PROGRESS 1N LOW TEMPERATURE PHYSICS
VII B
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CONTENTS O F VOLUMES I-VI
VOLUME I C.J. Gorter, The two fluid model for superconductorsand helium I1 (16 pages) R.P. Feynman, Application of quantum mechanics to liquid helium (37 pages) J.R. Pellam, Rayleigh disks in liquid helium I1 (10 pages) A.C. Hollis-Hallet, Oscillating disks and rotating cylinders in liquid helium I1 (14 pages)
E.F. Hammel, The low temperature properties of helium three (30 pages) J.J.M. Beenakker and K.W. Taconis, Liquid mixtures of helium three and four (30 pages)
B. Serin, The magnetic threshold curve of superconductors(13 pages) C.F. Squire, The effect of pressure and of stress on superconductivity (8 pages) T.E. Faber and A.B. Pippard, Kinetics of the phase transition in superconductors(25 pages) K. Mendelssohn, Heat conduction in superconductors(18 pages) J.G. Daunt, The electrodc specific heats in metals (22 pages) A.H. Cooke, Paramagnetic crystals in use for low temperature research (21 pages) N.J. Poulis and C.J. Gorter, Antiferromagnetic crystals (28 pages) D. de Klerk and M.J. Steedand, Adiabatic demagnetization (63 pages) L. Nkl, Theoretical remarks on ferromagnetism at low temperatures (8 pages) L. Weil, Experimental research on ferromagnetism at very low temperatures (11 pages) A. Van Itterbeek, Velocity and attenuation of sound at low temperatures (26 pages) J. de Boer, Transport properties of gaseous helium at low temperatures (26 pages)
VOLUME I1 J. de Boer, Quantum effects and exchange effects on the thermodynamic properties of liquid helium (58 pages)
H.C. Kramers, Liquid helium below 1 OK (24 pages) P. Winkel and D.H.N. Wansink, Transport phenomena of liquid helium I1 in slits and capillaries (22 pages)
K.R Atkins, Helium films (33 pages) B.T. Matthias, Superconductivity in the periodic system (13 pages)
CONTENTS OF VOLUMES I-VI
VOLUME I1 (continued)
E.H. Sondheimer, Electron transport phenomena in metals (36 pages) V.A. Johnson and K. Lark-Horovitz, Semiconductors at low temperatures (39 pages) D. Shoenberg, The De Haas-van Alphen effect (40pages) C.J. Gorter, Paramagnetic relaxation (26 pages)
M.J. Steenland and H.A. Tolhoek, Orientation of atomic nuclei at low temperatures (46 pages) C. Domb and J.S. Dugdale, Solid helium (30 pages) F.H. Spedding, S. kgvold, A.H. Daane and L.D. Jennings, Some physical properties of the rare earth metals (27 pages) D. Bijl, The representation of specific heat and thermal expansion data of simple solids (36 pages)
H. van Dijk and M. Durieux, The temperature scale in the liquid helium region (34pages)
VOLUME 111
W.F. Vinen, Vortex lines in liquid helium I1 (57 pages)
G. Carerj, Helium ions in liquid helium I1 (22 pages) M.J. Buckingham and W.M. Fairbank, The nature of the A-transition in liquid helium (33 pages) E.R. Grilly and E.F. Hammel, Liquid and solid 'He (40 pages)
K.W.Taconis, 'He cryostats (17 pages) J. Bardeen and J.R. Schrieffer, Recent developments in superconductivity (1 18 pages)
M.Ya. Azbel' and I.M. Lifshitz, Electron resonances in metals (45 pages)
W.J.Huiskamp and H.A. Tolhoek, Orientatien of atomic nuclei at low temperatures I1 (63 pages) N. Bloembergen, Solid state masers (34 pages)
J.J.M. Beenakker, The equation of state and the transport properties of the hydrogenic molecules (24pages) Z. Dokoupil, Some solid-gas equilibria at low temperature (27 pages)
CONTENTS O F VOLUMES I-VI
VOLUME N
V.P. Peshkov, Critical velocities and vortices in superfluid helium (37 pages) K.W. Taconis and R. de Bruyn Ouboter, Equilibrium properties of liquid and solid mixtures of helium three and four (59 pages) D.H. Douglas Jr. and L.M. Falikov, The superconductingenergy gap (97 pages) G.J. van den Berg, Anomalies in dilute metallic solutions of transition elements (71 pages) Kei Yosida, Magnetic structures of heavy rare earth metals (31 pages) C. Domb and A.R. Miedema, Magnetic transitions (48 pages)
L. Nkl, R. Pauthenet and B. Dreyfus, The rare earth garnets (40pages) A. Abragam and M. Borghini, Dynamic polarization of nuclear targets (66 pages) J.G. Collins and G.K. White, Thermal expansion of solids (30 pages) T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. Brickwedde, The 1962 jHe scale of temperatures (35 pages)
VOLUME
v
P.W. Anderson, The Josephson effect and quantum coherence measurements in superconductors and superfluids (43 pages) R. de Bruyn Ouboter, K.W. Taconis and W.M. van Alphen, Dissipativeand non-dissipative flow phenomena in superfluid helium (35 pages) E.L. Andronikashvili and Yu.G. Mamaladze, Rotation of helium I1 (82 pages) D. Gribier, B. Jacrot, L Madhavrao and B. Farnoux, Study of the superconductive mixed state by neutron-diffraction (20 pages)
V.F. Gantmakher, Radiofrequency size effects in metals (54 pages) R.W. Stark and L.M. Falicov, Magnetic breakdown in metals (52 pages) J.J. Beenakker and H.F.P. Knapp, Thermodynamic properties of fluid mixtures (36 pages)
CONTENTS OF VOLUMES I-VI
VOLUME VI
J.S. Langer and J.D. Reppy, Intrinsic critical velocities in superhid helium (35 pages) K.R. Atkins and I. Rudnick, Thud sound (40 pages) J.C. Wheatley, Experimental properties of pure He3 and dilute solutions of He3 in superfluid He4 at very low temperatures. Application to dilution refrigeration (85 pages) R.I. Boughton, J.L. Olsen and C. Palmy, Pressure effects in superconductors (41 pages) J.K. H u h , M. Ashkin, D.W. Deis and C.K. Jones, Superconductivity in semiconductors and semi-metals (38 pages) R. de Bruyn Ouboter and A.Th.A.M. de Waele, Superconducting point contacts weakly connecting two superconductors (48 pages) R.E. Glover, 111, Superconductivity above the transition temperature (42 pages) R.F. Wielinga, Critical behaviour in magnetic crystals (41 pages) G.R. Khutsishvili, Diffusion and relaxation of nuclear spins in crystals containing paramagnetic impurities (30 pages) M. Durieux, The international practical temperature scale of 1968 (21 pages)
P R O G R E S S IN LOW TEMPERATURE PHYSICS EDITED BY
D.F. B R E W E R Professor of Experimental Physics, Director of the Physics Laboratory, University of Sussex, Brighton
VOLUME VII B
1978 NORTH-HOLLAND PUBLISHING COMPANY NEWYORK OXFORD AMSTERDAM
0 North-Holland Publishing Company - 1978 AN 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 prior permission of the copyright owner
ISBN: Vol VIIB 0 444 85209 3 ISBN: SetNo. 0 444 85210 7
PUBLISIERS :
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM * NEW YORK * OXFORD SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
ELSEVIER NORTH-HOLLAND INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
Library of Congress Catalog Card Number 55-14533
Printed in Great Britain by Bell and Bain Ltd., Glasgow
PREFACE
The first six volumes in this series were published over the period 1957 to 1970 at the rate of about one every three years. There has now been a gap of eight years between volumes VI and VII, presenting certain problems of choice of material, particularly in deciding the claims of the currently most interesting research, the topics which have developed most significantly since 1970, and the balance of the volume as a whole. I have felt that one of the areas of greatest and growing interest - and not only in low temperature physics - is the behaviour of finite systems with limited dimensionality, to which is also attached the properties of surfaces. Hence the general articles by Kosterlitz and Thouless on two dimensional physics, and the more specific topics of “moderately small” superconductors by Fink, Bibby and McLachlan, quasi-one-dimensional superconductors by Gor’kov, and the surface of liquid helium by Edwards and Saam. The most exciting development of course, since 1970 has been the discovery of the new phases of liquid 3He in 1971, now known to be superfluid, and I have yielded to the temptation to give this what may be regarded as excessive attention in the three chapters by Wheatley, by Brinkman and Cross, and by Wolfle. To counteract the tendency towards concentration on superfluids, the articles by Flouquet and by Griiner and Zawadowski describe progress in the area of magnetism and nuclear orientation that has commanded interest for a long time and continues to develop. It is interesting to compare the present volume with earlier ones in the series. The first few volumes contained about 30 per cent each of liquid 4He, superconductivity and magnetism, the remaining 10 per cent being concerned with other topics such as molecular physics, thermometry and semiconductors. (See the preface to volume VI.) Volume I (1957) contained 18 chapters of average length 23 pages with a shortest paper of 8 pages and a longest of 63. Volume IV (1964) had declined to 10 chapters, but of average length 51 pages (shortest 30 pages, longest 97). Following changes were less marked. Later articles became less general, more specific ix
X
PREFACE
and, as noted in the preface to volume V, had relatively less text, about the same number of figures and tables, and more formulae and references. The tendency towards longer articles has continued in the present volume, which has 9 chapters averaging 81 pages within a range of 54 to 103 pages. The authors’ enthusiasm has necessitated dividing volume VII into two parts, A and By thereby somewhat impairing the balance within each but making them more manageable in size. No doubt the tendency to write more and more about narrower topics reflects the natural development of any science, regrettable as it may be. It becomes more difficult, especially for experimentalists, to be thoroughly au fait with detailed developments in particular areas of, say, both superconductivity and liquid 4He - much less superfluid 3He despite its bond of BCS theory with superconductivity. Nevertheless, ground of common and fertile interest does exist for low temperature physicists, and for this reason the “LT” series of conferences (now triennial instead of biennial as before 1975) continues to be scientifically profitable. It is also the reason for the publishers’ decision to revive this series of books after a long gap. To assume responsibility for Progress in Low Temperature Physics from the distinguished editorship of C. J. Gorter is a somewhat formidable task. I shall welcome comments on the structure of this volume as well as suggestions for any future volumes that may appear. I am grateful to Dr. and Mrs. W. S. Truscott for assembling the name and subject indexes, and to the publishers for their help in various ways. Sussex 1978
D.F. Brewer
CONTENTS VOLUME VIIA
Preface i x Contents xi Ch. 1. Further experimental properties of superfluid 3He, J.C. Wheatley
1
1. Experiments on orbitally related dynamic phenomena in superfluid 3He 3 2. Experiments on spin dynamics 34
3. Experiments of thermodynamic significance 75 4. Recent developments 96 References 101
Ch. 2. Spin and orbital dynamics of superfluid 3He, W.F. Brinkman and
M.C.Cross 105 Introduction 107 Energetics and superllow 110 Singularitiesand textures 134 Spindynamics 148 Orbitaldynamics 163 Appendix 184 References 188
1. 2. 3. 4. 5.
Ch. 3. Soundpropagation and kinetic coeficients in superfluid 'He, P. WGMe 191 Introduction 193 Normal Fermi liquid 195 Sound propagation in superfluid 3He 209 Kinetic coefficients of superfluid 3He 256 Conclusion 278 References 279 1. 2. 3. 4. 5.
Ch. 4. Thefree surface of liquid helium, D.O. 1. Introduction 285
Part 1. Experimental survey 286 2. Surface tension measurements 286 3. The surface tension of pure 3He and 4He 287 4. The surface tension of mixtures of 3He and *He 297 5. Capillary waves. Light scattering at the surface 308
xi
and W.F. S a m
283
xii
CONTENTS
6. Surface second sound. 3He surface currents 310 7. Evaporation and condensation 314 8. The scattering of free atoms at the surface 318 Part 11. Theoretical survey 324 9. The 4He ground state 324 10. 4He excited states 330 11. Theory for dilute solutions of 3He in 4He 347 12. Surface tension of 3He and dilute solutions of 4He in 3He 361 Appendix 363 Notes added in proof 362 References 365 Author index xv Subject index xxix VOLUME VII B
Preface ix Contents xi Ch. 5 . Two-dimensionalphysics, J.M. Kosterlitz and D.J. Thouless 311 1. Introduction 373 2. Examples of two-dimensional systems 378 3. Phase transitions in two dimensions 389 4. Metal-insulator transitions 420 References 429
Ch. 6. First and second order phase transitions of moderately small superconductors in a magnetic field, H.J. Fink, D.S. McLachlan and B. Rothberg Bibby 435 1. Introduction 437 2. Theoretical results 443 3. Experimental techniques 474 4. Second order phase transitions and the Landau critical point 479 5. First order transitions - superheating and supercooling 489 Appendix 511 List of symbols 512 References 51 3
Ch. 7. Properties of the A-15 compounds and one-dimensionality, L.P. Gor’kov 517 1. Short survey of the properties of the A-15 compounds. Some basic theoretical ideas 519 2. The choice of the electronic term for the linear chain. Instability of the spectrum 529 3. Interactions. Connection between structural and superconductive fluctuations in the linear chain. The role of the ‘three-dimensional effects’ 533 4. Peculiarities of the three-dimensional electron spectrum in the A-15 lattice due to the interchain tunnelling. Fine structure of the density of states 544
CONTENT3
xiii
5.Structure properties of the A-15 compounds in the simplest model. Phonon spectrum 552
6. Some results concerning the superconductivity of A-15 compounds 568 7. Discussion 579 8. Summary.Some concluding remarks 584 References 588
Ch. 8. Low temperature properties of Kondo alloys, G. Griiner and A. Zawadowski 591 1. Introduction 593 2. Basic models 595 3. The Kondo effect 604 4. Properties below the Kondo temperature 615 5. Phenomenological approaches to the general case 638 6. Conclusions 644 References 645
Cfi. 9. Application of low temperature nuclear orientation to metals with magnetic impurities, J. Flouquet 649 Introduction 651 1. Comments on nuclear orientation 652 2. Magnetism of an impurity: Kondo effect 669 3. The hyperfine field. The origin of magnetism 689 4. Single impurity effects 698 5. Interaction effects 713 6. Other applications 731 7. Conclusion 734 Appendix 737 References 741 Author index xv Subject index xxix
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AUTHOR INDEX TO VOLS. W - B * Abel, W. R., 732, 741 Abelh, B., 422,429 Abragarn, A., 655, 741 Abraham, B. M., 3,4,5,7,9,204,205,211, 102, 279
Abraham, E., 386,429,432 Abrikosov, A. A., 208, 535, 611, 614, 676, 112,279,588.645, 741 Adarns, E. D., 29,30,31,102 Adarns. G., 384,431 Adkins, C. J., 385. 424, 425, 429, 432 Ahonen, A. I.. 31, 33, 36, 37, 38, 39, 40, 64,65, 68, 69, 71, 12, 73, 74, 84, 85, 86, 88,89,142,383,101,188,429 Alexander, S., 419,429 Ali, M., 693, 108, 741 Allen, J. F., 287, 288, 296, 365 Allen, S. J., 385,424,429,433 Alloul, H., 631, 632, 637, 689, 709, 717, 735, 736, 645, 741, 744 Alquie, G., 694, 704, 105, 741 Als-Nielsen, J., 404. 407, 429 Alvesalo, T. A., 6, 7, 8, 9, 12, 19, 20, 80, 81, 82, 83, 94, 265, 267, 268, 101, 102, 279 Arnamou, A., 693, 741 Arnbegaokar, V., 50, 110, 112, 113, 130, 131, 158, 422, 429, 625, 101, 188, 429, 433,646 Ambler, E., 659, 746 Arnit, D., 328, 392, 365, 429 Anderson, A. C., 732, 741 Anderson, P. W., 14, 16, 20, 36, 39, 64, 80, 81, 112, 113, 114. 121, 122, 126, 127. 137, 139, 148, 168, 169, 170. 172, 176, 177, 181, 184, 195, 211, 224, 318, 324, 399, 423, 521, 561, 595, 613, 615, 619, 625, 626, 629, 673, 714, 101, 102, 103, 188, 189, 279, 365, 429, 588, 645, 647, 741, 742 Andrew, A. F., 285, 298, 310, 343, 347, 352, 356, 359, 365, 365
Andres, K., 318, 484, 664, 669, 731, 365, 513, 741, 742
Appleyard, E. T. S., 485, 513 Archie, C. N., 7, 9, 12, 18, 36, 80, 81, 82, 83, 84, 85, 87, 102
Arnold, E., 423,429 Arp, V., 442, 714, 726,515, 744 Ashkin, J., 408,429 Aslamazov, L. G., 386,429 Atkins, K. R., 285,287,288, 291, 293,296, 308, 314, 332, 334, 365, 366, 367, 368 Avenel, O., 63, 96, 143, 732, 101, 188, 741 Axe, J. D., 523,524,553,564,568,588,589
Babic, E., 645 Bagguley, D. M.S., 691, 694, 697, 741 Baldwin, J. P., 473,480,486, 488,513, 516 Balian, R., 159, 188 Balibar, S., 324, 366 Band, W. T., 4, 6, 9, 21, 28, 31, 92, 265, 267,268,102,280
Baratoff, A., 472,473,516 Barber, M. N., 411,430 Barclay, J. A., 698, 706, 741 Bardeen, J., 299,437,442,468,469,366,513 Barham, B., 692,728, 744 Barham, D. C., 688,736, 742 BarisiE, S., 577,588 Barnes, L. J., 441,513 Barrett, C . S., 521,522,588 Bartoff, A., 441, 457, 471, 476, 417, 491, 496,497,498,502,510,516 Barton, G., 78,101 Bassett, A., 643, 645 Batterman, B. W., 521, 522, 559. 562, 588, 589 Baxter, R. J., 419,429 Baym, G., 195,208,299,301,357,279,366 Beaglehole, D., 673,645 Real-Monod. M. T.,383, 732, 429, 744 Bean, C. P., 442,504,513 Beasley, M. R., 473,516
*Volume VII is published in two parts: page numbers 1-370 in the index refer to volume VII A and page numbers 371-746 refer to volume VII B.
xvi
AUTHOR INDEX
Bell, A. E., 645 Beloerskij, G. N., 693, 745 Beni, G., 384,432 Benoit, A., 665, 666, 668, 681, 687, 693. 698, 693, 708, 710, 711, 712, 725, 726, 731,734,736, 741, 745 Berezinskii, V. L.,381, 390, 392, 393. 407, 409,410,429 Berezniak,N. G., 287,298,367 Berglund, M. P.,732,741 Berlin, T. H., 390,430 Berman, H. S., 581,589 Bernas, B., 509,513 Bernier, M. E., 63,96,143,101, 188 Berthel, K. H., 578, 589 Berthier, C., 636,646 Berthold, J. E., 92, 93, 94, 382, 101, 430 Betbeder-Matibet, O., 182, 211, 188, 279 Betts, D. D., 390,430 Beun, J. A., 659,743 Bhatnager, A. K.,386,430 Bhatt, R. N., 588, 589 Bhattacharyya, P., 3, 6, 7, 8, 21, 28, 56, 63, 256, 267, 268, 273, 277, 101, 103, 279,280 Binder, K.,404,405,428, 430,433 Birgenau, R. J., 388, 404, 407, 523, 564, 429,430,589 Birks, A. R., 207,280 Birman, J. L.,557,589 Bishop, D. J., 382,429,433 Black, W.C., 732, 741 Blaha, S., 21, 139, 101, 188 Blair, D. G., 316,366 Blandin, A., 597, 601. 669, 670, 691, 698, 714,717,646,741,742 Blank, A. Ya., 392, 393,407, 429 Blatt, J. M., 186, 188 Blaugher, K. D., 586,589 Bleaney, B., 655, 741 Blin Stoyle, R. J., 652, 658, 741 Bloembergen, P.,389,406,430 Bloomfield. P.E., 612, 646 Blot, J., 438, 477, 480, 486, 489, 491, 509, 516 Blount, E. I., 56, 64,65, 71, 112, 113, 114, 141,143,521,561,101.188.588 Blumberg, R. H., 472,514 Boata, G., 504,692,513, 744 BObel, G., 504,613 Bogoliubov, N. N., 224,279
Boldarev, S. T., 287, 296, 297, 299, 300, 305,308,309,366,369 Bonsall, L., 427, 430 Borg. R. J., 714, 717. 745 Boucal, F.,692,728,730, 741 Bowley, R. M., 34, I01 Boyce, J. B., 688,696,709, 741 Boysen, J., 692,727,729,730, 741 Bozhkov, A. I., 309,366 Bozler, H. M.,10, 210, 102, 280 Bozorth, R. M., 693, 741 Bradley, P.E., 475,513 Brandt, D., 315,366 Brandt, E. H., 474,513 Breed, D. J., 389,430 Brenig, W., 611, 612,646 Bretz, M., 382,413,430 Brewer, D. F., 286, 382, 383,726, 366,430, 432, 745 Brewer, W. D., 669,683,692,693,706,708, 725, 126, 727, 729, 730, 741, 742, 745 Brezin, E., 392, 393,402,405, 430 Brinkman, W. F., 3, 20, 42, 44, 45, 46, 56, 59, 64,65, 66, 68, 69, 70, 71, 1.12, 113, 114, 115, 116, 119, 121, 122, 126, 127, 128, 130, 132, 134, 137, 139, 141, 142, 143, 148, 150, 151, 153, 155, 156, 157, 168, 184, 195, 211, 101, 102, 188, 189, 190,279 Bristow, J. R., 485,513 Brodale, G. E., 733,742 Brooker, G. A., 202,209,279 Brout, R., 714, 744 Brouwer, W., 293, 334,366 Brown, T. R., 384,421,431 Browne, M. E., 714,726, 744 Buchholz, F. I., 315,366,369 Bucholtz, L. J., 134, 146, 188 Bucker, E., 520, 552, 557, 558, 589 Budnick, J. I., I 14, 742 Burger, I. P. 442, 477, 486, 490, 496, 509, 694,704,705,513,514, 741 Burns, M.J., 384,486,431, 516 Butler, D. M., 420,430 Bychkov, Yu. A., 533,534, 543,589 Callinaro, G., 504,513 Cameron, J. A,, 662,692, 741 Camp, W.J., 403, 430
AUTHOR INDEX
xvii
Campbell, I. A., 662, 673, 691, 692, 694,
Cornbescot, R., 20,110,118,154,158,163,
697, 706, 707, 709, 724, 741, 742, 743, 744,745,746 Cannella, V., 714, 742 Caplin, A. D., 608, 630, 692, 645, 646, 742 Cardona, M., 441,457, 471,476,477,485, 486,491,496,497,498,502,510,514,515, 516 Carneiro, K., 378,433 Caroli, B., 674,718, 742 Caroli, C., 318, 324,441,366,514 Caroline, D., 287,296,305,366 Carroll, K. G., 475, 514 Casimir, H. B. G., 442,515 Caswell, H. L., 476,488,514 Caudron, R., 693, 741 Chambers, R. G., 469,514 Chm, S.-K., 425,430 Chang, C. C. 321, 325, 328, 332, 338, 341, 351, 362,366 Chapellier, M., 693, 724, 725, 731,741,742 Chechetkin, V. R., 269,279 Cheng, D., 205,279 Chernikova, D. M., 315,366 Chester, G. V., 321, 327, 328, 368 Chester, M.,382, 412,430 Chock, E. P., 698, 699, 700, 735, 745 Chouteau, G., 724, 742 Christiansen, P. V., 441,502,514 Chui, S. T., 429,433 Cipolla, J. W.,316, 366 Cladis, P. E., 125, 188 Clark, R. C., 307,366 Clark, W. B., 388,433 Clarke, J. S., 207,279 Clogston, A. M., 521,526,589 Cochran, J. F.,483,514 Cody, G. D., 477, 485, 486, 496, 521, 523, 552, 582,514,515,589 Cohen, M., 321, 325, 328, 332, 338, 341, 351, 362,366 Cohen, R. W., 521, 523, 552,582,589 Cole, M. W., 318, 328, 384, 412, 366, 430 Coles, B. R., 735,743 Collan, H. K., 6, 8, 9, 94, 265, 267, 268, 101, 279 Collier, R. S., 477,480, 514 Collings, E. W., 692, 742 Combescd, M., 180, 181, 234, 251, 252, 254,188,279
169, 171, 174, 179, 180, 181, 182, 183, 188, 234, 251, 252, 254, 276, 101, 188, 279 Compton,J. P., 662,692,709,720,721, 741, 742 Compton. V. B., 693, 741 Cooper, J. R.,608,645 Cooper, L. N., 442,468,469,513,514 Coqblio, B., 681, 691, 698, 711, 742 Corak, W. S., 483,514 Cornut, B., 681,711, 742 Corruccini, L. R., 42.43, 51, 54,55,56, 60, 65, 66, 69, 70, 71, 83, 89, 90, 121, 141, 150, 153, 163, 207, 101, 103, 189, 279 Costa Ribeiro,P., 693,726,730,741,742 Cox, M. A. A., 422,430 Craig, P. P.,693, 742 Crangle, J., 693, 707, 742 Cross, M. C., 3, 14, 16,20, 26,28, 56,110, 114, 117. 118, 120, 134, 139, 169, 170, 174, 175, 178, 179, 180, 181, 183, 185, 223, 296, 101, 188, 279, 368 Crow, J. E., 386,430,432 Crum, D., 287,297,300,301,302,305,312, 313,366 Cunsolo, S., 308,366 Czenvonko,J., 90,101
Dalmasso, C., 460,504,514 Dalton, N. W., 422,430 Daniel, E., 656, 670, 742 Daniels, J. M., 659, 742 Darken, L.S., 475,514 Dash, J. G., 361, 378, 381, 382, 383, 409, 412, 413, 414, 419, 420, 366, 430, 431, 432,433 Daunt, J. G.,307, 308,367 Davidov, D., 698,699,700, 735, 745 Davis, D. D., 693, 741 Davis, J. H., 475, 514 Daybell, M.D., 692, 742 DeBlois, R. W., 442, 504,514 d&nnes,P. G., 110, 112, 113, 124, 130, 131, 137, 144, 170, 181, 389, 398, 461, 464,465,466,467,577,188,189,430,516, 588 De Groot, S. R., 167,652,189, 743 de Jongh, L.J., 388,404,408,430 De la Cruz, P.,476,485,486,496,514,515
xviii
AUTHOR
Delaplace, R.,668, 698, 711, 712, 741 Delrieu, J. M., 31, 125, 173, 174, 102, 189 Denestein, A., 308,367 De Rosa,F., 385,429 de Shone, C., 351,368 DeSorbo, W., 442,504,514 Deutscher, G., 477,486, 509,513,514 Devaraj, N., 287,288,296,366 De Waele, A. Th.A. M.,664, 746 De Wijn, H. W., 388,430 Dickson, D. P. E., 287,296,305,366 Di Crescenzo, E., 476, 485, 486, 487, 489, 514
Dieterich, W., 557,558,589 Di Salvo, F. J., 386,425,433 Disatnik, Y.,300,366 Ditzian, R. V., 390,430 Dmitrenko, I. M.,502,516 Dobbs, E. R., 207,280 Doezma, R. E., 598,646 Do Kim Chung, 733, 745 Doll, R.,475,477,479,491,514 Domb, C., 422,430 Doniach, S., 383, 429, 731, 429, 433, 743 Donze, P.,735,743 Doran, J. C., 710, 742 Dorokhov, 0. N., 548, 550, 552, 562, 565, 566, 568, 581, 584, 586, 587, 589
Douglass, Jr., D. H., 472, 477, 480, 481, 483,488,514
Drew, H. D., 598,646 Druiker, A. K.. 509,514 DUM,A. G., 422,430 Dworin, L., 694, 697, 742 Dy, K. S.. 263,279 Dyatlov, I. T.,542,589 Dynes, R.C.,318,365 Dzyaloshinskyi, I. E., 533, 534, 535, 543, 544,580,588,589
Ebisawa, H., 20, 151, 152, 209, 211, 227, 233,276,102,190,279,280
Ebner, C., 294,295,301,321,328,357,366, 367 Echenique, P. M., 319, 320, 342, 366 Eckardt, J. R.. 287,288,289,292,293,294, 295, 296, 299, 300, 310, 316, 317, 318, 323, 324, 334, 335, 338, 347, 366, 367 Eckert, D., 578,589
INDEX
Edwards. D.O., 287, 288, 289, 291, 292, 293, 302, 313, 324, 350,
294, 295, 296, 297, 299, 300, 301, 303, 304, 305, 307, 308, 310, 312, 316. 317, 318, 319, 320, 321, 323, 328, 334, 335, 338, 342, 343, 347, 360, 361, 366, 367, 368 Edwards, J. T., 423,430 Edwards, S. F., 714, 742 Ehrenfest, P., 482,514 Eilenberger, G., 110.188 Einzel, D., 259,267,271,273,277,279,281 Elgin, R. L.,413,417,430 Elleman, D. D., 316,368 Elliott, C. J., 390, 430 Emerson, D. J., 726, 745 Emery, V. J., 205,279 Engelsberg, S., 56, 64, 65, 66, 68, 69, 70, 71,116,119, 126,127,128,132, 134.141. 142, 143, 156, 157, 102, 103, 189, 190 Englert, Th.,385,430 Ericson, T.,693, 742 Esel’son, B. N., 287,298, 307,367 Esfandiari, M. R.. 439, 452, 453, 454, 462, 464,465,466,467, 500,514 Eska, G., 669,742 Essam, J. W., 421,422,430,433 Everitt, C. W.F., 308, 367
Fa&, T. E., 479,490,508,514 Falicov, L. M.,426,431 Fatouros, P. P., 287, 292, 299, 300, 301, 310, 312, 318, 319, 320, 321, 322, 323, 324,342,343,347,350,363,366,367 Feder, J., 301, 324, 438, 441, 475,476, 477, 478, 490, 491, 492, 493, 494, 496, 497, 498.499.502,507,508,510,367,514,515, 516 Felsch, W., 742 Ferrell, R. A., 386, 430 Fert, A., 694, 697, 736, 742, 745 Fertig, W. A., 712, 743 Fetter, A. L., 75,88,92,124, 126,128, ,129, 134, 144, 145, 146, 101, 102, 188, 189 Feynman. R. P., 322, 329, 338. 340, 347, 381,367,430 Fields, J., 731, 743 Fine, H. L., 386,432 Finger, M.,733, 745 Fink, H. J., 439, 441, 452, 453, 455, 456, 457, 461, 464, 465, 466. 467, 506. 510, 513,514,516
AUTHOR INDW
Fiory, J. A., 717, 744 Fisher, K. H., 717, 742 Fisher, D. S., 395, 430 Fisher, I. Z., 293, 368 Fisher, M. E., 389, 390, 406,409, 411,430, 431,432,433
Fisher, R. A., 733, 742 Fisher, R. M., 475,514 Fishman, F., 146,189 Flint, E. B., 29, 30, 31, 102 Flouquet, J., 668, 681, 683, 687, 692, 693, 698, 699, 706, 708, 709, 711, 712, 724, 725, 726, 727, 729, 731, 734, 736, 741, 742,745 Follstaedt, D. M., 688, 736, 742 Fomin, I., 56,65, 66,73,116,132, 134,141, 142, 154, 361, 102, 189, 367 Ford, P. J., 608,646 Fowler, A. B., 385,424,431 Fowler, M., 614, 615,646 Fox, R.A., 659,698,733,736, 745 Frankel, R. B., 652, 742 Franks, J., 475,513 Freeman, A. J., 652, 690, 742, 743 Fremlin, D. H., 422,430 Frenkel, J., 291,367 Friedel, J., 398, 415, 417, 418, 425, 595, 598, 656, 670, 673, 679, 714, 430, 646, 741, 742, 743 Friedrich, A., 736, 742 Friedt, J. M., 693, 741 Frossati, G., 664,701, 702, 718.736, 743 Fu, H. H., 301,367 Fujiki, K., 269, 280 Fukuyama, H., 427,428,432 Fulde, P., 557, 558, 589 Fung, H. K., 557, 558, 589
Gainon, D., 735, 743 Gallop, J. C., 693, 707, 708, 709, 724, 741, 743
Ganguly, A. K., 557, 589 Garfunkel, M. P., 490,514 Gasparini, F. M., 287, 288, 289, 292, 293, 294, 295, 296, 299, 300, 310, 316, 317, 318,323,324,334,335,338,342,347,366, 367 Gaunt, D. S., 422,431,433 Gautier, F., 673, 693, 741, 743 Gayda, J. P., 422, 432 Geilikman, B. T., 269,279
Xix
Gergely. L., 608,646 Giaever, I., 412,514 Giannetta, R., 92,93,94,97, 99, 101, 102 Giauque, W. F., 733, 742 Gibbons, D. F., 484,514 Gibbs, J. W., 363,367 Gijsman, H. M., 664, 746 Gilbard, P. N., 643,646 Gilijamse, K., 389,430 Ginzburg, V. L., 438, 440, 442, 443, 446, 461, 462, 463, 464, 465, 466, 467, 515
Gittleman, J. I., 422,429 Gladstone, G., 580, 588. 589 Glasser. M. L., 318,367 Glen, M., 422,433 Glover, R. E., 386,431 Goldberg, I. B., 519, 521, 526, 527, 528, 548,549,578, 579,589
Goldstein, S., 20, 209,102, 280 Gomes, A. A., 673,743 Gongadze, A. D., 129,189 Gonzalez Jimenez, F., 663, 689, 699, 700, 735, 743
Goodkind,J. M., 95,96, 102, 103 Goodman, B. B., 496,515 Goodstein, D. L., 413,417,430 Gor'kov, L. P., 442,444,497,528,533,534, 535, 543, 544, 548, 550, 552, 562, 565, 566, 568, 576, 581, 584, 586, 587, 712, 515,588,589,741 Gorter, C. J., 442,655,659,515,743 Gorzkowski,W., 529,589 Gossard, A. C., 728, 744 Giitze, W., 611,612, 709,646, 743 Gould, C. M., 32, 63, 64, 85, 86, 96, 97, 98,126,143,184,102,189 Grace, M. A., 652,658,659,662, 741, 742, 743 Graebner, J. E., 731, 741 Gra!', P., 475,477,479,491,514 Graham, R., 154, 170, 173, 174, 253, 256, 189,279 Grant, M. F., 662,692,741 Grassie, A. D. C., 639,646 Grassman, P., 484,515 Greaves, N. A., 77,127,102,189 Greytak, T. J., 29, 86,87, 102 Griffin, A,, 318,367 Griffiths, D., 735, 743 Grimes, C. C., 384,430 Gross, E. P.,328,329,365,367
XX
AUTHOR INDEX
Griiner, G., 595, 597. 602. 604, 609, 616, 625, 629, 636, 639, 640,641, 643, 644, 710, 645, 646, 647, 743 Gubbens, P. C. M., 606,646 Guernsey, R. W., 4, 7, 92, 265, 102, 279 Guggenheim, H. J., 388,404,407,429,430 Guillou, J. C., 392,405,430 Gully, W. J., 20, 32, 63, 64, 85, 86. 126, 148,,184,209, 102, 189, 190, 280 Gumprecht, D., 693, 745 Gunther, L., 381,431 Guo, H. M., 287, 291, 296, 297, 299, 300, 302, 303, 304, 305, 307, 328, 360, 361, 367 Gurgenishuili, G. E., 129, 189 Guttreund, H., 540,589 Guyon, E.,477,486,514 Gylling, R. G., 732, 741 Hagn, E., 669, 742 Hake, R. R., 484,515 Hall, H. E.,4, 6, 9, 21, 26, 28, 31, 92, 109, 120, 164, 165, 173, 174, 176, 265, 267, 268,102,189,280 Halliday, W., 318, 368 Hallock, R. B., 382, 433 Halperin, B. I., 410,422,429,429,430,433 Halperin, W. P., 7, 9,12, 18, 36,80, 81, 82, 83, 84, 85, 87, 102 Hamann, D. R., 399, 612, 613, 619, 429, 645,646 Hamilton, W. D., 733, 745 Hanak, J. J., 521, 522, 559, 562, 581, 589 Handstein, A., 578,589 Hanson, M.,692,693,710,743 Harden, J. L., 442,515 Hartmann, F., 683,743,745 Hartstein, A., 385,424,431 Hasegawa, H., 689,743 Hauge, E. H., 401,431 Hauser, J. R.,460,515 Haya, J., 267,268,280 Hayano, R.S., 725, 744 Hayward, R. W., 659,746 Hebral, B., 701,703,736,743 Hedgcock, F. T., 692,742 Heeger, A. J., 594,646 Heine, V., 425, 430 Hemmer, P. C., 401,431 Henkel, P. P.,382, 431 Herb, J. A., 382,412,430,431
Hering, S. V., 413,431 Hertz, J. A., 391,431 Hickernell, D. C., 413, 430 Hill, R. W.,662,692, 741 Hippert, F., 735, 736, 741 Hiraki, T., 724, 743 Hirschkoff, E.C., 692,710,718, 743 Hirst, L. L., 696, 735, 743 Hiwonen, hl. T., 693, 742 Ho, T. L.,20,26,28,49,109, 114,117,118, 137, 165, 166, 168, 173, 174, 175, 185, 102,189,190 Hockney, R. W., 384,427,431 Hohenberg, P. C., 289, 328, 376, 386, 409, 410,367,431 Holliday, R. J., 662, 692,727, 729, 743, 744 Hollis-Hallett, A. C., 287, 288, 296, 366 Holtzberg, F., 715,730, 743 Hook, J. R..4.6,9,21,28,31,92.164,165,
l73,174,176,265,267,268,102,I89,280 Hoppes, D. D., 659, 746 Hordb, M., 608,646 Hornung, E. W., 733, 742 Horovitz, B., 540,589 Houghton, A., 14, 18, I01 Hu, C.-R., 50, 56, 60,139, 151, 152, 166, 172, 173, 174, 175, 102, 189, I90 Huber, J. G., 712,743 Huberman, B. A., 429,433 Hudson, R. P., 659, 733, 743, 746 Huff, G. B., 378,420,430,431 Hufner, S., 693, 709, 745 Hughes, H. P., 388,432 Huiskamp, W. J., 652, 662, 692, 706, 744 Hulm, J. K., 586,589 Hunter, G. H., 315, 367 Huntziger, J. J., 662, 733, 743 Huray, P. G., 725, 745 Hyman, D. S.. 318, 323, 324, 367, 369 Iche, G., 626,646 Ihas, G. G., 287, 292, 299, 300, 301, 312, 318. 319, 320, 321. 323, 324, 342, 343, 350,367 Imbert, P., 663, 689, 699, 700, 735, 743 Imry, Y.,381,409,431 Indovina, P. L., 476,485,486,487,489, 514 Ishii, H., 638,709,646, 743 Ishikawa, M.,20, 168, 186, 102, I89 Ittner, 111, W. B., 473, 515
AUTHOR INDEX
xxi
M., 173, 208, 257, 308, 330, 346,356, 189, 279, 367 Kharadze, G. A., 129,189 Khmelnitzkii, D. E. 716, 744 Jacarino, V., 521,526, 589 Khor, K. E., 423,431 Jacobs, I. S., 714, 745 Khukhareva, I. S., 485,486,515 Jacucci, G., 308,366 Khutsishvili, G. R., 662, 743 J&n?g, F., 167, 189 Kiewiet, C.W.,4, 6, 9, 21, 28, 31, 92, 265, Jasnow, D., 409 411,430,431 267,268,102,280 Jensen, H. H., 8, 202, 209, 102, 279 Kikuchi, M., 423,431 Jensen, M. A., 580, 588,589 King, J. G., 318,367 Johnson, C. E.,662, 743 King, P.J., 287, 308, 367 Johnson, R. T., 7, 29, 30, 86, 87, 91, 209, Kinsel, T.,506, 515 244, 245, 251, 252, 265, 267, 268, 102, Kip, A. F., 714, 726, 744 103,279,280 Kirkpatrick, S., 421,422,428,431,432,433 Johnston, W. D., 318, 367 Kiser, S. R., 477, 490, 496, 498, 507, 514 Jones, D. P.,441, 502, 503,515 Kitchens, T. A., 693, 743 Kittel, C., 460,602,670,674,515,744 Jose, J. V., 428,433 Kjems, J. K., 378, 413, 414, 431, 433 Joseph, A. S., 442, 504,515 Josephson, B. D., 48,289,102,367 Klein, E., 693,708, 729,730, 733, 741, 744 Joyce, G. S., 390,406,431 Klein, M. W., 714, 744 Jullien, R., 731,743 Kleinberg, R. E., 18,36,47,103 Kleinberg, R. L., 4, 5, 7, 14,20,29, 30, 36, 87,211,245,249,250,251,252,265,267, 268,102,103,279,280 Kac, M., 390, 430, 431 Kadanoff, L. P., 392, 408, 428, 429, 431, Klbman, M., 134,398,399,190,433 433 Klenin, M.A., 386,430 Koch, V. E., 238,248,249, 281 Kaeser, R. S., 733, 743 Kaiser, A. B., 643,646 Kodama, T., 383,429 Kogut, J., 392, 433 Kalfas, C. A., 733, 745 Kohn, W.,328,367 Kalos, M. H., 321,327,328,368 Kojima, H.,26, 29, 91, 102 Kalvius, G. M., 663, 743 Kamerlingh Onnes, H., 285, 287, 368 Kok, J. A., 483,515 Kammerer, 0. F., 386, 432 Kokko, J., 31,33, 65, 101 Kamper, R. A., 477,480,514 Kompaneets, D. A., 310,314, 343, 356, 359, Kaplan, J. I., 318, 367 365,365,368 Kaplan, T. A., 390,403,432 Kon, L.Z., 570,589 Karlsson, A., 598, 646 Kondo, J., 594, 604, 690,646, 744 Kasuya, T., 602, 646 Konter, J. A,, 724, 744 Katila, T. E., 693, 742, 743 Kossler, W.J., 717, 744 Kawabata, C., 428,433 Kosterlitz, J. M., 381, 386, 395, 397, 398, Kawamura, K., 267,268,290 399, 400, 401, 402, 407, 409, 415, 417, Keck, K., 315,366 419,429,431,433 Kedves, F. J., 608, 646 Kouvel, J. S., 714, 744 Keesom, W. H., 285, 287, 483, 368, 515 Kramer, L., 440,510,515 Keller, W. E., 287, 367 Krane, K.S., 698, 734, 744 Kellers, C. F., 318,368 Kreissler, A., 694,704,705, 741 Kelly, M. J., 426, 431 Krishnamurthy, H.R.,620, 621, 679, 680, Kessinger, R. D., 456, 514 646, 744 Ketterson, J. B., 3, 4, 5, 7, 9, 99, 100, 204, 205,207,211,234,246,102,103,279,280 Kr06,N.,646 Ivantsov, V. G., 287,307,367 Izumov, Yu. A,, 519, 579, 589
Khalatnikov, I.
xxii
AUTHOR INDEX
Levitt, D. A., 717,744 Levy,M., 50, 158, 101,188 Lhuillier, D., 173,175,190 Li, P. L., 692, 744 Liang, N.T.,422,431 Licciardello, D.C.,385,423,424,431 Lieb, E.H.,390,402,431 Lifshitz, E. M., 351, 357, 414, 368, 431 Lines, R.A. G., 662, 692, 741, 742 Liniger, W.,472,473,515 Lipson, S. G.,305,368 Liu, K.S.,321,327, 328,368 Labro, M., 692,746 Liu, L. L., 406,431 Liu, M., 172, 173,190 Lagendijk, I., 692,706,744 Livingston, J. P., 442,504,513 Laheurte, J. P., 287,307, 308,362,368 Loegel, B., 693, 741 Landau, D. P., 404,405,430 London, F., 442,492,515 Landau, J., 305,368 Landau, L. D., 195, 200, 250, 298. 351. London, H.,442,485,513,515 357,414,438,442,443,280,368,431,515 Loponen, M. T.,6, 8, 9,94,265, 267, 268, 101,279 Landauer, J. K.,484,515 Loram. J. W., 608,639,707, 646, 744 Landauer, R.,422,431 Landwehr, G., 385,430 Lounasmaa, 0.V., 6, 8, 9, 31, 33, 65, 94, 265, 267. 268, 732, 101, 279, 743, 744 Lang, H.,316,366 Lovejoy, D.R.,287,296,368 Langer, J. S.,412,422,625,429,431,646 Langmuir, I., 373,431 Loyalka, S.K.,316,366 Langreth, D., 624,646 Lubbers, J., 662,744 Larkin, A.I., 386, 533, 534, 543, 716, 429, Luders, G.,497,515 Lutes, 0.S.,475,485,488,692,515,744 589,744 Lasjaunias, J. C., 701,736,744 Luther, A., 402,431 Last, B.J., 422,431 Luttingcr, J. M., 172, 188 Lawson, D. T.,10,20,209,210,102,280 Lyden, J. K.,265,279 Lea, M. J., 207,280 Lynton, E. A., 506,515 Leath, P. L., 422,433 Lecoanet, B.,692,728,730,741 Ma, S. K., 392,429 Le Dang Khoi, V., 691,694, 744 MacLaughlin, D.F., 717, 744 Lederer, P., 707, 744 Magerlein, J. H.,287,288, 289,290,368 Lee, D. M.,10, 20, 32, 63, 64,85, 86, 96, Mahajan, S.,387,425,433 97, 98, 99, 126, 143, 148, 184, 207, 209, Mailfert, R.,521,522, 559, 562,589 Main, P.C.,4.6,9,21,28,31,92,265,267 210,102,189,190,280 268,102,280 Leggett, A. J., 3, 9, 14, 16, 19, 20, 26, 34, 35, 40, 56, 59, 60, 61, 62, 63, 65,66, 68, Maita, J. P., 520, 552, 557,558,589 74,75, 76,77, 78, 79, 81, 82, 92,97,115, Maki, K., 14, 18, 20, 32, 50, 56, 60, 126, 138, 139, 147, 151, 152, 154, 156, 157, 120, 121,122, 123, 125, 127, 148, 158, 209, 211, 218, 227, 233, 386, 102, 189, 163, 169, 177, 178, 179, 180, 182, 183, 190,279,280,431 184, 185, 186, 187, 195, 211, 223, 258, 275,276,277,278,732,102,189,190,280, Maley, M. P., 693,744 Maloney, M. D., 476, 485, 486, 496, 514, 744 515 Leiderer, P.. 305, 306, 309,368 Manchester, F. D., 307,368 Lekner, J., 347,368 Lenglart, P., 673,743 Mandelstam, L., 291,368 Levin, K.,79,80, 88,102 Manhes, B., 692,710, 744
Krusius, M., 4,9,10,11, 12, 13, 14, 15, 17, 19,20, 21, 24, 25, 26, 35, 36, 37, 38, 39, 40, 64,65, 68, 69,71,72, 73, 74, 84, 85, 86, 87, 88, 89, 142, 143, 175, 265, 383, 101,103,188,189,190,280,429 Kumar,P., 138,139,147.156,157.189,190 Kuper, C. G., 318, 324,368 Kurmaev, E. Z.,519, 579, 580,589 Kuroda, A., 428,433 Kuroda, Y.,93,102 Kurti, N.,662,743
AUTHOR
Maple, M. B., 712, 743, 744 Mapother, D. E., 493, 515 Maradudin, A. A., 427,430 Maraviglia, B., 351, 368 Marcus, P. M., 482,516 Margenau, H., 354,368 Marsh, J. D., 659, 692, 698, 706, 736, 742, 744, 745
Marshall, W.,714, 744 Martin, P. C., 166, 167,410, 190,431 Martinet, A., 477, 486, 514 Matheson, C. C., 316,366 Matho, K., 701,703,716,736,743,744 Mattheiss, L. F., 387,528,431,589 Matthias, E., 662, 744 Mattis, D., 624, 647 Maxwell, E., 475, 515 Mazur, P., 167,189 McCoy, R. J., 4, 7, 92, 265, 102, 279 McEvoy, J. P., 441, 502, 503, 515 McLachian, D. S., 438, 445, 446, 450, 451, 464, 478, 489, 500, 516
465, 479, 491, 501,
466, 480, 493, 504,
467, 481, 494, 505,
474, 482, 496, 506,
475, 484, 497, 510,
476, 485, 498, 514,
477, 488, 499, 515,
McLean, E. O., 382, 409, 413, 430, 432 McMillan, W.L., 588, 625,589,645 McTague, J. P.. 378, 379, 431, 433 McWane, J., 318,367 Megerle, K., 472, 514 Meinhold-Heerlein, L., 315, 368 Mendelssohn, K., 382,430 Mendoza, E., 287,296,305,366 Menyhard, N., 543,621, 626,628,589,646 Mermin, N. D., 20, 26, 28, 49, 109, 114, 117, 118, 137, 139, 165, 168, 176, 185, 186, 207, 328, 376, 380, 390, 392, 414, 102, 190, 279, 368, 432 Meyer, D. T., 318,368 Meyer, H., 318,368 Meyer, W. J., 688, 736, 742 Mezei, F., 636, 646 Michael, P., 475, 478, 479, 485, 489, 491, 500,501,515 Miedema, A. R., 388, 389,404,430 Migdal, A. .A, 392,428,614,432,645 Mignot, J. M., 701,702,703,736, 743, 744 Mihara, N., 329,368 Miljak. M., 608,646 Miller, R. E., 477, 485, 486, 496, 514, 51
Imex
xxiii
Mills, D. L.,707, 732, 744 Mineev, V. P., 120, 134, 137, 138, 139, 171, 183, 184, 185,190
Minier, M., 636, 646 Minugerode, G. V., 742 Misener, A. D., 287,288,296,485,365,513 Miyashita, S., 428,433 Mizuno, K.,603,646 Monarka, Y.P., 286,368 Moore, M. A., 78, 390,406,101,432 Morel, P., 113,168, 184, 188 Moriya, 673, 707, 728. 744 Morris, D. E., 481,515 Moskalenko, V. A., 570,589 Mott, N. F., 385, 422, 423, 424, 425, 432 Mrozinski, P., 287, 292, 318, 319, 320, 342,367
Mueller, R. M., 29, 30,31, 102 Muir, W.R., 692, 744 Muller-Hartmann, E., 612, 712, 646, 647, 746
Murani, A. P., 699,736, 744 Murnick, D. E., 717, 744 Murphy, G. M., 354,368 Muto, Y.,692, 742 Mydosh, J. A., 714, 742 Myers, H. P., 598,646 Myerson, R. J., 429,433 Nabarro, F. R. N., 398,415,417,418,445, 446, 450, 451, 464, 465, 466, 467, 474, 475,480,481,482,484,485,488,432,515, 516 Nagai, K., 267,268,280 Nagamine, K., 725, 744 Nagaoka, Y.,611,646 Nagasawa, H., 639,646 Nagi, A. D. S., 93,102 Nagle. D. E., 693, 742 Narahara, Y.,287,288,296,366 Narath, A., 659, 688, 692, 694, 695, 697, 728, 735,736, 742, 744, 746 Narayanamurti, V., 318,365 Nave, S., 725, 745 Nayak, V. S., 301,324,367 N&l, L.,714, 744 Nelson, D. R.,394,395,404,428,429,430, 432,433 Nemnonov, S. A., 580,589 Newrock, R. S., 692, 744 Nichols, T.L.,725, 745
xxiv
AUTHOR INDEX
Nielsen, M., 378, 379, 431 Niesen, L., 692, 706, 744 Nishida, N., 725, 744 Novaco, A. D., 378,413,414, 431 Nozieres, P., 182, 195, 211, 616, 621, 633, 677, 735, 188, 279, 646, 744
Paulson. D. N., 4, 5, 7, 9, 10. 11, 12, 13, 14, 15, 17, 19, 20, 21, 24, 25, 26, 29, 35, 86, 87, 89, 91, 143, 175, 209, 211, 244, 245,249,250,251,252,265,102,103,190, 280
Peierls, R. E., 373, 375, 376, 380, 386, 428, 432 Pelcovitz, R. A., 394, 404,432 Pellan. Y., 438,477,480,486,489,491,509, Obsen, C. E., 698,734,744 516 Odeh, F., 472, 473, 515 R., 711,712, 745 Pena, Okazaki, M., 423,433 Pendry, J. B., 319, 320, 342,366 Olli, E. E., 316, 368 Penrose, O., 409,432 Olsen, J. L., 484, 515 M., 385,424, 425,429,432 Pepper, Ono, K., 727, 743 B., 698,706,734, 741, 745 Perczuk, Ono, Y. A., 267,268,280 J. G., 693, 709, 744, 745 Perez-Ramires, Onori, S., 476, 485, 486, 487, 489, 514 Pershan, P. s.,166, 167, 190 Onsager, L., 373, 375, 390,409, 432 Orbach, R., 689,698,699,700,735,744,745 Peshkov, V. P., 287, 296, 305, 308, 309, 366,368 Orsay Group on Superconductivity, 441, M.,689, 744 Peter, 464, 465, 466,467,515 Pethick, C. J., 3, 6, 7, 8, 14, 17, 21, 28, 56, Osborne, D. V., 315,367,368 63, 171, 195, 204, 208, 256, 263, 267. Osheroff, D. D., 36, 39, 42, 43, 51, 54, 55, 268, 273, 277, 101, 103, 190, 279, 280 56, 60, 64, 65, 66, 69, 70, 71, 80, 81, 83, Pettersen, G., 502, 516 89, 90, 121, 127, 128, 130, 141, 142, 143, 148, 150, 153, 156, 157, 163, 296, 101, Philipps, N. E.. 606, 607, 732, 735, 647, 741, 745 103,188,189,190,368 Pickar, K.A., 308,368 Ott, H. R., 484,731,515, 741 Pinch, H. L.,422,429 Ottavi, H., 422, 432 Pineau, J. C., 438, 477, 480, 486, 489, 491, Owen, J., 714, 726, 744 509,516 Pines, D., 299,366 Pipman, J., 305,368 Paalanen, M. A., 31, 33, 36, 37, 38, 39, 40, Pippard, A. B., 469,482,484,515 64, 65, 68, 69, 71, 72, 73, 74, 84, 85, 86, Platzman, P. M.,384, 385, 427, 428, 432, 88,89,142,383,101,188,429 433 Padmore, T.C., 328,412,368,432 Pleiner, H., 154, 256, 189,279 Pagiola, E., 460, 504,514 Poisel, H., 305, 306, 309,368 Palmer, R. G., 139, 176, 188 Pokrovsky, V. L., 407,569,432,589 Papoular, M.,306,368 Pollak, R. A., 388,432 Park, J. G., 441, 502, 503, 515 Pollitt, S., 385, 424, 425, 429, 432 Parks, R. D., 731, 745 Polyakov, A. M.,392. 393, 402, 404, 432 Parodi, 166, 167,190 Pomeranchuk, I., 298,368 Parr, H., 456, 473,476,478,479,491, 492, Poppema, J. O., 659, 743 493, 495, 496, 498, 499, 502, 506, 507, Potts, R. B., 408, 419,432 508,516 Poulsen, R. S., 441, 457, 464, 465, 466, Parry, G. S., 388,433 467,516 Passell, L., 378, 413, 414, 431, 433 Pound, R. V.,655, 745 Patani, A., 399,432 Pratt, W.P., 692, 732, 742, 745 Pathria, R. K., 293, 334, 366 Presson, A. G., 439,455,457, 506, 510,514 Patton, B. R., 180, 190 Privorotskii, I. A,, 146, I89
AUTHOR INDEX
Pryce, M. H. L., 655, 741 Puff, R. D., 329,368 Quimby, S. L., 484,516 Rainer, D., 110, 112, 113, 117, 124, 130, 131, 144,211, 188, 189, 280 Rasmussen, F. B., 7, 9, 12, 18, 36, 80, 81, 82, 83, 84, 85, 87, 102 Ratto, C. F., 504, 513 Rayl, M., 521, 523, 552, 582,589 Reckers, A. B., 664, 746 Regge, T., 328,368 Rehwald, W., 521, 523, 552, 582, 589 Reivari, P., 693, 742 Renard, J. C., 504,516 Renton, C. A., 484,514 Reppy, J. D., 4, 7, 21, 28, 31, 92, 93, 94, 382,412,429,101,103,429,431,432,433 Reut, L. S., 293,368 Reuter, 0.E. H., 469,516 Ribault, M., 693, 724, 725, 731, 741 Rice, T. M., 388,426,432 Richardson, R. C., 7, 9, 12, 18, 20, 31, 32, 33, 36, 63, 64, 65, 80, 81, 82, 83, 84, 85, 86, 87, 126, 148, 184, 209, 383, 101, 102, 103,189,280,429 Richer, Y. A., 504,516 Ritchie, D. S., 390, 406, 422, 430, 432 Rivier, N., 627, 644,646, 647 Rizzuto, C., 504,595,608,630,513,646 Roach, P. D., 3, 4, 5, 7, 9, 204, 205, 211, 102, 279 Roach, P. R., 3,4, 5, 7,9,99,100,204,205, 207, 211, 234, 246, 102, 103, 279, 280 Robertson, J. A., 691, 694, 697, 741 Robinson, F. N. H., 659, 742 Robinson, G., 477,488,516 Rogani, A., 476,485,486,487,489,514 Rohrer, H., 484, 515 Rolt, J., 383, 432 Romagnan, J. P., 287, 368 Rose, M. E., 655, 745 Rosenbaum, B., 314,366 Rosenblatt, J., 438, 477, 480, 486, 489, 491, 509,516 Rothberg, B. D., 475, 488, 516 Rothberg, Bibby, B., 437, 445, 446, 450, 451, 464, 465, 466, 467, 474, 475, 480, 481, 482, 484, 485, 488, 515, 516 Rothwarf, F., 477, 490, 496, 498, 507, 514
xxv
Roulet, B., 318, 324,366 Rudermann, M. A., 602, 670,674, 647, 745 Rudnick, I., 382, 409,432 Rush, P., E., 475,513 Saam, W.F., 287, 294, 295, 304, 307, 308, 321, 328, 333, 335, 340, 341, 349, 350, 360, 362, 363, 366, 368 Sacli, 0.A., 726, 745 Saffren, M. M., 316,368 Sager, R. E., 39, 40, 41, 47, 49, 50, 51, 52, 56,57,58,63,68,73,90,91,150,103,190
Sager, R. W., 265,277,278,280 Saint-James, D., 318, 324, 442, 460, 461, 464, 465, 466, 467, 490, 500, 501, 504, 366,514,516 Saint Paul, M., 726,730, 742 Sakurai, A., 615,647 Samoilov, B. N., 656, 662, 745 Sanchez, J., 668, 693, 698, 699, 712, 724, 725,726,734,736, 741, 742, 745 Sanctuary, S., 692,706, 746 Sanders, T. M., 287, 288, 289, 290, 368 Sandiford, D. J., 4,6,9,21,28,31,92,265, 267,268,102,280 Sanvinski, R. E.,287, 291, 296, 297, 299, 300, 301, 302, 303, 304, 305, 307, 312, 313,328,360,366,367,368 Saslow, W. M., 110, 113, 166, 172, 173, 174,175,188,189,190 Satterthwaite, C. B., 483, 514 Scalapino, D., 402,431 Schechter, H., 378,432 Schick, M., 381,432 Schieffer, J. P.,580, 588, 589 Schlindwein, M., 399,432 Schlottmann, P., 709, 743 Schmid, A., 386,432 Schmidt, H., 119, 167, 171, 386, 189, 530 Schmidt, J. L., 692, 744 Schmidt, R. W.,714, 745 Schoepe, W., 31, 33,65,383, 101,429 Scholtz, J. H., 382, 409,432 Schrieffer, J. R., 442, 468, 469, 602, 624. 711,513,647,742,745 Scott, G. K.,388,426,432 Scott, W.R., 693, 707, 742 Scruby, C. B., 388,433 Scully, M. O., 318, 324, 367 Seiden, J., 269,360,280,368 Seki, H.. 314,366
xxvi
AUTHOR INDEX
Seligmann, P., 303, 368 Senoussi, S., 694, 697, 745 Seraphim, D. P., 482,516 Serene, J. W., 117, 180, 211, 212, 218, 227, 190,280 Serin, B., 386, 490, 506, 692,432, 514, 515, 744 Serwlock, R. G., 662, 743 Sevast'yonov, B. K., 485,486,516 Shablo, A. A., 502, 516 Shafi, Q., 399, 432 Shahzamanian, M., 21, 28,269, 102,280 Shaltrel, D., 689, 744 Shan, Y.,422,431 Shanabarger, M. R., 692,710, 718, 743 Shante, V., 421, 432 Shen, S. Y., 287, 288, 289, 292, 294, 295, 296, 299, 300, 310, 316, 317, 318, 319, 320, 323, 324, 342, 347, 366,367 Shermer, R. I., 692, 732, 745 Shields, S. E., 95, 96, 102, 103 Shih, Y. M., 321, 325, 328, 351,368 Shikin, V. B., 286, 368 Shirane, G., 388, 404, 407, 523, 524, 553, 564, 568,429, 430, 588, 589 Shirley, D. A., 652, 662, 733, 743, 745 Shumeiko, V. S . , 269, 280 Shvets, A. D., 287, 307, 367 Siddon, R. L.,381,432 Sierro, J., 735, 743 Siggia, E. D., 429,433 Simons, A., 385,433 Singh, A. D., 296,368 Skalyo, J., 388, 404, 407, 430 Sklijarevskii, V. V., 656, 662, 745 Skocpol, W. J., 386, 432 Skove, M. J., 475,514 Slichter, C. P., 688, 696, 709, 741 Smith, E. N., 92, 93, 94, 97, 99, 382, 101, 102,431 Smith, F. W., 441, 457, 471, 476, 477, 491, 496, 497,498, 502, 510,516 Smith, H., 3, 6, 7, 8, 14, 17, 21, 28,42, 44. 45,46, 56, 63, 64,65, 66, 68, 69, 71, 115, 116, 121, 128, 132, 134, 141, 142, 143, 150, 153, 154, 155, 156, 157, 202, 209, 256, 267, 268, 273, 277, 441, 502, 101, 102,103, 188,190, 279, 280, 514 Smith, P. V., 423,431 386.432 Smith, R. 0.. Sobyanin, A: A., 289,368
Soda, T., 34, 50,269, 103,280 Sblyom, J., 543, 613, 589,647 Sondheimer, E. H., 469,516 Souletie, J., 639, 641, 710, 714, 730, 647, 742, 745 Spanjaard, D., 659, 698, 736, 745 Stakelon, T., 688, 745 Stanley, H. E., 390,403,406,408,430,431, 432 Star, W. M., 630,631, 639,640,647 Steenberg, N. R., 734, 745 Steenland, M. J., 659, 743 Steinbeck, M., 47, 92, 265, 102, 279 Steiner, P., 693, 709, 744, 745 Stepanov, E. P., 656, 662, 745 Stephen, M. J., 445, 446, 450, 451, 464, 465, 466, 467, 474, 475, 480, 481, 482, 484,485,488,516 Stephens, J. B., 412,430 Stern, F., 426, 432 Stewart, G. A., 381,417,420,430,432 Steyert, W. A., 692, 693, 698, 732, 734, 742, 743, 744, 745 Stillwell, E. P., 475, 514 Stinchcombe, R. B., 422,432 Stone, N. J., 659, (198,736, 745 Strongin, M., 386, 432 Sudakov, V. V., 542, 589 Suhl, H., 611, 647 Suzanne, J., 378,432 Suzuki, M., 428, 433 Swallow, G. A., 707, 744 Sykes, J., 202, 209, 279 Sykes, M. F.,389,421,422,430,431,433 Symko, 0. G.. 692, 710, 718, 742, 743 Szentumay, Z., 646 Tai, P. C. L., 473,516 Takagi. S., 14, 16, 20, 26, 56, 59, 60, 63, 74, 158, 163, 169, 177, 178, 179,180, 182, 183, 185, 223, 275, 276, 277, 102, 103, 189,190, 280 Takano, Y.,31,33,65,383,101,429 Tam, C. P., 299. 300, 301, 312, 318, 319, 320,324, 342, 350,367 Toa, L. J., 698,699,100,735,745 Taub, H., 378,413,414,431,433 Taurian, O., 692, 693, 708, 719, 723, 724, 725,726, 741, 742, 745 Taylor, R.D., 662,693, 742, 743, 744 Teller, E., 408, 429
AUTHOR INDEX
Telschow, K. L., 382,433 Ter-Martirosyan, K. A., 542,589 Testardi, L. R., 519, 521, 523, 525, 526, 527,528,572,575,578,579,582,587,589 Tholence, J. L., 631, 635, 692, 693, 701, 702, 714, 715, 716, 718, 724, 725, 726, 728, 730,732,647, 741, 742, 743, 745 Thomas, L. K., 693, 745 Thompson, R. S., 386, 472, 473, 430, 432, 433,516 Thomson, J. O., 692, 693, 724, 725, 745 Thomson, J. R., 692, 693, 724, 725, 745 Thouless, D. J., 381, 385, 386, 395, 397, 398,409,415,422,423,424,430,431,433 Thoulouze, D., 664, 701, 702, 718, 726, 730, 736, 742, 743, 744 Tilley, D. R., 488, 516 Ting, C. S., 557, 558,589 Tinker, R., 318, 367 Tinkman, M., 386, 472, 473,481, 432, 515, 516 Tissier, B., 693,697,707, 708, ;56,727, 745 Tolhoeck, H. A., 652, 743 Tomasch, W. J., 442, 504,515 T o m , S.,125,188 Toth, R. W., 420, 430 Tough. J. T., 287, 291, 296, 297, 299, 300, 302, 303, 304, 305, 307, 328, 360, 361, 367,368 Toulouse, G., 121, 134, 137, 176, 398, 399, 425,188,190,433 Tournier, D., 701,702, 718, 743 Toumier, R., 631, 635, 692, 693, 701, 702, 707, 708, 710, 711, 712, 714, 715, 718, 724, 726, 727, 728, 730, 736, 647, 741, 742, 743, 745 Toxen, A. M.,472,485,486,516 Triplett, B. B., 606,607, 735,647, 745 Tsui, D. C., 385,424,433 Tsuneto, T., 50, 154, 157, 102, 190 Twose, W. D., 423,432 Typpi, V. K., 693, 742 Tzoar, N., 385,433 Uimin, G. V., 407,432 Uluer, I., 733, 745 Valette, C., 496. 498, 507, 508. 509, 513, 514,516 Valls, O., 79, 80, 88, 102 van Dam, J. E., 606,646
xxvii
van den Berg, 0.J., 594,606,646 Van Dyke, J. P., 403,430 Van Scriver, S. W., 413,431 Van Urk, A. T., 285, 287,368 Varoquaux, E. J., 63,96,143,101,188 Veillet, P., 691, 694, 744 Vetleseter, A., 732, 741 Veuro, M. C., 6, 8, 9, 94, 265, 267, 268, 101,279 Vibet, C.,63, 96, 143, 101, 188 Vieland, L. J., 578, 583, 589 Vig, J., 692, 744 Vilches, 0. E., 413,430,431 Villain, J., 403, 433 Violet, C.E.,714, 717, 745 Vochten, M., 692, 746 Volovik, G. E., 20, 120,'134, 137, 138, 139, 171,183.184,185,103,190 von Roosbrocck, W., 121, 141, 142, 143, 156, 157, 190 Vuorio, M., 53, 56, 65, 66, 73, 116, 120, 132, 133, 134, 141, 142, 732, 102, 103, 189,190. 741, 744 Vynchier, S.,692, 746 Wagner, H., 376, 390,392,368,432 Walker, L. R., 388,430 Wallace, D. J., 405, 433 Wallden, L., 598, 646 Wallis, R. H., 385,424,432 Walstedt, R. E., 388,659,717,430, 744, 746 460,515 Wang, J. S. -Y., Wang, S., 422,431 Wang, T. G., 316,368 Wanner, M., 305,306,309,368 Warner, D. D., 733,745 Wassermann, E. F., 693,725,726,732. 741, 745 Watson, B. P.,422,432,433 Watson, R. E.,690, 743 Waysand, G., 508,509,514,516 Webb, R. A., 18, 29, 30, 36, 39, 40,41, 47, 49, 50, 51, 52, 56, 57, 58, 63, 68, 73, 87, 90, 91, 150, 265, 267, 268, 277, 278, 102, 103,190, 279,280 Weeks, J. D., 429,433 Weger, M., 519, 521, 525, 526, 527, 528, 540,548,549,578,579,589 Wegner, F., 309,390, 405, 408, 431, 433 Welber, B., 484, 516
xxviii
AUTHOR INDEX
Wernick, J. H., 693, 741 Werthamer, N. R., 110,159,172,443,188, 190,516 Weyhmann, W., 692,727,729, 743 Whall, T. E., 608, 646 Wheatley, J. C., 3, 4, 5, 7, 8, 9, 10, 11. 12, 13, 14, 15, 17, 18, 19, 20, 21, 24, 25, 26, 29, 30, 32, 34, 35, 36, 39, 40, 41,44. 45, 47, 48, 49, 50, 51, 52, 56, 57, 58, 63, 65, 68, 73, 74, 77, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 95, 97, 143, 150, 175, 195, 204, 209, 211, 244, 245, 249, 250, 251, 252, 265, 267, 268, 277, 278, 692, 710, 718, 732, 102, 103, 190, 279, 280, 741, 743 White, G. K., 484,516 White, R. J., 639, 646 Wicklund, A. W., 578, 583,589 Widom, A., 318, 323, 324,367,369 Widom, B., 287,289,367,369 Wiechert, H., 315, 316,366,369 Wigner, E. P., 427,433 Wilkins, J. W., 8, 202, 207, 209, 620, 621, 679,680,102,279,646, 744 Will, H., 643, 645 Williams, G., 707, 735, 743, 744, 746 Williams, I. R., 664, 692, 706, 709, 720, 721, 742 Williams, P. M., 388, 433 Williamson, J. S., 509,513 Williamson, S. J., 557, 558, 589 Wilson, G. V. H., 692, 706, 709, 720, 721, 742, 744, 746 Wilson, J. A., 386,425,433 Wilson, K., 677, 679, 746 Wilson, K. G., 392, 619, 620, 621, 679, 680,433,646,647, 744 Winzer, K., 742 Wolff, P. A., 602, 693, 647, 741, 745
Wolfle, P.,3,4, 5, 7, 9, 11, 27, 75, 112, 113, 114, 180, 182, 183. 185, 209, 212, 217, 218, 221, 222, 227, 234, 235, 237, 238, 243, 244, 248, 249, 259, 267, 269, 270, 271, 213, 277, 103, 190, 279, 280, 281 Woo, C. W., 321, 325, 328, 351, 368, 369, 429 Woo, J. W. F., 386,429 Wu, C. S., 659, 746 Wu, F. Y., 390,402,431 Wyatt, A. F. G., 287, 308,367 Yamada, K., 616, 621, 626, 627, 628, 633, 678,735,647, 746 Yamazaki, Y., 725, 744 Yang, L. C., 382,412,430 Ying, S. C., 379, 433 Yogi, T., 509, 510, 516 Yoshida, K., 602, 615, 616, 626, 674, 677, 678,735,647, 746 Yoshimori, A., 615, 629, 677, 647, 746 Yoshino, S., 423,433 Yuval, G., 399,613, 619, 626,429,645, 647 Zavaritskii, N. V., 485, 516 Zawadowski, A., 595, 602, 613, 614, 616, 625, 626, 629, 639, 640. 643, 710, 646, 647,743 Zdrojewski, W., 693, 745 Zeger, R., 557,589 Zener, C., 669, 746 Zia, R.K. P.,405, 433 Ziman, J. M., 422, 433 Zinn-Justin, J., 392, 393, 402, 405, 430 Zinov'eva, K. N., 287, 288, 296, 297, 299. 300,307, 369 Zipfel, C. L., 384, 431 Zittartz, J., 403, 612, 712, 433, 647, 746 Zlatic, V., 644.647
SUaJECT INDEX TO VOLS. VIIA-B* Bulk properties of helium 3; soperfluid phases unless otherwise stated
bending energies 1 12-1 16,118 boundary conditions, A phase 74.114.131 boundary energies, B phase 67, 132f boojums 139f collective modes of order parameter 4f, 7, 11,27, 210,215,-219 221ff, 230, 239, 243,278 collision integral 208, 234-238, 26Of, 263f, 272 -, normal 3He 197f, 202f, 208 critical flow velocities 29ff dipolar healing length 67.70, 141 144 dipoldipole energy 32f, 35, 122f, 127, 148,154
magnetic susceptibility, anisotropic A phase 18f, 33 -,Bphase 89ff mass supercurrent, expression for 111, 113f, 116ff, 120
N.M.R., linewidths 63f -, longitudinal ringing frequencies 35-41
-, nonlinear effects
4242,149-153
-, relaxation in 52-55, 158-162
-, satellite resonances 96ff normal locking effect 26,169
orbital dynamics 163-171,173-176
-, collisionless 177-181
fountain pressure 95f free energy 76,81
orbital motions, field induced 14ff -, persistent 2&25,28 orbital viscosities 170f order parameter notations 75, 108,217ff orientation cnergks 123-130
hydiodynamic equations 173ff, 182R', 256ff
phase diagram 84-87, 89
intrinsic angular momentum density 26, 168,184-188 ion mobility 33f
quasiparticle scattering amplitude 197f 235
electric field orientation 31ff
kinetic equation, collisionless 212-215, 259 -, hydrodynamic 2652 269f. 275 -, normal 3He 196, 199,201,206 - , s ~ a l a r 258ff
relaxation time, attenuation, t. 6f Cooper pairs, h T , TCE 277f -, kinetic equation, iK, 7, 237f. 241f, 276 -, quasiparticle, z. (0) 6f, 9, 18, 63, 262 -, orbital, tl lSff, 26f -, viscous, r, 6-9 -, normal 3He,quasiparticle, r ~ ( 0 ) 203-2O8,238,245,25Oy262,277f
-,
Landau parameters 5, 7,90 Landau theory, normal jHe 195f kggett equations 60, 148f
*Volume VII is published in two parts: page numbers 1-370 in the index refer to volume VII A and page numbers 371-746 refer to volume VII B.
