Table Of ContentCAMBRIDGE MONOGRAPHS ON PHYSICS
GENERAL EDITORS
M. BERRY
Professor of Theoretical Physics, University of Bristol
P. C. W. DAVIES
Professor of Theoretical Physics, University of Newcastle upon Tyne
C. J. ISHAM
Reader in Theoretical Physics, Imperial College, London
M. J. RYCROFT
Head, Atmospheric Sciences Division, British Antarctic Survey
Magnetic oscillations in metals
Some pioneers
L. V. Shubnikov W. J. de Haas P. van Alphen
(1901-1945) (1878-1960) M(1.9 06-1967)
R. E. Peierls (b. 1907) L. D. Landau (1908-1968)
L. Onsager (1903-1976) Lifshitz (1917-1982)
I. M.
Magnetic oscillations
in metals
D. SHOENBERG, F.R.S.
Emeritus Professor of Physics, Cambridge University
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CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE
LONDON NEW YORK NEW ROCHELLE
MELBOURNE SYDNEY
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/978052 l 878 l
l l
Cambridge University Press 1984
©
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 1984
This digitally printed version 2009
A catalogue record for this publication is available from the British Library
Library of Congress Catalogue Card Number: 82-19762
ISBN 978-0-521-22480-2 hardback
ISBN 978-0-521-11878-1 paperback
Contents
Preface
xi
List of symbols and abbreviations
xvn
1 Historical introduction 1
1.1. Early history 1
1.2. 1947-60: The rise of Fermiology 9
1.3. After 1960: Fermiology comes of age 17
1.4. Plan of the book 20
2 Theory 22
2.1. Preliminary qualitative treatment 22
2.2. Calculation of energy levels 25
2.2.1. Electron dynamics in a magnetic field 25
2.2.2. Quantization of the electron motion 32
2.3. Calculation of the free energy 36
2.3.1. Calculation at T 0 for a two-dimensional slab of
k-space 37 =
2.3.2. Magnetization and number of electrons for a two
dimensional slab of k-space 41
2.3.3. Calculation for constant number of electrons
(a digression) 44
2.3.4. Application to real 2-D systems (digression con-
tinued) 48
2.3.5. 2-D results for parabolic band 50
2.3.6. Integration over 53
K
2.3.7. Phase smearing 57
2.3.8. Discussion of the Lifshitz-Kosevich formula for
n
and 66
M
2.4. Oscillations of Fermi energy 67
2.5. Oscillations of density of states 69
2.6. Many-body interactions 73
2.6.1. The self energy concept 73
2.6.2. Summary of §2.6 81
v
Contents
3 Observation of the de Haas-van Alphen effect 83
3.1. Introduction 83
3.2. Orders of magnitude of de Haas-van Alphen ampli
tudes 83
3.3. Static methods 86
3.3.1. Faraday-Curie method 86
3.3.2. Torque method 87
3.3.3. Foner method 95
3.3.4. Miscellaneous static methods 96
3.4. Dynamic methods 96
3.4.1. Pulsed field technique 96
3.4.2. Modulation methods 102
3.5. Dependence of frequency on stress 129
4 Other oscillatory effects 133
4.1. Introduction 133
4.2. Oscillatory thermal properties 133
4.2.1. Magnetothermal oscillations 134
4.2.2. Oscillations in specific heat 138
4.3. Oscillatory mechanical effects 140
4.3.1. Oscillatory magnetostriction 140
4.3.2. Oscillatory elastic properties and velocity of
sound 144
4.3.3. Extraction of strain dependence of the Fermi surface
from oscillatory data 146
4.4. Oscillations of the Fermi energy 150
4.5. Shubnikov-de Haas effect 153
4.5.1. Experimental methods and results 155
4.6. Oscillations in ultrasonic attenuation; giant quantum oscil
lations 160
4.6.1. Conditions for occurrence of the giant quantum
oscillations 161
4.6.2. Field dependence of the oscillations 164
4.7. Oscillations of other physical properties 173
4. 7.1. Optical properties 173
4.7.2. Miscellaneous transport properties 174
4.7.3. The Knight shift in nuclear magnetic resonance 174
4.8. Magnetic oscillations of other kinds 174
4.8.1. Azbel-Kaner cyclotron resonance 174
4.8.2. Magnetoacoustic effect ('geometric resonance') 175
4.8.3. The r.f. size effect (Gantmakher effect) 175
vi
Contents
4.8.4. The d.c. size effect (Sondheimer effect) 175
4.8.5. Magnetic surface states 176
4.8.6. Magneto-phonon effect 176
5 Fermi surfaces and cyclotron masses 178
5.1. Introduction 178
5.2. General principles 179
5.2.1. Determination of a Fermi surface from de Haas-van
Alphen frequencies 179
5.2.2. Determination of differential properties from cyclotron
masses 183
5.3. Results for some metals 185
5.3.1. The alkalis 185
5.3.2. The noble metals 196
5.3.3. 'Simple' polyvalent metals 210
5.3.4. Transition metals 221
5.3.5. Bismuth 228
5.3.6. Compounds 234
