MODERN PROBLEMS IN CONDENSED MATTER SCIENCES Volume 27.1 V.M. AGRANOVICH Moscow, USSR A.A. M A R A D U D IN Irvine, California, USA Advisory editorial board F. Abeles, Paris, France F. Bassani, Pisa, Italy N. Bloembergen, Cambridge, MA, USA E. Burstein, Philadelphia, PA, USA I.L. Fabelinskii, Moscow, USSR P. Fulde, Stuttgart, FRG M.D. Galanin, Moscow, USSR V.L. Ginzburg, Moscow, USSR H. Haken, Stuttgart, FRG R.M. Hochstrasser, Philadelphia, PA, USA LP. Ipatova, Leningrad, USSR A.A. Kaplyanskii, Leningrad, USSR L.V. Keldysh, Moscow, USSR R. Kubo, Tokyo, Japan R. Loudon, Colchester, UK Yu.A. Ossipyan, Moscow, USSR L.P. Pitaevskii, Moscow, USSR A.M. Prokhorov, Moscow, USSR K.K. Rebane, Tallinn, USSR J.M. Rowell, Red Bank, NJ, USA NORTH-HOLLAND AMSTERDAM · OXFORD · NEW YORK · TOKYO LANDAU LEVEL SPECTROSCOPY Volume editors G. L A N D W E HR Wiirzburg, Germany E.I. R A S H BA Moscow, USSR 1991 NORTH-HOLLAND AMSTERDAM · OXFORD · NEW YORK · TOKYO © Elsevier Science Publishers Β .V., 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the Publisher, Elsevier Science Publishers B.V., P.O. Box 211, 1000 AE Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. ISBN 0 444 88874 8 (Set) 0 444 88535 8 (Vol. 27.1) 0 444 88873 X (Vol. 27.2) North-Holland Elsevier Science Publishers B.V. P.O. Box 211 1000 AE Amsterdam The Netherlands Sole distributors for the USA and Canada: Elsevier Science Publishing Company, Inc. 655 Avenue of the Americas New York, NY 10010 USA Printed on acid free paper MODERN PROBLEMS IN CONDENSED MATTER SCIENCES Vol. 1. SURFACE POLARITONS V.M. Agranovich and D.L. Mills, editors Vol. 2. EXCITONS E.I. Rashba and M.D. Sturge, editors Vol. 3. ELECTRONIC EXCITATION ENERGY TRANSFER IN CONDENSED MATTER V.M. Agranovich and M.D. Galanin Vol. 4. SPECTROSCOPY AND EXCITATION DYNAMICS OF CONDENSED MOLECULAR SYSTEMS V.M. Agranovich and R.M. Hochstrasser, editors Vol. 5. LIGHT SCATTERING NEAR PHASE TRANSITIONS H.Z. Cummins and A.P. Levanyuk, editors Vol. 6. ELECTRON-HOLE DROPLETS IN SEMICONDUCTORS CD. Jeffries and L.V. Keldysh, editors Vol. 7. THE DYNAMICAL JAHN-TELLER EFFECT IN LOCALIZED SYSTEMS Yu.E. Perlin and M. Wagner, editors Vol. 8. OPTICAL ORIENTATION F. Meier and B.P. Zakharchenya, editors Vol. 9. SURFACE EXCITATIONS V.M. Agranovich and R. Loudon, editors Vol. 10. ELECTRON-ELECTRON INTERACTIONS IN DISORDERED SYSTEMS A.L. Efros and M. Pollak, editors Vol. 11. MEDIUM-ENERGY ION REFLECTION FROM SOLIDS E.S. Mashkova and V.A. Molchanov Vol. 12. NONEQUILIBRIUM SUPERCONDUCTIVITY D.N. Langenberg and A.I. Larkin, editors MODERN PROBLEMS IN CONDENSED MATTER SCIENCES Vol. 13. PHYSICS OF RADIATION EFFECTS IN CRYSTALS R.A. Johnson and A.N. Orlov, editors Vol. 14. INCOMMENSURATE PHASES IN DIELECTRICS (Two volumes) R. Blinc and A.P. Levanyuk, editors Vol. 15. UNITARY TRANSFORMATIONS IN SOLID STATE PHYSICS M. Wagner Vol. 16. NONEQUILIBRIUM PHONONS IN NONMETALLIC CRYSTALS W. Eisenmenger and A.A. Kaplyanskii, editors Vol. 17. SOLITONS S.E. Trullinger, V.L. Pokrovskii and V.E. Zakharov, editors Vol. 18. TRANSPORT IN PHONON SYSTEMS V.L. Gurevich Vol. 19. CARRIER SCATTERING IN METALS AND SEMICONDUCTORS V.F. Gantmakher and LB. Levinson Vol. 20. SEMIMETALS - 1. GRAPHITE AND ITS COMPOUNDS N.B. Brandt, S.M. Chudinov and Ya.G. Ponomarev Vol. 21. SPECTROSCOPY OF SOLIDS CONTAINING RARE EARTH IONS A.A. Kaplyanskii and R.M. Macfarlane, editors Vol. 22. SPIN WAVES AND MAGNETIC EXCITATIONS (Two volumes) A.S. Borovik-Romanov and S.K. Sinha, editors Vol. 23. OPTICAL PROPERTIES OF MIXED CRYSTALS R.J. Elliott and LP. Ipatova, editors MODERN PROBLEMS IN CONDENSED MATTER SCIENCES Vol. 24. THE DIELECTRIC FUNCTION OF CONDENSED SYSTEMS L.V. Keldysh, D.A. Kirzhnitz and A.A. Maradudin, editors Vol. 25. CHARGE DENSITY WAVES IN SOLIDS L.P. Gor'kov and G. Gruner, editors