) is also a consequence of the T matrix renormalization [46]. One can see that the T -matrix depends slightly on U provided that the latter is larger than the widths of the main DOS

Nonquasiparticle States in Half-Metallic Ferromagnets

229

1.5

DOS(eV-1)

DMFT 1.0 LSDA

EF

0.5 0.0 -0.5 -1.0 -1.5

-8

-6

-4 -2 E(eV)

0

2

Fig. 5. Density of states for HMF NiMnSb in LSDA scheme (dashed line) and in LDA+DMFT scheme (solid line) with effective Coulomb interaction U = 3 eV, exchange parameter J=0.9 eV and temperature T = 300 K. The nonquasiparticle state is evidenced just above the Fermi level

Spectral weight

0.06

0.04

U

*

0.02

0.00

0

1

2

3

4

5

U(eV)

Fig. 6. Spectral weight of the nonquasiparticle state, calculated as function of average on-site Coulomb repulsion U at temperature T = 300 K

peaks near the Fermi level in an energy range of 2 eV (this is of the order of U ∗  1 eV). For the spin-up states we have a normal Fermi-liquid behavior −ImΣd↑ (E) ∝ (E − EF )2 with a typical energy scale of the order of several eV. The spin-down self-energy behaves in a similar way below the Fermi energy, with a slightly smaller energy scale (which is still larger than 1 eV). At the same time, a significant increase in ImΣd↓ (E) with a much smaller energy scale (few tenths of eV) occurs just above the Fermi level, which is more pronounced for t2g states (Fig. 7). The similar behavior of the imaginary part of electronic

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V.Yu. Irkhin et al. 0.0 -0.4

Σ

t2g

Σe

Im Σ (eV)

g

-0.8

Σt Σe

-1.2 -1.6 -8

-6

2g

EF g

-4 -2 E(eV)

0

2

Fig. 7. The imaginary part of self-energies ImΣd↓ for t2g (solid line) and eg (dotted line), ImΣd↑ for t2g (dashed line) and eg (dashed dotted line) respectively

self-energy and the DOS just above Fermi level is a signature of the NQP states and is also noticed in the model calculation (Fig. 4). Thus the main results of [16] are (i) the existence of the NQP states in the real electronic structure of a specific compound, and (ii) an estimation of their spectral weight in the LDA+DMFT approach. The temperature dependence of the NQP density of states in the gap, which is important for possible applications of the HMF in spintronics, can be analyzed with the present technique.

4 X-ray Absorption and Emission Spectra Resonant X-ray Scattering Now we discuss the manifestations of NQP states in the core level spectroscopy [50]. Various spectroscopy techniques such as x-ray absorption, x-ray emission, and photoelectron spectroscopies (xas, xes, and xps) give an important information about the electronic structure of the HMF and related compounds, such as ferromagnetic semiconductors and colossal magnetoresistance materials (see, e.g., [51, 52, 53, 54]). It is well known that the manybody effects, particularly dynamical core hole screening, may be important for a core level spectroscopy even in the case that a system is not strongly correlated in the initial state [55, 56]. Therefore it is very interesting to look on the interplay of these effects with the NQP states, which are of essentially many-body origin themselves. To consider a core-level problem in the HMF we use the same Hamiltonian of the s − d exchange model, (1) in the presence of the external potential U induced by the core hole:
† ckσ ck
σ f † f , (23) H  = ε0 f † f − U kk
σ

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231

where f † , f are core hole operators. It is useful to write down the equation of motion for the retarded two-particle Green’s function [57] Gσkk
(E) =

ckσ f |f † c†k
σ E ,

(24)

which determines x-ray absorption and emission spectra [55]. Using a magnon representation for the spin operators, we derive the following equation for two-particle Green’s function:  

σ Gσpk
(E) −I Fk−r,r,k (E −tkσ )Gσkk
(E) = (1−nf −nσk ) δkk
− U
(E) p

r

(25) where nf is the occupation number for the f -hole in the initial state, which is further on will be put to zero and E is the electron energy with respect to ε0 ). We will take into account the occupation numbers nσk in a simple ladder approximation which works well in the limit of small concentrations of mobile carriers, except for the immediate vicinity of the Fermi edge. Here, we do not treat the problem of the x-ray edge singularity where more advanced approaches are necessary [55, 56]. The following notation has been used in (25): σ 1/2

bσq ck−p,−σ f |f † c†k
σ E , (26) Fk−p,q,k
(E) = (2S) † − where b+ q = b−q , bq = bq are the Holstein-Primakoff magnon operators [18]. The Green’s function F satisfies the equation −σ σ σ (E − tk−p,−σ + σωq )Fk−p,q,k
(E) = −U (1 − n k−p )Ψq,k
(E)
σ σ Fk−p+q−r,r,k −I(Nqσ + σn−σ
(E)] , k−p )[2SGk−p+q,k
(E) + σ

(27)

r

where we have performed decouplings in the spirit of ladder approximation, σ
bσ−q b−σ q  = Nq = σN (σωq ), N (ω) is the Bose function, and Ψ is defined as σ Ψq,k
(E) =




σ Fk−r,q,k
(E)

(28)

r

For U = 0 we have Gσkk
(E) = (1 − nσk )δkk
Gσk (E), where Gσk (E) is the one-electron Green’s function of the ideal crystal (cf. (3)), Gσk (E) = [E − tkσ − Σkσ (E)]

−1

, Σkσ (E) =

2SI 2 Qσk 1 + σIQσk

(29)

Note that the (29) gives correctly the exact Green’s function in the limit of an empty conduction band at T = 0 [58, 59, 10]. In a general case, we have the three-particle problem (conduction electron, core hole and magnon) which requires a careful mathematical investigation. However, we can use the facts that the magnon frequencies are much smaller

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than typical electron energies and enrgy resolution of xas and xes methods. Neglecting spin dynamics, the equations (25), (27) can be solved exactly in a rather simple way for the case of zero temperatures (Nq+ = 0, Nq− = 1). σ Under these conditions, Q does not depend on quasimomenta, and Ψq,k
does not depend on q, since the electron and magnon operator should belong to the same perturbed site: σ σ 1/2 Ψq,k

bσ c−σ f |f † c†k
σ E
(E) = Ψk
(E) = (2S)

(30)

We find in this case 2ISQσ (E) Rσ
(E) 1+ + σIQσ (E) k

1 − nσ k Rkσ
(E) = Gσkk
(E) , P σ (E) = E − tkσ Ψkσ
(E) = −

U P −σ (E)

k

(31) (32)

k

After substituting (31) into (25) we obtain the following equation for the Green’s function G
σ (E) Gσpk
(E) (33) [E − tkσ − Σ σ (E)] Gσkk
(E) = δkk
− Uef p

with the renormalized core hole potential:   Σ σ (E)P −σ (E) σ Uef (E) = U 1 + . 1 + U P −σ (E) + σIQσ (E)

(34)

Here we neglect the factor (1 − nσk ), since the band filling is small. Therefore one has a standard result for the impurity scattering with renormalized energy σ (E). A spectrum Ekσ = tkσ + Σ σ (E) and the effective impurity potential Uef local DOS is given by the following expression: 1 σ (E) = − ImGσ00 (E) Nloc π with Gσ00 (E) =


kk


Gσkk
(E) =

Rσ (E) σ (E)R (E) 1 + Uef σ

(35)

(36)

 where Rσ (E) = k Gσk (E), and Gσk (E) is given by (29). Generally speaking, theoretical investigation of the core level spectra requires numerical calculations of realistic band structure. We restrict ourselves to simple model calculations for the bare semicircular DOS from the (21). The local Green function from (35) describes the absorption spectrum for E > EF and emission spectrum for E < EF . As follows from the (35), (36), the experimental spectra is given by somewhat different expression than the DOS in an initial state, and new effects can occur.

Nonquasiparticle States in Half-Metallic Ferromagnets

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For I > 0 the results of (34)–(36) provide full solution of the Kondo problem for an impurity in the ferromagnet, within the parquet approximation [60]. In the case of I < 0, the situation is complicated by the presence of the “false” Kondo divergence in the T -matrix [61]. However, this difficulty is not important for the x-ray problem where a large damping is always present, and experiments are performed at sufficiently high temperatures with rather poor resolution compared to a scale of the “Kondo temperature”. To a leading order in U and I we obtain   σ (E)/Σ σ (E)) Rσ2 (E)/Rσ (E) 1 − Re(Uef σ δN σ (E) δNloc (E) =  2   σ 1 + Uef (E)Rσ (E) σ (E)/P −σ (E)) |Rσ (E)| 1 Re(Uef ImP −σ (E) 2  π   σ 1 + Uef (E)Rσ (E) 2

−

(37)

The term in (37) with ImP −σ (E) has a smooth contribution to the spectrum. In particular, it is non-zero in the energy gap. Note that for the emission spectra such term is absent. The NQP contributions to the absorption (for I > 0) and emission spectra (for I < 0) are proportional to δN σ (E). One can see from Fig. 8 that the upturn of the NQP tail which occurs for I > 0 becomes sharper, although the jump near EF weakens. For I < 0 case, the spectral weight of NQP contributions also increases in the presence of the core hole (see Fig. 9). These effects have a simple physical interpreσ (E) > 0 and for small band filling Rσ (E) < 0 near EF , tation. Since Uef the denominator of the expression (37)gives a considerable enhancement of

↓ Fig. 8. The local density of states Nloc (E) (solid line) for a half-metallic ferromagnet with S = 1/2, I = 0.3 in the presence of the core hole potential U = 0.2; smearing E +iδ is introduced with δ = 0.01. The dashed line shows the DOS N↓ (E) for the ideal crystal with spin dynamics being neglected. The value of EF calculated from the band bottom is 0.15. The energy E is referred to the Fermi level

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↑ Fig. 9. The local density of states Nloc (E) (solid line) for a half-metallic ferromagnet with S = 1/2, I = −0.3, δ = 0.025 in the presence of the core hole potential U = 0.2. The dashed line shows the DOS N↑ (E) for the ideal crystal. The value of EF calculated from the band bottom is 0.15

the NQP contributions to the spectra in comparison with those to the DOS. However, effects of interaction U turn out to be non-trivial and do not reduce to a constant factor in the self-energy. Strong interaction with the core hole results in a deformation of conduction band. With increasing U the spectral density concentrated at bottom of the band. This effect is very important for the NQP states located in this region. Therefore the spectral weight of the NQP states increases. At very large, probably unrealistic values of U , a bound state is formed near the band bottom, and the NQP spectral weight becomes suppressed owing to factor of U in the denominator of (34). To probe a “spin-polaron” nature of the NQP states more explicitly, it would be desirable to use spin-resolved spectroscopical methods such as x-ray magnetic circular dichroism (XMCD, for a review see [62]). Owing to interference of electron-magnon scattering and “exciton” effects from interaction of electrons with the core hole, the NQP contributions to x-ray spectra can be considerably enhanced in comparison with those to the DOS of ideal crystal. Now we consider the NQP effects in resonant x-ray scattering processes. It was observed recently that the elastic peak of the x-ray scattering in CrO2 is more pronounced than in usual Cr compounds [53]. The authors of this work have put forward some qualitative arguments that the NQP states may give larger contributions to resonant x-ray scattering than usual itinerant electron states. Here we shall treat this problem quantitatively and estimate explicitly the corresponding enhancement factor. The intensity of resonant x-ray emission induced by the photon with the energy ω and polarization q is given by the Kramers–Heisenberg formula [63, 52, 64]  2


n|Cq
|l
l|Cq |0  δ(En + ω  − E0 − ω) , (38) Iq
q (ω  , ω) ∝   − E − iΓ   E + ω 0 l l n l

Nonquasiparticle States in Half-Metallic Ferromagnets

235

here q  , ω  are the polarization and energy of the emitted photon, |n, |0 and |l are the final, initial and intermediate states of the scattering system with the energies Ei , respectively, and Cq is the operator of a dipole moment for the transition, which is proportional to (f c + c† f † ). For simplicity we assume hereafter that Γl does not depend on the intermediate state: Γl = Γ , and take into account only the main x-ray scattering channel where the hole is filled by conduction electron. Assuming that the electron-photon interaction that induces the transition is contact, the expression for threshold scattering intensity has following form [65]  ∞
 ∞ dt1 dt2 exp [−i(ω  − ε0 )(t1 − t2 ) − Γ (t1 + t2 )] Iω
∝ σσ


0

0


0|cσ exp(iHf t1 )c†σ
exp[iHi (t2 − t1 )]cσ
exp(−iHf t2 )c†σ |0 , (39) where Hf and Hi are conduction-electron Hamiltonians with and without core hole, respectively. The complicated correlation function in (39) can be decoupled in the ladder approximation which is exact for the empty conduction band. Then one can obtain [65] Iω


 2 
   σ ∝ G00 (z) ,  

(40)

σ

where z = ω  − E0 + iΓ. Owing to a jump in the DOS at the Fermi level, the NQP part of the Green’s function contains a large logarithm ln(W/z) at small z, W being a bandwidth. It means that the corresponding contribution to the elastic x-ray scattering intensity (ω  = E0 ) is enhanced by a factor of ln2 (W/Γ ), which makes a quantitative estimation for the qualitative effect discussed in [53]. Of course, the smearing of the jump in the density of NQP states by spin dynamics is irrelevant provided that Γ > ω, where ω is a characteristic magnon frequency.

5 Transport Properties Transport properties of the HMF are a subject of numerous experimental investigations (see, e.g., recent works on CrO2 [66], NiMnSb [67], and the reviews [2, 68, 69]). At the same time, a theoretical interpretation of these results is still problematic. Concerning electronic scattering mechanisms, the most important difference between the HMF and “standard” itinerant electron ferromagnets like iron or nickel is the absence of one-magnon scattering processes in the former case [2]. Two-magnon scattering processes have been considered many years ago for both the broad-band case (a weak s-d exchange interaction) [70] and narrow-band case (a “double exchange model”) [71]. Obtained temperature dependence of resistivity have the form T 7/2 and T 9/2 ,

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respectively. At low enough temperatures the first result fails and should be replaced by T 9/2 as well [72]; the reason is a compensation of transverse and longitudinal contributions in the long-wavelength limit, which is a consequence of the rotational symmetry of the s-d exchange Hamiltonian [73, 74]. Recently a general interpolation theory has been formulated [75]. Here we discuss main results of this work with a special emphasize to the NQP effects. In the spin-wave region the Hamiltonian (1) can be rewritten in the form
† (ck↑ ck+q↓ b†q + h.c.) H = H0 − I(2S)1/2


+I

kq

σc†kσ ck+q−pσ b†q bp

(41)

kqpσ

Here the zero-order Hamiltonian includes non-interacting electrons and magnons:

H0 = tkσ c†kσ ckσ + ωq b†q bq , (42) kσ

q

with the spin splitting ∆ = 2IS being included in H0 . In the half-metallic case spin-flip processes do not appear in the second order in I, since the states with only one spin projection presented at the Fermi level. At the same time, we have to consider the renormalization of the longitudinal processes in higher orders in I (formally, we need to include all terms up to the second order in a quasiclassical small parameter 1/S). To this end we eliminate from the Hamiltonian the terms which are linear in the magnon operators by using the canonical transformation [73]. Then, the effective Hamiltonian has a following form
 = H0 + 1 (Aσkq + Aσk+q−p,q )c†kσ ck+q−pσ b†q bp , (43) H 2 kqpσ

where Aσkq = σI

tk+q − tk tk+q − tk + σ∆

(44)

is the s-d scattering amplitude, which vanishes at q → 0 and thereby takes properly into account the rotational symmetry of electron-magnon interaction. More general interpolation expression for the effective amplitude which does not assume the smallness of |I| or 1/S was obtained in [74] within a variational approach, but it does not differ qualitatively from simple expression (44). In the case of real itinerant magnets including the HMF, a k-dependence of s-d exchange parameter should be taken into account, similarly to the temperature dependence of spin polarization. However, here we restrict ourselves only to the rigid spin splitting model appropriate for degenerate ferromagnetic semiconductors. One can expect from phenomenological symmetry considerations that the temperature dependences of transport properties are rather universal.

Nonquasiparticle States in Half-Metallic Ferromagnets

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The most general scheme for calculating the transport relaxation time is the Kubo formalism for the conductivity σxx [76]  σxx = β



β

∞

dλ 0

dt exp(−εt)
jx (t + iλ)jx 

(45)

0

 where β = 1/kB T, ε → 0, j = −e kσ vkσ c†kσ ckσ is the current operator, vkσ = ∂tkσ /∂k is the electron velocity. Rewriting the total Hamiltonian in the form H = H0 + H1 , the correlator in (45) may be expanded in the perturbation H1 [77]. In the second order we obtain for the electrical resistivity the following expression  ∞ kB T −1 ρxx = σxx = 2 2 dt
[jx , H1 (t)][H1 , , jx ] , (46)
jx  0 where H1 (t) is calculated with the Hamiltonian H0 . In the HMF situation the band states with one spin projection only, σ = α = signI, are present at the Fermi level. Below we consider the case I > 0, σ = + and omit the spin indices in the electron spectrum. Then one can find an expression for the transport relaxation time τ defined as σxx = e2
(v x )2 τ π
x 1 = (v − vkx
)2 (A↑kq + A↑k
,q−k
+k )2 Nq (1 + Nq−k
+k )nk (1 − nk
) τ 4T
k kk q 
× δ(tk
− tk − ωq + ωq−k
+k ) (vkx )2 δ(tk ) (47) k

Averaging over angles of the vector k leads us to the final result 1/τ ∝ I 2 Λ with
β(ωp − ωq )|p − q| fpq (1 + Nq )(1 + Np ) , (48) Λ= exp βωp − exp βωq pq where fpq = 1 for p, q  q0 and fpq =

[p × q] (p −

2

2 q) q02

(p, q  q0 ).

(49)

The wavevector q0 determines the boundary of a region where q-dependence of the amplitude become important, so that t(k + q) − t(k)  ∆ at q  q0 and the simple perturbation theory fails.  In elementary one-band model of the HMF where EF < ∆ one has q0 ∼ ∆/W (where W is the conduction bandwidth, and the lattice constant is put to unity) [73]. Generally speaking, q0 may be sufficiently small provided that the energy gap is much smaller than W , which is the case for real HMF systems. The quantity q0 determines a characteristic temperature and energy scale T ∗ = Dq02 ∝ D(∆/W ), where D ∝ TC /S is the spin-wave stiffness defined by

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ωq→0 = Dq 2 , and TC is the Curie temperature. It is important that similar crossover temperatures appear in the temperature dependence of the spin polarization (see, e.g., (13), (17)). This means that temperature dependences of both spin polarization and transport properties can be changed at low enough temperatures within the spin-wave temperature region. One has to bear in mind that each power of p or q yields a T 1/2 factor for the temperature dependence of resistivity. At very low temperatures T < T ∗ small quasimomenta p, q < q0 give the main contribution to the integrals. Then the temperature dependence of resistivity is equal to ρ(T ) ∝ (T /TC )9/2 . Such a dependence was obtained in the large-|I| case where the scale T ∗ is absent [71], and within a diagram approach in the broad-band case [72]. At the same time, for T > T ∗ the function fpq in (48) can be replaced by unity, leading to ρ(T ) ∝ (T /TC )7/2 , in agreement with the old results [70]. According to the calculations presented here, the NQP states do not contribute to the temperature dependence of the resistivity for pure HMF. An opposite conclusion was made by Furukawa [78] and related to an anomalous T 3 dependence in the resistivity. However, this calculation was not based on a consistent use of the Kubo formula and, in our opinion, can be hardly justified. On the contrary, impurity contributions to transport properties in the presence of potential scattering are determined mainly by the NQP states (it has been shown first in [12], see also [2]). To second order in the impurity potential V we derive for the electron Green’s function
(0) (0) (0) G(0) Gkk
σ (E) = δkk
Gkσ (E) + Gkσ (E)V Gk
σ (E)[1 + V pσ (E)] , (50) p (0)

where Gkσ (E) is the exact Green’s function for the ideal crystal (see (2)). Neglecting vertex corrections and averaging over impurities, we obtain for the transport relaxation time in the following form
−1 (E) = −2V 2 Im G(0) (51) δτimp pσ (E) p

Thus the relaxation time is determined by the energy dependence of the density of states N (E) for the interacting system near the Fermi level. The most nontrivial dependence comes from the nonquasiparticle states with the spin projection α = signI, which are present near the EF . Close to the Fermi level the NQP contribution follows the power law (6). Therefore, the impurity contribution to the resistivity is equal to    ∂f (E) δρimp (T ) 2 = −δσ (T ) ∝ −V dE − δNincoh (E) ∝ T 3/2 (52) imp ρ2 ∂E The contribution of the order of T α with α  1.65 (which is not too far from 3/2) has been observed recently in the temperature dependence of the resistivity for NiMnSb [67].

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To calculate the magnetoresistivity we take into account a gap in the magnon spectrum induced by magnetic field, ωq→0 = Dq 2 + ω0 . For large external magnetic field H, in comparison with the anisotropy gap, ω0 is proportional to H . At T < T ∗ the resistivity is linear in magnetic field: 9/2

ρ(T, H) − ρ(T, 0) ∝ −ω0 T 7/2 /TC

(53)

The situation at T > T ∗ is more interesting since the quantity ∂Λ/∂ω0 contains a logarithmic divergence with the cutoff at ω0 or T ∗ . We have at T > ω0 , T ∗ : T 3 ω0 (54) δρ(T, H) ∝ − [max(ω0 , T ∗ )]1/2 Of course, at T < ω0 the resistivity is exponentially small. A negative Hlinear magnetoresistance was observed recently in CrO2 [66]. The incoherent contribution to magnetoresistivity is given by √ δρimp (T, H) ∝ ω0 ∂δNincoh (σT )/∂T ∝ ω0 T . (55) Another useful tool to detect the NQP states is provided by tunneling phenomena [79], in particular by the Andreev reflection spectroscopy for the HMF-superconductor tunnel junction [81]. A most direct way is the measurement of a tunnel current between two pieces of the HMF with the opposite magnetization directions. To this end we consider a standard tunneling Hamiltonian (see, e.g., [55], Sect. 9.3):
(Tkp c†k↑ cp↓ + h.c.) , (56) H = HL + HR + kp

where HL,R are the Hamiltonians of the left (right) half-spaces, respectively, k and p are the corresponding quasimomenta, and spin projections are defined with respect to the magnetization direction of a given half-space (the spin is supposed to be conserving in the “global” coordinate system). Carrying out standard calculations of the tunneling current I in the second order in Tkp we obtain (cf. [55])
|Tkp |2 [1 + Nq − f (tp−q )][f (tk ) − f (tk + eV )]δ(eV + tk − tp−q + ωq ) I∝ kqp

Here V is the bias voltage. For T = 0 one has dI/dV ∝ δNincoh (eV ).

6 Conclusions To conclude, we have considered the special properties of half-metallic ferromagnets which are connected with their unusual electronic structure. Further

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experimental investigations would be of a great importance, especially keeping in mind possible role of the HMF for different applications [2, 3, 4]. Several experiments could be performed in order to clarify the impact of the nonquasiparticle states on spintronics. Direct ways of observing the NQP states would imply the technique of Bremsstrahlung Isohromat Spectroscopy (BIS) [21] or the spin-polarized scanning tunneling microscopy (SP-STM) [15], since for the most frequent case of minority-spin gap the NQP states lie above EF . In contrast with the photoelectron spectroscopy of the occupied states (PES) which has to show a complete spin polarization in the HMF with minority-spin gap, the BIS spectra should demonstrate an essential depolarization of the states above the EF . For the majority-spin-gap HMF, vice versa, the partial depolarization should be seen in the PES. The I −V characteristics of half-metallic tunnel junctions for the case of antiparallel spins are completely determined by the NQP states [75, 80]. The spin-polarized STM should be able to probe these states by the differential tunneling conductivity dI/dV [55, 82]. In particular, the SP-STM with positive bias voltage can detect the opposite-spin states just above the Fermi level for surface of the HMF such as CrO2 . The Andreev reflection spectroscopy for tunnel junction superconductor-HMF [81] can also be used in searching for experimental evidence of the NQP effects. These experimental measurements will be of crucial importance for the theory of spintronics in any tunneling devices with the HMF. Since ferromagnetic semiconductors can be considered as a special case of the HMF, an account of these states can be helpful for the proper description of spin diodes and transistors [83]. The research described was supported in part by Grant No.02-02-16443 from Russian Basic Research Foundation and by Russian Science Support Foundation.