xxix
xxx
SUBJECT INDEX
specific heat 77-83 sound, hydrodynamic 253ff -, macroscopic regime 2 5 M -, normal 3He 200-207 -, zero, longitudinal 5, 2Wf, 218, 220, 224-228,239-248 -, zero, longitudinal, anisotropic 9-13, 227-232, 249f -, zero, longitudinal, attenuation 4, 225ff, 239,244ff spin relaxation, intrinsic 274-279 spin supercurrent, expression for 111, 114, 116ff, 120 spin wave spectra 154-1 57 strong coupling parameters 76-80
superfluid density, anisotropic 25, 33, 92ff, 114-118 temperature scales 84f textures, NMR observations of 64-74 thermal conductivity 271ff -, normal 3He 208 topology of textures 134-140 vortices, non-singular 137f viscosity, second 269ff -, shear 6R, 94f, 165-168, 279 -, shear, normal 3He 204, 207 wall pinned mode, relaxation 56-62, 162f
Free surface of liquid helium accommodation coefficient 314,316f
phonons at surface 293f, 332-335, 337f
capillary waves 308f, 345, 358f
reflection coefficient for atoms 319-323, 342 ripplons 287f, 290f, 293ff, 308, 324, 326, 332-335, 337-341, 344ff, 357, 362f
evaporation 314ff, 363 -, velocity distribution 316ff excited states with surface 330-341, 343-347 films 3He on 4He, thin 29W, 312. 347-351, 352-360 -,thick 302f ground state with surface 325-329, 362f hydrodynamics of surface 343-346, 356-360 interfacial tension, 3He-4He mixtures 305f nucleation of phase separation 360f
scattering at surface, atoms 318f, 324, 341ff, 363 -, light 309, 330 -, neutrons 330, 341 surface normal density 310f, 344, 346, 356f surface scattering of 3He 313f surface states of jHe 298-301, 347-351 surface second sound 310f, 344f, 358f surface tension 286ff, 363ff -, 'He 295. 361f -, 4 H ~288-296, 3275 330, 338 -, 4He with thin jHe film 297-301. 355 -, 4He with thick 3He film 302f -, mixtures of 4He in 3He 307, 361f
Two-dimensional physics (See also Surface of helium: Monolayer of 'He: Surface sound) absence of long range order 373, 376, 379f. 386, 394f, 409,414,428 adsorbed atomic films 373, 3781 413, 419
Anderson localisation 423f change density waves 387f, 425f crystalline order 3798,413-419
suBJEa INDEX
dislocations in 2D solids 41-19 electrons on surface of helium 373, 3831 427f electrons at surface of semiconductor 373, 384%423 epitaxial monolayers 419f king model solution 373, 375. 388, 390, 406f, 419
xxxl
phase transitions 388,39&394 planar rotator 39Of, 395400,402,405, 410,428f renormalisation group 392-395,401, 428 spherical model 376, 390 superfluidity of adsorbed *He 382, 408411,429
magnetic layer compounds 374, 3885 403f, 408 magnetic systems, anisotropies in 403-408 metallic layer compounds 374, 386f, 425, metal-insulator transitions 374, 377, 402426
variable range hopping 422f vortices in superfluid *He 410413,429
percolation model 420ff
Wigner lattice 427f
topological order 397, 398ff. 415
Properties of moderately small superconductors coherence length 437,446,452,459,462, Ginsberg-Landau equations 438,440, 469,470,472,475,490,494,497 445,457,468,472,474,502,510 critical fields, bulk specimens 439-442, Ginsberg-Landau theory 443,450,463, 468,470, 502ff, 507 444,454,456ff, 473 critical field summaries 464-471 Kappa (Ginsberg-Landau parameter) critical field, supercooling 439, 447f, 438ff,446,450,452456.458f. 461f. 450f, 459ff, 468-471,481,488,490ff, 471,485,490,492,496,502ff, 506, 497f, 500,503 508, 510 -, superheating 438,4475 49-53, -, supercooling 492499,501, 504, 468-471,481,488,49OlT, 4971 500, 507 502ff, 506,509,510 -, superheating 492499,507 -, surface nucleation 441,459ff; 488, 491,497,501ff, 506 Landau critical point 437,448-451, 459, -, thermodynamic 439,44741,462, 462,468,473f, 476,479,481,483f, 468471,481,485-489,492 487f, 490,499, 505 cylinder in axial field 445-451,459f, 463, 468,485,487,489 non-local superconductors 468472,497 cylinder in transverse field 453f. 461, nucleation centres 490,4941 463,469,489 penetration depth 437ff, 445,450,452& delayed flux entry 504 456f, 459,46lff, 471ff, 483,486ff, 490, 492,497,505ff experimental methods 476-479 quantum oscillations 501 fist order transition 437f. 474,480 fluxoids 460,503-506 Gibbs free energy 438f, 443-447, 452, 4555 462f, 474,48lf, 484
sample preparation 47Sf second order transition 437f, 441,447f. 459, 419, 483,485, 489, 499 spheres 453,462,471,476,490-496, 498%507ff
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SUBJECT INDEX
transition times 508f
whiskers 450f, 474f, 481f, 485,488f, 499f
Properties of A-15 compounds A-15 crystal structure 525, 529
Batterman-Barrett transformation 521ff 526,5335 540, 553, 556f, 561f, 578, %Of, 585 change density wave instability 533-537, 539,542f composition and defects, effects of 523, 561, 582ff Cooper instability 534-537, 540-543, 569, 571 electronic band structure 526-531, 533f, 544,548 electronic density of states 520f, 526f, 546f, 580 -, logarithmic singularity 547-551, 555, 585ff electron-electron interactions 534-538, 588 electron-phonon interactions 538f, 554, 569,586, 588
linear chains 525, 529, 546f, 555f, 559, 564f, 579f, 5841 'Parquet' equations 537, 543, 519 relation between structural and superconducting transformations 523, 534, 544, 564,578, 585 softening of phonon spectrum 523f, 5321 553f, 5645 579,582, 587 strains, effects of 523, 557, 575, 577 superconducting gap 570f. 573,575, 579 temperature dependence of magnetic susceptibility 519f, 550-553, 556,558f, 58lf, 587f tetragonal distortion parameter 553, 555, 5S8f, 562ff, 568.575 three-dimensional effects 540. 544-547, 553, 566,585 tunnelling between chains 540, 545,548f
Low temperature properties of Kondo alloys and applications of nuclear orientation to metals with magnetic impurities Abrikosov-Suhl resonance 612,628, 640f analysis of NO experiments 686f Anderson Hamiltonian 596, 670-673, 678,691,709 Anderson model 595-598, 602f, 610, 620,632ff; 642,644,671ff, 680, 713, 718 Anderson model at T =O 624-628.639 angular correlation methods 653, 663 charge perturbations 6352 642, 670
599, 601, 625,
Edwards-Anderson model 714 Fermi liquid theory 622,624,626, 629, 677 Friedel sum rule 598,625
gamma-rays, angular distribution 652ff, 660f, 737 gamma-ray anisotropy 656,658f, 683, 685, 688 gamma-ray detection and counting 664ff giant moment 701, 704f, 707,718, 726, 733 heat capacity of dilute alloys 603, 607, 618. 627,634,663 hyperfine coupling 655, 657,662, 680, 682,690-693,697f, 700,702,713, 717,724,727ff hyperfine field, effective 684, 686, 700, 707,72Off, 724.729 saturation 689f. 692f. 698,700. 703. 733
-.
SUBJECT INDEX
implantation techniques 667f Kondo effect 604f, 607, 612f, 630f, 651, 656, 675,731, 735 -, logarithmic approximation 611f Kondo temperature 593, 609$ 615, 619f, 628ff, 634, 676,679,681f, 706712,726f, 730f magnetic-nonmagnetic transition 594, 603407,609 magnetic susceptibility of dilute alloys 593ff, 606,608,618f, 621,627,630, 632, 6345 680 Massbauer experiments 653,663,687, 689. 699f, N.M.R. experiments 659,663,687f, 694ff, 7271 735, 736 Nozieres’ F e m i liquid theory 616, 621ff, 627,677f
‘Poor man’s’ derivation 613f
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radioactive decay schemes 6561 renormalisation group 614, 616 resistivity of dilute alloys 593f, 599ff. 603% 607ff, 612,628,630f, 634,6391 675,703 resonances in dilute alloys 596, S98f, 601,603, 623,634,639-643 RKKY coupling 602,638,670,674,713 sample cooling for NO 659, 664 sample preparation for NO 665-669 Schrieffer-WoH transformation 603, 620f, 624,643f s-d Hamiltonian 601f, 617, 619, 632, 634f, 644,669 spin glass 670,704, 713-716, 720ff, 724f, 735 spin polarization 602, 637f, 674 thermometry by NO 731f Wilson’s numerical solution 594, 615-618,622f, 645,677,679
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CHAPTER 5
TWO-DIMENSIONAL PHYSICS BY
J.M. KOSTERLITZ and D.J. THOULESS Department of Mathematical Physics, University of Birmingham, UK
Progress in Low Temperature Physics, Volume VZZB Mited by D.F. Brewer 8 North-HollandPublishing Company, 1978
Contents 1. Introduction 373 2. Examples of two-dimensional systems 378 2.1. Classicalfluids and solids 378 2.2. Quantum fluids and solids 381 2.3. Two-dimensional electron gas 383 2.4. Superconductivityin two dimensions 386 2.5. Metalliclayer compounds 386 2.6. Magnetic layer compounds 388 2.7. Smecticliquid crystals 389 3. Phase transitions in two dimensions 389 3.1. Introduction 389 3.2. Magnetic systems 391 3.3. The planar rotator case - topological order 395 3.4. Anisotropies in two-dimensional magnets 403 3.5. Superfluids 408 3.6. Crystalline order in two dimensions 413 3.7. Epitaxial monolayers 419 4. Metal-insulator transitions 420 4.1. The percolation model 420 4.2. Anderson localisation 423 4.3. Density waves 425 4.4. The Wigner lattice 427 Note added in proof 428 References 429 Referencesadded in proof 433
373
1. Introduction
There have been various important results for two-dimensional systems obtained over the years, such as Langmuir’s theory of gas adsorption on surfaces (Langmuir 1918)’ Peierls’ argument for the absence of long-range order in two-dimensional solids and isotropic ferromagnets (Peierls 1934, 1935), and Onsager’s solution of the two-dimensional Ising model (Onsager 1944). It is only in recent years that a wide variety of two-dimensional systems have become available for detailed experimental study so that the relevance of various theoretical ideas could be assessed. At the same time there has grown up a great interest in the intluence of dimensionality on the behaviour of physical systems, particularly, but not exclusively, in relation to critical phenomena near phase transitions. Various examples of systems which can more or less be regarded as twodimensional will be considered in this review. Monolayers of gas adsorbed onto some substrate provide obvious examples. It is important that the substrate should have a surface which is clean and homogeneous, and helpful if it has a large surface to volume ratio. Various sorts of finely divided powders, ‘vycor’, which is a porous glass, and ‘grafoil’, which is graphite expanded so that there are only a hundred or so atomic layers between each surface are all suitable substrates. Monolayers of inert gases such as helium, neon and argon can display solid, liquid and gas phases, with properties somewhat different from the bulk materials. With helium, if the coverage is greater than a single monolayer, superfluidity can occur at low temperatures, and films of 3He show the effect of Fermi statistics and the nuclear moments give rise to important magneticeffects. There are at least two ways of studying the two-dimensional electron gas. Firstly it is possible to trap electrons above the surface of liquid helium. The electrons cannot penetrate the helium because the conduction band is very high in energy and they cannot leave the surface region because the small polarizability of helium is sufficientto produce an image charge which attracts the electrons. It is also possible to produce an ‘inversion layer’ at the surface of a semiconductor such as silicon by coating the surface with an insulating oxide layer and then applying a gate voltage on a metal electrode beyond the
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insulating layer which has the effect of bending down the conduction band of the ptype semiconductor until close to the surface the band lies below the Fermi level. The same may be done to the valence band of an n-type semiconductor. The electrons either on the surface of liquid helium or on the surface of the semiconductor move freely parallel to the surface if the surface is sufficiently homogeneous and are bound in a potential well perpendicular to the surface. In many cases only one level in this potential well is occupied so the perpendicular degree of freedom is removed and the electrons behave as a two-dimensionalelectron gas. Electrons in metallic thin films do not usually behave two-dimensionally, since the films are generally much thicker than either the wavelength or the mean free path of the electrons. When such films become superconducting, however, they may behave two-dimensionally, since the coherence length of the superconductivitymay be greater than the film thickness. These films are often somewhat unsatisfactory to study since there may be uncertainty both in their physical conformation and in their chemical composition. It is also possible to study superconductivityin the surface of a bulk material in various ways. For example there is a range of magnetic fields for a type I1 superconductor, between H,,and H c 3 ,for which superconductivity occurs in the surface but not in the bulk. In the systems mentioned so far there is either a two-dimensional system attached to the surface of a bulk substrate or the surface of the bulk material itself forms a two-dimensional system. There are however various materials which are bulk materials consisting of layers of materials which behave more or less two-dimensionally. Typical layer compounds are niobium, and tantalum disulphide or diselenide, which behave electrically as twodimensional metals, with a high conductivity parallel to the planes of metal atoms, and a low conductivity in a perpendicular direction which can be further reduced by intercalating certain organic materials between the layers. These particular compounds show a number of interesting phase transitions, such as the onset of superconductivity and the occurrence of charge density waves which in some cases lead to a transition between a metallic and a semiconducting or semimetallic state. For such layer compounds there is always some coupling between the layers, and this is likely to dominate close to a phase transition, since the strong coherence within the layers enhances the coupling between the layers. There are also many magnetic layer compounds such as Rb,MnF, in which there is strong coupling within layers of magnetic atoms and only weak coupling between the layers, and the difference in these coupling strengths
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may be sufficient for two-dimensional behaviour to be apparent everywhere except in the immediateneighbourhood of the phase transition. The smectic phasesof liquid crystalsarealso to someextenttwo-dimensional, in that they consist of regularly arranged planes of molecules. Phase transitions occur between the various types of smectic phases, such as the smectioA phase with the molecular axis perpendicular to the planes and the smectic-C phase with a tilted axis. There is also in some cases a transition from A, with a liquid type of disorder in the plane, to the B phase with a solid type of order. Probably the main reason for the interest in two-dimensional systems is that, while they are broadly similar in many respects to three-dimensional systems, the theoretical analysis is somewhatsimpler. The geometry of a plane is simpler and more familiar than the geometry of a volume, integrals are easier to evaluate, and fewer particles need to be studied in a molecular dynamics or similar numerical computation. All this would be more true of one-dimensional systems, but they are known to have peculiar properties not shared by three-dimensional systems. There are particular theories that can be solved in two dimensions but not in higher dimensions, and problems of classical statistical mechanics in two dimensions can be related to quantum ground state problems for one-dimensional many body systems. Apart from these theoretical points, the study of these two-dimensional systems on surfaces may provide valuable information about the nature of a surface that is not readily available from microscopic measurements. For example the isotherms for gas adsorption give informationabout the area and homogeneity of the substrate surface. A major part of the theoretical interest in two-dimensionalsystems has been concerned with the possibility of phase transitions and the nature of the longrange order that may occur in these systems, and this subject is covered in detail in 0 3 of this review. It has been known for forty years (Onsager 1944; Peierls 1936) that the Ising model has a phase transition in two dimensions, and it is generally believed that in analogous cases, where the order is described by a single real scalar quantity, there will in most cases be it phase transition from an ordered state at low temperatures to a disordered state at high temperatures. In many systems, however, the order is described by a quantity with more than one equivalent degree of freedom, and we use n to denote the number of degrees of freedom. For example in superfluids and superconductorsthe ordered state is described by a complex number, so n is 2, while for the isotropic Heisenberg model of a ferromagnet or antiferromagnet the order is described by a vector with n = 3. Less obviously n is 2 for the
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two-dimensional solid since there are 2 degrees of freedom determining the absolute position of the lattice in space. Peierls (1934, 1935) argued, and later Mermin and Wagner (1966) and Hohenberg (1967) showed that for such systems there is no long-range order of the usual type except at zero temperature, essentially because the thermal motion destroys the order at an arbitrarily low temperature. The interpretation of this formal result is not so clear. It may be that at low temperatures the system is almost ordered, with, for example, a Debye TZspecific heat at low temperatures in the case of the solid, but that there is a continuous transition to the high temperature state with no abrupt change of phase. Measurements of conductivity in superconducting thin films suggest that for such systems at least this is what occurs. Evidence of various sorts, from experimental studies, from computer simulations, from high temperature power series expansions, and from low temperature expansions has accumulated to suggest that in other cases there may be a sharp transition despite the absence of long-range order of the type that occurs in three-dimensionalsystems. When the number of degrees of freedom n is 2 or more, the two-dimensional systems are of particular interest in that they are just on the edge of having sharp phase transitions, so we can speak of 2 as being the ‘lower critical dimensionality’ of such systems. This is particularly clear in the case of the spherical model, which is known to be like a magnetic system in which n tends to infinity, since the failure of the system to order is due to the logarithmic divergence of an integral for 2 dimensions, and this integral would converge for 2 + e dimensions however small E is. Although 2 is the lower critical dimensionality, further study is needed to determine whether or not there is a phase transition for 2 dimensions, when n is finite. In order to resolve this problem attention has been focussed on the behaviour of defects in the ordered phase, and the question of whether they will occur spontaneouslyat low temperatures. It has been argued on these grounds that the superfluid and the solid should have a sharp transition, but that the Heisenberg magnet and other systems with n greater than 2 should not. The problem of defects such as dislocations in solids and vortices in superfluids leads to an analogy with a two-component plasma in two dimensions, which somewhat surprisingly seems to have an abrupt transition between a neutral and an ionized state. In 9 4 we consider a differentclass of problems related to transitions between metallic and insulating states. The simplest model that displays this behaviour is the percolation problem, in which a medium is supposed to be made up randomly of conducting and insulating parts. If the concentration of con-
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ducting parts is high enough there will be continuous conducting paths across the system and the medium will conduct, while below a critical concentration there are no continuous conducting paths and the system is insulating. There has been considerable study of this problem in two dimensions. It is too simple a model to be directly relevant to metal-insulator transitions, but it is actually more relevant to the problem of magnetic alloys, where the magnetic atoms take the place of the conducting regions and nonmagnetic atoms of insulating regions; only if there are indefinitely large clusters of magnetic atoms can the alloy be magnetic. In the real systems that have been mentioned there are several possible mechanisms for metal-insulator transitions. In the case of electrons trapped on a liquid surface it is known that the state of lowest energy is the Wigner lattice, in which the electrons are confined near to sites on a triangular lattice. It is not obvious that as the temperature is lowered there will be a sharp transition to this state, but there is evidence from molecular dynamics calculations that the transition is sharp. In the inversion layer electron densities are higher and so the exclusion principle is important. Also the substrate introduces disorder in the potential in which the electron moves, and this results in localization of the lowest energy electrons even without the influence of electron-electron interactions. An important feature of localization produced by disorder in this way is that it is predicted that the conductivity of the mobile electrons cannot fall below a definite ‘minimum metallic conductivity’. In two dimensions this minimum metallic conductivity should be a universal quantity. There is some evidence both from experiment and from numerical calculations to support this idea. In the metallic layer compounds the low mobility of the electrons perpendicular to the layers results in a band structure that is almost independent of the z-component of momentum, so that it is essentially two-dimensional. The Fermi surface has regions that are almost flat, and this makes it energeticallyfavourable for a charge density wave to form, which can result in the production of a band gap resulting from the superlattice formation so that the system becomes semiconductinginstead of metallic. This review is not restricted to systems that are studied at low temperatures, although a number of the phenomena can only be studied at low temperatures. It is however restricted to problems that are of current interest in fundamental physics, and it may well be that the most important two-dimensional systems are excluded as a result. For example there are a number of important biological processes, such as photosynthesis, that occur on more or less regular surfaces, and these are not considered here.
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2. Examples of two-dimensional systems
2.1. CLASSICAL FLUIDSAND SOLIDS
One of the most important examples of two-dimensional systems are provided by very thin films of volatile atoms or molecules on the surface of a solid. There has been a recent survey of this topic in the book by Dash (1975), so here we just mention some of the main features of such systems and the theoretical problems which arise when they are regarded as two-dimensional systems. The usual way of forming and studying a thin film is to have a solid substrate with a large surface to volume ratio while the material to be studied exists primarily as a gas in the neighbourhood of the substrate and condenses onto the substrate because of the lower potential energy of the surface, It is obviously necessary for the substrate to bind the gas molecules more tightly than they are bound in the bulk liquid for a thin layer to form. The temperature and chemical potential of the molecules can be controlled by controlling the temperature and pressure of the gas. The number of condensed molecules can be deduced from the decrease in the volume of the gas from its bulk value as a result of the presence of the substrate. The area occupied by the surface layer is the area of the substrate, and this may be measured, at least in relative terms, if the properties of a standard surface layer are known. The heat capacity of the surface layer can be deduced from the heat capacity of the combined system of substrate and gas by subtracting the contributions of the bulk substances. If the amount of surface material in a small volume is sufficiently great it is also possible to make structural studies by neutron scattering or by using the Massbauer effect. Recent measurements of the heat capacity of Ne monolayers have been made by Huff and Dash (1976). Argon has been studied using neutron scattering by Taub et a1.(1975). Kjems et al. (1976)haveused neutron scattering for monolayers of N, and McTague and Nielsen (1976) for 0,. Schechter et al. (1976) have used the Miissbauereffect to study butadiene iron tricarbonyl on grafoil. Earlier measurementsare summarizedby Dash (1975). These measurements have shown that even the simpler materials such as the inert gases have a variety of phases, corresponding to gaseous, liquid and solid phases in the bulk. Various types of solid phase have been observed,some of which appear to have lattice constants commensurate with the substrate lattice, while others seem to be incommensurate. For example argon seems to form an incommensurate triangular lattice on graphite (Taub et al. 1975),
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and oxygen forms a triangular lattice which is distorted from the closepacked form below 10 K and is close packed above 10 K (McTague and Nielsen 1976). A number of interesting theoretical questions are raised by these results, but not yet answered by them. Does the existence of something like a gasliquid critical point indicate that the phases behave like two-dimensional gases and liquids? Is there a sharp distinction between phases commensurate and incommensurate with the substrate, and how are incommensurate phases modified by the substrate? Does Peierls’ argument for the non-existence of long-range order in two dimensions mean that the substrate plays an essential role in the formation of a crystal lattice ofadsorbed gases? A partial answer to the first question is that the phase diagram of a classical gas-liquid system tells us rather little about the dynamics of the molecules, since in classical statistical mechanics the kinetic energy and potential energy make independent contributions to the free energy. For this reason the thermodynamic properties of the lattice gas, which would be a realistic model if molecules were bound to particular lattice sites of the substrate, and of a model in which the molecules move freely on the surface, are quite similar to one another. This fact has of course been exploited in the comparison of the critical behaviour of fluids with the critical properties of the Ising model. To settle this question it would be necessary to make studies of the diffusion of molecules on the surface, which will be more or less independent of concentration if the molecules are tightly bound to substrate sites and will be more rapid in dilute systems (as in a real gas) if the molecules are free to move. Similarly a strong substrate interaction would be shown up in flow experiments by a frictional force between a flowing monolayer and the substrate. A detailed study of a model of a two-dimensional lattice interacting with a regular substrate has been made by Ying (1971). In this work it was found that the lattice may either adopt the periodicity of the substrate or form a lattice incommensurate with the substrate. Figure 1 shows possible arrangements of molecules in a periodic potential well due to a substrate. In fig. l a the molecules are shown at the minima of the potential, with a few molecules in interstitial positions; this is a first approximation to an arrangement of the molecules in registry with the substrate. Figure lc shows the molecules as having relaxed away from the minima in the neighbourhood of the interstitial without losing their long-range registry with the lattice. In such a case the modes of oscillation will be governed by the potential well, there will be a minimum of the frequency spectrum, and the lattice specific heat will be of the Einstein form, exponential at low temperatures. In fig. l b the molecules
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J.M.KOSTERLITZ AND D.J. THOULESS
[CH. 5, $ 2
are shown forming their own periodic lattice incommensurate with the substrate. In fig. Id it is shown how this arrangement may adjust under the influence of the substrate without losing its own incommensurate period. In this case there are equivalent arrangements with the same energy obtained by moving all the molecules shown in fig. l b by a small amount, and then allowing them to relax again. Because such configurations have the same energy the phonons have an energy proportional to wave number at low frequencies, despite the effect of the substrate, and so the specific heat is Debye-like, proportional to T 2 at low temperatures. Although there are structure measurements which seem to show incommensurate crystal lattice for adsorbed molecules, there do not seem to be clean cases of T 2 specific heats at very low temperatures for classical systems.
(01
Ibl
[Cl
Idl
Fig. 1. Possible configurationsfor atoms in a one-dimensional periodic substrate potential. In case (a) the atoms are in at the potential minima, except for one interstitial atom. In case (b) the atoms are shown at incommensurate lattice positions. In (c) the atoms have moved under the influence of interatomic forces away from the positions shown in (a), while in (d) the atom have moved under the influence of the substrate potential away from the positions shown in (b).
Peierls’ argument (1934, 1935) for the nonexistence of long-range order in two-dimensional solids depends on the effect of long-wave phonons, and the argument of Mermin (1968) depends on a related property of the Hamiltonian. If phonons of wave number q have an energy ACq then their thermal motion gives rise to a mean square displacement of the molecules proportional to k,T/C2q2,and if this is valid for values of q right down to a magnitude of the order of one over the linear dimensions of the system, the sum over q gives a mean square displacement which increases logarithmically with the area of the system. This argument would seem to remain valid in the presence of a substrate if the lattice is noncommensurateand the specificheat is proportional to T 2 at low temperatures. If the molecules are in registry with the substrate there are no low energy phonons and so the argument does not apply.
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Recently it has been argued that other properties characteristic of the solid state may exist, even if the type of long range order which gives rise to a sharp Bragg peak in X-ray scattering is forbidden in two dimensions (Berezinskii 1970; Kosterlitz and Thouless 1972; Feynman 1973; Imry and Gunther 1971). It is argued that the long range fluctuations that give rise to the smearing of the Bragg peak may still leave the structure of the solid distinct from the fluid, so that there is a quadratic specificheat at low temperatures,and so that the concentration of dislocations is characteristic of the solid; the film is rigid rather than fluid. These ideas are discussed in more detail in 6 3. 2.2. QUANTUM FLUIDSAND SOLIDS
When atoms of helium or molecules of hydrogen or deuterium are adsorbed onto a surface their small mass is likely to lead to a large zero-point motion, so that films of helium or hydrogen behave rather differently from films of heavier molecules. It may be less easy for the molecules to be attached to particular sites on the substrate so that they may be regarded, to a 6rst approximation, as moving freely on the surface. Also in the case of helium, which in the bulk phase remains liquid down to zero temperature,various phenomena characteristic of quantum fluids can be seen. These systems are also described in detail by Dash (1975). At temperatures above 2 K monolayer helium films have a specific heat of k per atom characteristic of a two-dimensional perfect gas. At lower temperatures the specific heat of 4He rises to a peak, and there appears to be condensation to a liquid phase with a critical temperature of 1.4 K. The specific heat of 'He however drops and there is no sign of condensation. These results have been explained in detail by Siddon and Schick (1974) using the quantum virial expansion. At higher coverages, still in the monolayer region, solid phases of helium can be formed. These appear to be really incommensurate with the substrate since the specific heat is quadratic in temperature down to quite low temperatures. When the temperature is raised the specific heat rises to a peak and then drops again to its value in the fluid phase. It is not however clear whether this specific heat peak represents a sharp phase transition, but there is a good deal of indirect evidence that it does correspond to a melting transition. The work with helium therefore has much more bearing on the question of the existence of a two-dimensional solid phase than work with classical fluids. Although there is some evidence from the specific heat measurements of Stewart and Dash (1970) that films with very low coverage of helium atoms
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behave like quantum fluids, with specific heats proportional to T2for 4He and T for 'He, there is no evidence for superfiuidity in monolayer films. Superfluidity can be observed in thin films a few atoms thick, and in such films there appears to be a monolayer of helium atoms forming a solid on the substrate, and superlluidity occurs above a layer which, according to Chester and Yang (1973), is about 13 atoms thick. Experiments have been done on vycor (Brewer and Mendelssohn 1961; Henkel et a]. 1969), which is a porous glass, on grafoil (Herband Dash 1972) and on the surface of glass (Scholtz et al. 1974; Telschow and Hallock 1976). Superfluid flow is detected in various ways, which may either involve observation of mass transport under more or less steady conditions or the observation of third sound in the film, which implies the existence of a rapidly alternating supercurrent. Under these conditions the superfiuid can be regarded as two-dimensional, since the coherence length is much greater than the film thickness, so that the phase of the supeduid wave function cannot vary appreciably between the substrate and the free surface of the film. It is important that the surface of a piece of Pyrex is simple, while in vycor there are multiple connections between the pores, since a simple film might be regarded as more truly two-dimensional than a multiply connected film and this influences what is observed (Berthold et a]. (1977). The onset temperature at which superfluidity is observed is lower in helium films than in the bulk, and decreases as the thickness is decreased. It is not clear from the experiments whether the ideal transition is a sharp phase transition or a more gradual quantitative change, since actual measurements almost certainly involve averages over an inhomogeneous substrate with a varying film thickness. One interesting feature of the results is that the superfluid density obtained from third sound measurements appeared to jump discontinuously to zero, but this may have been a spurious effect due to the large attenuation of third sound near the onset temperature (Scholtz et al. 1974; Telschow and Hallock 1976). Specific heat measurements have also been made in the neighbourhood of the transition. These also show a transition which occurs at a low temperature in thinner films and a rounding of the transition so that the high peak of the bulk specific heat becomes a low lump for films of five atomic layers or so (Bretz 1973). Between 2 mK and about 1 K bulk 'He behaves as a Fermi liquid, and the nuclear spin enables a number of detailed measurements to be made by using magnetic resonance techniques. In the same temperature range the solid is paramagnetic. Measurements on 'He films have shown a strong enhancement of the liquid magnetic susceptibility over the bulk susceptibility (Rolt
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and Brewer 1972; Ahonen et al. 1976). This seems to be due largely to the solid-like layer of 'He atoms on the surface and the next high density layer which are almost ferromagnetic and so enhance the paramagnetism of the rest of the film (Beal-Moood and Doniach 1977). This interpretation is borne out by the fact that a 4He impurity in the film, which goes preferentially to the surface layer because of the higher mass of the atom, cuts down this enhancement of the susceptibilityvery sharply. For all cases in which some sort of phase transition appears to occur the question arises of how far surfaces are homogeneous, and to what extent the observed rounded peaks in specific heats are a result of an average over different regions of the surface rather than a characteristic of a homogeneous film. This is a difficult question to answer, as most experiments measure order over a rather short range. There is some discussion of this problem in the book by Dash (1975). 2.3. TWO-DIMENSIONAL ELECTRON GAS
Electrons can be trapped on surfaces in many different substances. In most cases the surfaces are irregular and the electronic energy levels are dominated by impurities, but in some cases surfaces are sufficientlyuniform that electrons can move freely on them without being trapped, so that a two-dimensional electron gas can be formed. One example is the surface of liquid helium which is flat under the influence of gravity except for the thermal motion and zero-point motion of surface waves. Low energy electrons cannot penetrate into the helium, since the p-band, which is the conduction band, is about a volt above the vacuum level. Electrons are however attracted to the surface by their own image charges, and so they are bound in a potential which is inversely proportional to the distance from the surface, and are free to move parallel to the surface. Since the dielectricconstant of liquid helium is close to unity the image charge is small, and so the separation between the lowest level and the first excited level (for fixed momentum parallel to the surface) is 0.55 meV. At sufficiently low electron temperatures all the electrons will be in the lowest level, and a two-dimensional electron gas will be formed; the coupling between the electrons and the helium is quite weak, so the electrons can be at a higher temperature than the helium. The interaction between electrons is inversely proportional to distance at large distances, but is modified at short distances both by the effect of the helium and by the finite extent of the wave function in the perpendicular direction; both these effects c ~ t lbe altered by applying
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an electrical field perpendicular to the surface to alter the shape of the wavefunction. It is necessary to compensate the long range repulsion between electrons with a positively charged metal plate parallel to the surface, and it is hard to get a high density on the surface, so that most of the effects observed are due to the interaction of the electrons with helium rather than with other electrons. There is a review of this subject by Cole (1974), and more recent work has been reported by Grimes et al. (1976). Experiments by Grimes and Adams (1976), have shown that the twodimensional plasma mode, with a frequency proportional to the square root of wave number, exists, and its damping is due to scattering from surface waves in the helium (Platzman and Beni 1976). It would be interesting to observe the condensation of electrons to form a lattice at low temperatures, but calculations by Hockney and Brown (1975) show that this should not be expected at the temperatures and densities reached so far. The Si-Si02 interfaces used in MOSFETs can also be made sufficiently homogeneous to hold a two-dimensional electron gas. These are made by growing a thin oxide layer on a cleared silicon surface, and a gate voltage is then applied to a metal electrode on the other side of the oxide layer, as shown in fig. 2. If the silicon is p-type a positive gate voltage is applied, and
tp-Si
Fig. 2. The layout of a MOSFET constructed from p-type silicon. The positive gate voltage is applied to the metal electrode, and the surface current is passed between the two n+ regions.
if this is saciently strong it will bend down the conduction band so that its edge lies below the Fermi level, as shown in fig. 3. The electrons in this inversion layer are bound to the surface, but can move more or less freely parallel to it, and it can be arranged that the Fermi level lies above the lowest level of the electron bound in this potential, but below the first excited state. When motion parallel to the surface is taken into account the levels become sub-bands, and the lowest sub-band will be partly occupied at very low temperatures. On the (100) surface of silicon the effective mass is about 0.2 electron masses, the splitting between sub-bands is about 20 meV, and a degenerate two-dimensional electron gas with a density of about lot6 electrons/m2 can be formed. The number of electrons can be controlled by
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changing the gate voltage, which therefore changes the Fermi energy. Conduction in the inversion layer can be observed by using the two n+ regions shown in fig. 2 as electrodes. One of the most interesting features of the inversion layer is the low temperature behaviour of the conductivity. Charges trapped in the oxide layer act as scattering centres, so that the mean free path of the electrons is of the order of 10 nm. When the wavelength becomes as large as this the electrons are expected to become localized so that conductivity should be by activated hopping, with energy supplied by the phonons; Mott (1969) has shown that under these conditions the logarithm of the conductivity is proportional to
Acceptor levels
Fig. 3. The band structure in a ptype semiconductor when a voltage sutliciently strong to produce an inversion layer is applied. The distance into the semiconductor from the oxide surface i s denoted by z.
T - 1 / 3 .These localised states should occur at the bottom of the band, and above some mobility edge the electrons should be in extended states (Mott 1966), and the conductivity should be independent of temperature at low temperatures, as in a normal metal. It has been argued by Licciardello and Thouless (1975) that the conductivity just above the mobility edge for any system in which electron-electron interactions are not important should be about 3 x ohm-'. All these features have been observed in experiments carried out in Cambridge (Pepper et al. 1974; Mott et al. 1975; Adkins et al. 1976). There are however measurements on similar devices which give results in conflict with these, both by Tsui and Allen (1974), and by Hartstein and Fowler (1975). These results are due either to more significant effects from electron-electron interactions or to greater macroscopic inhomogeneities in the specimens. Measurements can also be made on many other properties, such as magneto-resistive oscillations (Englert and Landwehr 1976) and ac conductivity (Allen et al. 1975), which can be compared with theory (TZOW et al. 1976).
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2.4. SUPERCONDUCTIVITY IN TWO DIMENSIONS The arguments of Peierls (1934,1935) and of Hohenberg (1967) show that the long-range order characteristic of a superconductor should not exist in two dimensions, since the phase of the superconducting wave function has unbounded fluctuations, and the arguments of Kosterlitz and Thouless (1972) show that a supercurrent should be unstable, since there is only a finite energy barrier for the creation of a flux line. Nevertheless superconductivity is observed in thin films under conditions in which the coherence length is much greater than the film thickness, so that the behaviour is two-dimensional. This subject is included in a recent review of fluctuation effects in superconductivity by Skocpol and Tinkham (1975). The theory of the superconducting transition in a thin film has been worked out on the basis of Landau-Ginzburg theory by Ferrell and Schmidt (1967), Abrahams and Woo (1968) and Schmid (1968), and on the basis of microscopic theory by Aslamazov and Larkin (1968). In this theory there is an enhancement of the conductivity above the critical temperature because of the formation of small superconducting regions by thermal fluctuations. These lead to a conductivityproportional to (T- Tc)-', with a coefficient that can be calculated (Aslamazov and Larkin 1968). Measurements by Glover (1967) on amorphous bismuth films, by Strongin et al. (1968) on dirty aluminium films and by Smith, Serin and Abrahams (1968) on dirty lead films have confirmed this prediction. The transition to the superconducting state appears to be gradual, so that there is no temperature at which the conductivity suddenly becomes infinite, although since the resistance drops exponentially as the temperature is lowered it soon becomes unobservably small. There is a critical region in which the transition from the (T-Tc)-' behaviour to the exponential behaviour occurs. This is what would be expected from a theory which denies the possibility of a sharp transition to superconductivity in two dimensions,but may alternatively be a result of film inhomogeneities. For clean films there is an additional effect which enhances the conductivity predicted by Aslamazov and Larkin (1968) by a factor which is proportional to ln(T-T,). This was calculated by Maki (1968) and by Thompson (1970, 1971), and been observed in a number of systems (Strongin et al. 1968; Crow et al. 1970). 2.5. METALLIC LAYER COMPOUNDS Certain compounds, in particular the disulphides and diselenides of niobium and tantalum, crystallizein such a way that the metal atoms form close-packed
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layers separated from one another by two layers of the nonmetallic atoms. As a result the conduction electrons can move freely in the planes, but only with difficulty between the planes, so that the conductivity in the parallel direction can be greater by a factor of lo00 or so than the conductivity in the perpendicular direction, and these materials can be regarded as two-dimensional metals. The difference in conductivity in the two directions can be further increased by intercalation of various organic materials between the layers. Band structure calculations have been carried out by Mattheiss (1973) and by other people, and have shown that the Fermi surface is almost independent of the wave number component normal to the planes, so that the electron velocity is almost parallel to the plane, and the dynamics can be regarded as two dimensional. These materials have a number of interesting phase transitions. There are structural phase transitions involving a rearrangement of the close-packed planes relative to one another or distortions of the planes. The materials can become superconducting, thus giving another example of an almost twodimensional superconductor - however they are not truly two dimensional since the phase of the order parameter is coupled between the layers by the tunnelling of electrons from one layer to another. A particularly interesting feature of these materials is that the electrons can form charge density waves which may be either commensurate or incommensurate with the crystal lattice. A review of this topic has been given by Wilson et al. (1975). The types of transitions which occur depend on the basic type of crystal structure of the compounds; the symmetry may be either octahedral or trigonal prismatic, with various modifications. In the octahedral form (1T) of TaSz there are two first order phase transitions at 352 K and 200 K, with an increase in resistivity at each of them, so that at low temperatures the material has a resistivity of the order of 10- ohmem, which is characteristic of a semiconductor or semi-metal. The trigonal form (2m of TaSzhas a second order phase transition at 80 K at which the resistivity begins to decrease more rapidly, so that it reaches a value of lo-’ ohm-mat low temperatures. These resistivities are all measured parallel to the planes of metallic atoms. X-ray and electron diffraction studies show that in the octahedral (1T) forms the higher temperaturetransition corresponds to the establishment of an incommensurate charge density wave with superlattice parameters that may vary continuously with temperature. At the lower transition temperature the charge density wave locks into the crystal lattice forming a commensurate state. The lowered conductivity indicates that the charge density wave is reducing the area of the Fermi surface to form something like a semi-metal.
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In the trigonal(2H) forms the charge density wave does not occur until considerably lower temperatures, and seems to be commensurate with the lattice. Since conductivity increases the charge density wave is not reducing the amount of Fermi surface available (Rice and Scott 1975). One interesting point about the incommensurate charge density waves observed with X-ray and electron scattering (Williams et al. 1976) is that although one would expect the atoms to have Coulomb energies spanning the entire range between the maximum and minimum of the potential produced by the varying charge density, in fact X-ray photoemission experiments seem to show only two different values for the atoms (Pollak and Hughes 1976). This must mean that the sort of adjustment of the atoms illustrated in fig. Id occurs.
2.6. MAGNETIC LAYER COMPOUNDS A survey of both experimental and theoretical aspects of magnetic layer compounds is contained in the review by de Jongh and Miedema (1974). These are materials in which the magnetic ions are arranged in planes such that there is a strong coupling between the spins within a plane, but only a weak coupling between spins in different planes. In materials such as FeCl, and CoCl, there is a strong ferromagnetic coupling within the plane and a coupling between planes which is weaker by a factor of 10 or so and antiferromagnetic. In more complicated ferromagnetic layer compounds there may be a greater distance between layers, and, for example, in (CH,NH,), CuCl,, the interplanar coupling is down by a factor of more than a thousand. There are many materials, such as Rb,MnF,, in which the coupling within the plane is antiferromagnetic. This can result in a very much reduced coupling between the planes, since one spin may have four spins in the next plane which are at an equal distance, and their effects tend to cancel out. As a result the coupling between layers in this material is down by a factor of lo6. The sublattice magnetization of this material has been studied by Birgenau et al. (1970a) using neutron diffraction and at lower temperatures using nuclear magnetic resonance by De Wijn et al. (1971). The exponent j? relating the sublattice magnetization to T,-T comes out to be about 0.18, which is a fairly typical value for antiferromagnetic layer compounds, considerably higher than the value 0.125 known for the two-dimensional Ising model, but lower than values obtained in three-dimensional systems. The measured anisotropy of the interaction is quite small, but it may have an important determining effect in the critical region.
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Materials such as Rb,CoF4 and K,CoF, are strongly anisotropic antiferromagnets which behave like Ising systems. The susceptibility of these materials has been investigated in detail by Breed et al. (1969). The results with these materials are in good accord with the calculations for the twodimensionalIsing model made by Sykes and Fisher (1962). There has been an extensive investigation of layered copper compounds such as (CnH2n+lNH3)2CuC14 which has been analysed by Bloembergen (1975) to find out how an ideal Heisenbergferromagnetbehaves. By increasing the value of n the strength of the interlayer coupling can be reduced, and the results extrapolated to zero coupling. The critical temperature inferred for this system is quite low, and could possibly be zero. In these layered compounds the behaviour is only two-dimensional a certain distance away from the critical temperature. Close to the critical temperature there is a long-range correlation of the spins within the layer, and such correlated regions will interact between one layer and the next even if the coupling between individual spins is weak, since many spins can contribute coherently. This is a similar situation to that mentioned for the superconductivity of layered compounds. 2.7. SMECTIC LIQUID CRYSTALS
In the smectic phases of liquid crystals the molecules are arranged in regularly spaced parallel planes. These planes can slip more or less freely over one another, so that each plane of molecules is behaving in some sense like a twodimensional system. In the smectic A phase the molecules are irregularly arranged in the plane, as in a two-dimensional liquid, while in the smectic B phase there is a solid arrangement within the plane, although the positions of atoms in different planes are not strongly correlated. DeGennes (1974) has discussed the problem of whether the B phase is really a sort of three-dimensional crystal, or if the layers are really free to move across one another as if they were uncorrelated two-dimensional crystals, and he favours the former possibility.
3. Phase transitions in two dimensions 3.1. INTRODUCTION
As discussed in the previous section there are a number of systems which are quasi two-dimensional which can be realised experimentally. The problem of B
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J.M. KOSTERLITZ AND D.J. THOULESS
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phase transitions in two dimensions is not just a theoretical one. Of course, real systems are never purely two dimensional because they either are coupled together as in layered magnets or to a substrate as in thin films adsorbed on a surface. To have a description of such a system, theory must take the effect of this coupling into account or show that it is negligible. Not too much is known about this at present but as we shall see even the weakest interaction can have dramatic effects, in certain cases completely obscuring the ideal twodimensionalbehaviour. However, before attempting to describe a real system, we must understand how the ideal system behaves. There are a number of two-dimensional models which undergo a phase transition for which exact results exist. The best known of these is the famous Lenz-Ising model which Onsager (1944) showed has a continuous phase transition. Then there are the two-dimensional ferroelectric models (Lieb and Wu 1972) which can be regarded as generalisations of the Ising model. These models display a wide range of critical behaviour ranging from first to infinite order transitions. The spherical model (Berlin and Kac 1952; Joyce 1972) can be solved in any number of dimensions and in two has no transition at finite temperature. We shall not discuss such models any further here. We now turn to the problem of systems with an order parameter transforming according to a continuous symmetry. From a theoretical point of view there are two apparently conflicting pieces of evidence to be reconciled. The first is that it can be rigorously shown that a magnetic system described by a rotationally invariant Hamiltonian has no spontaneous magnetisation in two dimensions (Mermin and Wagner 1966; Mermin 1967) and the second is the numerical results from high temperature series expansions (Stanley and Kaplan 1966, 1967; Stanley 1967, 1968a; Ritchie and Fisher 1973; Moore 1969; Betts et al. 1971)which indicate a transition at a finite temperature. The evidence for such a transition is somewhat better for the planar rotator model (spins confined to a plane) than for the Heisenberg model. Low temperature expansions (Berezinskii 1970, 1972; Wegner 1967) give a magnetisation proportional to some power of the applied magnetic field between zero and unity and indicate the possibility of a sharp transition between such behaviour and the high temperature regime where the magnetisation is proportional to the field. The limit of the number of components going to infinity corresponds to the spherical model (Stanley 1968b; Kac 1971) which has no transition (except at zero temperature). Thus, two-dimensionalmodels range from Ising-like which have a transition at finite temperature to the spherical model which does not. One of the tasks of theory is to explain what happens in models between these extremes and
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especially to elucidate the nature of a transition in which there is no spontaneous magnetisation in the low temperature phase. It must also be able to answer the question: is there a special value of the number of components of the spin below which there is a transition at finite temperature or is the critical temperature just a decreasing function of the number of components which goes to zero in the sphericalmodel limit? 3.2. MAGNETIC SYSTEMS
We will restrict our discussion to classical systems for simplicity, although at very low temperatures quantum effects will be important in real systems. It has been shown that in certain models classical behaviour holds provided one is sufficiently close to the critical temperature (Hertz 1976). We will assume that the classical approximation is adequate to describe the behaviour of real systems in two dimensions. With this simplification in mind we consider a system on a hypercubiclattice in ddimensions with Hamiltonian
H
=
-3
J(lr-r’l)s(r)-s(r’), r, r’
.;
where s(r) = (si(r), s2(r), . ,s,(r)) is an n-component spin of unit length at a lattice site r, such that s,’(r) = 1. We shall restrict ourselves to ferromagnetic nearest neighbour interactions and approximate the Hamiltonian by
c:=i
H = ++Jdr(c7~)~.
(3.2)
The motivation for such a continuum approximation is that the existence of an ordered state is determined by the long wavelength fluctuationswhen only spin configurationswith neighbouring spins differing infinitesimallyfrom each other need be considered. As an example, let us consider the planar rotator (n = 2) model in two dimensions. We may parametrise s(r) by the angle +(r) it makes with some arbitrary axis, so that the partition function becomes
Z = j64 exp {-(1/2T)j dr(v#2),
(3.3)
where the temperature Tis in units of J/kB.Ignoring the fact that the original Hamiltonian of eq. (3.1) is periodic with period 27r, the partition function may be trivially evaluated. For example, the two-point correlation function is
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for all values of T. This implies that there is no phase transition in this model and the spontaneous magnetisation is zero for all finite temperatures, in accord with the Mermin-Wagner (1966) theorem. However, the power law decay of the correlation function is not what we expect in the high temperature regime. Moreover, the susceptibility is infinite for T < 8.n. This leads us to suspect that although the approximations we have made are very plausible, they are an over-simplification. The model described by eq. (3.2) is not soluble for general n because of the restriction Is(r)l = 1, except in the limit n + 00 when it becomes equivalent to the spherical model. Recently Polyakov (1975) has applied renormalisation group techniques to such systems near two dimensions. By exploiting the ideas of Berezinskii and Blank (1973) he developed a method of successively averaging out the short wavelength components of the spins in the spirit of Wilson (1974) while maintaining the restriction of unit length spins at each stage. Since the original work of Polyakov there have been a number of papers on the subject using different techniques and approximations (Migdal 1975; Kadanoff 1976; Brezin andZinn-Justin 1976a, b; Brezin et al. 1976; Amit and Ma). Brezin and co-workers have studied this problem by field theoretic techniques. They prove renormalisability of such theories in 2+ e dimensions thereby proving the scaling form of the thermodynamic quantities in the asymptotic critical regime, and derive approximate forms for the scaling functions. To leading order, the results are the same as Polyakov’s and the difficulties are also similar. Since Polyakov’s method is physically more obvious we will use this rather than field theoretic methods to construct the renormalisation group procedure for an isotropic system with the Hamiltonian of eq. (3.2). We take a hyperspherical Brillouin zone so that s(r) contains Fourier components sp with I q1 < A, where A is a cut-off of the order of an inverse lattice spacing. Berezinskii and Blank (1973) have shown that one can decompose s(r) into short and long wavelength parts as
where s’(r) is a slowly varying vector with Fourier components 0 < I q1
x
c:::
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and Blank f973) Ve, = s'(e; Vs').
(3.6)
Using eqs. (3.5) and (3.6), the Hamiltonian becomes
Choosing the new cut-off /r = A(l -N) we can average over the perturbation theory in the calculation of the partition function 2 = Trs, Tr+ exp( -H/T).
4s:
by
(3.8)
We find
Finally, we rescale the momenta q -,q(l +61) so that the range of q is restored to its original value and, since Is'(r)l = 1, s'(r) is dimensionless so that we must also rescale si + (1 dS1)si.After performing these calculations, we obtain an effective Hamiltonian of the same form as the original one and a rescaled temperature according to
+
dT/dl = -(d-2)T+ (n-2)T2/2~.
(3.10)
This procedure generates a systematic expansion in d-2 and T,so that, provided the critical temperature is small, it will give us a good approximation in the low temperature ordered phase. From eq. (3.10) we see that there are two fixed points T* = 0
T" = 2n(d-2)/(n-2).
(3.1 1)
The T = 0 fixed point is stable with eigenvalue 1, = -(d-2) and the other is unstable with eigenvalue 1, = d-2+0(d-2)2. We identify this as the critical point T, = T*, below which temperature there will be a non-zero magnetisation (Polyakov 1975; Brezin and Zinn-Justin 1976a). From eqs. (3.10) and (3.11) we see that provided, n > 2 and (d-2) < 1, T, is small and
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[a.3 5 9 9
so this procedure provides a reasonable approximation in the low temperature
regime. Unfortunately, these techniques do not allow us to extrapolate far into the zero field, high temperature regime because we have made the implicit assumption that we are in the ordered phase. When we try to use these recursion relations for the disordered phase (T > T,) we find the solution TJT(1) = 1 - (1 - T,/T) exp (d- 2)l
(3.12)
which clearly is nonsense when exp (d-2)l > (1 -T0/T)-’ since we expect that in an exact treatment as 1 + a,T(1) a. This problem is especially acute in two dimensions when the recursion relation becomes dT/dl = (n-2)T2/2n,
(3.13)
which has only one unstable fixed point at T = 0 and has the solution T(1) = T(l -T(n-2)1/2~)-’,
(3.14)
which again breaks down at 1 N 2n/(n-2)Te This does, however, show that there is no spontaneous magnetisation at any finite temperature in agreement with the Mermin-Wagner theorem. These considerations allow us to define a characteristic length scale - the correlation length r(T)
N
N
ll-T/Tcl-1’(d-2)
d>2
exp[2n/(n-2)Tl
d=2
(3.15)
and homogeneity relation for the momentum dependent susceptibility (Pelcovitz and Nelson 1976) (3.16) which implies the scaling laws for the susceptibility x(T)
-
t2-q
withq = T,/2n.
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Unfortunately one cannot push the calculations much further at present. All a renormalisation group transformation does is to relate a thermodynamic quantity calculated from the physical Hamiltonian H,, to one calculated from the effective Hamiltonian H(1). The hope is that one can do the calculations with H(1) by more conventional techniques (perturbation theory, high temperature expansions etc.), thereby deducing the physical quantity as in eq. (3.16).We can compute x(T(I), qe') when qe' is large, but not in the disordered phase for small qe'. This is equivalent to saying that we can calculate the two point correlation function (s(r).s(O))when r/< < 1 but not when r/( > 1. If the high temperature phase is truly disordered in the sense that there are no Goldstone-like excitations (spin-waves), the correlation function should fall off as exp{-r/r} for r B but we cannot prove this by these methods (see, however, Fisher and Nelson 1977). This technique therefore does not prove that there is not another phase transition at a temperature T > T,, and in particular that the Heisenberg model in two dimensions does not have a phase transition. It is, however, difficult to imagine what the broken symmetry phase would be if this were not true. In the following, we shall assume that for n > 2 there is no transition in two dimensions. This is supported by other arguments (Kosterlitz and Thouless 1973).
r,
3.3. THEPLANAR ROTATOR CASE* - TOPOLOGICAL ORDFB When n = 2, the situation is somewhat different because then eq. (3.10)reads dT/dl = -(d-2)T
(3.17)
which, in this approximation, implies that T, = co. In two dimensions the temperature does not change on iteration, but according to eq. (3.4) the correlation function falls off only as r - T / 2 nwhich is not characteristic of a true disordered phase so this treatment cannot tell if this physically important case has a transition or not. Two approximations were made: (i) we ignored the fact that 4(r)varies only between 0 and 2w and (ii) more importantly, the topology of the field is a circle. These approximations are also made in the n > 2 case, but the curvature of the sphere can be detected locally and this is done to some extent in the perturbative treatment of the Hamiltonian. The possibility that the angle 4(r) goes through a multiple of 27r was ignored in the previous formulation of the theory. For example, the con-
* See Kosterlitz 1974.
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figuration shown in fig. 4a corresponds to a change of +2n, while that of fig. 4b to a change of -2z. These are closely analogous to vortices in a superfluid. One may readily calculate the energy of such a configuration in a system of radius R. In the configuration of fig. 4a ql(r) = rJrI
c1 = 12,
(b)
Fig.4. Typical singular configurations with a phase change of (a) A d = +2n and (b) A 4 = -28. The dots mark the position of the singularity.
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whence, with the Hamiltonian of eq. (3.2)
E
= R log R / t + O ( l )
(3.18)
where z is the lattice spacing. The entropy of such a conliguration is log R 2 / t 2 since its centre can be placed anywhere in the system, so that the free energy is
AF
= ( R - ~ Tlog ) Rlt.
(3.19)
At sufficiently low temperatures (T-c +n) such singular configurations are forbidden, while at high temperatures (T > 3). they can be thermally activated, thus increasing the disorder of the system. Of course, there are fluctuations about such configurations and also pairs of singularities of opposite strengths which have a finite energy proportional to the logarithm of their separation. This leads us to the concept of topological order (Kosterlitz and Thouless 1972, 1973) whose presence or absence may be determined in the following manner. The total change of phase on going round a large closed path in the system will be determined by the number and strength of the singular points enclosed by the path. Thus, if there are isolated singularities present (high temperature regime), the number of these will be proportional to the area enclosed by the path. The average total phase change will be proportional to the length L of the contour. If, on the other hand, only pairs of singular points are present (low temperature regime), the total average phase change will be determined by the number of pairs cut by the path, and so will be proportional to (Ld)’’’ where dis the mean separation of pairs. Now consider a large system with periodic boundary conditions. The number of multiples of 2n by which the phase changes on a path going right round the system is a topological invariant so that
are numbers defining a particular state. Transitions between one state and another take place if a pair of singularities is formed, and recombine after one has gone right round the system, which will cause a change of one in either n, or ny. There is, however, a logarithmically large energy barrier which can be only overcome at sufficiently high temperatures.
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[CH. 5 , s 3
-
It is straightforward to show that for a power law interaction of the form J(r) r - ( z + O J the energy of an isolated singularity is proportional to log R/z when Q > 2, and the same arguments will hold, except that the critical value of Twill be changed. For 0 < Q < 2 the energy is proportional to (R/z)'-" so that such configurationscan be ignored. For the two-dimensional Heisenberg model, on the other hand, there is only one topological invariant (Kosterlitz and Thouless 1973), but the energy barrier for changing this is of order unity and so transitions can take place at any finite temperature. This supports the assumption of the previous section that there is no transition in this case. Now we will try to put these ideas on a more formal and detailed basis following the ideas of Toulouse and Klkman (1976) who discuss the classification of topologically stable defects. They show that in a system with an order parameter described by an n-component unit vector there are topologically stable singular configurations (defects) which have a surface of singularities of dimension d-n, where d is the spatial dimension of the system. There are many well known examples of this in physics - vortices in superfluid 4He, disclinations in nematic liquid crystals (DeGennes 1974), disclinations and dislocations in crystals (Friedel 1964; Nabarro 1967) and many more. An example equally well known but less obvious when looked at from this point of view is the king model. The only way of making deviations from the completely ordered state is to flip blocks of spins. The boundaries between the domains of spins of opposite orientation are the defects-points in one dimension, lines in two, and surfaces in three. By specifying aparticular configuration of defects one specifies the configuration of spins, and so, knowing the energy of the defect configurations, one can,in principle, do the statistical mechanics of the Ising model. There are two separate problems to solve before attempting the statistical mechanics, (i) the classification of the possible types of defect, which depends only on the topology and (ii) the energy of a specified configuration of defects which depends not only on the number and types of defect present but also on the particular Hamiltonian of the system. The latter problem is much the harder. As an example, let us consider a one dimensional Ising system with a Hamiltonian (3.20) which has a trivial topology. Imposing cyclic boundary conditions, the spin configuration is completely determined by the 2N point defects at rl , r z , ...,
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r2N.It can be shown that the energy of such aconfiguration is 2N
E2N
E(r,-rj)+2Nfl~
= i +I
where E(r, -rj) = (-)‘+’[a(l -a)]-‘(
Irl -rj I ’
-O-
+
1) O(l ri-rj
I -7
(3.21)
and fl is the ‘chemical potential’ of a defect. It is possible to do approximately the statistical mechanics of this system using a renormalisation group procedure (Kosterlitz 1976; Anderson et al. 1970; Anderson and Yuval 1971) but this is not the point here. We merely wished to illustrate that one can have a trivial topology but the energetics can be difficult. Returning to the planar rotator case in two dimensions, because n = d we have topologically stable point defects (Toulouse and Kleman 1976) corresponding to typical spin configurations shown in fig. 4. A ‘topological current’ can be defined (Patani et al. 1976)
whose divergence obeys (Patani et al. 1976)
j y ddr a,,j,
= Nad,
where N is an integer and adthe surface area of a d-dimensional unit sphere. This implies that a, j , has &function singularities a,j, = ad
C1 n, &-3,
(3.23)
where n, is an integer (the strength of the defect at rJ. Equation (3.23) has the solution
where J,(r) is a function of r such that aJ,, = 0. Up to now our discussion has been general, valid for any unit vector order parameter with n = d. We must now solve eq. (3.22) for J in terms o f j in order to write the Hamiltonian in terms of the defects. This is not so straightforward, and to the authors’ knowledge has been solved only for the special cases of one and two dimensions, which, fortunately, includes the case under discussion.
3.M. KOSTERLlTZ AND D.J. THOULESS
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[CH. 5,P 3
For n = d = 1, the solution is trivial with the expected result
Nr) = A 4 = Ci (-)%(r-ri).
(3.25)
For n = d = 2 we can parametrise s(r) by an angle 4(r) whenj,,(r) = c,va,~(r) and so
It is straightforward to see from eq. (3.24) that the Hamiltonian of eq. (3.26) can be written as
H = H,+Ho,
(3.27)
where H, is the part due to the defects and Ho describes the smooth fluctuations about these. An explicit calculation taking into account the underlying square lattice structure gives (Kosterlitz 1974)
where z is the lattice spacing and 2p N n2 is the energy of a configuration in which two defects of strength & 1 are at one lattice spacing apart. The term ninj log R / z imposes the restriction that nr = 0. Ho may be written
zii
Ho = f 1(V$)'d2r
(3.29)
where J/ is a single-valued field describing the deviations from the local minima of energy H,. Note that a specification of the defects is insufficient to completely describe the spins, since small fluctuations of the spins from their positions shown in fig. 4 will not change the topology. The angle $ (r), which obeys $d$(r) = 0, completes the description of the spin configurations. In what follows we shall assume that only defects of unit strength are present, since these are more favourable than those with n, 8-2. We have investigated the effect of keeping more defects and this does not change the qualitative aspects of the solution. In such a case, the partition function may be written as (Kosterlitz 1974)
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where Tr
=
Sa)O C
“ --iI:2rI 1
N=O
401
...d2rjN.
(N!)