Vol. 26. HELIUM THREE W.P. Halperin and L.P. Pitaevskii, editors Vol. 27. LANDAU LEVEL SPECTROSCOPY G. Landwehr and E.I. Rashba, editors In preparation HOPPING TRANSPORT IN SOLIDS B. Shklovskii and M. Pollak, editors NONLINEAR SURFACE ELECTROMAGNETIC PHENOMENA G. Stegeman and H.E. Ponath, editors MESOSCOPIC PHENOMENA IN SOLIDS B.L. Altshuler, R. Webb and P.A. Lee, editors ELECTRONIC PHASE TRANSITIONS W. Hanke and Yu. Kopaev, editors ELASTIC STRAIN FIELDS AND DISLOCATION MOBILITY V.L. Indenbom and J. Lothe, editors Oh, how many of them there are in the fields! But each flowers in its own way - In this is the highest achievement of a flower! Matsuo Β as ho 1644-1694 P R E F A CE TO T HE S E R I ES Our understanding of condensed matter is developing rapidly at the present time, and the numerous new insights gained in this field define to a significant degree the face of contemporary science. Furthermore, discoveries made in this area are shaping present and future technology. This being so, it is clear that the most important results and directions for future developments can only be covered by an international group of authors working in cooperation. "Modern Problems in Condensed Matter Sciences" is a series of con­ tributed volumes and monographs on condensed matter science that is pub­ lished by North-Holland Physics Publishing, a division of Elsevier Science Publishers. With the support of a distinguished Advisory Editorial Board, areas of current interest that have reached a maturity to be reviewed, are selected for the series. Both Soviet and Western scholars are contributing to the series, and each contributed volume has, accordingly, two editors. Monographs, written by either Western or Soviet authors, are also included. The complete series will provide the most comprehensive coverage available of condensed matter science. Another important outcome of the foundation of this series is the emer­ gence of a rather interesting and fruitful form of collaboration among scholars from different countries. We are deeply convinced that such international collaboration in the spheres of science and art, as well as other socially useful spheres of human activity, will assist in the establishment of a climate of confidence and peace. The publishing house "Nauka" publishes the volumes in the Russian lan­ guage. This way the broadest possible readership is ensured. The General Editors of the Series, V.M. Agranovich A.A. Maradudin IX L. D. Landau(1908-1968) Introduction by G. Landwehr and E.L Rashba After quantum mechanics was developed in the mid-twenties, the new theoret­ ical concept was subsequently applied to problems of solid state physics. Felix Bloch was the first to address the problem of free electrons in a periodic potential in 1928. Two years later L.D. Landau, a young theoretician from Leningrad, worked out in Cambridge the quantum theory of diamagnetism in metals (Landau 1930). Up to then it had been tacitly assumed that the magnetic properties of electrons in metals were determined by their spin and by the electron binding in atoms. According to a theorem by Bohr and van Leeuwen, based on classical physics, it was argued that free electrons did not contribute to the susceptibility because a magnetic field did not change the velocity and consequently the energy of the electrons. Landau showed that this approach was inadequate. By solving the Schrodinger equation, incorporating a magnetic field by a vector potential in a gauge which we now call the Landau gauge, he showed that the motion of the electron perpendicular to the magnetic field is quantized. The quantization is coupled with a change in the density of states and results in a non-zero diamagnetic susceptibility. Landau performed the calculation for a range of