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40. G. Kotliar, S.Y. Savrasov: in New Theoretical Approaches to Strongly Correlated Systems, edited by A.M. Tsvelik (Kluwer Acad. Publ., Dordrecht, 2001) 226, 227 41. A.V. Zarubin, V.Yu. Irkhin: Fiz. Tverd. Tela 41, 1057 (1999) [Phys. Solid State, 41, 963 (1999)] 226 42. M. Jarrell: Phys. Rev. Lett. 69, 168 (1992) 226 43. O.K. Andersen, O. Jepsen, G. Krier: in Lectures on Methods of Electronic Structure Calculations, edited by V. Kumar, O.K. Andersen, A. Mookerjee (World Scientific, Singapore, 1994), p. 63; O.K. Andersen, T. Saha-Dasgupta: Phys. Rev. B 62, R16219 (2000) 227 44. L. Vitos, H.L Skriver, B. Johansson, J. Koll´ ar: Comp. Mat. Science 18, 24 (2000) 227 45. L. Chioncel, L. Vitos, I.A. Abrikosov, J. Koll´ ar, M.I. Katsnelson, A.I. Lichtenstein: Phys. Rev. B 67, 235106 (2003) 227 46. M.I. Katsnelson, A.I. Lichtenstein: Eur. Phys. J. B 30, 9 (2002) 228 47. N.E. Bickers, D.J. Scalapino: Ann. Phys. (N.Y.) 193, 206 (1989) 228 48. V.M. Galitski: Zh. Eksper. Teor. Fiz. 34, 115, 1011 (1958); J. Kanamori: Prog. Theor, Phys 30, 275, (1963) 228 49. V.I. Anisimov, O. Gunnarsson: Phys. Rev. B 43, 7570 (1991) 228 50. V.Yu. Irkhin, M.I. Katsnelson: Eur. Phys. J. B 43, 479 (2005) 230 51. Yu. M. Yarmoshenko, M.I. Katsnelson, E.I. Shreder, E.Z. Kurmaev, A. Slebarski, S. Plogmann, T. Schlathoelter, J. Braun, M. Neumann: Eur. Phys. J. B 2, 1 (1998) 230 52. M.V. Yablonskikh, Y.M. Yarmoshenko, V.I. Grebennikov, E.Z. Kurmaev, S.M. Butorin, L.-C. Duda, J. Nordgren, S. Plogmann, M. Neumann: Phys. Rev. B 63, 235117 (2001) 230, 234 53. E.Z. Kurmaev, A. Moewes, S.M. Butorin, M.I. Katsnelson, L.D. Finkelstein, J. Nordgren, P.M. Tedrow: Phys. Rev. B 67, 155105 (2003) 230, 234, 235 54. O. Wessely, P. Roy, D. Aberg, C. Andersson, S. Edvardsson, O. Karis, B. Sanyal, P. Svedlindh, M.I. Katsnelson, R. Gunnarsson, D. Arvanitis, O. Bengone, O. Eriksson: Phys. Rev. B 68, 235109 (2003) 230 55. G.D. Mahan, Many-Particle Physics (Plenum Press, New York, 1990) 230, 231, 239, 240 56. P. Nozieres and C.T. de Dominicis: Phys. Rev. 178, 1097 (1969) 230, 231 57. D.N. Zubarev: Sov. Phys. Uspekhi 3, 320 (1960); D.N. Zubarev: Nonequilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974) 231 58. Yu.A. Izyumov, M.V. Medvedev: Sov. Phys. – JETP 32, 202 (1971) 231 59. M.I. Auslender, V.Yu. Irkhin, M.I. Katsnelson: J. Phys. C 17, 669 (1984) 231 60. A.A. Abrikosov: Physics 2, 5 (1965) 233 61. H. Suhl: Phys. Rev. 138, A1112 (1965) 233 62. H. Ebert: Rep. Progr. Phys. 59, 1665 (1996) 234 63. J.J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Reading, MA, 1967) 234 64. F. Gel’mukhanov, H. Agren: Phys. Rev. A 49, 4378 (1994) 234 65. O.B. Sokolov, V.I. Grebennikov, E.A. Turov: phys.stat.sol.(b) 83, 383 (1977) 235 66. M. Rabe, J. Pommer, K. Samm, B. Oezyilmaz, C. Koenig, M. Fraune, U. Ruediger, G. Guentherodt, S. Senz, D. Hesse: J. Phys.: Condens. Matter 14, 7 (2002) 235, 239 67. C.N. Borca, T. Komesu, H.-K. Jeong, P.A. Dowben, D. Ristoiu, Ch. Hordequin, J.P. Nozieres, J. Pierre, S. Stadler, Y.U. Idzerda: Phys. Rev. B 64, 052409 (2001) 235, 238 68. M. Ziese: Rep. Prog. Phys. 65, 143 (2002) 235

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69. E.L. Nagaev: Phys. Rep. 346, 388 (2001) 235 70. M. Roesler: phys. stat. sol.8, K31 (1965); F. Hartman-Boutron: Phys. Kond. Mat. 4, 114 (1965) 235, 238 71. K. Kubo, N. Ohata: J. Phys. Soc. Japan 33, 21 (1972) 235, 238 72. V.S. Lutovinov, M.Yu. Reizer: Zh. Eksp. Theor. Fiz. 77, 707 (1979) 236, 238 73. A.P. Grigin, E.L. Nagaev: phys. stat. sol. (b) 61, 65 (1974); E.L. Nagaev: Physics of Magnetic Semiconductors (Mir, Moscow, 1983) 236, 237 74. M.I. Auslender, M.I. Katsnelson, V.Yu. Irkhin: Physica B 119, 309 (1983) 236 75. V.Yu. Irkhin, M.I. Katsnelson: Eur. Phys. J. B 30, 481 (2002) 236, 240 76. R. Kubo: J. Phys. Soc. Japan 12, 570 (1957) 237 77. H. Nakano: Prog. Theor. Phys. 17, 145 (1957); H. Mori: Prog. Theor. Phys.34, 399 (1965) 237 78. N. Furukawa: J. Phys. Soc. Japan 69, 1954 (2000) 238 79. M.I. Auslender, V.Yu. Irkhin: Sol. State Commun. 56, 703 (1985) 239 80. E. McCann, V.I. Fal’ko, Phys. Rev. B 68, 172404 (2003) 240 81. G. Tkachov, E. McCann, V.I. Fal’ko: Phys. Rev. B 65, 024519 (2001) 239, 240 82. Y. Meir, N.S. Wingreen: Phys. Rev. Lett. 68, 2512 (1992) 240 83. M.E. Flatte, G. Vignale: Appl. Phys. Lett. 78, 1273 (2001) 240

Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy S.J. Jenkins Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom [email protected] Abstract. A framework for discussion of stoichiometry at the surfaces of ternary materials is presented and applied to the semi-Heusler alloy NiMnSb. On the basis of density functional theory calculations, it is evident that Sb is preferentially found in the outermost layer, while Mn avoids the surface layer. Thermodynamic considerations for the growth of an Sb overlayer are also discussed, leading to the conclusion that epitaxy is only possible for a cubic overlayer structure. The stoichiometry of the surface is shown to have a marked effect on the spin-polarised surface states of NiMnSb, with major implications for its half-metallic properties.

1 Theory of Spintronic Materials: A Surface Science Perspective The essence of spintronics is the development and exploitation of a capability to control the flow of electrons through a device on the basis of their intrinsic spin, predictably and without ambiguity. Crucial to that aim is the generation of perfectly spin-polarised currents, and the ability to inject these into welldefined bandstructures with minimal spin-flip scattering. To this end, much attention has been focussed upon so-called spin-valve devices, in which one spin species is transmitted and the other attenuated. Perhaps the simplest way in which to achieve such a device is through the use of an exotic material having the properties of a metal in respect of one spin species and those of an insulator in respect of the other. Just such a behaviour was predicted in 1983 by de Groot et al [1], on the basis of their density functional theory (DFT) calculations for the semi-Heusler alloy NiMnSb. Considerable attention has been lavished on this and other similar materials ever since (see [2] and references therein), but at least one critical issue remains unresolved. Most of the known half-metallic compounds have complicated surface structures, and little is so far known about how the stoichiometry of the surface may influence its spintronic properties, or indeed those of the interfaces one might hope to grow upon them The purpose of the present work is to indicate some of the ways in which that issue may be addressed.

S.J. Jenkins: Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy, Lect. Notes Phys. 678, 245–259 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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1.1 Spintronic Properties of Semi-Heusler Alloys The Heusler and semi-Heusler alloys are related classes of material having the general formulae X2 MnY and XMnY respectively, where X represents an element from the d-block and Y an element from the p-block of the periodic table. In the case of the prototypical example, NiMnSb, the halfmetallicity identified by de Groot and co-workers [1] is manifest in the existence of a roughly 0.5 eV band gap in the spectrum of minority-spin states. The Fermi level, at least in a zero temperature approximation, thus shows 100% majority-spin polarisation in the bulk phase. The surface and interface properties of NiMnSb have, however, only recently received careful attention [2, 3, 4, 5, 6, 7, 8, 9, 10]. One of the major challenges in experimental work on such systems is presented by the possibility of segregation in the surface region [4]. Deviations from the bulk composition are expected to be severely detrimental to halfmetallic properties. Achieving control over the stoichiometry of the selvedge is, however, far from trivial, and success in attaining a well-characterised surface at the atomic scale has been limited. Nevertheless, Dowben and coworkers [3] have shown that careful preparation can result in an ordered surface structure for NiMnSb{001}. It is to be hoped that similar techniques can be developed for other facets, placing the surface science of these complex alloys on the secure experimental footing their potential importance doubtless warrants. In the meantime, however, an unusually high emphasis is necessarily placed on theory to indicate which surfaces and interfaces may be of interest for possible spintronic applications. The main points to emerge from such studies are (i) that even stoichiometrically perfect surfaces may deviate from half-metallicity due to minority-spin surface states [5, 6], (ii) that these surface states may transform into interface states as overlayers are built up on the surface [2], and (iii) that such interface states are a common feature of a great many otherwise promising heterojunctions [5]. Before discussing how these findings relate to the surface stability and composition, however, let us take a brief diversion to review the way in which such calculations are typically carried out. 1.2 Density Functional Theory of Magnetic Surfaces The application of DFT to the study of magnetic surfaces dates to the pioneering efforts of Freeman and co-workers in the 1980s [11, 12, 13, 14, 15, 16, 17, 18]. Using an all-electron full-potential linearised augmented plane wave (FLAPW) approach, these ground-breaking calculations established much of the fundamental framework underlying our current understanding of the topic. The role of d-band narrowing in enhancing magnetic moments in the surface layer, for instance, was already evident in studies of Fe{001} and Ni{110} published as early as 1983 [11, 12]. Nevertheless, the routine application of

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DFT to spin-polarised surfaces became a reality only with the advent of ultrasoft pseudopotentials in the early 1990s [19]. These allowed calculations to be performed significantly more cheaply than with all-electron techniques, and as computer codes have been developed to take advantage of this fact, so magnetic surface studies have become an increasingly common feature of the theoretical literature (see the present author’s recent review [20]). Questions concerning the accuracy of ultrasoft pseudopotentials for ferromagnetic systems have been addressed by Kresse and Joubert [21], from whom one may reasonably draw the conclusion that carefully constructed pseudopotentials will typically perform almost as well as all-electron methods in all but the most subtle of cases. The practicalities of DFT, as applied to magnetic surfaces, vary according to the precise computer code employed, but the majority of current work is carried out utilising a slab/supercell geometry. This involves the use of a periodically-repeating cell whose dimensions are consistent with the 2D unit mesh in the surface plane, but are considerably elongated in the direction of the surface normal. Embedded within this cell is a slab of material, intended to represent the surface being studied, separated from its periodic images in the cells above and below by vacuum regions. The periodic boundary conditions allow an efficient representation of the electronic wavefunctions by means of an expansion in plane waves, and ensure that the model surface is infinite in extent. The artificial periodicity in the surface normal direction constitutes only a minor source of error, so long as (i) the slab thickness is sufficient to allow bulk-like properties to assert themselves in central atomic layers, and (ii) the vacuum region is sufficiently wide to prevent each surface being influenced by the surface immediately above or below. Typical supercell dimensions for metallic surfaces, at which the screening properties are excellent, are likely to be of the order of 10-15 ˚ A slab thicknesses, with similar widths for the vacuum regions. The results discussed here were all obtained within just such a scheme, employing the supercell approach and ultrasoft pseudopotentials. Details of the calculations are reported elsewhere [2, 6, 7] and need not be repeated in the present work. Instead, we will concentrate upon the post hoc analysis of DFT results and the valuable additional information that may, in principle, be extracted from them.

2 Surface Stoichiometries in a Supercell Approach We now consider how results of DFT supercell calculations may be analysed to reveal the stoichiometries of complex surfaces under different ambient conditions. The methodology employed here is broadly the same as is applied routinely to discussions of the surface stability of compound semiconductors [22], albeit with a convenient graphical twist that is particularly suited to ternary compounds of the type with which we are here concerned.

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2.1 General Considerations Theoretical analysis of the surface stability for structures of differing stoichiometry is immediately problematic, as calculated internal energies for supercells containing different numbers of atoms cannot be directly compared. Instead, one must consider a free energy for each structure, and this will of course depend upon the chemical potentials of the constituent elements. For a general case of a ternary compound, the free energy of a supercell, F , may be written in terms of the internal energy, E, as: F = E − nx µx − ny µy − nz µz

(1)

where nx , ny and nz represent the number of atoms of elements x, y and z present in the supercell, and the µ’s represent their respective chemical potentials. Clearly, the precise values taken by the chemical potentials will vary according to the prevailing conditions; typically, lower absolute values correspond to cases where the concentration of that species is low, and the potentials tend towards higher absolute values as the concentration of each species increases. It therefore follows that the free energy of any structure can itself vary considerably as the ambient environment of the system changes. It is consequently impossible to determine a definitive ground state structure that would be most stable under all imaginable conditions. What is viable, however, is to place bounds on the allowed combinations of chemical potential that may occur. In this way, it is possible to determine which set of structures are most stable under the various experimentally achievable conditions. Consider again the case of a ternary compound. Clearly the first kind of constraint we may place upon the chemical potentials is that each individually must remain below the corresponding bulk chemical potential. If such a condition were not satisfied for a certain element, we should expect the species in question to segregate out of the compound, forming crystallites of the pure element and leaving behind a deficient parent compound. We thus have the following three inequalities: µi ≤ µbulk i

i = x, y, z

(2)

Furthermore, for each element, we can also consider its complementary binary compound. Thus element x finds its complement in the compound yz, element y in the compound zx, and element z in the compound xy. In this case, the sum of the chemical potentials for the constituents of the complementary compound must not exceed the bulk chemical potential of that compound. If such a condition were not satisfied, segregation would lead to crystallites of the binary compound in question, and the residual parent compound would be unduly rich in its complementary element. Thus, we must have: ij = yz, zx, xy (3) µi + µj ≤ µbulk ij

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Note here that the bulk chemical potentials of the elements are approximated by the internal energies per atom calculated in the appropriate bulk structure; the bulk chemical potentials of the compounds, on the other hand, are approximated by the internal energies per structural unit. That is, the bulk chemical potentials of the binary compounds are approximated by the bulk internal energy per pair of heteronuclear atoms, while that of the ternary compound is approximated by the bulk internal energy per heteronuclear triad. The inequalities above place considerable contraints upon the possible values of the chemical potentials, but do not in themselves fully define the range of accessibility. In order to do that, we must satisfy one further requirement, which is that the sum of the three elemental chemical potentials must precisely equal the bulk chemical potential of the ternary compound itself. This last constraint expresses the condition that the surface be in thermal equilibrium with its bulk: µx + µy + µz = µbulk xyz

(4)

At this stage, it is customary to employ this last expression to eliminate one or other of the individual elemental chemical potentials. Subsequent manipulations are carried out in terms only of the two remaining potentials. The choice of which particular chemical potential to eliminate is, however, entirely arbitrary. An alternative approach, which does not discriminate against one element in favour of its colleagues, is to adopt the graphical methodology presented below. 2.2 Graphical Representations of Stability The key to the present approach is the recognition that although one is dealing with a nominally three-dimensional problem, the constraint provided by the necessity for thermal equilibrium with the bulk ensures that the space spanned by the ensemble of chemical potentials is essentially only two-dimensional in nature. We may thus represent all the pertinent data on a two-dimensional graph, albeit with three separate axes (see Fig. 1). The crucial point to note about such a representation is that although all possible values of the individual chemical potentials may be represented by points on such a graph, only certain specific combinations are allowed. In particular, if the graduations on all three axes represent identical increments in chemical potential, then the sum of the three individual chemical potentials is a constant, no matter what point on the graph one considers. By choosing the origins of the axes appropriately, therefore, it is possible to ensure that the condition expressed in equation (4) is automatically satisfied by all points within the graph, and this has been done for all NiMnSb phase diagrams in the present work. The inequalites (2) and (3) may now be plotted against the three axes, without loss of generality.

−1 35 8.5

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S.J. Jenkins

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Fig. 1. Conditions for stability of bulk NiMnSb against segregation of (a) bulk Ni or bulk MnSb, (b) bulk Mn or bulk NiSb, and (c) bulk Sb or bulk NiMn. Panel (d) shows the combination of these conditions. Chemical potentials are in units of eV throughout the present work

In the case of NiMnSb, we find that the conditions expressed by the six inequalites give rise to an allowed region taking the form shown in Fig. 1. The requisite bulk chemical potentials for Ni, Mn, Sb, MnSb, NiSb and NiMn are taken from bulk DFT calculations [7]. Note that the absolute values of the various chemical potentials are of no significance, since pseudopotential calculations cannot yield such information – the energy of the core electrons is not explicitly included. Nevertheless, so long as all calculations are performed with the same set of Ni, Mn and Sb pseudopotentials, the results are perfectly consistent within themselves. Whilst considering the possible structures and stoichiometries of NiMnSb surfaces, it is also an interesting diversion to take a look at the surfaces of the three binary compounds mentioned already. Since these compounds each contain only two elements, the free energies that must be calculated are part of only a two-dimensional system. There is, therefore, no inherent reason to

Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy

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plot them on a triangular graph, but we may nevertheless choose to do so for aesthetic and comparative reasons. The plots for these compounds are thus each presented on two of the same axes (i.e. with the same ranges and origins) as for the ternary compound, with the third axis simply suppressed. Just as the requirement for thermal equilibrium with the bulk reduced the dimensionality of our original problem from three to two dimensions, so the equivalent conditions here reduce the accessible space in each of our free energy plots from two dimensions to just one. That is, MnSb surfaces must satisfy: (5) µm + µs = µbulk ms while NiSb and NiMn surfaces must satisfy µn + µs = µbulk ns

(6)

µn + µm = µbulk nm

(7)

and respectively, where we have introduced the notation (n, m, s = Ni,Mn,Sb) in the interests of brevity. These conditions constrain the system to lie not within a two-dimensional region of the triangular phase diagram, but instead on a one-dimensional line, as indicated in Fig. 2. 2.3 Reference Results for MnSb{0001}, NiSb{0001} and NiMn{001} Having determined the regions/lines of stability for the various compounds to be discussed, it is now a simple matter to plot the regions for which one or other surface phase provides the lowest free energy. Consider, for example, the case of NiMn{001}, where we have modelled a Ni(Mn)-terminated surface by means of a slab containing six layers of Ni (Mn) and five of Mn (Ni). The free energies of the supercells in the two cases are determined according to expression (1), and equating these we find the line of the phase boundary between the Ni-terminated and Mn-terminated surfaces. Plotting this against the axes described in the preceding sub-section, we find that the allowed combinations of Ni and Mn chemical potentials fall entirely within the Niterminated region (Fig. 3.a). Not only that, but the graph provides a very visual impression of the extent to which the preference is robust: the line of stability approaches the Mn-terminated region no closer than around 0.2 eV, which should be well within the ability of DFT to predict. In a similar way, we can also consider the stoichiometries of the MnSb{0001} and NiSb{0001} surfaces (Figs. 3.b and 3.c). In both cases, it is the Sb-terminated surface that is preferred, and the results are, if anything, even more robust than that for NiMn. The surfaces of the three binary compounds are thus characterised by a general tendency for Sb to occur in the surface layer (where present), and for Mn not to occur in the surface layer

S.J. Jenkins a.

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Fig. 2. Conditions for stability of (a) bulk NiMn, (b) bulk MnSb, and (c) bulk NiSb, against segregation of the constituent pure elements. The allowed values of chemical potentials are indicated by the lines terminated with circles

(where present). The former tendency appears somewhat stronger than the latter. These findings will be of some interest in the discussion of NiMnSb surfaces to follow. 2.4 The Nature of NiMnSb Surface Stoichiometry Let us first consider the stoichiometry of the NiMnSb{111} surface. When viewed along {111}, the structure of NiMnSb reveals itself to comprise a series of trilayers, forming the pattern -]-0-[Mn-Ni-Sb]-0-[-, where “0” indicates an empty atomic layer (see Fig. 4). Cleavage in this plane might therefore be anticipated to generate two distinct surfaces, the first terminated by a complete trilayer ending on a layer of Mn atoms, and the second terminated by a complete trilayer ending on a layer of Sb atoms. We shall label the fomer the Aα surface, and the latter the Bα surface. The question as to which of these might be more stable is not valid, as they are necessarily complemen-

Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy

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0.1

−657.0

−1

Sb

−656.6

36 −1

h ric

or po

−656.2

−1

36

0.9

Mn

Ni

0.5

Sb

−655.8

−1

36

1.3

Ni

1.7

−655.4

−1

36

Mn −655.0

53 .2

.8 52

−1

−1

.0

.6

54

53

−1

−1

.8

.4

54

54

µ Sb

−1

.2 55 −1

Sb poor

−1

.0

.4

56

.6 55 −1

−1

.8

56

56

.8

.2

52

53

.6 53

−1

−1

−1

−1 Sb rich

−1

.4

.0

.2

54

54

−1

55

.8 54

5.6

.4

µ Sb

−1

−1

−1

.8

5 −1

56

.0

56

56

−1

−1

−1

Sb poor

Sb rich

Fig. 3. Surface phase diagrams for (a) NiMn{001}, (b) MnSb{0001}, and (c) NiSb{0001}, showing regions of preference for Ni, Mn or Sb terminations. The circle-terminated line in each case indicates the condition for stability of the bulk compound, as in Fig. 2

tary rather than competitive. What may reasonably be queried, however, is whether the atoms at each surface can rearrange themselves into some more favourable configuration than those we have just described. In particular, can Mn atoms on the Aα surface migrate elsewhere to leave behind a bilayer-terminated Aβ surface ending on a layer of Ni atoms, or might not these Ni atoms also migrate elsewhere to leave a monolayer-terminated Aγ surface ending on a layer of Sb atoms? And might not the Bα surface suffer questions, the threetodifferent A surfaces and Bγ above surfaces? haveTobeen answer modelled these similar segregation form analogous Bβ posited by slabs consisting of varying quantities of Ni, Mn and Sb atoms [7]. The free energies of the supercells were as follows: FAα = EAα − 6µn − 6µm − 6µs

(8)

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S.J. Jenkins

[001]

Ni Mn

Sb

[010]

[100]

Fig. 4. Bulk structure of NiMnSb. The [111] direction runs diagonally upwards from left to right

FAβ = EAβ − 6µn − 5µm − 6µs

(9)

FAγ = EAγ − 5µn − 5µm − 6µs

(10)

Similarly, the free energies of the three different B surfaces were: FBα = EBα − 6µn − 6µm − 6µs

(11)

FBβ = EBβ − 6µn − 6µm − 5µs

(12)

−1 35 8.5

−1 35 9.3

−1 36 0.1

−1 36 0.9

−1 36 1.7

−655.0

.8 52 −1

.6 53 −1

.2 53 −1

.4 54 −1

µ Sb

.0 54 −1

.8 54 −1

.2 55 −1

Sb poor

.6 55 −1

.4 56 −1

.0 56 −1

.8 56 −1

.2 53 −1

.8 52 −1

.0 54 −1

.6 53 −1

.4 54 −1

.2 55 −1

.8 54 −1

.6 55 −1

.4 56 −1

.0 56 −1

.8 56 −1

Sb rich

−1 36 1.3

−1 36 1.3

−655.8

−655.4

−1 36 1.7

Mn −655.0

Sb

−1 36 0.5

Mn ric h

−1 36 0.9

−655.8

−655.4

Sb poor

−1 35 9.7

n

µM

−1 36 0.1 −1 36 0.5

−656.6

−656.2

or po Ni

Sb

µ Sb

−657.4

−657.0

or po Ni

−656.2

−1 35 8.9

−1 35 8.9 −1 35 9.3

Ni −656.6

Mn ric h

Mn po or

−1 35 8.5

−1 35 9.7

n

−657.0

µ Ni

µM

−658.2

−657.8

µ Ni

−657.4

−658.6

h ric Ni

−658.2

−657.8

−1 35 8.1

−659.0

h ric Ni

−658.6

Mn po or

b.