Since there must be N defects and N anti-defects for the condition n, = 0 to be obeyed, we must include the weight factor ( N ! ) - 2 . The domain of integration is such that lri-r,l > T, since two defects cannot be less than a lattice spacing apart. In this approximation, the partition function for the magnetic system becomes exactly the grand partition function of a neutral Coulomb plasma of 2N charged particles of diameter z. This finite short distance cut-off prevents the collapse of pairs of oppositely charged particles into an infinitely deep potential well (cf. Hauge and Hemmer 1971). The treatment of this system by renormalisation group methods is described in detail by Kosterlitz (1974), so we shall merely outline the procedure. The cut-off, or lattice spacing is scaled to ~(1+62) and the pairs of defects lying between z and z(l+61) are averaged out. This procedure reproduces a Hamiltonian of the same form as eq. (3.28) but with rescaled parameters (according to Kosterlitz 1974) d(l/T)/dl = -4z312/T2 +O(A4) dI/dl
=
-A(n/T- 2) +0(13),
(3.31)
where I is the activity exp(-p/T) and eq. (3.31) is valid for small I. Various irrelevant terms describing interactions between more than two defects are generated by this procedure, but they do not affect the main results, and have been ignored. Equations (3.31) have solutions as shown in fig. 5. The I = 0 axis is a fixed line which is stable for T < +z and unstable for T > +a. We interpret this as follows. For T < T,, assuming that our initial Hamiltonian is within the range of validity of eq. (3.31) (or that there are no other fixed points inaccessible to our approximations), which implies that ignoring defects of more than unit strength is not unreasonable, I will scale to zero. Thus our system is equivalent to one in which there are no defects, and the naive Gaussian form of the Hamiltonian is valid. For T > T,, the opposite is true because eventually on iteration the activity increases and we go outside the range of validity of the recursion relations. We interpret this as indicating that we are in the disordered phase because a large number of defects clearly corresponds to less order, provided, of course, there are no other fked points except the one at T = GO.In this regime we come up against
J.M.KOSTERLITZ AND D.J. THOULESS
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exactly the same problems as those discussed previously (Polyakov 1975; Brezin and Zinn-Justin 1976a). Within the approximations outlined above, we find the following scaling laws near the unstable fixed point T* = +n (Kosterlitz 1974): (i) T, is the solution of n/Tc-2 = 2n exp( -PIT,) +O(exp -2p/Tc);
(3.32)
(ii) Correlation length c(T)=
T
00
= exp a(T/Tc- 1)-lI2 T > T,;
(3.33)
Fig. 5. Trajectories in the (l/T,d) plane of eqs. (3.1). The broken line indicates the physical initial values of 1 = exp ( - p / T ) .
(iii) Susceptibility x(T)=
T
00
- t2-"
T > T,
(3.34)
with q = 4. This is in disagreement with a recent calculation by Luther and Scalapino (1977) who find q = 8-lI2; (iv) Free energy
AT)
-
A,
r 2+fo(T),
(3.35)
where fo(T) is analytic at T,, This implies that the specific heat exponent ct = -a,so that the specific heat is smooth at T,. We are unable to tell whether there is a bump in the specific heat at T, or some other value of T. In the F model, which also has a similar essential singularity in the free energy, the specificheat has a maximum below T, (Lieb and Wu 1972).
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(v) Correlation function (s(r)*s(O))
-
Irl-qfT)
T < T,
(3.36)
where q(T) N T/2n at low temperatures and as T + T,- we have q(T) N & -C(l -T/T,)'''. This treatment is good only for T < T,,and we have not been able to find the behaviour of the correlation function in the disordered phase when r % although, if there are no other fixed points, we expect it to decay exponentially. Of course this treatment can give only a very crude estimate of T, since we etc., which would have ignored terms in the initial Hamiltonian like ( arise from an expansion of si-sj = cos(4i-4,). Although such terms are irrelevant and do not affect the asymptotic critical behaviour, they will affect the value of T,. Villain (1975) has constructed a model in which the defect +Gaussian form of the Hamiltonian is exact. He chooses an interaction V(4) which keeps the important feature of the Hamiltonian, namely the periodicity, and adjusts the coupling constant so that exp{A(T) V(+)} is as good an approximation as possible to exp{cos+/T). Replacing 1/T by A ( T ) in eq. (3.32) he gets a T, which is in good agreement with the series expansion estimates of Stanley and Kaplan (1966, 1967). Further evidence for the susceptibility of eq. (3.34) is provided by the work of Camp and Van Dyke (1975) who re-analysed the high temperature series in terms of a susceptibility behaving as exp{u(T/T, - l)-'} and find some support for such a form but with v N 0.75. However, corrections to scaling are large which may explain this discrepancy. Zittartz (1976) has also looked at the two-dimensionalplanar rotator model from a different point of view, and reaches quite different conclusions from us. He finds a free energy in an external fieldf hl K ( T ) with K(T)= 4/(4- q(T)) and he gives various estimates of K(T).The essential difference between his results and ours is that he finds, on the basis of low temperature expansions, the spin wave stiffness constant vanishesat a higher temperature than that when q(T) = 4. At higher temperatures than this, he finds that all correlation functions are finite so that one may assume that the free energy is analytic. Thus, he identifies T, as the solution of q(T,) = 4. On the other hand, we find that the spin-wave stiffness constant vanishes discontinuously at a much lower temperature.
r,
-I
3.4. ANISOTROPIES IN TWO-DIMENSIONAL MAGNETS All real layered magnetic systems have both anisotropies within a plane due
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J.M. KOSTERLITZ AND D.J. THOULESS
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to crystal field effects etc. and weak coupling between the planes. These have very dramatic effects even for a ratio of interplanar to intraplanar couplings lJl/JllI as low as which is about the lower limit for real systems. In fact even such a small ratio can completely mask the critical behaviour of the ideal two-dimensional system. The same holds true for anisotropies in the plane where a ratio of anisotropy to exchange couplings of the same order of magnitude can completely obscure the critical behaviour of the isotropic system. Recent experiments on K,NiF4 and K,MnF4 (Birgenau et al. 1970b; Als-Nielsen et al. 1976) bear this out. These are nearly ideal two-dimensional Heisenberg antiferrornagnets in which there is a small Ising-like anisotropy of about lo-’ JIIand an interplanar coupling JI JII(de Jongh and Miedema 1974). In the temperature range I T/T,-1I 2 lo-’ an order parameter exponent fl II 0.138 is found (Birgenau et al. 1970b), although Binder and Landau (1976) point out that, by shifting T,,the data is quite consistent with the two-dimensional Ising value of 0.125. The staggered susceptibility exponent as measured by neutron scattering is consistent with the Ising value of 1.75 (Als-Nielsen et al. 1976). Thus, despite the small magnitude of the Ising anisotropy, behaviour characteristic of the two dimensional Ising model is observed. This is in complete contrast to the situation in three dimensions. Pelcovitz and Nelson (1976) have investigated the effect of anisotropies near two dimensions for Heisenberg-like models with the number of spin components n > 3 by Polyakov’s (1975) technique. They consider a Hamiltonian with both Ising-like and cubic anisotropy N
where oi denotes one of the components of si. They find that g2 and v are strongly relevant with eigenvalues about the isotropic fixed point (eq. 3.1 1) = 2, = 2 in two dimensions, while g, is marginally irrelevant. In the case where there is no cubic anisotropy, v = 0, this means that the coefficient g, of the o: terms grows very rapidly
Thus the effective Hamiltonian at the Zth stage of iteration has the same form as eq. (3.37) but with a large coefficient g,(Z). If g, < 0, the minima of the Hamiltonian are at oi = f l and so the system is equivalent to the Ising
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model. Similarly, if g , > 0, the system crosses over to rn = n- 1 behaviour. Specialising now to n = 3, the combination g , exp(4n/T) is a renormalisation group invariant, so according to conventional lore, the critical temperature will be given by (3.39) where z, is some constant, This has the solution
where g , is the strength of the anisotropy measured in units of the isotropic exchange coupling. Since the critical temperature of the two-dimensional model is finite, then z, is also finite, but a more detailed analysis is necessary to estimate its value. The same is true if g , > 0 since, according to the theory of the previous section, the two dimensional planar rotator model also has a finite T,. The lower limit in real materials is g , which means that the shift in T,is fairly large since log lo6 is still rather small. To see the pure isotropic Heisenberg critical behaviour two conditions must be simultaneously satisfied (i) T must be sufficiently far from T,(g,) so that the effects of the anisotropy are not felt too much and (ii) T must be close enough to T,(g, = 0) (which is zero in this case) so that we are still in the critical region of the pure system. For real materials, by virtue of eq. (3.40) these two conditions are impossible to satisfy, and so we are unlikely to see pure two dimensional Heisenberg behaviour. The same considerations hold for any relevant perturbation about the isotropic system (cubic etc.). Since in two dimensions, there are an infinite number of possible relevant perturbations (Brezin et al. 1976) (in contrast to the situation near four dimensions (Zia and Wallace 1975; Wegner 1972a,b, 1974) a real system is almost certain to have one or more of these present in the interaction due to crystal field effects etc. Binder and Landau (1976) have recently tested these ideas by Monte Carlo techniques and find consistency with the predictions T,(g2 = 0) = 0 and T,(g2)cc [logg,I-', although their data does not rule out a non-zero T,(g, = 0), and then they find consistency with a more conventional form N
with Q N 4. Assuming power law behaviour for the susceptibility and specific heat they find the ideal two dimensional Heisenberg exponents yH = 3,
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[CH. 5,$3
aH N -2, which are similar to the high temperature series estimates, (Stanley 1967, 1968a; Ritchie and Fisher 1973; Moore 1969) but their data are also consistent with an exponential susceptibility. In the presence of an Ising anisotropy, a two-dimensional Ising exponent of 1.75 is observed close to T,(g,) and a yclf N 2.2 further above T,. These results are not surprising if one considers the following simple picture. Suppose the ideal isotropic susceptibility behaves exponentially so that the effective exponent yerr(T)N 4n[T-T,(g,)]/T2 for T % T,. This must match on to the value yerr(Tc)= 1.75. Thus, the effective exponent will rise to a maximum value ymaxat some temperature above T, and then fall to its Ising value. The susceptibility data will have approximately the same slope, when plotted logarithmically, over the largest range of temperature, when yerr(T)= ymax.Thus, the data will appear to predict a definite value of ycft outside the Ising-like critical region. The situation in layered magnetic systems is even worse because there will not only be anisotropies within the plane which are themselves sufficient to obscure pure isotropic behaviour, but also the interplanar coupling JL which is at least 10-6J,lin real materials. It can be generally shown (Liu and Stanley 1972, 1973) that the exponent for cross-over from two to three dimensions is equal to the two-dimensional susceptibility exponent y. Thus, in purely isotropic exchange coupled layered Heisenberg magnets (n > 3), the interplanar couplingJLwill grow on iteration like
(3.41) Using the same arguments as for anisotropies in the plane, the critical temperature is shifted to
where x, is a finite constant and JL is measured in units of JII ,the intraplanar coupling. This form is verified in the spherical model where (Joyce 1972) Tc(JJ = n{log (32/JJ}
-I.
(3.43)
These results imply again that one is unlikely ever to see isotropic twodimensional critical behaviour experimentally. It is also likely that attempts (Bloembergen 1975) to estimate the ideal two-dimensional critical temperature from experiments on layered systems are doomed to failure at least until theoreticians produce a more detailed theory of layered systems, including
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407
both interplanar couplings and spin anisotropies, capable of predicting the functional form of the measured thermodynamicquantities. If the Ising anisotropy is much greater than the interplanar coupling, on the other hand, the two-dimensional behaviour will be Ising-like and the shift in T, due to Jl will be small T,(Jl) -T,(O)
N
JI",
which is small since 4 = 1.75. Thus the two-dimensional Ising behaviour will be observed, as it is in K2NiF4 and K,MnF4 (Birgenau et al. 1970b; AlsNielsen 1976), before the final cross-over to three-dimensional behaviour very close to T, . So far our discussion has been limited to Heisenberg models with n 2 3. The exceptionalcase of n = 2 has been studied by several authors (Pokrovsky and Uimin 1973, 1974; Berezinskii and Blank 1973*) in the low temperature region where the Gaussian approximation of eq. (3.3) to the Hamiltonian is valid. Consider the Hamiltonian H = 3 (84)2d2r+hp cosp4(r)d2r,
(3.44)
where the anisotropy cosp4 corresponds to a uniform magnetic field (p = l), an Ising ( p = 2), a cubic anisotropy (p = 4) etc. In the low temperature region, ignoring the effects of defects, for small h, we have (Berezinskii and Blank 1973; Birgenau et al. 1970b)
where
1, = 2-p2T/4~, If we take the defects into account this is modified to (3.46) which can readily be derived by the methods of Kosterlitz (1974), with q(T) N T/2n for low temperatures and q(T) N a-C(T,-T)''2 for T N T,.
* Note that these authors are correct only in the n = 2 case because they omitted to take into account the rescalingof the temperaturefor n 3 3.
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J.M.KOSTERLITZ AND D.J. THOULESS
[CH. 5,g 3
Since q(T,) = t , for p < 4 such a perturbation is relevant at the critical temperature of the pure isotropic model and so the system will cross over to some other critical behaviour - for p = 2 to Ising and p = 3 to the threestate Potts (1952) model. For p = 4 (cubic anisotropy) the perturbation is relevant below T,(h, = 0) and marginal at T,(O). It is not clear to what the system crosses over in this case, but when h4 is very large the system reduces to two Ising models with a coupling which vanishes as h4 -+ 00 which is very similar to that of the Ashkin-Teller model (Ashkin and Teller 1973; Kadanoff and Wegner 1971). Forp > 4, on the other hand, the perturbation is relevant for T ;5 16T,/p2 and irrelevant for 1 > T/T, > 16/p2.Thus there should be two transitions in such a system. At T N T,(O) there will be a transition of the type discussed in 0 3.3, with no spontaneous magnetisation but an infinite susceptibility, and at a lower temperature T,, N 8x1~'another transition to a phase in which there is a non-zero spontaneous magnetisation. It is amusing to speculate that traces of this low temperature transition may be detectable in layered systems with the appropriate symmetry. The effects of an interplanar coupling JLon an isotropic n = 2 system are almost as dramatic as for n > 3. By the same arguments as we used before, it can easily be shown that (3.47)
where z, is a finite constant and a is the constant which appears in eq. (3.33). An Ising-like anisotropy in the plane will lead to a shift in T, of the same form as eq. (3.47). Since the shifts in T, are large for realistic values of the anisotropy, the same pessimistic conclusions hold in this case as for the Heisenberg case. To conclude, in any real layered magnet, one is faced with a triple crossover phenomenon to analyse (de Jongh and Stanley 1976) since interplanar couplings and anisotropies as small as make themselves felt in a dramatic way. This leads the authors to think that, except in the Ising case, an analysis of experiments in the critical region in terms of exponents is not a fruitful approach but that the data should be fitted to some, as yet, non-existent functional form for the quantity under investigation. 3.5. SUPERFLUIDS
Superfluidity in thin films of 4He is now well established, so one runs up against the same sort of problem as in the two-dimensional planar model, but in a more acute form. One could argue in the magnetic case that it is the
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409
coupling between the layers in a real system which is responsible for the observed finite critical temperature, as theory predicts for the Heisenberg model. Superfluidity is observed in films about two atomic layers thick (Scholtz et al. 1974), which is as close to a two-dimensional system as it is possible to get. However, Hohenberg (1967) showed that for a two-dimensional Bose fluid, off-diagonal long-range order (Penrose and Onsager 1956) does not exist in the thermodynamic limit. Thus, if one insists that long-range order is essentialfor superfluidity,one has to look for other ways out. One possibility that has received a lot of attention (Imry 1969; Jasnow and Fisher 1969,1971) is that, although the theorem is rigorous in the thermodynamic limit, real systems are finite in extent and so a Bose-Einstein condensation might still take place. In the ideal Bose gas one finds a macroscopically occupied state for T < T,(A)
cy
T3/10g Ad,
(3.48)
where d is the thickness of the film, A the area of the system and T3the condensation temperature of the bulk. As the area A * 00, T,(A) + 0 consistent with the Bogolyubov inequality (Hohenberg 1967). However, for a realistic system, log Adis not very large, and so Bose-Einstein condensationis possible. The methods of Hohenberg (1967) have been extended to finite systems in which one dimension is small (Jasnow and Fisher 1969, 1971), and upper bounds established for a quantity analogous to the condensate fraction
where 4 is a slowly varying function of density p, temperature T and film thickness d. This again implies the possibility of having condensation in a large but finite region. A second way out along these lines is to argue that, since the film of 4He is adsorbed on a substrate, it is possible that inhomogeneities of the substrate cause spatial variations of the single particle energies. It has been shown that for certain forms of the spatial variation, Bose-Einstein condensation in an otherwiseideal Bose gas of infinite area can take place (Dash 1975). A third approach (Berezinskii 1972; Kosterlitz and Thouless 1973; Kosterlitz 1974) is to take the point of view adopted in 8 3.3 and ask the question - is superfluidity possible without long-range order? Let us suppose the condensation occurs in small regions of the system so that a condensate wave function can be defined
J.M. KOSTERLITZ AND D.J. THOULESS
410
+(r) = I +(r)l ei+(').
[CH.$ 0 3
(3.50)
Provided the length scale of spatial variations of the condensate wave function is sufficiently large, one should be able to write down a Ginzburg-Landau free energy functional to describe the system near T,,and so we will arrive at an effective Hamiltonian of the form
We assume that we are well below the mean field temperature, so that r,(T) -+ 0 and fluctuations in the amplitude I $1 will be small. These assumptions are supported by some recent work of Amit and Ma who showed that for n > 2 (Is1 - 1) is an irrelevant operator whose coefficient has a fixed point value of infinity in two dimensions, so that amplitude fluctuations are damped out on iteration. If the above assumptions are valid, the Hamiltonian for a two-dimensional Bose fluid can be written in an equivalent form to the planar rotator
'
H = h2n(r)/2m j(Vc$)'d2r,
(3.52)
where m is the effective mass of a helium atom and n(T)is the effectivenumber density of helium in the film. Berezinskii (1972) has derived a lattice gas description of a two-dimensional Bose fluid which is equivalent to eq. (3.52). Note that superfluid density p,(T) is equivalent to the spin-wave stiffness constant of a magnetic system, and may be defined as (Hohenberg and Martin 1965; Halperin and Hohenberg 1969a,b)
If the naive Gaussian form of the Hamiltonian of eq. (3.52) is used, ignoring any possible singular configurations
as expected.
The singular configurations in this case are the usual vortices in 4He, which in a two-dimensional system are defined by their position r , and vorticity n, Exactly the same arguments as for the magnetic system may be used. The energy of an isolated vortex in a system of radius R is (nh2n(T)/m)log (R/T), so that the critical temperature is approximately nh2n(T)/2m.So, at sufficiently
.
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411
low temperatures there will be no free vortices, only clusters of zero total vorticity and the system will show the characteristics of a superfluid, although there is no long-range order in the usual sense. If interactions between the vortices are taken into account, the critical temperature is lowered to nhZn(T,)/2mT,- 1 N 2n exp( -p/T,), where p is the vortex core energy. The superfluid density as defined by eq. (3.53) can be shown to behave close to T, as (see fig. 6 )
(3.55)
Fig. 6. The superfluid density p,(T) for a two-dimensional superfluid. Note the discontinuous drop to zero at To.
Note that p,(T,) is finite, so we predict that the superfluid density goes to zero discontinuously at T,.This is consistent with any scaling theory such as that based on the concept of a helicity modulus (Fisher et al. 1973)and does not mean that the transition is first order. The vortex model as formulated above predicts that the superfluid onset temperature and coverage are related by (n,)d/T,
N
2m/(nh2b)N 15 x 10’’ cm-’ K-’,
where d is the thickness of the film measured in layers, each layer being b N 3.6 A thick, and m is the mass of a 4He atom. (n,) is the average superfluid number density in the film. The experimental data seem to support a thickness versus onset temperature law of this form, but the experimental
J.M. KOSTERLITZ A N D D.J. THOULESS
412
[CH. 5,g 3
situation is still confused (Dash 1975). There is some disagreement in the interpretation of the average superfluid density (n,) (Padmore and Reppy 1973) and also the thickness d is presumably not completely uniform across the film due to substrate inhomogeneities (Dash and Herb 1973; Cole et al. 1975). Also the measured superfluid density will depend on the experimental technique. For example an experiment carried out at a finite frequency at fixed temperature shows that the apparent critical coverage decreases with increasingfrequency (Dash 1975; Chester et al. 1972; Chester and Yang 1973). A theory has recently been proposed (Dash and Herb 1973) to explain these variations - the connectivity theory, which exploits the substrate inhomogeneities. The similarity of most adsorption isotherms for different substrates is normally taken as an indication that the effects of heterogeneity are healed out as the thickness of the film is increased. However, this is not necessarily true if the thickness of the film is small compared to the scale of lateral variations of the binding energy to the substrate (Cole et al. 1975). In such a situation, it is argued that, because the vapour pressure must be constant over the film, the thickness dand the binding energy a are related 6d/d cc 6a/a. Thus the thicker the film, the greater the variations in thickness. When, however, the thickness is larger than the lateral scale of inhomogeneities the film thickness becomes more uniform. On an inhomogeneous substrate, there will be pools of thicker film which will be the first to become superfluid, and so the onset temperature will be technique dependent. In a quartz crystal microbalance type of experiment (Chester et al. 1972) which is carried out at a frequencyfin the megacycle range, superfluidity will be felt when the characteristic size of the pools of thicker film is of the order of the third sound wavelength at the given frequencyf, so that onset will depend on the frequency. There is still some controversy about this theory (Chester and Yang 1974). There is an alternative explanation of the increase in onset temperature with frequency which does not require inhomogeneity of the substrate. Suppose that in a uniform film, the superfluid velocity decays exponentially u&) exp( -t/z(T)) where T(T)is a relaxation time for T > T,,and a much slower decay for T < T,. Superfluid flow must decay even below T,owing to the presence of vortices (Langer and Reppy 1970). During each cycle, a flow relative to the substrate is induced and provided the relaxation time r(T) > l/f a superfluid-like response will be seen and will have the observed frequency N
CH.5 , # 31
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dependence. Also the apparent superfluiddensity will go continuouslyto zero, instead of discontinuouslyas in eq. (3.55). The true explanation probably lies somewhere between the two, but almost certainly substrate inhomogeneitiesare important in very thin films. There are many problems still outstanding in thin films. The theory of dynamics near onset is still an open one and an obvious problem is the anomalously high attenuation of third sound near onset. It is interesting to speculate that this may be due to vortices in the film. 3.6.
CRYSTALLINE ORDER IN TWO DIMENSIONS
We now turn to the problem of two-dimensional crystalline order and the melting transition. This has been observed in a number of gases adsorbed on various substrates. The structure of the adsorbed gas has been directly observed for certain combinations of gas and substrate by LEED and neutron diffraction (Dash 1975). For example for N2monolayers on graphite (Kjems et al. 1976) the measurements indicate that at high temperature (T> 78 K) there is short-range but no long-range order at low and high coverages. At intermediate densities a triangular lattice in registry with the substrate (a 1/3 x 4 3 epitaxial structure, Dash 1975) is observed, while at higher coverages and low temperatures a close packed triangular lattice not in registry with the substrate is also observed, with a lattice spacing very close to that of (1 11) planes in solid bulk Nz The data is consistent with a continuous transition to a liquid or gaseous state. No direct observations of the structure of 3He and 4He monolayers on various substrates have been made, but by comparison with the Nz observations, the specific heat data can be interpreted to indicate that at intermediate coverages on graphite there is a 43 x 1/ 3 epitaxial phase, and at near monolayer completion there is an ordered triangular crystal not in registry with the substrate. We will discuss the theory of the high density unregistered monolayer under the assumption that the substrate provides a uniform binding potential. Effects of the periodicity of the binding potential on the melting transition are not taken into account. This seems to be a reasonable approximation except at coverages very close to monolayer completion when additional complications arise due to promotion of some atoms to the next layer and vapour (Elgin and Goodstein 1973, 1974). The low temperature specific heat of such layers follows the Debye law for an ideal two-dimensional crystal especially for 3He and 4He on graphite (Bretz et al. 1973; Hering et al. 1976). The peaks in the specsc heat are identified with the melting transition. This
.
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J.M. KOSTERLITZ AND D.J. THOULESS
[CH. 5 , # 3
interpretation is consistent with the neutron diffraction peaks which gradually broaden (Kjems et al. 1976). The theory of the two-dimensional crystal near the melting temperature is based on the harmonic approximation, ignoring quantum effects, so that the Hamiltonian for an isotropic crystal is
wherepij is the stress and the strain uij is
with u the displacement of an atom from its equilibrium position. For an isotropic crystal
Equation (3.57) is also true for two-dimensional hexagonal and triangular lattices (Landau and Lifshitz 1959) the latter being relevant for helium on graphite. As discussed in 0 2, the mean square deviation of an atom from its equilibrium position is proportional to the logarithm of the area of the system, so there is no positional long-range order (Mermin 1968). However, consider the correlation function analogous to the superfluid or magnetic (V+ * V4} correlation function [eq. (3.53)]
which is a measure of directional long-range order. Here R = na+mb is the equilibrium position of an atom in the unstrained and r(R) the position in the strained lattice. In the harmonic approximation, taking into account only the long-wavelength phonon excitations, this quantity turns out to be equal to u z at all temperatures. Hence the crystal has long-range directional but not positional order (Mermin 1968). A picture of such a system may be obtained by imagining an unstretched rubber sheet with a lattice drawn on it. The long-wavelength excitations correspond to stretching without cutting the sheet, so that locally the lattice is perfect, but the displacements of the lattice points from their original positions can be very large. It is difficult to imagine how correlations between the crystal axes can be lost by such a process. If, however, we allow dis-
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Iocations and disclinations (Friedel 1944; Nabarro 1967) which correspond to cutting the sheet, taking out or adding a piece and gluing the edges together again in such a way as to locally restore the lattice structure, we can see that directional correlationscan also be lost (see fig. 7). The types of topological singularity,or defects,are slightlymore complicated here than in magnets or superfluids. A given perfect lattice is invariant under translation by R = na+rnb where u and b are the lattice vectors, but it is also invariant under rotations through multiples of 0 = cos%b, the angle between the lattice vectors, +n for a square and +a for a triangular lattice. Thus, a crystal can have both translational singularities - dislocations of the Burgers type-or rotational singularities-disclinations (Friedel 1964; Nabarro 1967). The latter are analogous to vortices in a superfluid and to disclinations in nematic liquid crystals, while the former do not exist in these systems. In smectic B liquid crystals, on the other hand, both can, in principle, exist. In two dimensions, the description of such singularities becomes simple since their position is specified by a point rather than a line as in three dimensions. If we take a closed contour around a disclination, a unit vector rotates by a multiple of +n for a square lattice, and the displacementu changes by an amount proportional to the radius of the coutour, while for a translational dislocation gdu = b,
(3.59)
where b is the Burger’s vector of the dislocation (Friedel 1964; Nabarro 1967). Having identified the types of singularity which can exist, we must ask about their energy. For a two-dimensional crystal an isolated disclination has an energy proportional to the urea of the system and so is most unlikely to occur since the energy will dominate the entropy for all temperatures. Note that in superfluids and nematic liquid crystals the energy of an isolated disclination (vortex) is proportional to the logarithm of the area. In what follows, we shall therefore ignore disclinations. Topological order in a two-dimensional crystal (Kosterlitz and Thouless 1972, 1973) can be defined in a very similar way to that of the magnetic or superfluid case. Suppose the system has enough short-range order so that a local crystal structure can be defined. Using the local order, we trace out a path from site to site which, in the perfect crystal, would be closed. If there are free dislocations present, the number of these within the contour will be proportional to the area enclosed by the path, and so the total average Burger’s vector will be proportional to the length L of the contour. If there
416
J.M. KOSTERLITZ AND D.J. THOULESS
[CH.5,s 3
are only bound pairs of dislocations the average Burger’s vector will be proportional to (JU)”~, where d is the mean separation. Note that even if we allow long wavelength excitations, such a procedure as outlined above is still possible because directional long-range order is not destroyed by these.
(b) Fig. 7. (a) A disclination in a square lattice of 4 x fx. (b) A translational dislocation with Burger’s vector b = (a, 0). The broken line is a typical contour round the dislocation which
would close in the perfect lattice.
If there are no free dislocations present, the system is rigid, while if free dislocations are present an arbitrarily small shear stress will cause the dislocations to move to the surface and so a response characteristic of a viscous liquid will ensue. Thus,the presence or absence of free dislocations determines whether the system behaves like a solid or a liquid. In the same way as for magnetic and superfluid systems, the rigidity modulus vanishes discontinuously
TWO-DIMENSIONALPHYSICS
CH.5 , g 31
417
at the melting temperature, but again this does not imply a first order transition. Since the energy of an isolated dislocation of Burger’s vector b in a system of area A is (Friedel 1964; Nabarro 1967) (3.60)
E = b2p(A+p) log A / A , + O ( l ) , 442p A)
+
and the entropy S N log A / A o ,the melting temperature is then approximately
T,
N
(3.61)
b2p(A+p)/4n(2p +A).
This can be written in terms of the transverse and longitudinal speeds of sound (Elgin and Goodstein 1973,1974)
T,
N
(b2p/4n) C:(C,?- C:)/C,?,
(3.62)
where p is the density of the layer. Since in helium layers C,?/Cf N 0.25 (Stewart 1974), we can rewrite eq. (3.62) as
T,
N
+
(3.63)
(b2p/4n) C:C,?/(C: C,?),
which introduces errors of not more than 5%. As pointed out by Elgin and Goodstein (1973, 1974) this is precisely the form derived by Feynman so that either form will give the same value for T, to within 5%. Finally eq. (3.63) can be expressed in terms of the measured value of the Debye temperature 0,
T,
N
1.15 (mkB/32x2h2)a0e&
(3.64)
where we have used b2 = 1.15ao where a. is the area of an elementary unit for a triangular lattice. The predicted dependence T, aoOi is well obeyed, but the values predicted by eq. (3.64) lie about 20% below the specific heat peaks (Elgin and Goodstein 1973, 1974). This simple theory is expected to overestimate the transition temperature since in the low temperature phase, thermally activated dislocation pairs will tend to reduce the shear modulus (Kosterlitz 1974). However, the underestimate may be caused by the use of classical elasticity theory ignoring quantum effects of solid two-dimensional helium. To take into account interactions between the dislocations within the ) is a scalar harmonic approximation, we introduce a stress tensor ~ ( rwhich N
J.M. KOSTEFUITZ AND D.J. THOULESS
418
[CH. 5 , $ 3
in two dimensionsfrom which the stress can be calculated
One can then define a source function q(r) describing the distribution of dislocations so that x(r) obeys the equation (Friedel 1964; Nabarro 1967)
V4x
= Kq
(3.65)
with
and q(r) =
c
a
e i j - by) 6(r -dU)).
ar,
Here r@)and b(@are the positions and strengths of the ath dislocation. The Hamiltonian eq. (3.56)becomes
(3.66) If we now write x = x d + x o , where x d is the stress tensor due to the dislocations and xo due to the phonons, where V4x0 = 0, we can easily see that the Hamiltonian becomes a sum of two terms
with Hd given by
(3.67) subject to cSu= 0 and where p is the ‘chemical potential’ of a single dislocation.
CH.5 , # 31
TWO-DIMENSIONAL PHYSICS
419
We can see from eq. (3.67) why the scaling procedure of Kosterlitz (1974) is more complicated than for the magnetic system. When two dislocations are very close to each other, they can be regarded as a single dislocation of strength 6, +b, . The square lattice is relatively simple (except that eq. (3.67) is not correct because of the lowered symmetry) because IS, +b2 I can take on onlv two values 0 or 2a, where a is a lattice spacing. In a triangular lattice, on the other hand, we must also allow for the possibility 16, +S2 I = a. This does not change the form of the scaling equations from those of eqs. (3.31) but the analysis is more complicated. The predictions are as in the magnetic system. There is a sharp transition at a definite temperature which is of infinite order. The specific heat is smooth at T, (a = -a)and as yet we cannot tell if the maximum occurs at T, or some other temperature. The rigidity modulus as defined as the response to an infinitesimal shear stress (3.68) vanishes discontinuouslylikethe superfluid density. Note that this is not inconsistent with a continuous transition. 3.7. EPITAXIAL MONOLAYERS The atoms of gas are preferentially adsorbed at the centres of the graphite hexagons (Dash 1975). Owing to the size of the adsorbate atoms, not every site can be occupied, but if every third site is occupied the adsorbate forms a regular triangular lattice, and so should be an excellent experimental realisation of a lattice gas, in which an ordering field produces deviations from this critical coverage. At the critical coverage (a 4 3 x 2/3 epitaxial structure) the specific heat anomaly for 3He and 4He on graphite is very pronounced, and in the range lo-' > I 1 -TIT, I > 5 x 10- fits well to a logarithmic CINk, N Alog I 1 -TIT, I +B with values of A quite close to the theoretical value for the two dimensional Ising model (Dash 1975). However, the critical coverage for the theoretical Ising lattice gas corresponds to a critical coverage of 0.5, rather than 0.33 for the experimental system. Alexander (1975) has suggested that the monolayer corresponds to a three state Potts (1952) model since there are three possible equivalent ordered states as shown in fig. 8. This model has a continuous transition (Baxter 1973) in two dimensions which is consistent with the experimental observations. For other gases on graphite, however, the transition appears
420
J.M. KOSTERLITZ AND D.J. THOULESS
[CH. 5 , s 4
first order. In the case of Xe there is evidence of two phase coexistence, while for N2there is still controversy (Dash 1975), although a recent paper (Butler et al. 1975) finds the data consistent with a clussicul transition! Clearly much work both experimental and theoretical remains to be done to clarify the effects of the substrate on different gases.
Fig. 8. The three possible equivalent orderings of 4He atoms adsorbed on graphite at critical coverage. The lattice sites are the centres of the graphite hexagons.
4. Metal-insulator transitions
4.1. THEPERCOLATIONMODEL
A simple model which can display either insulating or conducting properties is a system which is made up randomly of insulating and conducting material. This is the percolation model, which has been quite extensively studied as a model of random magnetic systems, composed of magnetic and nonmagnetic atoms. The model is usually formulated in terms of a lattice, and there are two versions of the model, bond percolation and site percolation. In the bond percolation model the lattice sites are connected by bonds which have a probabilityp of being available (conducting) and a probability q = 1-p of not being available. In the site problem the sites have a probabilityp of being available, and the bonds between two neighbouring available sites are always available. In fact the bond problem can be regarded as a special case of the site problem. When p is below a certain critical value pc there are only finite clusters of connected sites, but when p exceeds pc there will be an infinite cluster of connected sites, as well as finite unconnected to the main cluster.
TWO-DIMENSIONAL PHYSICS
CH. 5,# 41
421
There have been reviews of percolation theory by Shante and Kirkpatrick (1971), by Essam (1972), and by Kirkpatrick (1973). The two-dimensional percolation problem has attracted quite a lot of attention for various reasons. The one-dimensional problem is trivial, since there are no infinite clusters in that case unless p is unity, so the two-dimensional case is the simplest nontrivial case. There are some exact results available which were derived by Sykes and Essam (1964) who exploited a property of two-dimensionalspace which has no analogue in three dimensions. The simplest form of their argument can be used to show that the critical probabilityp, for site percolation on a triangular lattice is 3. This argument is illustrated in fig. 9. If the black circles denote available sites and the open circles unavailable sites, there is a finite cluster of available sites shown in the figure which is surrounded by a perimeter of unavailable sites. If there are
.
0
O
.
0
.
0
O
0
O
0
.
0 0 0 . 0 Fig. 9. Percolation model in a triangular lattice. The open circles denote unavailable sites and the filled in circles available sites. There is a cluster of five available sites shown in this diagram.
only finite clusters of available sites then the unavailable sites surrounding them must form an infinite cluster, but it is not possible in a random arrangement for there to be infinite clusters of both kinds (a non-random arrangement could have infinite strands of both available and non-available sites interleaved). A similar argument shows that pc for the bond problem on a square lattice is also 3, so that an infinite square network falls to pieces once half the bonds are broken. A more subtle version of the argument for the bond problem on a triangular lattice shows thatp, = 2 sin 10". For most other problems the critical probability has to be estimated by power series methods or Monte Carlo methods. The knowledge of exact values ofp, for certain lattices allows power series and Monte Carlo methods to be used to calculate certain critical exponents, such as the one giving the dependence of the number of sites in the infinite cluster on p-p, or the dependence of mean cluster size on pc-p, with conC
422
J.M. KOSTERLITZ AND D.J. THOULESS
[CH. 5 , s 4
siderable accuracy, and various estimates have been made recently (Dunn et al. 1975; Sykes and Glen 1976; Sykes et al. 1976; Gaunt and Sykes 1976; Cox and Essam 1976), all in fairly good agreement. The problem of the conductivityof a network in which resistors are removed at random, which is a percolation problem, has received some attention in recent years, because of its suggested relevance to a number of physical situations. Ziman (1968) proposed it as a model of the behaviour of electrons in a random medium, arguing that electrons of a certain energy would, if they behaved classically, be able to move in regions whose potential was lower than the electron energy, but would not be able to move where the potential was higher. If the fluctuations of the potential about the mean potential are symmetricalpercolation can occur in two dimensions once the energy is above the mean value, as can be shown by the argument used to show that p E = for the triangular site problem. Ambegaokar et al. (1971) have invoked percolation theory in their derivation of Mott's (1969) variable range hopping conductivity formula. In this problem the relevant critical probability is the minimum density of points p , which can be joined together by bonds whose length is less than ro to form an infinite cluster. Various estimates ofp, have been made (Dalton et al. 1964; Ottavi and Gayda 1974), and a recent value quoted is (4.4&0.2)/nr; (Fremlin 1976). The conductivity of a two-dimensional percolation system has been studied by physical simulation. Last and Thouless (1971) punched holes in conducting paper, and Watson and Leath (1974) cut links in chicken-wire. Kirkpatrick (1971) used a computer simulation to study the same problem. It was found that for p well above p , the conductivity can be fitted by an effective medium theory (Landauer 1952), but that as p approaches pc the conductivity curve flattens out, so that it is proportional to (p-pc)', where the exponent t is about 1.3. Stinchcombe and Watson (1976) have used real space renormalisationgroup methods to calculate this exponent, and obtained a slightly lower value. The value of this exponent is also of significance for magnetic systems, since the spin-wave stiffness near the percolation limit depends on it (Kirkpatrick 1973). Various physical systems which might be expected to have two-dimensional percolation properties have been studied. Abelks et al. (1975) have looked at heterogeneous films of tungsten and AI,O, and measured the conductivity; the results are not altogether in accord with the predictions of percolation theory. Liang et al. (1976) have studied the conductivity of bismuth films under conditions in which the films gave only partial coverage, and the coverage at which metallic conduction occurs is in accord with percolation
CH.5 , g 41
TWO-DIMENSIONAL PHYSICS
423
theory. Arnold (1976) has interpreted the conductivity of inversion layers in terms of percolation theory. 4.2. ANDERSON LOCALISATION
Since electrons are not classical particles but obey the laws of quantum mechanics, the percolation model is only directly relevant when the length scale of potential fluctuations is much longer than the wavelength of an electron. When the fluctuations have a short range the electrons can tunnel from one region of favourable potential through an unfavourable region to another favourable region, so one should not expect an abrupt change in the nature of the system at the percolation limit. However, Anderson (1958) has argued that in a system with a random potential electrons will be localised if the disorder is sufficientlygreat. These localised electrons form a continuous energy band, but the wave functions fall off exponentially from some maximum value, which may be in any region of the homogeneous system, and conductivity can only occur if energy is supplied by the phonons to carry an electron from one localised state to a neighbouring, which inevitably has a slightly different energy. At low temperatures this gives rise to Mott's (1969) variable range hopping with the characteristic exp -(T0/T)'l3 temperature dependence in two dimensions. When the disorder is less strong there may be extended Bloch-like states in the centre of the energy band, with a metallic conductivity, but in the tails of the band, outside some mobility edge, the states will be localised. The theory of localised electron states has been reviewed by Thouless (1974). As in the percolation model the two-dimensional system has received a lot of theoretical attention because it is the simplest non-trivial system. Mott and Twose (1961) showed that electrons are all localised in one dimension however small the disorder. Khor and Smith (1971) did numerical calculations for samples of two-dimensional lattices and showed convincingly that the states at the band edge were indeed exponentially localised. Further calculations have been carried out by Edwards and Thouless (1972), Kikuchi (1974, 1976), Licciardello and Thouless (1975b) and Yoshino and Okazaki (1976). These studies have all shown a transition between localised and extended states as expected, but have shown that Anderson's (1958) estimate of the degree of disorder needed to localise electrons was far too high; electrons are much easier to localise than his theory predicts. The two-dimensional localisation problem became particularly interesting when it was realised that electrons in an inversion layer could provide an
424
J.M. KOSTERLLTZ A N D D.J. THOULESS
[CH. 5, # 4
example of this phenomenon (Pepper et al. 1974; Mott et al. 1975). Mott (1970, 1974) has argued that the wavelength of an electron cannot be substantially less than the mean free path A, the distance in which the electron loses phase coherence, and if the criterion kA > 1 is applied to the formula
u = ne2 A/mv
(4.1)
for the conductivity in two dimensions, using n = k2/2n,the result d
1 e2 2n A
> --
is obtained. Licciardello and Thouless (1975a) have argued that this minimum metallic conductivity has a universal value in two dimensions, and have estimated its value as 0.12f0.03 e2/A. The argument is that localisation will occur if the amount by which conditions at the boundary of an area of side L can shift energy levels, which we call AE, is much,less than the spacing between levels q, in this case states confined to the volume will be little perturbed by neighbouring volumes, and so will be exponentially localised. If the electrons are Bloch-like with a mean free path 1,AE is equal to A divided by the time it takes an electron to diffuse to the boundary, so AE = h D/L2,
(4.3)
where D is the diffusion constant, provided L is much greater than A. The condition for localisation is then AE hD = -n(E)L2 < 1. rl
L2
(4.4)
From the Einstein relation between the diffusion constant and the conductivity this is equal to 2uA/e2, and so the condition for the existence of extended states can be expressed in terms of the conductivity without reference to the details of the microscopic structure of the solid. This relation is a special feature of two-dimensional systems, and the observation of minimum metallic conductivitiesof this order of magnitude is an important verification of the theory (Pepper et al. 1974; Pepper 1977). The experiments of Tsui and Allen (1974) and Hartstein and Fowler (1975) show that not all the phenomena connected with the transition from local-
CH.5,B 41
TWO-DIMENSIONAL PHYSICS
425
ised to extended states can be understood in these terms alone. There is nothing in this argument that forces the metallic conductivity to reach this minimum value before the transition occurs. It has been suggested (Adkins et al. 1976) that when there are inhomogeneities on a scale large compared with the wavelength the metallic conductivity is higher at the transition, because the inhomogeneity forces the conduction into restricted channels so that the system behaves more like a one-dimensional system in which states are always localised. Although some attention has been paid to the cross-over from localisation under the influence of short-range fluctuations to the percolation regime (Toulouse 1975; Friedel 1976), there has not yet been a satisfactorystudy of the problem.
4.3. DENSITY WAVES
There are many mechanisms which may give rise to a metal-insulator transition in a solid, which are reviewed in the book by Mott (1974). The only mechanisms which seem to have been explored in detail for the particular case of two-dimensional systems are Anderson localisation, discussed in the previous section, and the formation of charge density waves or spin density waves. Charge density waves in metallic layer compounds have been discussed in the review by Wilson et al. (1975). The mechanisms producing charge density waves and spin density waves are rather similar, but charge density waves are favoured by the electron-phonon interaction, which acts as an effective attractive interaction between electrons, while spin density waves, which keep electrons of opposite spin apart, are favoured by the repulsive Coulomb interaction between electrons (Chan and Heine 1973). In one-dimensional systems these density waves occur however weak the interaction but in two or three dimensions thay can only be made at the expense of increasingthe band energy, and how much this increase is depends on the detailed shape of the Fermi surface. The formation of such waves is particularly easy if there are flat parallel regions of the Fermi surface, and the small dependence of the energy on the component of momentum perpendicular to the planes makes the layer compounds and other two-dimensional systems particularly suitable. When density waves are set up they will generally occur as a pattern of standing waves with two sets of nodal planes, and so their effect is quite complicated. It is easier to consider the stability of the state without density waves when subject to a perturbation of the sort that a density wave would produce. If the state is unstable then a density wave will be set up. We therefore
426
J.M. KOSTERLITZ AND D.J. THOULESS
[CH.5 , s 4
consider the effect of a perturbation of wavenumber on a Fermi gas and calculate the susceptibilityto such a perturbation. This is a standard problem in many-body theory, and for the case of a charge density wave driven by an attractive interaction whose matrix element is a constant - V the condition for stability is
wherefis the Fermi function, &k is the band energy and the factor 2 comes from the sum over spin states. If there are flat parallel planes in the Fermi surface, separated by wave number Q, then if k is at a distance x below the Fermi surface and k + Q is a distance x above it, the value of & k + Q - & k is h2Qx/m*. At low temperatures the integral over x in eq. (4.5) will then diverge logarithmically, and so the instability must occur at some temperature. In the trigonal(1T) phase of the layer compounds the Fermi surface is as shown in fig. 10a, and a value of Q as shown in this figure will carry electrons in several regions of the Fermi surface close to another region. When the instability sets in a superlattice is formed, which need not be commensurate with the crystal lattice and gaps open up on those parts of the original Fermi surface connected by a reciprocal lattice vector of the new superlattice, so that the system becomes insulating. Rice and Scott (1975) have argued that for the hexagonal (2H) phases the Fermi surface comes close to six saddle points, as shown in fig. lob. These are points of high density of states, and therefore a value of Q that connects pairs of saddle points can make the right side of eq, (4.5) large and produce an instability. If a superlattice is formed in this way it will not reduce the area of the Fermi surface significantly, but it will push the Fermi surface away from these regions of high density of states and so reduce the scattering of electrons. This is therefore an explanatioa for the enhanced conductivity of the 2H phases when a superlattice is formed. Kelly and Falicov (1976) have proposed that a superlattice is formed by charge density waves in an inversion layer. This is used to explain various anomalous results such as the fact that there only appears to be a two-fold orbital degeneracy instead of the six-fold degeneracy expected on the (1 11) surface of silicon. A summary of various other aspects of many-body effects in inversion layers and for electrons on helium surfaces has been given by Stern (1976) in his summary of a conference devoted to these materials.
CH.5.8 41
427
TWO-DIMENSIONAL PHYSICS
Ia).
b)
Fig.10. Schematic Fermi surfaces for layer compounds. (a) shows the Fermi surface for the 1T phase, and (b) for the 2H phase. The wave number for a possible charge density wave is shown in each case by Q.
4.4. THEWIGNER LATTICE Wigner (1938) pointed out that the electron gas should at sufficiently low temperatures condense to a solid-like lattice structure. It is likely that suitable conditions for the formation of such a lattice could be attained either for electrons on the surface of helium, or for electrons in an inversion layer and so there has been a considerableinterest in the theory of the Wigner lattice. Bonsall and Maradudin (1976) have shown that the lowest energy configuration of the electrons in two dimensions is a close-packed triangular lattice, and they have found the normal modes of vibration of the system, which are a longitudinal plasma mode whose frequency is proportional to q'/2 for small wavenumber q and a transverse sound wave with frequency linear in q. Platzman and Fukuyama (1974) have studied the stability of the system taking account of nonlinear effects in a self-consistent way and allowing both for thermal and quantum zero-point motion. In this way they have estimated the melting curve of the lattice in terms of the two dimensionless parameters e2n1/2/(4mOkBT) and e2m/(4mOh2n1/2) where n is the density of electrons. Hockney and Brown (1975) have carried out a molecular dynamics calculation for a system of lo4 electrons confined to a square region with periodic boundary conditions. They calculated both the specific heat and the structure factor over a range of temperatures. The area of the square was 10-"m2, so the density was 1014 electrons/m2. They found that at low temperatures a triangular lattice was formed, and as the temperature was raised it seemed to
428
J.M. KOSTERLlTZ AND D.J. THOULESS
break into a polycrystalline state, and there was a fairly sharp phase transition to a disordered state at 3.1 K. This transition has a well-defined specific heat singularity with a specific heat exponent equal to 0.08 below the transition and 0.14 above, so it is sharper than the two-dimensionalIsing transition. These results raise a number of interesting questions. In the first place the transition temperature is lower by a factor of 30 than the temperature calculated by Platzman and Fukuyama (1974) in the classical limit, so doubt must be cast on all results obtained by similar arguments. The temperature is so low that the disordered state just above the transition has only a small fraction less correlation energy than the ground state. The existence of an ordered state at low temperatures is somewhat odd since the Peierls (1934, 1935) argument would say that the transverse sound wave must destroy longrange order for a sufficiently big system, but since we know a number of other examples of systems with phase transitions to which the same argument can be applied it is not very surprising. If the breaking up of a single crystal into a polycrystalline form below the transition temperature is a real feature of system, and not just an artifact due to the difficultyof fitting a triangular lattice into a square area, it presents an interesting challenge to statistical mechanics. The question of how an electron lattice can be detected once it is formed is a challenging one, and it would be helpful to have a firm theoretical prediction before experiments are done. It can be argued that a lattice will be pinned by the substrate and so the system will be an insulator, which might be hard to distinguish from Anderson localisation or a Mott insulator. It can also be argued that the system will not be readily pinned, since the potential energy of the whole lattice, not just of individual electrons, is in question, and this varies by a relatively small amount (of order N'/2, where N is the number of electrons) from point to point, unless the electron lattice is commensurate with the substrate lattice. If this is right one might expect the conductivity to be high, with the lattice moving as a whole.
Note added in proof Since this article was written, there has been a great deal of activity in twodimensional physics, especially on the planar rotator model and its generalisations. Monte Carlo simulations have shown that vortex like codgurations do develop (Suzuki et al. 1977; Kawabata and Binder 1977). JosC et al. (1977) have made a study of the planar rotator model with symmetry breaking fields by the Migdal (1975) renormalisation group technique. They
TWO-DIMENSIONAL PHYSICS
429
also rederived the recursion relations (3.3 1) working on a lattice throughout via duality relations. On the way they showed the equivalence of the discrete Gaussian model of the roughening transition, which is important in theories of crystal growth, to the Coulomb gas and planar rotator models (see also Chui and Weeks 1976). They further generalised the method described in $3.3 to cope with symmetry breaking fields by the introduction of two different types of vortices. Kadanoff (1977, 1978) has taken this approach further and has shown that a number of two-dimensional models such as the %vertex and Ashkin-Teller models can be mapped on to a generalised Coulomb gas problem with interacting electric and magnetic charges. Nelson and Kosterlitz (1977) have shown that the quantity p,(T)/T, as T + Tc-, reaches the coverage independent universal value of 2rn2kBlnn2= 3.52 x g.cm-’K-’ as predicted from the naive theory of single vortex formation. This result is an inescapable prediction since, at T,, all other effects which could be present in a superfluid film of 4He are irrelevant. An experimental measurement of this quantity will provide a check on the theory. Unfortunately experiments on 4He films (Bishop and Reppy 1977) are done at finite frequency and the critical dynamics must be investigated. Some progress has been made very recently with encouragingresults by Ambegaokar et al. (1978), Huberman et al. (1978) and Chui and Weeks (1978).
References Abelks, B., H.L. Pinch and J.I. Gittleman,l975, Phys. Rev. Letters 35,247. Abrahams,E.andJ.W.F. Woo, 1968,Phys. Lett.27A,117. Adkins, C.J., S. Pollitt and M. Pepper, 1976,J. Physique37,W343. Ahonen, A.I., T. Kodama, M. Krusius, M.A. Paalanen, R.C.Richardson, W. Schoepa and Y. Takano, 1976, J. Phys. 0,1665. Alexander, S., 1975,Phys. Lett. H A , 353. Allen, S.J., D.C. Tsui and F. De Rosa, 1975, Phys. Rev. Lett.35,1359. L121. Als-Nielsen, J., R.J. Birgenau, H.J. GuggenheimandG. Shirane, 1976, J. Phys. a, Ambegaokar, V., B.I. Halperin and J.S. Langer, 1971,Phys. Rev. B4,2612. Amit, D. and S.K. Ma, to be published. Anderson, P.W.,1958, Phys. Rev. 109,1492. Anderson, P.W., G. Yuval and D.R. Hamann, 1970, Phys. Rev. B1,4464. Anderson, P.W. and G. Yuval. 1971,J. Phys. C4,607. Arnold, E., 1976,SurfaceSci. 58,60. Ashkin, J. and E. Teller, 1943, Phys. Rev. 64,178. Aslamazov, L.G. and A.I. Larkin, 1968, Sov. Phys-Solid St. 10,875. Baxter, R.J., 1973. J. Phys. C6, Id45. Beal-Monod, M.T. and S. Doniach, 1977, J. Low Temp. Phys. 28,175. Berezinskii, V.L., 1970, Sov. Phys. JETP32,493.
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Bcrezinskii, V.L., 1972, Sov. Phys. JEW 34,610. Berezinskii, V.L. and A. Ya. Blank, 1973,Sov. Phys. JETP37,369. Berlin, T.H. and M.Kac, 1952, Phys. Rev. 86,821. Berthold, J.E., D.J. Bishop and J.D.Reppy, 1977, Phys. Rev. Lett. 39, 348. Betts, D.D.,C.J. Elliott and R.V. Ditzian, 1971,Can. J. Phys. 49,1327. Birgenau, R.J., H.J. Guggenheim and G. Shirane, 1970a, Phys. Rev. B1,2211. Birgenau, R.J.. J. Skalyo and G. Shirane, 1970b. J. Appl. Phys. 41,1303. Binder, K.and D.P. Landau. 1976,Phys, Rev. B13,1140. Bloembergen, P., 1975,Physica79B,467. Bonsall, L. and A.A. Maradudin, 1976, Surface Sci. 58,312. Breed, D.J., K. Gilijamse and A.R. Miedema, 1969, Physica 45,205. Bretz, M., 1973,Phys. Rev. Lctt.31,1447. Bretz, M., J.G. Dash, D.C. Hickernell, E.O. McLean and O.E. Vilches, 1973, Phys. Rev. A8,1589. Brewer, D.F. and K. Mendelssohn, 1961,Proc. Roy. Soc.A260,l. Brezin, E. and J. Zinn-Justin, 1976a,Phys Rev. Lett. 36,691. Brezin, E. and J. Zinn-Justin, 1976b,Phys.Rev.B14,3110. Brezin,E., J. Zinn-Justin and J.C. Guillou, 1976, Phys. Rev. D14,2615; B14,4976. Butler,D.M.,G.B.HutT,R.W.TothandG.A.Stewart, 1975,Phys.Rev. Lett.35,1718. Camp, W.J. and J.P. VanDyke, 1975, J. Phys. C8,336. Chan, S.-K. and V. Heine, 1973,J. Phys. F3,795. Chester, M., L.C. Yang and J.B. Stephens, 1972, Phys, Rev. Lett. 29,211. Chester, M. and L.C. Yang, 1973, Phys. Rev. Lett. 31,1377. Chester, M. and L.C. Yang, 1974, Phys. Rev. A9,1475. Cole, M.W., 1974,Rev. Mod. Phys. 46,451. Cole, M.W., J.G. Dash and J.A. Herb, 1975,Phys. Rev. B11,163. Cox, M.A.A. and J.W. Essam, 1976, J. Phys. C9,3985. Crow, J.E., R.S. Thompson, M.A. Klenin and A.K. Bhatnager, 1970, Phys. Rev. Lett. 24,371. Dalton, N.W., C. Domb and M.F. Sykes, 1964, Proc.CambridgePhil. Soc.83,496. Dash, J.G. and J.A. Herb, 1973, Phys. Rev. A7,1472. Dash, J.G., 1975,Films on Solid Surfaces (Academic Press, New York). DeGennes, P.G., 1974, The Physics of Liquid Crystals (Oxford University Press, Oxford). de Jongh, L.J. and A.R. Miedema, 1974, Adv. Phys. 23,l. de Jongh, L.J. and H.E. Stanley, 1976,Phys. Rev. Lett. 36,817. De Wijn. H.W., R.E. Walstedt, L.R. Walker and H.J. Guggenheim, 1971, J. Appl. Phys. 42,1595. Dunn, A.G., J.W. Essam and D.S. Ritchie, 1975, J. Phys. C8,4219. Edwards, J.T. and D.J. Thouless, 1972, J. Phys. C5,807. Elgin, R.L. and D.L. Goodstein, 1973, Monolayer and Submonolayer Helium Films (J.G. Daunt and Lerner, eds.) (PlenumPress,New York). Elgin, R.L. and D.L. Goodstein, 1974,Phys. Rev. A9,2657. Englert, Th. and G. Landwehr, 1976, Surface Sci. 58,217. Essam,J.W., 1972, Phase Transitions and Critical Phenomena, Vol. 2 (C. Domb and M.S. Green, eds.) (Academic Press, New York)pp. 197-270. Ferrell, R.A. and H. Schmidt, 1967, Phys. Lett.25A, 544. Feynman, R.P., 1973, quoted by R.L. Elgin and D.L. Goodstein in: Monolayer and Submonolayer Helium Films (J.G. Daunt and E. Lerner, eds.) (Plenum Press, New York). Fisher, D.S. and D.R. Nelson, 1977, Phys.Rev. B16 2300.
TWO-DIMENSIONAL PHYSICS
431
Fisher, M.E., M.N. Barber and D. Jasnow, 1973,Phys. Rev. A8,1111. Fremlin, D.H., 1976,J. Physique37,813. Friedel, J., 1964,Dislocations(PergamonPress, London). Friedel, J., 1976, J. Physique Lett. 37,9. Gaunt, D.S. and M.F. Sykes, 1976, J. Phys. A9.1109. Glover, R.E., 1967. Phys. Letters 25A, 542. Grimes, C.C. and G. Adam, 1976, Phys. Rev. Letters 36,145. Grimes, C.C.,T.R. Brown, M.L. Burns and C.L. Zipfel, 1976, Phys. Rev. BW, 140. Halperin, B.I. andP.C. Hohenberg, 1969,Phys.Rev. 177,952; 188,898. Hartstein,A. and A.B. Fowler, 1975,J. Phys. C8, L249. Hauge. E.H. and P.C. Hemmer, 1971, Phys. Norveg. 5, 209. Henkel,P.P.,E.N.Smithand J.D.Reppy, 1969,Phys.Rev. Lett.23,1276. Herb, J.A. and J.G. Dash, 1972,Phys. Rev. Lett. 29,846. Hering, S.V., S.W. Van Scriver and O.E. Vilches, 1976,J. Low Temp. P h s . 25,793. Hertz, J.A., 1976,Phys.Rev.B14,1165. Hockney,R.W.andT.R.Brown,1975, J.Phys.C8,1813. Hohenberg,P.C. and P.C. Martin, 1965, Ann. Phys. 34,291. Hohenberg,P.C., 1967, Phys. Rev. 158,383. Huff, G.B.and J.G.Dash, 1976,J. LowTemp.Phys.24,155. Imry, Y.,1969,Ann. Phys. 51,l. Imry, Y.and L. Gunther, 1971, Phys. Rev. B3,3939. Jasnow, D. and M.E. FisherJ969,Phys. Rev. Lett. 23,286. Jasnow, D. and M.E. Fisher, 1971,Phys. Rev. B3,895. Joyce, G.S., 1972, in: Phase Transitions and Critical Phenomena (C. Domb and M.S. Green, eds.) (Academic Press,London and New York), Vol. 11, Ch.10. Kac, M., 1971, Phys.Norveg.5,163. Kadanoff, L.P. and F. Wegner, 1971,Phys. Rev. B4,3989. Kadanoff, L.P., 1976,Ann. Phys. (N.Y.), 100,359. Kelly, M.J. and L.M. Falicov, 1976, Phys. Rev. Lett. 37,1021. Khor, K.E. and P.V. Smith, 1971, J. Phys. C4,2029. Kikuchi, M., 1974, J. Phys. Japan 37,904. Kikuchi, M., 1976,J. Phys. Japan 41,1459. Kirkpatrick,S., 1971,Phys. Rev. Lett. 27,1722. Kirkpatrick,S., 1973, Rev. Mod. Phys. 45,574. Kjems, J.K., L. Passell, H. Taub, J.G. Dash and A.D. Novaco, 1976, Phys. Rev. B13,1446. Kosterlitz, J.M. andD.J. Thouless, 1972,J. Phys. C5,L124. Kosterlitz,J.M. and D.J. Thouless, 1973, J. Phys. C6,1181. Kosterlitz,J.M., 1974, J Phys. C7,1046. Kosterlitz, J.M., 1976. Phys. Rev. Lett. 37,1577. Landau, L.D. and E.M. Lifshitz, 1959, Theory of Elasticity(London, Pergamon). Landauer, R., 1952, J. Appl. Phys. 23,779. Langer, J.S. and J.D. Reppy, 1970, Progress in Low Temperature Physics, Vol. VI (G.J. Gorter, ed.) (North-Holland, Amsterdam). Langmuir, I., 1918, J. Amer. Chem. Soc.40,1361. Last, B.J. and D.J. Thouless, 1971, Phys. Rev. Lett. 27,1719. Liang, N.T., Y. Shan and S. Wang, 1976,Phys. Rev. Lett. 37,526. Licciardello, D.C. andD.J. Thouless, 1975,Phys. Rev. Lett.3,1475. Licciardello, D.C. and D.J. Thouless,1975,J. Phys. C8,4157. Lieb, E.H. and F.Y. Wu, 1972, in: Phase Transitions and Critical Phenomena (C. Domb and M.S.Green, eds.) (AcademicPress, London and New York) Vol. I, ch. 8.
432
J.M. KOSTERLITZ AND D.J. THOULESS
Liu, L.L. and H.E. Stanley, 1972, Phys. Rev. Lett. 29,927. Liu, L.L. and H.E. Stanley, 1973, Phys. Rev. B8,2279. Luther, A. and D. Scalapino, 1977, Phys. Rev. B16, 1153. Maki, K., 1968, Prog. Theor. Phys. 39,897,40,193. Mattheiss, L.F., 1973, Phys. Rev. B8,3719. McTague, J.P. and M. Nielsen, 1976, Phys. Rev. Lett. 37,596. Mermin, N.D., 1967, J. Math. Phys. 8, 1061. Mermin, N.D. and H. Wagner, 1966, Phys. Rev. Lett. 17,1133. Mermin, N.D., 1968, Phys. Rev. 176,250. Migdal, A.A., 1975, Zh. Eksp. Teor. Fiz. 69, 1457. Sov. Phys. JETP 42, 743. Moore, M.A., 1969,Phys. Rev. Lett. 23,861. Mott, N.F.,and W.D. Twose, 1961,Adv. Phys. 10,107. Mott N.F. 1966, Phil. Mag. 13,989. Mott, N.F., 1969,Phil. Mag. 19,835. Mott, N.F., 1970, Phil. Mag. 22,7. Mott, N.F., 1974, Metal-Insulator Transitions (Taylor and Francis, London). Mott, N.F., M. Pepper, S. Pollitt, R.H. Wallis and C.J. Adkh. 1975, ROC. Roy. SOC. A345,169. Nabarro, F.R.N., 1967,Theory of Dislocations (Clarendon Press, Oxford). Onsager,L.,1944, Phys. Rev. 65,117. Ottavi, H. and J.P. Gayda, 1974,J. Physique 35,631. Padmore,T.C.and J.D. Reppy, 1973,Phys.Rev. Lett. 33,1410. Patani, A., M. Schlindwein, andQ. Shafi, 1976,J. Phys. A9,1513. Peierls, R.E., 1934, Helv. Phys. Acta. 7, Suppl. II,81. Peierls, R.E., 1935, Ann. Inst. Henri Poincare 5,177. Peierls, R.E., 1936, Proc.Camb. Phil. SOC.32,477. Pelcovitz, R.A. and D.R. Nelson, 1976, Phys. Lett.57A, 23. Penrose, 0.and L. Onsager, 1956, Phys. Rev. 104,576. Pepper, M., S. Pollitt and C.J. Adkins, 1974,J. Phys. C7,1273. Pepper, M., 1977, Proc. Roy. Soc.353A, 225. Platunan, P.M. and H. Fukuyama, 1974, Phys. Rev. B10,4988. Platzman, P.M. and G. Beni, 1976, Phys. Rev. Lett. 36,626, Pokrovsky,V.L. and G.V. Uimin, 1973, Phys. Lett. 45A,467. Pokrovsky,V.L. and G.V. Uimin, 1974, Sov. Phys. JETP 38,847. Pollak, R.A.and H.P. Hughes, 1976, J. Physique37,C4-151. Polyakov, A.M., 1975, Phys. Lett. 59B, 79. Potts, R.B., 1952, Proc.Camb. Phil. Soc.48,106. Rice,T.M. andG.K. Scott, 1975, Phys. Rev. Lett.35,120. Ritchie, D.S.and M.E. Fisher, 1973, Phys. Rev. B7,480. Rolt, J. andD.F. Brewer, 1972, Phys. Rev. Lett.29,1485. Schechter, H., J. Suzanneand J.G. Dash, 1976,Phys. Rev. Lett. 37,706. Schmid, A., 1968, Z Phys. 215,210. Scholtz, J.H.,E.O. McLean and I. Rudnick, 1974, Phys. Rev. Lett. 32,147. Shante, V. and S. Kirkpatrick, 1971,Adv. Phys. 20,325. Siddon, R.L. and M. Schick, 1974, Phys. Rev. A9,907,1753. Skocpol,W.J. and M. Tinkman, 1975, Repts. Progr. Phys. 38,1049. Smith, R.O.,B.SerinandE.Abrahams, 1968, Phys. Lett. 28A, 224. Stanley, H.E., 1967,Phys. Rev. 164,709. Stanley, H.E., 1968a, Phys. Rev. Lett.20,150. Stanley, H.E., 1968b,Phys. Rev. 176,718.
TWO-DIMENSIONAL PHYSICS
433
Stanley, H.E. and T.A. Kapfan, 1966,Phys. Rev. Lett. 17,913. Stanley, H.E.and T.A. Kaplan, 1967,J. Appl. Phys. 38,975. Stem, F., 1976,Surface Sci.58,333. Stewart, G.A.and J.G. Dash, 1970,Phys. Rev. A2,918. Stewart, G.A., 1974,Phys. Rev. AlO, 671. Stinchcombe, R.B.andB.P. Watson, 1976,J. Phys. (3,3221. Strongin, M., O.F. Kammerer, J. Crow, R.S. Thompson and H.L. Pine, 1968, Phys. Rev. Lett. 20,922. Sykcs, M.F. and M.E. Fisher, 1962,Physica 28,919,939. Sykes, M.F.and J.W. Essam, 1964,J. Math.Phys.5,1117. Sykcs, M.F. and M. Glen, 1976,J, Phys. A9,87. Sykes, M.F., D.S. Gaunt and M. Glen, 1976,J. Phys. A9,97,715,725. Taub, H., L. Passell, J.K. Kjems, K. Carneiro, J.P. McTague and J.G. Dash, 1975, Phys. Rev. Lett. 34,654. Telschow. K.L. and R.B. Hallock, 1976,Phys. Rev. Lett 37,1484. Thompson, R.S.,1970,Phys. Rev. B1,327. Thompson, R.S.,1971,Physica 55,296. Thoulcss, D.J., 1974,Physics Repts. 13C,93. Toulouse, G.,1975,C.R. Hebd. Sean.Acad. Sci. 280B,629. Toulouse, G.and M. K16man, 1976,J. Physique Lett. 37,L149. Tsui, D.C. andS.J. Allen, 1974,Phys, Rev. Lett.32,1200. Tzoar, N., P.M. Platman and A. Simons, 1976,Phys. Rev. Lett.36,1200. Villain, J., 1975,J. Physique 36,581. Watson, B.P. andP.L. Leath, 1974,Phys. Rev. B9,4893. Wegner, F., 1967,Z. Phys. 206,465. Wegner, F., 1972,Phys. Rev. BS,4529;B6,1819. Wegner, F., 1974,J. Phys. 0,2098. Wiper, E.P., 1938,Trans.Faraday Soc.34,678. Williams, P.M., C.B. Scruby, W.B. Clark and G.S.Parry, 1976,J. Physique 37, (3-139. Wilson, J.A., F.J. Di Salvo and S. Mahajan, 1975,Adv. in Phys. 24,117. Wilson, K.G. and J. Kogut, 1974,Phys. Rep. C12,75. Ying, S.C., 1971,Phys. Rev. B3,4160. Yoshino, S.and M. Okazaki, 1976,Solid State Comm.20,81. Zia, R.K.P. and D. J. Wallace, 1975,J. Phys. A8,1084. Ziman, J.M., 1968,J. Phys. C1,1532. Zittartz, J., 1976,Z. Phys. UB, 55,63.