5.4. Strain dependence of the Fermi surface 237
5.4.1. Alkali metals 237
5.4.2. Simple polyvalent metals 239
5.4.3. The noble and transition metals 240
5.4.4. Ferromagnetics 244
5.5. Temperature dependence of the Fermi surface 245
5.6. Modification of Fermi surface by alloying 247
6 Magnetic interaction 254
6.1. Introduction 254
6.2. Justification of replacing by B in the formula for M
H
255
6.3. Theory of MI for a single frequency in absence of anisotropy
and shape effects 257
6.4. Crystal and shape anisotropy effects 263
6.4.1. Crystal anisotropy 264
6.4.2. Effect of shape 266
6.5. Qualitative verification of theory for a single fre
quency 277
6.6. Phase smearing in MI conditions 283
6.7. Magnetic interaction for several de Haas-van Alphen fre
quencies 290
6.7.1. General formulation for MI with two frequencies 290
6.7.2. MI of a single frequency and its harmonics 311
vii
Contents
6.8. MI in oscillations of other thermodynamic properties 312
6.8.1. MI in the magnetothermal effect 313
6.8.2. MI in oscillations of magnetostriction and elastic
constants 318
6.9. Other oscillations in MI conditions 322
6.10. How to avoid MI (within limits) 327
6.11. The LOFER state; MI in the stars? 327
7 Magnetic breakdown 331
7.1. Introduction 331
7.2. The probability of MB and the Blount criterion 333
7.3. The coupled network 337
7.4. Comparison of MB and MI 352
7.5. Galvanomagnetic effects and MB 353
7.5.1. One-dimensional network 354
7.5.2. Two-dimensional network with hexagonal sym
metry 359
7.5.3. The Stark quantum interferometer 364
8 The Dingle temperature 369
8.1. Introduction 369
8.2. Measurement of Dingle temperatures 371
8.3. Dilute alloys 374
8.4. Dislocations 386
r « d
8.4.1. Small orbits: 389
d
8.4.2. Large orbits:r 392
»
8.4.3. More detailed calculations 393
8.4.4. Some reservations 396
8.4.5. Experimental evidence 398
8.5. Mosaic structure 411
8.5.1. Orientation with F extremal 412
8.5.2. Orientation off extremal 415
8.5.3. A limit to the reduction factor? 417
8.5.4. Polycrystal as extreme example of mosaic struc
ture 418
8.6. Scattering of electrons by phonons 422
Phase and spin-splitting 425
9
9.1. Introduction 425
9.2. Ambiguity of g-factor determination 426
9.3. Phase determination: methods 429
9.4. Phase determination: some results 435
Vlll
Contents
9.5. Determination of the spin-splitting (g) factor 439
9.5.1. Direct observation of spin-splitting 439
9.5.2. g-factor from amplitudes of fundamental and
harmonics 444
9.6. Discussion of g value results 452
9.6.1. Spin-orbit splitting 453
9.6.2. Many-body interactions 456
9.7. Anomalous situations 462
Appendices 468
1. Ellipsoidal surfaces of constant energy 468
Al.1. Parabolic band 468
Al.2. Non-parabolic band 474
Al.3. The Lax model 475
2. Note on thermodynamics, illustrated by free electron model
example 477
3. Calculation by the Poisson summation formula 481
4. The steady susceptibility 483
A4.l. The spin susceptibility 483
A4.2. The steady diamagnetic susceptibility 486
A4.3. Steady susceptibility in the Lax model 490
5. Cornu spiral and related topics 493
A5.l. The Cornu spiral 493
A5.2. Nearly cylindrical FS 496
6. Electron-phonon interaction 500
A6. l. Properties of the self energy function 500
A6.2. The integral in (2.184) 503
7. Numerical estimates of IM/HI and ldM/dHI 507
A 7.1. Estimates for a 3-D system 507
A7.2. Estimates for a 2-D system 510
8. Calibration of field modulation system for finite sample 512
9. Magnitude of magnetothermal oscillations 513
10. Oscillations of velocity of sound 518
11. Damping of giant quantum oscillations by electron
scattering 521
12. Potential energy of array of strips of alternating charge 525
13. Field modulation when both HF and LF present 527
14. Conductivity for two-dimensional network 530
A14.1. Switching probabilities 530
A14.2. Terminal point of electron path 532
A14.3. Calculation of conductivity 532
ix
Contents
15. Improvement of Dingle plot procedure 537
16. Strain field of dislocations and estimate of din F/ds 540
17. Statistical analysis of some phase addition problems 544
Al7 .1. Addition of phases of equal magnitude, but of ran
dom sign 544
Al 7.2. Superposition of N oscillations with statistically distri
buted phases 545
Al 7.3. Variance of the reduction factor 547
18. Long beat in amplitude 549
19. Ambiguities in the HR method 551
Bibliography and author index
553
Notes added in proof 566
Subject index 568
x