−1 35 7.7

−659.0

−1 35 8.1

a.

−1 35 7.7

FBγ = EBγ − 5µn − 6µm − 5µs (13) Once again, we may equate free energies for pairs of structures, taken either both from the A paradigm or both from the B, to obtain the boundaries between the different phases. Plotting these in relation to the allowed region of stability for NiMnSb, we reach the conclusion that the Sb-terminated structure is preferred for both the A and the B surfaces (Fig. 5). In the case of

Sb rich

Fig. 5. Phase diagrams for NiMnSb{111}, showing the stability of Ni, Mn and Sb terminated phases for (a) the A surface, and (b) the B surface. The central polygon indicates the region within which bulk NiMnSb can be stable, as in Fig. 1

Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy

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the B surface, this is hardly surprising: the Sb-termination coincides with the “natural” weak-point associated with the empty layer of the bulk structure, and the preference is manifest by a considerable margin. In the case of the A surface, however, the expectation might have been for a Mn-terminated surface, based upon the “weak empty layer” argument, but in fact this is the least favourable termination. There is some room for the possibility that a Ni-terminated Aβ surface might be feasible in the extreme Ni-rich condition, if the DFT calculations are subject to a small degree of error, but the Sbterminated Aγ surface appears overwhelmingly more likely. In this respect, the results for NiMnSb{111} would seem to follow the same fundamental trend identified in the three binary compounds: namely, the affinity of Sb for the surface layer, the reluctance of Mn to occupy the same region, and the relative ambivalence of Ni. In light of this analysis, therefore, the stoichiometry of the NiMnSb{001} surface takes on an interesting quality. The possible surface terminations here are either an A surface ending with a mixed MnSb layer, or a B surface ending simply with Ni. The question must clearly arise as to whether the affinity of Sb for the surface layer is enough to stabilise the A termination, in the face of the opposing influence exerted by the presence of Mn atoms. A partial answer has already been provided by the work of Dowben and co-workers, who presented convincing photoemission evidence to suggest that the MnSbtermination was indeed favoured under the Sb-rich conditions they explored [3]. Theoretical analysis can now flesh out this information, as the phase diagram (Fig. 6.a) clearly shows the MnSb-termination to be favoured under all accessible conditions. As might have been inferred from study of the binary compounds, the structural preferences of Sb appear to dominate those of Mn.

2.5 Thermodynamics of Growth In addition to the insight into surface terminations provided by free energy analysis, one may also determine the nature of the growth modes for different overlayer materials (see [23] for an example). Such information is clearly of paramount importance if practical semi-Heusler interfaces are ever to be manufactured for use in spintronic devices. As an example, let us take the case of Sb overlayers on NiMnSb{001}, whose electronic and structural properties were described previously by the author on the basis of DFT calculations [2]; we now re-analyse the energetics of those calculations to assess the implications for growth. As before, we start with the free energies of the slabs used in the DFT work, equating these to find the boundaries between the different structural phases. We utilise the following expression: Fν = Eν − 5µn − 5µm − (5 + 2ν)µs

ν =0−3

(14)

−1 35 8.9

−1 36 0.1

−1 36 0.9 −1 36 1.3 .8 52 −1

.6 53 −1

.2 53 −1

−1 35 9.3 .4 54 −1

µ Sb

.0 54 −1

.8 54 −1

.6 55 −1

Sb poor

.2 55 −1

.0 56 −1

.4 56 −1

.8 56 −1

.2 53 −1

.8 52 −1

.6 53 −1

.4 54 −1

.0 54 −1

.2 55 −1

.8 54 −1

.6 55 −1

.4 56 −1

.0 56 −1

.8 56 −1

Sb rich

MnSb −655.0

−1 36 1.7

Mn ric h

−1 36 1.3 −1 36 1.7

−655.0

−1 35 9.7

n

µM

−1 36 0.1 −1 36 0.5

−1 36 0.9

−655.8

−655.4

−1 36 0.5

−1 35 8.9 −1 35 9.3

−655.8

−655.4

µ Sb

−1 35 8.5

−1 35 8.5

−1 35 9.7

n

MnSb

or po Ni

Ni

or po Ni

Mn ric h

−656.6

−656.2

µ Ni

−657.0

−656.2

Sb poor

... etc er er, lay ay no tril mo er, ay Sb bil

Ni

−656.6

−657.4

Sb

µM

−658.2

−657.8

µ Ni

−657.4

−657.0

−658.6

h ric Ni

−658.2

−657.8

−1 35 8.1

−659.0

h ric Ni

−658.6

Mn po or

b. Mn po or

−659.0

−1 35 7.7

a.

−1 35 7.7

S.J. Jenkins

−1 35 8.1

256

Sb rich

Fig. 6. Phase diagrams for NiMnSb{001}, showing the stability of Ni and MnSb terminated phases for (a) the clean surface, and (b) the clean and Sb-overlayer covered surfaces. The central polygon indicates the region within which bulk NiMnSb can be stable, as in Fig. 1

where the subscript ν refers to the number of monolayers of Sb deposited upon the clean MnSb-terminated {001} surface. The pre-factors for the chemical potentials are, of course, simply reflective of the number of each type of atom present in the supercells used for the original DFT calculations. Plotting these phase boundaries once more on the familiar triangular axes, we obtain the regions for which the clean surface, the monolayer-covered surface, the bilayer-covered surface and the trilayer-covered surface are variously the most stable structures (Fig. 6.b). Notably, the boundary between the first two of these regions falls almost precisely at the edge of the accessible zone for the chemical potentials; those between the monolayer, bilayer and trilayer structures actually fall beyond that edge. It follows, therefore, that the monolayer-covered structure is essentially isoenergetic (in terms of free energy, of course) with a putative system in which Sb segregates to form three-dimensional islands of bulk-like material in isolated locations on an otherwise clean surface. Furthermore, the bulk-like islands are actually preferred over the epitaxial bilayer and trilayer structures. This suggests the so-called Stranski–Krastanov growth mode, in which two-dimensional epitaxial growth for the first few layers is superseded by three-dimensional island growth in later stages. Experimental evidence, however, strongly suggests a fully epitaxial (or Frank–Van der Merwe) structure for Sb overlayers on NiMnSb{001}, but crucially the square symmetry of the substrate induces a cubic structure in the overlayer rather than its ground state hexagonal structure [3]. Presumably, the establishment of the cubic structure in the first layers creates a kinetic barrier that prevents the adoption of a hexagonal structure in subsequent growth. If one were to substitute the bulk chemical potential of cubic

Theoretical Stoichiometry and Surface States of a Semi-Heusler Alloy

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Sb into the stability conditions for NiMnSb, one would find that the sequence of boundaries between monolayer, bilayer and trilayer structures would very rapidly converge onto precisely that position on the phase diagram. In other words, the epitaxial growth becomes isoenergetic with the cubic structure at an early stage, and remains so as growth continues. Epitaxial growth to form a cubic overlayer is thus possible, whereas the epitaxial growth of a hexagonal overlayer is not.

3 Stoichiometry and Spintronic Structure The arguments presented above concerning surface stoichiometry are of more than purely academic interest. In efforts to create useful spintronic interfaces from semi-Heusler or other half-metallic materials, a deep understanding of the interplay between atomic and electronic structures is of fundamental importance. Indeed, the precise composition of the surface layers can have a profound influence on the potential spintronic properties, although calculations for the minority-spin states of the NiMnSb{001}/Sb system have shown that surface/interface states can destroy half-metallicity even in perfectly stoichiometric cases [2, 6]. The effect of varying surface stoichiometry is, however, best seen in the case of NiMnSb{111}, where one may recall two distinct types of surface must occur: the A surfaces for which a complete trilayer would terminate with Mn atoms, and the B surfaces for which it would terminate with Sb atoms. In the latter case, the preference for Sb termination is so strong that it would be perverse to expend much effort investigating the electronic structures of the alternatives, but for the former the margins are not nearly so large. We therefore present minority-spin bandstructures for all three of the A surfaces, but only the most stable of the B surfaces (Fig. 7). As can readily be appreciated, the nature of the minority-spin surface states is radically different in each of the four cases shown. It is worth noting that all of the possible terminations of the NiMnSb{111} surface shown feature minority-spin states at the Fermi level. In effect, therefore, none are truly half-metallic in the surface region. Crucially, however, the dispersion curves reveal that the metallic minority-spin states are located in very specific parts of the surface Brillouin zone. The surface state of the Aα termination, for instance, has a rather flat dispersion curve, and cuts the Fermi level close to the zone edge. The Aβ termination, in contrast, displays two parabolic surface states, which cut the Fermi level around halfway between the zone centre and zone edge, while the Aγ termination features two nearly degenerate parabolic states that cut the Fermi level close to the zone centre. In the Bα termination, a single parabolic surface state cuts the Fermi level midway between zone centre and zone edge. All the Fermi contours are roughly circular, with the exception of that corresponding to the higher-lying of the Aβ surface states, which cuts off the corners of the hexagonal surface Brillouin zone.

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Energy (eV)

Aα

Aβ

Aγ

Bα

2.0

2.0

2.0

2.0

1.0

1.0

1.0

1.0

0.0

0.0

0.0

0.0

M −1.0

−1.0

K

−1.0

−1.0

Γ −2.0

−2.0

M

Γ

K

−2.0

M

Γ

K

−2.0

M

Γ

K

M

Γ

K

Fig. 7. Minority-spin states at NiMnSb{111} surfaces. The lines common to all panels represent the projected bulk minority-spin bandstructure. Lines crossing the majority-spin Fermi level (i.e. 0 eV) are surface states

It has been shown that interface states localised at the junctions between semi-Heusler alloys and semiconductors also tend to destroy strict halfmetallicity [5]. As yet, however, no systematic study of the effect of interface stoichiometry on the location of such states within the Brillouin zone has been reported.

4 Concluding Remarks The foregoing discussion should have impressed upon the reader the fact that the detailed surface structure of complex alloy systems can have a profound influence on electronic and spintronic properties. In such circumstances, it would be useful to have access to a simple “rule-of-thumb” to guide one’s understanding of which surface terminations are likely to be more stable than others. In this regard, we here propose that the over-riding consideration is that the p-block element of a semi-Heusler alloy (Sb in this case) will tend to be located in the surface layer. A secondary consideration is that, all else being equal, Mn will tend to avoid the surface layer. The other d-block element (Ni in this case) shows no strong tendency either way. As for interface growth, the thermodynamic considerations are likely to be even more complicated than those for the clean surfaces. The interface between a semi-Heusler alloy and a binary semiconductor, for instance, may involve as many as five different atomic species. The stoichiometries that may

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arise in the vicinity of such an interface will undoubtedly be complex, and the resulting electronic structures may deviate considerably from those of the idealised cases already studied. Furthermore, theoretical insights into growth, along the lines described above, may yet prove invaluable in manufacturing the well-characterised interfaces essential to future spintronic devices.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

R.A. de Groot et al: Phys. Rev. Lett. 50, 2024 (1983) 245, 246 S.J. Jenkins, D.A. King: Surf. Sci. Lett. 501, L185 (2002) 245, 246, 247, 255, 257 D. Ristoiu et al: Europhys. Lett. 49, 624 (2000) 246, 255, 256 D. Ristoiu et al: Appl. Phys. Lett. 76, 2349 (2000) 246 G.A. de Wijs, R.A. de Groot: Phys. Rev. B 64, 020402 (2001) 246, 258 S.J. Jenkins, D.A. King: Surf. Sci. Lett. 494, L793 (2001) 246, 247, 257 S.J. Jenkins: Phys. Rev. B (in press) 246, 247, 250, 253 W. van Roy et al: Appl. Phys. Lett. 77, 4190 (2000) 246 W. van Roy et al: J. Cryst. Growth 227, 862 (2001) 246 W. van Roy et al: Appl. Phys. Lett. 83, 4214 (2003) 246 H. Krakauer et al: Phys. Rev. B 28, 610 (1983) 246 S. Ohnishi et al: Phys. Rev. B 28, 6741 (1983) 246 E. Wimmer et al: Phys. Rev. B 30, 3113 (1984) 246 C.L. Fu et al: Phys. Rev. Lett. 54, 2700 (1985) 246 S. Ohnishi et al: J. Magn. Magn. Mater. 50, 161 (1985) 246 A.J. Freeman, C.L. Fu: J. Appl. Phys. 61, 3356 (1987) 246 C. Li, A.J. Freeman: J. Magn. Magn. Mater. 75, 53 (1988) 246 C.L. Fu, A.J. Freeman: J. Phys. C 49, C8 (1988) 246 D. Vanderbilt: Phys. Rev. B 41, 7892 (1990) 247 S.J. Jenkins: Spin-polarised surfaces: Current state of Density Functional Theory Investigations. In: Computational Materials Science, vol 15, ed by J. Leszczynski (Elsevier, 2004) 247 21. G. Kresse, D. Joubert: Phys. Rev. B 59, 1758 (1999) 247 22. G.-X. Qian et al: Phys. Rev. B 38, 7649 (1988) 247 23. S.J. Jenkins, G.P. Srivastava: Surf. Sci. Lett. 398, L308 (1998) 255

Magnetization, Spin Polarization, and Electronic Structure of NiMnSb Surfaces Markus Donath1 , Georgi Rangelov1 , J¨ urgen Braun1 and Wolfgang Grentz2 1

2

Physikalisches Institut, Westf¨ alische Wilhelms-Universit¨ at, Wilhelm-Klemm-Str. 10, 48149 M¨ unster, Germany Kantonschule Z¨ urcher Oberland, 8620 Wetzikon, Switzerland

Abstract. NiMnSb, a half-Heusler alloy, is one of the local-moment ferromagnets with unique properties for future applications. Calculations for the bulk band structure predict exclusively majority bands at the Fermi level, thus indicating 100% spin polarization there. As one thinks about applications and the design of functional materials, the influence of reduced dimensions, e.g., in ultrathin films or at interfaces, must be considered. The magnetization, spin polarization, and electronic structure are expected to be sensitive to stuctural and stoichiometric changes. In this paper, we report on investigations of the spin-dependent unoccupied electronic states at NiMnSb surfaces by appearance potential spectroscopy and inverse photoemission. The data are discussed along with results on the surface structure and magnetic order.

1 Introduction NiMnSb belongs to an interesting class of materials, the half-metallic ferromagnets [1]. These materials are defined as magnetic materials with a band gap at the Fermi level for electrons of one spin direction. Band structure calculations for bulk NiMnSb show a gap of about 0.5 eV at T = 0 for the minority electrons, which means 100% spin polarization at the Fermi level EF . Taking spin-orbit interaction into account in the calculations results in a reduced but not vanishing gap: 99% spin polarization at EF is predicted from relativistic calculations [2]. NiMnSb crystallizes in the Heusler C1b structure, which is closely related to the zincblende structure. As in the case of the group III-V semiconductors, this crystal structure is the reason for the band gap. It is, therefore, called a covalent band gap [3]. As a consequence, the crystal structure and the site occupation within the given structure are important for the appearance of the gap. Atomic disorder, especially Ni-Mn interchange, results in a strong reduction of the spin polarization at EF [4]. However, this interchange costs approximately as much energy as the evaporation of the metallic constituents [3]. One of the reasons for the scientific interest in NiMnSb during recent years is the fact that one expects exclusively majority states at the Fermi level with a spin polarization of 100%. NiMnSb may play a role in spintronic devices, where spin-polarized currents are essential. However, spintronic devices will M. Donath et al.: Magnetization, Spin Polarization, and Electronic Structure of NiMnSb Surfaces, Lect. Notes Phys. 678, 261–273 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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not consist of bulk materials but rather structures of reduced dimensions. Because of this, the question of whether NiMnSb keeps its half-metallic behavior at surfaces, interfaces and in nanostructures is essential. A variety of experimental results for NiMnSb is available in the literature. While bulk measurements support the half-metallic behavior (see, e.g., [5, 6]), surface-sensitive techniques have failed so far to detect the energy gap for minority electrons or the 100% spin polarization at the Fermi level EF . Furthermore, there exist no systematic studies of the spin-resolved electronic structure E↑,↓ (k) of single-crystalline NiMnSb samples so far. Most of the studies were performed on polycrystals or thin films, single and polycrystalline, prepared under various growth conditions on a variety of substrates. Spin-integrated photoemission (PE) spectra obtained with a photon energy of 45 eV showed clear signatures of the Ni and Mn 3d contributions [7]. Angleintegrated PE with variable photon energies identified Ni and Mn 3d emission from the valence bands, yet emphasized the need for further careful investigations both experimentally and theoretically [8]. A spin-polarized PE study found values of the spin polarization up to 50% near photothreshold [9]. The authors concluded that, if an energy gap for minority electrons should exist, it is smaller than 0.5 eV. A further spin-resolved PE work, yet with photon energies between 38 and 76 eV, reported spin polarization values of at most 40% close to the Fermi level [10]. A reduced surface magnetization at remanence and/or a surface phase different from the bulk were proposed as possible explanations for the unexpected low polarization values. In another study, a temperature-dependent cross-over from half-metallic to normal ferromagnetic behavior at 80 K was reported and discussed as a possible reason for the detected low polarization [11]. It was suggested that, at room temperature, the spin polarization at EF may be considerably lowered due to a populated minority band, although the magnetization is not much reduced compared with the value at T = 0. (With a Curie temperature of about 730 K, the magnetization at room temperature amounts to ≈ 92% of the saturation value [12].) However, even point-contact Andreev reflection measurements at 4.2 K at the free surface of NiMnSb gave a maximal value of only 44%, independent of different surface preparations and magnetic domain structures [13]. This result is in line with former spin-polarized PE data, which did not find a higher polarization value at 20 K than at room temperature [9]. As a complement to the reports above, a room-temperature study by angle- and spin-resolved inverse photoemission reported close to 100% spin polarization at EF and Γ under some conditions [14]. All these results show that the current understanding of the electronic structure of NiMnSb in the bulk and at surfaces/interfaces is fragmentary. In this contribution, we will present spin-resolved data of the unoccupied electronic states in NiMnSb from two techniques, appearance potential spectroscopy (APS) and inverse photoemission (IPE). First, we will discuss

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the sample preparation and characterization because the sample condition is crucial for surface-sensitive measurements.

2 Sample Preparation and Characterization All surface-sensitive studies have in common that the sample surface must be prepared and characterized with care. Not only the chemical composition and crystallographic order but also the magnetic state and domain structure must be analyzed before any meaningful data interpretation. External magnetic fields to saturate the sample magnetization cannot be applied during electron spectroscopic measurements, especially those with angular resolution. An exception are PE measurements at normal electron emission with an external field normal to the surface, as it was used in an early study [9]. For all other experiments, the magnetic state in remanence must be known for quantitative data analysis. The preparation of single crystals of NiMnSb is not a trivial task. Even the best crystals available consist usually of several grains with slightly different crystal orientation. A high-quality crystal with (001) surface, showing sharp diffraction spots in LEED (low-energy electron diffraction), was used for spin-integrated PE studies [15]. In most studies on bulk samples, however, polycrystals have been used [5, 8, 13]. Alternatively, NiMnSb samples were prepared as thin films on, e.g., Si [10], Mo(100)/MgO(110) [16] or GaAs [17, 18]. The films are deposited in growth chambers different from the analysis chambers and some are protected with cap layers of Al or Sb before being transferred. However, removal of the cap layer to recover a stoichiometric surface is problematic. In the various experiments on bulk or thin-film samples, the surface with desired composition identical to the bulk stoichiometry was prepared by careful sputter-annealing cycles, filing or cleaving. Magnetically, the samples were characterized by magneto-optic Kerr effect [14, 10] or bulksensitive magnetometry (alternating gradient field magnetometer) [17, 18]. High remanent magnetization of almost saturation was observed. However, even in the case of MOKE, it is not clear whether the magnetization of the surface layers, relevant for (I)PE studies, is in fact represented by the MOKE measurements because of the much larger probing depth of MOKE compared with electron spectroscopies. For the spin-resolved APS measurements, we used a single crystal with (001) surface that was already used in a former study [15]. The surface was prepared in an ultra-high vacuum system (base pressure of about 5 × 10−11 mbar) by repeated cycles of sputtering with 1000 eV Ar+ ions and prolonged annealing at 700 K [19]. During sputtering, the surface becomes enriched with Ni due to preferential sputtering of Mn and Sb. Subsequent annealing enables Mn and Sb to diffuse towards the surface. Finally, we find a recovery of the bulk stoichiometry at the surface combined with a good surface ordering. This was monitored by Auger electron spectroscopy (AES) for the

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Fig. 1. LEED pattern of NiMnSb(001) at 99.8 eV primary electron energy

surface composition and by LEED for the surface order. A representative LEED pattern, taken at an electron energy of 99.8 eV, is shown in Fig. 1. It demonstrates a clear fourfold symmetry in correspondence with the cubic bulk lattice. The sharpness of the spots and the low background reveal a clean surface with a rather good surface ordering. Magnetically, the surface was characterized by extra situm Kerr microscopy and in situ as well as extra situm MOKE. Since the axes of easy magnetization in NiMnSb are of
111 type [20], none of the axes of easy magnetization lie in the surface plane. Kerr microscopy revealed a complex magnetic domain structure at remanence with domain sizes up to the (sub)millimeter range. Longitudinal MOKE measurements (Fig. 2) with a laser spot of ≈1 mm diameter confirm this situation. The external magnetic field was applied parallel or antiparallel to the [110] crystal direction. The arrows represent the direction of the magnetic field scan. The loops of Fig. 2 represent typical hysteresis curves from three different regions of the sample surface. The coercive field as well as the remanent magnetization in size and direction is not at all uniform over the surface. A comparable situation with a complex closure domain structure at the surface was observed on Ni(001) [21]. We attribute the different types of loops to a different averaging of the MOKE measurement over differently oriented domains. No quantitative result can be deduced from a single loop. By systematic scanning the surface, however, we were able to obtain a mean value of the remanent magnetization of ≈ 56% of the saturation

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Fig. 2. Magnetic properties of a NiMnSb(001) sample. Typical longitudinal MOKE hysteresis loops measured at three different locations of the crystal surface. The arrows represent the direction of the magnetic field scan (anti)parallel to the [110] crystal direction

magnetization. This number becomes important for analyzing the spinresolved APS spectra in Sect. 3. For the spin-resolved IPE measurements, a polycrystalline NiMnSb sample was used [22]. The surface was filed in situ with a diamond file, which results in a surface quality equivalent to the quality of a cleaved surface. During IPE data aquisition, the sample was filed every hour. No changes of the surface was detected either by AES or by IPE during this time. Extra situm MOKE measurements of the polished sample showed high remanence. However, it is not clear whether the magnetization within the surface layers

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after in situ filing remains aligned in the direction of the external field, which was used for magnetizing the sample.