References added in proof Ambegaokar, V., B.I. Halperin, D.R. Nelson and E.D. Siggia, 1978, Phys. Rev. Lett. in press. Bishop, D. and J.D. Reppy, 1977,Bull. Am. Phys. SOC. 22,638. Chui. S.T. and J.D. Weeks, 1976,Phys. Rev. B14,4978. Chui, S.T. and J.D. Weeks, 1978,to be published. Huberman, B.A., R.J.Myerson and S. Doniach, 1978,Phys. Rev. Lett. in press. Jose, J.V.,L.P. Kadanoff,S. Kirkpatrick and D.R. Nelson, 1977,Phys. Rev. 816,1217. Kadanoff, L.P.. 1977,Phys. Rev. Lett. 39,903. Kadanoff, L.P., 1978,J. Phys. A. in press. Kawabata, C. and K. Binder, 1977,Solid State Comm. 22,705. Nelson, D.R. and J.M. Kosterlitz, 1977,Phys. Rev. Lett. 39, 1201. Suzuki, M., S. Myiashita, A. Kuroda and C.Kawabata, 1977, Phys. Lett. 6OA 477.
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CHAPTER 6
FIRST AND SECOND ORDER PHASE TRANSITIONS OF MODERATELY SMALL SUPERCONDUCTORS IN A MAGNETIC FIELD BY
H.J. FINK Department of Electrical Engineering, University of California, Davis, California 95616, USA
D.S. McLACHLAN Department of Physics, University of the Witwatersrand, Johannesburg 2000, South Africa
B. ROTHBERG-BIBBY * Department of Physics, University of the Witwatersrand, Johannesburg 2000, South Africa
*Present address: 9 Walnut Way, Maungaraki, Lower Hutt, New Zealand. Progress in Low Temperature Physics, Volume VIIB Edited by D.F.Brewer Q North-Holland Publishing Company, 1978
Contents 1. Introduction 437 1.l.General considerations - Landau critical point 437 1.2. General considerations - thio superconductors 437 1.3. General considerations - bulklimit 439 1.4. Limitations of article 442 1.5. Objective 443 2. Theoretical results 443 2.1. Ginzburg-Landau theory 443 2.2. Phase transitions for Fa 1 446 2.3. Superheating for temperatures below the LCP 451 2.4. Supercooling for temperatures below the LCP 459 2.5. Thermodynamic critical field for temperatures below the LCP 462 2.6. Summary of various critical field approximations for other considerations 462 2.7. Penetration depth 463 2.8. Recent theoretical developments 473 3. Experimental techniques 474 3.1. Sample preparation 475 3.2. Measuring techniques 476 4. Second order phase transitions and the Landau critical point 479 4.1. Introduction 479 4.2. Behavior of the order parameter 480 4.3. First and second derivatives of the free energy 481 4.4. Critical field measurements in the second order region 485 4.5. Landau critical point 487 4.6. Thermodynamic critical field 489 5. First order transitions - superheating and supercooling 489 5.1. Introduction 489 5.2. Low K type I coated and uncoated superconductors 493 5.3. Intermediate K values and the metastable surface sheath 502 5.4. Low K type I1 superconductors and the observation of individual fluxoids 503 5.5. Measurement of 6(T,H) 506 5.6. Study of metastable phases of elements and alloys 507 5.7. Transition and nucleation times 508 5.8. Applications 508 5.9. Recent experimental developments 509 Appendix - derivation of equation (2.57) 511 List of symbols 512 References 513
+
1. Introduction 1.1. GENERAL CONSIDERATIONS- LANDAU CRITICAL POINT
As the temperature is increased from absolute zero to the transition temperature T, of a superconductor,* the penetration depth A(T) and the coherence length t ( T )increase from a finite value at absolute zero to infinity at T,. A moderately small superconductor is defined as one which has at least one dimension perpendicular to the applied magnetic field which is comparable to A(T) at a temperature T = TL which is defined below as the Landau Critical Point (LCP) temperature. This definition excludes a thin film in an applied magnetic field perpendicular to its surface. Depending upon the geometry, the temperature TL is directly related to the smallest dimension of the specimen, and the ratio of the critical field HL at which the LCP occurs to the bulk critical field H,(TL) is a constant which depends only on the geometry of the specimen. For temperatures below TL the superconductor makes a first order magnetic phase transition to the normal state in a large magnetic field and for T > TL it makes a second order phase transition. We assume that the smallest specimen dimension is considerably larger than A(0) such that TL is defined. This definition eliminates very small and very thin superconductors. For these Bardeen (1962) also calculates a critical point which, however, does not depend on the sample size but on the material parameters N(0) and V(BCS). The latter has not yet been experimentally found. The similarities between the Bardeen critical point and the LCP are discussed in the appendix of Rothberg Bibby (1975). On the other hand, the specimens under consideration should not be too large in comparison to L(T), otherwise the temperature range between TLand T,will not be readily accessible to experimentation. Typically, superconductors with at least one spatial dimension in the micrometer range have a LCP temperature TL around 50 millidegreesbelow T,.
1.2. GENERAL CONSIDERATIONS - THIN SUPERCONDUCTORS To illustrate the meaning of the LCP we consider fig. 1 which shows calcu-
* A list of symbols is printed after the appendix.
H.J.FINK ETAL.
438
[CH. 6,8 1
lations (solid lines) of three critical fields by Feder and McLachlan (1969) and Feder (1970) for a moderately thin film of thickness din a magnetic field parallel to its surfaces. The experimental points and the curves for K = 0.13 (broken lines) are from Pellan et al. (1973). Very close to T, the value of A(T) 2 d and the phase transition from the superconducting to the normal state and vice versa is of second order (Landau 1935). As the temperature is lowered (d is fixed) A(T)decreases and when I(T) = rZ(TL)= d/2/5 the LCP is reached. For values of I(T) < A(TL),that is for lower temperatures, there exist 3 critical fields for a superconductor whose Ginzburg-Landau (1950) K-value [K = I(T)/<(T)]is small compared to unity.
I "
'b I
1
5
10
50
100
d/X(T)
Fig. 1. Superheating field H.,,, thermodynamic critical field HT and supercooling field H,. for a film of thickness din a magnetic field parallel to its surfaces calculated from the GL theory in the approximation K 4 1. Solid lines are from Feder and McLachlan (1969). Experimental points of small indium squares, 1 .O pm thick, and curves for K = 0.13 (broken lines) are from Pellan et al. (1973).
The theoretical superheating field Hshis defined as the field up to which a metastable solution of a homogeneous superconducting state exists when calculated from the Ginzburg-Landau (GL) equations for a given geometry. At this field the Gibbs free energy of the superconducting state G, is larger than that of the normal state G, and the superconductor goes into the normal state by a first order magnetic phase transition as the magnetic field is increased.
CH.6,g 11
SMALL SUPERCONDUCI'ORS IN MAGNETIC
FIELD
439
The thermodynamic critical field HT is defined as the field at which the Gibbs free energy of the homogeneous superconducting state G, is equal to that of the normal state G,. If the homogeneous superconducting state is a lowest free energy solution up to HT, then the transition is usually a reversible first order phase transition. For moderately small specimens the homogeneous superconducting state is stable for H c HT and metastable for H > HT. The critical field HScis the supercooling field and can be observed when the applied magnetic field Ho is decreased from above HT while the superconductor is in the normal state. If at Ho = HT the specimen does not make a transition to the homogeneous superconducting state and remains in the then G, > G, and the normal state normal state as Ho is decreased below HT, is metastable. This state may persist until superconductivity nucleates by a second order phase transition. The smallest value of this field is H,, at which an infinitesimally small amount of superconductivity must nucleate. Once a nucleation center is established, superconductivity usually spreads quickly throughout the specimen which then makes an apparent first order transition to the Meissner state. We shall limit this article principally to type I superconductors. For these, as can be seen from fig. 1, H,,/Hc decreases as d/A(T)is decreased, reaches a minimum at d N 4.5A(T), then increases for smaller values of d/A(T) and merges at the LCP, located at d = &A(T), with the supercooling curve. Similar results were obtained by Esfandiari and Fink (1975) for a cylinder of radius a in an axial magnetic field where it was found that H,,/H, has a minimum at a N 4.2A(T) for all K values smaller than approximately 0.5. By contrast HT and H,, increase smoothly in fig. 1 when d/A(T>is decreased for all values of d/A(T). 1.3. GENERAL CONSIDERATIONS - BULK LIMIT
As the temperature is lowered further, 1(T) may become much smaller than d and the specimen tends to become a bulk superconductor. Whether this transition to the bulk configurationis accomplished for a given specimen with thickness d or not depends on the values of A(0) and t(0). If we limit our considerations to specimens with K ;5 1 the value of A(T) is the controlling characteristic length. When ((7") = A(T)/Kis of the order of the smallest sample dimension additional size effects occur. Figure 2 shows a number of critical fields for a bulk specimen with zero demagnetizing field. The curve for Hsh is from Fink and Presson (1969). For K < 1.1 the value of Hshis defined as above and for IC > 1.1 two curves
H.J. FZNK ETAL.
440
[CH. 6,s 1
are shown both of which are theoretical results. The upper is the maximum field the GL eqs. permit, which, however, is unstable and is indicated by the broken line. The upper limit of metastability for K > 1.1 is indicated by the solid curve Hsb. Neither curve has been confirmed yet by experiments. For K + co the value of Hsh + O.745Hc (Kramer 1968), where H, is the bulk thermodynamic field. Hshis most likely the upper experimental limit. 1.6
1.5 1.4
1.3 13 r" \ 0
r
1.1
1.0 0.9
0.8 0.7
0.3
0.5
3.0
1.0
5.0
10.0
a
Fig. 2. Critical magnetic fields of a bulk superconductor without demagnetizing field. is the superheating field. The lower curve for K > 1.1 is the upper theoretical limit of and the broken line is the maximum theoretical field which is physically metastability for HSh unstable. For K > 1.1 neither curve has been confirmed yet by experiments. Hc3 and Hc2 are the surface and bulk nucleation fields respectively. Hc3 is also equal to the supercooling = H., if the surface sheath field H, for a free surface for small K values (seetext) and HC2 is completely eliminated. He,is the lower critical field for flux entry into type I1 superconductors (K > 1/ 4 2 ) . Hmhis the lower limit of the metastable surface sheath.
Hshwas calculated first by Ginzburg (1958) for a bulk superconductor for K $ 1. He finds that
CH. 6,s 11
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
441
The constant 0.89 was obtained by numerical integration at K = 0.02. The Orsay Group (1966) and Caroli (1967) found by an analytical approximation forlc 4 1
Poulsen and Fink (1974) numerically calculated Ha,,for the semi-infinite superconducting half space for K values from above unity to K = 0.001. Their results are discussed below. In the extreme non-local limit, that is, when ic2 4 (1 -TITc) 1, the nonlocal bulk superheatingfield is (Smith et al. 1970)
+
'.
Hg,/Hc= 1. 3 6 / ~ ' / ~'(1~ ~ -/T/TC)'/' '
(1-3)
The value of x is defined by eq. (2.4). The surface nucleation field of a bulk superconductor Hc3
= 2.397uHC
(1.4)
is usually the same as the supercooling field provided K < 0.417 and the surface of the specimen is not in contact with a metal in the normal state. An exception to that rule is a metastable surface sheath which may surround a bulk specimen. The latter may exist for a small range of magnetic fieldsjust below Hc3for K values between 0.407 and 0.595 before a first order transition to the fully superconductingstate takes place (Feder 1967; Park 1967; McEvoy et al. 1967; Smith and Cardona 1967a,b; Christiansen and Smith 1968; McEvoy et al. 1969). If the surface of the superconductor is plated with a normal metal the surface sheath may be partially or totally destroyed if the metal is ferro- or antiferromagnetic (e.g. Barnes and Fink 1966). In the extreme limit, that is, when the sheath is completely eliminated by the proximity effect, the supercooling field is for K < 1/42 equal to the bulk nucleation field Hc2= d%Hc.
(1.5)
There is no supercooling for type I1 superconductors. At Hc3and Hc2the magnetic phase transition is of second order. If the proximity effect is not effective in completely eliminatingthe sheath, or if the normal metal makes poor contact with the superconductor, the supercooling field for a bulk specimen may be anywhere between Hc3and Hcz.
442
H.J. FINK ETAL.
[CH.6,s 1
Since the superheated Meissner state and the supercooled normal state are metastable, transitions to the normal and superconducting states may occur in an experiment before H , is equal to the theoretical values Hshor H,, respectively. The full range of superheating and supercooling is usually observed only in small spheres near T, where the coherence length or the size of the nucleation centers become comparable to the size of surface defects. H,, is the lower bulk critical field at which magnetic flux may enter the Meissner state in an increasing magnetic field. It is defined by a relation which equates the Gibbs free energy of the fully superconducting state in a magnetic field with no fluxoid present to that with one fluxoid present. In fig. 2 the values of H,, are from Harden and Arp (1963). Also solutions for K < 0.707 are shown by the broken line. At H,, the magnetic phase transition is not of second or higher order. Delayed flux entry into type I1 superconductors (Bean and Livingston 1964; Joseph and Tomasch 1964; DeBlois and DeSorbo 1964) when the applied field H,, is increased beyond H,, is of the same physical origin as superheating. The shielding currents at the surface of the specimen constitute an effective barrier to the magnetic field pressure acting upon the specimen from outside. 1.4. LIMITATIONS OF ARTICLE In practice all type I1 superconductorstrap flux and therefore have irreversible magnetization curves. Hysteresis due to flux trapping by pinning will not be discussed here. Furthermore, we shall not discuss hollow superconducting cylinders whose diameters would classify them as moderately small; current carrying characteristics of moderately small specimens, such as thin wires, whiskers and microbridges; films in a magnetic field perpendicular to their surfaces and the Josephson effect in moderately small specimens. Boundary Effects and Small Specimens are discussed in a review by Burger and SaintJames (1969). In this article we shall base our discussion primarily upon the GinzburgLandau theory (1950) which is relatively simple, conceptually, yet rich in its phenomenological insight and powerful in obtaining critical magnetic fields, although not always in a simple manner. This theory is based upon the electrodynamics of superconductors by London and London (1935) which, in turn, is based on the two-fluid model by Gorter and Casimir (1934a, b). Gor’kov (1959a, b) showed that the GL theory can be derived in the appropriate limits from the microscopic theory of Bardeen et al. (1957), thus
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providing the basic justification for the GL theory. For a review of the GL theory and a discussion of its limitations, see Werthamer (1969). The GL theory provides a convenient foundation for understanding the basic phenomena of size effects and magnetic phase transitions of superconductors and is used throughout this article.
1.5. OBJECTIVE The calculation and experimental verification of critical field diagrams such as figs. 1 and 2, and the accurate measurement of K values for the purpose of characterizingsuperconductorsis one of the primary objectives of superheating and supercooling studies. Superheating and supercooling measurements are the most reliable way of determining K values for type I superconductors. 2. Theoretical results
2.1. GINZBURG-LANDAU THEORY The central part of the GL theory is the complex order parameter
W )= W) exp (i@(r)),
(2.1)
which is analogous to a macroscopic wave function that describes the center-of-mass motion of the electron pairs (or superconducting electrons) which are the essence of superconductivity. Y is proportional to the local density of superelectrons which have an effective charge e* and effective mass m* which is twice that of the free electron. The phase @ of the order parameter describes the long range phase-coherence of the superconductingcondensate. The total Gibbs free energy difference between the superconducting and normal states consists of 3 terms: a negative contribution which arises from the pairing energy of the superconducting ground state, a positive contribution from the magnetic pressure difference between the local magnetic field H(v) and the applied field Ha, and a positive contribution from the kinetic energy of the superconducting electrons. It is in Gaussian units (Ginzburg and Landau 1950)
I
1’
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In eq. (2.2) the value of H,(T) is the bulk thermodynamic critical magnetic field and the applied field Ha is that field which would be present if the superconductor were removed. The integral extends not just over the volume of the superconductor but over all space. If the demagnetization coefficient is zero then the contribution of the volume integrals outside the superconducting body is zero; otherwise it is not, due to the magnetic pressure term. Y, is the value of Y in zero magnetic field. Near T, it is
I I’
1 I’
where n is the total number of conduction electrons per unit volume, T is the ambient temperature, T, is the transition temperature of the superconductor and x = x(I) is a function of mean free path 1 (Gor’kov 1959b). It is approximately
where 5(3) = 1.202, vF is the Fermi velocity and k is Boltzmann’s constant. A large number of investigators have calculated critical fields and have used various notations. In the interest of internal self-consistency,we shall use the following notation and approach. One usually assumes that the relative permeability of the metal is unity, hence V x A(r) = H(r) in c.g.s. Gaussian units. It is convenient to cast eq. (2.2) into normalized units defined by
Q(r) =
t( V@+$
A).
(2.1 1)
Likewise, all distances are measured in units of the GL low field penetration
CH. 6,B 21
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depth A = A(T) such that the normalized volume u becomes -F2+$F4+(VxQ-Ho)2+
E
445
V/A3.Then eq. (2.2)
(2.12)
F is the modulus of the normalized order parameter and Q is the normalized superfluidvelocity. If F and Q are equilibrium values then the first variation of g, namely
k = g(F+ JF, Q + JQ) -g(E Q),
(2.13)
defines a minimum of g(F,Q) if 8g = 0 provided the second variation J2g > 0 (positive definite). If 8F and 8Q are arbitrary but small deviations from the equilibrium values it follows from 8g = 0 that V2F = x2(F2+ Q 2- 1)F,
(2.14)
V X V X Q =- Q F 2 ,
(2.15)
VlsF = 0.
(2.16)
In eq. (2.15) we have put V x Ho = 0, and eq. (2.16) means that the gradient of Fis zero perpendicular to the free surface of the superconductor. Let us consider the case of an infinitely long cylinder in an axial magnetic field applied parallel to the z-direction. This case simulates experimental results by Rothberg Bibby et al. (1975) on long crystal whiskers of tin and is discussed in detail, as an example, in the subsequent theoretical sections. To obtain exact solutions for the critical fields one expresses eqs. (2.14) and (2.15) in cylindrical coordinates (2.17) (2.18) where by symmetry F = F(r) and Q,(r) is the #-directed superfluid velocity. Since the z-directed field is (2.19)
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one has to solve these equations with the boundary conditions dF/dr = 0 at r = a (radius of cylinder) and r = 0; Q = 0 at r = 0, and H, = H , at r = a. However, if a is of order unity (in conventional units of order A) and K Q 1 such that ( ( T )is much larger than the radius of the cylinder, one would expect that F(r) is approximately independent of the spatial coordinates. We therefore proceed similarly to Ginzburg (1958) by solving first eq. (2.18) with F = F, = constant and substituting the result (2.20) into eq. (2.12) with FF = 0. After integration of eq. (2.12) one obtains for the normalized Gibbs free energy differenceper unit volume (2.21) where I,, Zland I, are modified Bessel functions of the second kind. 2.2.
PHASE TRANSITIONSFOR F :
41
We investigate the magnetic phase transitions for a thin superconductor for which F i may become very much smaller than unity in eq. (2.21). We follow essentially, with some small modifications, the approach taken by Rothberg Bibby et al. (1975). We assume that F,a Q 1 is satisfied so that one can expand the last term in eq. (2.21) in a series. With the first 3 terms of the series one gets
0
= AF:-+~BF:++cF:,
(2.22)
where A = H;a2/8 - 1,
(2.23)
B = H,2a4/24- 1,
(2.24)
C = 1 1 H;a6/1024.
(2.25)
Depending on the values of A , B and C, that is, on the values of Ho and Q,
SMALL SUPERCONDUCTORS IN MAGNETIC mELD
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447
the function g/v has the limiting behavior shown schematically in fig, 3 for F: 4 1. We define the following critical fields (the prime is the derivative with respect to F,): (a) thermodynamic size dependent critical field HT : g = 0;
g' = 0;
g" > 0;
Fi > 0.
(b) superheating field (largest possible) Hsh: g
> 0;
g' = 0;
g" = 0;
g"'
> 0;
Fi >0
+ 0
Fig. 3. Schematic representation of eq. (2.22) for limiting values of the applied field H o .
(c) supercoolingfield (smallest possible) Ha, : g = 0;
g' = 0;
g"
< 0 for F i = 0;
simultaneously a stable superconducting state should be possible into which the metastable normal state can switch with the following properties: g
< 0;
g' = 0;
g"
> 0;
F,' > 0.
(d) If in (c) the latter state with F," > 0 does not exist, which is the case for very thin specimens, then the transition is of second order with: g = 0;
g' = 0;
g" > 0;
Fi + 0.
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Since g p = A - BF: g"/V =
+ CF;,
(2.26) (2.27)
-B+2CF:,
it follows from (d) and (c) that A = 0 and B > 0 for supercooling, and
A = 0 and B < 0 for a second order phase transition. The conditions A = 0 and B = 0 define the Landau critical point which (in GL normalization) for a cylinder in an axial magnetic field is H t = 813
(2.28)
a; = 3.
(2.29)
In the supercooling and second order phase transition regions, critical field, sample size and temperature are related by (A = 0)
Hi
(2.30)
= 8/u2 (GL normalized units),
where H , = H,,for a > 4 3 and H , = Hzp for a 5 43. The division between the second order phase transition field and the supercooling field is somewhat artificial, since superconductivitynucleates by a second order phase transition on the surface of the specimen at the supercooling field. Once nucleated, however, it switches to a superconducting state with F,' > 0 at the supercooling field, whereas the superconductor does not have this opportunity for HZp,that is, for a 4 4 3 . For the thermodynamiccritical field the following solution is valid
-
- 5>0 =
or
Bi
=
16ATCT/3, (2.31)
and for the superheating field
Thus, eqs. (2.31) and (2.32) are of the same form except for a numerical coefficient N(N,h = 4 and NT = 16/3).
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When eqs. (2.28) and (2.29) are cast into conventional units, the LCP of a cylinder of radius a in an axial field is defined by the following magnetic field HLand temperature TL H , = diZHc(0)A’(O)/a2,
(2.33)
TL/Tc= t , = 1 -3A2(0)/4a2,
(2.34)
where we have used the followingapproximations near T, H, = H,(o)( 1 -t 2,
A = A(O)/(l -t4)”2
N
~H,(o)(1 - i), N
(2.35) (2.36)
A(0)/2(1 --t)”2.
With the definitions
(2.37) (2.38) where H,(t) and HL are in conventional units, eqs. (2.30) to (2.32) are normalized with respect to the LCP and reduce to: = At
for
0 5 A t 5 1,
(2.39)
htc = A t
for
A t > 1,
(2.40)
h&
N(hi - At)99hi/ 128 = (hi - 1)2
for
A t > 1.
(2.41)
When N = NT = 16/3 the value of 11, is identified with hT and when N = Nsh = 4 the value of h, = h s h . Figure 4 shows a plot of eqs. (2.39H2.41) near the LCP. For an approximation to higher order in F, than given by eq. (2.22), one finds that near the LCP the order parameter of a cylinder in an axial field for a 2 2/3 is approximately F,Z = ~ ( 1 -3/a2)
(2.42)
with a2 = 3At.
(2.43)
y = yT = 32/33 forthe thermodynamic field and y = 7.h = 64/99 for the
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superheating field. From eq. (2.42) it is obvious that the range of applicability of eq. (2.41) for h, and hsh is A t - 1 4 1. Thus, it is necessary to obtain an exact solution of eq. (2.21) when the latter restriction is not satisfied. Exact solutions of eq. (2.21) near the LCP are shown in fig. 5 by the solid lines and should be compared to the results of fig. 4. The experimental points are those of Rothberg Bibby et al. (1975) replotted for the [Ool] tin crystal
I
I
1
2
0
At Fig. 4. Various critical fields near the Landau Critical Point (LCP) calculated from eqs. (2.39H2.41). ho and A I are defined by eqs. (2.37) and (2.38) respectively. Magnetic fields and temperature are normalized by the LCP.
whisker No. 21 at zero strain with HL = 10.18 G and Tc-TL 3 0.027 K. These values are consistent with Hc(0) = 305 G, A(0) = 5.1 x 10-%m and radius a = 5.2 x 10-%m. The different symbols of the points signify experiments (discussed in detail in 34) on different days. The agreement between the exact calculations and the experiments is then a confirmation of the validity of the exact GL theory near T, for type I superconductors at least for GL K values on the order of 0.1.
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451
A comparison with the exact solutions shows that the value of Fo given by eq. (2.42) is accurate to about 0.5% at the thermodynamic field for 4 3 5 a 6 10 and at the superheatingfield for 4 3 5 a I 3.
At
Fig. 5. Solid lines are exact critical fields of a cylinder in an axial magnetic field near the Landau Critical Point (LCP) calculated numerically from eq. (2.21). Magnetic fields and temperature are normalized by the LCP. ho and At are defined by eqs. (2.37) and (2.38) respectively. The experimental points of a tin whisker are from Rothberg Bibby et al. (1975). HL = 10.18 G, T,-TL = 0.027 K and T , = 3.723 K. Circles represent the differences between the fields at two observed transitions, one interpreted as the thermodynamic equilibrium field and the other as the supercooling field. Squares represent the difference between the only observed transition field, interpreted as the thermodynamic equilibrium field, and the smooth interpolation between the supercoolingfield in the first-order region and the unique transition field in the second-order region. The open and closed shapes represent results obtained on different days. For further details see text.
2.3.
SUPERHEATING FOR TEMPERATURES BELOW THE LCP
2.3.1, Cylinder in an axial magneticfield
In the limit that the radius a of a cylinder is very much smaller than the coherence length <(T)one may neglect the term (VF/x)' in eq. (2.12), and then
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one obtains eq. (2.21) which was solved numerically by Esfandiari and Fink (1975). This equation was minimized with respect to F,(dg/dF, = 0) and the maximum magnetic field was obtained when the minimum of g became an inflectionpoint as shown schematically in fig. 3 (d2g/dFi = 0). The superheating field Hsh,the corresponding order parameter F,, and the normalized Gibbs free energy differenceper unit volume g/u are shown in fig. 6. These results are valid when ax/A(T) < 1 (in conventional units). When I2/Z0in eq. (2.21) is expanded for Foa 9 1 one obtains from g' = 0 and g" = 0 an analytic approximation for Ha,,for large values of u/A which is eq. (2.61) in table 1 . 0.8
0.6
0.2
0 1
2
3
5
7
10
20
ah
Fig. 6. Exact solutions of eq. (2.21) at the superheating field If.,,. Also shown are the corresponding values for Fo . These results are valid for a 9 (.
If the above restriction u Q t(T)is not satisfied, then eqs. (2.17) and (2.18) must be solved with the appropriate boundary conditions. Numerical results has a minimum for IC 5 1 are shown in fig. 7 (Esfandiari and Fink 1975). value of about 1.58HJT) when a N 4.2A(T) for all values of K < 0.5. This behavior is similar to that of a thin film shown in fig. 1. The superheating
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field of a type I bulk specimen is reached when aK/A ;L 4.5. When K 4 1 rather large values of a/A are required before one is allowed to make a meaningful comparison between experimental values of Hahand theoretical values obtained from the semi-infinite half space such as eqs. (1.1) or (1.2). The minimum of H,,JH, as a function of a/A has its origin in the LCP at which a first order magnetic phase transition merges into a secondorder phase transition.
-0.1 1
2
3
5
10
20
Q Ih
Fig. 7. Superheating field as a function of cylinder radius a for various K values calculated from eqs. (2.17)and(2.18).The applied field is parallel to the axis of the cylinder From Esfandiari and Fink (1975).
2.3.2. Cylinder in transversefield
Figure 8 shows results similar to fig. 7 except that the magnetic field is applied transverse to the cylinder axis (Esfandiari 1976). F and Q are both functions of r and $ and their solutions are curved surfaces. For very large values of a/A, that is, in the bulk limit, the values of Hshl are one-half of HshII due to the demagnetizingfield of a long cylinder. 2.3.3. Films and spheres
The only published superheating fields for moderately thin films are those shown in fig. 1. No superheating fields have been published for arbitrary values of K for films and spheres of arbitrary thicknessesand radii respectively. D
[CH.6,4 2
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2.3.4. Bulk superconductors For sufficiently large specimens the superheating field of the semi-infinite half-space is the upper magnetic field limit of a long cylinder in an axial field for u/A > 1 and of a very thick film in a parallel field. Approximations are given by eqs. (1.1) and (1.2). An exact solution for the semi-infinite half-space is obtained by assuming that the surface is at x = 0, that the space for x > 0 is filled with the superconductor and that the magnetic field is parallel 3.0
2.5
2.0
9
\
=f
1.5
1.o
0.5
0
1
2
3
5 a/A
7 1 0
20
Fig. 8. Superheating field H.,,as function of cylinder radius a for various K values for cylinders in transverse. fields. From Esfandiari (1976).
to the positive z-direction. Equations (2.14) and (2.15) can be integrated once in closed form in Cartesian coordinates. One useful result of the integration is
Since H(x) = dQ/dx and the magnetic field dependent penetration depth
S (normalized by the low field penetration depth A) is
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455
(2.45) one obtains from eq. (2.44) at the free surface of a superconductor a relation between H(0) = Ho (applied magnetic field), F(0) and 6 which is (2.46)
Fig. 9. Order parameter F(0) at the surface of the metal is shown for the Meissner state of semi-infinite superconducting half-space as a function of Ho/H,where Ho is the applied field. The lower branch for Ho > H, is unstable and so is part of the upper branch near the peak for K > 1.10. From Fink and Presson (1969).
It is known from solutions for the semi-infinitehalf-space (Fink and Presson 1969) that
(2.47) at the maximum field which is equal to the superheating field for K < 1.1. This is shown in fig. 9. When a variational calculation on the free energy g is performed in the manner of the stability analysis by Fink and Presson (1969) at the maximum field (k, = 0) at which eq. (2.47) applies, one finds that the second variation 6'g is always zero. From this, one is not able to extract
H.J. FINK ET AL.
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information concerning the field dependent penetration depth 6. However, from the third variation Z3g = 0 one finds the following relation at the maximum field (Fink 1973)
6'H; =
[1 -FZ(O)]2 1 - ZFZ(0)
(2.48)
Therefore, the surprising result emerges from eqs. (2.46) and (2.48) that at the maximum field, regardless of the value of IC (conventionalunits)
611 = d2.
,
2.0 1.9
(2.49) I
I
I
-
1.8-
1.71.6-
------------------- ... 5.0 3.0 1.4-
.
K
1.5
=m
1.0
-
4
rg
1.3-
1.2-
HO'H,
Fig. 10. The dependence of 6/1(6 = maguetic field dependent penetration depth; 1 = low field GL penetration depth) on the applied field H,,for various values of the GinzburgLandau parameter K. From Fink and Kessinger (1967).
One can also show that the fourth variation of the Gibbs free energy b4g > 0 (positive definite). Equation (2.49) corroborates previous results by Fink and Kessinger (1967) shown in fig. 10 and recent experimental results by Parr (1975, 1976a). Equation (2.49) is valid for all K-values at the maximum (mathematical) field which for K < 1.I is the superheating field. For K > I . I
CH.6 , I 21
10
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
-. \, -
I
I
I
I
"
1
'
I
I
1
457
I
I
I
-
\
-
\
(1.1)
-.---Eq-
5-
-
I "
>
3-
X"
Eq. (l.Z)/
1
I
I
1
0.03
0.01
I
,
I
0.05
l
l
1
1
0.3
0.1
I
1
1
0.5
1
1
1
1.0
a
Fig. 11. A comparison of the exact solution of the superheating field of the semi-infinite superconducting half-space by Poulsen and Fink (1974) with approximationseqs. (1.1) and (1.2).
For IC -+ 0 the value of d6/dF(O) + -2 and for K + a0 it approaches zero at the maximum field. Equation (2.49a) is exact but differs considerably from that used by Smithet al. (1970) which is d6/dF(O) = -F-'(O). With eq. (2.49) and the auxiliary eq. (2.48), used as a consistency check, the Ginzburg-Landau eqs. (2.14H2.16) were solved for K 4 1 by Poulsen and Fink (1974) at the superheatingfield with the boundary condition (normalized units) -Q(O)
=
(2.50)
d T H 0 = dT(dQ/dx),=o.
The results are summarized in figs. 11-13. The values of H,JHc for various K--valuesare : 3 ~ 1 0 - lo-' ~
Ic
Hsh/Hc
26.71
15.47
8.496
3x10-'
lo-'
3x10-'
1
4.952
2.828
1.809
1.274
458
H.J.
FINK ET AL.
[CH.6, SI 2
X
Fig. 12. Solutions of the modulus of the order parameter F(x) and the superfluid velocity Q(x) at the superheating field for various K values between 1 .O and 0.03. The coordinate x is in units of the coherence length {(T)and Q(0)= H,,,/H,,
70.78
0.76
X
Fig. 13. Same as figure 12 except for K values between 0.03 and 0.001.
CH.6,f 21
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
An empirical fit through these points to better than
459
0.5% accuracy for
0 < K ;5 1is H,,/H, = (1 + 0 . 6 5 8 ~ - 0 . 2 3 7 ~+’ O . ~ ” ~ C ’ ) / ~ ~ ’ ~ I C ’ ’ ~ .
(2.51)
The normalized superfluid velocity at the surface of the superconductor is directly related to the (maximum) superheating field by Q(0)= -Hsh/Hc. Q(0)was found from the solutions shown in figs. 12 and 13. It required finding the functions Q(x) and F(x) simultaneously, which satisfied, in addition to’ (2.50), the boundary conditions dQ/dx = 0 at x = ao, M/dx = 0 at x = 0 and x = co,and F = 1 at x = 00. Only for K N 1.0 are the ranges of variation of F(x) and Q(x) approximately equal, about 5 to 10 e(T). For K < 1 the function Q(x), which is not an exponential function, varies very rapidly near x N 0. However, Q(0) also increases very rapidly for decreasing K values and the solutions had to be computed over distances of or larger to satisfy all the boundary conditions. The exact solution of fig. 11 is also shown in fig. 2.
2.4. SUPERCOOLING FOR TEMPERATURES BELOW TKE LCP 2.4.1. Cylinder in a magnetic$eldparallel to its axis Near the LCP the maximum supercooling field of a cylinder in an axial field is in conventionalunits H,,/H, = 4L/a.
(2.30)
Equation (2.30)is correct as long as r(T) $ a. When the radius of the cylinder is larger than both A and g, eq. (2.30)does not apply. For a bulk specimen the supercooling field is given for K < 0.417 by H,, = H,, = 2.397~H,.
(2.52)
For K > 0.417 supercooling does not take place since the second order phase transition field Hc3 > H,. This is strictly correct only if one neglects the possibility of a metastable surface sheath for IC values between 0.407 and 0.595, as discussed in $1.3. For moderately small superconductors, that is, in the region between where eq. (2.30)is valid and the bulk case, the supercoolingfield is the size-dependent surface nucleation field. The nucleation field Hc3(a,n)= HIIof a cylinder in an
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axial magnetic field is implicitly a function of the number of fluxoids n enclosed in the cylinder at HI,.Calculations by Saint-James (1965)are shown in fig. 14 where the bulk nucleation field Ifc2 = d%HC = 40/2nt2 is plotted as a function of the size-dependent surface nucleation field HII.Hr is a reference field defined by Hr = 40/2na2, where +o = hc/2e is the flux quantum. For n = 0 the solid curve in fig. 14 corresponds to eq. (2.30) which is valid in the range b
0 c a/( < 1.33
(2.53)
and is described by the equation (HI,= Hsc) (2.54)
HI14 Fig. 14. Bulk nucleation field Hct as a function of the surface nucleation field H , of a cylinder in a magnetic field parallel to its axis. The reference field H, is defined by H,= &/2nu2 where do is the flux quantum. n is the number of fluxoids enclosed by the cylinder at H , For n = 0 eq. (2.54) applies. For details see text. From Saint-James (1965).
.
The value of 1.33 in (2.53) was calculated from the intersection point of the curves for n = 0 and n = 1 of fig. 14. Similar results were obtained by Dalmasso and Pagiola (1965). In fig. 14 the value of HII+ Hc3 = 1.695HO2 when HII/Hr cv 1OOO. Hauser et al. (1974) obtained similar results for the surface nucleation field of a cylindrical cavity.
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2.4.2. Film in a magneticJieldparalle1to its surfaces
Ginzburg (1958) was the first to calculate the second order phase transition field of a thin film
(2.55)
HscIH, = d Z 1 / d ,
where d is the thickness of the film. A thick film makes a second order phase transition at Hc3 = 1.695 H c 2 . As for a thick cylinder Hc3 = H,, for K < 0.417. For K > 0.417 thick films do not supercool,provided one neglects the metastable surface sheath for K values between 0.407 and 0.595 as discussed in 5 1.3.
65-
I
3-
0
2
4
6
10
8
12
H/// H,
Fig. IS. ParaIleI nucleation field Ho = H I as a function of bulk nucleation field He, = 4,,/2nC2, normalized by the reference field H, = 1),/2nd', where d is the thickness of the superconducting layer. Note that Hc2/H,= ( d / 0 2 where is the coherence length as measured in the bulk. Equations which are good approximations are given for d / t < 1.7 and > 2.4. From Fink (1974).
<
Saint-James and deGennes (1963) were the first to calculate the size dependent surface nucleation field of a film Hc3(d)= H I , which approaches eq. (2.55) in the thin limit. It is shown in detail in fig. 15. 2.4.3. Cylinder in a magneticjeld transverse to its axis
Exact calculation of supercoolingand surface nucleation fields for arbitrary K
H.J. FINK ET AL.
462
[CH.6,O 2
values and radii have not yet been published, but one would expect the critical fields at which a second order phase transition occurs to be similar to those of a film discussed in 5 2.4.2. 2.4.4. Sphere in a magneticfield
Surface nucleation and supercooling fields of spheres as functions of radius for arbitrary K values have not yet been published. One expects results similar to those of a cylinder in an axial field discussed in 5 2.4.1. CRITICAL FIELD FOR TEMPERATURES BELOW THE LCP 2.5. THERMODYNAMIC
The thermodynamic critical field HT is defined as that field at which the Gibbs free energy difference between the superconducting state and the (nonmagnetic) normal state is zero. This definition applies to all sample shapes. HT = Hc(T)by definition when all dimensionsof a sample under consideration are very large compared to A(T) and ( ( T ) .For a cylinder in an axial magnetic field it follows from eq. (2.21) with g = 0 and g’ = 0 when Z2/Z0is expanded for values of F,a $ 1 that (Esfandiari 1976) (2.56)
HT/Hc = 1 +A/a.
Near the LCP the appropriate solution for HTobtained from an approximation to higher order in Fo than given by eq. (2.22), and consequently from a more complicated equation than eq. (2.41), is (2.57)
Pr = 8/33 for a cylinder and eq. (2.57) is valid for 1 5 A t 5 33, or 2 / 3 5 a/A 5 10, to about 5 % or better accuracy, and for 4 3 5 a/A 6 4 to better than 1% accuracy. Figure 16 summarizes the various critical fields of a cylinder in an axial fieldfora 4 <. 2.6. SUMMARY OF VARIOUS CRITICAL FIELD APPROXIMATIONS FOR OTHER CONFIGURATIONS
The Gibbs free energy difference of a cylinder of radius a in a magnetic field transverse to its axis is for a -4 (Ginzburg 1958)
<
CH.6,g 21
SMALL SUPERCONDUCTORS IN MAGNETIC F’IELD
463
(2.58)
The term connected with the magnetic field in eq. (2.58) is twice as large as that in eq. (2.21), partially because the volume integral of the term lH-Hal in eq. (2.2) outside the cylinder is not zero due to the nonuniform field distribution in a transverse field. The Gibbs free energy difference of a film of thickness din a magnetic field parallel to its surface is ford < (Ginzburg 1958)
’
<
(2.59)
<
and that of a sphere of radius r 4 (Ginzburg 1958)
!!= - F i + + q + f H : [ V
-
1- 3 coth (For)+----q . (For) 3 1 For
Eqs. (2.21), (2.58H2.60) are written in GL normalized units. The critical fields of a cylinder in a transverse field, of a film in a parallel field and of a sphere are calculated from eqs. (2.58X2.60) in the same manner as that of a cylinder in an axial field as shown above. All the pertinent fields are summarized in tables 1 4 and in figs. 16-19. The figures show also the exact solutions of eqs. (2.21),(2.58X2.60). 2.7. PENETRATION DEPTH For a bulk superconductor the measured penetration depth is defined by eq. (2.45). Within the context of the GL theory 6(Ho) approaches A, defined by eq. (2.7), as the applied magnetic field Ho approaches zero. Otherwise 6 is larger than 1but at most d21 at H,,, for a bulk superconductor [eq. (2.49)]. In general, H(x) does not fall off exponentiallyexcept in the limit that Ho + 0, in which case F(x) + 1 and eq. (2.15) is simplified. For a slab of finite thickness din a field parallel to its surfaces, it is proper to define an effective penetration depth 6(Ho,d) by
6(d) =
Ho
H(x)dx,
(2.88*)
0
where the surfaces of the slab are located at x = 0 and x = d. The superfluid *Eqs. (2.61)-(2.87) are listed in tables 14.
Table 1 Critical fields of a cylinder in an axial magnetic field
Eq.no.
Equation
Ref.
Remark
(2.29) (2.34 (2.33) (2.30) (2.52) (2.61) (2.62) (2.63) (2.56) (2.57)
[I] Ginzburg (1958). [2] Saint-James and DeGemes (1963). [3] Orsay Group on Superconductivity (1966). [4] Poulsen and Fink (1974). [51 Rothberg Bibby et d. (1975). 161 Esfandiari (1976). [7] to about 5 % or better accuracy within stated interval; for ref. see $2.8. [8] replace H, by Hzpfor T > TL.
E
7c
Table 2 Critical fields of a cylinder in a transverse magnetic field Eq. no.
Equation
(2.29)
(2.34) (2.64) (2.65) (2.52)
(2.66) (2.67) (2.68) (2.69) (2.70)
~~
'For references see table 1.
Ref.'
Remark
466
N
.
H.J. FINK ETAL.
II
.-
[CH.6,# 2
Table 4 Critical magnetic fields of a sphere
E.q. no.
Equation
(2.80)
r/AL = J21/2 TL/Tc= 1 -21A2(0)/16r2
(2.81) (2.82) (2.52) (2.83) (2.84)
H,./Hc(0) = ( a / 2 ) ( l ( 0 ) / r ) f ; HL/Hc(TL)= 1Hs.IHc(T) = J%A(T)lr HJH, = 2 . 3 9 1 ~ HJHC = [4/5(15)1/4](r/A)1/z HJHC = JZb{(A/r)'+(4b/21)[1-(21/4)
(2.79)
Ref.'
[l]
Remark r-radius; Lcp
TL T.; r S €(O N
[1,8] [2] [I]
TL-Tc;r5€ T c > T > T L a n d T S T L ; r / € 52 K < 0.4172; r S C r/A 9 1; r < €
x (A/r)2]2}1/z
H*/Hc = 0.561/J~ HT/& = (2/3)'"(1+ 3A/2) RT/Hc same as eq. (2.84) except B = 15/63 HT = He; usually enters intermediate state at WJ3
'For references see table 1. P
9
H.J. FINK ET AL.
468
[CH. 6,5 2
velocity in the center of the slab, Q(+d), must be zero by symmetry and therefore it follows that (in GL normalization) S(d) = - Q(O)/H,. This is formally the same as eq. (2.45), however, the appropriate Q(0) value must be obtained from an exact solution of the GL eqs. (2.14H2.16) for a slab of thickness d.* No solutions of 6(d) have been published yet for arbitrary values of dexcept those shown in fig. 10 which are valid ford $ 6.
3 .O
I
I
I
CYLINDER
r"
2.5
-
2.0
-
1.5
-
0
1
I
l
l
"
- A X I A L FIELD
/.
1
I
I
1.o 1 1
1
I
2
3
1 5
1
10
20
ah Fig. 16. Shown are various critical field approximations for a cylinder of radius a in a magnetic field parallel to its axis when a < <(TI.Numbers in parentheses are equation numbers in the text and in table 1 ;sh means superheatingfield, T thermodynamic field, sc supercooling field and LCP Landau critical point. The minimum is at all. = 3.8 and H,,/H. = 1.59.
The GL theory, which is local, can be modified to take non-local effects into account. Non-local effects will become of importance in a bulk superconductor when 6(H,,t) < 5, = fiuF/nd(0)where d(0) is one-half the BCS (1957) energy gap at T = 0 K. For S 2 to,the local theory is applicable. In
* Boundary conditions:dFfdx = 0 at x = 0 and x = )d, and Q = 0 at x = )d.
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
CH.6,s 21
469
particular, very close to T, all superconductors become local since A(?) approaches infinity. Non-locality means that the current density i(r) at some point r depends on the gauge invariant superfluid velocity Q(r') throughout a volume of approximate radius toaround the point located at r. Pippard (1953) first introduced this length by an analogy to the mean free path 1 as used in the 2.5
2 .o
\ <"
EXACT ($h)\
1.5
0
I
1.o
EXACT ( T ) (2.65) sc
(2.69) T
0.5 1
2
3
5
1
10
28
=/A
Fig. 17. Similar to fig. 16 except for a cylinder of radius a <( e(T) in a magnetic field transverse to its axis. Equation numbers refer to table 2. The minimum is at all. N 3.8 and HSh/H, 1.12.
work on the anomalous skin effect by Reuter and Sondheimer (1948) and Chambers (1952) and Chambers' non-local extension (Pippard 1953). tois the extent of a Cooper (1956) pair. It can be thought of as the smallest size (-ti) which can be formed by a wave packet of superconducting carriers in a bulk superconductor. More precisely, tois defined in the BCS (1957) theory by the integralofthe kernelJ(Ir'-rl, t ) (2.89)
H.J.FINK ET AL.
470
[CH. 6, !j 2
At r' = r the value of J is unity at t = 0 and decreases smoothly towards zero as 1 r' -r is increased. Jvaries only slowly with temperature. Consider some volume near the surface of a bulk superconductor. We know that the superfiuid velocity Q,which is the response to a magnetic or electromagnetic field, varies significantly only over the distance 6(Ho) near the surface when a magnetic field is applied parallel to the surface. If 6(Ho) 6 to
I
3 .O
I
I
I
I
I I I I I
FILM 2.5
2
1
2.0
1.s 2.77) T
/EXACT
(T)
1.o 1
2
3
5
7
10
20
d /A
Fig. 18. Similar to 6g.16 except for a film of thickness d Q C(T) with the magnetic field parallel to its surfaces. Equation numbers refer to table 3. The minimumis at d / l = 4.55 and H,,,lH. = 1.533.
then approximately only a fraction alto of the paired electrons within the volume of the smallest-sized wave packet can contribute directly to the supercurrent density. Thus, in the extreme non-local limit the effective number density of carriers neftcontributing to the electromagnetic response in a bulk superconductor is very approximately
For a non-local superconductor one can modify the GL theory simply by replacing n in eq. (2.3) by some nea < n such as eq. (2.90). Alternatively, one
CH. 6,421
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
471
can derive from eq. (2.7), in analogy with the anomalous skin effect, an effectivepenetration depth Jeff N
(A2tO)'?
(2.91)
I
-
-
lz/
where 6(Ho) N A,, was used. Smith et al. (1970) "scale" n also by Y(0) YmIZ= Fz(0) such that their n,, nFz(0)S/t. This leads to A,, (A2to/ F2(O))'I3 and to eq. (1.3) for IC Q 1.
I
1
2
3
5
7
10
20
r/X
Fig. 19. Similar to 6g. 16 except for a sphere of radius r Q {(T).Equation numbers refa to table 4. The minimum is at r/A = 5.24 and H&f, = 1.323.
From a practical point of view, when parameterizing non-local superconductors by experiments, it is probably simplest to retain definition (2.7) for A and reduce !P,l by an appropriate amount in relation to its local value. This, from eq. (2.7), leads to larger values of A(0) than one would obtain otherwise from the local theory and most likely to other temperature dependences than given by eq. (2.36). This increase in A(0) is observed for some materials, such as aluminium, whose measured A(0) value, as defined by eqs. (2.7) and (2.36), is by a factor of 3.3 larger than the calculated local value.
I
H.J. FINK ET AL.
472
[CH.6,$2
For moderately thin non-local films to which 6(Ho) < d << toapplies, we suggest, when applying eq. (2.90) to modify the GL equations, that the value of T o in eq. (2.90) be replaced by +d. Due to the boundaries of the film, the range function J i n eq. (2.89) is spatially reduced. The smallest size of the wave packet the superconductingpairs can form in one dimension is controlled by the width of the film. Correspondingly more effective reduction of J should occur in a thin cylinder and a small sphere. The effective penetration depth corresponding to eq. (2.91) would then be thickness dependent and its value would be approximatelyA,, (+A’d)”’. We suggest that for very thin non-local films to which d 6(Ho) 4 to applies, both 6(Ho)and tobe replaced by +din eq. (2.90), so that in this case neff n which is the approximate relation for a local superconductor. Thus, in the very thin limit, all dirty superconductors whose mean free paths are not limited by d should behave similarly to local superconductors. The function x in eq. (2.3), defined by eq. (2.4), contains the mean free path I and toso that other thickness dependences of 6(d) can be obtained. For example, for very thin films to which 1 d 6 4 toapplies, x a d/to, n,, n, so that eq. (2.7) yields Ae, A((o/d)*’2,where A is the uncorrected local value. Non-local effects [eq. (2.90)], mean free path effects [eq. (2.4)], and effects related to the reduction of the mean free path by boundary scattering (size effects) are often difficult to separate clearly from each other in thin specimens. In the latter case it is not possible to define a penetration depth as a single material parameter, in general, which is independent of thickness and magnetic field. For example, for clean, very thin films in which I is limited by d, eq. (2.74) for T > TLcan be written with ,Iecff = sA(
N
-
--
-
N
H’JH, = fi4Aeff/d= s4zA(t0/d3)1t’,
(2.74a)
where the constant of proportionality s depends on whether diffuse or specular electron scattering is dominant at the specimen surfaces. Toxen (1962) obtained s 4 4 2 , and Thompson and Baratoff (1968) s = 7z2/2/3 for specular scattering, and Liniger and Odeh (1963) obtained s = 8 4 3 and Thompson and Baratoff (1968) s = $11’ for diffuse scattering. Tinkham (1958) suggested an effective penetration depth, which is related to eq. (2.4) and takes empirically the mean free path and the thickness dependences into account and which is to be substituted into the London eqs.
-
CH. 6,g 21
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
473
(2.91a) Justifications for this approach are given by Ittner (1960) and a discussion is given by Tinkham (1975). A discussion in terms of Gor’kov’s Green’s function formalism is given by Thompson and Baratoff (1968) who obtain results for extreme type I superconductors (K 4 1) which are very similar to those of the GL theory if 1 is replaced by 6. Assuming specular scattering at the surfaces, they obtain for the LCP of a thin film d/dL n/1/2, and for T > TL the critical field equation H2,/Hc(T) z n26/2d. These results are similar to eqs. (2.71) and (2.74) within 0.7 % accuracy. Numerical calculations by Baldwin (1964) based on non-local calculations by Liniger and Odeh (1963) lead to similar results for diffuse scattering (H2,/HC(TL) = 2.27). Furthermore, due to the anisotropy of the Fermi surface the penetration depth of a single crystal is also anisotropic (e.g. Tai et al. 1975). Unfortunately, none of the experimental data yet obtained is sufficiently accurate and reliable to distinguish between the coefficients of the equations for H2,/Hc(T)[eq. (2.74)] and d/6, [eq. (2.71)] given in previous sections and those given in this section. The interpretation of experimental results has usually been done in terms of the formulae summarized in tables 1-4, determining where necessary a size dependent penetration depth, e.g. d(d, 0), and this is the approach that will be used in 8 4. N
THEORETICAL DEVELOPMENTS 2.8. RECENT
Parr (1976c) calculated the superheating field for the semi-infinite half-space by an analytic approximation. His result is H,,/H, = (1 + 1 5 d ~ ~ / 3 2 ) / ( 1 / % ) ’ ~ ~ .
(2.92)
Equation (2.92) is 2 % larger for K = 0.3 and 10% larger for K = 1.O than the exact results. Equations (2.57), (2.62), (2.67), (2.70), (2.76), (2.78), (2.84) and (2.87) near the LCP were calculated by the authors as well as the exact solutions of Hsh in figs. 18 and 19 and of HTin figs. 16-19. The latter solutions were obtained from eqs. (2.21) and (2.58)-(2.60) by numerical procedures similar to those used in obtaining the results of fig. 6 and discussed in detail in 9 2.3. Equation (2.49), the data after eq. (2.50), eq. (2.51) and the theoretical results of figs. 5,12 and 13 are unpublished.
H.J. FINK ET AL.
414
[CH. 6,#3
Rothberg Bibby et al. (1975) interpret their recent experiments on tin whiskers (discussed in experimental detail in 8 4) by using terms up to order F6 in the field-dependent term of the Gibbs free energy, and in doing so point out that the zero-field GL equations should also contain a term of this order for consistency in describing first order transitions. In the present normalization there should thus be added to the right-hand side of eq. (2.22) a term -3c I$ such that eq. (2.25) changes into
C
= 11Hi~~/1024-~
(2.93)
and eq. (2.41) into
N(hi - dt)(99hi/128 - c/dt)
= (h; - 1)’.
(2.94)
The value of c can then in principle be obtained by a study of (h;-l)’/ @,-At). This was in fact found to approach zero at the LCP and hence c was 99/128 at the LCP. This is surprisingly large for a term which is usually neglected when compared, for instance, with Brandt’s (1973)series expansion for the coefficient connected with this term. (His value of c is [31(5)/ 4A2(3)](1-t)=0.681 (1 - t ) and in the experiments (1 - t ) N lo-’.) Since the variables in eq. (2.94) are normalized with respect to HL and TL, the accuracy with which c can be measured depends strongly on the accuracy with which the LCP can be determined. Rothberg Bibby et al. (1975) used the criterion of the presence or absence of hysteresis, and estimated relatively small errors on their value of c. But if this criterion is doubted (as discussed in $4) then the errors in c as determined in this way become far larger. For the value of TL shown in fig. 5, for instance, the corresponding value of c is perhaps zero, but probably negative. Further detailed experimental results of H,(T) and HJT) near the LCP would be desirable. 3. Experimental techniques In this section the preparation of moderately small superconductors and the experimental methods used to detect their superconducting to normal phase transitions will be discussed. If the preparation method or experimental technique is widely known, the reader will be referred to the original paper for details, and it will only be where the technique is peculiar to this subject, or where knowledge of it is helpful in understanding the experiments described in $94, 5 that any details will be given.
CH.6,O 31
SMALL SUPERCONDUCTORS IN MAGNETIC FlELD
475
3.1. SAMPLE PREPARATION 3.1.1. CyIindersand whiskers
Doll and Graf (1967) prepared their microcylinders by extruding indium from glass capillaries at a temperature several degrees below the melting point in a helium atmosphere. The surfaces obtained were such that no defects could be detected by a light microscope. These cylinders were too large (7-122 pm) to be used for a study of the second order region, and their superheating fields were lower than those subsequently found in spheres (Feder and McLachlan 1968, 1969). McLachlan (197Oa, b) as well as Michael and McLachlan (1974), prepared samples (2.8-10 pm) by forcing the molten metal into glass capillary tubes. While the type I indium samples (Michael and McLachlan 1974) also exhibited poor superheating, the type I1 indium bismuth sample (McLachlan 1970a, b) may well have shown ideal superheating. This is probably because a defect, in order to act as a nucleation centre, must have dimensions of the order of e(T) and t(T)is much smaller in type I1 materials than in type I. Many experiments (Lutes and Maxwell 1955; Lutes 1957; McLachlan 1972; Rothberg Bibby et al. 1975) have been done on whiskers, which are needle-like single crystals, capable of standing elastic strains of over 1 %. They have the correct dimensions for doing size effect work as their mean diameters are typically 0.5 to 2 pm. The only successful method to date for preparing whiskers from superconducting materials such as tin, indium and their alloys is the extrusion technique of Fisher et al. (1954), which results in whiskers of irregular cross section (Bradley et al. 1957; Rothberg et al. 1971). The result of this is that, although the cross section of the whiskers may be nearly constant along their lengths, the theoretical formulae for circular cylinders are not strictly obeyed. Ideal superheating is not observed in whiskers because the sharp edges or protuberances resulting from their irregular cross sectionscanact as nucleation sites. Such a sharp protuberance has even been observed to have a phase diagram different from that of the rest of the whisker (Rothberg et al. 1971). Whiskers have also been used to study dT,/dq (Davies et al. 1966) and dH,/&, (Rothberg 1972) as they can stand elastic strains el of several percent. 3.1.2. Films
Thin films are prepared either by vapour deposition in a vacuum or by rolling
476
H.J. FINK ETAL.
[CH. 6 , $ 3
foils from the bulk material. The vapour deposition of thin films is a vast field and as the techniques are well known they will not be gone into here. It is, however, interesting to note that Caswell (1965) was able to vary the amount of hysteresis observed by controlling the microstructure of his films. For thicker ‘films’ (several microns and above), foils are to be preferred as the mean free path in a rolled foil is generally much larger than in an evaporated film of the same thickness. Also, a foil does not have the backing which, due to differential thermal contraction, tends to distort and strain films as they are cooled down. 3.1.3. Spheres All the spheres made from low-melting point metals and alloys (Pb, Cd, TI, Sn, In, Ga, Hg, SnIn and InBi) were prepared by the ultrasonic dispersion of the molten metal in a suitably heated fluid. The resulting spheres, typically 1 to 100 pm in diameter, were separated into various size ranges usiug the fact that they settle at different rates in a suitable fluid. Single spheres (see for instance: Feder and McLachlan 1969; Parr and Feder 1973) were selected from a given size range, whereas the powders were generally placed in an inert dispersant so that the powder occupied some 5-10% of the totalvolume (see for instance: Smith et al. 1970). A different technique had to be developed for Zn and A1 because of their high melting points (de la Cruz et al. 1971b). The molten metal was collected at the end of a ceramic tube which was closed except for a small hole approximately 0.5 mm diameter. Drops of molten metal were then blown through the hole into a dewar of liquid nitrogen where they solidified. The size spread was 10 to 100 pm and the powders were sorted and dispersed as before. 3.2. MEASURING TECHNIQUES
To detect the combination of magnetic field and temperature at which the superconducting to normal transition takes place in the second order region and at or near the LCP, a wide variety of techniques are possible. These include resistive, tunnelling, magnetization, a.c. susceptibility methods and microwave absorption. In the microwave absorption method (Di Crescenzo et al. 1973) the surface-resistance derivative dR/dH was measured as a function of the magnetic field parallel to the sample surface. The resistive method, used by authors too numerous to mention, is the simplest and most common, but as the electrical contacts can act as nucleation sites it is not suitable for the study of ‘ideal’ superheating and supercooling. Tunnelling
CH.6,# 31
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
477
as used by Giaever and Megerle (1961), Douglass (1961a, b, 1962) and Collier and Kamper (1966) is certainly the least ambiguous way of studying the second order region and the LCP. However, the preparation ofthe samples is more difficult and the technique has so far been limited to thin films. It is also apparently not suitable for the study of superheating and supercooling as the junctions and/or strains in the sample act as nucleation sites. In order to study superheating and supercooling well below the LCP, it is necessary that the sample be mounted in a strain-free manner and that no contacts be made to it. This necessitates the use of a magnetic or susceptibility method. The disadvantage of these techniques is that the signal becomes extremely small near the LCP, or in the second order region where u/A is of the order of 1 or less. This means that accurate measurements of T,are not possible. For this and other reasons many authors have used a combination of techniques such as tunnelling, resistance and magnetization (Burger et al. 1965); resistance and magnetization using a torque magnetometer (Robinson 1966); a.c. susceptibility, magnetization and resistance (Miller and Cody 1968); a.c. susceptibility and resistance (Pellan et al. 1973). All the recent experiments designed to study ideal superheating and supercooling, with the exception of Doll and Graf (1967) (who use a He magnetometer), use an a s . susceptibility method. In the original powder work of Feder et al. (1966a) the sample (a 1-5 pm indium powder) was placed in the r.f. coil of the tank circuit of a marginal oscillator and the change in the oscillator frequency was plotted on an XY recorder as a function of applied field at various temperatures. This method has been successfully used in many subsequent experiments.The Brown University group (see for instance: Smith et al. 1970) measured the change in the low frequency magnetic susceptibility in superconducting powders using an electronic mutual inductance bridge. As a considerable part of $5 will be devoted to superheating and supercooling in single micro-particles (spheres, whiskers and microcylinders), the little known method developed for and used in virtually all experiments of this type will be briefly discussed here. The sample is mounted in as strain-free a manner as possible in the secondary of a micromutual inductor. The voltage output from the secondary is stepped up by a factor of about 700 by a transformer located in the helium bath, and the output of the transformer is connected to the input of a low noise lock-in amplifier. In a typical setup (McLachlan and Feder 1968) the residual resistance of the secondary inductor was of the order of 0.1 and hence the resistance seen at the input of the lock-in amplifier was of the order of 50 OOO 61, which is about optimum
478
H.J. FINK ETAL.
[CH.6,# 3
at the frequency of 140 kHz used in these experiments. In this way a noise level of 2 . 0 ~lo-" V was achieved (using 8 0.3 sec time constant), which enabled the change in signal of about lo-'' V,caused by the transition of a single microparticle, to be observed (McLachlan and Feder 1968, 1969). The pick-up coils (secondaries) had to be designed so that the sample had a reasonable fill factor. For spheres (Feder and McLachlan 1969; Parr and Feder 1973) this was done by twisting an insulated piece of 15 pm copper wire in the fingers until a vanishingly small (H 50 pm dia.) loop was obtained at the end. This was then lacquered down onto a plexiglass disc, as was the primary or drive coil, which consisted of a pair of parallel wires passing close to the pickup coil. On the other hand, whiskers (McLachlan 1972) were laid in the groove formed by two touching parallel 15 pm wires and the drive wires crossed the pickup wires at right angles near both ends of the 1.0-2.0 mm long whiskers. The microcylinders (McLachlan 1970a, b; Michael and McLachlan 1974) were long enough to place in one secondary coil of a conventional concentric mutual inductor, the secondary of which consisted of two nearly equivalent coils (I.D. 200pm) connected in series opposed. However, as the cancellation of the series opposed coils was insufficient, a bridge circuit had to be used to further cancel the background signal. The drive or a s . magnetic fields at the surface of the superconductor usually of the order of 0.1 G which, if parallel to the static or swept magnetic field, can have considerable influence on the observed and H,, as H,,,= H(static) -k H(pcak a.c.) and H, =H(static) - H(peaLa.c.). This can, of course, be taken into account by measuring Hubor H,, for a number of different a.c. fields (currents) and extrapolating to zero field. To overcome this the static and a.c. magnetic fields are sometimes placed at right angles to each other, which considerably reduces the vector sum of the two fields and hence the effect on Hnhand Hat. The signal S(T, H), in a reversible region, measures the derivative of the magnetization M-HV,,,, where V,,,is an effective volume, and hence
If the amplitude of the a.c. field is small compared with the static field and they are at right angles to each other, dV,,,/dH=O to first order, because the scalar value of the total field does not change. Hence
CH.6,# 41
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
479
As the second term dominates, especially near Hsh,it is necessary to use a parallel field geometry when attempting to observe the change in 6(T, H) with magnetic field (Parr 1975). The above considerationsapply to both single particle and powder samples. In order to measure the various critical fields of interest, a hysteresis loop at constant temperature is obtained by slowly sweeping the magnetic field up and down. In order to obtain good supercooling the magnetic field must be taken to a high enough value to drive the entire sample normal, as tiny residual pockets of superconductivity can act as nucleation sites when the field is decreased. Similarly, to obtain good superheating, the field must first be decreased to a point where the entire sample is superconducting. The temperature is usually stabilized, using an electronic feedback device, and measured using the vapour pressure of the helium bath or a germanium thermometer. In order to enhance the superheating and supercooling observed in cylinders or whiskers, an inhomogeneous d.c. field is often used to eliminate nucleation at the ends of the specimen. This has been done by Faber (1957), Doll and Graf (1967), McLachlan (1972) and Michael and McLachlan (1974). In all cases considerable enhancement of the superheating or supercooling field was observed, but except for supercooling very close to To,the ideal values of the superheating and supercooling fields were not observed. For the exact coil configurationsused, the reader is referred to the above papers. 4. Second order phase transitions and the Landau critical point
4.1. INTRODUCTION
In this section we deal primarily with the experimental predictions of $2.2, that is the second order region and the LCP. The experimental verification of some of the thermodynamic predictions as well as tunnelling results to show directly the variations with temperature and magnetic field of ILII’, which is proportional to l+lz, will also be discussed. The variation of H,JH, with sample size is dealt with at the end of this section. The following section ($5) deals primarily with superheating and supercooling in the region where the moderately small specimens are behaving as bulk or nearly bulk specimens. (It is only in moderately small specimens
400
H.J. FINK ETAL.
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that significant superheating and supercooling have been observed.) This division into two sections is not purely arbitrary, since specimens which are small enough to have an experimentally accessible second order region do not exhibit optimum superheating at lower temperatures even in experiments specificallydesigned to observe this (see McLachlan 1972; Pellan et al. 1973). The samples which exhibit the largest superheating and supercooling are in the 10-50 pm range and their second order regions extend over too limited a temperature range to be accurately studied. The criterion usually used to distinguish between first and second order transitions, that is the presence or absence of hysteresis in the phase transition, needs clarification as it is a source of ambiguities and possible errors. The problem does not lie in the methods used to distinguish between the normal and superconducting states (see 4 3), but rather in the fact that most, if not all, samples do not exhibit the maximum predicted superheating and supercooling in the first order region (Baldwin 1963 and fig. 5). An example of the type of data obtained is shown in fig. 20 which shows the hysteresis in the resistive transitions obtained by Rothberg Bibby et al. (1975). These data have been chosen for presentation as they are the clearest of the very few cases where all three fields (HT, H,,, Hsh)are seen in the same sample. However, it will be seen there was no superheating field just below the LCP (fig. 20), although as fig. 21 shows there was probably ideal supercooling as the supercooling field is an extension of the second order critical field [eq. (2.30) for axial cylinder]. On the other hand Baldwin (1963) concluded that he was only observing of the predicted supercooling. These observations illustrate the fact that the hysteresis observed may be ambiguous and depends on the sample. Further, it must always be borne in mind that the lack of hysteresis does not guarantee a second order transition.
