THE HALL EFFECT IN METALS AND ALLOYS THE INTERNATIONAL CRYOGENICS MONOGRAPH SERIES General Editors Dr. K. Mendelssohn, F. R. S. The Clarendon Laboratory Oxford, England Dr. K. D. Timmerhaus University of Colorado Boulder, Colorado H. J. Goldsmid Thermoelectric Refrigeration, 1964 G. T.Meaden Electrical Resistance of Metals, 1965 E. S. R. Gopal Specific Heats at Low Temperatures, 1966 M. G. Zabetakis Safety with Cryogenic Fluids, 1967 D. H. Parkinson and B. E. Mulhall The Generation of High Magnetic Fields, 1967 W. E.Keller Helium-3 and Helium-4, 1969 A. J. Croft Cryogenic Laboratory Equipment, 1970 A. U. Smith Current Trends in Cryobiology, 1970 C.A. Bailey Advanced Cryogenics, 1971 D. A. Wigley Mechanical Properties of Materials at Low Temperatures, 1971 C.M.Hurd The Hall Effect in Metals and Alloys, 1972 THE HALL EFFECT IN METALS AND ALLOYS Colin M. Hurd Division of Chemistry National Research Council of Canada Ottawa, Ontario, Canada c:t? PLENUM PRESS· NEW YORK-LONDON· 1972 Library of Congress Catalog Card Number 76-157936 ISBN 978-1-4757-0467-9 ISBN 978-1-4757-0465-5 (eBook) DOI 10.1007/978-1-4757-0465-5 © 1972 Plenum Press, New York Softcover reprint of the hardcover 1st edition 1972 A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N.Y. 10011 United Kingdom edition published by Plenum Press, London A Division of Plenum Publishing Company, Ltd. Davis House (4th Floor), 8 Scrubs Lane, Harlesden, NWI0 6SE, London, England All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher To Elvira and Ph. Preface I hope this book will be useful to at least two groups of individuals: the nonspecialist reader with a general knowledge of solid-state science and seeking an introduction to the theory and practice of the Hall effect in metals, and the specialist seeking a contemporary review of the relevant literature. The literature has been surveyed thoroughly up to the middle of 1970, while the more accessible journals have been followed to late 1970. I have been selective in cases where there is a great volume of literature, particu larly in the case of old or obscure measurements of low accuracy, but in all cases I have tried to present the reader with sufficient information to judge whether a particular reference matches his interest and is therefore worth tracing. I compiled the book from reading the original publications, but inevitably there will be errors arising in transcription or inadvertent omissions. I hope the reader finding these will be charitable enough to write to me. lt is a pleasure to acknowledge the numerous useful discussions I have had at various times with associates and colleagues, particularly Drs. Mme M. T. Beal-Monod, J. E. A. Alderson, R. D. Barnard, T. Farrell, and P. Monod. Their influence appears at various points in the text-although, of course, they must not be held responsible for anything I have written. I also wish to acknowledge the help I have received from many people in various departments of the National Research Council of Canada, particularly Miss M. Green and Mrs. I. Delaney of the National Science Library, Mrs. A. Lapp, Mrs. H. S. Cuccaro, and their staff of the Steno Pool, and Mr. J. Novak who prepared most of the illustrations. Finally, it is a pleasure to acknowledge the indulgent cooperation of my colleagues Dr. J. E. A. Alderson and Mr. F. Turner during the preparation of this book, and the National Research Council of Canada for its continuous support. C. M. HURD Division of Chemistry National Research Council of Canada Ottawa KIA OR9 vii Acknowledgments The author acknowledges the permission granted by the following publishers and societies to make use of figures that have appeared in their pUblications: Masson et Cie (La surface de Fermi des metaux), Pergamon Press, Ltd. (Journal of Physics and Chemistry of Solids), American Institute of Physics (Physical Review), Taylor and Francis, Ltd. (Philosophical Magazine), and the National Research Council of Canada (Canadian Journal of Physics). Ix Contents Introduction ........ ........................................ . Chapter 1 The Dynamics of Electrons in Metals: Low-Temperature Effects .. 5 1.1. Theoretical Conditions ................................ 5 1.2. Definition of the Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3. Semiclassical Electronic Motion with Electric and Magnetic Fields ............................................... 10 1.3.1. Compensation in Metals .. .. .. .. .. .. .. .. .. .... .. 17 1.4. Quantum Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . .2 1. . . . . . . . . . 1.4.1. Quantum Oscillations . . . . . . . . . . . . . . . . . . . . 2.4 . . . . . . 1.4.2. Magnetic Breakdown ........................... 28 1.5. Size Effects .......................................... 32 1.5.1. Classical Size Effects.... .. .. ... . .. .. .. .. .. .. .. .. 33 1.5.2. Quantum Size Effects ...................... .... 44 Chapter 2 The Hall Effect in Limiting Cases . . . . . . . . . . . . . . . . . . . . . . .5 1 . . . . . . 2.1. The Hall Effect in the Classical High-Field Limit ....... 51 2.1.1. The Hall Effect with No Open Orbits. . . . . . . . . . . . 55 2.1.2. The Hall Effect with Open Orbits ............... 62 2.i.3. Chambers' Expression for the Conductivity Tensor 66 2.2. The Hall Effect in the Low-Field Limit ............... . 69 2.2.1. The Case with Localized Magnetic Moments ..... 78 2.2.2. The Low-Field Spin Component ................ 81 2.3. The Hall Effect in the Two-Band Model 87 xi xii Contents Chapter 3 The Hall Effect in Nearly-Free-Electron Metals. . . . . . . . . . . . . . . . . 93 3.1. Introduction.......................................... 93 3.2. The Harrison Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3. The Examples of Al and In ... . . . . . . . . . . . . . . . . . . . . . . . . 97 3.3.1. Case of the High-Field Limit. . . . . . . . . . .. . . . . . . . . 98 3.3.2. Case of the Low-Field Limit............. . . . . . .. 100 3.3.3. Case of the Intermediate-Field Region ........... 102 Chapter 4 The Hall Effect in Group 1B Metals 113 4.1. Introduction.......................................... 113 4.2. The Hall Effect in the Pure Group 1B Metals........... 118 4.2.1. The Dependence of R on Magnetic Field ........ 120 4.2.2. The Dependence of R on Temperature .......... 123 4.3. The Hall Effect in Very Dilute Alloys. . . . . . . . . . . . . . . . . . 137 4.4. The Hall Effect in Alloys with B-Metal Solutes.. . . . . . . . . 140 Chapter 5 The Hall Effect in Magnetic Metals. . . . . . . . . . . .. . . . . . . . . . . . . .. 153 5.1. Phenomenological Aspects. . . . . . .. . . . . . . . . . . . . . .. . . . . . . 153 5.2. Some Conduction-Electron Spin Interactions ............ 162 5.3. Survey of the Theoretical Developments. . . . . . . . . . . . . . . . . 166 5.3.1. Theories with Itinerant Magnetic Carriers ........ 167 5.3.2. Theories with Localized Magnetic Carriers ....... 173 Chapter 6 Experimental Techniques and the Hall Effect in Unusual Conditions 183 6.1. Introduction.......................................... 183 6.2. Sources of Error ..................................... 184 6.3. Methods with Electrical Contact to the Sample ......... 189 6.3.1. dc Methods ................................... 189 6.3.2. ac Methods ................................... 192 Contents xiii 6.4. Methods without Contacts..... . . . . . . . . . . . . . . . . . . . . . . .. 195 6.5. Measurements under Unusual Conditions. . . . . . . . . . . . . . .. 200 6.5.1. Measurements at High Pressures ................ 200 6.5.2. Measurements on Liquid Metals ................ 201 6.5.3. The Hall Effect in Superconductors.... . . . . . . . . .. 207 Chapter 7 A Guide to the Literature for Metallic Elements and Binary Metallic Alloys .................................................... 219 Appendix A Derivation of Expression (2.59)................................ 373 Appendix B Units and Conversion Factors 375 Notes Added in Proof ....................................... 376 Author Index. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 377 Subject Index ............................................... 387