3 Spin-Resolved Appearance Potential Spectroscopy APS is a surface sensitive tool to study the unoccupied density of states (DOS) with elemental resolution [23]. For spin-resolved APS, the sample is bombarded with a spin-polarized electron beam of variable energy while the total yield of emitted x-rays or electrons is monitored [24, 25]. At energies high enough to excite a core electron into empty states above the Fermi level, the yield of emitted particles increases due to recombination of the created core hole via x-ray or Auger electron emission. In our case, we detect the emitted x-rays [26]. Potential modulation together with lock-in techniques are used to separate the small signal from the dominating background. Since both the exciting and the excited electron are scattered into empty states, the rate of possible excitations and, thereby, of detected recombinations depends on the DOS above EF . The spin-polarized electron beam used for excitation is emitted from a GaAs photocathode irradiated with circularly polarized 830 nm laser light. This arrangement provides about 30% spin polarization of the emitted electrons [27]. The spectra shown have been renormalized to 100% hypothetical beam polarization. The APS signal depends on the spin of the exciting electron in the case of ferromagnets because of the spin-dependent unoccupied DOS. The elemental resolution comes from the fact that core levels are involved whose energies are characteristic of the various elements. Figure 3 shows spin-integrated APS data for NiMnSb(001) (dots) in comparison with APS data for polycrystalline Mn from the literature (solid line) [23]. Both spectra show pronounced structures at 639.7 eV from the Mn 2p3/2 and at 650.5 eV from the Mn 2p1/2 thresholds. These features originate from the self-convoluted local density of unoccupied 3d states. The spectral features at the high-energy side of the main lines correspond to maxima of the sp-like DOS, whose appearance and energetic positions are sensitve to the short-range crystallographic order around the atom where the local excitation occurs. The larger linewidth in the case of elemental Mn reflects the wider Mn unoccupied DOS compared with the local DOS of Mn in NiMnSb. The structure-sensitive regions show significant differences, which indicate the different crystal structures of Mn (body-centered A12 structure) and NiMnSb. For a more quantitative analysis, we performed calculations based on self-consistent electronic structure results for the bulk. Theoretical APS spectra have been obtained in a fully relativistic framework as a self convolution of the matrix element-weighted, orbitally resolved unoccupied DOS [19]. A comparison between the experimental (dots) and theoretical spectra (solid line) is shown in Fig. 4a. The vertical line marks the threshold energy. The almost quantitative agreement suggests that the DOS as calculated resem-

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Fig. 3. Spin-integrated APS data for NiMnSb(001) at the Mn 2p threshold (dots). For reasons of comparison, APS data for polycrystalline Mn are shown as solid line[23]

bles the Mn local DOS in NiMnSb quite well in the spin-integrated case (for details of the calculation, see [19]). By adding spin resolution (see spectra in Fig. 4b), we are able to obtain information about the spin-dependent DOS. The experimental data for majority (↑) and minority (↓) spins are presented by filled and open circles, respectively. First of all, the APS line shows a clear spin asymmetry, A, between the spin-dependent intensities I↑ and I↓ [A = (I↑ − I↓ )/(I↑ + I↓ )]. With the complex magnetic domain structure in mind, it is not surprising that we observed different asymmetry values in the spectra, depending on the particular sample region probed. The spin-resolved spectra of Fig. 4b represent an average from many different surface regions. By taking into account the reduced remanent magnetization as described in Sect. 2, we end up with an extimated spin asymmetry of A = −0.115 ± 0.012 for a NiMnSb sample with saturated magnetization. The negative asymmetry reflects the high density of unoccupied minority states in NiMnSb, i.e. the unoccupied minority Mn 3d states. The theoretical spectra, however, predict a higher negative spin asymmetry in the main APS peak of A = −0.25. This discrepancy between experiment and theory by a factor of two and more is in line with the findings of earlier spin-resolved electron spectroscopic experiments. Possible reasons will be discussed in Sect. 5. APS measurements at the Ni 2p3/2 threshold were performed as well but show much lower intensity due to the smaller local density of unoccupied states there. This is in agreement with calculations that predict a negligible contribution of Ni to the unoccupied minority d DOS.

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Fig. 4. Comparison between experimental (dots) and theoretical (lines) spinintegrated (a) and spin-resolved (b) APS data for NiMnSb(001)

Concerning the spin polarization at the Fermi level, one must note that APS is not particularly sensitive in that region. The nature of the APS signal doesn’t allow one to resolve small energy gaps of the order of 0.5 eV giving rise to a positive spin asymmetry at EF . The APS signal is rather dominated by the minority d states above EF leading to a negative spin asymmetry. Therefore, we will now discuss an IPE experiment with more direct access to the electronic states at the Fermi level.

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4 Spin-Resolved Inverse Photoemission IPE is a powerful tool to study the unoccoupied electronic states in the valence band region. With the use of spin-polarized electrons, the minority and majority states can be probed separately [28]. Compared with APS, IPE has the advantage that the unoccupied states are measured directly, not as a self convolution of the matrix element-weighted DOS. In addition, angleresolved measurements provide k resolution, provided the sample is a single crystal. Alternatively, APS has the advantage of elemental resolution due to the excitation of core levels. This is of particular importance for the study of multi-component systems. Therefore, APS and IPE complement each other for studying unoccupied states. In the described spin-resolved IPE experiment, low-energy spin-polarized electrons from a GaAs photocathode impinge on a polycrystalline NiMnSb sample. The electrons have a transversal spin polarization of about 35% [29]. Some electrons decay through radiative transitions into lower-lying unoccupied states of the sample. These transitions are detected via the emitted photons. The photon detector is an iodine-filled Geiger-M¨ uller counter with a CaF2 entrance window [30, 31]. The detection energy is 9.7 eV with a resolution of 0.8 eV (full width at half maximum). The angle of electron incidence was chosen to Θ = 45o . Since the sample is polycrystalline, only k-integrated information can be obtained from the measurements. Figure 5 presents IPE spectra for excitation with minority (closed circles) and majority (open circles) spin electrons. The IPE spectra are normalized to hypothetical 100% spin polarization of the electron source. The angle Θ between electron spin polarization and sample magnetization has been taken into account as well. Two structures dominate the spectrum: a weak structure just above the Fermi energy and a peak at about 1.9 eV. The spin asymmetry at EF amounts to about 10%. The peak at 1.9 eV exhibits spin asymmetry as well, yet with opposite sign. From the calculated spin-dependent DOS (see, e.g., [32]), one expects a structureless spectrum for majority spin but a peak structure from the Mn 3d holes in the minority spectrum with no minority intensity right at EF , i.e. 100% spin asymmetry there. At first glance, there is no agreement between experiment and calculated DOS. However, a careful analysis of the data shows that the peak at 1.9 eV appears at the same energy in both spin channels. This is an indication that the minority and majority partial spectra might not be well separated even after the normalization procedure. We know that magnetization within the surface layers may be reduced considerably compared with saturation. We expect the Mn 3d intensity exclusively in the minority channel based on the theoretical prediction that all majority 3d states are occupied. In addition, there is no structure in the majority DOS above the Fermi level. Therefore, we introduce a parameter to account for the presumably incomplete magnetization at the surface. By chosing a factor of four for increasing the spin asymmetry, the spectra in Fig. 6 are obtained. Now, within statistical limits, the majority spectrum

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Fig. 5. Spin-resolved inverse photoemission data of a polycrystalline NiMnSb sample with filed surface

shows a step-like increase at EF and no further structure. The minority spectrum resembles the Mn 3d holes and very little intensity at the Fermi level. The spin asymmetry at EF reaches about 40%. This value is still less than half of the expected 100% but it is in excellent agreement with most other spin-resolved electron spectroscopic results, which report values between 40 and 50%. The described experiment succeeded in detecting the minority Mn 3d holes of NiMnSb in agreement with the APS results presented above. It also showed very low minority intensity at the Fermi level, indicative of low density of minority states there. However, the minority Mn 3d holes were not observed in another spin-resolved IPE experiment on NiMnSb thin films [14]. In addition, the reported spin asymmetry values at EF are significantly different in both experiments. We can only speculate about the reason for the different results. The surface structure and/or stoichiometry of the thin film investigated there might be different from the surface of the bulk crystal in the experiment described here. It should be noted that spin-resolved IPE measurements on a sputtered and annealed NiMnSb single crystal with (110) surface showed a sign change of the spin asymmetry at EF , possibly due to

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Fig. 6. Spin-resolved inverse photoemission data of Fig. 5 but normalized by accounting for a noncomplete magnetization (25% of saturation)

Ni enrichment at the surface [22]. Minority d holes of Ni appeared above the Fermi level. All these results emphasize how critical the sample conditions are. The few experiments available so far lead to the conclusion that further experiments on well-defined samples are needed to understand the electronic structure of NiMnSb in the vicinity of the Fermi level.

5 Conclusion We have presented spin-resolved appearance potential and spin-resolved inverse-photoemission experiments for the unoccupied electronic states of NiMnSb. Both experiments detected the minority 3d holes of Mn. In addition, the observed low minority intensity at EF combined with a structureless majority intensity support qualitatively the half-metallicity of NiMnSb. Quantitatively, however, the expected 100% spin polarization at the Fermi level is not verified, a result that has been found in a number of spin-polarized electron spectroscopic experiments before. The following issues must be examined in future studies on well-defined samples: (i) the magnetization within

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the surface layers compared with the bulk, (ii) surface phases with respect to composition and/or crystallographic order different from the bulk, (iii) the position and width of the band gap in the calculations. Even with a stoichiometric surface, the number of nearest neighbors is changed at the surface and surface/interface states may form and influence the gap [33, 34]. In the case that the free surface will not provide 100% spin polarization, there is still hope that specific interfaces may open the way for 100% spin-polarized charge-injection [35] in spintronic devices. Other materials with half-metallic behavior must be considered as well in order to design functional materials for future applications [36].

Acknowledgements We thank M. Neumann for lending the NiMnSb(001) single crystal and for many helpful discussions. Enjoyable collaboration with H. Kolev, Ch. Eickhoff, and W. van Roy is gratefully acknowledged. We are indebted to J.S. Correa and D.H. Yu for critical readings of the manuscript.

References 1. R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow: Phys. Rev. Lett. 50, 2024 (1983) 261 2. Ph. Mavropoulos, K. Sato, R. Zeller, P.H. Dederichs, V. Popescu, H. Ebert: Phys. Rev. B 69, 054424 (2004) 261 3. C.M. Fang, G.A. de Wijs, R.A. de Groot: J. Appl. Phys. 91, 8340 (2002) 261 4. D. Orgassa, H. Fujiwara, T.C. Schulthess, W.H. Butler: Phys. Rev. B 60, 13237 (1999) 261 5. P.A.M. van der Heide, W. Baelde, R.A. de Groot, A.R. de Vroomen, P.G. van Engen, K.H.J. Buschow: J. Phys. F: Met. Phys. 15, L75 (1985) 262, 263 6. K.E.H.M. Hanssen, P.E. Mijnarends, L.P.L.M. Rabou, K.H.J. Buschow: Phys. Rev. 42, 1533 (1990) 262 7. S.W. Robey, L.T. Hudson, R.L. Kurtz: Phys. Rev. B 46, 11697 (1992) 262 8. J.-S. Kang, J.H. Hong, S.W. Jung, Y.P. Lee, J.-G. Park, C.G. Olson, S.J. Youn, B.I. Min: Solid State Commun. 88, 653 (1993) 262, 263 9. G.L. Bona, F. Meier, M. Taborelli, E. Bucher, P.H. Schmidt: Solid State Commun. 56, 391 (1985) 262, 263 10. W. Zhu, B. Sinkovic, E. Vescovo, C. Tanaka, J.S. Moodera: Phys. Rev. B 64, 060403 (2001) 262, 263 11. C. Hordequin, D. Ristoiu, L. Ranno, J. Pierre: Eur. Phys. J. B 16, 287 (2000). 262 12. L. Ritchie, G. Xiao, Y. Ji, T.Y. Chen, C.L. Chien, M. Zhang, J. Chen, Z. Liu, G. Wu, X.X. Zhang: Phys. Rev. B 68, 104430 (2003) 262 13. S.K. Cloves, Y. Miyoshi, Y. Bugoslavsky, W.R. Branford, C. Grigorescu, S.A. Manea, O. Monnereau, L.F. Cohen: Phys. Rev. B 69, 214425 (2004) 262, 263 14. D. Ristoiu, J.P. Nozi`eres, C.N. Borca, T. Komesu, H.-K. Jeong, P.A. Dowben: Europhys. Lett. 49, 624 (2000) 262, 263, 270

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15. S. Plogmann: Untersuchung der elektronischen und magnetischen Eigenschaften manganhaltiger Heusler Legierungen mittels Photoelektronen- und R¨ ontgenspektroskopie, PhD Thesis, Universit¨ at Osnabr¨ uck (1999) 263 16. D. Ristoiu, J.P. Nozi`eres, L. Ranno: J. Magn. Magn. Mat. 219, 97 (2000) 263 17. W. Van Roy, G. Borghs, J. De Boeck: J. Crystal Growth 227-228, 862 (2001) 263 18. W. Van Roy, G. Borghs, J. De Boeck: J. Magn. Magn. Mat. 242-245, 489 (2002) 263 19. H. Kolev, G. Rangelov, J. Braun, M. Donath: Phys. Rev. B, in press (2005) 263, 266, 267 20. R.M. Bozorth: Ferromagnetism (IEEE Press, New York, 1993), p. 15 264 21. K. Starke, K. Ertl, V. Dose: Phys. Rev. B 46, 9709 (1992) 264 22. W. Grentz: Spin- und winkelaufl¨ osende Inverse Photoemission an NiMnSb, PhD Thesis, Freie Universit¨ at Berlin (1991) 265, 271 23. R. Park, J.E. Houston: Phys. Rev. 6, 1073 (1972) 266, 267 24. K. Ertl, M. Vonbank, V. Dose, J. Noffke: Solid State Commun. 88, 557 (1993) 266 25. J. Reinmuth, F. Passek, V.N. Petrov, M. Donath, V. Popescu, H. Ebert: Phys. Rev. B 56, 12893 (1997) 266 26. G. Rangelov, K. Ertl, F. Passek, M. Vonbank, S. Bassen, J. Reinmuth, M. Donath, V. Dose: J. Vac. Sci Technol. A 16, 2738 (1998) 266 27. U. Kolac, M. Donath, K. Ertl, H. Liebl, V. Dose: Rev. Sci. Instrum. 59, 1933 (1988) 266 28. M. Donath: Surf. Sci. Rep. 20, 251 (1994) and references therein 269 29. W. Grentz, M. Tschudy, B. Reihl, G. Kaindl: Rev. Sci. Instrum. 61, 2528 (1990) 269 30. V. Dose: Rev. Sci. Instrum. 39, 1055 (1968) 269 31. V. Dose: Appl. Phys. 14, 117 (1977) 269 32. K.E.H.M. Hanssen, P.E. Mijnarends: Phys. Rev. B 34, 5009 (1986) 269 33. S.J. Jenkins, D.A. King: Surf. Sci. 494, L793 (2001) 272 34. S.J. Jenkins, D.A. King: Surf. Sci. 501, L185 (2002) 272 35. G.A. de Wijs, R.A. de Groot: Phys. Rev. B 64, 020402 (2001) 272 36. M. Fonin, Yu.S. Dedkov, U. R¨ udiger, G. G¨ untherodt: Lect. Notes in Phys. 678, 291 (2005) 272

Spin Injection Experiments from Half-Metallic Ferromagnets into Semiconductors: The Case of NiMnSb and (Ga,Mn)As Willem Van Roy IMEC, Kapeldreef 75, B-3001 Leuven, Belgium [email protected] Abstract. Materials with 100% conduction electron spin polarization are possible candidates for electrical spin injection into non-magnetic semiconductors, as they do not suffer from the conduction mismatch problem encountered by ferromagnetic metals with incomplete spin polarization. However, I show that truly half-metallic contacts are extremely challenging to fabricate, and will in many cases suffer from other problems such as the presence of a Schottky barrier that prevents the injection of any carriers at all. The suggested solution is to insert a tunnel barrier to accommodate both the Schottky barrier and any deviations from a true halfmetallic nature. The largest spin injection we have obtained using NiMnSb is 6% for polycristalline films on AlOx tunnel barriers, and 2.2% or less for epitaxial single crystalline films. This is much below what is obtained using traditional contacts such as (Co)Fe, and suggests that the experimental NiMnSb/GaAs interfaces have a low spin polarization. On the other hand, using a Zener tunnel junction to convert polarized holes into polarized electrons, we have obtained 80% spin injection from the ferromagnetic semiconductor (Ga,Mn)As into non-magnetic GaAs, showing that this material possesses a very high spin polarization.

1 Introduction The spin degree of freedom is the subject of intense interest, with the aim of adding new functionality to traditional charge-based electronics. Halfmetallic materials are of particular importance, because of their predicted 100% spin polarization. They provide model systems to study spin-dependent effects, and at the same time represent the highest level of spin-sensitive functionality that can be obtained in device applications. In particular for electrical spin injection into non-magnetic semiconductors, a conductivity mismatch problem has been pointed out that suppresses the spin injection through an Ohmic contact [1]. The most successful solution so far is the insertion of a tunnel barrier [2, 3, 4], which has led to injected spin polarizations up to 32% at low temperature [5] and 16% at room temperature [6] for amorphous AlOx barriers, and 52% at 100 K and 32% at 290 K for epitaxial MgO barriers [7]. A different option makes use of highly polarized contacts. The requirements are such that even a spin polarization of 98% in the ferromagnet would lead to negligible (<1%) polarization in the semiconductor [1], hence only truly half-metallic contacts (100% polarization) are acceptable. In this chapter we W.Van Roy: Spin Injection Experiments from Half-Metallic Ferromagnets into Semiconductors: The Case of NiMnSb and (Ga,Mn)As, Lect. Notes Phys. 678, 275–287 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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will describe our experimental attempts towards realizing such spin injectors based on NiMnSb and (Ga,Mn)As. 1.1 Halfmetallic Contacts to a Semiconductor A fully spin-polarized Ohmic contact to a semiconductor has to fulfill a number of requirements. The half-metallic properties that have in most cases been calculated or predicted for the bulk phase need to be preserved at the interface. This implies that the contact has to be deposited on the semiconductor in such a way that the material quality (stoichiometry, crystal structure, . . . ) is already sufficiently high in the initial nucleation layer. The deposition of the contact should not introduce damage to the semiconductor, such as oxidation, formation of interface compounds, or the indiffusion of magnetic impurities that may scatter the injected spins. In addition, the contact needs to be electrically transparant: there should be no band offsets that lead to a large Schottky barrier. For practical applications the half-metallic nature of the contact should be preserved up to room temperature. Many materials have been predicted to possess half-metallic properties, including full- and half-Heusler alloys, simple ferromagnetic oxides such as CrO2 and Fe3 O4 , complex oxides such as La2/3 Sr1/3 MnO3 , ferromagnetic semiconductors, and many others. In Sect. 2 we will focus on NiMnSb, which has a crystal structure that is compatible with the zinc-blende crystal structure of GaAs or Si, and can be grown epitaxially. Even under these circumstances we will conclude that half-metallic Ohmic contacts cannot be obtained, and that it is better to introduce a tunnel barrier and aim at highly-polarized contacts. Section 3 will discuss spin injectors based on the magnetic semiconductor Ga1−x Mnx As.

2 NiMnSb-Based Spin Injectors 2.1 Crystal Growth and Structural Issues The Heusler crystal structure is shown in Fig. 1. It is closely related to the zincblende crystal structure of GaAs, with a lattice mismatch of only 4%. This similarity has allowed us to grow single crystalline NiMnSb films on GaAs [8, 9]. All films discussed here were grown by molecular beam epitaxy (MBE) using individual Knudsen sources for Ni and Mn, and a cracking source for Sb2 . The GaAs substrates were prepared either in the same MBE chamber, or were transferred from a second chamber under vacuum or protected with an Sb cap. When different substrate orientations such as (001), (111)A and (111)B are used, the NiMnSb film takes on the same orientation as the substrate. The growth is sensitive to the substrate temperature; in

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Fig. 1. The Heusler and zincblende crystal structures consist of 4 interpenetrating fcc sublattices A, B, C, and D. In a full-Heusler material such as Ni2 MnSb all sublattices are occupied. This structure can also be viewed as a simple cubic structure with Ni atoms on positions A and C interpenetrating an ordered NaCl structure with Mn and Sb on positions B and D. In a half-Heusler material such as NiMnSb, the C sublattice is unoccupied (or “occupied” by a vacancy
). In a zincblende semiconductor such as GaAs the C and D sublattices are empty. The diamond structure of Si is identical, with Si occupying both sublattices A and B

particular, the incorporation of Sb into the films decreased strongly for substrate temperatures of 320–400◦ C, 1 and no Sb at all was incorporated for Tsub ≥ 480◦ C. Although epitaxial growth was possible for temperatures as low as 80◦ C, most films reported here were grown and optimized at 240◦ C. The crystal structure is also one of the major weaknesses of NiMnSb as a half-metallic material. As discussed by de Groot et al. in part III, chapter 1 (see also [17]), the gap in the minority spin band of NiMnSb is a symmetry effect resulting from the periodic nature of the crystal structure. As a consequence, any disruption to the regular order may result in defect states in the minority spin gap and the loss of half-metallicity. Calculations have shown that half-metallicity is destroyed by less than 5% chemical disorder (atoms occupying the wrong sublattice) [10]. The same effect can be expected for other types of point defects, such as deviations from stoichiometry, or the presence of impurity atoms. Even more disturbing is the fact that a mere interruption of the crystal (at a surface or interface) may result in the formation of surface or interface states that are not half-metallic [11]. This is the case for all free surfaces of NiMnSb, 1

As in most MBE applications the absolute substrate temperature is difficult to determine. All values reported here were deduced from thermocouple (TC) readings, by linearly interpolating between the GaAs desoxidation temperature (585◦ C, TCreading of 740◦ ), and an assumed real temperature Tsub = 0◦ C at TC-reading 0◦ (an arbitrary but reasonably good estimation).

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and is very similar to the formation of surface states responsible for the Fermi level pinning and the associated Schottky barrier at a III-V semiconductor surface. In the contact between NiMnSb and a zinc-blende semiconductor, two out of the three sublattices are continuous across the interface, and one might expect that the formation of interface states is suppressed. This is not the case, except for some very specific (111) interfaces. Viewed along the {111} direction, NiMnSb consists of trilayers with the stacking order Mn– Ni–Sb–
–Mn–Ni–Sb–
, where
stands for the vacancy site. GaAs consists of bilayers with stacking order Ga–As–
–
–Ga–As–
–
. In the simplest approximation, one expects that both crystals are cut at a vacancy plane, with a Ga or an As termination for GaAs, and a Mn or Sb termination for NiMnSb. This results in four different interface polarities, and calculations for the lattice matched semiconductors CdS and InP showed that only one of these combinations is half-metallic, specifically the Sb–S or Sb–P interface [11]. This result can presumably be generalized and suggests that the Sb–As interface between NiMnSb and GaAs should be half-metallic. Similarly one expects that also other extended defects such as dislocations and stacking faults introduce defect states that locally destroy half-metallicity. Experimentally we could show that the similar crystal structure allows epitaxial growth of NiMnSb on both the Ga-terminated GaAs(111)A surface and the As-terminated GaAs(111)B surface. In atomic layer epitaxy experiments where the elements were supplied sequentially at Tsub = 240◦ C, both the sequences Mn–Ni–Sb and Sb–Ni–Mn resulted in epitaxial growth. Reflection high energy electron diffraction (RHEED) showed very slight roughening at layer thicknesses of 3–7 ˚ A with full recovery of the smooth surface within 15–20 ˚ A, suggesting that both Sb-polar and Mn-polar NiMnSb can be grown on As-polar GaAs(111)B. However, we did not have the means for checking the resulting stacking order after the growth, and it is conceivable that differences in interface energy force the atoms to swap places during the initial stages of the growth. Cross-sectional transmission electron microscope (TEM) images showed abrupt interfaces, but could not elucidate the stacking order. The second type of defect, chemical disorder or point defects in general, cannot easily be detected by common techniques such as TEM or x-ray diffraction (XRD) that reveal the presence of atoms, but do not identify them. Techniques such as extended x-ray absorption fine structure (EXAFS) probe the local structural environment around selected atoms, and are sensitive to point defects. However, the signal from the defects is superimposed on the signal of the atoms with a regular environment. An EXAFS study on point defects in the full Heusler alloy Co2 MnSi [12] showed Mn–Si disorder of (0 ± 5)%, and Mn–Co disorder of (7 ± 21)%. The large fitting uncertainties are clearly not sufficient for NiMnSb, where we need a sensitivity well below 5%. We have used a different technique, zero-field nuclear magnetic resonance (NMR) [13], to probe the local magnetic environment through the transferred hyperfine fields from nearby magnetic ions. This technique has

Mnplanar def.