+
4.2. BEHAVIOR OF THE ORDER PARAMETER
I
I
Since the square of the modulus of the energy gap A is proportional to the square of the modulus of the order parameter $ a measurement of Id by tunnelling provides the most direct experimental evidence of a second order phase transition.The tunnellingexperiments of Douglass (1961a,by1962) and Collier and Kamper (1966) distinguish clearly between first and second order transitions. Figure 22 shows how the energy gap for a 3000 A aluminium film,in aparallel field,wasobserved to go continuouslyto zero asHapproached H,, while in a 4OOO A film there was an abrupt change in A * at H, indicating a first order phase transition. Unfortunately, none of these authors made a
I I
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CH. 6,841
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
48 I
detailed study of the validity of the equations given in $2. But Douglass' (1961) data are consistent with a LCP at a film thickness which has the predicted value of 2 / 5 times the penetration depth. 3.866 K
I
0
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8
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1
1
16
H (gauss 1 Fig. 20. Recorder tracings showing the measurement of the various critical fields at a fixed strain (0.35%) at gradually reduced temperatures near T,(e) for a [OOI] whisker. The double-headed arrows ($) indicate the reversible transitions corresponding to Hz,(7'. e) above the LCP and HT(T,e) below the LCP. The single headed arrows indicate the irreversible superheating(t ,H.h) and supercooling(1,H,J transitions which occur below the LCP. (Reproduced from Rothberg Bibby et al. (1975) by courtesy of The Royal Society.)
Thermal conductivity measurements can also be related to Id I * and hence to I '. The experiments of Morris and Tinkham (1961) show, for example, how the thermal conductivity was observed to go smoothly to its normal state value as H tended to H, for a second order transition in a 700 A tin film, while the transition was abrupt for a first order transition in a 2800 A film.
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4.3. FIRST AND
SECOND DERIVATIVESOF THE FREE ENERGY
The change in the Gibbs free energy of a superconductor with temperature T,
482
H.J. FINK ETAL.
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applied magnetic field Hand uniaxial stress o1is given (Seraphim and Marcus 1962; Rothberg Bibby et al. 1975) by
1 (K)
Fig. 21. Graphs of Hi, (2'. e) [a], H z (T,e) ( 0 )and H i (T, e) as functions of the temperature T and the strain e for a [Ool J whisker. Note that the same straight line can be drawn through the supercooling and second order critical field points and that To(e) i s a linear function of the strain for this orientation. Data from Rothberg Bibby et al. (1975).
(a
The last term becomes + Y dP if hydrostatic pressure P is applied, where V is the volume. The entropy S,magnetization M and the generalizeddisplacement Vq (q being the ith component of the strain) are thus first derivatives of the Gibbs free energy, and the latent heat is related to S.Derivatives of these quantities are the second derivatives of the Gibbs free energy, and are quantities such as the specific heat, thermal expansion and elastic moduli. The order of a thermodynamic phase transition is defined in terms of the behaviour of these derivatives of G (Ehrenfest 1933; Pippard 1957). Accordingly, as the temperature is increased through the LCP the changes, A"" (at the
CH. 6,o 41 SMALL SUPERCONDUCTORS IN MAGNETIC mELD
483
normal-superconducting transition) of first derivatives of the Gibbs free energy are no longer finite but are zero; but the second derivatives of the Gibbs free energy now show finite discontinuities. Due to the divergence of A(T) at To all superconductors, no matter how large, pass through their LCP as the temperature is increased, and hence undergo a second order phase transition before reaching To.That the superI.Og,
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conducting transition at T, is of second order was in fact first suggested from measurements of the specific heat (Keesom and Kok 1932) since these show a finite discontinuity and there is no latent heat associated with phase transition at T,.In general, it is the absence of a latent heat which makes second order phase transitions reversible, while first order phase transitions, which have a latent heat associated with them, may be hysteretic. Measurements at T, have been made on the specific heat discontinuity A " T H = 0 by Corak and Satterthwaite (1956) and Cochran (1962) and on the
484
H.J. FINK ET AL.
[CH.6,# 4
thermal expansion discontinuity dn*aby Andres (1964) and White (1964) while changes in the elastic moduli for polycrystalline and single crystal samples have been observed by Landauer (1954), Grassman and Olsen (1955), Welber and Quimby (1958) and Gibbons and Renton (1959). These quantities (second derivatives) are interrelated because of the Ehrenfest relations, as shown for a superconductor under hydrostatic pressure by Pippard (1957). For single crystals under elastic strain Rothberg Bibby et al. (1975) found that the Ehrenfest relation
where s,, denotes the compliance tensor, was satisfied within the experimental errors quoted by the individual authors. Unfortunately there were insuillcient experimentaldata available to test the other Ehrenfest relations. By contrast, when the transition is of first order there is a change in first derivatives of the Gibbs free energy at the transition, such as the volume (Olsen and Rohrer 1956), or the length (Ott 1974). Since 'An'G = H,?/8n, the thermodynamic approach also predicts relations between first derivatives and some features of the critical field curve; for example volume changes are related to dHc/dP. It is therefore possible in principle to observe the change of order of the transitions of a moderately small superconductor at the LCP by measurement of any of the first and second derivatives of G; unfortunately, except in the case of magnetization and susceptibility, such measurements have not yet been performed. If pressure P or uniaxial stress oi is regarded as a third thermodynamic variable in addition to H and T then the phase diagram becomes a second order transition surface, bounded by T, (al), by H,(T,o,), and by a Landau critical line. Such a surface was proposed by Hake (1968, 1969) for the second order transition at Hc2 of a type I1 superconductor but has not yet been experimentally observed. However, Rothberg Bibby et al. (1975) have observed (fig. 21) a similar surface in the transitions of elastically strained tin single crystals (whiskers) and have, where possible, applied the Ehrenfest equations with two independent variables, which relate the surface to the second derivativesof the Gibbs free energy. Magnetization M is a first derivative of the Gibbs free energy so that the behaviour of M and dM/dH, often used in studying the superconducting transition, is different at first and second order transitions. A formula for dM/dH for a cylinder, derived from the GL theory in the second order region,
CH.6,# 41 SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
485
is shown to be reasonably well obeyed by Michael and McLachlan (1974). 4.4.
CRITICAL FIELD MEASUREMENTSIN THE SECOND ORDER REGION
The thermodynamic critical field in the second order region is given by eqs. (2.30), (2.65), (2.74), (2.82) for axial and transverse cylinders, films and spheres respectively, but as no thorough measurements have been made of second order transitions of spheres or transverse cylinders this article will concentrate on axial cylinders and films. For these the equations are Axial cylinders: Hzp/Hc(t)= 4L(t)/a, Films :
Hzp/Hc(r)= dZA(r)/d.
(2.30) (2.74)
Using the temperature dependences H,(T) = H,(O) (1-tZ) and Az(t) = Az(0)/(l- t 4 ) the expected temperature dependence of Hzp is [(l - t 2 ) / (1 + t z ) ] ’ l z which becomes (I - t)”’ very close to T,. At this stage it should be pointed out that another type of second order superconducting-normal transition exists for “bulk” samples with K greater than 0.417. (See for instance Cody and Miller 1968; Di Crescenzo et‘al. 1973; Maloney et al. 1972). This transition is the appearance of the surface sheath at H c 3 . However, it is not easily confused with Hzpas it has a different temperature dependence, namely (1 - t Z ) / ( l + t 2 ) , and can usually be distinguished from Hzp by the experimental procedure. Although this H,, is related to If,, below K = 0.417, it will not be dealt with in this article. Lutes (1957) and Rothberg Bibby et al. (1975) (fig. 21) have 3bserved in whiskers with the magnetic field parallel to the axis that H2pdoes indeed have a temperature dependenceof (1 - f)’I2 close to T,.The latter authors have also shown (figs. 5 and 21) that at temperatures immediately below the LCP, H,, has the same temperature dependence as Hzp,as predicted in § 2. As whiskers are not circular in cross section, eq. (2.30) cannot hold exactly. This, coupled with the fact that the small and irregular cross section make it impossible to determine an exact value for a, rules out the possibility of obtaining exact values of &a, 0) or A(0)from these measurementsalthough the values obtained by Rothberg Bibbyet al. (1975) are reasonable. For films in parallel fields, where eq. (2.74) should hold exactly, Hzp has been studied or observed by many workers (see for instance: Zavaritskii 1951; Sevast’yonov 1961; Appleyard et al. 1939; Khukhareva 1962, 1963; Toxen 1962; Baldwin 1964; Burger et al. 1965; Cody and Miller 1968, 1972; B
486
[CH.6 , 8 4
H.J. FINK ET AL.
Miller and Cody 1968; Maloney et al. 1972; Di Crescenzo et al. 1973; Pellan et al. 1973). Many report, directly or indirectly, observing the temperature dependence [(l- t 2 ) / (1 + t 2 ) ] 1 / 2for H2p.However, the values of A(0) obtained from eq. (2.74) were in all cases different from that for the bulk material. For instance, Sevast’yonov’s (196 1) torque magnetometer measurements on tin and indium films (thicknesses 300-5000 8) show qualitative agreement with eq. (2.74) but thevalues obtained for A(0)are higher than that for the bulk material. Khukhareva (1962, 1963) found similar results for freshly deposited and annealed aluminium and mercury films. These and other more recent results (see below) show that the correct penetration depth to use in interpreting these results is not A(0) but the size dependent 6(d, 0) discussed in 3 2.7. In evaporated films the shorter mean free path is at least partially due to the evaporation process, but this will be treated as a “bulk” property, as we are interested in the effect that the size of the specimen has on the mean free path and hence on 6 (d, 0). In the extreme case, discussed in 3 2.7, when d I 6 < to5 I the effective penetration depth becomes 6 (d,O) = sA (50/d)112(see 3 2.7), and H Z p should be proportional to d-’I2 as predicted by eq. (2.74a). Experimental confirmation of this thickness dependence was found by Toxen (1962) in pure indiumfilms,whose thicknesseswerelessthanabout 20008. A d- 3/2dependence for HZpwas also found by Burger et al. (1965) for tin films in this size range, whereas Baldwin (1964) found H Z pvaried as d - P where IpI 2 1.25. Toxen’s (1962) work was extended by Toxen and Burns (1963) to include impurity mean free path effects as well as size effects and was applied to the critical field data measured on In Sn alloys. In thin films of lead (Cody and Miller 1968,1972) and tin (Miller and Cody 1968) eq. (2.74) was found to be obeyed quantitatively only if a size dependent penetration depth was used. The size dependent penetration depth used was a modification of that given in 5 2.7, namely N
N
where l/leff= 1/1+ l/d and lis the “bulk” mean free path. A similar increase in the penetration depth with decreasing thickness has been observed in the recent microwave measurements by Di Crescenzo et al. (1973). The results of Di Crescenzo et al. (1973) for H2,/HCand HT/Hc are shown over a large range of A(t)/d,or rather 6(d, t)/d,in fig. 23. The agreement with theory in the second order region is excellent. The results of Pellan et al. (1973), shown in fig. 1, also show good agreement with theory.
CH. 6,841
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
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4.5. LANDAU CRITICAL POINT
As discussed in 3 2 the LCP is given by the conditions specified in eqs. (2.29) (2.71) and (2.79) for axial and transverse cylinders, films in parallel fields, and spheres respectively. As virtually all meaningful work has been done on axial cylinders and films, attention will again be focussed on these geometries. The conditions for the LCP are: Axial cylinders: a/l(T,) = a l l L = 4 3 ,
(2.29)
d/l(T,) = d / l L = 4 5 .
(2.71)
Films: ,
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Fig. 23. H2,/Hcabove the LCP and HT/Hcbelow the LCP plotted as a function of d/d(r,d). The experimental results are those as shown in fig. 7 of Di Crescenzo et al. (1973)and the theoretical curve is that given in fig. 18. HT/Hcas given by the exact theory is slightly larger than the interpolation used by Di Crescenzo et at. (1973) over most of the range of dl6 0 , d ) .
When eqs. (2.29) and (2.71) are combined with eqs. (2.30) and (2.74) respectively the following equations valid at the LCP,are obtained: Axial cylinders: HL/Hc(TL)= 2.3 1,
(4.1)
HL/Hc(TL)= 2.19.
(4.2)
Films :
Experimentally the LCP is usually obtained by extrapolating H,,, Hsh and/or HT back to the point where they intersect, that is, where hysteresis is first observed. As previously mentioned the hysteresis observed is seldom
488
H.J. FINK ETAL.
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ideal which can make this way of identifying the LCP questionable. A further problem, as mentioned in $2.7 and 54.4, is that the penetration depth A(0) appearing in the equations is affected by the thickness of the samples so that it is difficult to make a quantitative test of the formula for the LCP. However, since A(or 6 ) has been eliminated in eqs. (4.1) and (4.2) some of the predictions of GL theory can be tested without knowing exactly what A(0) or 6 (d, 0) is under these circumstances. As previously mentioned, Douglass’ (1961a, b) data are consistent with a LCP at a film thickness which has the predicted value of 4 5 times the penetration depth [eq. (2.71)], although his experimental points only bracket this value and none are taken very close to the critical conditions. For films, Baldwin (1964) and Caswell(l965) found no hysteresis if H / H c was greater than N 2.19 and for whiskers Lutes (1957) found no hysteresis for H/H, greater than N 2.31. Although the scatter of the data points makes exact determination of the LCP impossible, these measurements may be taken as qualitative verification of eqs. (4.1) and (4.2). Rothberg Bibby et al. (1975) also working on whiskers, were able to distinguish between superheating, supercooling and thermodynamic transitions using a switching technique described by Rothberg et al. (1971). They also knew that they had maximum supercooling since (as shown in fig. 21) eq. (2.30) was obeyed; their measurements were close enough to T, that supercooling at H c 3 ,if relevant, would give the same temperature dependence as that at H,, (see Tilley et al. 1966). The results of Rothberg Bibby et al. (1975) were analyzed using as a criterion for TL the presence or absence of hysteresis if T c T L or T > TL respectively. The behavior of HT was discussed using terms up to order F6 in the GL equations, as discussed in 4 2 above, and a refinement, using terms up to order F * , was discussed by Nabarro and Rothberg Bibby (1975). The major problem which led to this new analysis was that the observed HT(T) values seemed to approach Hsc(T)at a finite angle at this value of TL. An alternative analysis of the same data has been carried out by one of the present authors (HJF) with a higher value of T L ; that is, making the assumption that there was no hysteresis at temperatures just below this value of TL even though the transition was of first order. Behavior of this kind might perhaps arise because whiskers are far from ideal cylinders. Excellent results are obtained by such an analysis, as shown in fig. 5, and it should also be noted that the values of(HL, TJ used in this analysis then obey eq. (4.1). As shown in fig. 20, Rothberg Bibby et al. found that the onset of hysteresis was usually in the form of supercooling; superheating began at lower temperatures.
CH.6,O 51 SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
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The non-existence of a superheatingfield just below the LCP has also been observed in whiskers by McLachlan (1972) and in axial and transverse cylinders (Michael 1973; Michael and McLachlan 1974). The reason for this is at the moment unknown. Michael (1973) has also observed that the onset of second order phase transitions in both axial and transverse cylinders occurred at the same reduced temperature as predicted by eq. (2.29). Similar results are reported by McLachlan (1972) for whiskers but as the cross sections of whiskers are irregular these results are less conclusive. Di Crescenzo et al. (1973) have obtained TL in three different ways from their results on Pb films and found the resulting TL’sto be in excellent agreement. Their methods are: Directly from eq. (2.71), using a value of 6(d, 0) obtained from other measurements. (ii) From the temperature at which the condition given by eq. (4.2) is satisfied. (iii) From the point where their microwave absorption peaks change shape indicating a shift from a first to a second order transition. (i)
The internal self-consistency of their results is an excellent verification of the theoretical predictions made in $2. 4.6. THERMODYNAMIC CRITICAL FIELD
The best experimental results for the thermodynamic critical field as a function of size are those of Pellan et al. (1973) shown in fig. 1 and Di Crescenzo et al. (1973) shown in fig. 23. In fig. 23 the results of Di Crescenzo et al. (1973) are plotted together with the results from 5 2 shown in fig. 18 and not the interpolation formula used by these authors. As can be seen in both cases the agreement between the theory and experiment is excellent, even in the intermediate ranges of d/d. 5. First order phase transitions- superheating and supercooling 5.1. INTRODUCTION
In this section the objectives will be firstly to evaluate the results obtained in superheating and supercooling experiments in terms of the equations given in
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tj 2, which are summarized in the phase diagram shown in fig. 2, and secondly, to examine the change in the superheating and supercooling fields as the LCP is approached. It is necessary to discuss superheating and supercooling first, as one cannot expect results like those shown in fig. 1 unless the sample is exhibiting nearly ideal superheating and supercooling. In fig. 2 it can be seen that the most important parameter in superheating and supercooling is K , and from figs. 1,7 and 8 it can be seen that this is also a dominant parameter in the determination of how the superheating and supercoolingfields approach the LCP. For instance, if the K value becomes large Hs,/Hc should not exhibit a minimum as a function of a/A. We will therefore examine the ‘bulk’ superheating and supercooling results first and then the evidence for size effects in each range of K. However, it is first necessary to make some general observations regarding superheating and supercooling. Previous review articles on this subject are Burger and Saint-James (1969) and Burger (1969). Experimentally, supercooling is relatively easy to observe, whereas large superheating is difficult to observe in bulk specimens. Supercooling was first studied systematicallyby Faber (1957), who found that by locally lowering the magnetic field applied parallel to a cylinder of a type I material such as Sn, In, or Al, he could keep the specimens in the normal state at fields much lower than the bulk critical field Hc(T).The relative amount of ‘supercooling’thus obtained varied significantlyas a function of the position of the field coil along the specimen, and it was proposed that certain flaws and defects near the surface of the specimen acted as nucleation centres for the superconducting phase. It was also observed that the effect of the flaws seemed to decrease as the critical temperature was approached. This is to be expected if one assumes that the flaws have dimensions such that they become of negligible size compared to r(T) and A(T) near T,. Therefore, very close to T, the supercooling observed should be typical of the material and not of the defects, and useful information can be expected from the results. The first results on superheating were obtained by Garfunkel and Serin (1952), but large superheating, as well as supercooling, was first observed (Feder et al. 1966a) in samples consisting of a large number of indium spheres having diameters in the range 1-5 pm. Similar experiments have since been performed on other materials. The reason for this success is that the chance of having a defect that will act as a nucleation centre decreases with sample size. Also, when a phase is nucleated in one sphere, it will not propagate to the others. The drawback with experiments on powders lies in the fact that one observes relatively smeared-out transitions, which makes the final interpretation of superheating and supercoolingfields somewhat arbitrary. This non-
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ideal behavior is caused by variations in size, shape, and quality of the spheres, as well as clustering and other effects that can distort the magnetic field seen by some spheres. Most of the above disadvantages were first overcome by Feder and McLachlan (1968, 1969) who studied superheating and supercooling in selected single spheres. The advantagesare: (i) The sample can be selected for size and quality under a microscope and in the case of a metal coated sphere, used in experiments to study the suppression of the surface sheath, the coating can be checked to see if it is continuous. (ii) The superheatingand supercooling transitions are sharp and unambiguous; indeed it was only after the publication of results for single spheres that it became clear which points on the hysteresis curves for powders actually corresponded to the superheating and supercooling fields, so that the powder data could be correctlyinterpreted. (iii) An accurate demagnetizing coefficient taking into account the finite penetration depth 6(T, H ) can be obtained. (iv) The variation of 6(T, H ) with magnetic field can also be measured and the only measurements of 6(T, H ) in strong (H -,Hsh)magnetic fields have been obtained from single spheres (Parr 1975,1976a). (v) The magnetic field and sample can be rotated with respect to each other and information can be obtained about the nucleation sites causing premature transitions from the metastable states. As better and sometimes unique data can be obtained from single sphere measurementsthis review will where possible concentrate on the data obtained in this way. For more details on the results obtained from powders the reader is referred to Smith et al. (1970). In spite of the use of inhomogeneous magnetic fields to eliminate nucleation at the ends of the cylinders, really good superheatinghas never been observed in type I microcylinders (Doll and Graf 1967; McLachlan 1972; Michael and McLachlan 1974). Greater success has been obtained with a type I1 microcylinder (McLachlan 1970b). The superheating results from films (fig. 1 and Pellan et al. 1973) also do not exhibit anything like ideal superheating. The results of superheating and supercooling experiments are usually given in terms of Ginzburg-Landau bulk formula derived in 6 2, i.e., H,,(T) = H,,(T) = 2.397 KH,(T)
(2.52)
492
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H.J. FINK ETAL.
and H,,(T) = Hc(T)/(K1/2)"2
K
4 1.
(2.63)
Although these results strictly apply only for local electrodynamics near T,, these formulae are used to parameterize the results over the entire temperature range and we define
For a spherical sample it has been customary to multiply the observed superheating field by +,to take into account the fact that on the equatorial band of the sphere, the field is times larger than the applied field. Hence for a sphere K,h is defined as
+
K&")
= 0.3143Ht(T)/Hfh(T).
(5.3)
However, close to T,,Parr and Feder (1973) have found it necessary to take into account the finite penetration depth of the magnetic field. The local equatorial field He, parallel to the surface of a sphere (radius r) as given by London (1950) is
Therefore, for small A/r Parr and Feder (1973) defined
as
where A(0) is the appropriate penetration depth at zero field and y = [l -(T/Tc)4]-1/2 is the well-known phenomenological temperature dependence of A(T). In order to calculate K,h or K,, it is necessary to know H,(T). As it is in most cases impossible to measure H,(T) in the same sample, it is normally calculated from the formula
CH.6,O 5 J
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493
where H,(O) is the bulk critical field at absolute zero and D is the function givingthe deviation from parabolicity(e.g., Mapother 1962). Asmanymeasurements are made extremely close to T,, the value of H,(T) and hence K(T)is extremely sensitive to the exact value of T, used in the above formula. As T, cannot usually be obtained accurately from the experimentalresults it must be chosen so as to give self-consistent results for ‘csh and icSc. The value chosen is invariably, within the experimental error, consistent with the accepted value of T, . For details as to how T, is actually determined see Feder and McLachlan (1969) and Parr and Feder (1973). 5.2. Low K TYPE I COATEDAND UNCOATED SUPERCONDUCTORS
A typical curve for the susceptibilityof an isolated low K single sphere is shown in fig. 24. The depression in the diamagneticsusceptibility as Hshis approached is due to the increase of 6(T,H ) with magnetic field and is only readily observed when the small alternating magnetic field is parallel to the static field. The variation of K,,, and ci, with the angle between the magnetic field and the sample is shown for an indium sphere in fig. 25. Virtually no correlation was found between the directions of greatest superheating and greatest I
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Fig. 24. The off-balance signal from the detection system plotted as a function of the magnetic field for a 15.4 ,urn diameter So sphere. The a.c. field is parallel to the static field and the depression of the signal close to H,,, is due to the field dependence of the penetration depth. The effect is almost undetectable if the a.c. field is at right angles to the static field. The positions of H.h and H,, are clear and unambiguous.After Pam (1976a).
H.J. FINK ETAL.
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supercooling, showing that the defects limiting superheating and supercooling were not necessarily the same and that the limitation of superheating by defects was far more severe than that of supercooling. In tin, however, there was a far stronger correlation between the directions of greatest superheating and greatest supercooling (Feder and McLachlan 1969). The directions giving
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0.08
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0.06
O0
Fig. 25.
K&
. .
:.
,980
900
I
1
-4
. .-.
1800
and K,, as a function of magnetic field direction and temperature for a 16 pm indium sphere. After Feder and McLachlan (1969).
the highest superheating and the lowest supercooling fields were used in the calculation of K , ~ ( Tand ) K,,(T), which are plotted against T/T,. Such a plot is shown for a 18.6 pm In sphere in fig. 26. The apparent increase in K,,,(T) and ic,,(T) close to T, was due to the fact that t ( T ) = A(T)/Kbecame comparable with r and a size effect set in. Size effects such as this have been observed in all single spheres (which have diameters between about 10 and
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50 pm) and powders (ranging in mean diameter from 1 to 50 pm, though usually sorted into particular size ranges) studied. These size effects make it almost impossible to get a good extrapolation to rcSh and K,, for samples smaller than 10 pm, while for the larger spheres (40-50pm) size effects are e.,
---' 71
C
I
0 349
I
-
0 e
-c
X
0
0
0
lo
O
0.155
0
1
I 0
*.
I J
l
.
l
.
l
.
0.061 I
1
0.5 0 6 0.7 0.8
0.9
I
I 1.0
T/TC
Fig. 26. Supercooling data, given as K,, ( t ) . The temperature dependence near T, is weak which indicates 'ideal' supercooling close to T,. Notice size effects near T, starting around = 0.99.The GL parameter K is determined by extrapolation to t = 1. For pure In x,,,, given by eq. (5.3). is also shown. After Parr (1976b).
barely visible on the scale shown. In practice the 'best' spheres are usually in the 20 pm range which represents a compromise between large size effects and the probability of getting a sphere relatively free of defects. This probability is proportional to r 2 for surface defects and to r 3 for bulk defects.
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However, if the correct choice is made for T, and if the region where size effects occur is neglected, it is found that K,h(T) and K,,(T)extrapolate to the same value of K at the transition temperature. In order to get this coincidence of K,h(T) and K,,(T)for gallium, Parr and Feder (1973) found it necessary to use eq. (5.9, but when eq. (5.5) was used for tin and indium (Parr 1957,1976b) the resulting K values were virtually no different from those found using eq. (5.3). In table 5 are shown the values of K obtained from supercooling Table 5 Values of the GL parameter K for various elements and alloys obtained from supercooling experiments
A1 0.015‘
Cd Ga, Gaa Ga, 0.012b ~ 0 . 0 8 2 ~~ 0 . 1 3 ~21.66’ O.14ld In 0.061”
InBi0.19% 0.155” 0.141”
In Pb Sn Zn Hg 0.115’ 0.062r 0.240b 0.0926O 0.015” 0.137‘ 0.060” 0.087‘ 0.061” 0.061’
InBi0.395% 0.2w 0.217”
InBi0.60% 0,349” 0.302”
Ode la Cruz et al. (1971b). ”de la Cruz et al. (1971a). ‘Feder et al. (1966b). dParr and Feder (1973). ‘Burger et al. (1967). ‘Smith et al. (1970). gFeder and McLachlan (1969). ”Parr (1976b). ‘Pam (1975). The measurements (c) should be regarded as preliminary. GaB and Ga, are crystallographic modifications of the usual orthorhombic a phase. Typical accuracy for the better experimentsis 1.0% to 2.0%. The second result for the alloys is obtained from the Goodman equation.
experiments for various elements using both powder and single sphere results. Also shown are the results for a series of In Bi single spheres. The increase in K with increasing bismuth content or resistivity for the In Bi alloys (Parr 1976b) is in reasonable agreement with the Goodman (1961) formula,
where K, is the value of K for the pure element, y is the electronic specific heat coefficient and p is the resistivity of the alloy. In most cases the same or a very similar value of K is obtained by extrapolating K J T ) to T = T,. The values of K obtained from the transitions of thin films in transverse magnetic field have always been found to be higher than those shown in table 5. Cody and Miller (1972) have shown that, at least in the case of Sn and Pb,
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the KSh(T)values obtained from these and supercooling experiments differ by a constant factor (2.53 for Sn and 1.33 for Pb) over a considerable temperature range. They also show that this factor increases steadily from Pb to Sn to In to Al, i.e., with the degree to which the electrodynamics are expected to be non-local. Their conclusion is that H J T ) (or HC3(T))is independent of the non-locality of the electrodynamics, whereas the transitions in perpendicular field (H,(T) or H,,(T) are sensitive to it. We feel that, pending further theoretical investigation of this discrepancy, the K,, results are to be preferred as they are independent of the type of electrodynamics and are supported by the K,h results. As can be seen from fig. 26 the value for K extrapolated from the KJT) values for the In Bi alloys is unambiguous, and it should be noted that the temperature range over which K,,(T) is relatively temperature independent increases with alloying. This is to be expected as the temperature range over which the GL theory is valid increases with K. The temperature dependence of K,, stems from the temperature dependence of HC2/H,= z / z ~ , ( T where ) the parameter K~ increases with decreasing temperature (Gorkov 1959c), and the temperature dependence of the ratio Hc3/H,2, which also increases with decreasing temperature (Luders 1967). An increase in K,,(T) with decreasing temperature is thus expected and is always observed experimentally. However, as the increase is more rapid than expected, because the enhanced nucleation at defects at lower temperatures gives an apparent increase in K (Feder and McLachlan 1969), an exact comparison with theoretical predictions is not worthwhile. The K,,, formulae given earlier hold only in the local limit, i.e., close to T, where (1 - t ) < K’ and A(T) > to.In the extreme non-local limit 1 % 1 --t 9 K’, the non-local superheating field is (Smith et al. 1970)
where C = 1.36. Smith et al. (1970) have shown that the (l-t)-1/12 temperature dependence does indeed hold for their superheating results using tin and indium powders in the temperature range t = 0.8 to nearly 1, but that the constant C required to fit these results is N 0.9 and not 1.36. This may be somewhat fortuitous as the rotation diagrams (fig. 25) show that for superheating there is considerable nucleation at defects for t < 0.9. In practice this equation has not yet been used to find K,h for any element. As shown in fig. 26, K , ~ ( T for ) indium has an anomalously large temperature dependence even when compared with K,,(T).Similar effects are observed for
H.J. FINK ETAL.
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Sn. For the strong coupling superconductors Hg and Pb (Burger et al. 1966; Smith et al. 1970)the anomalous temperature dependence of HSh/Hcor K s h ( T ) found for Sn and In is not observed. Indeed, for Hg and to a lesser extent for Pb, ic,h(T) and K,,(T)are in reasonable agreement for temperatures well below T,, showing that the region where non-local electrodynamics is applicable is much wider in these materials than in Sn or In. As is to be expected, the superheating results of Parr (1976b) show that the effect of non-local electrodynamics decreases rapidly with alloying. Indeed, H,,/H, or K,h(T) for the most concentrated alloy (In Bi 0.60%) is almost temperature independent over the range t = 0.85 to 1. Parr (1976b) found that if the eqs. (5.2) or (5.3) are used to calculate K , h ( T ) and hence K by extrapolation, the result for the most concentrated alloy is about 30% below the value found from the supercooling results. However, good agreement, especially in the case of the alloys, is found between the K(T,)values if eq. (2.92) is used to calculate K,h(T). This is to be expected as eqs. (5.2) and (5.3) are only true for K < 1. The supercooling field for a superconductor plated with a high conductivity metal is reduced due to the complete or partial suppression of the surface sheath; the lowest possible supercooling field being the bulk nucleation field
H,, = d / 2 ~ H , .
(1.5)
Feder and McLachlan (1969) found that single gold-plated indium spheres supercooled to the bulk nucleation field near T,. Figure 27 is typical of the results they obtained using eq. (1.5) to obtain K : ~ . The supercooling field is lower than in the unplated case especially near T,. The superheating and supercooling properties were found to change with time when the plated spheres were left at room temperature. This was probably a result of interdiffusion of the gold and indium changing the boundary conditions. Therefore, the spheres were stored in liquid nitrogen and aged at room temperature. Small spheres were found to have a lower T, when coated, as did large spheres when sufficientlyaged. Size effects were also larger in coated than in uncoated spheres, which is not surprising, as the order parameter is drastically lowered at the surface of a coated superconductor. In fig. 27 one finds by extrapolation that K:(T,) = 0.0685 after 1: h of ageing and that K%(T,)= 0.0620 after both 6 and 20 h of ageing. For lower temperatures the surface superconductivityis not completely suppressed since F2(0) is larger at lower temperatures and the result of using eq. (1.5) is that rcL(T) > K,,(T)for T < T,. Smith and Cardona (1967a) have found similar
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results for copper coated powder samples but as their transitions are more smeared out, their results are more ambiguous and difficult to interpret. The results for JC:~,calculated from eq. (5.3), are not fully understood, and as their interpretation depends heavily on proximity theory, they will not be discussed here. Although small spheres have in general proved to be the best samples in which to observe superheating and supercooling in pseudobulk materials, the
A 35pm
I
0.4
indium sphere
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
1
T/Tc
1 .o
Fig. 27. K:, and K : ~plotted as a function of reduced temperature for a 65 pm gold-plated indium sphere aged 1.75, 6 and 20 h at room temperature after plating. T. = 3.404 K. The dotted line shows for a 35 pm indium sphere for comparison.After Feder and McLachlan (1969).
single spheres studied to date have not been sufficiently small or the powders sufficiently homogeneous to study the LCP and the second order region. The sharp increases in KSh(T)andrcs,(T)observed close to T,are, however, evidence for size effects. Parr (1976b) has plotted some of his results for single spheres as in fig. 1 and they do indeed show the predicted increase in H,,IH, and decrease in H,,/H, as r/A(T)decreases. On the other hand, an array of films (fig. l), indium microcylinders and single whiskers (assumed to be circular in cross section), have been studied over a wide range of d/A(T) and u/A(T).
H.J. FINK ET AL.
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As an example the type of critical field diagram obtained for whiskers in axial magnetic fields is given in fig. 28. The second order region is not as clearly shown as it is for the films in fig. I , but the superheating field does show a very decided minimum at about the value predicted for cylinders in recent calculations by Esfandiari (1976). Neither whiskers nor films showed good superheatingproperties for large a/A (say a/A > lo), in spite of the fact that
2.6
-
2.0
-
1.5-
1.0 -
0.5
I
1
I
I
I
2
3
I,
5
I I I I I 6 1 0 9 1 0
I
I
20
alh Fig. 28. The normalized superheating fields H J H . (0) and the normalized supercooling for tin whisker field HJH, (0)are plotted as a function of a/rl(T)= a[l -(T/Tc)4]1'2/A(0) no. 2 of McLachlan (1972). H2p/Hc(.) in an axial magnetic field. The theoretical results are those of Esfandiari (1976). In order to get the fit between experiment and theory shown above the transition temperature of the whisker has to be taken as 3.692 K and A(0) as 4.85 x lom6cm, in moderate agreement with the results for the whisker shown in fig. 5.
in the case of the whiskers an inhomogeneous magnetic field was used to inhibit nucleation at the ends of the whiskers. A study of size effects in indium cylinders was made by Michael and McLachlan (1974). They discovered that their K,, values were larger than those obtained for bulk specimens unless a size correction for the value of H,,/H,, or H,,/H,, was made using the theoretical calculations of Saint-James (1965) shown in fig. 14. When this was done excellent agreement with the bulk
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value of xsC(Tc) = 0.062 was obtained for a cylinder with a diameter of 7.6 pm. There appears to be an overcorrectionfor smaller cylinders (2.6-4 pm) as the K~,(T,)value was found to be less than the accepted value. (There is no theoretical reason or other experimental evidence for lower K,, values in very small samples.) Not only do the results of Saint-James (1965) predict a size dependent Hc,/Hc, but they predict quantum oscillations in H,, . These oscillations
-
h
0.12-
0.10-
0.06.
0.04-
2
4
6
8
10
12
16
14
18
"sc
Fig. 29. Shows the quantum oscillations in the supercooling field (ITc3) predicted by Saint-James (1965) and shown in fig, 14. On the lower X axis is plotted the supercooling field, on the upper X axis Saint-James normalized supercooling field h. To correspond to Saint-James' Y axis ( a 2 / t 2 ( t ) ) 1, --t which has the same temperature as a2/C2(r)close to T, is plotted. The points marked are obtained using the inhomogeneous field technique (see !j3); those marked 0 are obtained from a uniform field. The two dashedlines show the region where, from the measurement of the radius a, the change in quantum number n from zero to one is expected to occur. After Michael and McLachlan (1974).
have been observed by Michael and McLachlan (1974), as is shown in fig. 29, where the superheating field is plotted against (1 - r). (1 - t ) is plotted on the Y axis as it is the temperature dependence of Saint-James(1965) Y axis, which is n 2 / t 2 ( t ) ,close to T, . The first oscillation (n = 0) can clearly be seen and the second (n = 1) just discerned, and further oscillations are lost in the scatter of the data points. The dashed lines show the region within which, due to the experimental error in the measurement of the radius, the transition from n = 0
H.J. FINK ETAL.
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to n = 1 is expected to occur. Here n is the number of flux quanta enclosed by the surface sheath at the moment of nucleation. Not only is this an interesting observation of macroscopic quantum effects but it also provides further proof that superconductivity in the bulk nucleates from the surface. Similar oscillations, but in the resistivity, have been observed in indium and indium alloy cylinders by Shablo and Dmitrenko (1970). 5.3. INTERMEDIATE IC VALUES AND THE METASTABLE SURFACE SHEATH Detailed analysis of the one dimensional GL equations has shown that the superconducting surface sheath can exist on a metastable sample below the bulk critical field (Feder 1967; Park 1967). The limiting field for this state in decreasing fields, HsEh,has been calculated as a function of ic and it has been predicted that Hssh< Hc3 for 0.407 c ic < 0.595 with Hssh= Hc3 for ic = 0.407 and Hssh = H , at ic = 0.595 (Park 1967). The ratio Hssh/Hc is predicted to vary from 0.976 to 1 as K increases from 0.407 to 0.595. The value of K = 0.407 for H,, = Hsshwas confirmed by the calculations of Christiansen and Smith (1968). Effects similar to those predicted above have been seen in bulk single crystals of Pb and Ta by McEvoy et al. (1967, 1969), and in Pb and Sn In powders by Smith and Cardona (1967b). As the theory is for a semiinfinite plane we have chosen to present the results of McEvoy et al. (1969) on a bulk tantalum cylinder, as a bulk cylinder is an excellent approximation to a semi-infinite plane if r % ( ( T ) and I(T). These results are shown in fig. 30 where Hssh/H,and H,,/Hc are plotted versus T. Also K is shown, as obtained from K = 0.417 Hc,/H,. In order to obtain the results for H,,,/H, ancillary coils had to be placed around the ends of the cylinder to prevent nucleation there. It can be seen that these results are in qualitative agreement with the predictions of Park (1967); however, Ifssh= Hc3 at K = 0.39 and Hsshis not even rapidly approaching H, for ic = 0.48. Smith and Cardona (1967b) found I f s s h = H,, at ic = 0.39 and 0.396 for Pb and Sn In (1.99 at respectively, and an extrapolation of their results gives Hslh = H, for K of the order of 0.707. From this one may be tempted to conclude that Hssh< H, as long as the material is type I (H, > Hc2).However, as these measurements were made on spheres and well below T, ,the situation at the moment is anything but clear. Smith et al. (1970) have observed that the superheating fields of Sn In powders tend to be higher than predicted by the GL theory in the K range 0.4 to 0.5. Preliminary measurements by Pettersen and Parr (1976) on a 18.9 pm In Bi (1.24 at %) sphere give K 1: 0.65 at t = 1 from measurements of H,, . In
x)
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the same sample ‘ideal’ superheating is observed with a ratio &/H, at t = 1 of about 1.6 & 0.1, which is higher than predicted by GL theory. Both these results indicate a trend for the measured values of Hsh/Hcto be higher than predicted by the GL theory, at least in this range of K. There are no specific experimental studies of size effects in this K range. 1
I
\
1
0 Hs,/Hc
40.50
with auxillary field
X
0
0
0.LO
1.35 1
3.0
I
I
3.5 L.0 TEMPERATURE K
Fig. 30. H,,/H, (0)and H.,/H, (0)are shown as a function of temperature for an electropolished single-crystalcylinder of tantalum. HC3is the field where the surface sheath nucleates and H.. is the field where superconductivity in the bulk nucleates from the surface sheath. The Y axis on the right in conjunction with the straight line shows the temperature dependence of K for this specimen. Note that for K 5 0.39, HC3= H, while for 0.39 < K .c 0.48 the value of H,, ,or H,,hin this case, is lower than H.. After McEvoy et al. (1969).
5.4. LOW
K TYPE
11 SUPERCONDUCTORS AND THE OBSERVATION OF INDIVIDUAL
FLUXOIDS
From fig. 2 it can be seen that a superheating field exists and is greater than H, even for some type I1 superconductors. At first many workers thought that ~ ~ (1.2)] ’ * must hold for all K because it gives the formula Hth= H , / K ~ ’ ~[eq.
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Hsh= H, for K = 0.707, which means that no superheating would be possible for higher values of K as the surface energy is negative. The phenomenon of superheating in type I1 superconductors was first discussed by Bean and Livingston (1964) in terms of a surface energy barrier delaying the entry of the first fluxoid(s) above H,, . They were unable to give an exact value for Hshbut thought it to be of the order of H,. Delayed entry of flux was then observed by two groups (Joseph and Tomasch 1964; De Blois and De Sorbo 1964), using samples where K > 0.707. The observed barrier field was found to be close to the thermodynamicfield. Later GL calculations (see !j 2.3), not subject to the earlier restriction that K < 1, resulted in the superheating curves such as those shown in figs. 2 and 1 1. These calculations also showed that superheating was basically the same physical phenomenon in types I and I1 materials and that Hsh> H, even for low K type I1 superconductors. As discussed in !j 2.3 and shown in fig. 2 for K > 1.1 there are two superheating fields; these are an upper mathematical field and a lower field &,where the mathematical solution becomes unstable (see fig. 2). Because the theoretical and experimental situation for K > 1.1 is at the moment unclear it will not be further discussed here. Renard and Richer (1967) were the first to observe superheating above H , in a sample of pure niobium but as they do not specify their K value, exact comparison with theory is impossible. Delayed entry of flux into type I1 microcylinders was also seen by Boato et al. (1965) and their results were in qualitative agreement with a calculation by Bobel and Ratto (1965). An interesting feature of these results was the observation of tiny blips on their oscilloscope curves correspondingto the entry and exit of single fluxoids. McLachlan (1970b) observed the entry of single fluxoids at a field of Hsh/Hc = 1.23 which could be identified with the superheating field given in 3 2.3 and fig. 2 for a sample with a K value of approximately 1.1. Figure 31 shows some of the results of McLachlan (1970a, b) for an In Bi (1.84 at %) microcylinder (K N 1.l) in a parallel field. Here the off-balance signal of the microsusceptibilityapparatus,described in § 3 is plotted as a functionof the magnetic field Ha for various reduced temperatures. Close to the transition temperature (fig. 31a) where a < d?A(T) [eq. (2.29)], the transition was reversible. Some of the observed minima correspond to the entry and exit of single flux quanta. The points at which the fluxoids enter and leave are in reasonable agreement with the theoretical predictions of Saint-James (1965), fig. 14 and Dalmasso and Pagiola (1965). The transitions in a perpendicular field were also reversible (second order) in the same temperature range as those in a parallel field, in agreement with the GL theory.
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Below the LCP, as expected, the transition became irreversible. From fig. 31b it is seen that the diamagnetic susceptibility originally decreased as the increasing field caused an increase in the penetration depth. At the point marked H, = Hshthe first fluxoid entered and the diamagnetic susceptibility increased. This process was observed for the next four fluxoids but the sixth
-1
0
I
I
I
I
5
10
15
20
“m-
-I
I
11nV t z.932
a
m LL LL
Hc3
0 0
50
25
75
100
125
150
EslZ!= 1 nV t z.932
. 5 HCI
0
25
H (gauss) 50
75
Fig. 31. The off-balance signal from the detection system is plotted as a function of the magnetic field for a low K type I1 superconductor at various fixed temperatures.A description of the various phenomena observed is given in the text. Data from McLachlan (1970a, b).
fluxoid was unpinned and driven in and out of the sample by the alternating field, giving rise to a differential paramagnetic effect. This interpretation is supported by the fact that when the alternating magnetic field was reduced by a factor of ten, only the eighth fluxoid was unpinned and gave rise to a differential paramagnetic effect. When the field was reversed the susceptibility
506
H.J. FINK ET AL.
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curve was completely different with a change in slope marked by the value of Hc3and a return to the original diamagnetic line marked by H,, . At still lower temperatures (fig. 31c) individual flux quanta could not be observed, but a small increase in the diamagnetism marked the irreversible entry of the first fluxoids or the superheating field H, = Hsh.There then appeared to be a surface sheath which made the sample appear diamagnetic until the field labelled HBin the diagram was reached (the surface sheath may be related to the ‘giant vortex’ postulated by Fink and Presson (1968)). At this point the differential susceptibility became paramagnetic as expected for a type I1 superconductor between H,, and H,, (or IT,,). In decreasing the field from above H,, changes in the slope of the susceptibility curve mark H,, and Hc2 while H,, is identified as before. Further hysteresis loops given by McLachlan (1970a’ 1970b)show in more detail how the major hysteresis loops depended on temperature and also show the entry and exit of individual fluxoids in minor hysteresis loops. Also shown in fig. 31d is a hysteresis loop in a transverse field. When the values of H,, and H, = Hsh in transverse fields were multiplied by 2, to take into account the demagnetization coefficient, it was found that the values obtained agreed with H,, and Hsh,found for parallel fields, within the experimental error. Using values of H, obtained from the results of Kinsel et al. (1964), McLachlan (1970b) showed &/H, = 1.23 for a/A > 20 in agreement with the results of § 2.3 and fig. 2 for a superconductor with a K of s 1.l. The expected IC value for In Bi (1.82 at %) is about 1.1 and a similar IC value was also derived from measurements of t ( T )and A(T)in the latter reference. The values of H,, derived from the lattter work were, however, lower than expected for the bulk material, thus indicating supercooling. Another somewhat surprising and unexplained result was the observation of a minimum in H,h/Hc as a/A(T) decreased towards the LCP, which, although predicted and observed for type I superconductors, is not predicted for a material with a K value of this magnitude (see fig. 7). 5.5. MEASUREMENT OF 6 (T, H )
Recently, Parr (1975, 1976a) has made measurements of A(T,O) over a wide range of temperatures and of 6 (T,H) for temperatures just below T, in magnetic fields approaching Hsh on Sn,In and In Bi single spheres. The values of A(T,O) were obtained by observing the change in the off-balance signal S(T,O)at the H,, field, where His relatively small, as a function of temperature. The normalized temperature dependence of this signal ( P a r 1975) is
CH.6,g 51 SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
S(T, 0) -= S(0,O)
507
+
1- 31(0)y/r 3(1(0)y/r)’ 1 - 31(0)/r+ 3(A(O)/r)’
’
which if y is known enables A(O)/r and hence A(0) to be determined from the data. It was found that the temperature dependence of 1 in all cases was equally well described by I(T) = 1(O)y, where y = 1/(1-t4)”’, or 1(T) = IBcS(O)ZBcs,where Z,, is the BCS temperature dependence. The values of 1(0) found in this way agreed with other experimental values for A(0) found from the temperature variation of A@). To determine 6(T,H)for a fixed temperature, the change in the off balance signal between the superconducting and normal state was measured as a function of field. As the variation was small the measurements had to be repeated and averages taken. Because of the demagnetizing field, the total field varied over the surface of the superconducting sphere and the data therefore contains information about an averaged penetration depth d(T,H). A procedure was developed for extracting the bulk field dependence of 6(T,H) from the data (Parr 1975). It was found that the field dependence of 6 came close to being a universal function of H/H,,. At &,y 6(TyH,,)/1(T,0)N 4 2 in agreement with eq. (2.49) and d6/dH diverged as shown in fig. 10, while at which is all in good agreelow magnetic fields 6(T,H)/1(t)N 1+a(H/H,,)z, ment with predictions made using the GL theory. 5.6.
STUDY OF METASTABLE PHASES OF ELEMENTS AND ALLOYS
The possibility exists that metastable phases of elements or alloys may form in small spheres due to rapid quenching or other factors and these phases may remain quenched-indue to the lack of a nucleation site for the crystallographic phase transition. So far, this phenomenon has only been observed and investigated in gallium. In the original work by Feder et al. (1966b), gallium after being sonicated in ethyl alcohol saturated with sodium oleate, was quenched into liquid nitrogen after (i) being cooled to room temperature, (ii) being quenched to 0°C and stored for 24 h. The first sample was found to contain Gas and Ga, in the ratio 5:l while the second contained Ga, and Gas in the ratio 4: 1 plus a trace of Ga,. From an examination of their powder type hysteresis loops Feder et al. (1966b) were able to conclude that: (i) For Ga,, K,, = 0.085 and qh= 0.082 at 0.94 K (T, was not measured but is known to be 1.082 K). (ii) For Gas, T, = 6.20 0.10 K and also K,, = 0.13 and K,h = 0.19 at 4.2 K.
+
H.J. FINK ET AL.
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+
(iii) For Ga,, T, = 7.62 0.10 K and Ga, is probably a type I1 superconductor with a K of about 1.66 fairly near T, . When Parr and Feder (1973) studied carefully selected single spheres, which had been slowly cooled and frozen in a stream of cold nitrogen gas, it was found that virtually all the spheres were in the p phase. When held just below room temperature for extended periods only a few of the larger spheres with obvious defects made the transition into the stable a phase. Extensive measurements on fl phase single spheres enabled the following parameters to be measured: T, = 6.07 0.03 K, H,(O) = 538 L- 5 G, K,, = 0.141 & 0.002, and A(0) = 880 f 100 A. From the shape of the H,(T) curve it was deduced that p gallium is a weak superconductor, as is a gallium. There are some errors in the interpretation in the original paper due to a faulty calibration of the germanium thermometer. This was corrected for by Parr (1974) who studied the isotope effect in p gallium and found that for 69Ga, T, = 6.10 L- 0.02 K and for "Ga, T, = 6.02 & 0.02 K which yields an a of 0.43 f 0.02 in the equation T, M-'. N
5.7. TRANSITION AND NUCLEATION TIMES Early work by Faber (1957) showed that the transition time in bulk superconductors of the metastable superheated superconductingstate to the normal state was limited by eddy current damping. Valette and Waysand (1975) have recently performed experiments where the flux pulse due to the transition of single (10-40 pm) Hg spheres is led along a transmission line into a special low noise amplifier and the integrated pulse signal of the amplifier output is fed into an oscilloscope provided with a switchable storage screen. The pulse shapes showed that the transition times of particles of even this small size are limited by eddy current damping. The nucleating process could not be observed directly as it is shorter than the 6 nsec dead time of the amplifier. Combining their own and previous results, Valette and Waysand (1975) concluded that eddy current dampingwas probablythe limiting factor in observing the nucleation process, even for a region as small as approximately T(T)in diameter. At this point the transition time would be N 4 x lo-" sec which must be taken as a provisional upper limit for the nucleation time. 5.8.
APPLICATIONS
One possible application of superheating is the detection of charged nuclear particles using a superheated powder.
CH.6,§51
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
509
A moderately small superconductor in a field H such that H, < H < Hsh can jump into the normal state under the influence of an external perturbation. Therefore, when a superheated superconducting powder is traversed by a particle beam the resulting destruction of superconductivity can be monitored, using the methods already described, as a function of the number of incident particles. Experiments by Valette (1971) showed that the condition for a superheated granule to be driven into the normal state is that the energy loss in the granule be sufficient to raise the temperature of the granule by an amount
Experiments have been done using 8- irradiation emitted from radioactive 7 9 A ~9 81amalgamated in the Hg spheres of the superconducting powder (Bernas et al. 1967), and using a van de G r d electron beam (Rosenblatt et al. 1973). Experiments have also been done using high energy 0.5-7 GeV electrons from an electron synchrotron (Druiker et al. 1975). Here the destruction of superconductivityis not primarily due to the high energy electrons themselves but rather to the photoelectrons created by the transition radiation which is due to the relativistic electrons crossing the dielectric (paraffin wax) to metal (Hg) interfaces. The latter application in particular shows some promising results. As small superconducting grains are bistable they can naturally be used as computer elements. However, as even 40 pm Hg spheres have switching times of the order of 400nsec they would not be very fast. 5.9.
RECENT EXPERIMENTAL DEVELOPMENTS
Very recently, Yogi (1977) has made a study of superheating in In, Sn and the In Bi and Sn In alloy systems in a rf magnetic field. In this method the sample, a sphere of approx 1-2 mm in diameter, was placed in a helical resonator at the point where the magnetic field was a maximum and the Q of the resonator was monitored as a function of the peak rf field. A noticeable change in the Q-value of the resonator marked the superconducting to normal transition, and the peak rf field at this transition could be determined to within an accuracy of & 10%. Whilst considerable superheating was observed in the rf magnetic fields, the same samples showed very little superheating in a quasi static magnetic field.
510
H.J. FINK ET AL.
For In and Sn the observed superheating fields at t N 1 were very similar to those which had been found earlier by Feder and McLachlan (1969) and Smith et al. (1970). At lower temperatures, H,h/Hc was found to vary as 1/(1 - t ) l I 6 which is characteristic of line nucleation (Yogi 1970). Therefore it was inferred that the nucleation centres for the normal phase were in the form of a line in the above experiments. Whether this temperature dependence is characteristic of transitions in a rf magnetic field is not yet known, as this type of nucleation may well be caused by inhomogeneities at the surface, such as grain boundaries. The latter are not present in single crystal microspheres whosevalues of HSh/Hcwere found to vary as 1/( 1 - t)‘” by Smith et al. (1970), which is characteristic of plane nucleation. For the alloys whose K values were as large as 1.75, the results were consistent with the theoretical predictions for K < 1.1 discussed in $ 2 and shown in fig. 2. Unfortunately, the few experimental results for K greater than 1.1 are such that it is impossible to decide if the upper limit of the superheating field is determined by the solutions to the GL equations in $2.3 or the theoretically smaller field due to instabilities determined by Fink and Presson (1969). However, Yogi’s (1977) results and those of McLachlan (1970b) are in disagreement with Kramer’s (1968) theoretical predictions for the fluctuationlimited superheating field.
Acknowledgements The authors would like to thank H. Parr for makinga number of his papers available to us prior to publication. One of the authors (H.J.F.) would like to thank M.R. Esfandiari and R.S. Poulsen for their assistance with the numerical solutions and permission to use their unpublished results. Another of the authors (B.R.B.) would like to thank the PEL Library, DSIR, Lower Hutt, New Zealand for making library facilities available to her. We all thank Professor F.R.N. Nabarro for his interest in this work and his constructive criticism. The manuscript was typed at the opposite ends of the globe by the following people: Miss Susie Butz, Miss Tamara Gibbons, Miss Aelise HOUX, Mrs. Nora Main and Mrs. Gwen Murphy, while most of the figures were drawn by Mrs. Katherine Peach; the authors would like to acknowledge their debt to all of them.
SMALL SUPERCONDUCTORS IN MAGNETIC FIELD
511
Appendix - Derivation of equation (2.57) Starting with eq. (2.21) and substituting for Fou = X , the first derivative of g is
Assuming that a is known, g = 0 and dg/dX = 0 are two equations with two unknowns whose solutions for Xand Ho 3 HT(X)are
With B
3
Z2(X) Zo(X)/Zf (A’) eq. (A.2) transforms into
X2(3B- 1) = 2U2(2B- 1).
(A.4)
Expanding B in a series for X 2 eq. (A.4) becomes
< 1 and neglecting powers of X ” and higher,
1 -a2/3+ 1 1X2/32+7X4/15360 = 0.
(A.5)
A solution ofeq. (A.5) near the LCP where (1 -u2/3) 4 1 is
(a)’
=
Fi
=~ ( 1 - ~ 2 ) [ 1 - ~ ( l - ~ ) ] .
Equation (A.6) reduces to eq. (2.42) when the higher order term (1 -3/u’)’ is neglected. Substituting eq. (2.42) into eq. (A.3) and neglecting higher order terms in (1 - 3/a’) leads to
Near the LCP a’ N 3 so that the term in brackets [ J is 3 for u2 = 3. With this substitution eq. (A.7) becomes eq. (2.57) in conventional units.
512
H.J. FINK ET AL.
List of symbols vector potential coefficient of free energy (2.22), (2.23)
radius of cylinder coefficient of free energy (2.22), (2.24)
specific heat; coefficient of free energy (2.22), (2.25) velocity of light; coefficient of free energy (2.93), (2.94) thickness of film superconducting electron pair charge = 2e modulus of GL normalized order parameter, r dependent (2.6)
modulus of G L normalized order parameter, independent of r Gibbs free energy of normal state Gibbs free energy of superconducting state normalized Gibbs free energy difference (2.5) applied magnetic field, conventional units bulk thermodynamic critical field (2.35) lower critical field for flux entry into bulk superconductor bulk nucleation field (1 5 ) surface nucleation field (1.4) applied field at some value 0, conventional units; G L normalized applied field = H./d\/z x Hc local magnetic field on equatorial plane of a superconducting sphere superheating field conventional units; superheating field, GL normalized = H , h / d i H, supercooling field, conventional units; supercooling field, GL H, normalized = H.&
magnetic field at Landau Critical Point (LCP), conventional units magnetic field corresponding to the lower limit of the metastable surface sheath thermodynamic critical field applied field normalized by LCP = HolHL superheating field normalized by LCP supercooling field normalized by LCP thermodynamic critical field normalized by LCP second order phase transition field normalized by LCP current density Fourier component mean free path mass of superconducting electron pair = 2m(m = free electron) magnetization coefficient coefficient = 4, for superheating field coefficient = 1613, for thermodynamic critical field electron density of states at the Fermi surface for one spin conduction electron density pressure superfluid velocity coordinate (cylindrical); radius of sphere resistance; surface resistance signal from detection apparatus temperature, ambient transition temperature temperature of LCP TIT. TLITc volume; BCS electron phonon interaction constant volume of sample which excludes magnetic field GLnormalized volume = V/A3
SMALL SUPERCONDUCTORS IN MAGNETIC Fermi velocity coordinate temperature dependence of the penetration depth thermal expansion coefficient coefficient electronic specific heat coefficient; coefficient (2.42). one-half of the BardeenCooper-Schrieffer energy gap (Tc-T)/(T,-Td = ( I - t ) / (1 - f L ) field and/or mean free path dependent penetration depth first variation of GLnormalized Gibbs free energy difference second variation of GL normalized Gibbs free energy difference third variation of GL normalized Gibbs free energy difference fourth variation of GL norrnalized Gibbs free energy difference strain, ith component zeta function of 3 = 1.202 (Riemann) GL K value = 1(T)/ C(T) temperature dependent supercooling K, value defined by eq. (5.1)
P . C
4 I I
K9h
FIELD
temperature dependent supercooling K, value for coated superconductors derived from eq.(1.5) temperature dependent superheating K, value defined by eq. (5.2) temperature dependent superheating K, value derived from eqs. (5.2), (5.3) m
4n)
513
= C (X+I)-";n = 2, 3, k=O
...
A, A(T),A ( f ) low field penetration depth (2.7)s(2.36) LL
W'L)
t, W),C(r) coherence length (2.9) 1.
I I
01
@(r)
I X P
Wr)
IW lYml
Jl(r)
0.18fmp/kTo, BCS (Pippard) coherence length, fiuF/wA(0) 3.1416 stress Ith component phase of order parameter, r dependent coordinate, cylindrical function of I(2.4) resistivity complex order parameter, r dependent absolute value of order paramter absolute value of order parameter in zero field, independent of r modulus of complex order parameter, r dependent
References Andres, K., 1964, Phys. Kond. Mat. 2,294. Appleyard, E.T.S.,J.R. Bristow, H. London andA.D. Misener, 1939, Proc. Roy. Soc (London) 172A, 540. Baldwin, J.P.,1963, Phys. Lett. 3,223. Baldwin, J.P., 1964, Rev. Mod. Phys.36,317. Bardeen, J., L.N. Cooper and J.R. Schrieffer, 1957, Phys. Rev. 108,1175. Bardeen, J., 1962, Rev. Mod. Phys. 34,667. Barnes, L.J. and H.J. Fink, 1966, Phys. Lett. 20,583. Bean, C.P. and J.P. Livingston, 1964, Phys. Rev. Lett. 12,14. Bernas, B., J.P. Burger, G. Deutscher, C. Valette and J.S. Williamson, 1967, Phys. Lett. MA, 721. Boata, G., G. Callinaro and C. Rizzuto, 1965, Solid State Comrnun. 3,173. Babel, G. and C.F. Ratto, 1965, Solid State Commun. 3,177. Bradley, P.E., J. Franks and P.E. Rush, 1957, Proc, Phys. SOC.(London) B70.889.
514
H.J. FINK ET AL.
Brandt, E.H., 1973,Phys. Stat. Sol. (b)57,277. Burger, J.P., G. Deutscher, E. Guyonand A. Martinet, 1965, Phys. Rev. 137,853. Burger, J.P., J. Feder, S.R.Kiser, F. Rothwarf and C. Valette, 1966, Roc. 10th Inter. Conf. on Low Temp. Phys., M.P. Makov ed. (Viniti, Moscow, 1967) Vol. IIB,p. 352. Burger, J.P., 1969, Superconductivity, P.R. Wallace ed. (Gordon and Breach, New York) Vol. 1, pp. 463-488. Burger, J P. and D. Saint-James, 1969, Superconductivity, R.D. Parks ed. (Marcel Dekker Inc., New York) pp. 977-1005. Caroli, C., 1966, Ann. Inst. Henri Poinark A4,159. Caswell, H.L., 1961, J. Appl. Phys.32,105. Caswell, H.L., 1965, J. Appl. Phys. 36,80. Chambers, R.G., 1952, Proc. Roy. SOC.A215,481. Christiansen, P.V. and H. Smith, 1968, Phys. Rev. 171,445. Cochran, J. F., 1962, Ann.Phys 19,186. Cody, G.D. and R.E. Miller, 1968, Phys. Rev. 173,481, Cody, G.D. and R.E. Miller, 1972, Phys. Rev. B5,1834. Collier, R.S.and R.A. Kamper, 1966, Phys. Rev. 143,323. Cooper, L.N., 1956, Phys. Rev. 104.1189. Corak, W.S.andC.B. Satterthwaite,1956, Phys. Rev. 102,662. Dalmasso, C. and E. Pagiola, 1965, Nuovo Cirnento 35,811. Davis, J.H., M.J. Skove and E.P. Stillwell, 1966, Sol. Stat. Comm. 4,597. DeBlois, R.W. and W. DeSorbo, 1964, Phys. Rev. Lett. 12,499. De la Cruz, F., M. Cardonaand M.D. Maloney, 1971a, Phys. Rev. B3.3802. De la Cruz. F., M.D. Maloney and M. Cardona, 1971b, Physica 55,749. Di Crescenzo, E., P.L. Indovina, S. Onori and A. Rogani, 1973, Phys. Rev. B7,3058. Doll, R.and P. Graf, 1967, Phys. Rev. Lett. 19,897. Douglass, Jr., D.H., 1961a, Phys. Rev. Lett. 6,346. Douglass, Jr., D.H., 1961b, Phys. Rev. Lett. 7,14. Douglass,Jr., D.H., 1962, IBM J. Res. Dev. 6,44. Douglass,Jr., D.H. and R.H. Blumberg, 1962, Phys. Rev. 127,2038. Druiker, A.K., C. Valette and G. Waysand, 1975, Low Temperature Physics LTl4 M. Krusius and M. Vuorio eds. (North-Holland, Amsterdam) Vol. 4, p. 278. Ehrenfest,P., 1933, Commun. Phys. Lab. Univ. Leiden Suppl. 75b. Esfandiari, M.R.and H.J. Fink, 1975, Phys. Lett. 54A, 383. Esfandiari, M.R., 1976, Thesis, University of California, Davis, USA. Faber, T.E., 1957, Proc. Roy. Soc.(London)A241,531. Feder, J., S.R. K i m and F. Rothwarf, 1966a, Phys. Rev. Lett. 17,87. Feder, J., S.R. Kiser, F. Rothwarf and C. Valette, 1966b, Solid State Commun. 4,611. Feder, J., 1967, Solid State Comrnun. 5,299. Feder, J. and D.S. McLachlan, 1968, Solid State Commun. 6,23. Feder, J., and D.S. McLachlan, 1969, Phys. Rev. 177,763. Feder, J., 1970, Thesis, University of Oslo, Norway. Fink, H.J., 1973, unpublished. Fink, H.J., 1974, J. Low Temp. Phys. 16,387. Fink, H.J. and R.D. Kessinger, 1967, Phys. Lett. =A, 241. Fink, H.J. and A.G. Presson, 1968, Phys. Rev. 168,399. Fink, H.J., and A.G. Presson, 1969, Phys. Rev. 182,498. Fisher, R.M., L.S. Darken and K.G. Carroll, 1954, Acta Met. 2,368. Garfunkel, M.P. and B. Serin, 1952, Phys. Rev. 85,834. Giaevcr. I. 2nd K. Megerle, 1961, Phys. Rev. 122,1101.
SMALL SUPERCONDUCTORS IN MAG"IC
FIELD
515
Gibbons, D.F. and C.A. Renton, 1959, Phys. Rev. 114,1257. Ginzburg, V.L.andL.D. Landau, 1950,Zh. Eksp.Teor. Fiz.20,1064. Ginzburg, V.L., 1958,Zh. Eksp. Teor. Fiz. 34,113. (Engl. trans., 1958,JETP7,78). Goodman, B.B., 1961,Phys. Rev. Lett. 6,597. Gor'kov, L.P., 1959a, Zh.Eksp. Teor. Fiz.36,1918 (Engl. trans., 1959,JETP9,1364). Gor'kov, L.P., 1959b, Zh. Eksp.Teor. Fiz.37,1407 (Engl. trans.,1960,JETP10.998). Gor'kov, L.P., 1959c, Zh.Eksp. Teor. Fiz. 37,833. (Engl. trans., 1960, JETP 10,593). Gorter, C.J. and H.B.G. Casimir, 1934a, Physik. Z.35.963. Gorter, C.J. and H.B.G. Casimir, 1934b.Z.Tech. Phys. 15,539. Grassman, P. and J.L. Olsen, 1955, Zeit. Phys. 28.24. Hake, R.R., 1968, Phys. Rev. 166,471. Hake, R.R., 1969,J. Appl. Phys.40,5148. Harden, J.L. and V. Arp, 1963, Cryogenics 3,105. Hauser, J.R ,J.S.-Y. Wangand C. Kittel. 1974, Phys, Lett. 47A,34. Ittner, 111, W.B, 1960,Phys. Rev. 119,1591. Ittner, 111, W.B, 1963, Physics of Thin Films, G. Hass ed. (Academic Press, New York and London) Vol. 1,pp. 233-273. Joseph, A.S. and W.J. Tomasch, 1964,Phys. Rev. Lett. 12,219. Keesom, W.H. and J.A. Kok. 1932,Comrnun. Phys. Lab. Univ. Leiden No. 221e. Khukhareva, I.S., 1962, Zh. Eksp. Teor. Fiz. 41,728 (Sov. Phys. JETP 14,526). Khukhareva, IS., 1963, Zh. Eksp. Teor. Fiz. 43,828 (Sov. Phys. JETP 16,828. Kinsel, T., E.A. Lynton and B. Serin, 1964, Rev. Mod. Phys. 36,105. Kramer, L., 1968,Phys. Rev. 170,475. Landau, L.D., 1935,Phys. Z. Sowjet. Union 8,113. Landauer, J. K., 1954,Phys. Rev. 96,296. Liniger, W. and F. Odeh, 1963, Phys. Rev. 132,1934. London, F. and H. London, 1935, Proc. R. SOC. A149,71. London, F., 1950,Supeduids, Wiley, New York,1950, p. 35. Luders, G., 2.Physik 202,8. Lutes, O.S. and E. Maxwell, 1955, Phys. Rev. 97,1718. Lutes, O.S., 1957, Phys. Rev. 105,1451. Maloney, M.D., F. de la Cruz and M. Cardona, 1972, Phys. Rev. B5,3558. Mapother, D.E., 1962,IBM J. Res. Develop. 6,77. McEvoy, J.P., D.P. Jones, and J.G. Park, 1967. Solid State Commun. 5,641. McEvoy, J.P., D.P. Jones and J.G. Park, 1969,Phys. Rev. Lett. 22.229. McLachlan, D.S. and J. Feder, 1968, Rev. Sci. Instr. 39,1340. McLachlan, D.S., 1970a, Solid State Commun. 8,1589. McLachlan, D.S., 1970b, Solid State Commun. 8,1595. McLachlan, D.S., 1972,J. Low Temp. Phys. 6,385. Michael, P., 1973,Thesis, University of the Witwatersrand,Johannesburg, South Africa. Michael, P. and D.S. McLachlan, 1974, J. Low Temp, Phys. 14,607. Miller, R.E. and G.D. Cody, 1968,Phys. Rev. 173,494. Morris, D.E. and M. Tinkham, 1961,Phys. Rev. Letters 6,600. Nabarro, F.R.N. and B. Rothberg Bibby, 1975, Phil. Trans. Roy. SOC.A278,343. Olsen, J.L. and H. Rohrer, 1956, Helv. Phys. Acta. 29,426. Orsay Group on Superconductivity, 1966, Quantum Fluids. Proceedings of the Sussex University Symposium, Sussex, 1965, D.F. Brewer ed. (North-Holland, Amsterdam) pp. 26-67. Ott, H.R., 1974, Thesis, Eidgenossische Technische Hochschule, Ziirich, Switzerland. Park, J.G., 1967,Solid State Commun. 5,645.