magnetic fields and temperatures in which the difference in energy between two subsequent magnetic sub-bands hw is small compared with the thermal energy /cT, with ω = eB/m (h = Planck's constant/271, ω = cyclotron frequency, Β = magnetic field, m = electron mass). This condition is not satisfied at low temperatures and high magnetic fields. Landau noted that under these circumstances no linear dependence of the magnetic moment on Β could be expected and that a strong periodic variation in Β should occur. He concluded that it should hardly be possible to observe the periodic effects experimentally, because they would be XI xii G. Landwehr and E.I. Rashba averaged out due to inhomogeneities of the magnetic field. This estimate was based on the electron rest mass. Referring to Bloch's theoretical work (Bloch 1928) Landau noted that his calculations should, in principle, also be valid for crystal electrons although the quantitative application of the results should not be possible. At the same time at the University of Leiden, the susceptibility and the magnetoresistance of bismuth single crystals were investigated at temperatures between 14 and 20 K. De Haas and van Alphen studied the susceptibilities in magnetic fields up to 15 kG and found an oscillatory behaviour in the high-field range (De Haas and van Alphen 1930). In the same year, Shubnikov and de Haas found an oscillatory magnetoresistance (Shubnikov and de Haas 1930). The new experimental findings showed that Landau's estimates about the observability of the periodic variations in the susceptibility had been too pessimistic. It was Rudolf Peierls, then working with a Rockefeller fellowship as a guest of Enrico Fermi in Rome, who made the first detailed calculations of the oscillatory susceptibility, which we now call the de Haas-van Alphen effect (Peierls 1933). Due to the quantum effects in the regime hco > kT (high magnetic fields, low temperatures) it is not possible to extend the rather general method used by Landau for the calculations into the low-field range. Therefore, a model calculation was performed assuming that the number of electrons was so small that only the lowest quantized energy bands were occupied. In order to empha­ size the essential features of the quantization caused by a magnetic field, Peierls discussed a two-dimensional model for Τ = 0, which he considered as physically meaningless, but instructive. It is interesting to note that nowadays we have been able to realize semiconductor heterostructures, which really behave like two-dimensional systems. Peierls showed that oscillatory behaviour of the susceptibility, which is periodic in l/B, is expected not only in two dimensions, but also for three- dimensional systems. In order to allow a comparison with the experiments by de Haas and van Alphen, he performed rather tedious calculations for finite temperatures. It turned out that there was qualitative agreement between theory and experiment. Peierls recognized that bismuth has a particular band structure with a very small but anisotropic effective mass. The carrier concentration he estimated was about two orders of magnitude too small. However, in sub­ sequent years the band structure of bismuth was studied in some detail and reasonable agreement between theory and experiment was obtained (Mott and Jones 1936). Further work by Shoenberg (1939) on the de Haas-van Alphen effect revealed a wealth of detailed information about the electronic band structure of bismuth close to the conduction-band edges. The experiments by Shoenberg were very successful, because they were performed at liquid-helium temperatures, whereas the original measurements had been done with liquid hydrogen. Whereas a theory was at hand for the interpretation of the oscillatory