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0

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Frequency (MHz) Fig. 2. Zero-field NMR spectrum taken at 4.2 K on a Ni1.00 Mn0.99 Sb1.00 film grown on GaAs(111)B at a substrate temperature Tsub = 240◦ C [16]. Solid lines show the low intensity part of the spectrum, revealing the defect lines on a very small background. Filled spectra show the main Heusler peaks. The 55 Mn and 121 Sb overlap at 300 MHz but they can be deconvoluted based on the 123 Sb line at 164 MHz

the major advantage that the resonance lines of the regular and modified environments are well-separated, such that very low levels of disorder can still be clearly identified. In the experiments spin echo 55 Mn, 121 Sb and 123 Sb NMR spectra were recorded at 4.2 K and room temperature. The resonance conditions 61 Ni fell outside of the range of the available spectrometers. Initial NMR experiments [14] on films with large deviations in Sb-content showed large chemical disorder (poorly defined lines) for Sb-poor films, while excess Sb resulted in the formation of MnSb inclusions. XRD measurements on Sb-rich films grown on GaAs(111) [9] confirmed the presence of MnSb, and showed that also NiSb is formed. 2 A reduction in growth rate from 300 to 40 nm/h resulted in a strong improvement in chemical order [16]. The NMR spectrum in Fig. 2 shows very sharp lines, indicating a well-ordered material, and only very small defect-related features. The line at 274 MHz corresponds to 1.1% of all Mn-atoms involved in a planar defect, the line at 211 MHz corresponds to AsSb substitutional defects involving 0.5% of all Sb lattice sites, and the integrated intensity below 150 MHz is most likely related to 0.2% of all Sb atoms having a modified environment. These low disorder values are in agreement with earlier NMR work on bulk NiMnSb [15], and indicate that the presence of a vacancy in the crystal structure does not necessarily lead to large disorder. The total disorder is below the limits given by [10], for a film grown on GaAs(111)B, suggesting that a half-metallic interface might have been realised.

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2.2 Ohmic Contacts for Spin Injection and Band Offsets The fabrication of a spin-polarised light-emitting diode (spin-LED) is at present the easiest way to check for spin-polarized carrier injection into a semiconductor: electrons3 are injected from the polarized contact into the semiconductor and collected in a quantum well or a bulk-like region confined between cladding layers with a higher bandgap. When unpolarized holes are supplied from the semiconductor substrate, electron-hole recombination leads to the emission of light with a circular polarization that reflects the polarization of the injected electrons.4 A naive attempt to build a spin-LED based on the epitaxial contacts described in Sect. 2.1 results in a device that emits light at high bias voltages, but without any polarization. The easiest interpretation would be that the contact is not truly half-metallic, and that the resistivity mismatch is suppressing the spin injection. However, the high threshold voltage for luminescence suggests that other effects are involved as well. Indeed, a characterization of a similar contact on n-GaAs(001), i.e., without the LED, reveals that a Schottky barrier is formed with a height of 0.8 eV. Under these conditions the injection of electrons from the metallic contact into the semiconductor is impossible, irrespective of their spin, and the current through the device consists entirely of holes flowing from the substrate into the metal. However, the device still emits light, indicating that electrons have to be present in the system. The source of these electrons is revealed by simulations of the device under bias using a semiconductor simulation tool such as MediciTM , which allows to switch on and off individual processes in the calculations. Figure 3(a) shows the results of a simulation at a bias voltage of 3.5 eV with the impact ionization module switched off. The quasi-Fermi level of the holes QF p closely follows the valence band edge, indicating a high hole density throughout the structure. The electron quasi-Fermi level QF n , on the other hand, is found far below the conduction band edge, indicating that virtually no electrons are present. With the impact ionization model switched on, however, QF n lies close to the conduction band edge and the electron concentration is high (Fig. 3(b)). These electrons are generated when the holes flowing through the device acquire sufficient energy to excite electron-hole pairs. The electrons find their way to the active region, where they are responsible for the unpolarized light emission. These results show that even if a half-metallic contact can be formed, this is not guaranteed to be Ohmic. A difference in work functions can result in a 3

Usually one wants to inject spin-polarized electrons, as the hole spin lifetime is extremely short in GaAs and related semiconductors 4 Most magnetic thin films have in-plane magnetization, while the light in a surface-emitting LED carries information of the perpendicular spin component only. We use the oblique Hanle effect to create an out-of-plane spin component, and as a bonus obtain information about the spin lifetime [6].

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Energy (eV)

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optional tunnel barrier AlGaAs GaAs

(a)

281

(b) CBE

p-AlGaAs

QFn QFp

2 1

VBE

0 0

100

200

0

100

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Depth (nm) Fig. 3. Simulated band diagrams of a NiMnSb-based AlGaAs/GaAs/p-AlGaAs spin-LED with Fermi-level pinning, at a bias of 3.5 V, (a) without impact ionization, and (b) with impact ionization. Full lines are the conduction and valence band edges (CBE and VBE), dashed lines are the quasi-Fermi levels for electrons and holes (QF n and QF p )

potential step (the contact potential [22]) that prevents the injection of electrons. Indeed, the calculated density of states for the half-metallic NiMnSbCdS interface suggests that the Fermi level of NiMnSb is situated around the center of the CdS bandgap, with metal-induced gap states extending down from the conduction band edge of CdS to the NiMnSb Fermi level [11]. 2.3 Tunnel Contacts Taking into account the technological challenges in realizing a truly halfmetallic contact, the presence of band offsets that may prevent any electron injection, and also the concern that finite-temperature effects limit the halfmetallic state to zero temperature [18, 19], one reaches the conclusion that an Ohmic contact is in general not useful for spin injection. The insertion of a tunnel barrier solves all of these problems. It can accommodate the voltage drop needed to align the Fermi level of the contact with the conduction band of the semiconductor. It also removes the requirement of strictly 100% spin polarization in the contact, and softens the target from half-metallic to highly polarized materials. 2.4 AlOx Tunnel Barriers These (Figure 4(a)) are well understood from earlier work on magnetic tunnel junctions, and this technology has already been successfully transferred to tunnel spin injectors on GaAs using traditional ferromagnets such as CoFe [20, 21, 6]. To fabricate the NiMnSb/AlOx /(Al,Ga)As tunnel junctions, the (Al,Ga)As spin LEDs were transferred through air from the MBE system to a sputter chamber where the AlOx tunnel barrier was created by two sequences of Al deposition followed by oxidation in a controlled O2 atmosphere. The junctions were then reloaded into the MBE system for the deposition of the

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AlOx AlGaAs NiMnSb GaAs

(a)

(b)

AlGaAs n++ / n-

p+ AlGaAs p-GaAs

active region

substrate

NiMnSb

GaAs

(epi)

active region

p+ AlGaAs p-GaAs substrate

Fig. 4. Schematic band diagrams of tunnel contacts between NiMnSb and GaAs with (a) an AlOx and (b) a highly-doped Schottky tunnel barrier

NiMnSb injector. As the AlOx barrier is amorphous, epitaxial growth was no longer possible. The NiMnSb films were polycrystalline (no RHEED of XRD signals), but the magnetization loops remained relatively good. Spin injection experiments showed an injected electron spin polarization of ≥ 6% at 80 K (Fig. 5) [31]. This is less than what we obtain with CoFe injectors on similar tunnel contacts (24% at 80 K), suggesting that the low crystal quality results in a severe loss of polarization. 2.5 Highly Doped Schottky Tunnel Barriers

Circular polarization (%)

These (Figure 4(b)) offer an alternative: by introducing a thin, highly doped semiconductor layer at the interface with the magnetic contact, the depletion region becomes so narrow that electrons can tunnel through, and spin injection becomes possible [23, 5]. As the surface remains crystalline, epitaxial

2 1 0 -1 -2 -0.5

0.0

0.5

Oblique magnetic field (T) Fig. 5. Hanle-type spin injection measurement on a NiMnSb/AlOx spin-LED showing an injected electron spin polarization of (6.2 ± 0.1)% at 80 K. Open circles show the raw data, closed circles the data after subtraction of the magnetic circular dichroism signal, and the full line the fitted curve [31]

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growth of NiMnSb is possible, and one expects to be able to combine the best of two worlds: the presence of a tunnel barrier removes the requirement for 100% spin polarization, while the epitaxial growth should allow highly polarized NiMnSb films to be obtained. The results have been disappointing so far. We have obtained spin polarizations of 2.2% and <1% at 80 K for contacts on (001) and (111)B GaAs. This is less than for polycrystalline NiMnSb on AlOx , suggesting that the origin of the low polarization is not to be found in the bulk polarization of NiMnSb, but is an interface issue. For Fe/(Al,Ga)As spin-LEDs with a similar structure, an expansion of the GaAs lattice constant near the interface has been observed using TEM [24]. The amount of expansion correlated with a reduction in spin injection efficiency, and it was suggested that the diffusion of Fe into the semiconductor substrate is responsible for both. To check for interface effects we have grown NiMnSb films on GaAs(001) LEDs at reduced growth temperature. RHEED and XRD indicated that the Heusler phase was formed down to Tsub = 10◦ C, with relaxation (formation of misfit dislocations) suppressed in 10 nm thin films at 100◦ C and below. At 60◦ C roughening occurred for films thicker than 25 nm, and at 10◦ C there were some indications in RHEED of a polycrystalline amorphous phase superimposed on the Heusler (001) phase. Presumably the disorder is increased in all the low-temperature films, but this was not verified directly by NMR. Spin injection signals from these films were always 1% or smaller, indicating that interface mixing does not play a major rule, and suggesting that the polarization of the experimental NiMnSb/GaAs(001) and (111)B interfaces is very small. Finally we note the recent results by X. Y. Dong et al. [32], who obtained 27% spin injection at 2 K from epitaxial Co2 MnGe contacts on GaAs(001). The growth temperature (175◦ C) and tunnel Schottky barrier were similar to ours. The higher result may be due to the better lattice matching, or to a higher spin polarization at this particular interface. However, it remains far below the expected 100%, and underlines once again the need for a tunnel barrier to compensate for imperfect half-metallicity of the contacts.

3 Ga1−xMnxAs-Based Spin Injectors Another class of half-metallic materials that can be grown epitaxially on zincblende semiconductors are magnetic semiconductors. Using the II-VI diluted magnetic semiconductor (Zn,Mn,Be)Se as a spin aligner, electrons with 90% spin polarization have been injected into GaAs [33]. The spin splitting in (Zn,Mn,Be)Se is the result of the giant Zeeman splitting and requires low temperatures and large magnetic fields. Ferromagnetic semiconductors such as (Ga,Mn)As have also been predicted to be halfmetallic [25, 26]. However, their p-type character poses a problem because of the short hole spin lifetime

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GaMnAs n-AlGaAs

(a)

p+ AlGaAs GaAs

p-GaAs

active region

substrate

Circular polarization (%)

in a zincblende semiconductor. The obvious solution is to use interband tunneling in a highly doped p-n junction to convert the holes into electrons using a process called Zener tunneling,5 see Fig. 6(a). Early attemps at realizing (Ga,Mn)As spin injectors based on Zener tunneling resulted in relatively low spin polarization values of 6.5% and 1% at 4.2 K [27, 28].

20

(b)

10 0 -10 -20 -0.5

0.0 B45 (T)

0.5

Fig. 6. (a) Schematic band diagram under bias of a Zener tunnel junction between p-(Ga,Mn)As and n-GaAs; (b) Hanle-type spin injection measurement at 4.2 K, showing an injected electron polarization of (82 ± 10)%

We have optimized the semiconductor design using the MediciTM simulation tool, to prevent electrons from spending too much time in the n-AlGaAs transport layer, and losing their spin before they reach the active region. In addition, we used (Ga,Mn)As films that have been optimized using a lowtemperature annealing step. This eliminates Mn interstitial atoms from the film and results in increased carrier concentrations and an increased Curie temperature TC [29, 30]. Our films had a TC = 120 K, and Hanle-type measurements (Fig. 6(b)) showed that the electrons injected into the GaAs active layer had a spin polarization of (82 ± 10)% at 4.6 K [34]. A device from the same wafer that had not been subjected to the low-temperature annealing delivered a much smaller spin polarization of (47 ± 5)%, showing the importance of the (Ga,Mn)As material quality for its electronic properties and half-metallic nature. The injected spin polarization decayed slowly with temperature, and disappeared at 120 K. This is the same temperature where the in-plane remanent magnetization of the (Ga,Mn)As film vanished, however, the shape of both curves is different with the spin polarization decaying more rapidly than the magnetization. The injected spin polarization also decreases dramatically with the bias over the LED. Both effects are related to the fact that the elec5

The term Zener tunneling is used for current flowing in the reverse direction of the p-n junction: electrons tunneling out of the valence band of the p-type layer into the conduction band of the n-type material. Esaki tunneling is essentially the same process but in the opposite direction: electrons tunneling from the conduction band of the n-type material into the valence band of the p-type layer (i.e., in the forward direction of the p-n diode).

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trons tunnel out of the interfacial (Ga,Mn)As layer which is partly depleted [34]. At high bias, electrons can tunnel from a larger energy window inside the (Ga,Mn)As valence band, and from a spatial region closer to the n-side of the Zener junction. Both of these reduce the polarization. An in-dept theoretical analysis can be found elsewhere [35]. At higher temperatures the resistance of the p-n diode between the substrate and the active layer decreases, while the resistance of the Zener tunnel junction remains nearly constant. As a result, a larger fraction of the bias is applied to the Zener junction, which causes a drop in injected spin polarization.

4 Conclusions Our experiments suggest that it may be technologically feasible to fabricate half-metallic NiMnSb/GaAs contacts, i.e., contacts that combine a low chemical disorder with the correct interface polarity to suppress the formation of normal-metallic interface states. However, it is not straightforward to do so, and routine contacts may not be half-metallic. The contacts have a potential step of 0.8 eV, which prevents the injection of all electrons into the semiconductor, irrespective of their spin. In addition there are concerns about finite-temperature effects that limit the existence of a true half-metallic state to zero temperature. All these issues can be circumvented by introducing a tunnel barrier, which accommodates the potential difference between both materials and allows the use of contacts with less than 100% spin polarization. Polycrystalline NiMnSb films on an amorphous AlOx tunnel barrier resulted in (6.2 ± 0.1)% spin injection at 80 K. Higher quality epitaxial films on tunneling Schottky barriers resulted in lower spin polarizations. This is tentatively attributed to the diffusion of magnetic elements into the semiconductor, and experiments at lower deposition temperatures are in progress to circumvent this problem. Ferromagnetic semiconductors can also be considered as half-metallic materials, and we demonstrated (82 ± 10)% spin injection at 4.2 K using (Ga,Mn)As in combination with a Zener tunnel junction to convert holes into polarized electrons. Acknowledgements It is a pleasure to acknowledge the contributions from, and discussions with, V. F. Motsnyi, P. Van Dorpe, Z. Liu, B. Brijs, S. Degroote, W. van de Graaf, M. W´ ojcik, E. J¸edryka, S. Nadolski, R. A. de Groot, G. A. de Wijs, G. Borghs, and J. De Boeck, and the financial support from the EC project FENIKS (G5RD-CT-2001-00535) and a Postdoctoral Fellowship from the Fund for Scientific Research Flanders – Belgium (F.W.O.).

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References 1. G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip, B.J. van Wees: Phys. Rev. B 62, R4790 (2000) 275 2. E.I. Rashba: Phys. Rev. B 62, R16267 (2000) 275 3. A. Fert, H. Jaffr´es: Phys. Rev. B 64, 184420 (2001) 275 4. H. Jaffr´es, A. Fert: J. Appl. Phys. 91, 8111 (2002) 275 5. A.T. Hanbicki, O.M.J. van ’t Erve, R. Magno, G. Kioseoglou, C.H. Li, B.T. Jonker, G. Itskos, R. Mallory, M. Yasar, A. Petrou: Appl. Phys. Lett. 82, 4092 (2003) 275, 282 6. V.F. Motsnyi, P. Van Dorpe, W. Van Roy, E. Goovaerts, V.I. Safarov, G. Borghs, J. De Boeck: Phys. Rev. B 68, 245319 (2003) 275, 280, 281 7. X. Jiang, R. Wang, R.M. Shelby, R.M. Macfarlane, S.R. Bank, J.S. Harris, S.S.P. Parkin: Phys. Rev. Lett. 94, 056601 (2005) 275 8. W. Van Roy, J. De Boeck, B. Brijs, G. Borghs: Appl. Phys. Lett. 77, 4190 (2000) 276 9. W. Van Roy, G. Borghs, J. De Boeck: J. Magn. Magn. Mat. 242–245, 489 (2002) 276, 279 10. D. Orgassa, H. Fujiwara, T.C. Schulthess, W.H. Butler: Phys. Rev. B 60, 13237 (1999) 277, 279 11. G.A. de Wijs, R.A. de Groot: Phys. Rev. B 64, 020402 (2001) 277, 278, 281 12. B. Ravel, M.P. Raphael, V.G. Harris, Q. Huang: Phys. Rev. B 65, 184431 (2002) 278 13. All NMR measurements courtesy of M. W´ ojcik, E. J¸edryka, S. Nadolski, Institute of Physics, Polish Academy of Sciences, Warsaw 278 14. M. W´ ojcik, W. Van Roy, E. J¸edryka, S. Nadolski, G. Borghs, J. De Boeck: J. Magn. Magn. Mat. 240, 414 (2002) 279 15. J. Schaf, K. Le Dang, P. Veillet, I.A. Campbell: J. Phys. F 13, 1311 (1983) 279 16. W. Van Roy, M. W´ ojcik, E. J¸edryka, S. Nadolski, D. Jalabert, B. Brijs, G. Borghs, J. De Boeck: Appl. Phys. Lett. 83, 4214 (2003) 279 17. C.M. Fang, G.A. de Wijs, R.A. de Groot: J. Appl. Phys. 91, 8340 (2002) 277 18. R. Skomski, P.A. Dowben: Europhys. Lett. 58, 544 (2002) 281 19. P.A. Dowben, R. Skomski: J. Appl. Phys. 93, 7948 (2003) 281 20. V.F. Motsnyi, J. De Boeck, J. Das, W. Van Roy, G. Borghs, E. Goovaerts, V.I. Safarov, cond-mat/0110240; Appl. Phys. Lett. 81, 265 (2002) 281 21. P. Van Dorpe, V.F. Motsnyi, M. Nijboer, E. Goovaerts, V.I. Safarov, J. Das, W. Van Roy, G. Borghs, J. De Boeck: Jpn. J. Appl. Phys. 42, L502 (2003) 281 22. S.M. Sze: Physics of Semiconductor Devices, 2nd edn (Wiley, New York 1981) p 246 281 23. A.T. Hanbicki, B.T. Jonker, G. Itskos, G. Kioseoglou, A. Petrou: Appl. Phys. Lett. 80, 1240 (2002) 282 24. R.M. Stroud, A.T. Hanbicki, G. Kioseoglou, O.M.J. van’t Erve, C.H. Li, B.T. Jonker, G. Itskos, R. Mallory, M. Yasar, A. Petrou: Spintech II conference, Brugge (Belgium), 4-8 August 2003 283 25. T. Ogawa, M. Shirai, N. Suzuki, I. Kitagawa: J. Magn. Magn. Mat. 196-197, 428 (1999) 283 26. L.M. Sandratskii, P. Bruno: Phys. Rev. B 61, 214402 (2003) 283 27. M. Kohda, Y. Ohno, K. Takamura, F. Matsukura, H. Ohno: Jpn. J. Appl. Phys. 40, L1274 (2001) 284

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Growth and Room Temperature Spin Polarization of Half-metallic Epitaxial CrO2 and Fe3 O4 Thin Films M. Fonin1 , Yu. S. Dedkov2 , U. R¨ udiger1 and G. G¨ untherodt3 1 2

3

Fachbereich Physik, Universit¨ at Konstanz, 78457, Konstanz, Germany Institut f¨ ur Festk¨ orperphysik, Technische Universit¨ at Dresden, 01062 Dresden, Germany II. Physikalisches Institut, Rheinisch-Westf¨ alische Technische Hochschule Aachen, 52056 Aachen, Germany

Abstract. Potential applications of systems based on half-metallic ferromagnets (HMFs) in magneto- and spinelectronics including spin injectors, magnetic field sensors and magnetic memory devices stimulated the investigation of the electronic properties of HFMs by a multitude of experimental and theoretical approaches. The main question remains however if the 100% spin polarization at the Fermi level (EF ) theoretically predicted for HFMs can be confirmed experimentally. This article gives a short overview on the half-metallic ferromagnets with special emphasis on magnetite (Fe3 O4 ) and chromium dioxide (CrO2 ). The spin-resolved electronic structure of thin epitaxial Fe3 O4 (111) and CrO2 (100) films has been investigated at 293 K by means of spin-resolved photoemission spectroscopy (SP-PES). Epitaxial Fe3 O4 (111) films have been grown on different substrates by oxidizing epitaxial Fe(110) films. High surface quality and chemical homogeneity as well as high crystalline order in the bulk of Fe3 O4 (111) films were confirmed by means of STM and LEED. The Fe3 O4 (111) epitaxial films show a maximum spin polarization value of −(80±5)% near EF at 293 K confirming the half-metallic nature of Fe3 O4 in the [111] direction. High-quality epitaxial CrO2 (100) films have been prepared by a chemical vapor deposition technique. Near EF an energy gap was observed for spin-down electrons and a spin polarization of about +(90 ± 10)% was found at 293 K by means of SP-PES. This value and the magnitude of the gap in the minority spin channel are in good agreement with the prediction of the half-metallic nature of CrO2 .

1 Introduction The intriguing feature of the material class of half-metallic ferromagnets (HMF) is metallic conductivity for one spin component and insulating behavior for the other one. The theoretically predicted 100% spin polarization at the Fermi level EF of HMFs [1, 2, 3, 4, 5] makes them promising materials for magnetoelectronic devices [6, 7, 8]. According to Julli`ere’s model [9] the tunnel magnetoresistance (TMR) of ferromagnet/insulator/ferromagnet tunnel junctions depends on the spin polarization of the ferromagnetic electrodes used. The TMR increases with increasing spin polarization of the electrode materials. This fact has revived an intensive research interest on the M. Fonin et al.: Growth and Room Temperature Spin Polarization of Half-metallic Epitaxial CrO2 and Fe3 O4 Thin Films, Lect. Notes Phys. 678, 289–308 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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class of half-metallic ferromagnets like Heusler alloys [10], manganites [11], and transition metal oxides [1, 2, 3, 12, 13, 14, 15, 16, 17]. Theoretical calculations based on the local spin-density approximation (LSDA) to the density-functional theory have predicted only minority spin states at EF for Fe3 O4 [4, 5] and only majority spin states at EF for CrO2 [1, 2, 3].