516
H.J. FINK ET AL.
Pam, H. and J. Feder, 1973, Phys. Rev. B7,166. Parr, H., 1974,Phys. Rev. B10,4572. Pam, H., 1975, Phys. Rev. B12,4886. Parr, H., 1976a, Phys. Rev. B14,2842. Parr, H., 1976b, Phys. Rev. B14,2849. Pam, H., 1976c, Z. Physik B25,359. Pellan, Y., J. Blot, J.C. Pineau and J. Rosenblatt, 1973, Phys. Letters 44A,415. Pettersen, G. and H. Parr, 1976, to be published. Pippard, A.B., 1953, Proc.Roy. Soc.(London)A216,547. Pippard, A.B., 1957, Elements of Classical Thermodynamics, The University Press, (Cambridge)pp. 136146. Poulsen, R.S. and H.J. Fink, 1974, unpublished. Renard, J.C. and Y.A. Richer, 1967, Phys. Letters MA, 509. Reuter. G.E.H. and E.H. Sondheimer, 1948, Proc. Roy. Soc.A M , 336. Robinson, G., 1966,Proc.Phys. Soc.89,633. Rosenblatt, J., J. Blot, Y. Pellan and J.C. Pineau, 1973, C.R. Hebd. Seances Acad. Sci. B277,527. Rothberg, B.D., F.R.N. Nabarro and D.S. McLachlan, 1971, J. Low Temp. Phys. 5,665. Rothberg, Barbara D., 1972, Thesis, University of the Witwatersrand, Johannesburg, Rothberg Bibby, B., F.R.N. Nabarro, D.S. McLachlan and M.J. Stephen, 1975, Phil. Trans.Roy. Soc.A278,311. Saint-James,D. and P.G. DeGennes, 1963. Phys. Letters 7,306. Saint-James,D., 1965,Phys. Letters 15,13. Seraphim,D.P. and P.M. Marcus, 1962, IBM J. Res. and Dev. 6,94. Sevast’yonov, B.K., 1961, Zh. Eksp. Teor. Fiz. 40, 52 (Sov. Phys. JETP 13, 35). Sevast’yonov, B.K. and Y.A. Sokolina, 1962, Zh. Eksp. Teor. Fiz. 42, 1212 (SOV.Phys. JETP 15,840). Shablo, A.A., and I.M. Dmitrenko, 1970, Proc. 12th Inter. Cod. on Low Temp. Phys. Eizo Kanda ed. (Academic Press of Japan, 1971)p. 407. Smith, F.W. and M. Cardona, 1967a, Solid State Commun.5,345. Smith, F.W. and M. Cardona, 1967b, Phys. LettersZA, 671. Smith, F.W., A. Bartoff and M. Cardona, 1970, Phys. Kondens. Marterie 12,145. Tai, P.C.L., M.R. Beasley and M. Tinkham, 1975, Phys. Rev. B11,411. Thompson, R.S. and A. Baratoff, 1968, Phys. Rev. 167,361. Tilley, D.R., J.P. Baldwin and G.Robinson, 1966,Proc.Phys. Soc.89,645. Tinkham, M., 1958,Phys. Rev. 110.26. Tinkham. M., 1975, Introduction to Superconductivity(McGraw-Hill, New York) pp. 72-86. Toxen, A.M., 1962,Phys. Rev. 127,382. Toxen, A.M. and M.J. Burns, 1963, Phys. Rev. 130,1807. Valette, C., 1971, Thesis, Universitd Paris-Sud, Orsay, France. Valette, C. and G. Waysand, 1975, Low Temperature Physics-LT14, M. Krusius and M. Vuorio eds. (North-Holland,Amsterdam) Vol. 2, p. 183. Welber, B. and S.L. Quimby, 1958, Acta Met. 6,351. Werthamer, N.R., 1969, Superconductivity, R.D. Parks ed. (Marcel Dekker. N.Y.) pp. 321-370. White, G.K., 1964, Phys. Letters 8,294. Yogi, T., 1977,Thesis, CaliforniaInstitute of Technology, Pasadena, USA. Zavaritskii, N.V., 1951,Dokl. Akad. Nauk SSSR 78,665.
CHAPTER 7
PROPERTIES OF THE A-15 COMPOUNDS
AND ONE-DIMENSIONALITY BY
L.P. GORKOV The Landau Institutefor TheoreticalPhysics, Academy of Sciences of the USSR, Moscow, USSR
Progress in Low Temperature Physics, Volume VIIB Edlted by D.F. Brewer 0 North-HollandPublishing Company, 1978
Contents 1. Short survey of the properties of the A-15 compounds. Some basic theoretical ideas 519 2. The choice of the electronic term for the linear chain. Instability of the spectrum 529
3. Interactions. Connection between structural and superconductive fluctuations in the linear chain. The role of the ‘three-dimensional effects’ 533 4. Peculiarities of the three-dimensional electron spectrum in the A-15 lattice due to the interchain tunnelling. Fine structure of the density of states 544 5. Structure properties of the A-15 compounds in the simplest model. Phonon spectrum 552 6. Some results concerning the superconductivity of A-15 compounds 568 7. Discussion 579 8. Summary. Some concluding remarks 584 References 588
1. Short survey of the propertiesof the A-15compounds. Some basic theoretical ideas
Since the discovery in 1954 of the high T, for V3Si(T, = 17.0 K) and Nb3Sh (T, = 18.2K) materials with the A-15 structure were intensively investigated. Such interest, continuing for more than twenty years, was connected partially with the importance of their practical applications for high field magnets. However, the real reason for the numerous, very difficult experimental studies was the hope of achieving still higher temperatures for superconductivity. In fact, the highest critical temperatures up to date have been obtained in Nb,Ge (T,N 22.2 K) which again is a member of the A-15 structure family. There are still no ‘room’ (nitrogen) temperature superconductors. Meanwhile as a result of these efforts a new and interesting physical field has appeared. This review paper will be devoted to this latter side of research in the A-15 type materials which is of general physical interest. The problem of obtaining high T, superconductors will not be discussed. From the very beginning it should be emphasized that the character of the experimental results in this field, their accuracy and the unambiguity of the interpretation are still very far from the level which is achieved now in the physics of ordinary metals. The cause lies in the numerous difficulties of a metallurgical and technological character concerning the preparation of perfect samples. Despite these complications the main features of the phenomena are now well established. A few excellent reviews (Testardi 1973; Weger and Goldberg 1973; Izumov and Kurmaev 1974) exist to which I shall refer the reader in the cases when details are needed. Nevertheless it seems desirable to summarize briefly the basic experimental facts to define the problem which is discussed theoretically in the main part of the paper. Among the first unexpected results were the observations of the anomalous temperature dependence of the magnetic susceptibility and the Knight shift in the A-15 compounds. (The latter is usually proportional to the magnetic susceptibility.) It is well known that for the ordinary metals both these quantities are almost independent of temperature. The diamagnetic part of the susceptibility is connected, as a rule, with inner electronic shells of the
L.P. GOR'KOV
520
[CH. 7,$ 1
atoms. As for the paramagnetic contribution, it is commonly ascribed to the Pauli spin susceptibility of the conduction electrons and can be expressed in terms of the electronic density of states, v(E):
x
= pB!v(&)ch- '(&/2T)ds/4T.
-
Due to the large energy scale (the Fermi level EF 5-10eV) of electron bands in metals, V ( E ) changes slowly (v(E) w v(0);E = E-E,). The order of magnitude of v(0) varies from the value v(0) 0.3 states/eV.atom (for two spin directions) for non-transition metals to v(0) 2-5 states/eV -atom for the typical transition metals. There are several simple qualitative explanations of the enlargement of v(0) for transition metals based either on the relative narrowness of the d-electron bands for these elements or on the role of the Coulomb interactions. A striking peculiarity in the behaviour of the susceptibility of the A-15 compounds is its remarkable temperature variation, extending over the whole room temperature interval. This can be seen from fig. 1 which reproduces the recent data on the susceptibilityof V3Si. The change of the magnetic susceptibility with temperature reaches about 30% of its value which, in turn, is rather large in these compounds. The specific heat at low temperatures is usually divided into a sum of electronic and lattice vibration contributions N
N
(1)
c = aT+bT3,
T(K)
Fig. 1 . Plot of the temperature dependence of the magnetic susceptibility, ,y, for V3Si (Maita and Bucker 1972).
CH. 7,811
PROPERTIES OF A-15 COMPOUNDS
521
where the electronic part, linear in T, can again be expressed in terms of v(0): a = *x’v(O).
(2)
(The values of v(0) estimated from the magnitude of the magnetic susceptibility and from the specific heat measurements, differ from each other.) The temperature dependence of the magnetic susceptibility and rather large values of v(0) [eqs. (I), (2)] have been considered together by Clogston and Jacarino (1961) in the phenomenological ‘density of states peak‘ model for conducting electrons. Of great importance is the fact that the width of the peak in the electronic density of states at the Fermi surface introduced to fit the observed temperature dependences has to be of the order of a few hundred Kelvins (Clogston and Jacarino 1961). Meanwhile a rough estimate of the d-electron band width AE based on photoemission data gives for V,Si AE % 3.4 eV. (See Testardi 1973 and Weger and Goldberg 1973 for more details.) The most important feature of the A-15 compounds is the phenomenon of the low-temperature structural transformation observed for Nb,Sn and V,Si (the Batterman-Barrett ‘martensitic’ transformation). This transition from the cubic to the tetragonal phase occurs at T, N 43-50 K for Nb,Sn and T, N 20-25 K for V,Si. In figs. 2 and 3 the results of X-ray studies of the lattice parameters are given for Nb,Sn (Mailfert et al. 1967) and V,Si (Batterman and Barrett 1966). According to the symmetry theory analysis (Anderson and Blount 1965) this transformation is expected to be a phase transition of the first kind. This question will be essential below. Figures 2 and 3 show that the correspondingdiscontinuities of the lattice parameters are in any case not well pronounced for V,Si. (For discussion of this question see Testardi 1973.) No hysteresis effects have been observedwithcertainty. The nature of the structural transformation became clear after the ultrasonic velocity measurements for V3Si had been performed by Testardi’s group (see Testardi 1973). They studied the temperature behaviour of the elastic moduli and observed for the elastic shear modulus C, = +(C,, - C12) a remarkable temperature softening (at cooling) down to zero values at T = T,. A similar behaviour was later found for Nb,Sn (Rehwald et al. 1972). On fig. 4 the shear moduli C, for Nb,Sn and V3Si are plotted as a function of In T. It seems that the Batterman-Barrett transformation should be accepted as the basic property of these materials. From this point of view the anomalous
522
[CH. 7,g 1
L.P. GOR'KOV
---
5.300
I---
Nb,Sn
..../
L
al
-
al V .r
4.J, c
1
0
8
. I
I
I
I
20
I
I I,
I
I
I
40
60
I
I
I
I
80
, ---_ % 0
T(K)
Fig. 2. X-ray data for the temperature dependence of the lattice parameters in Nb,Sn (Mailfert et al. 1967).
4.728 c/a = 1.0022 C
4.724
\ I
4.720
I
I
Cubic
I
Tetragonal phase
4.716 i
I
I
p-u4a
I
8
*Bil T"a20.5 K
I
4.712
n
I
12
I
I
I
16
20
24
6
T(K) Fig. 3. X-ray data for the temperature dependence of the lattice parameters in VISi (Batterman and Barrett 1966).
PROPERTIES OF A-15 COMPOUNDS
CH.7.0 1J
523
temperature dependences of the magnetic susceptibility and elastic modulus, responsible for the transformation mentioned above can be considered as a ‘precursor’ behaviour above the temperature of the structural transition. Rather close critical values for the structural and superconductive transition temperatures for Nb,Sn and V,Si immediately put forward the question of whether both phenomena are connected. There are numerous pieces of evidence in favour of this suggestion. For instance, both T, and T,,, are strongly dependent on deformation and composition; T, depends on whether the cubic-tetragonal transformation in the given sample occurs, or not. The ‘saturation’ of the growing tetragonal deformation at Toin the V3Si is easily seen on the corresponding curve in fig. 3 [‘arrest of the structural instability’, according to the terminology of Testardi (1973)l.
25
I
I
50
75
I
100
I
I
150
200
I
I
250 300
T(K)
Fig. 4. The temperature dependence of the elastic shear modulus, C., in VsSi (Testardi 1973) and Nb,Sn (Rehwald et al. 1972) plotted as a function of lnT.
However, the question is whether this interdependence between both phenomena results from the occasional closeness of the critical temperatures (the same characteristic energies) for both transitions or whether both of them are a consequence of some intrinsic physical mechanism. Perhaps the most important and strange experimental fact which forces us to consider this last point of view, is the softening of the phonon frequencies in Nb,Sn and V,Si discovered in the inelastic neutron scattering experiments (Shirane et al. 1971; Axe and Shirane 1973). The striking feature of the results is that the softening (at cooling) of the phonon frequency concerns phonons with large wave vectors, of the order of the reciprocal lattice vector. These data (Axe and Shirane 1973) for Nb,Sn are shown in fig. 5.
L.P. GOR’KOV
524
[CH.7,# 1
Very often the suggestion is discussed whether high Tc’s in these compounds are due to some hypothetical mechanism of ‘enhancement of superconductivity by lattice instability’. As for the existence of this enhancement, the point of view accepted below would give a negative answer. 14
-
12
10
-
-0-T-295
K
--e-T=120
K
*T=
62K
*T=
46K
-
8w
Q
E
-
h
b 0-
3
6-
4-
I
0. 0 Fig. 5. Dependence of the dispersion law for the transverse mode (q = (Zs/a)[C, C, 01, ell [lTO]) on temperature, according to the neutron inelastic data in Nb3Sn (Axe and
Shirane 1973).
In what follows we shall try to answer the question whether it is possible to unite the main physical facts within the framework of a certain theoretical idea. Therefore many of the experimental results, mentioned above, will be discussed again in more detail in connection with their theoretical interpretation.
CH. 7,8 11
PROPERTIES OF A-15 COMPOUNDS
525
At first sight the crystal structure of the A-15 materials looks rather complicated. The unit cell for the chemical composition A,B is shown in fig. 6a and contains two ‘molecules’. Non-transition atoms B occupy the body centered cubic (b.c.c.) positions. The transition element atoms A are situated on the faces of the cube. The space group of the cubic structure is Of (Pm’n) and contains non-trivial translations. In fig. 6b some details of the tetragonal phase are shown, namely, that the tetragonal distortion is accompanied by the sublattice displacements of transition atoms. The space group for this phase is DZ,,[see Testardi (1973) for more details].
Fig. 6. (a) Crystal structure of the A-15 lattice compounds, A3B. Circles correspond to non-transitionatoms in the b.c.c. positions. Crosses are used for transition element atoms. The dashed line is to emphasize the idea of the three orthogonal sets of linear transition atoms chains. (b) The displacement of sub-latticesof the transition atoms in the tetragonal phase is shown by arrows.
The dashed lines in fig. 6a for the cubic phase are to demonstrate the geometrical image of the A-atoms, lying along the three different sets of the orthogonal linear chains. The importance of this picture was first emphasized by Weger (1964), who pointed out that due to the small overlap between d-orbitals of the transition atoms from different chains, d-electrons of the single chain in the first approximation move only along the chain’s direction. We will discuss the reasons for such a suggestion later. At present we mention only that this observation is the underlying idea for all the current theoretical models, including the model to the discussion of which the main part of the present paper is devoted. In this section it is reasonable to give also a short survey of the two main previous theories of the A-15 compounds properties and their starting microscopic suggestions concerning the electronic energy spectrum.
526
L.P. GOR’KOV
[CH. 7,9 1
The first one was the Labbe-Friedel-Barisic (LFB) model. (See Testardi (1973) and Weger and Goldberg (1973) for references.) They have suggested for Nb,Sn and V3Si wide and strongly overlapping s-p electron bands with a large number of current carriers, as is commonly assumed for the transition metals. At the same time they claimed that some of the d-electron bands are almost empty but their bottoms* are very close to the position of the Fermi level in the s-p bands, p. If the d-electrons of the chain were one-dimensional, the density of states, v(E),near the edge of the band, c 0 , had to be of the form:
It seems almost evident that this singularity would be located at the point r ( k = 0) of reciprocal space. However, other possibilities for the location of the Van Hove type singularities (3) exist, and they are enumerated in Weger and Goldberg (1973). When introducing this suggestion about the position of the Fermi level, the peak in the density of states accepted in the article by Clogston and Jacarino (1961) on a phenomenological basis, can be connected with the square root singularity (3). The angular momentum degeneracy of the d-electron levels in the atom is partially lifted by the lattice field. Hence, different d-sub-bands can be taken. We prefer not to go into all the details of the LFB model, limiting ourselves to qualitative ideas only. The contribution of the d-electrons to the energy of the crystal (at T = 0) is equal to
(The sum over the index i corresponds to the three different sets of chains.) This contribution could be very essential for the elastic properties. In fact, in the presence of deformations the bottom, cOi, of each of the electron bands shifts in some way (different for each of the three orthogonal sets of chains). The electronic contribution to the elastic moduli, which are proportional to the second derivatives of expression (4)with respect to strains, can be made negative and large enough to provide the instability of the cubic phase, if p is chosen close enough to the position of the singularity (3). The BattermanBarrett transformation is therefore interpreted in this model as a sort of Jahn-Teller effect -the degeneracy in the electron numbers in different *Or the tops of the filled d-band.
CH. 7,§11
PROPERTIES OF A-15 COMPOUNDS
527
chains is removed by the redistribution of electrons between chains accompanied by spontaneous distortion. In the case of a singularity (3) at the point r of the reciprocal lattice symmetry considerations give the softening of the shear modulus C,. To summarize, the LFB model with a suitable choice of parameters gives a reasonable account of most of the A-15 material’s structural properties. Its vulnerable points are the following (a) the one-dimensionality manifesting itself in eq. (3) for the density of states; (b) the necessity to choose the Fermi level very close (-20 K) to the edge of the d-band. Let us discuss both these difficulties. A very crude (in the tight binding approach) estimation of the values for transfer integrals, B, between different sets of chains gives the order of magnitude, By a few tenths eV. It should be mentioned that the packing of the transition element atoms in the A-15 structure is usually even denser than in the corresponding metals. At the same time the tunnelling could be even more intricate due to the presence of the large s-p shells of the non-transition atoms. However, in any case one should expect that the three-dimensional corrections to the onedimensional electron spectrum are too large to take literally the one-dimensional Van Hove singularity (3). An attempt to improve the Labbe-Friedel picture in this particular point was the so-called ‘RCA model’ (see Testardi 1973; Weger and Goldberg 1973). The idea was essentially the same; the bottoms of the d-electron bands are close to the Fermi level position of the s-p electrons. However, even with the three-dimensional corrections, the density of states for the d-bands, v&), should be larger than that for the s-p band (the assumed narrowness of the d-band). Therefore in the RCA model one considers again three sets of chains with steplike densities of states:
-
(a 9 1, the edges of d-zones, E O i ,
are strain dependent). As in the LFB model case, the appropriate parameters can again be chosen to fit most of the experimental data in a rather satisfactory way. The distance of the Fermi level from the edge of the d-band is now about 100 K. Therefore one sees that in any case the second objection against the LFB model also concerns the RCA model. It seems difficult to point out any reasons why at the overlapping of the very wide (- 10 eV) s-p electron
528
L.P.GOR’KOV
[CH. 7,g 1
bands with the d-band, the bottom of the latter has to be so close to the Fermi level. This could simply be an occasional numerical coincidence for Nb,Sn and V3Si. However there is evidence that some characteristic features of the best investigated Nb,Sn and V3Si are also inherent in other members of this class of material (Testardi 1973). A few words should be said about the attempts to get an answer by the straightforward computer calculations of the band structure for these rather complicated compounds (see for results Weger and Goldberg 1973 and Matheiss 1975). Starting from first principles, one has to pass through about 20 bands before the position of the Fermi level is reached. (The latter is defined by the whole number of electrons on the A and B atoms per unit cell.) Different authors estimate the accuracy of their calculations in different ways. We believe it would be very optimistic to state that the relative positions of bands are now defined better than within the limits of a few eV. The large difference between the results of Weger and Goldberg (1973) and Matheiss (1975) confirms this suspicion. The main objection against this ‘first principle approach’ consists in the neglect of many essential features of the Fermi liquid theory, i.e. interactions and scattering effects. Therefore we believe that the question of the band structure cannot be resolved now in a straightforward way and is open for phenomenological suggestions. In concluding this section let us formulate the starting points for the following consideration. The most important suggestion we will accept (Gor’kov 1973, 1974), is that there exist some d-bands which do not overlap with the s-p electron valence bands. More exactly, it is enough for the latter to be completely occupied. In the real crystal the wave functions of the d-electrons are strongly different from the d-functions of the free atoms. Designating for the sake of simplicity the corresponding zones as d-electron ones we at the same time emphasize their origin from the transition atoms in the chains. However, it is supposed that these bands are formed selfconsistently, with all the lattice field and interaction effects taken into account. In our picture the d-electrons are the only carriers in the system. In the next section we consider these electrons in the first approximation as one-dimensional, taking only their motion along the chain into account. The interchain transfer will be considered as a perturbation; it is responsible for many details of the physical results. It is supposed below that the reader is familiar with the basic ideas and results of the theory of phase transitions and the BCS theory of superconductivity. Therefore many references which have already become ‘classic’ will be omitted.
CH.7,#21
PROPERTIES OF A-15 COMPOUNDS
529
2. The choice of the electronic term for the linear chain. Instability of the spectrum
The simplified theory of metals and dielectrics connects the conduction properties with the number of electrons per unit cell. Substances with an odd number of electrons have half-filled energy bands and have to be metals. If the band is filled up to the top, the corresponding electrons do not contribute to conductivity. Hence, metals with an even number of electrons per unit cell exist only if there is an overlap between some bands. This picture is widely known. However it is worth emphasizing that the physical consequences are independent of the initial band model. One of the main results obtained in the framework of the Fermi liquid theory consists in the proof of the correctness of this concept even in the presence of interactions between electrons. The famous Luttinger theorem argues that the whole number of states per unit cell (for two spin directions)inside the volume in the reciprocal unit cell limited by all Fermi surfaces, always corresponds to an odd or even integer. For a single chain of transition atoms there are two A-atoms in the unit cell, i.e. an even number of electrons. Therefore one should understand whether it is possible to reconcile the concept of the non-overlapping bands with the experimental evidence of a large number of carriers in the A-15 compounds. We have mentioned that the assumed electron band comes from d-electrons moving in the crystal field of the surrounding atoms. Therefore let us investigate in more detail what sort of limitations on the zone structure arise due to the special symmetry group of the A-15lattice. In fig. 7 the crosses represent the transition atoms in the chain, the circles for the B-atoms reflect the symmetry of the field of the host lattice matrix. The point group is D4,,, however the full space group contains non-trivial translations (the screw axis). The group of symmetry for the single chain coincides with the small group of the wave vector at the point X of the simple cubic Brillouin zone (fig. 8). The corresponding representations {XI, X,, X , , X4} for electron wave functionsat the point Xwere known (Gorzkowski 1963). All representations are twofold degenerate. To get the electron energy spectrum near the zone boundary (near the point X)it is necessary to expand the product of representations X:.X, on the representations of the point group D,,,,. This is a standard group theory approach for the selection of all the non-zero matrix elements between the wave functions of the representation X i . Only X , ,X4will be needed.
530
L.P. GOR'KOV
[CH. 7,$ 2
The distribution of the matrix elements for some combinations of physical parameters over representations A: ,A , , B:, B i of the group D4,, = D4xCi (which are the only ones to appear in the products X , * - X , and
Fig. 7. Symmetry of the crystal field for an electron moving along the chain direction.
Fig. 8. The simple cubic Brillouin zone.
X,'.X4), are listed in table 1. For twofold degenerate representations it is convenient to write down the electron Hamiltonian near the point X in the matrix form using the Pauli matrixes 8,, 8,, 8,:
PROPERTIES OF A-15 COMPOUNDS
CH.7,821
531
Table 1 The distribution of the matrix elements of the strain tensor components, elk,the sublattice displacements, u l , momenta and the magnetic field, H, over some representations of the group Da.
The factors A , and B, contain, in principle, all combinations from table 1 with their own coefficients. First we note that the componentsp x ,p y (for the chain along the [Ool] direction) can appear only in the presence of threedimensional effects and should be omitted. All spin-&pendent terms, other than (u-H ) , are proportional to the spin-orbital interaction constant. At least for Nb,Sn and V,Si the latter is small and can be neglected. Therefore one sees from eq. (6) that in the absence of strains, &ik and sublattice distortion, u,, the electron term (6) corresponds to the following two branches of the spectrum: e = f ul"1;
(6, = n/a-p.).
0
Therefore the energy band as a whole for a single chain has the structure shown in fig. 9. For an even number of electrons from the two transition atoms (6 in V,Si and 10 in Nb3Sn) a band arises filled as shown in fig. 9. In other words the position of the Fermi level (Fermi surface) is fixed automatically at the point X. The finite slope for both branches, &/*, = fu, provides the metallic properties of these compounds, The other two representations XI, X , have deldp, = 0 at the point X and will not be considered further.* In the presence of the distortions e l k and u, eq. (6) is of the form:
*These representations could give a base for the LabbbFriedel singularity (3) in tho density of states.
532
L.P. GOR'KOV
[CH.7,8 2
The deformation potentials d, and d3 do not remove the two-fold degeneracy (7). Actually, as will be discussed below, these terms are responsible for the temperature variation of the bulk modulus. The latter varies very slowly. Therefore both d, and d3 can be neglected in comparison with dl We will also omit yu, in (6') for simplicity in all the expressions. Now it is seen that in the presence of strains the energy spectrum (7) splits into two branches:
.
Fig. 9. Electronic band supposed for terms X2,X., at the point X.The lower branch is occupied. The shadowed region corresponds to other filled bands (s-p bands).
with the gap A, = dl(&,,-&yy). Calculating in the usual way the electronic contribution to the free energy of the crystal
(where the s u m is taken over three sets of chains and two branches (8), and the potential 60, is given for unit volume), one can easily obtain for its elastic part (
PROPERTIES OF A-15 COMPOUNDS
CH.7,831
533
Logarithmic integration over the interval T/v Q IS,] n/a has appeared in eq. (10). Introducing the elastic modulus C,,as usual
we get the logarithmic temperature-dependent contributions to the moduli C,,and C,,:
SC,, = -(2d$naZu) In (AEIT); where AE
-
SC,, = +(d:/zazv) In (AEIT),
(12) v/a is of the order of the band width. For the shear modulus
SC, = -(3d,2/na2v) In (AEIT).
(13)
Equation (13) corresponds to the softening of the C,at cooling. The meaning of these results becomes clear from a closer look at expression (8) for the split spectrum. The deformation now results in the appearance of a gap in the energy spectrum of fig. 9. Therefore the energy of the whole system decreases at deformation. Similar considerations were first exploited by Peierls in his famous proof of the instability of the periodic one-dimensional array of metallic atoms with respect to the alternating of its period. In our case the symmetry of the A-15 lattice reduces this instability to the instability with respect to acoustic shear waves (and/or the sublattice optical phonon mode at k = 0, if the term yu, is included).
3. Interactions. Connection between structural and superconductive fluctuations in the linear chain. The role of the ‘three-dimensionaleffects’ The results obtained above are incomplete in two respects. The obvious one is that electrons are not one-dimensional. However, for the time being the discussion of the three-dimensional effects will be postponed. One has to recognize first that the problem of the Peierls structural instability in itself is more complicated than it looks in the light of the arguments of the preceding section. Historically this fact was first discovered in the theory of ‘one-dimensional metals’ (Bychkov et al. 1966; Dzyaloshinskyi and Larkin 1971). One can verify that a one-to-one correspondence exists between the problem mentioned and that for the A-15 compounds. Qualitatively one can understand this fact noting that the band structure in fig. 9 looks like the
534
L.P. GOR’KOV
[CH. 7,s 3
band structure for a chain with the period a* = -)a reduced to one-half of the corresponding Brillouin zone. In this section we have no intention of going into all the details of the mathematical side of the problem. However it would be desirable to explain its most prominent features to get a deeper insight into the consequences. The physical reasons why some doubts are relevant concerning the Peierlslike arguments given above, become clearer with a more attentive look at eqs. (8) and (13). According to eq. (13), for any arbitrary weak interaction dl there is a temperature, Tp, at which the electronic contribution cancels the ‘bare’ lattice part of the elastic modulus, C:’).The developing spontaneous distortion leads to a gap in the energy spectrum (8). However, from the naive point of view it seems possible, at least in principle, to reduce the whole energy of the system due to superconductivity (i.e. if the energy gap in the eq. (8) is of ‘superconductive’ origin). Indeed, both phenomena arise due to the electron-phonon interaction and both consist in the enhancement of roughly the same weak electron-phonon coupling constant by a logarithmic factor which becomes large at low temperatures. The difference between the Cooper effect and the Peierls instability lies in the cut-off value 63 entering into this enhancing factor ln(iS/T). For the Cooper effect 6 is of the order of some averaged phonon frequency (-0.01 eV). The cut-off AE in eq. (13) is supposed to be about several eV. Therefore with weak interactions one could expect for T,(T,J values much larger than T,. Meanwhile, these two temperatures are rather close to each other both for Nb,Sn and V3Si and are m a l l enough for us to believe that the interaction is still a weak one. For many of the A-15 compounds the structural transformation has never been observed. Certainly one could point out many reasons explaining the absence of this latter (three-dimensionalcorrections, for instance). However, we shall investigate the question in more detail. For a moment let us forget about phonons and consider only short-range and weak electron-electron interactions. In our model there are a number of interaction constants, g,, corresponding to different electron scattering processes between electrons from ( 2 )bands (7). They all are shown in fig. 10. Let us just state without proof, that according to the results (Bychkov et al. 1966; Dzyaloshinskyi and Larkin 1971) the physical peculiarities of the onedimensional system are defined mainly by g,. Therefore in the following qualitative discussion only this scattering constant will be mentioned. The first step would consist in an attempt to get some first order corrections to the scattering processes of fig. 10. There are two contributions represented by the diagrams in fig, 11. If we omit the numerical coefficients these matrix
CH. 7,s 31
PROPERTIES OF A-15 COMPOUNDS
535
...
Fig. 10. Possible interactions between electrons describing different scattering processes between the branches (+) of the zone on fig. 9 near the point X = n/a.
elements can be written down using Green’s functions (Abrikosov et a]. 1965)
where + ( p ) are the two electron branches (7) and w, = (2n+ 1)aT. For the diagram (a) of fig. 11 one has
Fig. 11. The 6rst order perturbation corrections to the scattering amplitude g1 of fig. 10: (a) Cooper-type channel, (b) Peierls-type, or the charge density wave channel.
and
for the diagram (b). In (a) and (b) we used the fact that both points f a / u at the zone boundary differ by the vector of the reciprocal lattice K = 2a/a and are equivalent.
L.P. GOR'KOV
536
[CH. 7,$3
Hence, the integration over 6 can be extended over the interval ( - n/a, ./a). The frequency sum gives the well-known relation:
Substituting it into expressions (a) and (b) one finds out that both of them are proportional to
s
d6
US
S
2T
k g: -th - = k 2g: In 6 T
(here we have introduced a certain cut-off value, 5, for the range of integration). The magnitude of g , for both corrections is enhanced by a factor ln(G/T), which at low enough temperatures can compensate the weakness of interaction g , + 1 :
Therefore to get the correct expression for the scattering amplitude between in the temperature range (16), one should sum all the correcelectrons (i-) tions of the form g,[g, ln(6/T)]". Keeping this in mind, we have to investigate which class of diagrams gives contributions of this order of magnitude. In the three-dimensional BCS theory of superconductivity one collects all the ladder diagrams (a). As a result, the familiar expression for the corresponding 'scattering amplitude' - the vertex part r - is of the form:
This pole which exists at attraction g , < 0, defines the onset of superconductivity i.e., the transition temperature T, . The scattering channel (a) of fig. 1 1 ought to be called the Cooper-type channel. Diagram (b) corresponds to the dielectric electron-hole pairing, i.e. to charge density waves (CDW). This channel can be called the Peierlstype channel. (Note the different signs in expression (15) for (a) and (b).) Therefore the important question is whether any competition between the two channels could exist in the subsequent corrections to the vertex part.
CH. 7,G 31
PROPERTIES OF A-15 COMPOUNDS
537
Straightforward calculation of the diagram shown in fig. 12 immediately gives the corrections
which at the temperatures defined by eq. (16), has exactly the same order of magnitude as the second approximation for the ladder Cooper-type diagrams. The necessity to include the corrections from one type of channel into the diagrams for the other, even with weak interactions, leads to a rather complicated set of equations usually named ‘parquet’ equations. It is useful to
Fig. 12. Insertion of the Peierls-type channel contribution into the Cooper-type diagram.
consider their structure in more detail; however, we will postpone this to the end of the section and start with the discussion of the physical consequences obtained. The striking difference between the ladder approximation and the more exact approach becomes evident if one compares the corresponding results for the possibility of the Peierls-type pairing. In the ladder approximation for the Peierls channel one would get the pole-type behaviour (17) of the vertex r for the repulsive sign of g, (the opposite sign in (15) if compared with Cooper pairing). The ‘parquet’ calculations reproduce expression (17), which has no pole at g, > 0. Therefore the metallic state is stable for repulsive interactions between electrons. If g, < 0, the same pole (17) appears simultaneously in both parts of the vertex r responsible for the BCS and CDW types of pairing. In other words, with weak attractive electronelectron interaction the instabilities with respect to the structural and superconducting transformations correspond to identical critical temperatures, Torit.It should be mentioned that in reality there are no phase transitions in the one-dimensionalcase and this statement means only that the fluctuations corresponding to both types of ordering, develop simultaneously beginning with this characteristic temperature, Tcrit.As far as only the electron-
538
L.P.GOR’KOV
[CH. 7,j 3
electron interactions are considered, any instability means the instability of the electron subsystem. In our case this instability is characterized by two electronic order parameters. Let us now consider how these resuIts will be altered if phonons are ncluded (interaction with the lattice). In addition to the short-range vertexes g, of fig. 10, phonon-induced scattering processes are now possible, which are the result of an exchange of virtual phonons between electrons. The corresponding scattering amplitude is proportional to
where w1 - w 3 and p 1 - p 3 are the corresponding changes of the frequency and momentum of the upper electron in fig. 10. We have written the interaction (18) in a completely schematic form. Actually the corresponding D-function reflects the symmetry properties of the active phonon mode (gephis connected either with acoustic modes (being proportional to dl , d2), or with the optic ones, y, etc.). The frequency Q is of the order of a typical frequency of the phonon mode w,(q). The latter generally speaking, has a three-dimensional behaviour since the d-electrons are not the only source of elastic forces in the lattice. The important feature of the interaction (18) consists in the retardation effects. At low temperatures w1 - w3 T < coo, SZ and this effect can be neglected. In this case eq. (18) reduces to the usual attractive interaction of the BCS theory. In the opposite limiting case (at T % wo) the interaction (18) is small (- (f2/T)’). If only the electrons of one chain are considered, expression (18), in general, has to be averaged over the qL, i.e. over the dependence of w,(q) on q1 along the flat (one-dimensional) Fermi surface. The electron-phonon interaction changes the phonon D-function. The corresponding Dyson equation:
-
contains the self-energy part for phonons, n (the polarization operator). Its meaning, if eq. (19) is rewritten in the form: D-’(q) = W(q)-n(q).>,
(19’)
consists in the renormalization of the phonon frequency. If we neglect all other corrections, the first approximation for n has the form of the bubble
CH. 7,831
PROPERTIES OF A-15 COMPOUNDS
539
shown in fig. 13a. As a matter of fact, in the end of the previous section we have calculated the corresponding contribution of fig. 13a into Z7 for the acoustic modes. It was found to be proportional to ln(dE/T). Hence, one can interpret the results (12), (13) as a renormalization of the elastic moduli due to the first-order electronic corrections. The dimensionless electron-phonon coupling constant in the case of a shear wave is equal to geph 2 = 3d:/aazvCio).
(+I
0<:D(-1
(-1
(a)
(b)
Fig. 13. Schematic representation of the polarization operator: (a) The first order electroniccontribution, (b) General form of the polarization operator. Trianglescorrespond to the electron-phonon vertices improved by various corrections.
The first approximation of fig. 13a should be, in principle, improved by the electron-electron and electron-phonon interactions. This is schematically shown by triangles in fig. 13b, which represents the change of the electronphonon interaction due to electronic corrections. The effective electronphonon interaction is determined by many details of the real system. The essential point is that if T B w o , SZ, one can neglect in the expression for n(q)any insertion of internal phonon D-lines. Therefore let us determine the temperature T p ,at which the renormalized frequency w,,,(q, T ) in (19') tends to zero:
If Tp found as the solution of eq. (20) is large, Tp p wo, all the calculations above are self-consistent and the Peierls instability (or the giant Kohn anomaly) occurs in our system.* *As it was mentioned, wo(q) has well pronounced threedimensional features, therefore the temperature Tp could be. the temperature of a real three-dimensional structure transformation.We will come back to this point a little later.
L.P. GOR’KOV
540
[CH.7,g 3
However it could turn out that the ‘bare’ electron-phonon constant g.bhis small enough and, as a result, the solution T, of eq. (20), is such that Tp coo, 8. This case needs an additional investigation. At T 4 wo the phonon induced electron4ectron interaction (18) reduces to the shortrange BCS attraction -gZphand for the one-dimensional system has to be
+
-
included into the whole scheme described above. As far as phonons are supposed to be the only source of attractive interaction, the critical temperature, Tcri,,if it exists, is smaller than wo. As before, at Tcrltthe system manifests instability with respect to the coexisting structural and superconductive fluctuations. These arguments were based on the one-dimensionalpicture (in particular, the dispersion oo(q,)has been neglected). The next question concerns the role of the three-dimensional effects. In the framework of the one-dimensional theory it was possible to ascribe the fact that the structural transformation temperature T, is sufficiently low, to the absence of solutions of eq. (20) for Tp > coo. The three-dimensional electron tunnelling between different chains could independently reduce T,. In fact, if the electron spectrum is of a three-dimensional character there is no more exact ‘nesting’ of the Fermi surfaces which has been the cause for the appearance of the factor ln(dE/T) in the Peierls channel. If the energy scale for the three-dimensionality is a certain T*, the temperature dependence of the renormalized frequency would saturate at T T*.t The general speculations about the role of the three-dimensional effects become clearer if one considers in more detail the structure of the mathematical equations corresponding to the described phenomena. No attempts have been undertaken to solve these equations even numerically owing to the evident reasons of their extremely complicated structure and the great number of unknown factors. Therefore in what follows we again use the schematic language of diagrams omitting almost all details of the crystal structure, the specific form of interactions etc. For the beginning some convenient form of the exact equations for the vertex part, r, will be derived. Perhaps, the following comments could clarify the physical reasons why this quantity is one of the most important in the theory. Let us, for instance, introduce some external ‘force’, fi,, with a definite space symmetry acting on the electronic density. The corresponding Hamiltonian will be of the form ~ , f , , ,where 8, is the Fourier component N
?For an array of weakly interacting parallel chains some other possibilities exist for the exact ‘nesting’ of the slightly curved Fermi surfaces (see, for example, Horovitz et al. 1975). These considerations are not applicable to the three orthogonal sets of chains in the A-15 compounds.
CH.7, 8 31
PROPERTIES OF A-15 COMPOUNDS
541
of the electron density operator. The external small perturbation, fi, gives a linear response for the electronic density (Di,). The latter can be written as follows:
This relation is shown in fig. 14. The brackets used here for simplicity of notation mean the usual integrations and summations over the internal momenta and frequencies (p,p’). One sees therefore that if r contains the solution of the pole type for some T and some g the response (a,,) can
Fig. 14. Schematic representation of linear response for the electronic density to an external ‘force’ he.It is Seen that this response could get infinite if the vertex part, r,has pole behaviour (see text).
become infinite. This fact corresponds to the instability of the electronic system with respect to an arbitrary small external ‘force’&. At this point one sees the analogy between the ‘soft vibration mode’ concept for some lattice distortion and the instability with respect to an electronic order parameter. In particular it is possible to consider a more generalized form of the perturbating Hamiltonian, including the ‘Cooper-pairing forces’, $ + $ + A , , and define the temperature of the onset of superconductivity T, in the same way. It is also useful to rewrite eq. (19) in the absence of retardation effects (T < wo)in the form which is shown diagramatically in fig. 15. One sees that if the vertex part tends to infinity, the D-function will be infinite, too. D(q)
Db(q)
D,(q)
-=-t
Fig. 15. The phonon Green’s function, D, expressed in terms of the ‘bare’ Dco,function and the electron vertex part. Pole behaviour for r a t some q means softening some active phonon mode.
[CH. 7,I 3
L.P.GORKOV
542
From the point of view of eq. (19’) it means the softening of the renormalized frequency for one of the phonon modes considered. Returning to the derivation of the exact equations for r let us introduce, Pl(pl , p 3 ; following Dyatlov et a]. (1957), three blocks: C(p, ,p2;p3,p4), p 2 , p4), P2(pl p4;p 2 ,p3) whose definition is seen from fig. 16. Each block is the sum of all diagrams which are reducible with respect to the division of the diagram into parts connected only by two G-lines. Therefore the whole vertex r can be written as the sum:
where R(p, ,p 2 ;p 3 ,p4) denote all the diagrams which are irreducible in the sense explained above. Several examples for R are shown in fig. 17 beginning (a)
(b)
9 C(..
3!
1 p4
Fig. 16. The definition of blocks for the Cooper-type (a) and Peierls-type (b) channels. The dashed lines show the corresponding cross sections giving in the one-dimensional case the logarithmic enhancement of the interaction constants at low temperatures.
with the ‘bare’ interaction* g. For each of the three blocks there is an evident relation which is a generalization of the Dyson equations for D- or Gfunctions
*The only specificity of the phonon-induced interaction is that it becomes effective in the narrow range of energies wo < EF very close to the Fermi surface.
7s.
0 31
PROPERTIES OF A-15 COMPOUNDS
543
P I ( P I , P I - K ; P ~ , P ~ += K )([R(pI,l ; p i - ~ ,I+K)+C(pI, Z ; P ~ - K , Z+K) +P2(Pi I+ K ; 1, Pi - rc)lG(0W+ 4 (21 "1 x rb2,I + K ; r, p Z + m . 9
The system of eqs. (21) (21') and (21") is exact and it is needless to say that it is also correct for the three-dimensional case. For ordinary threedimensional metals the approximation of the BCS theory consists in neglecting (for weak interaction) all diagrams which do not introduce the logarithmic enhancement factor ln(&/T). Therefore in the BCS theory two blocks PI,P2 remain small at any temperatures and thus R can be reduced to the 'bare' interaction constant. The critical temperature is determined as the pole of the vertex part corresponding to the Cooper block and can be
+HRt
,l-TP3 PL-
--P4
t
Fig. 17. A few examples of nonparquet diagrams.
found by solving the homogeneous equation (21') where for r (21) only its part Cis to be substituted. In principle, similar solutions of the homogeneouequation are possible even for the P-block; however, the logarithmic enhances ment appears in the Peierls-type channel only for the one-dimensional case. In the parquet approximation which has been mentioned so often before all the non-trivial diagrams for R (which, as can be shown, do not exceed g3ln(&/T)) and all the corrections to the Green's functions are omitted. As for the latter, they give a contribution of the order of g2In(G/T) and can be taken into account as corrections to the first parquet approximation by means of the procedure elaborated in Menyhard and S6lyom (1973). For mathematical details we refer the reader to the original papers (Bychkov et al. 1966; Dyaloshinskyi and Larkin 1971; and Menyhard and S6lyom 1973). The main result of the parquet approximation which was stressed all the time, is that in the one-dimensional case the system of equations for Cooperand Peierls-type blocks can not be decoupled. The critical temperature, TCl,(, of the form Twit
= ( 6 9 dE)
lgl"z exp (- l/lsl)
(22)
determinesthe starting point for the developmentof both typesof fluctuations
544
L.P. GOR’KOV
[CH.7,s4
Now we return to the question how the fluctuation regime becomes a real phase transition due to effects of three-dimensionality. Among the latter the interchain electron tunnelling seems to be the most important for the properties of A-15 compounds. We have already mentioned it as the cause which destroys the mechanism of the logarithmic enhancement of interactions in the Peierls channel due to cancelling the exact ‘nesting’ of flat electron and hole Fermi surfaces. However the solution for the homogeneous part of eq. (21”) for the Peierls block could still exist if the energy scale, T*, characteristic for ‘three-dimensionality’, is not too large in comparison with Tcrlt(22). In fact, the integration in eq. (21”) extends over the region of momenta and frequencies which is not essentially altered by a change of the electronic spectrum, partly due to the assumption T* < AE, and partly to the slow logarithmic variation of all inner blocks when the lower cut-off T is substituted by T*. In other words, the solution of (21”) could exist as a result of the compensation of the main one-dimensional contributions in both parts of this equation. As for superconductivity,the origin of the Cooper factor ln(G/T) is a consequence of time reversal invariance only, and we believe that if an attraction provides the existence of Tcrit(22) in the framework of the strictly one-dimensional theory, superconductivity always appears at some T, < Tcritwhen the electron interchain transfer is included. Among other three-dimensional effects, as was pointed out in Gor’kov and Dzyaloshinskyi (1974), there are the three-dimensional dependences of the phonon frequencies on their wave vectors. The same applies to the ‘bare’ constants gi which are of Coulomb origin. The influence of this effect on the Peierls (structural) pairing is not so drastic. We will not deal with these problems any further. The aim of this discussion was to point out a very attractive possible explanation why T, and T,,, in the A-15’s are not far from each other. The idea is that both transitions are driven by electronic order parameters. Onedimensionality as the first approximation is essential in order to shift both temperatures to rather low values of the order of TCrit < coo. The large tunnelling could be the factor explaining the absence of the structural transformation in most of the A-1 5 materials.
4. Peculiarities of the threedimensional electron spectrum in the A-15 lattice due to the interchain tunnelling. Fine structure of the density of states Although we have stressed that the electronic band of fig. 9 can be considered as a general form of solution for the electron moving along the chain in the
CH.7.8 41
PROPERTIES OF A-15 COMPOUNDS
545
crystal field of the A-15 lattice, for the beginning in what follows an even more simple picture will be chosen. The three-dimensional features of the electronic spectrum will be considered in the tight-binding approximation by introducing two transfer integrals: A, for the motion along the chain, and B, the transfer integral between nearest neighbouring transition atoms of two orthogonal chains (A % B). Further we will point out which of the results obtained are sensitive to the tight binding model. If a:’, af’ are the amplitudes of the electron wave functions for the two sublattices of the transition atoms of the x-chain, the electronic spectrum is defined by the equation
where the column {@} is constructed from the three subcolumns
Let us write down only the equation for a:), at). The remaining equations can be obtained by a cyclic permutation of (x, y, z):
In this approximation we have for the one-dimensional band
when the positioning of the Fermi level is close to the point X,E in eqs. (23), (23’) is small and the Fermi surfaces are not too far from the corresponding faces of the simple cubic Brillouin zone. Therefore almost everywhere on the zone face the three-dimensional corrections for the spectrum of the electrons on the x-chains can be obtained by substituting into (23’) the perturbation expressions for a:’, a t ) proportional to ( B / A ) ~ : ; . After some calculation one gets instead of eq. (7):
~l‘fi= T*(sinz~p,a+sin2~pya-2)
+
-
[ ( O S , ) ~ T * 2(sin2+p,a sin’ fpya)211’z.
(25)
L.P. GOR’KOV
546
[CH.7,g 4
Here we have introduced T * = 4B2a/v = 16B2/AE.
(25’)
Both solutions for the electronic spectrum (25) have the form of two slightly shifted and split electron and hole branches and for many phenomena studied below, it can again be considered as a ‘three-dimensional spectrum’ for each of the three independent sets of chains. In general eqs. (25) are incorrect at the edges and in the comers of the cubic zone where the degeneracy in eq. (23) becomes fourfold and sixfold respectively (see below). It is easy to check the following property of solution (25):
The density of states V ( E ) can be defined for each of the two branches (25). The number of electrons occupying all possible states with energies lower than E , N(E),is equal to (per unit volume)
The expressionsfor the numbers of electrons and holes can be easily calculated as the corresponding volumes restricted by the energy surfaces (25). For instance, for the electron pockets
In expression (28) E is measured in units of T’(2 = &/TI)and E, K are the standard notations for the elliptic integrals. The relation (26) results in the identity
Therefore from eq. (29) one sees that the stoichiometric composition (the numbers of electrons and holes are equal) corresponds to the position of the Fermi level at
PROPERTIES OF A-15 COMPOUNDS
CH.7, $41
547
The density of states, v(E), is determined by differentiating eq. (28). Its electronic contribution is
[-2
= v(0)(2/7c2)In (32T*/lsl).
(32)
It is instructive to understand in more detail the origin of these singularities. Let us consider expression (25) near the edge of the cubic face (at small 6, = n/a-p,). One of the roots (25) looks now as E(1X) N
-2T*(ts,U)z+(v632/[2T* cos2 ($p,a)].
(33)
Denoting 1’ = ~ ( S , U ) ~5’ , = (v6,)2/[2T*cos2(~p,a)] and introducing hyperbolic variables : r2-1’
=
E;
(1 = s
to perform the integration over d j p / ( 2 ~ for ) ~ the evaluation of the number of states, N(E), one immediately obtains the logarithmic singularity (32) in the density of states. Therefore this singularity arises due to the saddle-like form (33) of the energy surfaces near the edge of the cube along the whole length of the latter. The origin of the second singularity at E = -2T* is the same but it is connected with the lines pa = 0 or pp = 0. (Note the absence of any small prefactor in eq. (32), which could arise, at first sight, as the result of three-dimensionality!)
[CH. 7.8 4
L.P. GOR’KOV
548
The logarithmic singularity (32) is smoothed by many physical factors like defects or impurities. Putting this last question aside we discuss in this section only the limitations connected with the tight binding approach. First of all, there is no exact equivalence between the lines py = 0 (or pz = 0) and the edges of the cube even in this approximation. The evident reason lies in the fact that the edges of the Brillouin zone are the lines of matching of the energy spectrum for the two orthogonal sets of chains. Therefore near these lines one should investigate in a more careful way the splitting of the fourfold degeneracy of energy levels of the unconnected chains. Restricting ourselves to the case of small 6, = z/u-p,, 6, = n/a-p,, it is possible to simplify the fourth order determinant in the system of eqs. (23), (23‘) for ( x ) and (y) chains. After simple but rather tedious calculations it is possible to show that the corresponding solutions have two low-lying branches (8 4 B) which have to be determined from the equation (Gor’kov and Dorokhov 1976a): &2[v2(6: +6,2)+4B2~vz~2]+2eB~~v,~2av(s: +6,2)-u463,2 +4B41~,12~4(6,4+614) = 0.
(34)
For an arbitrary E T* the fourth term is small and can be omitted. In this case eq. (34) describes the smooth transition of the Fermi surface from the x-chain zone face to the y-chain face. At small E 4 T* and 6, $- 6,: N
2vasB21vz124 4 6 ; +4B41vz12a46,2 = 0, with the definition (25’) for T* and Ivz12 = 4cos2(#p,u) this equation can be shown to be identical to eq. (33). The second pair of roots for the equation given by this fourth-order determinant near the edge of the cubic face, is of the form:
I
The gaps f4B Icos(+p,a) (far from the corners [l 111) are of the order of T*’I2. The energy band constructed in this model can be identified as the o-band in the numerical calculations for V,Ga (Weger and Goldberg 1973). Comparing the results for the value of splitting at the point M with eq. (36) * N 300 K. (In Weger and Goldberg (1973) the overlap between one gets Tvl0, various d-bands was obtained. Nevertheless, we believe that the above estimate provides a correct order of magnitude for T* a few hundred Kelvin.)
-
CH.7,§41
PROPERTIES OF A-15 COMPOUNDS
549
The tight-binding model described above has been introduced, partly, to get the possibility for comparison of the results with the numerical calculations (Weger and Goldberg 1973), partly for convenience to deal in what follows with analytical expressions. However, it should be emphasized that the only physical approximation of the theory is the assumption that the transfer integrals for electrons on the orthogonal chains are small in comparison with the width of the one-dimensional band. There are no other physical reasons but simplicity to construct the latter in the way it has been done above (i.e. using in eq. (23) the transfer integral A between only the nearest neighbours along the chain). Therefore an important question arise, whether the results will change essentially if the one-dimensional electron band is of a more complicated form than the cosine law (24), and in particular whether the singularity (32) in the density of states exists in the general case. It appears that the symmetry of the density of states with respect to the point (30) is destroyed. This is due, first of all, to the annihilation of one of the logarithmic singularities. In fact, it is easy to verify that the form (33) of the energy spectrum at the lines p,, = 0 or pz = 0 is a consequence of (24) only. In the general case one obtains:
wheref, and fi,, are some functions of p z . Hence, the range of the saddlelike form of the spectrum (33) is limited in pz (by the condition that the left side has to be small: E < T*). As to the singular logarithmic peak (32) at E = 0, it always exists, though the coefficient before it is, in general, numerically different. Indeed, as has been mentioned, at the edge of the cube the energy levels are fourfold degenerate in the one-dimensional chain approximation. A more detailed investigation which we will not carry out here shows that, if the interaction between neighbouring orthogonal chains is small (B Q d E ) the twofold degeneracy of the electronic spectrum is still present along the whole edge of the cube (p, = n/a,p y = ./a).* For the low-lying branches ( E ;5 T* 4 B), instead of (34), we have a new equation *The two-fold degenerate term along the whole edge of the cube is a consequence of the . logarithmicsingularityinthe density of states is due to the special symmetrygroup O h 3The relationship between the coefficients in (34’) which is a consequence of the fact that the fourfold degeneracy of onadimensional chains at the cube edge is broken by the small overlapping between chains. 0
L.P. GOR’KOV
550
[CH. 7,g 4
containing some new functions q1,.’ q2,,. At small E < T * and at appropriate signs of q1,=and q z , z it again gives the singularity:
m = v(o)(2/n:z)rIn (T*/l+,
(32’)
where
<
1 and the position of this singularity with respect to The magnitude of the chemical potential, corresponding to the stoichiometric composition (i.e. pStoich T*) cannot be defined in the general form. According to (28) a change 6p T* corresponds to the stoichiometry of the order of T*/AE (a few percent). The fine structure of the electron density of states of necessity leads to a temperature-dependent contribution to the paramagnetic susceptibility N
N
N
The experimental data for V3Si and Nb3Sn show a remarkable temperature dependence of the magnetic susceptibility. We note a rather important detail that in the strictly one-dimensional approach the only (logarithmic) temperature dependent contribution to x comes from the representation B2- of table 1. However it is proportional to the square of the interchain transfer integral value (Gor’kov 1973, 1974). Hence, the temperature dependence of the susceptibility for V3Si and Nb3Sn must be attributed to some structure of the electronic density of states. Figure 18 shows the results (Gor’kov and Dorokhov 1976a) of numerical calculations for the magnetic susceptibility (37) in which the density of states (31) has been used (at various positions of the Fermi level). It can be seen that both increase and decrease of x at cooling are possible for the twopeak structure. As far as we know, the correlation of the temperature behaviour of the magnetic susceptibility with composition has never been studied. One can find some data showing that the difference x(T,)-x(300) changes for samples with different T,. However to the author’s knowledge there is no data indicating that x(TJ ~(300).Therefore this could be an
-=
PROPERTIES OF A-15 COMPOUNDS
CH. 7,8 41
551
additional argument in favour of the fact that the peak at e = -2T” is actually smeared out according to the previous discussion.
5% F 0 ‘ 7
t
~
x2.
\
0
------------‘.---------;/-4)
/ /
2.1
-
0’
//
//
/ / /
,/-
R~ 4 ,//
-0.05
-0.20 ?-.0.40 -1.00
,--gS~----.
r 2.3’ n
/ /
/
1
1.9 -*’
/
/
/ /
1.7
/
/
/
/
/
--/’ 1.5-
I
I
I
I
I
I
If only the peak (32’) at E = 0 survives, as seems most probable, the temperature dependence x ( T ) for the corresponding position of the Fermi level is of the form: X ( T )= p:v(0)(2/n2){ In (T*/T)+const.
(38)
L.P. GOR’KOV
552
[CH. 7,O 5
The plot of the magnetic susceptibility, 1, for V3Si and N3Sn versus 1nT is drawn in fig. 19. These results look like strong evidence in favour of the existence of the well pronounced singularity (32’) in the best samples of Nb3Sn and V3Si.
25
50
75
100
150 200 250 300
T(K1 Fig. 19. The magnetic susceptibility of V3Si (Maita and Bucker 1972) and Nb3Sn (Rehwald et al. 1972) plotted in Gor’kov and Dorokhov (1976a) as a function of 1nT. The numerical values for xv3si have to be multiplied by the factor 181/51 to correct the arithmetic error of Gor’kov and Dorokhov (1976a).
It is also worth pointing out that the singularity in the density of states can change the ordinary linear in T contribution to the specific heat caused by electrons. In full correspondence with (38) one would have
where C,,(T) is the linear in T term C,,(T) = +n2v(O)T
for the flat dispersion law. 5. Structure properties of the A-15 compounds in the simplest model. Phonon spectrum
We have seen that because of considerable complications in the mathematical part of the problem [see eqs. (21), (21’) and (2191 any attempts to develop
CH.7.8 51
PROPERTIES OF A-15 COMPOUNDS
553
a completely microscopic theory for the A-15 compounds look rather hopeless. Therefore in our analysis of the structural and superconductive properties we shall oversimplify the problem, considering it from the ‘threedimensional’ point of view. We suppose that the 3D effects have restored the Fermi liquid picture of well defined quasiparticles which is usual for the description of the ordinary metal properties. If Tcrit, eq. (22), is of the order of the characteristic phonon frequency (200-300 K), as could, in principle, be expected, then at T, N 20K and T, N 20-50K one still has a temperature range where such an approach seems to be selfconsistent at least qualitatively. The basic role of the initial one-dimensional picture is reduced only to the essential renormalization of interaction-blocks in eqs (21) and (21”). Generally speaking, the structure transformation is possible as the ‘remains’ of the one-dimensional instability - the initial coincidence between flat electron-hole Fermi surfaces. The basic assumption for the following treatment is that the structural instability is enhanced if the Fermi level position lies in the immediate vicinity of the peak (32’). As was shown above, the existence of the peak at compositions not very different from the stoichiometric one is a specific property of the A-15’s which stems from the choice of the term X, or X, for the electrons on weakly overlapping linear chains. It is evident that this singularity would appear in any diagram of the Peierls channel. The experimental results for the magnetic susceptibility give a justification of this suggestion. The well known strong dependence of the structural transformation temperature on the minor details of sample preparation (including its strong dependence on the composition) gives an additional evidence in favour of this point of view. According to 53, the structural transformation can be driven either by the electronic density parameter, or by some lattice distortions of the same symmetry. The last concept is usually elaborated in terms of the soft phonon mode. For the A-15 lattice there are a few modes which have the same symmetry representation, I‘:2. One of them is the acoustic tetragonal distortion, the others are connected with the optical phonons at k = 0, responsible for the displacements of the transition atom sublattices. Hence, as was noticed by Shirane and Axe (1971), one should take the coupling between these modes into account. According to Shirane and Axe the softening of the elastic modulus can take place even in this case. We will now briefly repeat their argument, considering three order parameters: the tetragonal distortion a = a/c- 1, the sublattice displacement, u,
554
L.P. GORKOV
[CH.7,f 5
and a parameter, c, describing the electronic density symmetry change. In the harmonic approximation one has for the free energy an expression of the form:
If only tetragonal strains are applied, the equilibrium values for u and [ are to be defined from the conditions:
After calculation one gets the expression for the effective shear modulus:
and it is seen that (suggesting (40) is positive at high temperatures)
changes the sign, if any of the two quantities, a(T) or Oz(T),pass through zero. In other words, Cmffbecomes soft even if C,(T)by itself does not. The softening of the elastic modulus depends quantitatively (and sometimes qualitatively) on the choice of the driving order parameter. In what follows, in order to simplify the calculations and to make more apparent the peculiarities connected with the peak structure (32’), we assume that the structural transformation is caused by the softening of C,(T) by itself and neglect all contributions which come from coupling with the other degrees of freedom (u and 0. In particular in the Hamiltonian (7) for the onedimensional electron and its three-dimensional version (23), (23’) and (25) :
in the general expression for the electron-phonon interaction
we will omit all terms with y, d, and d3 (we discuss this approximation below). All corrections to the electron-phonon interaction of fig. 13b are also neglected. Certainly if the quantitative side of the matter and comparison
PROPERTIES OF A-12 COMPOUNDS
CH. 7,5 51
555
with experiment is kept in mind, taking account of all these factors could be very important. If we restrict ourselves to the tetragonal distortion a = a/c- 1 along z: E,,
=
E~~
= 3a;
E,,
= -$a,
(42)
then the ‘lattice’part of the elastic energy
can be represented in the following way
The electronic contribution to the elastic energy comes from the terms quadratic in strains in the expression for the electronic thermodynamic potential:
In this expression the sum is taken over all three sets of chains (i), the index o! labels the two branches of the spectrum (25). If the chemical potential lies in the neighbourhood of one of the peaks (32), one can use the simplified expression for the electron energy (33):
which is responsible for the singularity in the density of states. In this way one gets the i-chain contribution to the potential ~562~:
here for the x-chain, for instance, h(,) = d , ( e y y - ~ , , ) + ~ B ( ~=- H d,a+ ) pB(a-H)and includes the terms with the magnetic field. The integration
L.P. GORKOV
556
[CH.7,O 5
over E is defined schematically in the sense that we will be interested in the elastic contributions which are not dependent on the exact energy limits of integration. For the same reasons we shall not define the constant in (44) under the sign of the logarithm. After straightforward differentiation of expression (44) for the x- and y-chains and combining the result with the lattice contribution one obtains for C,(T) in the cubic phase (at H = 0)
where Go)is the transformation temperature at p = 0 (i.e. when the Fermi level is exactly at the position of the peak (32)). TLo' is defined by the condition
and a function s(x) has been introduced
The asymptotic behaviour of s(x) looks as follows: s(x) =
-7<(3)2/4n2; -In(2yx/a);
x
< 1,
x $ 1.
(46')
Using eq. (37) for all three sets of chains we get for the magnetic susceptibility
Therefore in the cubic phase within the framework of this approximation there is a relation between the temperature dependence of both quantities:
CH. 7,$ 5 1
PROPERTIES OF A-15 COMPOUNDS
557
Let us accept the notation A(T) = dlct(T) and expand the thermodynamic potential in the deformations up to terms of the fourth order: 651 = -${[In
&}.
(~)+s(~)]A’+sw(~)
(48)
Equation (48) is the expansion for the thermodynamic potential usual in the Landau theory of second order phase transitions. In our case 662 (48) defines the growth of the spontaneous tetragonal distortion a(T) in the immediate vicinity of T < T,. For p = 0 (when the singularity (32) is most pronounced) one has
and 4 T:)-T c, = dfV(0) -a2 . TE) Expression (45) also defines the dependence of T, on the composition. For smallp/T 6 1 AT,/T&” = -(71;(3)/4d)@/TLo))’.
(51)
From eq. (48) the change of T, can be found when external deformations are applied:
A T,/TE) = -(75(3)/4~’)(d~ct/T~’>~.
(49’)
It is known (Maita and Bucker 1972; Dieterich and Fulde 1971;Williamson et al. 1974; Ting et al. 1973) that the temperature of the martensitic transformation is rather sensitive to the external magnetic field. Using (44)with H # 0 it is easy to derive the following expression for the shift of martensitic transformation temperatures in the presence of a field: ATmITm = -P(pBH/Tm)’, with
(51’)
L.P. GOR’KOV
558
[CH. 7,9 5
At p = 0 we have /? = 7C(3)/4nZ N 0.21, which is close to B = 0.217 of the Labbe-Friedel model (Dieterich and Fulde 1971). If 1p\ increases, the value of /? decreases:
B = 0.21[1 - 0 . 1 6 ~ / T ~ ’ ) 2 ] ; (PIT 4 1). There are few data for V,Si: /? = 0.15f0.01 (Maita and Bucker 1972) and /? = 0.101f0.02 (Williamson et al. 1974). The dependence of /? on the quality of the samples has been confirmed in Williamson et al. (1974). Let us turn to the tetragonal phase. Instead of expansion (48) one has now d ( T ) TZ’. Straightforward calculations of (44)are still possible. In this way one gets for the shear modulus corresponding to the tetragonal deformation (42) below T,,, the following expression: N
C, = -dlv(0)- f2{ In-++ T ;
[-
s( “ y ) + s ( q J .
(45’)
The expression for the magnetic susceptibility becomes:
As was already mentioned, both these expressions are also correct at all temperatures if the tetragonal distortion is caused by some external strains. In the absence of stresses the magnitude of d ( T ) in these equations should be defined from the equilibrium conditions :
which can be expressed in the form
i m
=4
1- m deln-
T’:
I4
E-P-A E-P+A P T - t h 2T
+-AT ch-’ 2T (52)
CH. 7,g 51
PROPERTIES OF A-15 COMPOUNDS
559
(The right-hand side of eq. (52) actually does not depend on the cut-off value under the sign of the logarithm.) In fig. 20 the result of the numerical solution of eq. (52) is shown (at p = 0). The dashed line represents the experimental X-ray data for Nb,Sn. At T = 0:
C,(O) = (2/d)d:v(o).
(53')
1.0 0.8
0.6 h
0
v
\ 5
iz
0.4
Y
5
0.2
0.2
0.4
0.6
0.8
1.0
TIT,
Fig. 20. The square of tetragonal distortion az(T) = (a/c- 1)2 calculated using eq. (52) at p = 0. The dashed line represents normalized experimental data for Nb3Sn (Maiifert et al. 1967).