2 Half-Metallic Ferromagnets A half-metallic ferromagnet is a solid with a special band structure where for electrons of one spin component it is a normal metal, but the electrons of the other spin component have a gap in the density of states at the Fermi energy, making this component insulating [18, 19, 20]. As only one spin component is present at EF , half-metallic ferromagnets are 100% spin-polarized. There are no pure elements which are half-metallic ferromagnets. For example, cobalt and nickel have fully spin-polarized 3d bands at EF , however the almost unpolarized 4s bands also cross EF which means that both spin up (↑) and minority (↓) electrons are present at EF . In order to achieve the full spin polarization at EF , it is necessary either to push the bottom of 4s band above EF or to depress EF below the bottom of 4s band. This can be done by forming an alloy or an oxidic compound. Thus all half-metallic ferromagnets contain more than one element. Historically, the first compound which was identified as a “half-metallic ferromagnet” was NiMnSb by de Groot et al. [10]. Subsequently, this concept was applied to other Heusler alloys: PtMnSb [21], CoMnSb [22] and FeMnSb [23]. Besides Heusler alloys, the majority of HFMs was identified among transition metal oxides. Band structure calculations on the basis of local spin density approximation (LSDA) to the density functional theory predict a HMF state for CrO2 [1], Fe3 O4 [4, 5], Sr2 FeMoO6 [24], and mixedvalence manganites [25, 26]. Following the classification of half-metallic ferromagnets given in overviews by Coey et al. [19, 20], the type IA HMFs are metallic for majority electrons but semiconducting for minority electrons. The opposite situation arises for the type IB half-metallic ferromagnets. Half-metallic oxides where the 4s states are pushed above EF by s-p hybridization are either of type IA when there are less than five d electrons or of type IB when there are more than five d electrons. Chromium dioxide (CrO2 ) is a type IA HMF with only majority electrons present at EF [1, 2, 3]. Otherwise, so-called Heusler alloys containing heavy p elements like Sb tend to have the 3d levels depressed below the 4s band edge by p-d hybridization. The best-known example is the NiMnSb alloy [10] which is type IA HMF. A type IB half-metallic ferromagnetic oxide is Sr2 FeMoO6 where only minority electrons are present at EF [24]. In the type II HMFs electrons lie in a band that is sufficiently narrow for them to be localized. The localized carriers form polarons with low mobility and high effective mass. Conduction in these materials is by hopping from one

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site to another with the same spin. Similarly, disorder may induce localization below or above a mobility edge when EF lies near the bottom or top of the band. Magnetite (Fe3 O4 ) is a type IIB HMF with a spin gap in the majority density of states [4, 5]. In the type III half-metallic ferromagnets the majority electrons are localized and minority electrons are delocalized or vice-versa. The density of states is finite for both sub-bands at EF , but the electrons in one band have a larger effective mass than in the other leading to a difference in the mobilities. Thus, only one sort of carriers contributes significantly to the conduction. A prominent example of the type IIIA HMF is the optimally doped lanthanum strontium manganite (La0.7 Sr0.3 MnO3 ) with mobile majority electrons and immobile minority electrons at EF [27]. Magnetically ordered semimetals with a large difference in effective mass between electrons and holes represent type VI half-metallic ferromagnets. For example, Tl2 Mn2 O7 where a few heavy majority Mn holes lie at the top of Mn(eg ) band, while there is an equal number of light minority electrons of mainly Tl (6s) character [28]. Ferromagnetic semiconductors can also exist with fully spin-polarized valence, conduction, or impurity bands. They represent type V half-metallic ferromagnets. There is no distinctive property allowing to identify a material to be a half-metallic ferromagnet [19, 20]. The best indication of a type I or a type II HMF is metallic conduction in a solid with a spin moment at T = 0 which is precisely an integer number of Bohr magnetons per unit cell. However, the integer spin momentum criterion is a necessary but not a sufficient condition of half-metallicity. Another way to find appropriate candidates is based on band-structure calculations. However, the experimental determination of the spin polarization remains the key problem. The spin polarization P of a material is defined at EF as: P = [N↑ (EF ) − N↓ (EF )]/[N↑ (EF ) + N↓ (EF )]

(1)

where N↑ and N↓ are densities of states of majority and minority bands, respectively. The determination of P requires either a direct, spin-selective measurement of the density of states or a spin-dependent transport measurement. The experimental methods which allow determination of P including spin-polarized photoemission spectroscopy, point-contact spectroscopy and spin-polarized tunneling are discussed in detail by Coey et al. [19, 20]. It is important to mention that spin-resolved photoemission spectroscopy is the most direct technique for measuring spin polarization which is, however, limited to the surface region of the material only.

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3 Magnetite Magnetite has been the first known magnetic material (discovered ∼1500 B.C.), however many of its electronic and magnetic properties are still a subject of intense discussion. Magnetite (Fe3 O4 ) crystallizes in the cubic inverse spinel structure (space group F d3m, a = 8.396 ˚ A) where the oxygen anions (O2− ) form a closepacked face-centered cubic (fcc) sublattice with Fe2+ and Fe3+ cations located in the interstitial sites [29]. Two different kinds of cation sites exist in the crystal: the first (A) is tetrahedrally coordinated to oxygen and occupied only by Fe3+ ions, and the second (B ) is octahedrally coordinated to oxygen and occupied by equal numbers of Fe2+ and Fe3+ ions (Fig. 1).

Fe tet (Fe

3+

)

3+

2+

Fe oct (Fe , Fe ) O

2-

Fig. 1. Perspective side view of the Fe3 O4 crystal structure

Below TN = 858 K, magnetite is a ferrimagnet with the A and B sites having opposite spin directions. The d -orbital occupation can be represented as (t32g e2g )↓ at A sites and (t32g e2g )↑(t0.5 2g )↓ at B sites resulting the net magnetization of 4 µB per Fe3 O4 formula unit. This magnetic structure was first proposed by N´eel [30] to explain the magnetization data and was subsequently established in neutron scattering experiments [31]. Fe3 O4 undergoes a first order phase transition (Verwey transition) on cooling below TV  120 K, at which the resistivity of magnetite decreases sharply by two orders of magnitude [32, 33]. The Verwey phase transition below TV is accompanied by a crystallographic structural change, which reduces the symmetry from cubic to monoclinic [34, 35]. Verwey and coworkers [32, 33] proposed that this phase transition is associated with an electron localization-delocalization transition on the B sites. The Fe2+ ion at the B site can be described as an “extra” electron plus an Fe3+ ion. The high electrical conductivity of magnetite at room temperature is due to the hopping of the “extra” electron between equivalent B sites. Cooling below TV freezes these “extra” electrons and causes their ordering at the Fe2+ B

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sites. The interaction of the “extra” electron with the B site ions is described by a nearest-neighbor Coulomb repulsion U and the tight-binding hopping integral t for the “extra” electron. The ratio U to t determines whether the insulating or metallic state is obtained. On the other hand, Mott suggested that electron-phonon interaction should be taken into account [36]. Above the Verwey transition, according to Mott, Fe3 O4 is considered as a Wigner glass in which “extra” electrons randomly occupy one-half of the B sites. Due to the disorder-induced randomfield potentials the carriers are localized. The electrical properties of such glass are described by a narrow polaron band. The most probable conduction mechanism is small polaron hopping at B sites of Fe3 O4 [37]. The electronic structure of Fe3 O4 was recently studied theoretically by several groups [4, 5, 38, 39, 40, 41]. Zhang and Satpathy [5] performed band structure calculations using the LSDA to the density-functional theory and the “constrained” density-functional methods. A self-consistent linear muffintin-orbitals method in the atomic-spheres approximation (LMTO-ASA) was used in these calculations. It was found that the O 2p orbitals lie well below EF , with the electron bands near EF consisting primarily of Fe 3d orbitals. This result is in agreement with the earlier augmented plane wave (APW) calculations [4]. The most interesting feature of the calculated spin-polarized band structure is that the majority spins (↑) are semiconducting while the minority spins (↓) are metallic [5]. Only minority electrons were found to be present at EF leading to the 100% spin polarization and moreover, the d electrons at EF have predominantly B -site character. The minority character of the electrons at EF was found to be in a good agreement with earlier APW calculations [4] as well as with the spin-polarized photoemission experiments [42]. The experimental studies concerning the spin-resolved electronic structure of Fe3 O4 were performed mostly by means of photoelectron spectroscopy [42, 43, 44, 45, 46, 47, 48]. It is worth to mention that due to a very low conduction of Fe3 O4 below TV the spin polarization of Fe3 O4 can not be measured by any superconducting method, thus spin-resolved PES [42, 43, 44, 45, 46, 47, 48] and point-contact spectroscopy [49] are the most appropriate measurement techniques. Alvarado et al. performed the first spin-resolved photoelectron spectroscopy measurements on Fe3 O4 (100) single crystals [42, 43, 44]. The spin polarization P (hν) of the photoelectrons with the energies up to 11.2 eV was measured by Mott scattering. P (hν) = (n↑ − n↓ )/(n↑ + n↓ ), where n↑ (n↓ ) is the yield of majority (minority) photoelectrons obtained at the photon energy hν. The maximum spin polarization value of −60% near EF was measured at around 5 eV photon energy at 10 K [43]. The experimentally obtained spin-polarization values were compared with those predicted by the single-ion-in-a-crystal-field (SICF) model. As derived in this model the maximum obtainable spin polarization at T = 0 is P = −66.6% [43, 44] which is in agreement with the measured value of –60%. However, this spin polarization value is sufficiently lower than –100% predicted on the basis LSDA

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calculations [5] and as the experimentally observed –80% [48]. This discrepancy in spin-polarization values will be explicitly discussed later on. Until recent times, spectroscopic studies of magnetite were limited to bulk samples. The problem of preparing Fe3 O4 thin films with well-defined stoichiometry (and, even more, stoichiometric surfaces) seemed to be unsolvable. However, recent work demonstrated the possibility to stabilize Fe3 O4 in form of epitaxial thin films on different single-crystal substrates [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. Spin-resolved photoemission measurements of epitaxial Fe3 O4 (111) thin films grown on the Fe(110) surface were performed by Kim et al. [45]. A larger photoemission intensity at EF for the Fe3 O4 (111)/Fe(110) films was found comparing to bulk Fe3 O4 samples. This higher photoemission intensity was attributed to the possible formation of an oxygen deficient Fe4 O4 surface with respect to the bulk-terminated one. The observation of relatively small spin-polarization values about +16% at EF supports the fact that the films do not have the pure Fe3 O4 stoichiometry. As for the (100)-oriented Fe3 O4 surface the spin-resolved photoemission measurements of thin epitaxial films grown on MgO(100) substrates yielding spin polarization values of about –(40–55)% near EF [46, 47] which are still significantly lower than theoretically predicted by LSDA calculations. This large discrepancy was claimed to result either from the sample surface imperfection [46] or from strong electron correlation effects [47]. The spin polarization of Fe3 O4 (111) thin films grown on Fe(110)/W(110) and Fe(110)/Mo(110)/Al2 O3 (11¯20) was studied recently by means of spinand angle-resolved photoemission spectroscopy [48, 62]. Here, high-quality epitaxial Fe3 O4 (111) thin films were prepared by oxidation of Fe(110) thin films [48, 62]. Figure 2 shows the characterization of in situ prepared Fe3 O4 (111) films performed by means of XAS, LEED and STM. The left-hand panel presents the Fe L2,3 -edge XAS spectrum of the epitaxial Fe3 O4 (111) thin film (Fig. 2(a)) in comparison with that of the epitaxial Fe(110) thin film (Fig. 2(b)). The characteristic multiplet structure of the XAS spectrum at the Fe L2,3 - and the O K-edges (not shown here) of Fe3 O4 agrees with earlier experiments [64, 65, 63]. The corresponding LEED patterns of the Fe3 O4 (111) and Fe(110) surfaces are shown in the insets. The hexagonal (2 × 2) LEED patterns indicate the formation of a Fe3 O4 (111) surface [53, 63, 45, 48, 62]. The right-hand panel of Fig. 2 presents the STM characterization of the Fe3 O4 (111) surface. Large epitaxial islands with lateral extensions of more than 100 nm have been formed (Fig. 2 (c)). The islands are monatomically flat with step heights of approximately 5 ˚ A which corresponds to the distance between equivalent Fe3 O4 (111) surface terminations [58]. The vertical peakto-peak roughness of the surface on a lateral scale of 1000 nm is about 60 ˚ A. Fig. 2 (d) shows a STM image with atomic resolution of the regular defectA in-plane periodicity free Fe3 O4 (111) surface. A hexagonal lattice with a 6 ˚ and a corrugation amplitude of about 0.5 ˚ A was observed. The value of the

Total Electron Yield (arb. Units)

Half-Metallic Ferromagnets CrO2 and Fe3 O4

a

Fe3O4

295

c

50 nm

b

Fe

d

705 710 715 720 725

Photon Energy (eV)

5 nm

Fig. 2. Fe L2,3 -edge XAS spectra of (a) an epitaxial Fe3 O4 (111) film and (b) an epitaxial Fe(110) film. Insets show corresponding LEED patterns. In (c) a 200 × 200 nm2 STM image of the epitaxial Fe3 O4 (111) film surface and (d) an atomically resolved regular Fe3 O4 (111) surface (20 × 20 nm2 )

6˚ A in-plane periodicity fits well with the Fe3 O4 (111) in-plane lattice constant of 5.92 ˚ A. The observed structure of the Fe3 O4 (111) surface is an good agreement with LEED measurements as well as with previous studies [58]. These data confirm the high bulk and surface quality of the investigated Fe3 O4 (111) films. Figure 3 shows spin-resolved photoemission spectra recorded near EF together with the total photoemission intensity (left-hand panel: a, c, e) and the resulting spin polarization values (right-hand panel: b, d, f) as a function of the binding energy relative to the Fermi level of Fe3 O4 (111) films as well as of Fe(110) films. In Fig. 3 a, c, e the spectra of a pure Fe(110) film [48], a Fe3 O4 (111) layer on the Fe(110)/W(110) system [48], and of a Fe3 O4 (111) layer on the Fe(110)/Mo(110)/Al2 O3 (1120) system [62] are presented, respectively. The solid circles in (b) correspond to the spin polarization of the pure Fe(110) film, in (d) of the Fe3 O4 (111)/Fe/W system, and in (f) of the Fe3 O4 (111)/Fe/Mo/Al2 O3 system. The spin-resolved spectra of the valence band of Fe(110) (Fig. 3 (a)) 3 1 ↓⊕ ↓ states near 0.25 eV and from the show the emission from the 4 1 ↑⊕ ↑ states near 0.7 eV. The spectra are in agreement with previous measurements [66, 67]. The Fe(110) films show a spin-polarization value of about −(80 ± 5)% at EF at 293 K.

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80 40 0

Intensity (arb. units)

-40 -80 80

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a) d)

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b)

a)

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a) e)

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Binding energy (eV) Fig. 3. Left-hand panel : Spin-resolved photoemission spectra of (a) the Fe(110)/W(110) system, (c) the Fe3 O4 (111) on Fe(110)/W(110) sytem, and (e) the Fe3 O4 (111) on Fe(110)/Mo(110)/Al2 O3 (1120) system for hν = 21.2 eV in normal emission. (Spin down: triangles down; spin up: triangles up; total photoemission intensity: solid circles). Right-hand panel : spin polarization as function of binding energy of (b) the Fe(110)/W(110) system, (d) the Fe3 O4 (111) on Fe(110)/W(110) system, and (f ) the Fe3 O4 (111) on Fe(110)/Mo(110)/Al2 O3 (1120) system

The spin-resolved spectra of the valence band near EF clearly show a dominant emission from the minority Fe-3d states (≤1.5 eV) (Fig. 3 (c)) and from the O-2p states (≥1.5 eV) (not shown here) for Fe3 O4 (111) films grown on Fe(110)/W(110) substrates. The Fe3 O4 (111)/Fe(110)/W(110) system shows a negative spin polarization at EF of about −(80 ± 5)% with a parallel magnetic coupling between Fe3 O4 (111) and the underlying Fe(110), contrary to the antiparallel coupling in the Fe3 O4 (111)/Fe(110) system observed by Kim et al. [45]. In order to get a rough estimate of the maximal achievable spin polarization at 300 K the spin-polarization value of −(80 ± 5)% of the epitaxial Fe3 O4 films in Fig. 3 (d) is compared in first approximation with the temper-

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ature dependence of the magnetization M (T ) of an epitaxial film of Fe3 O4 obtained by pulsed-laser deposition on MgO [68]. When extrapolating M (T ) to zero temperature and normalizing it to P = −100% we find by scaling M (T )∼P (T ) that the experimental value of P (300)K is by 6% lower than M (300)K. Such a reduction of the spin polarization near surfaces can be expected due to the excitation of spin waves [69, 70]. In [45] the rather high photoemission intensity observed near EF , even at the highest oxygen exposure of 1500 L, was not attributed to the Fe metal underneath, because of the absence of the Fe 3p core level spectra. Therefore, as proposed by the authors, it may indicate the formation of an oxygen deficiency near the Fe3 O4 surface. This may also be responsible for the small positive spin polarization of +16% [45]. Here, it was not possible to obtain a reduced emission intensity at EF upon exposing the sample to additional oxygen at various temperatures and in a wide oxygen pressure range. In previous spin-resolved threshold photoemission measurements, i.e. without any energy analysis [42, 43, 44], a spin polarization of −60% was found for Fe3 O4 single crystals. However, the spin polarization value by itself is no proof of a half-metallic state as is evident by the above example of Fe(110). Therefore, the features in the spin-resolved photoemission spectra have to be compared with spin-polarized band-structure calculations as discussed below. In the photoemission spectra in Fig. 3 (c), only features of Fe3 O4 are present with a negligible contribution from the Fe(110) substrate. This is further corroborated by a comparison with the spin-split electron band structure of Fe3 O4 based on the LSDA to the “constrained” density-functional method [5, 48]. As demonstrated in Fig. 3 for oxidized layers a reduced photoemission intensity near EF is observed comparing to the pure Fe(110) film: for majority electrons more than 20 times and for minority electrons more than 8 times. This can be interpreted as a band gap formation for majority electrons near EF for Fe3 O4 (111)/Fe(110) (Fig. 3 (c). Such a majority gap is also consistent with the band structure calculation [5], which shows a 0.45 eV gap for the majority electrons below EF . The spin-resolved photoemission spectra of Fe3 O4 (111) for binding energies 1.5 eV ≤ Eb ≤ EF support the band-type description of the electronic structure of Fe3 O4 . Within this energy interval the photoemission spectra were previously attributed to the ionic configuration-based transition 5 T2 (3d6 )→6 A1 (3d5 )+e− of Fe2+ , where e− denotes a photoelectron. The maximum obtainable spin polarization at T = 0 of P = −2/3 or −66.6% derived in this model [44], however, cannot account for our experimental finding of P = −(80 ± 5)% at room temperature. Hence, despite the importance of electron correlation effects in transition metal oxides, a band-type description of the electronic structure of Fe3 O4 seems to be appropriate. The features of Fe 3d bands in the range of 2 eV below EF in the PES spectra of the Fe3 O4 (111) film on the Al2 O3 (1120) substrate (e) do not differ from these of the film on W(110) (c), but a significant decrease of the

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Fe 3d photoemission intensity has been observed for the films grown on the Al2 O3 (1120) substrate [62]. The Fe3 O4 (111) films on the Al2 O3 (1120) substrates show a maximum negative spin polarization at EF of about −(60±5)% at 293 K (Fig. 3 (d)). However, this polarization value also cannot be related to a contribution from Fe(110) underlayer, as the thickness of the Fe3 O4 (111) film determined by TEM (not shown here) is about 150 ˚ A. Comparing the Fe3 O4 (111) films on W(110) and Al2 O3 (1120) substrates the reduction of the observed spin polarization for the Fe3 O4 (111) films on the Al2 O3 (1120) substrate may be caused by the cleaning procedure after the ex situ sample transfer or by strain in the Fe3 O4 (111) surface layers caused by a lattice mismatch between the Mo(110) and Fe(110) layers as well as Fe(110) and Fe3 O4 (111) layers [62]. As reported by Jeng et al. the resulting strain can lead to a reduction of the spin polarization value [39]. In this case the presence of uniaxial strain leads to a broadening of the Fe 3d bands reducing the insulating band gap of the majority spin. As a consequence the halfmetallic behavior of cubic magnetite is reduced and in the high-strain regimes Fe3 O4 (111) can eventually turn into normal-metal behavior. Another reason for the photoemission intensity decrease can be the cleaning procedure after the ex situ sample transfer into the PES chamber. As reported before [46] the cleaning procedure can crucially influence the surface structure leading to a decrease or a total loss of the spin polarization. In conclusion, the measured spin polarization value of −(80±5)% at EF at 293 K [48], rules out the ionic-configuration-based approach setting the upper limit of P = −66.6% at T = 0. The agreement of the photoemission spectroscopy in [111]-direction with density functional calculations, predicting an overall energy gap in the majority electron bands in high symmetry directions, provides evidence for the half-metallic ferromagnetic state of Fe3 O4 .

4 Chromium Dioxide Chromium dioxide (CrO2 ) is a well-known magnetic material which is widely used in magnetic recording media. CrO2 is the only stoichiometric binary transition metal oxide that is a ferromagnetic material with the Curie temperature of about 393 K. The magnetic saturation moment of CrO2 is approximately 2µB per Cr4+ ion. CrO2 crystallizes in the rutile structure (space group P 42 /mnm) in which the unit cell consists of two formula units (Fig. 4). The Bravais lattice is tetragonal (c/a = 0.65958) with a lattice constant of a = 4.421 ˚ A. It consists of Cr4+ ions octahedrally coordinated by O2− ions where the oxygen octahedra are arranged in “ribbons” running parallel to the tetragonal c-axis. Adjacent octahedra on the same ribbon share a common edge, whereas octahedra on adjacent ribbons share a common corner and are situated relative to each other according to a four-fold screw axis with non-primitive translation equal to half the c-axis.

Half-Metallic Ferromagnets CrO2 and Fe3 O4

a

O

299

2-

4+

c

a

Cr

Fig. 4. Perspective side view of the CrO2 crystal structure

Several band structure calculations were performed to reveal the picture of the CrO2 electronic structure. In the early work of Schwarz [1] the band structure of CrO2 was calculated with the augmented spherical wave method by performing self-consistent spin-polarized calculations. The majority spin electrons were found to be metallic, whereas the minority spin channel was found to be semiconducting, leading to a 100% spin polarization of the conduction electrons at EF . Thus, on the basis of LSDA band structure calculations CrO2 was predicted to be a half-metallic ferromagnet [1]. According to Schwarz [1], the electronic structure of CrO2 consists of the low-lying O 2s bands (binding energy in the range of −18 eV) which are separated by a gap from the next mainly originating from O 2p states. These O 2p states interact with the exchange-split Cr 3d states and lead to bonding and antibonding states. In the case of majority electrons the O 2p and Cr 3d states are so close in energy that a relatively strong covalent interaction occurs, while for spin down electrons the separation between these states is large enough to open an energy gap between the O 2p and Cr 3d bands. This mechanism leads to the half-metallic character of CrO2 . The Fermi energy is located within the energy gap of the minority density of states but is also in a local minimum of majority density of states. Recently, LSDA+U band structure calculations of CrO2 were performed by Korotin et al. [3]. The results of these calculations were compared with the earlier LSDA band structure [1] and revealed some differences, mostly amounting to an upward shift of the unoccupied d states by approximately 2 eV [71]. The results of Korotin et al. [3] mainly differ from those of Schwarz [1] in the position of the unoccupied minority Cr d band. Compared to the LSDA results of Mazin et al. [71] this Cr d band is shifted by about 1.4 eV towards higher energies (less than half of the U value, due to the Cr-O hybridization [3]). Thus the Fermi level appears to be at a somewhat lower position than in the middle of the gap, while in the LAPW calculations reported by Mazin et al. [71], as well as in [1], it is closer to the top of

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the gap. The question is to what extent this difference reflects real physics, that is, renormalization of the bands due to electron-electron correlations beyond LSDA, and to what extent it reflects the difference in the computational technique and associated approximations. If the electron-electron correlations dominate, full potential calculations are preferred over atomic-sphere approximation (ASA) calculations [3]. On the other hand, if the correlations beyond the LSDA are very strong, the LSDA+U calculations may be superior regardless of the spherical approximation. It is worth noting that although the statement of Korotin et al. [3], that “LSDA+U is superior to LSDA since it can indeed yield a gap if the local Coulomb interaction is large enough”, is often true, it is not so clear when the local interaction is small compared to the band width, and the material is actually a metal. In case of CrO2 , when U /W ∼ 0.5, it is unclear whether the crude way in which LSDA+U includes correlations is better or worse than LSDA, which includes them roughly in a mean-field manner. The early experiments aiming to determine the spin polarization of CrO2 performed by surface sensitive techniques did not give a convincing proof of the half-metallic nature of CrO2 . Wiesendanger et al. [72] performed spinresolved STM measurements of a Cr(001) single crystal surface using CrO2 STM tips. The magnetic contrast in STM images observed on Cr(001) was attributed to spin-polarized tunneling from the CrO2 STM tip. In this experiment a spin polarization value of about 20% was determined for CrO2 which was in contradiction to the electronic band-structure calculation predicting a 100% spin polarization at EF . Direct measurements of the spin polarization performed on polycrystalline CrO2 films by means of spin-resolved photoelectron spectroscopy gave high spin-polarization values of about 95% at a binding energy of 2 eV [17], but an extremely low photoemission intensity was observed at EF . These early experiments showed that a careful preparation of well-defined epitaxial CrO2 samples is crucial for proving the half-metallic state of CrO2 . The electronic structure of compacted CrO2 powder samples was studied by specific-heat measurements as well as by photoemission (UPS) and inverse photoemission spectroscopy (BIS) [13]. The UPS spectra exhibited a weak but finite intensity at EF , consistent with the metallic behaviour of CrO2 . Most of the spectral weight of the Cr d band is distributed in broad peaks well above and below EF , reminiscent of the upper and lower Hubbard bands, respectively, which would be consistent with the LSDA+U band structure calculations. The obtained UPS and BIS spectra were compared with the theoretical spectra deduced from LSDA [1] and LSDA+U [3] electronic band structure calculations. The most remarkable difference between the LSDA and LSDA+U calculations is the position of the main Cr 3d peak above EF . The large (∼5 eV) splitting between the prominent Cr 3d peaks in the combined UPS-BIS spectra [13] is closer to the value (∼4.5 eV) deduced from the LSDA+U calculation than that (∼1.6 eV) deduced from the LSDA