The relation (47') between x(T) and C,(T) is violated in the tetragonal phase. In this connection it is interesting to point out that at low temperatures (if p = 0) the expression for the magnetic susceptibility takes the form
The logarithmic term in this expression has its origin in the fact that at the tetragonal transformation (42) only two chains are distorted. Therefore a similar term enters the elastic modulus which is responsible for the deformation E,, = -eYy in the tetragonal phase.
L.P. GOR’KOV
560
[CH.7,O 5
Let us once more emphasize the essential difference between the results obtained above and all the Labbe-Friedel type models and their picture of the Jahn-Teller effect consisting in the redistribution of electrons between different chains. Though the density of states peak (32), (32’) plays an important role in our treatment of the structural transformation, there is no considerable redistribution of the occupation numbers of electrons between the three sets of chains (in comparison with the initially large number of carriers on each chain). In all the previous calculations we have neglected the change of the chemical potential caused by spontaneous (or by external) distortions. As a matter of fact at this transformation the new position of the chemical potential should be defined from the condition that the whole number of carriers which is determined as
(where the sum of the contributions from all chains is to be taken) has to remain unchanged after distortions. The investigation of the equation
N
= const+ I
vI(e, A, p)n,(&)ds,
where n&) are the Fermi functions for the electron occupation numbers on different chains, shows that the corresponding change of the chemical potential position has the form
and is comparatively small due to the presence of the large logarithmic factor ln(T*/T,,,) % 1. However the change of the chemical potential must be included into the calculations pretending to describe the quantitative behaviour of A-15 compounds. For instance, the curve
shown in fig. 21, has been calculated as a function of the Fermi level position, po, in the cubic phase at T = 0; namely this po is the true characteristic of the chemical composition. In the cubic phase the function ?(T/po)is responsible for the temperature variation of both the magnetic susceptibility and the shear modulus C,.For this last quantity the dashed line represents the logarithmic contribution
PROPERTIES OF A-15 COMPOUNDS
CH. 7,451
561
which is the only important one in the case of small po (if T/po+ 00, i.e. the chemical potential coincides with the position of the singularity (32)). The horizontal line contains the ‘bare’ lattice contribution at a fixed composition, po , and is normalized in the appropriate way [see eq. (45)] :
/I
-0.51
I
I
I
-1.01
1
Fig. 21. Graphical definition of the martensite transformationtemperature. The function ij(T/p,,) is defined by eq. (49, the dashed line corresponds to its logarithmic asymptotic
I). behaviour at large T. The horizontal line is -n2C.(0)/2v(0)d~’+ln(~~*const/2ylpo Note the minima on the curve for f ( v p o ) which makes possible the first order regime of the phase transition for some interval of compositions.
The point of intersection graphically defines the transition temperature, Tm(po), for the structural transformation at a given composition, po The minimum on this curve is an interestingproperty of the C,dependence on T and p which immediately raises the question about the thermodynamic character of the phase transition that has been tacitly supposed to be of second order. Before discussing the question in more detail it should be mentioned that the common group theoretical approach (Anderson and Blount 1965) leads to the conclusion that the corresponding transformation in V3Si and
.
L.P. GOR’KOV
562
[CH.7, § 5
Nb,Sn has to be of first order. This fact immediately follows from the X-ray data (Mailfert et al. 1967) in fig. 2 for the temperature dependence of the lattice parameters in Nb,Sn;* however the question is not so clear in the case of VgSi where the two temperatures T, and T, are rather close to each other. Starting with symmetry considerations it is easy to construct the following invariant of the point group Oh,of the third order in deformations
This invariant appears in the Landau expansion for the thermodynamic potential (48) if one takes into account the terms proportional to d2 in eq. (41) for the electron Hamiltonian. The detailed investigation in the article by Gor’kov and Dorokhov (1976a) has shown that the electronphonon interactions d, and d3 are responsible for the temperatnre variation of the bulk moduli which are known to change very slowly for Nb,Sn and V,Si. Hence, we suppose that the ratio 9 = d 2 / 4 is small: 9 -4 1. After simple calculation one gets instead of eq. (48):
At small p (p < T z ’ ) one has the following jump of the spontaneous distortion, asp,for the martensite transformation : asp= 29p/d,.
(55)
This mechanism of the d, interaction is responsible for the first order character of the phase transition only at sufficiently small p. In fact, one sees that s”(p/T) in the third term of eq. (48’) changes its sign as lpl increases (s’’(x0) = 0 at xo N 1.91). The cause for this is clear from fig. 21 where the function i @ “ / ~ p o is ~ ) plotted. If 1p1 increases, the horizontal line lies lower and the point of the intersection which defines the instability of the lattice, moves to the left. Near the minimum the existence of the two-valued solution of the graphic equation of fig. 21 points to the first order type of the phase transition. This is the second mechanism causing the first order transition *In our plot of aa(T) in fig. 20 we have actually defined T,-T part of the curve above the jump a2(T = T , .
T, extrapolating the linear in
CH. 7,851
PROPERTlES OF A-15 COMPOUNDS
563
which is not connected with the symmetry arguments. For this case there are no simple expansions like eq. (48‘) since the ‘jump’, asp, is not small. The temperature of the structural transformation and other characteristics can be found only by numerical calculations. Only the simplest limit of T = 0 has been considered. In this case the complete expression for 6 0 can be written down in the closed form:
The solution of the equilibrium condition %S2/8d = 0, obtained numerically, is shown in fig. 22. The point of the ‘transition’ (with changing composition) is defined by SS2(p*) = 0 after substitution of A@) into expression (56). At p > p* = 1.32(nT2’/2y) only the cubic phase is possible. The corresponding ‘jump’ (the spontaneous distortion) at T = 0,
dla,,(0) = 2.27(n TE)/2y)
/
1-
/
/ /
I I I
I I
01 0
II
1
0.5
I 1.32 1 . 5
I
111 /11:
Fig. 22. The equilibrium solution A = dla(0) at T = 0 as a fundon of p. Point
1.32(nTm/2y) corresponds to the boundary composition for the existence of the tetragonal phase. At p > p* only the cubic phase is possible at all temperatures (the role of superconductivity is not considered).
p* =
564
L.P. GOR’KOV
[CH.7,$5
is not small in comparison with the value of eq. (53). No attempts to get the corresponding transition diagram on the phase plane (temperature-composition) were undertaken for reasons which will be discussed below. The interesting result following from eq. ( 5 9 , is the possibility for the tetragonal deformation a/c- 1 to change sign with the change of p. Such phenomena have been reported for the alloys Nb3Sn, -,Sb,. In concluding this section we shall consider some other properties of the phonon spectrum in these compounds. The inelastic neutron scattering experiments (Shirane et al. 1971;Axe and Shirane 1973) for V3Si and Nb3Sn have discovered a remarkable temperature softening (at cooling) of the frequencies of phonons with wave vectors running over almost the whole Brillouin zone. This fact is commonly believed to be closely connected with the high T,’s in these compounds and therefore it drew considerable attention. Simple considerations show that this softening can be understood qualitatively as a manifestation of the initial picture of quasi-one-dimensional chains. The important fact is that the softening mentioned above has been observed for phonons with wave vector lying in the plane q = v, 0). To clarify the basic physical idea let us return to the approximation of onedimensional chains considered in 52. The essential feature we used in that section was that each of the three orthogonal sets of chains provides an additive electronic contribution to the elastic energy. Altogether these contributions respond to the softening of C , , and C, with decreasing temperature (at q = 0). If q is not equal to zero, the logarithmic terms (12) and (13) would be of the form (for the z-chain)
(r,
depending on which is larger: T or vq,. Hence, (1) only the component of the wave vector along the chain direction is relevant for the magnitude of the contribution from the given chain, (2) this contribution is saturated with the value of uq, instead of T,if uq, is larger than T. Roughly speaking, at q = 0 all three sorts of chains take a part in the corresponding change of the elastic energy. However, if the phonon acquires a wave vector directed along the chain, the contribution to the harmonic part of the elastic energy is ‘switched off. The neutron data (Shirane et al. 1971; Axe and Shirane 1973) were obtained for phonons with propagation directions along [lo01 and [110]. Therefore even if the magnitude of this wave vector is large enough to switch off the contributions from two chains
CH. 7,§ 51
PROPERTIES OF A-15 COMPOUNDS
565
(x and y ) the corresponding contribution from the third chain perpendicular to them would remain unchanged in the one-dimensional model. It is reasonable to believe that in a certain sense the same qualitative explanation of the phenomena would still be correct, at least as long as each of the three orthogonal sets of chains gives an additive contribution to the elastic energy. The simplest verification of this suggestion would be the experimental study of the temperature dependence of the frequencies of phonons with wave vectors having three non-zero components. As far as it is known, there are no data for phonons with propagation direction along [lll]. Evidently, these qualitative arguments have to be confirmed by straightforward calculations based on our theory taking the three-dimensional character of the electronic spectrum into consideration. It means that, as a result of calculations, we should obtain the right sign of the effect (lowering of frequencies at cooling) and, on the other hand, the effect of softening if it exists, has to be most pronounced in the vicinity of the peak in the density of electronic states. We shall not go into all the details and limit ourselves to a description of the main steps in the calculations (Gor'kov and Dorokhov 1976a). It is supposed that the temperature is low (T << T*), therefore in what follows we look for the major terms in T/T*. In correspondence with the comments in $ 3 we need the expression for the polarization operator, n(q,T), defining renormalization of the phonon frequency. For the general case it can be written in a closed form using the electronic Green's function
= (const/4ni) j j j j ( & ~ ) ~ d z th d ~(2/2T) p x HTr [WR(p+ z)WR(p- z>-c.c.l>,
(57) where p+ = pkjq. Integration over d3p, owing to the pole behaviour of both Green's functions, is actually restricted to the momenta region close to the corresponding Fermi surfaces. The last ones, in turn, are situated in the neighbourhood of the cubic Brillouin zone faces. Integration in eq. (57) over the three different pairs of zone faces corresponds now to the above mentioned fact that the contributions from the three sets of orthogonal chains are addi(41). tive. The 2, matrices have appeared due to the interaction term c& in Peph
L.P. GOR’KOV
566
[CH. 7,$5
Before writing down the explicit expression for the electron Green’s functions it is convenient to simplify the notation. For the x-chain we use: a = T*(cos2q+cosz@)+p;
c = ~*(cos~q-cos~~),
(58)
and
The Green’s functions on the respective faces are the solutions of the matrix equation
with the Hamiltonian
A
= -a2+ud,ty-ct,.
The solution is of the form:
6 ( p , z ) = [(z+4 2 + us,.ey - ct,][(z+
- (udJ2 -CZ]- l .
(60)
The simplest case in the general expression (57) corresponds to phonons with wave vector q along one of the cubic edges. For instance, let us take q 11 [Ool]. In this case the contribution to the elastic energy proportional to EZ,(q) in the harmonic approximation comes from both x- and y-chains. The result of the calculation has been represented in the article by Gor’kov and Dorokhov (1976a) as a dispersion law for the elastic modulus Cll(qZ):
This modulus is responsible for the propagation of longitudinal phonons along [Ool]. The integral factor in eq. (61) varies slowly from 1 at qz = 0 to, approximately, 0.75 at q, = n/a. The softening of these phonons at all qz’s has the same order of magnitude as the softening of the acoustic shear modulus C, at q = 0. It is easy to explain the mathematical reason for this large contribution, noticing that in this special case the poles of both Green’s functions can be taken in the form (33) along the whole edge of the cube
CH.7,g 51
PROPERTIES OF A-15 COMPOUNDS
567
because the wave vector q = (Ooqz) enters only the function cos(+p,a) in this last expression. If the wave vector q is moved from the [Ool] direction, the logarithmic contribution to eq. (61) disappears due to the fact that now in the Green’s functions G(p+)and G(p-) two poles cannot be chosen on one and the same line connected with singularities in the density of states. From eq. (33) it is seen that for instance, the x-chain contribution behaves differently in dependence on whether this move occurs in the plane (z, x) or (2, y). In the first case a drastic decrease of the contribution from the x-chain takes place at q,u > (TT*)”~
-
In the second case the corresponding criterion is qya 2 (T/T*)”’.
The first condition determines a rather narrow cone (- 1”) of q-directions along [Ool] beyond which the x-chain temperature dependent contribution is completely switched off. However, if the vector q lies in the (z, y)-plane, then even though the temperature dependent elastic contribution becomes smaller in full correspondence with the criterion (627, and the frequency softening must be weaker, it still exists and is quite remarkable. In this case it is described by a more complicated expression. Therefore, let us consider a phonon with wave vector in the (2, y)-plane: q = (O,qyyqz)with qy,qz > qcrit, the latter being defined by eq. (62’). Substitution of eq. (60) into (57) gives
n(q,T)= (const/4ni) JJJJ dqd$dp,dz th (z/2T)
+
+
x ([(z a+)(z a - ) - (YSx)2
+c+cJ
x {[(vSX)’-(z+p+2T* cos’ ‘p+)(z+p+2T* cos’ x [(v6J2-(2+p+2T*
+C.C.).
COS?
$+)I
p-)(~+p+2T* COS’ $-)I}-’ (63)
It can be verified that in this case the most important term in T/T* comes from those regions of integration in (63) where two square brackets in the denominators are small (for example, p+2Tcos2 ‘p+ and p +2Tc0s2$- 4 1).
568
L.P. GOR’KOV
[CH. 7.8 6
This condition defines two points again lying on lines connected with singularities in the density of states; however, this time these lines are two different edges of the Brillouin zone face. Considering small z (i.e. assuming T Q T*) after some calculations one gets the following expression for the temperature dependent contribution to the elastic deformation energy connected with this phonon
where
and u = sinrc,, /!I= sinx,. At qz = qyl(u, /3) = 0 and one should look for even higher terms in the expansion of eq. (63) in T/T*. The result obtained in Gor’kov and Dorokhov (1976a) for this case, has the form of a temperature correction to the dispersion law of the shear modulus
This modulus enters in the common way into the expression for the velocity of the transverse phonon mode (q = (2n/u)[C,5,OI; e II [lTO]) investigated in the neutron measurements (Axe and Shirane 1973) for Nb3Sn. Expressions (61), (64) and (65) are the main terms connected with the singularities of the density of states. The corresponding coefficients are undoubtedly exact only for the model equations (23’). At finite p the character of the temperature dependence changes at T p. Apart from the main contributions considered above, there always must be temperature corrections of the order of (T/T*)2.
-
6. Some results concerning the superconductivityof A-15 compounds In accordance with the concluding remarks of $ 3 and with the approach which we have used for the description of the structural transformation, superconductivity in the A-1 5 compounds will also be considered from the
PROPERTIES OF A-15 COMPOUNDS
CH. 7.8 61
569
three-dimensional point of view taking the curvature of the Fermi surfaces into account. In what follows we accept the scheme of the BCS theory. It is known that the BCS theory in spite of its mean field theory character, has a very high accuracy. The temperature interval IT- T, I where the thermodynamic fluctuations regime would be essential is extremely narrow: ATIT,
N
(TJAE)’.
We cannot point out with certainty the corresponding criterion in our case. A likely estimate can be obtained using the value of T* instead of AE as a measure of three-dimensionality in the A-15 materials. After this short comment let us turn to the Cooper effect in systems under consideration. In the common approach to the problem one usually leaves only the phonon-induced part of the electron-electron interaction. The reason for doing so is that this interaction in some sense is a ‘long range’ interaction.* As a consequence, the repulsive interactions enter into the expressions for T, weakened by a factor lnAE/wo. Keeping to this scheme we will, however, suggest that the phonon-induced interaction is already renormalized by all the effects of softening considered above. The matrix element in the right side of eq. (21’) has the schematic form:
where the integration over d3p at the last step is separately performed over the energy variable, the distance u = e(p’)-p from the Fermi surface (0 (wChar)),and over the Fermi surface. The former is just responsible for the Cooper effect, i.e. the logarithmic enhancement of the interaction constant &,at low temperatures. (The cut-off procedure coincides with that acc‘epted in the BCS theory.) The second integration becomes nontrivial for anisotropic systems. As was pointed out in Pokrovskii (1961), for the last case the critical temperature has to be defined from the solution of N
*The retardation effect in eq. (18) results in correlation between electrons after the electron-phonon scattering which holds for time intervalsof the order of ti/wo. In the case of short-range interactions of a Coloumb nature this time scale would correspond to atomic units.
570
L.P. GORKOV
[CH.7 , $ 6
the homogeneous integral equation for a small anisotropic energy gap A,(p) with the kernel of the general form
and A,@) is dependent only on the position of the momentum p on the Fermi surface. Equation (66) represents the pole-like solution for the vertex r in the anisotropic systems. The complications in the solution of the problem in our case arise on two accounts : the non-trivial dependence of the integrals over the constant energy surfaces on the given distance from the Fermi surface (fine structure of the electron spectrum) and the large number of different branches of the spectrum (three sets of chains). The second question has no principal significance (‘two band’ models have been studied by many authors; see, for example, Moskalenko 1966); however, it makes the resulting equations quite cumbersome. As for the shape of the Fermi surface, we have already mentioned that its main parts are located close to the corresponding faces of the Brillouin zone (the characteristic distance US T* ) . The behaviour of the Fermi surface near the edges of the cubic zone is described by eqs. (34), (34‘). The schematic equation for the pole in the Cooper block shown on fig. 23, contains matrix Green’s functions which are matrices of the sixth order. We will label the single elements of the latter, G$’, by two pairs of indices. The upper pair represents the type of chains (i.e. along (x, y, z)), the pair of Greek indices below appears as a consequence of the twofold degeneracy of the electron representation at the point X.On the main part for each of the three pairs of zone faces only the corresponding terms, G$) are large. To simplify the problem we suppose that the vertex of the dashed line in fig. 23 is diagonal on the chain index. In other words, we neglect the phononassisted tunnelling of electrons between chains: after the process of electronphonon scattering the electron is left on its own chain. Therefore the two ingoing lines (p, - p ) in fig. 23 have identical indices (i), and this gives the possibility of introducing three matrices of the ‘superconducting gaps’: N
In the absence of non-diagonal contributions from G$)(i # k) to the internal part of fig. 23 one would get the picture of three independent superconducting parameters for each of the three orthogonal sets of chains. The straight-
CH. 7,5 61
PROPERTIES OF A-15 COMPOUNDS
571
forward calculation of the contribution from the non-diagonal elements shows that the coupling between chains is actually not a weak one. The superconductivity in these compounds has all the essential three-dimensional features. It is well known that the phonon-mediated electron-electron interaction is short range in space. Nevertheless, the vertices of the dashed lines, which represent the corresponding deformation potentials, are dependent on the phonon wave vectors (on the atomic scale). The Hamiltonian Pcph (41) contains the interaction of electrons with phonon modes at q = 0. The generalization of (41) is self-evident:
where thefi are linear in distortions of the lattice (tz would correspond to the representation A; of table 1 which changes sign when t + - t and therefore is not present in eq. (67)).
d
Fig. 23. Diagrammatic form of the equation for the pole solution in the Cooper-type channel, defining the critical temperatures of the superconducting phase transition. Only a phonon-mediated electron-electron interadon is included.
Let us now write down the right-hand side of the equation shown in fig. 23:
and carry out some simplifications making possible its calculation. If the singlet-type of Cooper pairing is chosen, the superconductive gap has to contain only combinations symmetric in a, fl:
512
[CH.7,5 6
L.P.GOR’KOV
Before substituting (69) into (68), one should accept an additional assumption concerning the averages ( f f ( q ) ) proportional to the corresponding D-functions. Namely, we will neglect their q-dependence, i.e., both interacting electrons are to be on the same chain and, hence, A,@) = const. With this assumption we lose an essential part of the cubic anisotropy of the real crystal. Nevertheless this seems to be the only possibility in order to get any results without numerical calculations. For the beginning we start with the calculation of the diagonal terms in eq. (68). After substitution of (69) and the expression (60) for the Green’s functions, G$’, and after some simple matrix calculations, the diagonal part of (68) can be represented as follows
The denominators of the Green’s functions in the last expression, according to the BCS theory (see the derivation of (66)), provide the logarithmic dependences on temperature near the corresponding Fermi surfaces. The combinations A, B, C have the following form:
First of all one sees that the equation for #’ is split off from those for (a!), 8:)). An interesting question arises in connection with different signs before ( f f ) in the square brackets of eq. (70). Let us recall that for the phonon-induced interaction the (+)-sign at (f corresponds to attraction [(ff) < 0, see eq. (18)].Therefore, if only the phonons are taken into consideration, then most likely yo = 0. It is difficult to estimate the relative magnitudes of (ff). However, comparing (at small q ) eqs. (67) and (41) one obtainsf, = 0 andf, %. f 2 . Further we will omitf,. As for the cut-off value in eq. (68), it is known from neutron and tunnelling experiments, that the frequencies of the effective phonons usually never exceed 200 K (see Testardi 1973). This suggests the idea that 8 Q T*. Expressions (70) for A, B contain the dependence on p and A = +d,a. At an arbitrary position of the Fermi level the terms with A and p enter everywhere together with the only characteristic scale T * . This would
:)
CH.7.8 61
PROPERTIES OF A-15 COMPOUNDS
573
correspond to a weak dependence of T, on ~5pand deformations (an effect of the order of (T,/T*)’). If the position of the chemical potential is chosen near one of the peaks (32), (32’)* the characteristic scale for the variations of 6p and A is of the order of T, Q T*,8. Let us now rewrite the equation of fig. 23 in a new general form
c’-
where {JI) is the ‘column of gaps’ constructed from the three submlumns a:), for each of the chains. We will write down only the row of eq. (71) for the chain along x :
The other two equations can be obtained by cyclic permutation of (x, y, z). The following notation and abbreviations are used: X = ln(2ye/xT), (the combination 2y/n N 1.22 is the same as in the BCS theory for a chosen cut-off procedure); R includes various numerical factors and the interaction (f:)+ The matrix R‘”’is the result of integration in (70):
(fi).
where P(x) = X’+2XIn(T)+1.32, 32T*
F(z) = ~ ~ ~ t h ~ z ~ u l n / l - u - z ~ ,
(74) and G = 0.916. We shall also need the asymptotic expressions for F(z):
*In what follows p is the position of the Fermi level with respect to this singularity.
[CH. 7,4 6
L.P. GOR’KOV
574
Two ‘coupling constants’ kl ,k , come from the non-diagonal terms e(xy) in eq. (68). The main contribution to kl and k, comes from the vicinity of the edges of the Brillouin zone. Both kl ,k , are proportional to (T*/dE)li2Q 1. However, in our model:
kl = (cl-In (AE/T*))(T*/dE)’’’; k , = (In (AE/T*)-c,)(T*/dE)’”
,
-
contain numerical factors between 2-5 (at T* ki ++). A more attentive examination makes it possible to simplify (72) even further. In fact, the second row of in (73) contains the term -8G.X. Therefore, at the compensation between left and right sides of eq. (72) for a:) one gets N
or
Within this accuracy (better than -10%) the whole system (71), (72) can be reduced to a system of three equations. Among all the solutions for T, only that giving the highest T, is important. Now suppose this temperature is T , b , A ) and the vector {$,} is known. The change of T, under some perturbation l? = RO-tRlcan be obtained with the help of the usual orthogonality condition :
+w:,
Rl@,) = 0.
(77)
Differentiating (73), one gets for aKfi/aT:
The two first terms in (78) are proportional to the elastic modulus C, (50) and at T T k T, are small (of the order of unity) in comparison with the third logarithmic contribution. For simplicity we shall write all results obtained below within this logarithmic accuracy. N
N
CH. 7,g 61
PROPERTIES OF A-15 COMPOUNDS
575
We shall determine the dependence of T, on deformations in a cubic sample (a:' = a:' = a:)). For a tetragonal distortion A, = fdla which touches only the chains along x and y, one gets from (77) and (73)
(here we have also used the relation aF(z)/az = -(2/z)s(&)). enough 5 TZ), then AT, < 0. For p = 0 (T,(O) = Tco):
If p is small
Comparing eq. (79') with the corresponding dependence of T, (49') one sees that the factor 31n(64yT*/nT',) in the denominator of eq. (79') reflects in the correct way the weaker sensitivity of T,to deformation. The quadratic dependence of T, on a was observed for V3Si (see Testardi 1973, eq. (11)). Using his experimental data and the value dl 1 eV one obtains the following estimates: ln(MyT*/nT<,0))N 7-7.4 and T* 500-800 K. In the tetragonal phase or for samples deformed by an external uniaxial stress, the new symmetry of the crystal is reflected in the symmetry of superconducting gaps:
--
in the limiting case of large applied strains the highest T, corresponds to the z-row of the matrix R, i.e. to the z-chain where the dependence on A = fdla is absent. Using the asymptotic expression for F(z) (75) at A % Tone gets A = klX/F(A/2T,)
N
k, In (2yO/xT,)/ln2 (2yA/xT,) 4 1.
(81)
- -
To understand the order of magnitude of external deformations needed to get a strong enough inequality (81) we shall take d = d1q,, 8 (8 UH)K), kl 0.1 (at T*/AE 1/50-1/80for V3Si). Onegets A ;5 0.1 at or, 2 x lo-' (compare this with the spontaneous deformations ay,sl = 2.5 x and aNbrSn =6.5~
-
-
N
Let us now turn to the problem of the very high upper critical fields, H c z , in these materials. Using eq. (77) we shall restrict ourselves to the interval IT-T,I 4 T,. To find the value of H,, in this temperature range one should derive the corresponding Ginsburg-Landau equation for the
[CH. 7,fi 6
L.P.GOR’KOV
576
coordinate dependent parameter $(r). The procedure (Gor’kov 1959) consists in the substitution of
into eq. (68) for the Fourier component, [$,I, and the subsequent expansion of the right side up to terms of the order of (qu/T,)’. Simple calculations give the contribution to KiY of eq. (73):
(The last coefficient comes from the integration over the whole Fermi surface, far from edges. Its exact expression is therefore dependent on the choice of the model spectrum (25). At the same time it has not to be sensitive to small deformations and to the variation Sp.) The Ginsburg-Landau equation for the wave function [$(r)] = $(r)[A, A, 11 can be immediately obtained from (77):
-)$(r) aKp) aT
LIT(T+U’ aKl‘i’
{(
- 7GC(3)u2 4n2T:
-1
.a -2e A , az
c
>I
+A’
(- i V - - ’r A)2} *
$(r) = 0.
(82)
(We use (”) to label the vectors in the (xy)-plane.) This equation can be compared with the one for the BCS theory (Gor’kov 1959):
Equation (82) being rewritten in the same form, gives for the cubic phase
481c’T: 7f;(3)$
48n’T:[ 1 64yT*] -1n7[(3)vz 2G nTko’
+-
The factor in the square brackets reflects the peak (32) (or (32’))in the density of states. It is interesting to compare the values of H,, for pure Nb or V and Nb3Sn and V3Si. Due to differences in T, and larger densities of states (smaller Y’ and the large logarithmic factor) one gets an enhancement of He’by a factor of about 20-30 if compared with the transition metals.
CH.7.8 61
PROPERTIES OF A-15 COMPOUNDS
577
A very interesting situation corresponds to the limiting case (81) of large external deformations (A Q 1). Instead of (82) and (83) one has the equation
-
which reflects the anisotropy of strains in the form of the anisotropy of the G-L effective mass tensor* mtfl m$rA-2. Since the critical field, H c 2 , has the anisotropy HLk' (mlmk)l'', one sees that the enhancement factor is proportional to A-' for H i and to This could be the likely for explanation for the large critical fields observed in real crystals of A-15 materials. In commonly occurring samples large enough internal stresses exist. The mean critical field is mainly defined by the component H:2. In any case this mechanism can compete with the commonly accepted impurity induced increase of If,, . The results obtained above contain the well-known experimental fact that though T, is sharply dependent on the composition, the correspondingchange of T, is not catastrophic. At A t = 0 (no deformations) we get the equation for T,Q
-
At small p one again obtains the same large logarithmic factor in the denominator :
*The picture under discussion has some features in common with what was obtained for weakly coupled linear chains in the Labbe-Friedel-Barisit model (BarisiE and DeGennes 1968) though there are many differences in the underlying ideas. We only note that in our case the anisotropy of the effective mass tensor in the G-L equations is due to the coupling between chains and not to the intrinsic anisotropy of the three-dimensional electron masses (Barisit and DeGennes 1968). This anisotropy is too large (- T*/AE to be taken into account in eq. (84).
[CH.7,66
L.P. GOR’KOV
518
Let us also discuss in a few words what happens with T, at structural transformations. In fig. 24 both T, and T, are schematically plotted as functions of the composition y. The dashed parts for each curve are supposed to have been calculated in the absence of the second phase (for instance, eq. (85) for T,). To the left of the point of intersection, y < yS,, the developing spontaneous deformation will suppress T, in accordance with eq. (79).
I P
I-1 s i n
Fig. 24. Behaviour of T, and To is shown schematically as a function of electron concentration (with chemical potential as a measure of doping). The dashed parts of both curves have no physical meaning as long as competition between two phases is neglected. At p < p,,,, a small but sharp drop of T, is expected as a result of the finite spontaneous deformations developed after the structural transformation (see eq. (55)).
We have already shown that the martensitic phase transformation is actually of the first kind. Therefore for y = psm the jump in the deformation will result (neglecting fluctuations) in a jump of To.This sharp decrease of T, at the boundary of the cubic phase has been observed (Berthel et al. 1975) for V3Si (AT, 0.3K).A similar behaviour (a maximum T, just at the boundary between tetragonal and cubic phases) has been reported for Nb,Sn, -xAlx I K, see Testardi 1973). alloys (Vieland and Wicklund 1971) (AT, The discussion above has shown that all these effects can be explained on the basis of modified BCS theory. There is no need to introduce new mechanisms of ‘the enhancement of superconductivity by the structural instability’. This question is often discussed by many authors (see Testardi 1973; Wcgcr and Goldberg 1973). N
N
CH.7,g 71
PROPERTIES OF A-15 COMPOUNDS
579
It is reasonable, however, to put forward the question what conditions are the most favourable for superconductivity in this or that compound. The results obtained above show first of all that any internal strains have to be removed (by heat treatment) and the composition should be as close as possible to that corresponding to the boundary of the cubic phase. In the preceding section we have discussed the remarkable softening of the phonon modes which has to occur for some phonons with wave vectors lying in the plane (I 10). According to the familiar McMillan interpolating expression for the effective interaction constant of the BCS theory, interactions (or c f 2 ( q ) ) in our equations) are proportional to the mean value of (0-’(p-p’)) where ( ...) means an average over the positions (p; p’) on the Fermi surface. The quasi-one-dimensional character of our model immediately means that the Fermi surface is very close to the planes (110) where the softening of phonon modes just occurs. As we have proved, a considerable magnitude of this softening should be expected in the narrow strip (62’) near the edges of the zone, if the position of the Fermi level is close to the peaks (32), (32’). Being unable to give a numerical account for the contribution of this effect to the value of To we note only that the mere existence of this effect points in the right direction. This effect, by itself, is a manifestation of the ‘parquet’ peculiarities discussed in 5 3. In conclusion of this section we point out that within the framework of the suggestions accepted above (namely, that the superconducting ‘gaps’ a:”(p) = cry’ are not dependent on the position p on the Fermi surface) a complete theory of the superconducting phase can be developed. However, the considerable anisotropy of the energy gap for these compounds (see Testardi 1973; Weger and Goldberg 1973; Izumov and Kurmaev 1974) perhaps cannot be adequately described in the approximation considered above being the result of the anisotropy of interactions in eq. (66).
7. Discussion
It is interesting to compare some physical consequences obtained in the preceding sections with the available experimental data. We do not pretend to do a quantitative comparison since many of the factors have been omitted. However, one will see that qualitatively the theory in its main aspect gives a self-consistent interpretation for the unusual properties of these compounds. Tn the range of coexistence of superconductivity and tetragonal distortions the derivation of any analytical expressions in the tetragonal phase below To
580
L.P. GOR’KOV
[CH. 7, # 7
becomes rather difficult.* Therefore most of our results concern the structural properties. Strictly speaking only Nb,Sn provides a sufficiently wide temperature range for the existence of the non-superconducting tetragonal state. Even in this case we will not try to get any complete and general results, since it would involve us in the study of interactions and competition between different order parameters: tetragonal distortions, sublattice displacements and electronic density waves. Such a general formulation of the problem would make tedious the analytical approach (in any case this goes beyond the scope of our review paper). It seems that owing to the large number of free parameters it would be easier to develop a quantitative comparison with the experimental data on the basis of a more phenomenological description. Instead of doing this, we will concentrate our attention on the physical manifestation of the density of states peak (32), (32’) which, providing it exists, could play an important role in any choice of the driving order parameter. We have already pointed out in 0 4 that for strictly one-dimensional chains there is no logarithmic enhancement of the magnetic susceptibility within the first approximation in the interaction constants. Therefore the observed temperature dependences in Nb,Sn and V3Si have to be ascribed to the varying density of states in eq. (37). An alternative (or additional) explanation which in our A-15 structure is connected with the unusual role of impurities by themselves will be discussed briefly below. The plots of the magnetic susceptibility for Nb,Sn and V,Si versus 1nT in fig. 19, if compared with eq. (38), give the densities of states for uncoupled chains, respectively, v(0) = 6.7 states/eV.atom and v(0) N 13 states/eV*atom (at 6 = 1). These are rather large values, as are the values v(0) N 6.5 states/ eV-atom and v(0) N 9.2 states/eV.atom for pure niobium and vanadium (according to the magnetic susceptibility data; see Gladstone et al. 1969), and would correspond to rather narrow energy bands. For the widths of the latter an estimate using the photoemission data (Kurmaev and Nemnonov 1971) gives for V,Si AE N 3-4 eV. One can reconcile the large values v(0) obtained from the data of fig. 19 with these estimates of the band widths *As long as the superconducting gap does not depend on the point on the Fermi surface, the extension of the corresponding formulas of the BCS theory presents no particular complications. It is known, however, that the superconductivity in these compounds is quite anisotropic. As was mentioned,the gap anisotropy is connected with the dependence of the interactions on momenta along the F e d surface. In the light of the comments in the preceding section concerning the role of the phonon modes ‘softening’ and its sharp dependence on wave vectors this fact could be. of principal significance.
CH.7,s 71
PROPERTIES OF A-15 COMPOUNDS
581
introducing as a possible explanation either the ordinary Coulomb mechanism for the enhancement of the magnetic susceptibility, or the contribution from the additional peak in the density of states connected with impurities (see below). In the simplest version of the theory (Gor'kov and Dorokhov 1976a) the deformation potential dl can be estimated from eq. (45) if we take the temperature behaviour of C, at temperatures higher than. '7 In this way one gets dl = 2.5 eV for Nb,Sn and dl = 1-1.5 eV for V3Si. Other independent posibilities are available in principle, for the definition of d l . For example, in the cubic phase eq. (49) defines the change of T, under applied external stresses. Expression (48) for the thermodynamic potential leads to the anharmonic correction to Hooke's law. The accuracy of the existing data is still insufficient for a successful comparison with the theory. However, for instance, for V,Si, with the help of eq. (49), describing the temperature dependence of distortion near T,, a rather close value of dl N 1.1 eV can be obtained. One of the main uncertainties is that in all the previous expressions the parameters 7 ' : ) and p are actually unknown, being dependent on the quality of the sample. (For instance, the procedure of heat treatment determines the distribution of internal strains which are very important in many respects). This fact finds its reflection for example, in the irreproducibility of the results of experiments devoted to the dependence of the martensitic transformation temperature on the applied magnetic field. According to eq. (51') the coefficient jl depends on p. At the same time B also changes strongly in the presence of any internal stresses. In our opinion, any detailed comparison with experiment is justified only if the complete set of data for the magnetic susceptibility, elastic moduli, tetragonal distortions, phonon spectrum etc. is obtained from measurements on one and the same single crystal. The only example known is the various results obtained for the single crystal of Nb3Sn grown by Hanak and Berman (1967). In this case we have data for the temperature behaviour of the tetragonal distortion obtained by X-ray measurements (see fig. 3). These data are also plotted as a dashed line in fig. 20 together with the theoretical curve calculated at p = 0. From eq. (53) it is easy to get the independent estimate of dl N 1.6 eV. This smaller value of dl obtained in this way, could be understood as the result of the arrest of the growingtetragonal deformation by the onset of superconductivity. One should also keep in mind that the phase transition at T, is actually of the first order (see, however, the discussion below of the low temperature behaviour of the elastic moduli for Nb,Sn and V3Si). H
L.P. GOR'KOV
582
'
[CH.7,s7
For the magnetic susceptibility a sharp decrease below T, is typical. Using eq. (54) for x at T N T, one succeeds in getting a value for this jump quite close to the result observed in Nb,Sn (Rehwald et al. 1972). A more interesting fact concerns the difference in the low temperature behaviour of the elastic properties for V,Si and Nb,Sn. For the former the shear modulus becomes finite below T, but is still small at all lower temperatures. Leaving aside the question of the role of the superconductivity which has approximately the same characteristic energy scale, let us point out that such behaviour is typical of the concept of the 'soft phonon mode'. In this concept the initially unstable phase with a small negative 'bare square of frequency', oi < 0 at the low temperature transition goes over to the stable phase with the positive value w& > 0. This last quantity is, as before, of the order of Ioi(T T,)I. To illustrate this important idea, we take as an example the expression for the renormalized shear modulus C,[eqs. (457, (53')l. At T < T, the logarithmic dependence is saturated by developed deformation d ( T ) = +d,a(T) T,. Hence, at T < T,, C,""(T) Crb(T,) < C, (T = 300 K). The softness of the elastic shear modulus C, for V3Si at low temperature appears therefore as evidence that in this compound the martensitic transformation is driven by tetragonal distortion. The fact that both transitions in V3Si have almost identical critical temperatures (T, N 20-25 K ; T, N 17.5 K) seems in this scheme rather incidental since the transition temperatures were not calculated. In the Nb,Sn case the shear modulus C, at low temperature increases again in magnitude, exceeding the room temperature value (Rehwald et al. 1972). If this result is not caused by some experimental uncertainties, the experimental data are in apparent contradiction with the predictions of eqs. (45') and (53'). The large value of C,at T = 0 cannot be explained either by introducing a finite p or by taking the first order type of the structural phase transition into account. The restoring of C, at low temperatures can be explained in terms of eq. (40). Since it is experimentally known that optical phonon modes do not become soft, the natural explanation consists in the suggestion that for Nb,Sn the ruling structural order parameter is of electron origin, in accordance with the discussion of 5 5. The softening of C, is forced and takes place only in the temperature interval where the denominator in eq. (40) tends to zero. A very important problem for the understanding of the properties of the A-1 5 structure materials is the role of contaminations and defects. Numerous investigations (see Testardi 1973) have shown that almost all properties are extremely sensitive to alloying or violation of the stoichiometry. The results N
-
N
CH.7,871
PROPERTIES OF A-15 COMPOUNDS
583
can be summarized as follows: at substitutional alloying the properties are much more sensitive to the substitution of transition atoms than nontransition ones. This conclusion is perhaps, the only straightforward indication of the unique role of the transition atom chains. From the point of view of the theory which starts with the picture of one-dimensional chains, one has to distinguish between two different limiting cases. If in the A3B compound the atom B is substituted (like Nb3Sni-,Sb, or Nb,Sn,-,Al,) the main effect is an increase or partial decrease in concentration of conduction electrons in the chain. A secondary effect would consist in the backward scattering of the conducting electrons. If the latter is small the description of alloys using only the parameter p - the chemical potential position would be adequate. Let us check that the existing experimental data for this type of alloys are consistent with our picture. In fact, for Nb3Snl-,Al, the boundary of the tetragonal phase existence corresponds to x S 0.075 (Vieland and Wicklund 1971). The corresponding change, dp, of the chemical potential is estimated from
-
at x = 0.075it would be dp 30 K T,. We have already had a chance to mention that for Nb,Sn,-,Sb, the sign of tetragonality changes at x = 0.15. From one side it is again a reasonable estimate for dp. On the other hand, the estimate (86) is, to some extent, uncertain. Indeed, it is only slightly changed, if we substitute for the density of states the expression (32) in the neighbourhood of the peak, since the prefactor (2/nZ)compensates in a considerable way the magnitude of the logarithm. The second limiting case in the one-dimensional approach corresponds to a substituted atom or defect in the positions of the transition element atoms. Such a configuration could 'lock' the electron motion along the chain and would therefore demand a three-dimensional mechanism for the electric current. The backward scattering effects would be the most important in this case. Even in the first case the main question remains whether at x = 0 p lies close to the peak (32). It is possible to get some idea about how p is changing in dependence on stoichiometry violation, if we again use the estimate (86) and substitute for its right-hand side 6xp instead of 2x. Here,p is the number of electrons supplied by the transition atom to the conduction band. Hence, in this case the change of p is several times faster. It is worth mentioning that in the process of preparation of A-15compounds their chemical comN
584
L.P. GOR’KOV
[CH.7,9 8
position is known rather approximately. The common practice for the selection of the so-called ‘stoichiometric’ samples consists in the measurements of the residual resistance. Unfortunately we are not able to suggest any theoretical expression for this quantity. This problem is especially di5cult for one-dimensional objects. Beside the famous Mott localization in one-dimensional metals it is interesting to note an additional phenomenon in the A-15 systems. The linear chain of Nb atoms, for example, has an electronic band which is similar to the half-filled electron zone for the chain with a period a* = $a (i.e. k = n/2a*). In the process of solution the substitutional atoms take random positions. However, these positions, z,,, are correlated along the chain due to the periodicity and symmetry of the host matrix: z,, = a*n where n are random integers. Hence, after averaging over impurity positions the electron spectrum retains some memory of this correlation, which is left in the form of an additional peak in the density of states near the point X in our system (Gor’kov and Dorokhov 1977). Actually we are dealing here whith a phenomenon resembling a sort of parametric resonance and consisting in the appearance of Van Hove type singularities in the density of states at ka* = in. The height of this peak is determined by the parameters of the impurity atom only, its width is proportional to the concentration. Apparently, such a peak would appear in all the physical effects considered above thus making ambiguous their interpretation. One can hope that the height of this peak is not large numerically. To what extent this peak is smoothed out due to three-dimensionality of the electron spectrum and how it can be distinguished in the experimentally observed quantities is now completely unknown.
8. Summary. Some concluding remarks
In this section we would like to list again the main steps we have taken in the course of our attempts to explain the peculiarities of the A-15 structure materials. We have used Weget’s suggestion that in these compounds some unexpected small parameter exists. This parameter allows us to consider electrons of the transition atom chains in the first approximation as onedimensional. The next basic assumption concerning these one-dimensional electrons consists in the introduction of non-overlapping conduction bands of fig. 9.
CH. 7.8 81
PROPERTlES OF A-15 COMPOUNDS
585
This suggestion automatically fixes the position of the Fermi level. The band constructed in this way immediatelyled us to the problem of stability familiar from the theory of one-dimensional metals. The corresponding argument reproduced in 5 3 shows that at sufficiently weak ‘bare’ electron-phonon interaction the instability with respect to fluctuations of structural parameters in the strictly one-dimensional case is inseparable from the superconducting fluctuations. In the one-dimensional approximation there is no real phase transition. Therefore the next step would be to include some three-dimensionalfeatures which have to restrict the regime of growing one-dimensional fluctuations. We have considered only electron tunnelling between chains since it is evident that the transfer integrals are to be large in the A-15’s. Even in this approximation we were unable to get a quantitative description for this transition from the one-dimensionalfluctuations regime to the three-dimensional phase transformation. The structure fluctuations (and/or the corresponding electron density fluctuations) are most sensitive to the three-dimensionality of the Fermi surfaces. In any case, the electron tunnelling decouples the structural transition temperature, T,, and the temperature of onset of superconductivity, T,. The question whether the three-dimensionality of the electron motion destroys the structural phase transition or leaves T, finite, cannot be resolved without calculations based on microscopic principles. Needless to say, this programme in its general form looks unreal. Therefore these suggestions show only the possibility of the structural transition being driven by the electronic order parameter. Assuming one-dimensionality of chains in the first approximation one has to consider the next correction. We have done this in a straightforward way in 0 4 by introducing an overlap between neighbouring orthogonal chains (and neglecting all interactions). The fine structure of the density of states arising in the tight binding model has two logarithmic peaks which actually are non-equivalent in a more correct approach. In the general case only the peak at E = 0 survives [see eq. (32‘)l. We would like to emphasize that if we accept Weger’s arguments in favour of the quasi-one-dimensionality of the d-electron motion and our choice of the electron band structure we must also accept the peak structure which is their immediate consequence. The structure in the form of the peak (32’) must exist (with a reservation concerning the signs of functions v ~ ,qz,* ~ ,in eq. (34’)); the only question is how narrow it can be in real materials and how distant it is from the stoichiometric composition. In the presence of
586
L.P. GOR’KOV
[CH.7.8 8
strains and scattering defects the peak (32’) is smoothed out (i.e. in eq. (32‘) I E l > W). The only result which could be interpreted as evidence in favour of the two-peak structure consists in a ‘crater-like’ dependence of T, in the system of alloys of NbjSn doped with elements of the IVth and Vth group (Hulm and Blaugher 1974). In Gor’kov and Dorokhov (1976a) we kept to this interpretation and had supposed that the T, maxima are caused by the maximum of the BCS interaction constant due to the phonon spectrum softening near the peaks (32). However, the results of $ 6 have shown that this ‘crater-like’ behaviour can be explained with the help of fig. 24 where the small decrease of T, in the interval of the existence of the tetragonal phase is actually caused by spontaneous deformations. At the same time there are no serious reasons to expect that the electron motion along chains can be described in the tight-binding approximation which would be the necessary condition for the well pronounced peak at E = -2T’. In $ 4 we have already revised our previous point of view concerning this question (Gor’kov and Dorokhov 1976a). We have made the peak structure (32’) one of the centres of attention in our review. One of the reasons for this, as was mentioned, consists in the existence of the ‘precursor’ temperature behaviour of the magnetic susceptibility for most of the A-15compounds. The fine structure in the density of states due to impurities, briefly described at the end of the preceding section, obscures the question to some extent. It has been shown above that the existence of the logarithmic peak could be an additional mechanism for the martensitic transformation in Nb,Sn and V3Si because it brings the fast temperature dependences of the electronic contribution into the elastic moduli. Formulas of $ 5 5 and 6 have only a qualitative character since such essential features as, for example, the dependence of the effective electron-phonon interaction on temperature (see fig. 13b), coupling between various order parameters etc. have not been considered. Moreover, the logarithmic peak (32’) can be considerably weakened by the factor (2/n2)c. Nevertheless, the results concerning the dependence of T, on strains, the higher sensitivity of T,,, to various factors and, in particular, the enhancement of values of the critical magnetic fields (see eqs. (82), (84)) are in correspondence with experiment and can be therefore considered as a manifestation of the peak-like singularity. As a whole the theoreticalpicture described above looks quite self-consistent. The crucialexperimentto test the basic idea concerningthe one-dimensionality of the transition atom chains could be the investigation of the temperature
CH.7, 8 8)
PROPERTIES OF A-15 COMPOUNDS
587
behaviour of the phonon modes with wave vectors in the [l 111 direction. If it happens that the softening of the phonon spectrum abruptly disappears when the wave vector goes out of the plane (110) it would be strong evidence in favour of the picture suggested above. A very serious difficulty for the further development of theory and its comparison with experiment is the lack of experimental data extending over the whole set of physical parameters and obtained from one and the same single crystal with a well-defined composition. Some other methods of composition control independent of the resistance ratio measurements are desirable. Investigation of the magnetic susceptibility behaviour as a function of composition could provide very important information concerning the electronic density of states. Probably the study of Nb3Sn could be preferable since in this compound the temperature interval between T, and T,is large enough to investigate its electronic properties without additional complications connected with superconductivity. Many A-15 compounds like, for instance, V3Ga show a considerable increase of the magnetic susceptibility on cooling, Ascribing this fact to the fine structure in the density of states it is reasonable to expect according to the foregoing, some manifestation of the peak (32’) in the structural parameters. Certainly, the structural transition by itself could be absent since this is only a question of the magnitude of the interaction constant d , , on the one hand, and of the peak location on the other. (We have also mentioned that impurities smooth the peak (32’) out due to scattering effects.) Nevertheless, a more careful study of structural properties (and in particular the temperature behaviour of the moduli) for single crystals of these compounds would be interesting. If the peak position corresponds to a rather large distance from the stoichiometric composition or if the peak is smoothed out the structural parameters (elastic constants) can still have some anomalies in their temperature behaviour. This is possible if the three-dimensional character of the spectra does not eliminate completely the initial instability of the one-dimensional approximation (some effects of the order of (T/T*)’). We have pointed out this possibility in $j 3. Perhaps this is just the case for Nb,Al where some small temperature variation of the sound velocities has been observed at almost constant magnetic susceptibility. [See also the corresponding results for V,Ge (Testardi 1973).] In conclusion we shall mention a certain incompleteness of the theory. Everywhere, beginning with the question about the singularities in the density of states and ending with the results of $j 6* concerning the super*These results are obtained in Gor’kov and Dorokhov (1976b).
588
L.P. GOR'KOV
conducting properties, the structure of the electronic spectrum has been considered in the absence of electron4ectron and electron-phonon interactions. The importance of interactions in the formation of the band structure and in enhancement of the paramagnetic susceptibility is well known [see the review paper (Gladstone et al. 1969)l. In the estimate of v(0)for Nb,Sn and V,Si obtained from the magnetic susceptibility the large values of v(0) also attract some attention. In 9 3 it has been shown that in our quasi-onedimensional model for the properties of A-15 compounds interactions play an even more important role. Unfortunately any three-dimensionalapproach like the random phase approximation is not applicable here even qualitatively since the three-dimensionality parameter is supposed to be small (T* < AE). It would be very interesting to investigate (even numerically and in an oversimplified model) the mutual influence of the scattering processes [eqs. (21'), (21")] and of the tunnelling between chains and, in particular, their role in restoring the Fermi liquid concept of quasiparticles at sufficiently low temperatures. In the foregoing we have noted several times that due to the large number of possible parameters it seems preferable to compare the theory with experiment at the current level of the latter using a more phenomenological picture. This approach has been suggested recently by Bhatta and McMillan (1976). They have used the ideas described above concerning the electronic nature of the structural order parameters and written down the LandauDevonshire expansion for the free energy of these compounds. Their general conclusion is that on this basis it is possible to get excellent agreement with the existing experimental data. A vulnerable point of their theory was the neglect of the temperature variation of the magnetic susceptibility.
AcknowIedgements The author would like to thank O.N. Dorokhov for essential assistance in the preparation of the manuscript and A.F. Dite for much editorial advice. References Abrikosov, A.A., L.P. Gor'kov and I.E. Dzyaloshinskyi, 1965, Quantum Field Theoretical Methods in Statistical Physics (Fkxgamon P m s . Oxford). Anderson, P.W. and E.I. Blount, 1965, Phys.Rev. Lett. 14,217. Axe, J.D. and G. S h e , 1973, Phys. Rev. B8,1965. BarisiE, S. and P. DeGennes, 1968, Solid State Corn. 6,281. Batterman, B.W. and C.S. Barrett, 1966, Phys. Rev. 149,296.
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Berthel, K.H., D. Eckert and A. Handstein, 1975, 4th Int. Symp. Reinstoffe in Wissenschaft and Technik Dresden. Bhatt, R.N. and W.L. McMillan, 1976, Phys. Rev. B14,1007. Bychkov, Yu.A.,L.P. Gor’kov,I.E. Dzyaloshinskyi, 1966,ZhETP50,738. Clogston,A.M. and V. Jacarino, 1961, Phys. Rev. 121,1357. Dieterich,W. and P. Fulde, 1971, Z. Phys. 248,154. Dyatlov, I.T., V.V. Sudakov and K.A. Ter-Martirosyan, 1957,ZhETP32,767. Dzyaloshinskyi, I.E. andA.1. Larkin, 1971,ZhETP61,791. Gladstone, G., M.A. Jensen and J.P. Schrieffer, 1969, in: Superconductivity,ed. R.D. Parks (Marcel Dekker, New York). Gor’kov, L.P., 1959,ZhETP36,1918. Gor’kov, L.P., 1973,ZhETP60,1965. Gor’kov, L.P., 1974,ZhETP Letters20,571. Gor’kov, L.P. and I.E. Dzyaloshinskyi, 1974,ZhETP 67,397. Gor’kov, L.P. and O.N.Dorokhov,l976a,J. Low Temp. Phys. 22,l. Gor’kov, L.P. and O.N. Dorokhov, 1976b, Solid State Comm. (in press). Gor’kov, L.P. and O.N. Dorokhov, 1977,preprint. Gorzkowski,W.. 1963,Phys. StatusSolidi 3,910. Hanak, J.J. and H.S. Berman, 1967, J. Phys. Chem. Solids, 28,249. Horovitz, B., H. Guttreund and M. Weger, 1975, Phys. Rev. B12,3174. Hulm, J.K. and K.D. Blaugher. 1974, Low Temperature Physics-LTI3, eds. K.D. Timmerhaus, W.J. OSullivan and E.F. Hammel (New York). Izumov, Yu.A.and E.Z. Kurmaev, 1974,UPhN113,193. Kurmaev, E.Z. and S.A. Nemnonov, 1971. Phys. StatusSolidi (b), 43,K49. Mailfert,R.,B.W. Battermanand J.J. Hanak, 1967, Phys. Lett. MA,315. Maita, J.P. andE. Bucker, 1972, Phys. Rev. Lett.29,931. Matheiss, L.F., 1975,Phys. Rev. B12,2161. Menyhard, N. and J. S6lyom, 1973,J. Low Temp, Phys. 12,529. Moskalenko,V.A. and L.Z. Kon, 1966,ZhETP 50,724. Pokrovskii,V.L., 1961,ZhETP40,641. Rehwald, W., M. Ray1,R.W. C0henandG.D. Cody, 1972,Phys.Rev.B6,363. Shirane,G. and J.D. Axe, 1971, Phys. Rev. B4,2957. Shirane, G., J.D. Axe and R.J. Birgeneau, 1971,Solid State Comm. 9,397. Testardi, L.R., 1973, Physical Acoustics, eds., W.P. Mason and R.H. Thurston (Academic Press, New York) 10.4. Ting, C.S., A.K. Ganguly, R. Zeger and J.L. Birman, 1973, Phys. Rev. B8,3665. Vieland, L.J., 1970, J. Phys. Chem. Solids31,1449. Weger, M., 1964, Rev. Mod. Phys. 36,175. Weger, M. and I.B. Goldberg, 1973, Solid State Physics, eds., H. Ehrenreich, F. Seitz and D. Tumbull28,l. Williamson, S.J., C.S. Tingand H.K. Fung, 1974, Phys. Rev. Lett. 32,9. Vieland, L.J. and A.W. Wicklund, 1971,Phys. Lett. MA,43.
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CHAPTER 8
LOW TEMPERATURE PROPERTIES OF KONDO ALLOYS BY
G. GRUNER and A. ZAWADOWSKI Central Research Institute for Physics, Budapest, Hungary
Progress in Low Temperature Physics, Volume VIIB Edited by D.F. Brewer Q North-Holland Publishing Company, 1978
Contents 1 . Introduction 593 2. Basicmodels 595
3. The Kondo effect 604 3.1. Experimental evidence for the magnetic-nonmagnetic transition 605 3.2. Theory of the Kondo e!€ect above TK 610 4. Properties below the Kondo temperature 615 4.1. Wilson’s numerical solution and Fermi liquid theories 615 4.2. Experimental results for T Q TIC 629 5. Phenomenologicalapproaches to the general case 638 6. Conclusions 644 Note added in proof 645 Referenoes 645
1. Introduction
Sincethe early days of metal physics it has been a well established experimental fact that some impurities with an unfilled 3d shell have well-defined magnetic moments when dissolved in simple metals, such as copper, whereas others are nonmagnetic. A striking phenomenon, the resistance minimum, was also discovered more than forty years ago, and its connection with the magnetic impurities has been emphasized. The systematic study of dilute alloys of 3d elements in simple hosts however, started only around the 'Fifties, when the consequences of magnetic moments in metals became clarified by experiments which measure local properties of alloys. Magnetic and nonmagneticimpurity states were first accounted for by Friedel and Anderson; the Hartree-Fock solution of the proposed model led to a clear distinction between the magnetic and nonmagnetic states. The explanation of the resistance minimum by Kondo in 1964 started an enormous activity in this field. The resistance minimum signals the onset of a smooth transition to a nonmagnetic state at low temperatures; subsequent experimental and theoretical efforts concentrated on the nature and details of this transition. Properties well below the characteristic temperature, called the Kondo temperature TK,were firmly established only later, and simple power laws of temperature found by very careful experiments are now interpreted by a Fermi liquid theory. The renormalization group treatment of the models accounts properly for the relationship between this low temperature state and the high temperature behaviour. Experiments on a wide range of alloys demonstrate a common behaviour, with characteristic temperatures ranging from the mK region to above room temperature. The temperature dependence of the impurity resistivity Rimpand of the susceptibilityXimp characteristic of a typical Kondo alloy are schematically plotted in fig. 1. At high temperatures Ximp exhibits a Curie-Weiss behaviour, whereas Rlmpapproaches a constant value which can be accounted for by the Hartree-Fock (HF) treatment of models discussed in 52. The logarithmic increase of Rimpwith decreasing temperature is due to the nonperturbative nature of the problem, and was first discussed by Kondo. The flattening off both of the resistivity and of the susceptibility at lower tern-
G. GRUNER AND A. ZAWADOWSKI
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[CH. 8, J 1
peratures is associated with the disappearance of the effective moment of the impurity - as evidenced also by specific heat experiments. The basic features of the transition to a nonmagnetic ground state, summarized in 0 3 are well accounted for by theories which start from the high temperature, weak coupling limit; they are however insufEcient to describe the properties at T Q TK.The resistivity saturates at zero temperature and is determined by the charge neutrality of the singlet ground state. Near to T = 0 the properties are similar to those of a Fermi gas. Fermi liquid theory works well in this region; this, and evidence for the internal structure of the singlet state will be discussed in 54, together with the relationship between the
sirrplepowcr
L
crossover region
logarithmic r q o n
law region ~
1
-_ T
Fig. 1. Schematic temperature dependence of the impurity resistivity R,, and inverse impurity susceptibility~,,,,,-l.Characteristic temperature dependences and various regions are indicated in the w e .
low and high temperature properties, established by Wilson by numerical calculation. Phenomenological approaches which endeavour to account for general features for arbitrary alloy systems will also be discussed in this section. Since this paper focuses only on the basic developments in this field it is by no means exhaustive. Details and elaborated discussions of various properties can be found in several reviews. The experimental situation has been discussed by Van den Berg in Volume IV of this series (Van den Berg 1964). This was followed by the review of Kondo (1969) and that of Heeger (1969). The theoretical and experimental developments are summarized in
CH.8, 0 21
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595
Magnetism V (1973) and in the review papers of Rizzuto (1974), Griiner and Zawadowski (1974). 2. Basic models
The static magnetic susceptibility associated with the 3d atoms dissolved in simple metals is generally used to establish the magnetic character of the impurities. Various impurities in different hosts display widely different properties. Manganese has a well-defined magnetic moment in copper; at low impurity concentrations, where magnetic interactions between the impurities are absent, the susceptibility is given by
where 8 is negligibly small, peff 5pB close to that corresponding to s = $, the free spin of manganese. Chromium and iron also possess a magnetic moment in copper, but 8 is larger; in the case of nickel impurities however only weak Pauli paramagnetism is observed. Vanadium and cobalt are borderline cases; for these impurities 8 is of the order of room temperature. The situation is rather similar in a gold matrix, but in aluminium neither of the 3d impurities shows clear-cut magnetic behaviour (Rizzuto 1974). The appearance of a magnetic moment at the impurity site thus depends both on the host matrix and on the 3d element. In a free 3d atom Coulomb and exchange forces are responsible for the magnetic moment. The admixture of the 3d wave functions with the conduction electron band of the host leads, however, to delocalization of the impurity electron states, and, in case of strong admixture, to the destruction of interactions which favour magnetic behaviour. These ideas were first applied by Friedel (1956) to account for magnetic and nonmagnetic impurities. The model Hamiltonian applied by Anderson (1961) leads to results in the HF treatment which are closely analogous to those derived by Friedel using scattering theory. In the so-called non-degenerateAnderson model orbital degeneracyofthe d states is neglected, and the impurity is represented by a localized orbital at energy ed. The Coulomb interaction is given by the term Undgd-o where n d o = ad&,# + *is N
the occupation number of the d level for spin o and U is the Coulomb coupling. The admixture of the d-level with the conduction band states is represented by a vkd transition matrix element. The Hamiltonian operator of
596
G. GRUNER AND A. ZAWADOWSKI
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the model is given by
where the first term represents the host states, with u& and aka the creation and destruction operators of the conduction electrons. The mixing of the d and s states leads to a broadening of the d-level. Via the ‘golden rule’ one obtains
where I Vl is the appropriate average of vkd, and P,,(&~)is the density of host states at the Fermi level for one spin direction. The density of d-states has a Lorentzian form (2.4)
and the occupation number
With Coulomb interaction absent the position of the d-leveland the occupation number are the same for’o and -cr and the impurity is obviously nonmagnetic. In the HF approximation the Coulomb interaction is taken into account by an effective field, which shifts the position of the d-level, and
The shape of the resonance is, however, unchanged. Equation (2.6) together with (2.5) leads to two coupled equations
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PROPERTIES OF KONDO ALLOYS
597
Depending on U, A and on &d one obtains one solution, for which (n,,) = (nd-,), or two symmetrical solutions with (&) # (nd-,). The former corresponds to the nonmagnetic, the latter to the magnetic impurity. The phase boundary separating the two solutions is given by
that is the usual Stoner condition for magnetism in metals. The system is magnetic if UPd(&F)> 1, which occurs for large density of d-states at the Fermi level. From eq. (2.4) it follows that magnetism is favourable if the d-states are half filled and in this case the condition reads U/nA = 1. For less or more than half filled d-states, the condition for the appearance of magnetic moment requires larger U or smaller A. When the orbital degeneracy of the d-states is also included, the intraatomic exchange 9 between the different orbitals m also appears, and the number of d-electrons ranges from 0 to 2(21+ 1) = 10. The Hamiltonian for orbital degeneracy is given by
where only terms containing occupation numbers have been retained. The HF treatment leads to the condition for the appearance of spin magnetism (Blandin 1967) (2.10)
(v+21F)pd(EF) = 1.
An orbital magnetic moment can also develop in the degenerate case, the condition for this being given by
-
--
For 3d elements in a metallic host U 3 eV and .F 0.5 eV are the generally accepted values, for noble metals A 0.5 is obtained from various estimations (Griiner 1974). Thus ( U + 4 9 ) / n A 3 and the magnetic behaviour of manganese impurities, where N = 5 and the d-level is half filled in noble metals is explained. Chromium and iron have nearly half filled N
G. GRUNER AND A. ZAWADOWSKI
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[CH.8, $ 2
d-levels and are also magnetic; at the end of the series the nearly filled d-level of nickel leads to small Pd(&F), and so (U+49)PQ(&F) 1. Indeed, nickel is nonmagnetic in noble metals. Aluminium has a larger host density of states, and so A is larger; early estimates led to A 2 eV (Friedel 1956). With this value even for the half filled d-level the condition for magnetism is never reached, thereby explaining the nonmagnetic behaviour of 3d impurities in an aluminium matrix. With the above values of U,9 and A it is apparent, that the condition (2.11) is rarely fulfilled and so the orbital moment is generally quenched. The HF analysis of the degenerate Anderson model leads to the condition for spin magnetism which is determined by the joint effect of Coulomb and exchange forces. This may be an artifact of the HF approximation, and will be discussed later. It is however useful to analyze the various experimental quantities by replacing U by U + 4 9 and then using the results obtained for the nondegenerate case. In the following therefore U + 4 9 will be called U unless stated otherwise. The best confirmation of the Friedel-Anderson picture comes from optical and photoemission experiments. Detailed measurements on a wide range of alloys are available, these also give accurate experimental values for the basic parameters involved. In accordance with the suggestion of the model, two optical absorption peaks were observed in AgMn, one below, one above the Fermi level. A 0.5 eV and from the poxion of the absorption peaks one arrives at U 5 eV (Myers et al. 1968), in agreement with the theoretical estimates. Contrary to this, only one absorption peak was found in CuNi, with A = 0.3 eV and Ed = -0.75 eV (Drew and Doezema 1972). ByTntegrating the density of states up to E~ the number of electrons on the d-level is 8.9, in excellent agreement with the number of d-electrons of the nickel atom N = 9. The positions of the d-resonances in AgMn and CuNi obtained by optical experiments are shown in fig. 2. In s p z of consixrable efforts no clear-cut experiments are available in aluminium-based alloys. The HF solution of the Anderson model can be related to the phase shift formalism used by Friedel (1956). The phase shifts qu(w) of the scattered conduction electrons are given, for angular momentum 1 = 2, by
-=
N
-
N
where ( n u ) is the occupation number of the d orbital for spin Q. This relation also follows from the Friedel sum rule (2.13)
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PROPERTIES OF KONDO ALLOYS
599
if the nonresonant qo and q l , etc. phase shifts are neglected, and only the resonant q2 phase shift is considered. N is the total number of d-electrons of the impurity and N = (21+ l)C,(n,) expresses the charge neutrality. The formalism is particularly useful to express various physical quantities in terms of phase shifts, as only the values of phase shifts at the Fermi level appear in the macroscopic parameters.
Fig. 2. Position of the d-resonances in &Mn and @Ni obtained by optical experiments. The parameters of the Anderson model appropriate for the two alloys are also given.
Thus, keeping only the 1 = 2 phase shift, the impurity resistivity is given by
+
Rfmp= Ro5(sin2~,(eF) sin2q-,(EF)),
(2.14)
with q,(eF) = (n,)n. For magnetic impurities,
0. GRUNER A N D A. ZAWADOWSKI
[CH.8, 8 2
Au host
0 Cu
10
host
"
5 '*
4-
Ti
Cr
V
Mn
Co
Fe
NI
Fig. 3. Impurity resistivities in copper and gold at room temperature.
7 ~
5T 4 -I i
3-
*I I
11
1 j
.
TI
~
I
t
V
Cr
I--+
Mn
-+---------t-----
Fe
Co
NI
+ a Cu
Zn
Fig. 4. Impurity resistivities in aluminium at room temperature.
PROPERTIES OF KONDO ALLOYS
CH. 8, $ 2
601
at large distances r from the impurities. This perturbation can be measured by nuclear quadrupole effect experiments, which give essentially the same general picture as that obtained from the resistivity. The amplitude of the perturbation (Blandin 1967) is A = 5(sin q,(eF)+sin q - , ( ~ ~ ) ) ,
(2.16)
and displays the same characteristics as those shown in figs. 3 and 4. The magnetic moment of the impurity is given by the difference of occupation numbers for D and - D electrons. In terms of phase shifts
Equations (2.14) and (2.17) can be used to construct appropriate phase shifts the significance of which will be discussed later. When the impurities are magnetic, a spin perturbation develops around the impurity moments. In terms of the H F analysis, this is given by the difference of the charge perturbations for opposite spin electrons. In particular in the strongly magnetic limit when one resonance is well below, the other well above the Fermi energy, sinq,(EF) q,(eF) and sinq-,(E,) = n-q,(eF) and so
-
= (2/npO(EF))[q~(eF)-(n-
~ C ( ~ F ) )cos I (%Fr)lr3*
(2.18)
In this strongly magnetic limit the magnetic moment of the impurity is only slightly modified by the interaction with the conduction electrons [see eq. (2.17)], and then it is appropriate to represent the impurity by a welldefined spin. The Hamiltonian of this so-called s-d model is given by
H
= -2
j Y C ( ~ ) Sd3r, -~(~)
(2.19)
where 9 ( r ) is the distance dependent s-d coupling constant, S the impurity and 9 the conduction electron spin. When the s-d coupling is weak, perturbation theory can be applied to calculate various physical quantities. The impurity resistivity can be obtained using the ‘golden rule’. Both spin-flip and non spin-flip scattering contributes to the scattering amplitude, and the resistivity is given by Rimp= 5 R 0 ~ 2 & , p ( ~ i m1) p+
(2.20)
with S ( k ) taken to be constant. One-third of the resistivity comes from spin conserving, two-thirds from spin-flip scattering.