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calculation. The underestimate of the splitting in the LSDA calculation is due to the neglect of the Hubbard splitting of the d band by ∼U. The value of U estimated from the experiment [13] is about 3.4 eV (U ∼ 5 − 1.6 = 3.4 eV) which is in a good agreement with the value of U = 3 eV used in the LSDA+U electronic band structure calculations [3]. Recently, single crystal CrO2 (100) films were fabricated using chemical vapor deposition with chromyl chloride (CrO2 Cl2 ) as a liquid precursor [16]. The surface quality of the prepared CrO2 films was probed by atomic force microscopy (AFM) and verified the single crystalline state of the films. The CrO2 (100) films were atomically smooth with a rms roughness of less than 5˚ A for 1000-˚ A thick films. The atomic planes of CrO2 were found to be separated by 4.4-˚ A or 8.8-˚ A high steps which corresponds to a single or double unit-cell length along the a-axis of the CrO2 lattice. Magnetization and resistivity measurements on these films were in good agreement with those measured on films made with the CrO3 solid precursor. The spin-polarization measurements on these CrO2 samples were performed by means of the pointcontact Andreev reflection (PCAR) technique. In these experiments the value of P was determined to be as high as 98.4%. This is the largest spin polarization ever determined in any ferromagnetic metal, affirming convincingly the half-metallic nature of CrO2 . The most recent study of the spin-dependent electronic structure of epitaxial CrO2 (100) films was performed by means of spin- and angle-resolved photoemission spectroscopy [73]. In this case the epitaxial CrO2 (100) films were prepared on isostructural TiO2 (100) substrates by a chemical vapor deposition (CVD) using CrO3 as a solid precursor. The crystallographic quality of the CrO2 (100) films was characterized by x-ray diffractometry (XRD). A typical x-ray diffraction pattern (Fig. 5 (a)) shows only dominant CrO2 (200) and CrO2 (400) peaks indicating a preferred a-axis growth. No impurity phases in the CrO2 film, including Cr2 O3 , were found. Figure 5 (b) shows longitudinal magneto-optical Kerr effect (MOKE) hysteresis loops of CrO2 (100) films with the applied magnetic field along the in-plane easy (c-axis) and hard (a-axis) magnetic axes. The coercive field along the easy magnetic axis which can be extracted from the plot is around 160 Oe. The surface quality of the freshly prepared CrO2 (100) films was probed by LEED and STM under UHV conditions. Figure 6 (a) shows a typical LEED pattern obtained on the CrO2 (100) surface. A clear two-fold symmetry observed in the LEED patterns can be attributed to an unreconstructed (1 × 1) surface structure of CrO2 (100) (Fig. 6 (b)). The lattice parameters of the (100) plane calculated from the LEED images have been determined to be 2.74 ± 0.20 ˚ A for the [010] direction and 4.36 ± 0.20 ˚ A for the [001] direction. These values are in good agreement with those taken from crystallographic bulk data (2.91 ˚ A, 4.41 ˚ A).

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Intensity (arb. units)

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Fig. 5. (a) A x-ray diffraction pattern of an epitaxial CrO2 (100) film deposited on a TiO2 (100) substrate. The peaks labled with an asterisk (∗) are due to the underlaying TiO2 (100) substrate and the sample holder made of brass. (b) In-plane magnetic hysteresis loops of CrO2 (100) film taken at 293 K

a)

b)

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a

Fig. 6. (a) LEED pattern of a CrO2 (100) film (the electron beam energy is 112 eV); (b) schematic top view of the bulk-terminated (100) surface of CrO2

Large epitaxial truncated pyramidal-like islands of CrO2 with average heights of 10 nm and lateral dimensions of about 500 × 600 nm2 , were observed by STM (Fig. 7 (a)). The island edges are preferably oriented along the [001] and [010] in-plane crystallographic directions of the CrO2 surface. On a lateral length scale of 1000 nm the vertical peak-to-peak roughness of the prepared CrO2 (100) film was found to be 15 nm, which is mostly due to the steps. Figure 7 (b) shows a STM image obtained on top of such a pyramid with the edges preferably parallel to the [001] direction of the CrO2 surface. The atomically flat terraces of CrO2 are clearly visible on this image. The step height of each individual step was determined to be about 4.4 ˚ A (see height profile A in Fig. 7). This value is in good agreement with the CrO2 lattice constant (4.41 ˚ A) as well as with that measured before (4.4 ˚ A) [16]. The small circularly shaped islands observed on the surface are attributed to the contaminations due to the ex situ film preparation and transfer.

Half-Metallic Ferromagnets CrO2 and Fe3 O4

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Distance (nm) Fig. 7. (a) a 1 × 1 µm2 STM image of the CrO2 (100) surface showing truncated pyramidal islands. (b) 110 × 110 nm2 STM image obtained on top of a pyramid together with a line profile corresponding to the STM line scan A over the CrO2 (100) surface. Scanning parameters: UT = 1 V; IT = 0.12 nA

Figure 8 shows a series of angle-resolved PES spectra recorded near EF and the resulting spin polarization as a function of the sputtering time (inset in Fig. 8), calculated from the spin-resolved PES spectra of the CrO2 (100) films for a variety of sputtering and annealing periods [73]. The position of the Cr-3d band in the range of 2 eV in the PES spectra changes considerably with increasing sputtering time from approximately 2.3 eV below EF for nonsputtered CrO2 (100) film to 2 eV below EF after 750 sec of sputtering. At the same time an increase of the intensity of the shifted Cr-3d band has been observed. These effects can be due to an increasing structural disorder of the CrO2 (100) surface produced by the sputtering process. Annealing of the sputtered CrO2 (100) film at 150◦ C for 12 hours in UHV leads to a restoring of the peak at 2.3 eV below EF and of its former intensity (spectrum B in Fig. 8). We conclude that the annealing leads to a complete recovery of the crystalline properties of the CrO2 (100) surface layer structure. Directly after the introduction of the sample into the UHV a spin polarization value of +(80±10)% at EF has been found (Fig. 8). After several sputtering cycles (total sputtering time 210 sec) the spin polarization increases up to +(90 ± 10)%. This effect can be explained by an improving of the surface quality by removing contaminations that has also been observed by a STM surface analysis. After additional sputtering the sample’s spin polarization decreases continuously and approaches a value less than 10% after 750 sec. This effect can be attributed to an increasing destruction of the surface order

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Binding Energy (eV) Fig. 8. ARPES spectra as a function of binding energy recorded near EF and spin polarization (inset) at EF (solid circles) and at 1 eV (solid squares) of a CrO2 (100) film vs sputtering time. A and B are the spectra measured after annealing of the sputtered (sputtering time: 750 sec) CrO2 (100) film at 100◦ C and 150◦ C for 12 hours, respectively

with increasing sputtering time, i.e. the sputtering process eventually produces an amorphous non-magnetic topmost layer of CrO2 [74]. A following annealing process of the sputtered CrO2 (100) film at 150◦ C for 12 hours in UHV leads to a restoring of the high spin polarization up to +(85±10)% (see point B in Fig. 8). Annealing seems to heal the surface layers from defects produced by the sputtering without the formation of a Cr2 O3 overlayer. Figure 9 (a) presents the spin- and angle-resolved photoemission spectra together with the total angle-resolved photoemission intensity and the spin polarization as a function of the binding energy of a CrO2 (100) film after sput-

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Binding Energy (eV)

Intensity (arb. units)

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Cu(100) CrO2(100)

1.0 0.8 0.6 0.4 0.2 EF -0.2

Binding Energy (eV) Fig. 9. Spin polarization as a function of binding energy (right-hand panel in (a)) of an epitaxial CrO2 (100) film after sputtering for 210 sec at 500 eV and an additional annealing treatment at 150◦ C for 12 hours together with the corresponding angleresolved spin-polarized photoemission spectrum (spin down: triangles down, spin up: triangles up) and the total photoemission intensity (solid circles) near EF (lefthand panel in (a)). In (b) a zoom of the photoelectron spectrum of an epitaxial CrO2 (100) film in the energy range near EF in comparison with the spectrum of copper single crystal (s-metal) is presented

tering for 750 sec and subsequent annealing for 12 hours at 150◦ C [73]. The spin-resolved spectra of the valence band of the CrO2 (100) near EF clearly show a dominant emission from the majority Cr 3d states (<3 eV) and the O 2p states (>3 eV). The spin-resolved spectra show a clear half-metallic feature, i.e. a metallic Fermi cut-off for the majority spin with the disappearance of spectral weight near EF -reflecting the energy gap – for the minority spin. A maximum positive spin polarization of +(90 ± 10)% was observed for such a CrO2 (100) film at 293 K. Thus the measured maximum spin polarization is close to the theoretically predicted value of 100% [1, 3]. Consistent with

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the band-structure calculations, a weak photoemission intensity due to the presence of the majority states at EF has been observed [13, 73]. In previous spin-resolved photoemission measurements on polycrystalline CrO2 samples an extremely low photoemission intensity at EF was observed and a high spin polarization of approximately +95% could be measured only for binding energies of about 2 eV below EF [17]. The position of the Cr 3d band has been estimated to be about 2.5 eV below EF [17]. In contrast, in the present study a weak but finite intensity was observed at EF (see Fig. 9 (b)), which allows the determination of the spin polarization at EF . The position of the d band was found to be at 2.3 eV below EF . The observation of a finite intensity at EF is in agreement with recent photoemission and inverse photoemission experiments, where, however, the position of the Cr 3d band has been found to be around 1.3 eV below EF [13]. Compared to our study the difference in the positions of the Cr 3d band [13] and the lack of the photoemission intensity near EF [17] can most probably be attributed to the polycrystalline nature of the former CrO2 samples. In conclusion, the spin-resolved photoemission study of CrO2 (100) films yields a spin polarization of approximately +90% at EF at 293 K [73]. This value and the magnitude of the gap in the minority states are in good agreement with the prediction of the half-metallic nature of CrO2 .

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On the Importance of Defects in Magnetic Tunnel Junctions P.A. Dowben and 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 Abstract. The interpretation of junction magnetoresistance results of ferromagnet/insulator/ferromagnet structures must consider a number of complications. These complications include spin polarized defects within the nonmagnetic dielectric layer, and the recognition that many metal to metal oxide interfaces often involve further oxidation and reduction making such interfaces very heterogeneous. The possibility of spin-polarized defects in the barrier layer is important as large values of magnetoresistance may result that have little to do with the ferromagnetic polarization. Interlayer exchange coupling between two ferromagnetic films, separated by a nonmagnetic nonmetallic spacer (semiconductor and insulator materials) can also occur. This coupling is distinct from the very low temperature tunneling phenomena between two ferromagnets, through a dielectric spacer layer, as the coupling is sometimes oscillatory. As an illustration, the ferromagnetic coupling between Co and CrO2 , through an insulator (Cr2 O3 ) may be related to defect states in the insulating barrier layer.

1 Introduction Defects, in thin insulating oxides, play a key role in influencing electronic interface states, electric transport, and magnetic coupling between magnetic thin films. Full understanding of the basics of the physics involved is essential for development of new devices for applications in spin electronics, in particular for tunnel magnetoresistance involving a thin insulator sandwiched between two ferromagnetic films. There is an increasing body of evidence that impurities in the insulating layer will “dope” this layer and alter the net polarization of electrons injected into the insulating layer [1, 2, 3, 4, 5], with strong temperature effects [3, 5], most recently demonstrated in Cr2 O3 [5]. Our studies concentrate on chromium oxide films, made of ferromagnetic metallic CrO2 phase, covered with a thin native layer of the more stable Cr2 O3 phase which is insulating and nominally (not necessarily) antiferromagnetic. A more complete experimental picture of the importance of defects is obtained through photoemission studies combined with electrical and magnetic characterization. This review shows how these complementary experimental techniques provide insight into the presence of defect states in the oxide barrier, and how device properties are influenced. P.A. Dowben and B. Doudin: On the Importance of Defects in Magnetic Tunnel Junctions, Lect. Notes Phys. 678, 309–329 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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CrO2 is a high-electron-spin polarization material, both in theory [6, 7, 8, 9, 10, 11] and in experiment [12, 13, 14, 15, 16, 17, 18, 19] and considered a possible spin injection material [19, 20, 21]. Imbalance between the two spin populations around the Fermi energy is expected to be near its maximum with the result of 100% spin polarization, and values above 90% have been observed in spin polarized photoemission [13, 14], Andreev scattering [15, 16, 17, 18] and from a superconducting tunnel junction [12]. Table 1 summarizes the electron spin polarization results for CrO2 , along with other estimates of polarization extracted from the published literature [19, 22, 23, 24, 25, 26, 27]. Indeed, it is considered one of the most likely candidates for a half-metallic ferromagnet [19]: a metal in one spin channel and insulator in the other spin direction. Table 1. Experimental polarization values for CrO2 taken from the literature. Some values are estimated based on applying a modified Juliere’s model [19] to published magneto-resistance data (a) and tunnel junction data assuming a polarization of 34% for the cobalt side of the tunnel junction (b) Polarization 97% 98.4± 4% 96±4% 90±3.6% 81±3% >82%(a) >62%(a) <66%(a) <94%(a) 1.3%(b) 20±10% 95% 95%

Temperature 1.2 K 1.6 K 1.6 K 1.6 K 1.6 K 4.2 K 5K 4.2 K 4.2 K 77 K 300 K 293 K 300 K

Technique superconducting tunnel junction Andreev scattering Andreev scattering Andreev scattering Andreev scattering magnetoresistance magnetoresistance magnetoresistance magnetoresistance tunnel magnetoresistance spin-polarized tunneling spin-polarized photoemission spin-polarized photoemission

[12] [18] [17] [15] [16] [19,24] [19,23] [25] [26] [22] [27] [14] [15]

As discussed extensively elsewhere [28, 29, 30, 31, 32], the high polarization values determined for CrO2 , including the spin polarization data obtained from spin-polarized photoemission [13, 14], are not compelling proof of “pure” half-metallic character. In a strict sense, half-metallic ferromagnetism is limited to zero temperature since magnon and phonon effects lead to reductions in polarization at finite temperatures. Indeed, magnetoresistance properties related to spin polarization of CrO2 exhibit a dramatic drop with increasing temperature [19, 23, 24, 25, 26]. Associated with the compound structure of all the postulated half-metals, there are typically a number of low-energy transverse and longitudinal optical modes. These optical phonon modes can couple to spin-wave modes and reduce the net polarization [29]. This magnon-phonon coupling [29, 30] has been shown to be directly applicable to the magnetore

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sistance obtained with magnetic tunnel junction geometries [31]. In a strict sense, this thermally activated spin mixing means that half-metallic ferromagnetism is limited to perfect crystals at zero temperature. While finite-temperature spin disorder destroys the complete spin polarization of half-metallic ferromagnets, this is not the only source for limited [19, 22, 33, 34] magnetoresistance in chromium oxide junctions. The temperature dependent decrease in magnetoresistance may be strongly influenced by defects in the barrier layer (typically Cr2 O3 ), turning on and off Coulomb blockade, resonant scattering, and inelastic spin flipping scattering [19, 33, 34]. Imperfect interfaces between the ferromagnets and the dielectric Cr2 O3 barrier layer also present a problem. This chapter summarizes the information, obtained through a variety of experimental techniques, all leading to the conclusion that imperfections at interfaces or inside the thin insulating layer exist, and lead to significant changes in magnetoresistance properties. In the first section, we describe how photoemission techniques provide insight into the ferromagnet/ferromagnetic oxide interface, showing that metal/insulator/metal systems can show complicated interface structures. In the second part, we show that a combination of electronic transport and photoemission reveal that impurity states exist in CrO2 / Cr2 O3 / CrO2 systems, and are detrimental for magnetoresistance properties at higher temperatures. This conclusion is reinforced by experimental evidence that defects can become magnetically polarized at higher temperatures, and can provide a magnetic coupling between the two magnetic electrodes.

2 Chromium Oxide Interfaces and Surface Composition Photoemission, and in particular angle resolved X-ray photoemission spectroscopy (ARXPS), are tools providing unique insight into the composition of the few nanometers near the surface of materials, allowing understanding of the chemical states in an insulator whose thickness is relevant for tunneling properties. One major advantage in exploiting CrO2 as a spin injector to a magnetic tunnel junction is that the stable surface is Cr2 O3 [35, 36], although this immensely complicates the study of CrO2 using electron spectroscopies. Using angle-resolved photoemission, we have been able to characterize the Cr2 O3 oxide surface of CrO2 thin films. The ratio of Cr2 O3 intensity to CrO2 intensity for each emission angle was derived by decomposing every O 1s spectrum into two peaks, corresponding to the Cr2 O3 and CrO2 oxide phases respectively. This intensity ratio is shown in Fig. 1. This analysis of angle resolved X-ray photoemission intensities (ARXPS) is now fairly common. The basic idea of the analysis is to account for each layer contributing to the photoemission signal. So that for constituent component A, with concentration fj (A), the normalized core-level intensities as a function of angle can be written with respect to a reference constituent component B, as:

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Fig. 1. The room-temperature XPS data (left) and intensity ratio of Cr2 O3 peak to CrO2 peak vs. different emission angle (right) for sputter deposited films. Taken from [63] ∞ 

−λ

fj (A) e

jd A cos(Θ)

j=0

R(Θ) =  ∞

−λ

fj (B) e

jd B cos(Θ)

,

(1)

j=0

where λA,B are the effective mean free paths. For determining the thickness of a slab of material A on B, based on the same core level (and hence the same cross-section, and nearly the same free path and analyzer transmission function) this reduces to: 1

ICr2 O3 /ICrO2 = e λcos(Θ) − 1 ,

(2)

where λ is the effective mean free path (about 1 nm) and Θ is the emission angle with respect to the surface normal. Since the Cr2 O3 signal increases relative to the CrO2 signal at the higher emission angles, it is clear that Cr2 O3 dominates the surface as the effective probing depth decreases with increasing emission angle. The thickness of this insulating Cr2 O3 layer is typically twice the photoelectron escape depth or about 2 nm thick, using a summation modeling analysis described elsewhere [37, 38, 39, 40, 41], while the thinnest Cr2 O3 layers are about 0.5 nm thick. While tunneling magneto-resistance in ferromagnetic/insulating/ferromagnetic (FM-I-FM) systems has attracted a lot of attention since 1995 [42, 43, 44], existing tunneling magneto-resistance data are mostly restricted to Al2 O3 oxide as the insulating barrier layer, though a number of other oxides barrier layers [44], including Cr2 O3 [19, 22, 33, 34, 45] have been recently explored. The importance of the oxidation of the magnetic layer in contact with an oxide has received little attention, even though Co oxidation at NiO oxide [46, 47, 48] and Cr2 O3 [49] interfaces has been clearly established.

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We have examined the reduction and oxidation (REDOX) reaction that accompanies the deposition of both a superconductor (Pb) and a transition metal ferromagnet (Co) on the stable Cr2 O3 insulating surface of CrO2 . Metal thin-film deposition over the Cr2 O3 surface of CrO2 thin-film substrates exhibits a redox reaction at the interface. The transition metal forms an oxide in combination with the reduction of the near-surface chromium oxide to Cr2 O3 . The insulating barrier layer Cr2 O3 increases with the formation of Pb3 O4 in Pb/Cr2 O3 /CrO2 and CoO in Co/Cr2 O3 /CrO2 junctions respectively [49]. The ARXPS Cr 2p data for different Cr2 O3 /CrO2 samples, following the preparation of the clean stable Cr2 O3 surface oxide, and following the deposition of Co, are plotted in Fig. 2(a). The Co was evaporated on epitaxial single crystal Cr2 O3 /CrO2 samples on TiO2 . As expected, Cr 2p core level spectra

Fig. 2. (Left) Cr 2p core level X-ray photoemission spectra taken on different sample surface at normal emission angle [49]. The literature values of Cr 2p3/2 binding energy for both Cr2 O3 and CrO2 are marked on the top. The upper panel also shows the data for clean CrOx surface and the right lower panel shows the data for the CrOx with Co evaporated on the top. (Right) The angle-resolved Xray photoemission of Co 2p at room temperature are shown after deposition on the native Cr2 O3 surface of CrO2 [49]. The black curve is the spectrum taken at normal emission angle while the red (light gray) curve is the spectrum taken at 60◦ emission angle and the difference of these two spectra is plotted in the bottom. Literature values are marked at the top

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show that the clean surface is dominated by the stable Cr2 O3 surface oxide [35, 36]. The binding energy of the major Cr 2p 32 core level contribution in most of our thin film samples is about 576.8 ± 0.2 eV which generally corresponds to the accepted binding energy for Cr2 O3 oxide [50]. The other Cr 2p 32 core level peak contributions are generally somewhat lower (but very strong in the CrO2 grown on TiO2 ), at about 576.3 ± 0.3 eV, and this suggests the presence of the CrO2 oxide [51]. After the deposition of the equivalent of 3 monolayers of Co on the Cr2 O3 /CrO2 oxide, the Cr 2p 32 binding energy increases to a binding energy which is more representative of the Cr2 O3 phase (576.7±0.2 eV). Either some CrO2 has been reduced to the native Cr2 O3 increasing the amount of Cr2 O3 , or because of the limited mean free path, the deposition of Co diminished the sampling of the CrO2 that lies below the native Cr2 O3 oxide at the metal/oxide interface [49]. A combination of both effects is also possible. Evidence of CoO formation at the Co/Cr2 O3 interface, for CrO2 epitaxial samples, is found after the deposition of Co [49]. The Co 2p 32 XPS spectra, for both normal emission angle (black) and 60◦ emission (red), plotted in Fig. 2(b), exhibit two strong overlapping features. The binding energy of Co 2p 32 major peak, at 776.8 ± 0.2 eV, corresponds to the metallic Co binding energy [52]. The 2p 32 core level feature at 780.2 ± 0.2 eV is generally associated with the binding energy of CoO [53]. The relative intensities (the ratio of intensities) of these two features depends upon the emission angle. As the emission angle increases the photoemission becomes more surface sensitive increasing the high binding energy Co 2p 32 core level intensity relative to the other Co 2p core level features. Since the Co 2p 32 core level feature at 780.2 ± 0.2 eV is suppressed at the larger emission angles, it is clear that CoO resides below the metallic Co, or rather, Co oxidizes at the interface between Co and Cr2 O3 /CrO2 . For Co/Cr2 O3 /CrO2 junctions, chemical mass balance requires that the oxidation of the Co must be accompanied by either reduction of CrO2 to Cr2 O3 , with an associated increase in the Cr2 O3 layer thickness and oxygen transport to the interface, or reduction of the Cr2 O3 to a suboxide. There is little evidence to support the latter redox reaction, but it cannot be wholly excluded by the data [49]. Clearly nominal CrO2 /Cr2 O3 /Co magnetic junctions are, in fact, more complex multilayers systems akin to a CrO2 /Cr2 O3 /CoO/Co system [49]. It must be recognized that many metal-to-metal oxide interfaces involve further oxidation and reduction, making such interfaces very heterogeneous. The interpretation of tunnel magnetoresistance results must now assume that ferromagnetic metals will NOT generally form abrupt interfaces with transition metal oxide dielectric barriers. This raises significant questions about the composition of insulating oxide-barrier layers, but is not the only potential complication hindering understanding tunnel magnetoresistance devices.

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3 Intermediate States in the Barrier For ferromagnet/insulator/ferromagnet tunneling devices, the change of resistance when changing the magnetic electrodes alignment can be approximately (an over simplification to be sure) related to the spin polarizations of the magnetic entities [42, 43, 44]. Very large magnetoresistance ratios are expected for tunnel junctions made of high-spin-polarization materials. First results on chromium oxide powders showed very encouraging large magnetoresistance at low temperatures [24, 25, 26]. In two reports, however, a very small magnetoresistance ratio (1%) at 70 K [22], or even negative magnetoresistance (–8%) at 4.2 K [54] was found for CrO2 . There is a clear need of clarification of the electric transport in chromium oxide junctions. CrO2 thin films with large crystallites of several microns size permit investigation of intergrain tunneling through 1–2 nm Cr2 O3 film involving only a few crystals. The maximum (Rmax ) of the magnetoresistance curve corresponds to when the total magnetization of the film is close to zero (the anti-parallel alignment of neighbor crystallites is highest). Applied field larger than 0.12 T saturate the magnetoresistance at the minimum values and the magnetization of our samples, as seen in Fig. 3. The magnetoresistance ratio, defined as (Rmax – R)/Rmax , had values of typically 15% to 25%, when there is tunneling through just a few CrO2 crystallites (with the junctions between crystallites made of Cr2 O3 ) [33, 34]. Similar values were also observed on thick chromium oxide films, on samples of typically few ohms resistance values. The magnetoresistance ratio decreases exponentially with increasing temperature

Fig. 3. Magnetoresistance curve at 3 K of (1) a chromium oxide sample made of hundreds of connections between CrO2 grains and (2) a sample made with an electric percolation path through only a few CrO2 crystallites [33, 34]

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to values lower than 1% at temperatures higher than 100 K, consistent with the results observed on CrO2 powders [19, 23, 24, 25, 26]. A pronounced zero-bias anomaly of the conductance is found at low temperatures. The low-bias conductivity is strongly reminiscent of the so-called “giant resistance peak” observed by Rowell and Shen on Cr-I-Ag tunnel junctions [55]. In this case, the insulator “I” was fabricated by oxidation of a Cr film, and the anomaly attributed to the presence of magnetic CrO2 and Cr2 O3 , making this very similar to our own example [56]! In our magnetoresistance measurements [33, 34], samples with a high resistance (several hundreds of kΩ at room temperature) have magnetoresistance ratios decreasing by a factor two with increasing voltage bias. Figure 4 indicates that a low-bias magnetoresistance of 30% decreases down to 15% at higher bias voltages. For the low resistance samples (R RQ = h/2e2 ), the intergrain resistance is much smaller than the quantum of resistance (RQ = h/2e2 ∼ 13.8 kΩ), which is classified as a strong tunneling regime. The magnetoresistance, at

Fig. 4. Normalized resistance as a function of voltage for samples involving just a few CrO2 crystallites (top), made of 10–15 junctions in series. The weak tunneling sample (—) has a large bias resistance of more than 1 MΩ (40 RQ ). The strong tunneling case (....) corresponds to a sample of resistance smaller than a few kΩ (0.5 RQ ) at large voltage bias. Bottom panel shows the magnetoresistance as a function of voltage bias. For a weak tunneling sample, it decreases as a function of voltage bias for temperatures lower than 10 K. For the strong tunneling case, the decrease as a function of bias is more pronounced down to lower magnetoresistance values at high bias [33]

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large bias, is even smaller (reduced to a few percent, as seen in Fig. 4), than is observed for the large resistance samples. Possible explanations for these observations take into account a model of Coulomb blockade of the electric transport by electrostatic interactions with impurity electronic states, playing the role of Coulomb islands in a dielectric. The Coulomb blockade electrostatic energy exponentially increases the resistance at low temperatures in a sequential tunneling model. Averin and Nazarov [57] showed that inelastic co-tunneling, where the electrons tunnel through a virtual state in an island within the dielectric barrier, becomes a dominant current path at low temperatures and low bias. The data of Fig. 4 corresponds to the predictions of the Takahashi and Maekawa model of enhancement of magnetoresistance due to co-tunneling effects [54]. This co-tunneling event has a probability proportional to the product of the probability to add an electron to the island with the escape probability of an electron out of the island. The magnetoresistance ratio for such fully correlated events is therefore twice the magnetoresistance ratio of uncorrelated events, or sequential tunneling, occurring at high voltage bias. For the strong tunneling case, the magnetoresistance is expected to decrease strongly with bias [54], following our observations. Defects in the tunneling barrier can also be related to a low-density of states within the bandgap, similar to localized states in disordered materials [58, 59]. A model of resonant tunneling [60] can be put forward to explain the electric transport in such materials, without the need of electron-electron interactions. Such a model predicts a significant reduction of the magnetoresistance ratio [61], and even a sign reversal of the magnetoresistance if the impurities are very asymmetrically positioned in the barrier [62], as found experimentally in asymetric CrO2 /Cr2 O3 /Co junctions [54]. The systematic observation of current anomalies at zero bias only, as well as the factor two of TMR changes with bias in our intergrain junctions, are not directly reconciled with a model resonant tunneling. The occurrence of significant inelastic scattering makes a definitive choice between the models difficult, but it is clear from the experimental results that defect states play a crucial role for TMR properties. Combined photoemission and inverse-photoemission studies for different temperatures confirm the occurrence of such intermediate states, becoming available with activation energy values of several meV. These techniques, complementary to transport studies, allow us to discard a model where heating of the junctions plays a role and is a possible source of strong nonlinear resistance curves. Spectra of the density of states near the Fermi level of the Cr2 O3 surface of the CrO2 sample showed a shift with temperature, largely at the conduction band edge, much greater than 3kB T (Fig. 5) [5, 33, 34]. At temperatures lower than 200 K, a gap of 3.5±0.15 eV was found, confirming that the Cr2 O3 coverage of a CrO2 grain provides a wide bandgap tunnel barrier, as summarized in Table 2. Heating the sample to room temperature showed a significant decrease of the conduction band edge

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Fig. 5. Combined photoemission (left) and inverse photoemission (right) of the Cr2 O3 surface layer of CrO2 . There is a clearly evident shift of the conduction band edge of the Cr2 O3 overlayer on CrO2 with decreasing temperature from room temperature (dark grey (red)) to low temperatures (190 K, black (blue)). Intentionally introduced defects appear to suppress Coulomb blockade effects as denoted by the curve (grey (green)) taken at 190 K. After argon ion sputtering the surface, while appearing to preserve the native Cr2 O3 overlayer chemical state, the conduction band edge is relatively temperature insensitive over the range 180–320 K. The inset shows the estimated band gap from combined photoemission and inverse photoemission studies. Taken from [33]

energy, without other significant changes in the spectra. An unambiguous temperature-dependent change of the valence band conduction band gap by 0.7 eV was found between 190 to 300 K. Surface charging effects were excluded by the absence of peak and core-level shifts and the absence of any power dependence (fluence) on the binding energies. The hypothesis of surface phase transition is very unlikely as no changes in the spectrum and no discontinuity in the bandgap values were observed when varying the temperature. With an appropriate surface treatment, by sputtering, the position of the conduction band edge is largely temperature independent, as indicated by the green curve in Fig. 5. While the core level photoemission provided no evidence that such a defective surface is anything other than Cr2 O3 , the number of intermediate states must increase in the Cr2 O3 barrier layer. This deduction, combined with the absence of a characteristic temperature related to charging effects, suggest that the band edge position is dominated by the ability to populate defect states. In the native Cr2 O3 surface layer, it appears that at low temperature the conduction band edge electrons are trapped or immobile, and at high temperature there is greater mobility. This effect could well extend to defect states well in the gap that occur at such low populations that they cannot be

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Table 2. The Band Gap(s) of Cr2 O3 Band Gap of Cr2 O3 (eV) 2.8–3.4 3.2 3.2 1.5 0.9

Method combined photoemission and inverse photoemission optical absorption optical absorption theory theory

[33,34] [63] [64] [65] [66]

observed in either photoemission or inverse photoemission. We claim therefore that there is the availability of donor states within 0.7 eV of the conduction band edge of the Cr2 O3 barrier layer and that a thermal energy overcoming 10–20 meV relates to impurities in the Cr2 O3 barrier. This energy magnitude also corresponds to the width (5 mV) of the resistance peak found by Rowell and Shen [55], consistent with our own measurements [33, 34].

4 Polarizable Defects in Cr2 O3 ? Spin-resolved inverse photoemission investigations provide insight into the spin polarization of the electronic states of the surface layers. In this section, we show that the native Cr2 O3 surface is antiferromagnetically coupled to the CrO2 thin film substrate, with a temperature dependent induced polarization [5]. The Cr2 O3 exhibits the characteristic behavior of a rigid-band/spinmixing behavior (non-Stoner) of a local moment paramagnet. This, combined with the strong shifts of the conduction band edge from room temperature to low temperature, just discussed, suggest that the extent of the induced polarization of the Cr2 O3 oxide surface, by the CrO2 substrate, may be partly related to impurity mediated transport effects identified in CrO2 /Cr2 O3 /CrO2 junctions [5]. The spin-resolved electronic structure of the (unoccupied) conduction bands for Cr2 O3 overlayers, on epitaxial CrO2 thin films, are shown in Fig. 6, for both room temperature (about 300 K) and low temperature (190 K). A measurable negative spin asymmetry is observed in spin-polarized inverse photoemission 3 eV to 4 eV above the Fermi energy. Large spin asymmetries at the Cr2 O3 surface are expected to be enhanced in the unoccupied bands [67], but the observation of net polarization indicates a net magnetic ordering at the surface of this insulating layer. The corresponding positive spin asymmetry, at 4 to 5 eV above the Fermi level, is quite weak, as is typical in experiments where the electron mean free path is spin dependent [68, 69, 70, 71, 72].

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Fig. 6. Spin-polarized inverse photoemission spectra show the conduction band of the Cr2 O3 overlayer on epitaxial CrO2 thin films at room temperature (300 K) and low temperature (190 K) respectively [5]. The spin-up () and down () density of states separately indicated. The inset shows the in-plane hysteresis loop, along the magnetic easy axis, obtained from MOKE

This enhancement of the spin-minority density of states, and the spin asymmetry in spin-polarized inverse photoemission at room temperature, are suppressed at 190 K. This effect cannot be directly related to the magnetization of the CrO2 substrate, which is observed to significantly decrease in this temperature range [73, 74]. Rather, these results suggest that the native Cr2 O3 surface of CrO2 is polarized by the highly polarized ferromagnet CrO2 “substrate”. This induced polarization, reflected in a spin asymmetry favoring spin-minority occupation, is not commonly seen in spin polarized inverse photoemission. Rather, the spin-majority component tends to be favored [70, 71], as the majority-intensity component is greater due to a variety of factors, including differences in the spin-dependent mean free paths of low energy electrons. The unoccupied spin-minority asymmetry suggests that the Cr2 O3 is “antiferromagnetically” aligned with respect to the CrO2 . This is consistent with x-ray magnetic circular dichroism (XMCD) measurements of

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the CrO2 /Cr2 O3 mixed phase system [75]. This antiferromagnetic moment alignment between CrO2 and Cr2 O3 is much like the predicted Ti induced moment in SrTiO3 at the Co/SrTiO3 interface [94]. The higher polarization asymmetry requires that the polarization is mediated by gap states (possibly metal-induced gap states [77, 78]) or defect states [1, 2, 3, 4, 78, 79] whose mobility increases with increasing temperature. These findings indicate that the magnetic coupling exist between the bulk ferromagnetic metal and the dielectric surface. This has direct implications for a trilayer tunneling system, where understanding of the magnetic properties is essential for designing a spin-electronics element.

5 Defect Mediated Coupling? Interlayer coupling between two ferromagnetic films, separated by a nonmetallic space layer, has been given increasing attention. While most of the experiments have focused on nonmagnetic metal spacers, which shows an oscillatory exchange coupling known to be related to the RKKY interaction, exchange coupling also exists for semiconductor and insulator spacer materials where the RKKY model of interaction should be inapplicable. This coupling through nonmetallic spacers [79, 80, 81, 82, 83, 84] sometimes appears to be distinct from the very low temperature tunneling phenomena between two ferromagnets, through a dielectric spacer layer [21, 42, 43, 44], as the coupling is sometimes oscillatory [79, 80, 81, 82, 83, 84, 85]. Temperature dependent coupling in Fe/ZnSe/Fe [82, 86, 87], Fe/Si/Fe [3, 80, 87, 88, 89, 90], Fe/SiO/Fe [79, 88], Fe/Ge/Fe [88, 91, 92], Fe/FeSi [93] and MnGa/Ga,As,Mn/MnGa [94, 95] as well as photo-induced coupling in Fe/FeSi [96] and Fe/SiO [97] have been observed. Several models were proposed to account for the coupling, for example: thickness fluctuations [98], loose spins [99] and quantum interference [100]. Oxide magnetic/insulator superlattices consisting of Fe3 O4 /NiO [101, 102], and NiO/CoO [103, 104] have been fabricated. Superexchange appears to be suppressed in these insulating systems and the interlayer coupling is confined primarily to the interface region and therefore sensitive to interface morphology [46, 47, 48]. There is an increasing body of evidence that impurities, in the insulating layer, will “dope” the insulating layer and alter the net polarization of electrons injected into the insulating layer [1, 2, 3], with strong temperature effects [3, 5], as we have just indicated for Cr2 O3 on CrO2 [5]. It has been proposed that impurity or defect states within the mobility gap of the insulator are the origin of the magnetic interaction across the insulating or semiconducting barrier and that increasing temperature enhances the coupling due to increased mobility [1, 2, 3, 79, 81, 88]. The conduction bands in insulators are expected to be polarized by exchange coupling with highly polarized states [77]. Because of the interest in both superconductor/insulating/ferromagnetic S/I/FM [12] and ferromagnetic-insulating-ferromagnetic FM/I/FM [42, 43,

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44, 84, 85, 112] junctions and more specifically, because of the interest in CrO2 -insulating-ferromagnetic FM/I/FM junctions [19, 22, 33, 34, 45, 105], we have examined the magnetic coupling properties between Co and CrO2 ferromagnetic layers “through” the stable insulating Cr2 O3 surface of CrO2 . Because the Co oxidizes at the interface between Co and Cr2 O3 /CrO2 , following Co evaporation, we have actually formed Co/CoO/Cr2 O3 /CrO2 multilayers, and an insulating barrier layer of larger effective thickness. The hysteresis loops of the Co/CoO/Cr2 O3 /CrO2 multilayer samples show step-like behavior, as shown in Fig. 7. These steps in the hysteresis loop indicates that the Co and CrO2 layers flip at different fields [45]. In order to get a clear picture of the magnetic coupling, if any, between these two ferromagnetic layers, we measured the minor loops. First, we applied a magnetic field at 200 Oe or –200 Oe to saturate the sample, then swept the field in a range smaller than the coercivity of the “complete” hysteresis loop. Figure 7 shows the complete hysteresis loop for this Co(40˚ A)/CoO/Cr2 O3 /CrO2 multilayer FM/I/FM “trilayer” like system and the minor loops, taken at 338 K. The coercivity of the Co top layer is far larger than the coercivity of the CrO2 layer for all samples, in spite of the considerable difference in thickness. The hysteresis loops for CrO2 alone (inset to Fig. 7) are similar to the minor loops in this Co/CoO/Cr2 O3 /CrO2 multilayer system, and it is clear that these minor loops are the consequence of reversal in the CrO2 underlayer alone. There is a shift between the centers of two minor loops, denoted as 2Hex , which is the external field required to cancel out the magnetic interlayer coupling. The small value and positive sign of Hex in our data, as illustrated by the data obtained Co(40˚ A)/CoO/Cr2 O3 /CrO2 in Fig. 7, compellingly suggests that the Co layers and the CrO2 layer are weakly ferromagnetically coupled through the CoO/Cr2 O3 barrier, with weak temperature dependence similar to that suggested by the model of Bruno [106]. Weak ferromagnetic coupling was also observed for thicker insulating barriers in Fe/MgO/Fe/Co multilayers [84]. If the two ferromagnetic layers, with uniaxial anisotropy, are coupled ferromagnetically, the total energy of this system can be expressed [107, 108] as: E = −MCrO2 tCrO2 Hcos(θ − β) + K1CrO2 sin2 β − MCo tCo Hcos(θ − α) +

K1Co sin2 α

(3)

− J1 (T )cos(α − β)

where, MCrO2 , tCrO2 and K1CrO2 are the magnetization, thickness and first order anisotropy constant of CrO2 layer respectively, while MCo , tCo and K1Co are the magnetization, film thickness and first order anisotropy constant for Co layer respectively, and α, β and θ are the angle of MCrO2 , MCo and H with respect to the easy axis, respectively, H is the applied field and J1 is the coupling constant between the two ferromagnetic layers. CrO2 and Co are ferromagnetically coupled when α = β. Both the magnetic state of the trilayer system and the coercivity can be determined from this energy

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Fig. 7. The complete hysteresis loop and minor loops measured at 338 K for the sample Co(20˚ A)/Cr2 O3 /CrO2 . The minor loops were obtained after saturation and 2Hex is the shift between the centers of the minor loops. The hysteresis loop for the CrO2 thin film substrate (alone) is shown as the inset, also at 338 K [45]

expression. The value of Hex , in Fig. 7, is related to the coupling constant J1 by J1 = Hex MCrO2 tCrO2 . From the hysteresis loops for the Co(40 ˚ A)/CoO/Cr2 O3 /CrO2 and Co(60 ˚ A)/CoO/Cr2 O3 /CrO2 multilayer FM/I/FM “trilayer” like systems, A we obtained Hex as a function of temperature [45]. It is evident that for 40 ˚ Co and 60 ˚ A Co Co/CoO/Cr2 O3 /CrO2 samples, the measured Hex increases slowly with increasing temperature. The magnetization of CrO2 decreases very fast as the temperature approaches the Curie temperature of CrO2 , which is 390 K [109]. As a result, the coupling constant J1 must decrease with increasing temperature. The picture of coupling between the Co and CrO2 layers is clearer when we plot the coercive switching fields for the Co and CrO2 layers in the Co/CoO/Cr2 O3 /CrO2 multilayers, as a function of temperature, depicted in Fig. 8. We define the coercive switching field of each layer, according to the step-like complete hysteresis loops, which, although offset in magnetization, provide an indication of the individual layer coercive field. These “coercive” fields, as a function of temperature, differ for different Co thickness Co/CoO/Cr2 O3 /CrO2 multilayers samples, as seen in Fig. 8. It is obvious that with a Co layer (forming the Co/CoO/Cr2 O3 /CrO2 multilayers instead of the Cr2 O3 /CrO2 bilayer), the coercive switching field of CrO2 underlayer is larger than for the bare CrO2 layer. The differences between coercive fields for bare CrO2 layer alone, without any Co coverage (plotted in each panel of

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Fig. 8. Switching field as a function of temperature for a variety of Co/Cr2 O3 /CrO2 multilayers samples with different Co thickness. () shows the switching field of Co layer in the Co/CoO/Cr2 O3 /CrO2 multilayers, (•) shows the switching field of CrO2 underlayer in the Co/CoO/Cr2 O3 /CrO2 multilayers, and () shows the switching field of CrO2 without any Co overlayer coverage, for reference. Panel (a) shows the data for the sample with 20 ˚ A Co, panel (b) shows the data for 40 ˚ A Co and panel (c) shows the data for 60 ˚ A Co. Adapted from [45]

Fig. 8 for reference), and in the Co/CoO/Cr2 O3 /CrO2 multilayers, increases with increasing Co coverages. This increase in the effective CrO2 coercive field is the result of the ferromagnetic coupling between Co and CrO2 . Strong ferromagnetic coupling would tend to make the Co and CrO2 ferromagnetic layers switch together. While the coercive field of Co is larger than the CrO2 layer by itself, in spite of the disparity in film thickness, these two layers still do not reverse magnetization “together”. Rather, there is an increase in the coercive switching field of the CrO2 layer, in the Co/CoO/Cr2 O3 /CrO2 multilayer, compared to the CrO2 alone. This perturbation of the CrO2 coercive field by a cobalt layer, and the existence of a nonzero Hex make clear that while there is coupling between the ferromagnetic layers, the coupling must be weak. As temperature increases towards the Curie temperature of CrO2 (390-397 K), the coercive switching field of CrO2 , in the Co/CoO/Cr2 O3 /CrO2 multilayer, decreases and approaches the coercive switching field of CrO2 alone, while the coercive switching field of Co top layer increases. This supports the contention that the CrO2 layer tends to be the ’spectator’, while cobalt layer tends to be the “actor” or “driver”, but this spectator behavior of CrO2 is more extreme near the CrO2 Curie temperature. At temperatures above the Curie temperature of CrO2 (390-397 K), there is a critical temperature where the Co layer, in the Co/CoO/Cr2 O3 /CrO2 multilayer, exhibits decreasing coercivity with increasing temperature. This latter critical temperature in the cobalt layer behavior increases with increasing thickness of the Co layer in

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˚ the Co/CoO/Cr2 O3 /CrO2 multilayer from 390 K for the sample with 20 A Co coverage to 405 K for 40 ˚ A Co. Since the coupling between the Co and the CrO2 layers is stronger with increasing Co thickness, it should not be too surprising that the induced polarization of “paramagnetic CrO2 ” also increases with increasing Co-layer thickness. As temperature increases and approaches the Curie temperature of CrO2 , MCrO2 and K1CrO2 decrease and eventually CrO2 becomes paramagnetic. From the energy equation for a trilayer system it is clear that Co becomes the driving layer near the Tc of CrO2 , but this does not mean that the ferromagnetic Co cannot continue to weakly polarize the CrO2 layer above the Curie temperature of CrO2 . There is evidence of both phenomena in our MOKE data. The mechanisms for the weak ferromagnetic coupling between Co and CrO2 , in the Co/CoO/Cr2 O3 /CrO2 multilayer, have not been precisely identified. Simple RKKY coupling [110], perpendicular coupling [110, 111], tunneling [84, 85] or orange peel effect (correlated roughness) coupling [84] of the ferromagnetic layers isolated by a nonmagnetic, nonmetallic barrier layer, as suggested for other oxide barrier layers, are not completely appropriate models for the coupling of Co and CrO2 through the CoO/Cr2 O3 barrier, and do little to explain the polarization of the CrO2 layer above the CrO2 Curie temperature. Models based on tunneling of the wave functions of each ferromagnet through the insulating barrier [77, 106], applied elsewhere to the Fe/MgO/Fe/Co system [84], require unrealistically small barrier heights. We have observed weak coupling above the antiferromagnetic polytype N´ eel temperature of Cr2 O3 through a dielectric barrier layer material with a band gap well above 2 eV and without an appreciable density of states at EF (Table 2).

6 Conclusion: Defects May Be Important Chromium dioxide CrO2 is a ferromagnetic metallic material that has been of great interest for spin electronics, because of its potentially high degree of spin polarization. This high polarization is not realized at elevated temperatures, but we can take advantage of the naturally occurring stable Cr2 O3 surface, providing a few-nanometer-thick dielectric layer on the ferromagnet CrO2 . The electronic and magnetic properties of the interface and top thin film provide evidence that defects exist in this thin dielectric and influence significantly the electric transport properties in ferromagnetic/insulator/ferromagnetic structures. By combining photoemission and electric transport properties, we can show that intermediate states exist in the Cr2 O3 barrier separating CrO2 crystallites. Such defects have a profound influence on the tunnel magnetoresistance properties and may lead to coupling between ferromagnetic layers at higher temperatures. Cr interstitial defects in Cr2 O3 are known from other work [113].

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Polarization of defects within the barrier layer could be one possible mechanism for coupling. The fact that both Cr2 O3 and CoO are insulators does not alter the fact that these barrier layers will weakly polarize. This polarization, surprisingly, increases with increasing temperature. This review emphasizes the need for combining several experimental techniques to get understanding of these complicated magnetic oxide materials. It also shows that surface-sensitive techniques are ideal for characterizing tunnel junction systems, allowing insight into composition and electronic properties of interfaces buried under a few nm.

Acknowledgements The support of the Office of Naval Research, the NSF MRSEC (DMR 0213808), the Center for Materials Research and Analysis (CMRA), and the W.M. Keck Center for Mesospins and Quantum Information Systems are gratefully acknowledged. The authors would like to acknowledge a number of helpful conversations with E. Tsymbal.

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