[CH.8, 0 2
G . GRUNER AND A. ZAWADOWSKI
602
The interaction between the impurity spin and conduction electrons leads to an antiferromagnetic polarization of the host states, and the effective spin is reduced. This reduction is proportional to .F in the first order of perturbation,
This polarization, however, is not homogeneous, but decays as r - j with increasing distance r. The perturbation called the RKKY perturbation was first derived by Rudermann and Kittel (1954), by Kasuya (1956), and by Yoshida (1957); it has the form (2.22)
h ( r ) = (Ftp0(~F)/87t)cos (2kFr)/r3.
A more elaborate analysis in the framework of the Anderson model shows that this reduction appears rather on the d-level than in the conduction band (see e.g. Zawadowski and Griiner 1974). Looking at expressions (2.22) and (2.18) and at (2.21) and (2.17) it is clear that the Anderson model in the strongly magnetic limit and the s-d model lead to similar expressions for the various physical quantities, but with parameters of the particular models. The comparison of relevant formulae can, in fact, be used to derive the correspondence of the parameters of the two models. This correspondence, was, however, derived by Schrieffer and Wolff (1966) using a canonical transformation. When U and &d are much larger than A, the Anderson model reduces to the s-d model, with (2.23) When Ed is below the Fermi level, and Eda+ U above, F is negative. With parameters, appropriate for CuMn, i.e. U = 5 eV and A = 0.5 eV one obtains F&) 0.05 eV. T h z the weak coupling limit is appropriate. Strictly speaking equation (2.23) is not appropriate for cases where the d-level is not half filled, as then, due to the charge neutrality, Edu A. Nevertheless, F is expected to increase by going away from the symmetric case,
-
N
N = 5. 9 can, in fact, be measured very accurately by the nuclear magnetic resonance (NMR) method. The RKKY spin perturbation, eq. (2.22), gives rise to a distribution of hyperfine field shifts at the host nuclear sites and leads to a broadening of the host NMR line. This has been widely used to
CH.8, 8 21
PROPERTIES OF KONDO ALLOYS
603
determine 9,and the observed values are collected in fig. 5 for noble metal hosts. 9 has a minimum in the middle of the series and increases with increasing or decreasing N,which can be taken as a direct consequence of the Schrieffer-Wolff transformation, given by eq. (2.23).
'ti-
,___-L_I_-.I---I-
V
Cr
Mn
Fe
Nt
Fig. 5. s-d couplings in noble metal hosts, derived from NMR experiments ( M i m o 1971).
The HF solution of the Anderson model can also be used to account for other parameters, such as thermoelectric power and specific heat. In both cases the width of the d-resonances appears: the electronic specific heat coefficient is given by (2.24)
and the thermoelectric power reads
s = n(ki/3e)(T/A)(sin 2qu(eF)+ sin 2q-#(&F)).
(2.25)
With parameters such as resistivity or effective magnetic moment taken from other experiments, reasonable values for the density of states or for the width A are obtained in cases where the impurity is strongly magnetic or weakly magnetic. Discrepancies have however been found for alloys which are near to the magnetiononmagnetic boundary. AuV or CuCo have already been mentioned as borderline cases, and AlMnTso fallsinto this
G. GRUNER AND A. ZAWADOWSKI
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[CH.8, 3
category (Griiner 1974). Here the analysis of y and Sleads to a large density of states at cF and so to a narrow effective width of the d-states suggesting that the HF approximation breaks down in these cases. Indeed one does not expect a sharp boundary between magnetic and nonmagnetic impurities suggested by the HF approximation; due to the low dimensionality of the problem, fluctuations are expected to smear out the sharp phase transition between the magnetic and nonmagnetic limit. This transition moreover, occurs also when the temperature is changed; the impurity which has a well-defined magnetic character at high temperatures may display properties characteristic of nonmagnetic impurities below certain temperatures. The transition as a function of temperature is smooth, but leads to drastic temperature dependences of various physical properties. 3. The Kondo effect
The first evidence for strongly temperature dependent effects in dilute alloys came from resistivity experiments in the early ’Thirties. In cases when the impurity is magnetic, the resistivity starts to rise at low temperatures, which, together with the phonon resistivity, yields a resistance minimum. Subtracting the phonon part, the impurity resistivity was shown to obey the empirical law Rimp
N
log T+const.
(3.1)
The explanation of this logarithmic temperature dependence is due to Kondo (1964), who in the framework of the s-d model showed that in third order of
perturbation the scattering amplitude of the conduction electrons diverges as T -,0. The first Born approximation leads to the temperature independent term given by eq. (2.20). In the next order, depending on the intermediate state, both the direct and the exchange scattering are operative. For the direct scattering the electron with momentum k is scattered from the initial state to the intermediate state with k“ and is then scattered to the final state k‘. In the exchange process an electron-hole pair is created first, then the hole annihilates the electron with momentum k. In both cases the scattering rates contain the term
where f k is the Fermi distribution function. The spin-flip scattering gives
CH.8, Q 31
PROPERTIES OF KONDO ALLOYS
605
S’S- for the direct and S-S+ for the exchange process, and thus the total scattering amplitude is proportional to
The spin conserving scattering rates proportional to S‘ do not contribute to this term. The integral diverges for T -t 0 or for w + 0. Taking the simple form for the density of host states PO(&) =
0
-D<&
(3.4)
one obtains a logarithmic temperature dependence, and the resistivity is given by Rimp= Ro4FZS(S+1)[1 +49p0(eF) In k T / D ] .
(3.5)
The logarithmic behaviour of the resistivity is in accordance with the experimental observations, and is the consequence of the Fermi statistics of the conduction electrons and of the commutation rules for the spin operators. The impurity spin represents an internal degree of freedom, which provides an indirect coupling between electrons scattered successively on the impurity spin. The situation is reminiscent of superconductivity where the phonon operators provide the effective coupling between the conduction electron states. 3.1. EXPERIMENTAL EVIDENCE FOR
THE MAGNETIC-NONMAGNETIC TRANSITION
A diverging scattering amplitude at T = 0 is clearly not a final solution but merely signals the breakdown of perturbation theory. Subsequent theoretical efforts concentrated on better solutions of the problem. Experimentally, the nature of the anomalies and, in particular, the low temperature properties have been studied intensively on a wide range of appropriate alloy systems. The anomaly in the behaviour of the resistivity was found to be accompanied by anomalies in the thermoelectric power and specific heat and these were shown to be the consequence of a smooth transition to a nonmagnetic impurity state, as evidenced by the magnetic susceptibility. The finite Weiss constant 8 is a property of single impurities and is not due to interaction effects. 8 can, in fact, be taken as evidencefor the disappearance
[CH. 8, 8 3
G . GRuNJ3R AND A. ZAWADOWSKI
606
of the effective spin at low enough temperatures. Allowing perfto be temperature dependent, the susceptibility is written as
A Curie-Weiss form, and thus a finite susceptibility a t zero temperature suggests a disappearing effective moment as T --t 0. This was first demonstrated by Van Dam et al. (1972) on AuV; the temperature dependent effective moment in this alloy is s h o w n 3 fig. 6. Magnetic susceptibility experiments measure the z-component of the magnetization, thus the above
50
100
150
200
250
300 T
[OK]
Fig. 6. Temperature dependenteffective moment in AuV alloys, derived from the magnetic susceptibility.
experiment demonstrates that (S') = 0 a t T = 0. The nonmagnetic nature of the impurities is further demonstrated by the temperature dependence of the impurity specific heat. Careful experiments, performed by Triplett and Philipps (1971) in CuCr alloys are shown in fig. 7. A broad anomaly is observed at temperatures around 8, and the area under the anomaly leads to the entropy AS=
C,(T)/TdT = R In (2S+ 1)
(3.7)
with S = 3, which is the high temperature spin of chromium impurities in copper. The impurity spin has therefore been removed from the system a t low temperatures, and the total spin of the impurity conduction electron system is zero at T = 0.
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PROPERTIES OF KONDO ALLOYS
607
The logarithmic increase of the resistivity provided early evidence for anomalous transport properties. Subsequent experiments demonstrated that this phenomenon is related to the smooth magnetic-nonmagnetic transition detected by susceptibility and specific heat experiments. Following the logarithmic behaviour at high temperatures, the resistivity flattens off somewhat below 8 and saturates well below this temperature. Evaluation of Rlmp is particularly difficult due to phonon effectsand deviationsfrom Mathiessen's rule. However, by changing the host properties, e.g. by alloying, various parts
IY
- 5 1 at pprn
0
A33.6 pprn
A A S
021.2
0-0,.
ppm
A
"pa 2.0
A U
m
0
I
.
l
o 'A
0.1
tb
1:o
T
[OK]
Fig. 7. Impurity specific heat as a function of temperature in S C r alloys (Triplett and Philipps 1971).
of the resistivity curve can be investigated. Figure 8 gives a summary of such xAu, -,Fe alloys, with characteristic temperatures, called TK, studies for -Cu also indicated. The Kondo temperatures obtained by fitting the experimental points to a calculated curve, given by eq. (3.14), with TKas a free parameter. The thermoelectric power shows a broad hump around 8 giving further evidence for a strongly energy and/or temperature dependent scattering amplitude. This parameter is also strongly iduenced by the background potential scattering, and direct comparison with theoretical expressions is difficult; it can however be used to evaluate the characteristic temperature of the magnetiononmagnetic transition.
[CA.8, 8 3
G . GRUNER AND A. ZAWADOWSKI
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The above examples were taken from alloy systems, where the transition from the high temperature magnetic behaviour to the nonmagnetic low temperature state occurs at low temperatures. It is emphasized that alloys viewed traditionally as 'nonmagnetic' can also retain magnetic properties at high temperatures. Early indications for this in the case of AlMn alloys (Caplin and Rizzuto 1967) have been followed by careful susceptibility +
r f
.-,
- .
f
-.
.
, ----.--
A 6
Cu -0.01 at % Fe 24
6
at% Fe13
C ~ o A u , o - O O Oat% Fe 8 6 4
0
Au -0.0025 at% Fe +------t----t-
0.01
01
0 24 f
10
,
~
f
10
.
T/T~
+.-
-
100
Fig. 8. Impurity resistivity versus temperature in Cu,Aul -xFealloys (Loram et al. 1970).
experiments in this system, and indeed the high temperature behaviour of ximp(T)can be described by a Curie-Weiss law with 8 somewhat above room temperature (Miljak and Cooper 1976). In addition to this the impurity resistivity of A1 3d alloys at high temperatures display a behaviour rather similar to t h a t found in copper alloys as shown in fig. 3, suggesting that at least in the middle of the series 3d transition metal impurities develop a magnetic moment above room temperature (Kedves et al. 1973).
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PROPERTIES OF KONDO ALLOYS
609
The characteristic temperatures where the magnetic-nonmagnetic transition occurs thus span a wide range, from well below 1 K to the lo3 K region. The characteristic temperature is not well-defined as various physical quantities are smooth functions of temperature. Rough estimates can be obtained on purely experimental grounds, leading to Kondo temperatures depending on the parameter measured. For the susceptibility 0 is usually taken as TK, for the resistivity the temperature where Rlmprises to half of the T = 0 value, for the specific heat and thermoelectric power, the temperature where the maxima are found. Also, comparisons of the measured temperature dependences with theoretical expressions containing TK as a free parameter can be used to evaluate the Kondo temperature. TK values for gold, copper and aluminium alloys are collected in fig. 9 (for details of evaluation of TK see Griiner 1974). The wide range of Kondo
CuMn AuMn
I TI
V
Cr
Mn
Fe
Co
Ni
Fig. 9. Kondo temperatures of 3d transition metal impurities in gold, copper and aluminium.
G. GRUNER AND A. ZAWADOWSKI
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[CH. 8, 0 3
temperatures, when correlated with changes of the basic parameter of the Anderson and s-d models, suggests a non-analytic dependence of TK on these parameters. Comparing the overall behaviour of TK with 9 inferred from NMR experiments and shown in fig. 5, an exponential relation
appears to hold, where TFis the Fermi temperature; ambiguities in 9 and TK do not allow the exact proportionality and the pre-exponential factors to be determined. F is given by the parameters of the Anderson model, and it is evident from eq. (2.23) that TK depends in a nonanalytical way on the strength of the Coulomb interaction and on the position of the single particle resonances. In the symmetrical case, for U/nA 3 which is appropriate to CuMn, TK K. For U/nA 1, the case of AlMn, TK lo3 K. Also, when U is the same, but Ed,,is near to the Fermi leFl, TK is higher. It is evident from fig. 9 that alloys which are clearly in the nonmagnetic limit of the Anderson model can also be incorporated into the general picture-with kTK of the order of the single particle width A. Thus, the Kondo effect is not a purely low temperature phenomenon, and is characteristic in general of dilute alloys. Properties below the Kondo temperature TK, analyzed in detail in $4, supply additional evidence for this, and point to a unified behaviour in the function of TIT,.
-
-
N
-
3.2. THEORY OF THE KONDO EFFECT ABOVE TK
The first improvement of Kondo’s result came from calculations of further terms of the perturbation series. The first correction to the resistivity given by eq. (3.5) or to the scattering amplitude is proportional to fln(kT/D) or 9ln(w/D), where w is the energy which becomes large as Tor o goes to zero. Thus it is better to organize the perturbation result for any correction term according to the coupling 9 and to the logarithm in the following scheme a, +u,F In ( k T / D ) + u z F
Inz ( k T / D )+. ..
+b 1 9 +bzF2In ( k T / D )+ ... +c l F +c z F In (kT/D)+ ... + ....
(3.9)
CH. 8,
31
PROPERTIES OF KONDO ALLOYS
611
If the logarithmic terms are large one may keep the first line and drop the others (logarithmic approximation). The leading logarithmic terms were first summed by Abrikosov (1965). This approximation leads to spurious divergency due to the appearance of the term [1 - 2 F p 0 In (kT/D)]-’
(3.10)
in the scattering amplitude which diverges for $r < 0 at Tg = D exp (+ 1/2Fp0)
(3.1 1)
called the Kondo temperature. This divergency cannot be physical as no real phase transition is expected and is due to the following two limitations of the leading logarithmic approximation: (i) the result is always real, as the imaginary parts contribute to the next approximation ( b i ,c i , ... may be complex), thus the analytical properties of these results are wrong; (ii) only those diagrams give contributions to the scattering amplitude which may be cut into pieces by cutting only one electron line and one spin line in each step (‘parquet’ diagrams). Further attempts set out to avoid at least one of these two restrictions. Enormous mathematical efforts have been made to drop the first restriction thereby keeping ‘parquet’diagrams but retaining correct analytical behaviour. The solution was manifested in the Nagaoka-Suhl integral equations for the spin conserving (t) and spin dependent ( 2 ) part of the scattering amplitude T = t+ra.S
(3.12)
which has been derived by various methods. These methods, Nagaoka’s decoupling scheme (Nagaoka 1965) for the Green’s function equation, Suhl’s equation (Suhl 1965) derived on the basis of Chew-Low scattering equations and the summation of the ‘parquet’ diagrams (see e.g. Brenig and Gotze 1968) are essentially equivalent. These equations have been solved exactly and compared with experiments. The prediction was a logarithmic temperature dependence for different quantities (resistivity, susceptibility, etc.) both at high and at low temperatures.
612
G. GRI)NER AND A. ZAWADOWSKI
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According to Hamann's solution (Hamann 1967) the spinconserving part of the scattering amplitude exhibits a peak around w = 0, t(w+ie) = (1/2ip0){l-In (w/g)[ln2(T/g)+S(S+ 1 ) ~ ~ - ~ / ' }(3.13a) while the spin-flip part has a maximum around t ( w ) = (1/2po)[ln2(w/T:)+S(S+ 1)n2]-'l2
(3.13b)
for T = 0. Similar formulae are valid for w = 0, but for T = 0 only w needs to be replaced by kT under the logarithms. The resonance formed symmetrically around w = 0 is called the Abrikosov-Suhl resonance. Similar solutions have been obtained by others (e.g. Brenig and Gotze 1968, Zittartz and Miiller-Hartmann 1968). The resistivity increases with decreasing temperature (Hamann 1968) and has the form = 1-In ( T / ~ ) [ l n 2 ( T / T+S(S+ ~) Rimp(T)/Rimp(0)
The heat capacity exhibits a peak around T = 1967), while the magnetic susceptibility is
(3.14)
+e(Bloomfield and Hamann
(3.15) for lnT/Tf 1. A decrease in the effective moment by lowering the temperature is the consequence of a correlation between the spin polarization of the conduction electron gas and the impurity spin, i.e. the impurity spin is gradually screened by the conduction electrons (Miiller-Hartmann 1969). The above expressions of the various physical quantities are in fair agreement with experiments at temperatures T > TK but this agreement breaks down progressively with decreasing temperatures. This is demonstrated in fig. 8, where the observed temperature dependence of the impurity resistivity is fitted to eq. (3.14). The fit is appropriate in the high temperature region but the theoretical curve deviates sharply at low temperatures. In the temperature region T < TZ the calculated resistivity is logarithmic while experimentally a simple power law behaviour is observed. A similar discrepancy was found for other quantities, with the low temperature logarithmic behaviour predicted being found to disagree with experimentally observed power dependences.
CH. 8 , 0 31
PROPERTIES OF KONDO ALLOYS
613
This disagreement between theory and experiment initiated another line of approximations, in which the effect of a large number of electron-hole pairs was included. It was noted first by Anderson in a series of papers (Anderson 1967a, b, 1970, 1972; Anderson et al. 1970; Yuval and Anderson 1970) that the Kondo effect is a typical example of infrared divergency, because the energy spectrum of the electron-hole pairs created by the spin-flip processes does not contain a gap. As a next step Anderson and Yuval(l971) noted that the spin-flip process in the Kondo problem can be regarded as a sudden change in the spin dependent mean field potential acting on the electrons and caused by the rigid impurity spin. The time dependence of this potential acting on, e.g. the spin up electron is shown on fig. 10, where the sudden changes represent
Fig. 10. Time dependence of the impurity potential acting on a spin-up electron.
the spin-flips. Thus, the Kondo effect can be regarded as a series of subsequent X-ray absorption problems which have been solved exactly in the large time limit. Anderson and his co-workers tried to solve the problem by combining this idea with scaling. However, as the strength of the spin-flip processes increases as a result of scaling the time differences between spin-flip processes liecome shorter, thus the asymptotic solution breaks down. Anderson (1970) also called attention to the scaling properties of the problem, characteristic of situations where the infrared divergencies dominate the dynamical behaviour. Such scaling was first discussed by Anderson (1970), and by Anderson et al. (1970). It was simplified considerably by Anderson (1970) using the ‘poor man’s’ method. S6lyom and Zawadowski (1974) slightly modified this method to make it capable of accounting for higher order corrections. If, for example, the one particle scattering amplitude T‘ is kept invariant under the scaling, and the cut-off D is changed by AD and the coupling 9 J
614
0.GRUNER AND A. ZAWADOWSKI
[CH.8, g 3
by A.F, then the compensation can be formulated as (aT’/aD)AD+(aT’/aF)AF= 0.
(3.16)
To have a definite scaling formula the coefficients aT‘/aD and aT’1a.F must be known. However these can be calculated only by using perturbation theory. If these calculations are performed up to third order the scaling equation has the following form (3.17)
Tp’
T
Fig. 11. Solution of eq. (3.13, F e a ( T )= *(a = kT)keeping the first (1st) and fist two (2nd) terms in the bracket of eq. (3.17). The dotted line represents the expected exact result.
This scaling agrees with those obtained by more sophisticated renormalization group calculations by Abrikosov and Migdal (1970) and by Fowler and Zawadowski (1971) where the Gell-Mann and Low version of the renormalization group known from field theory has been applied, The first order scaling, when only the second order term is retained on the right-hand side of eq. (3.17), leads to a solution which contains the factor given in eq. (3.10), thus it diverges at o T i similarly to Abrikosov’s original solution. The solution of the differential equation given in eq. (3.17) is shown in fig. 11, with initial parameters o = D and P = So;the scaled value is called .Feff. At small D this solution saturates at Saff = 1, called the fixed point. The fixed point occurs in the intermediate coupling region where it is not sufficient to keep only the first terms in the perturbation series. The fixed point therefore may be spurious, nevertheless eq. (3.17) leads to a crossover N
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
615
temperature between the weak and strong coupling regions, given by TK = D(21Flp0)1’2exp (1/29tp0).
(3.18)
The appearance of the factor (2191p0)1/2indicates a slight modification from the previous expression of the Kondo temperature given by eq. (3.11). Sakurai and Yoshimori (1973) were able to show that the binding energy of the singlet state is also proportional to TK given by eq. (3.18). The above theoretical treatments clearly demonstrate that the Kondo temperature represents a crossover between the weak coupling and strong coupling limits. Starting from the high temperature region all the known analytical methods break down near TK. The reason for this is the lack of typical dominating diagrams in the strong coupling region, thus the full series on the right-hand side of eq. (3.17) must be evaluated. The exact scaling curve is therefore not known, but Anderson (1972) and Fowler (1972) argued, on the basis of dimensionality, that the scaling curve must go to infinity in the way indicated in fig. 11, because no singularity is expected at finite 9. The trajectory, however, cannot be determined by the methods summarized before. We note finally that all the results obtained in the weak coupling region have also been derived by constructing a trial wave function for the ground state (for a review see Yosida and Yoshimori 1973). 4. Properties below the Kondo temperature
In the low temperature region, at temperatures well below TK the strong coupling between the impurity and the conduction electrons leads to properties drastically different from those observed above and around the Kondo temperature. Recent theoretical predictions valid in the T 4 TK region are verified by findings arrived at by very careful experiments performed in the last couple of years. We review first the theoretical achievements and then the experiments will be analyzed in the light of available theoretical results. NUMERICAL SOLUTION AND FERMI LIQUID THEORIES 4.1. WILSON’S
The theoretical treatment of the Kondo Hamiltonian, discussed in 5 3, acted as a bottleneck for some time until Wilson dropped completely the analytical approach and produced a numerical solution. The solution is a perfect
616
G. GRUNER AND A. ZAWADOWSKI
[CH. 8, 8 4
combination of the use of a computer to diagonalize a Hamiltonian and of the renormalization group concept and is valid in the whole temperature range including the spin compensated state at very low temperatures, the paramagnetic behaviour at high temperatures and the crossover region. The method was first applied to the Kondo Hamiltonian and was extended to the Anderson model later. After Wilson's solution, many of his low temperature results were derived in other ways. In the framework of the Kondo Hamiltonian, Nozieres (1974) was the first to suggest a simple, phenomenological Fermi liquid theory. In the Anderson model the behaviour of a Fermi liquid type had already been demonstrated in the late 'Sixties and by combining these results an almost exact description has been obtained for T = 0 (see e.g. 43.5 in Gruner and Zawadowski 1974). Further developments are due to Yosida and Yamada (Yosida and Yamada 1970, 1975; Yamada 1975a' b), who treated the complete perturbation series for the symmetric Anderson model.
4.1 . I . Wilson's numerical solution of the s-d model Theories discussed in the previous section demonstrate that the summation of a certain class of diagrams cannot give a final solution of the Kondo problem around and below the Kondo temperature. On the other hand, different scaling approaches strongly indicate the existence of scaling, thus a renormalization group method appears to be an adequate approach even if the original Gell-Mann and Low version fails. Wilson developed a new formulation of the renormalization group theory which does not rely on perturbation theory nor on diagrammatic calculations. The advantage of the method is a general procedure for reducing the degrees of freedom and for finding the new Hamiltonian by computer. In the present problem the computer is used to diagonalize Hamiltonians, and naturally the number of states cannot exceed a few hundreds. Considering the impurity and a few one-electron states of the conduction band the total number of the combined states is especially large due to the spin variable. As the Kondo problem is associated with the region of the Fermi level Wilson used the following scheme. First, he considered a couple of conduction electron states and the impurity and diagonalized the Hamiltonian. Next he included one more one-electron state from the low energy region near to the Fermi level, and to keep the total number of the st,ates constant before diagonalizing the Hamiltonian he dropped the states with highest energies. The procedure turned out to be adequate to find the character of low-lying excited states, and the temperature dependences of the various physical quantities.
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
617
At first the Kondo problem was transformed to a form suitable for scaling. The original model with uniform energy distribution for the conduction electrons was not adequate. Consequently Wilson introduced a model Hamiltonian with discrete energy levels proportional to *A" where A is an arbitrary constant in the range of two and n is an integer. In this model there are only a few states in the cut-off region, but the density of states is infinite at the Fermi level. The model is given by the Hamiltonian
H =
1n-"(u:u,,-b,+b,)-~(A+uA).S
(4.1)
n
where a,, and b, are annihilation operators for electrons above and below the Fermi energy, S is the operator for the impurity spin, u is the Pauli operator. The operator A, defined as A = CP,oA-"'2(a,+b,,), is introduced to represent the exchange interaction. It replaces the electron field operator on the impurity site $(O) appearing in the s-d interaction $'(O)a$(O). The modification in A by the factor is necessary to compensate for the effect of the increasing number of electrons around the Fermi energy in order to keep the rate of the spin-flip processes invariant. This Harniltonian is transformed into a hopping Hamiltonian of a semiinfinite one-dimensional hopping chain by a canonical transformation. The impurity is coupled to the end of the chain to site 0, and the Hamiltonian has the form
whereS, is the electron annihilation operator on site n and it is, furthermore, a linear combination of different a, and b, operators. The hopping matrix elements vary along the chain and E, -+ 1 as n -ia.Let us denote by HN the Harniltonian obtained from that given by eq. (4.2) by cutting the upper limit of the summation at n = N. The procedure of increasing the number of states is given by going from HNto HN+1.The recursion form is
Wilson first diagonalizes IfN in some approximation then in the numerical approximation he drops all the states of H N except for the first 100 to 300 eigenstates before introducing H N + + . The essence of the calculation is to show that a spin-compensated state is formed in the original model. This can be done by proving that the impurity
G. GRUNER AND A. ZAWADOWSKI
61 8
[CH. 8, $ 4
spin is infinitely strongly coupled to the single electron at site n = 0 to form a spin singlet. The only effect of the presence of the singlet on the other electrons is that a weak interaction is induced between electrons on the first neighbouring sites next to site n = 0. The computer calculation is designed as follows. At the beginning it is assumed that the exchange is very weak, thus the electron gas on the hopping chain is almost free. By carrying out many steps of the renormalization group procedure the spin compensated state is found, where the singlet is formed at the end of the chain, and thus the site n = 0 can be dropped, because an infinitely large energy is necessary to polarize this singlet. What remains is the weakly interacting chain but starting with site n = 1. Thus it must be shown that the treatment of the interaction must actually result in losing site n = 0. In order to signal this transition one should look for the energy spectrum of the lowest-lying states without interaction. The situation is somewhat different depending on whether the number of sites is even or odd. The energy levels of the noninteracting system in rescaled energy units for even and odd N values are Wad,
0.6555
1.976
4
8
1.297
2.827
442.
Let us assume that we start with a weak coupling problem with N even, N can then be increased step by step all the time by one. In every second step an energy spectrum is recovered which is similar to the original one until at around N = 20 a systematic strong change occurs and in the further steps the spectrum tends to converge into a spectrum characteristic of odd N. The change of the first excitation energy is shown in fig. 12. Thus, one electron tied together with the impurity spin is gradually decoupled from the chain, and the infinitely strong binding energy indicates an effective new exchange coupling Feff = 00. The success of Wilson’s theory is that it can be applied to calculate the susceptibility and heat capacity. If the new excitation spectrum is known these quantities can be evaluated by computer. It should be mentioned that the low temperature calculations can be simplified if the excitation spectrum obtained by computer is fitted to the spectrum of a model problem in which a weak pseudopotential with two adjustable parameters is acting on sites n = 1 and 2. By the appropriate choice of these parameters a fairly large
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
619
number of excitation energies are described with excellent accuracy. We restrict ourselves only to the final results and for mathematical details we refer to the original paper (Wilson 1975).
I
06
+ 0.11
-t
+t
+ t
++
+
+
N
I
22
18
U
24
*
28
Fig. 12. The behaviour of the first excitation energy in Wilson's solution.
O
b f 1
2
3
4
Pig. 13. Temperature dependence of the magnetic susceptibility of the s-d model (Wilson 1973).
The susceptibility multiplied by temperature Tximpis a universal function of TIT, and does not depend directly on the bare exchange coupling as was suggested by Anderson and his co-workers (Yuval and Anderson 1970; Anderson et al. 1970). The impurity susceptibility is shown in fig. 13. The Kondo temperature TK has been determined numerically and the best analytical fit for its dependence on the bare coupling was obtained by the
G. G R m E R A N D A. ZAWADOWSKI
620
[CH.8, 8 4
following mathematical expression TK
& F ~ ~ O ) ( ~ ~ F I ~P O X ){P -” ~1/2191~0+ 13312F~01 +O(S2p;)} (4.4)
where E(.Fpo) is a regular function of J representing the cut-off. The nonanalytical part of this formula agrees with the form obtained from the analytical renormalization group calculation [see eq. (3.18)] ; however the exponent contains some polynomial which can be neglected in the very weak coupling limit. The susceptibility for T < T K approaches a constant,
thus the ground state is a singlet. For T > TK,a Curie-Weiss behaviour is found which can be very accurately fitted by the formula
Finally, the ratio of the coefficient of the linear term in the specific heat and of the susceptibility can be very well fitted by
This formula can also be derived by analytical methods and will be discussed in the next subsections. The calculations have been extended by Krishnamurthy et al. (1975) to the symmetric Anderson model with one localized impurity orbital. The result for the parameter value 8A/nU = 0.064 is shown in fig. 14 where the dotted line represents the computer result for the s-d model with an adjusted Kondo temperature TK(d, W = (1/12)u[l +f(p0P,ff)I(2po~ect)”~ exp{l/2po@ef,+ ..-1, (4.8) N
where 2po.Feff = -8A/nU andf@,.@’,,,) is a smooth function. Thus the expression for the Kondo temperature agrees with the previous result with geff determined on the basis of the Schrieffer-Wolff transformation [see eq. (2.23)]. The numerical results show that the mapping of the two problems on each other is very accurate, exceeding all expectations. It is pointed out that the effective bandwidth &U[l +f(pPCrf)] before the expression (4.6) is
CH.8, 4 41
PROPERTIES OF KONDO ALLOYS
621
proportional to U ; this may be explained by the fact that according to the Schrieffer-Wolff transformation the approximation #eff = const. is correct only in the energy range characterized by U and perf gradually going to zero beyond this range. Calculations have been performed for the nonmagnetic limit aA/U 1 as well, and the susceptibility obtained can be expressed by
as was to be expected on basis of the analytical calculation (e.g. Menyhhrd 1973; Yamada 1975a).
Fig. 14. Temperature dependence of the magnetic susceptibility of the Anderson model (Krishnamurthy et al. 1975).
For more details we refer to Wilson’s original work (Wilson 1975) and to a brief discussion by Nozitres (1975). 4.1.2. Noziires’ Fermi liquid theory Wilson’s numerical solution (Wilson 1973, 1975) proved that a singlet ground state is formed at T = 0. The polarization of the conduction electrons is described by the ground state of the hopping model. This state has a very complicated structure in terms of the unperturbed states because in each diagonalization more and more electron-hole pairs are mixed in. Using the fact that the spectrum of the excited states is very like the free electron spectrum, Nozibres has proposed a Fermi liquid picture. The theory is formulated in the framework of the scattering theory and uses phase shifts.
[CH. 8, 9 4
G. G R m R A N D A . ZAWADOWSKI
622
The essence of Wilson's numerical solution is built in the theory in the form of the following assumptions: (i) The elastic scattering is spin-conserving. The spin-flip process breaks up the ground state and as in this spectrum a threshold is expected at an energy around kT,, the amplitude of these processes must show a strong exponential decay as T = 0 is approached. (ii) The elastic scattering can be described by a phase shift q&) fully determined by the occupation numbers of the single electron states, and deviations from the ground state values are given as a function of energy E
(4.10) = 9" +(ao')V(~); this assumption is characteristic of Landau's where $,JE) Fermi liquid theory. The phase shift can be expressed by the magnetization of the conduction electrons nt -nl = rn as
(4.11)
If,(&) = q o + u E + a p n y
where a is defined by the following expansion $(&)
(4.12)
= qo+u&.
In this way the impurity specific heat Cimpand the magnetic susceptibility ximpdue to the impurity can be easily expressed (4.13)
CimplC = UhPOY
and Xtotal
=
x + Ximp
(4.14)
= x[1 + a l v o +24"hl,
where C is the specific heat of the conduction band, and the Pauli term. Thus
x
= 2p0(gp0)2 is
(4.15)
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
623
Furthermore, it is shown that if the Kondo resonance is tied to the Fermi level, a simultaneous shift in E and p (the chemical potential) by the same amount leaves the phase shift invariant, thus a+2p0$' = 0.
(4.16)
A further assumption made on the basis of Wilson's numerical result is that I+Ttl e I$Tll and so $"$"
= 0.
(4.17)
The last two equations combined with eq. (4.15) give (4.18)
in agreement with eq. (4.7). (iii) In order to calculate the electrical resistivity it is assumed that at T = 0 the phase shift ~(0)= 3. The first term of the temperature dependence of the resistivity comes from the energy dependence of the phase shift, thus after taking the thermal average over 6n one gets
(iv) The inelastic scattering near to T = 0 is dominated by the creation of one electron-hole pair, which can be given by a scattering amplitude depending only on the spin variables. Thus, depending on whether the two electrons have parallel or antiparallel spin this amplitude is A,, or A + + respectively. From simple phase space arguments it follows that the inelastic scattering is proportional to T2.Thus the total conductivity is (4.19)
Considering a very weak electron-electron interaction the electron+lectron scattering amplitude has already appeared in terms of the phase shift on Sn, thus
G . GRUNER AND A. ZAWADOWSKI
624
[CH. 8, 8 4
From the previous assumption given by eq. (4.17)it follows that I A, IA , I and thus eq. (4.19)has the simpler form
,
~ imp (T)/~ im= p (10+nZTZaZ ) = 1
+ Tzn4p$(Cimp/C)*,
I&
(4.20)
where in the second form, a is expressed by the coefficient of the specific heat. In order to bring these results into agreement with Wilson's numerical solution one must assume that @' l/kT,. A more precise comparison can be made by correlating the two forms for the susceptibilities given by eqs. (4.5)and (4.14); this gives
-
TKa = 0.324.
(4.21)
We emphasize that the theory is valid for S = 3: generalizations for arbitrary spin have not been performed yet. 4.1.3. Anderson model at T
=
0
In the Anderson model the only interaction involving the s-electrons is the s-d mixing. The effect of s-electrons can therefore be incorporated exactly into the width of the d-electrons, thus only the Coulomb interaction acting between electrons on the d-levels is left for further treatment. Schrieffer and Mattis (1965)showed that the problem of d-electrons is essentially equivalent to a many-body problem of an energy band with band shape similar to a d-level and with Coulomb interaction between the electrons. It differs, however, from band magnetism, as the momentum conservation characteristic to a real band does not enter. The effect of the Coulomb interaction is smooth a t U = 0 and no singular behaviour isexpected. (In the Kondo model the singular behaviour in the exchange interaction can be characterized by the non-analytical expression exp($.Fpo), but in the Anderson model it becomes analytical as exp { - U/nA} if U and J are connected by the Schrieffer-Wolff transformation.) It has been shown using phase space arguments similar to Landau's Fermi liquid theory that the imaginary part of the self-energy correction &(w) due to the Coulomb interaction disappears at the Fermi level. This observation permitted Langreth (1966) to conclude that at the Fermi level the scattering of the conduction electrons on the d-level is elastic, thus at w = 0 the scattering amplitude can be exactly expressed by a phase shift given as
(4.22)
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
625
This connection gradually breaks down as 101 increases because ImC does not remain negligible. Furthermore, Anderson and McMillan (1967) and Langer and Ambegaokar (1961) showed that the Friedel sum rule holds exactly at T = 0, thus the charge of the impurity ion can be given by the phase shifts qlms(m = 0) as (4.23)
where 1, m,0 are the orbital and spin quantum numbers. In the case of impurities with an open 3d shell the dominant contribution comes from the 1 = 2 scattering channel, so in the first approximation v ~ , , , , ,for ~ 1# 2 can be neglected. In the absence of a magnetic field a singlet ground state is expected - in agreement with the experimental result discussed previously; thus q l = 2 , m , ashould be independent of rn and 0. Keeping only the dominant 1 = 2 phase shift in the Friedel sum rule and assuming that the impurity is completely screened (charge neutrality) at T = 0, the phase shift at the Fermi level is determined as ql=2(0 =
0) = nN/lO,
(4.24)
so the forward scattering amplitude is fkk(W
= O+iS) = (5/npo)exp {i(nN/lO)}sin nN/lO.
(4.25)
This simple and - in the dominant phase shift approximation - exact result has several important consequences. It leads to the resistivity formula given by eq. (5.1) and to the amplitude of the Friedel charge oscillation in the asymptotic, large distance region. As the scattering amplitude is simply related to the d-electron Green’s function Yd,
For T = 0, the density of states at the Fermi level pd(w = 0) is also determined by the charge neutrality and by singlet behaviour of the ground state. It is obvious that the high temperature density of states must be roughly Hartree-Fock in character, thus in the case of magnetic impurity, it must exhibit two peaks representing the two d-levels. As has been emphasized by Griiner and Zawadowski (1972), this high temperature result is expected to be correct on a large energy scale since strong distortion in the density of
G. GRuNER AND A. ZAWADOWSKI
626
[CH. 8, g 4
states would lead to a larger change in the d-electron energy than the energy associated with the temperature range discussed. Therefore, only one possibility remains for explaining the large d-electron density of states at w = 0, namely, to assume a narrow resonance around the Fermi level whose height is determined by the phase shift at T = 0 and washed out at high temperature. Further discussion of this point is left to $ 5 .
4.1.4. Fermi liquid theory for the Anderson model From the previous discussion it is obvious that a Fermi liquid theory can be applied to the d-electrons even for finite Coulomb interaction. Summation of a certain class of diagrams is, however, not sufficient (Tche and Zawadowski 1972; MenyhCrd 1972, 1973). Recently, Yosida and Yamada have published a series of papers (Yosida and Yamada 1970, 1975; Yamada 1975a, b) in which a perturbation expansion has been given for the symmetric nondegenerate Anderson model in a closed mathematical form with the aid of determinants built up from the unperturbed Green’s function. The success of this method is due to the fact that it is not restricted to summation of a certain class of diagrams, and therefore this method is capable of determining general relations between different quantities. The main point is that the coefficients of the power expansion of different quantities such as free energy, susceptibility, self-energy are given in terms of determinants like
D”(1, ..., n) =
0
B12
9 2 ,
0
... ... ... YnI
%13
...
... ...
91,
(4.27)
9112
...
0
where Q,, is the unperturbed Green’s function $i,
=
- (T,{adu(ti>adu(z,)}),
(4.28)
with w, = (n/b)(2n+ 1). Similar determinants have already been used by Yuval and Anderson (1970). As was first noticed by MenyhCrd (1973) not only the free energy but also the even and odd parts of the static impurity susceptibility can be expressed by these determinants
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
627
(4.29)
x [D2"+2(1,2, . . . I 2n+2)D2"(ly2,
... 2n)]con,
and
x [Dz"+z(l,2, ..., 2n+2)02"+2(2,3,
..., 2n+3)],,n,
(4.30)
and the subscript 'con' means that only those products which correspond to connected diagrams should be retained. Yosida and Yamada could express all the different physical quantities by fey,. and i d d . Comparison of the free energy and susceptibility at low temperature shows that the heat capacity Cimp= yT for low temperature is determined completely by Ti,, as
Thus from eqs. (4.29) and (4.31) it follows that
In the large U/nA limit, where the s-d Hamiltonian can be applied, j&, = jiodd (see Yamada 1975a), and so
This result is different from that obtained in the U = 0 limit where jiodd = 0, thus the interaction produces a universal enhancement by a factor two. This result was also derived by Anderson and Armytage (Armytage 1974) by scaling the problem to an exactly soluble limit (Toulouse limit, see also Rivier 1976). The first manifestation of this result, was, however, given by Wilson's numerical solution, which was followed by Nozi6res' Fermi liquid theory. The one-particle d-electron Green's function is given as (4.34)
G. GRUNER AND A. ZAWADOWSKI
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[CH.8, 8 4
with
c(0)=
-W(jeven-
1)-i~d~~dd[(XZ~Z/d2)+(WZ/Az)].
(4.35)
d
Thus the self-energy shows the character of Fermi liquid theory and I m x d contains the contribution of the creation of an electron-hole pair, as well. The resistivity can be also obtained (4.36)
By comparisori of these results with those valid for the s-d model -where the only characteristic energy is the Kondo temperature TK- one finds that xevenand Xodd take over the role of the Kondo temperature for large U, so that kTK
A/x"even
(4.37)
A/l?odd-
Yamada (1975a) following Menyhhrd's work (1973) calculated many quantities up to the fourth order of perturbation theory. The convergence is surprisingly good even, for instance, for U/nA 2. The d-electron density of states obtained in second order of perturbation theory indicates a HF-like d-resonance and the central peak called the Abrikosov-Suhl resonance (see fig. 15). This will be discussed in Q 5. A comparison of these theoretical results with experiments ought to provide important information for the internal structure of the resonance and the validity of strong or weak correlation limits for various alloys. This is, however, not the case, and the reason is that orbital degeneracy may strongly
-
Fig. IS. Density of states obtained from perturbation treatment of the Anderson model (Yamada 1975a).
CH.8, 841
PROPERTIES OF KONDO ALLOYS
629
influence the above results. The Anderson model has been generalized for the orbital degenerate case and the consequences of the model have been worked out by Yoshimori (1976) using a technique based on Ward identities. The basic difficulty, however, is connected with the appropriate form of the generalized Anderson model. To discuss orbital degeneracy, Anderson (1961) suggested a two-level model which was designed for the HF approximation. Therefore, the interaction on the d-level was expressed in terms of occupation number operators. This restricted Hamiltonian has been generalized in several steps (for references, see $2.2 of the review by Griiner and Zawadowski 1974) to reestablish correct symmetry properties, but the matrix elements with four different quantum numbers have been neglected. In general, the Coulomb interaction in second quantized form must be written as
(4.38) where the summation on the quantum numbers is restricted only by the conservation law m +m' = A +A'. Thus, to get correct results for orbital degeneracy, one should start with a Hamiltonian containing all terms, and the matrix elements (mm'l f i ' f i ) must be parametrized taking into account the symmetry of the d-orbitals by using the results known from atomic physics. Such a theory probably contains more than three Fermi liquid parameters and it is expected to give relations between various physical quantities different from those arrived at by using the restricted form of the Anderson model.
4.2. EXPERIMENTAL RESULTS FOR T 4 T K
In spite of intensive efforts, the low temperature transport, magnetic and thermal properties of various alloys have only recently been clarified. Experimental evidence available at present suggests that different alloys ranging from the nonmagnetic limit of the Anderson model to the strongly magnetic limit display basically the same behaviour as a function of the reduced temperature T/T,. At low temperatures the temperature dependences are completely different from that found at temperatures above TKand are also in disagreement with those suggested by theories briefly summarized in $ 3 . Besides the investigation of the temperature dependences of the macroscopic quantities, long-range correlation effects, implied by the non-
G. GRUNER AND A. ZAWADOWSKI
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[CH.8. 8 4
perturbative aspect of the problem, have also been sought but have not yet been explored in full detail. 4.2.1. Temperature dependence of the macroscopic parameters Experimental work on various dilute alloys demonstrates that the logarithmic temperature dependences observed at temperatures T > TK are modified, and at temperatures well below the Kondo temperature analytical dependences on the temperature are observed. Experimental work in this temperature range is particularly difficult due mainly to interaction effects between the impurities. In some cases single impurity effects can be masked by interactions even for concentrations as low as 100 ppm. Simple power law behaviour in the Kondo-region was first observed by Caplin and Rizzuto (1967) in AlMn, - where the impurity resistivity has the form Rimp(T) = R,rnp(O)[I -(T/edzI,
(4.39)
with OR = 530f30 K. The characteristic temperature when interpreted in terms of an energy dependent scattering potential gave a width significantly smaller than the single particle width A. This temperature dependence was related to the specific heat and susceptibility. Subsequent experiments on .-CuFe demonstrated that this is also the case for this alloy (Star 1971) and that non-analytical behaviour, in apparent agreement with early theories, is due to impurity interaction effects. This is demonstrated in fig. 16, where Rimpis plotted for low and high impurity concentrations (Star 1969). The T 2 temperature dependence, found also in other systems, suggests a Fermi liquid behaviour, with characteristic temperatures close to TK values given in fig. 9. This conclusion is reinforced by the behaviour of the thermoelectric power which is a linear function of temperature in the low temperature region. The specific heat can also be interpreted in terms of Fermi liquid theory: in alloys ranging from CuCr to AlCr cimpis proportional to the temperature. The magnetic susceptibility which exhibits a Curie-Weiss behaviour at temperature T > T, saturates at low temperatures and the leading temperature dependence is given by ~imp(T) Ximp(0)tl -(T/81>21 =I
(4.40)
for CuFe, with 8, similar to that found for the resistivity. The change-over
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
631
from the Curie-Weiss behaviour is particularly important as this parameter has also been evaluated theoretically. The temperature dependence of the impurity susceptibility (Alloul 1976) is shown in fig. 17, the insert demonstrating the low temperature behaviour given by eq. (4.40). The data were obtained by Mossbauer and NMR techniques, and they are in broad agreement with susceptibilitydata obtained by macroscopic experiments (Tholence and Tournier 1970), but allow a more accurate evaluation of the temperature dependence. The significance of the agreement between macroscopic and local experiments will be discussed later.
0.52 6
0.524
0.522
Fig. 16. Temperature dependence of the impurity resistivity in G F e alloys of various impurity concentrations (Star 1969). The logarithmic expression is derived on the basis of theory described in 9 3.
The various macroscopic parameters are thus determined by simple power dependences on the temperature. This has been used by experimentalists to argue that, as a consequence of the Kondo effect, a simple, Lorentzian resonance appears at the Fermi level. The various thermal, magnetic and transport properties follow immediately from such an assumption, the prediction of which agrees with those made by the HF approximation to the Anderson model in the U + 0 limit.
[CH. 8, fi 4
G. GRUNER AND A. ZAWADOWSKI
632
Recent theories of the s-d model and the Anderson model in the strong correlation limit also lead to Fermi liquid behaviour but with relations between the various quantities somewhat different from those given by the simple resonance model.
-u 45
-12
-10
AK K
0
-8
10 LO
30
j:
a -0.2
4
-2
J
0 Fig. 17. Temperature dependence of the impurity susceptibility in G F e . Kd("Fe) refers to the iron impurity hyperfine field shift, AK/K to the temperature dependence of the shift at near neighbours of the impurity (Alloul 1976). The orbital contribution Korb has been subtracted from the susceptibility.
The low temperature experimental data are in accordance with Fermi liquid behaviour suggested both for the weak and strong correlation limit, but strict comparison with theory is not possible at present. This is due to the fact that only the non-degenerate Anderson model was considered, and this is clearly inappropriate for 3d impurities. Assuming that the scattering channels corresponding to different invalues are independent, the various macroscopic parameters are multiplied by
CH.8, 5 41
PROPERTIES OF KONDO ALLOYS
633
21+ 1 ;in the following we use this factor to account for the low temperature experimental results. This does not modify the relation between the various parameters and leads to a relation Ximp/cimp the same as that obtained for the non-degenerate case. It is, however, important when the characteristic energy which determines the low temperature fluctuations is compared with TK, the transition temperature. Including the orbital degeneracy is this way, the assumption of a simple Lorentzian resonance at the Fermi level, having a width A’ leads to
-(T/8R)2] = Rjmp(0)[I - ( R ~ ~ ~ T ~ / ~ (4.41) A ’ ~ ] ,
Rimp(T)= &,,(0)[1 with the relations
Yimp/Ximp =
2
2
2
(kB/h),
YimpeR = 2(21+ 1)kB/‘d3*
(4.42)
These agree with those obtained in the limit U + 0 by Yamada (1975a). In the strong correlation limit U -, 00, in terms of a phenomenological parameter ct introduced by Nozieres (1975) (but for 1 = 0)
Yimp = +&(21+
1)~s
Ximp = (4~b/n)(21+l)a,
R,,p(T) = R,mp(0)[1 -(T/8R)’] = Rimp[l -kZk2T2tlZ]
(4.43)
YimplXimp = Bk2(kb/~3, YirnpeR = f(21+ 1)kB
(4.44)
and
could be expected on the basis of a naive generalization. In table 1 the specific heat, the zero temperature magnetic susceptibility and the coefficient of the T2 resistivity for CuFe, AuV and AlMn alloys are collected, together with the phenomzologizl paramzers tl determined by using eqs. (4.43). There is broad agreement between values determined from different physical quantities, and a similar overall agreement can be achieved using the simple resonance concept with A‘ defined in a phenomenological way. The crucial difference between the two concepts is expected to show up in the relation between the various parameters, shown in the last two columns of table 1. It has been emphasized before, that predictions valid for the orbital non-degenerate case have little relevance to the actual situation, and therefore it is not possible to test their validity. We note only, that yimp81mp is near to that predicted by the strong correlation case [see eq. (4.44)] while
[CH. 8, 9 4
0.GRUNER AND A. ZAWADOWSKI
634
Table 1 Macroscopic parameters in the low temperature region for G F e , &V and g M n . The phenomenological constants a-I were obtained from eq. (4.43) where the subscripts refer to experiments from which a was derived.
CuFe &V YMn
CuFe -
&V AlMn -
-
800 68 44
4.9x
22
43 x 10-4 23 x 10-4
400 630
9 . 6 ~ 5= 48 1 1 0 ~ 5 =550 206 x 5 = 1030
69
0.75
17.6 = 2.0 8.2
250x5 = 1250
1240
0.63
24 E~ = 2.9
1950
1980
0.85
2 1 x 5 = 110
27.7
8 i 8= 3.3
Y , ~ ~ / 0.7 x ~for~ all~ three alloys, which is smaller by about a factor of 3
than that predicted. It must also be emphasized that both models assume that the only effect concerning the various parameters is the appearance of a single, narrow resonance at the Fermi level. This can be realized only for the symmetrical Anderson model N = 5 , since in other cases the single particle resonances are near to E~ and certainly have a strong influence. Both a-' and A' are defined as phenomenological parameters, and both are of the order of the Kondo temperature as determined from the high temperature properties, or the temperature dependences around TK . Beside this broad overall correlation it is of fundamental importance to establish the proper relation between the characteristic energies which determine the high energy T > TK and low energy T 4 TK properties. This relationship can, in principle, be obtained by analyzing the high temperature experimental data in terms of perturbation treatments of the s-d or Anderson model, leading to T K as given by, for example eq. (4.4), and then correlating TK with a as determined before. Another way to examine this relation is to analyse experimental data in the whole temperature region from T > TKto T < TKin the light of available theoretical results.
CH.8, 8 41
PROPERTIES OF KONDO ALLOYS
635
Only the magnetic susceptibility has been calculated over a wide temperature range. xtmp(T)calculated from the s-d model and from the Anderson model in the strongly magnetic limit are in good agreement, and are shown in figs. 13 and 14, the zero temperature value being given by eq. (4.6). For the CuFe alloy, on which the most detailed experimental data are available, 8 = 30 K (Tholence and Tournier 1970) and thus from eq. (4.6) TK = 15 K. Extrapolating the Curie-Weiss relation eq. (4.6) to T = 0 leads to ximp(T+ 0) = 0.O85g2&kTK, which is somewhat smaller than that evaluated from eq. (4.5) in the T = 0 limit. Experimental evidence for this is supplied by detailed susceptibility data on CuFe, presented in fig. 17, which indeed strongly suggest that the Curie-Wzs fit extrapolates to lower values than that actually measured at T near to zero. A computer fit of the experimental data to the calculated temperature dependence has not been performed, but nevertheless the above analysis strongly indicates that Wilson’s numerical calculation represents well the experimental data in the whole temperature region. The transport properties have not been calculated in the crossover region, and therefore no comparison with experiments is possible at present. The evaluation of a is subject to ambiguity due to the orbital degeneracy, and therefore the relation aT,, for which the theory predicts eq. (4.21) cannot be established on experimental grounds. It is evident nevertheless that the Fermi liquid properties predicted by recent theories are well confirmed experimentally, and this is also the case for the transition region as far as the magnetic susceptibility is concerned. The orbital degeneracy appears to be an important factor, the role of which should be clarified. Also the alloys investigated experimentally have less or more than a half-filled d-shell, and this also appears to be important both at low and high temperatures. 4.2.2. Correlationefects in the Kondo state
The appearance of the narrow resonance at the Fermi level and the manybody screening aspect suggest that a coherence length
r=
VFIkBTK,
(4.45)
where vF is the Fermi velocity, has a fundamental importance at T 4 T K . A naive approach to the problem would suggest that the coherence effect appears in all physical quantities like charge perturbation, spin correlation functions or spin polarizations (see 5 3). In that this has not been confirmed
[CH.8, 8 4
G. G R m E R AND A. ZAWADOWSKI
636
by experiments, it supports the view that the Kondo effect is more complicated than that of the appearance of a structureless resonance at cF. The coherence effect for charge perturbation was first calculated by Mezei and Griiner (1972), using a simple Lorentzian resonance for the scattering amplitude. A strong depression of the oscillation amplitude A was found for distances r < t where = v,/2r and r is the width of the resonance. Nuclear quadrupole experiments performed on A1 3d transition metal alloys indeed show a strong depression of the chargeperturbation in the case of AlMn. In fig. 18 recent experimental data on AlSc and AlMn are reproduced. -
<
q(
1
I
1 I
A l Mn 2c
10
1
t 2.8
2
3
4
5
6
7
8 9
t
t
5.6
8.4
shelln’ r(A)
Fig. 18. Experimental values of the electric field gradient around Sc and Mn impurities
in aluminium, The full curve is the envelope of the charge oscillation calculated from the asymptotic expression, the dotted line is calculated by taking into account the preasymptotic behaviour (Berthier and Minier 1973a, b).
CH. 8. $41
PROPERTIES OF KONDO ALLOYS
637
The full line is the asymptotic expression, given by eq. (2.15) with the oscillations neglected. The asymptotic form describes well the experimental points for nonmagnetic Sc impurities; for AlMn there is a clear depression of the charge perturbation near to the impzities. Analysis in terms of the theory leads to r 0.5 eV (see the dotted line in fig. 18). Thus r is definitely smaller than the single particle width A and shows the importance of the many-body resonance in the behaviour of the charge perturbation. r,on the other hand, is larger than the width kT, obtained from the analysis of the macroscopic properties of this alloy (see table 1); this may be due to the internal structure of the resonance, or to the fact that both single particle and many-body effects contribute to the depression of the charge perturbation. This is indeed a possibility for AlMn which is near to the magnetienonmagnetic boundary, and will be discussed later. Similar experiments have not been performed on systems where the difference between kT, and A is larger; in CuFe or in similar alloys this would enable a clear distinction to be made between the effect of the two resonances. In spite of considerable effort, the distance dependence of the spin correlation functions of the type (S+ * a - ( r ) )has not been clarified experimentally. Recent neutron experiments performed on CuFe are subject to ambiguities due to large impurity concentrations used (c> 400 ppm); indeed, impurityimpurity interactions have always played a significant role in this particular system. The spin polarization around impurities has been extensively studied by NMR techniques. The host NMR is sensitive to spin polarization, which gives rise to a broadening of the central line and also to the appearance of distinct satellites due to the hyperfine interaction between spin density and host nuclei. The line width is subject to impurity-impurity interactions, but the satellite positions are independent of impurity concentrations and reflect single impurity behaviour. The temperature dependences of satellite shifts in Cu 3d transition metal alloys were first investigated by Boyce and Slichter (1974) and later by Alloul(l976) in considerable detail. In CuFe a comparison of satellite shifts with the macroscopic susceptibility demonstrates (Alloul 1976) that the perturbation is given by
-
Aa(r, T ) = A(T)B(r),
(4.46)
where A has the same temperature dependence as zimp(T). The form of the perturbation is thus not modified by the Kondo effect. The spin polarization
638
G. GRUNER A N D A. ZAWADOWSKI
[CH. 8, !j 5
has no long-range nonperturbative part, it has the same form as that given by the RKKY result [eq. (2.22)].In the HF analysis the spin perturbation is given by the difference of the charge perturbations for spin-up and spindown conduction electrons to du(r, T ) = dp,(r, T)-dp-,(r, T ) [see eq. (2. I5)] and therefore the behaviour of charge perturbation and spin polarization would be essentially similar. However, spin-flip scattering and the internal structure of the resonance at eF strongly modify this result. The essential point is that the charge perturbation reflects the properties of the bound state, and the spin polarization that part which has been lifted by the external magnetic field. No straightforward relationship is thus expected between the two quantities. This has been borne out by the recent calculation of Ishii (1976) who showed that in the Kondo state (4.47)
where (S') represents the local spin polarization, at distances
D kFrc < - In -. kDT.1 k TK
(4.48)
Thus da(r) is not modified at distances smaller than r compared with the perturbative form. At distances r > r, du(r, T ) = -(1/2n) (S') cos (2kFr)/r3.
-
(4.49)
For CuFe, with D 10eV and TK 20 K, r, lo3 A, a distance well beyond the experimentally accessible range. Thus, long-range nonperturbative spin polarization is neither expected theoretically, nor observed experimentally in dilute alloys in the Kondo state. N
N
5. Phenomenological approachesto the general case Theoretical solutions of the Anderson model and of the s-d model discussed briefly in the previous section have been obtained for certain values of the basic parameters which describe the impurity stdes in a metallic host. In the weak correlation limit U -+ 0 the HF solution is appropriate for an arbitrary hopping matrix element V,, and position of the d-level ed. Correlation effects have only been treated in the symmetrical case, N = 5,
CH.8, 0 51
PROPERTIES OF KONDO ALLOYS
639
and in the limit U -, co : the s-d model is also appropriate only in this limit. The basic concepts, the development of the singlet ground state, and longrange correlation effects, as well as the crossover from the high temperature weak coupling to the low temperature strong coupling limit are well understood. Nevertheless, it is desirable to use these concepts, and the general results obtained in the case of Anderson model, to describe the properties of alloys (which do not fall within the limits where the exact solutions are available), in a semi-empirical way. Different phenomenological approaches are available. They are successful in accounting for certain experimental quantities, including relationships between them and also temperature dependences. They emphasize various aspects of the problem and visualize the many-body effects in different ways. Star (1969) postulated that a narrow Lorentzian resonance (with a width A') develops at eF; the macroscopic, thermal and transport properties can then be derived using standard formulae. Souletie (1972) and others (Loram et al. 1972; Nagasawa 1972), on the other hand, assumed that the magnetio nonmagnetic transition can be described within the framework of the HF solution with U 4 0 as T -P 0. A unified point of view has been presented by Gruner and Zawadowski (1972). This approach incorporates both the many-body and the single particle aspects of the dilute alloy problem. Certain properties at T = 0 can be derived using the exact results valid for arbitrary parameters of the Anderson model, in particular the scattering amplitude, from which the resistivity can be evaluated. Arguments given for the validity of the phase shift approach at cP (4 4.1) lead to Rimp(T= 0) = Ro sin2 (Nn/lO).
(5.1)
It is essential to emphasize that this result, although having the same form as that obtained from the HF solution in the nonmagnetic limit, is generally valid for arbitrary strength of the Coulomb interaction. The T = 0 resistivities, presented in fig. 19, can indeed be described by eq. (5.1) not on$ for aluminium-based, but also for copper- and gold-based alloys, in spite of the double peaked structure shown in fig. 3 having been taken as evidence for the magnetic behaviour in the last two cases. It is also important to realize that the T = 0 resistivities reflect the scattering amplitude at ep, and are independent of the structure of the resonance. Therefore the behaviour displayed in fig. 19 cannot be taken as evidence for a Lorentzian resonance with a single particle width A as suggested by the nonmagnetic HF solution;
G. GRUNER AND A. ZAWADOWSKI
640
Au k
[CH.8, 8 5
t
o Cu host
V
Ti
V
Cr Mn Fe Co NI
Cu Zn
Fig. 19. R,,, versus the 3d atomic number in the T + 0 limit for Cu, Au and Al based alloys (Gruner and Zawadowski 1972).
with t(eF) determined by the charge neutrality, and different assumptions can be made concerning the energy and temperature dependences.
5.1. NARROW RESONANCE LEVEL CONCEPT Star (1969) who first showed experimentally that the temperature dependences of the various macroscopic parameters are determined by simple power laws in CuFe, postulated that a simple resonance width of the order of kT, appears atthe Fermi level. The scattering amplitude has the form t(o)
-
1/(d
+Ar2).
(5.2)
The resonance itself has no temperature dependence, but thermal smearing results in strong temperature dependences a t temperatures kT A'. At low temperatures the Bethe-Sommerfeld expansion gives eqs. (4.41)and (4.42) and analysis of various thermal, magnetic and transport properties leads to the values close to a-' given in table 1. At higher temperatures the calculated behaviour faithfully describes the overall temperature dependences obtained experimentally and shown in figs 6, 7 and 8. Clearly, the Abrikosov-Suhl resonance provides the theoretical background of this phenomenological description which has been proved to be N
CH.8, 8 51
PROPERTIES OF KONDO ALLOYS
641
extremely useful for relating various physical quantities. The simple form of the resonance excludes, however, the possibility of logarithmic behaviour. In fact at high temperatures, the various parameters derived on the basis of eq. (2) are given by the inverse power of temperature. Also, the approximation over-emphasizes the energy dependences, the temperature dependence of t(o) being neglected. 5.1.1. Description in terms ofphaseshijts Several workers have used the phase shift formalism to describe the temperature dependences of various quantities, an approach which has been elaborated in considerable detail by Souletie (1975). The magneticnonmagnetic transition is visualized as a smooth transition from the magnetic HF solution to the nonmagnetic limit. At high temperatures the phase shifts are different for Q and - 0 while at T = 0, q,(eF) = q-,(cF). Different parameters thus can be correlated by evaluating q,(T) and q-,(T) using expressions valid in the H F treatment. The magnetic susceptibility, for example, is given by
while the impurity resistivity is
Both can be described adequately with the same parameters qa(w) and q-,(o). The method has been proved to be extremely successful in accounting
for a broad range of experimental parameters for widely different alloys. The reason for this, however, is not entirely clear, the main objection being perhaps, that HF treatment neglects spin-flip scattering, which is essential in the Kondo problem (Stewart and Gruner 1973). Souletie’s approach also over-emphasizes the temperature dependences. Energy dependences, which are essential in the narrow resonance level concept, are completely ignored.
5.1-2. Phenomenologicalpicture of resonanceformation Each of the two alternative approaches discussed earlier proposes a widely different picture of the resonance formation in dilute alloys. The narrow resonance level concept relies heavily on the Abrikosov-Suhl resonance which arises as a consequence of many-body effects; the phase shift approxi-
642
G. GRUNER A N D A. ZAWADOWSKI
[CH. 8, Q 5
mation, on the other hand, is based on the Friedel-Anderson single particle description. Both approaches have been designed to account for the temperature dependences of various macroscopic quantities, and do not aim to discuss long-range correlation effects. Long-range correlation effects, are, however, fundamental in the low temperature limit and are closely related to resonance formation. It is not surprising therefore, that the two models predict entirely different behaviour. The coherence length for long-range correlation effects is given by g = v,/2A' with A' kT, in the narrow resonance level model, while Souletie's approach gives ( = vF/2A.For alloys with kT, < A, this difference is substantial. Correlation effects were discussed in $4.2.2 where experiments related to the charge perturbation around the impurities were summarized. The depression of the charge perturbation in AlMn was shown to be compatible with an overall energy dependence havinga width r 0.5 eV. The significance of this result is clear. r is larger than the width of the narrow, manybody resonance, which in the case of AlMn is about 0.2 eV (see table 1) but is significantly smaller than the singleparticle width A which is expected to be larger than 1 eV. The reason for this discrepancy is that various experiments are sensitive to different parts of the scattering amplitude, or of the density of d-states. Macroscopic experiments sample the energy (and temperature) dependence within the energy region kT around E ~ as, opposed to optical experiments which are sensitive to states far from the Fermi level. The charge perturbation is sensitive to the behaviour of the scattering amplitude averaged over a broad energy range. The value of r determined from the behaviour of the charge perturbation then suggests that the density of states has a sharp top at E~ and broad tails: the former determining the low temperature macroscopic properties. Correlating this finding with the theoretical results obtained, it is evident that the sharp top of the density of states can be associated with the many-body resonance developing at the Fermi level at temperatures below T K .The broad tails on the other hand correspond to the single particle resonances, not affected by the low energy processes associated with the Kondo effect. The positions of the single particle resonances are determined by U,A and by N,and the HF approximation of the Anderson model accounts we 11 for these resonances depicted in fig. 2 for AgMn and CuNi. The narrow central peak which develops at E~ has a m a p x d e determxed by the charge neutrality, and has a width of the order of kT,. It has a Lorentzian top and logarithmic tails, as evidenced by the behaviour of the macroscopic properties and by theories summarized in $4.1.
-
N
CH.8, 8 51
PROPERTIES OF KONDO ALLOYS
643
In the strongly magnetic limit U/A9 1 the single particle peaks are at energy U apart, the central peak is narrow and well resolved. The situation appropriate for CuFe is shown in fig. 20. Here A and kTK differ by more than an order ofmagnitude. With a decreasing U ratio the broad FriedelAnderson resonances move towards E~ and the central peak broadens rapidly. For AlMn, U/nA 2 1 and indeed kTK is only an order of magnitude smaller thand. Finally, for U + 0 the HF picture for a truly nonmagnetic impurity, i.e. one Lorentzian resonance with a width A, is recovered. The
A4 M Y (Abrikawv-SuH) mMy
CF
5 CV kTfWl CV N.6
w
Fig. 20. Schematic density of d-states a t T Q T,, appropriate for 9 F e . The width of the single particle resonances is A and that of the central peak kT,. The shape of the central peak at high and at low energies is also indicated (GrUner and Zawadowski 1972).
relation between the width of the many-body peak and the single particle resonances is given by the expression for TK in terms of 9 (eq. 4.4) with 9 given by the Schrieffer-Wolff transformation. Various macroscopic properties, early optical experiments on strongly magnetic alloys, together with results on local properties, are in accordance with this description. Recent optical experiments also support this point of view. Optical properties of AlMn (Beaglehole and Will 1972) have been interpreted in terms of this mTdel (Griiner 1974); in CuNi and AuNi alloys careful experiments (Bassett and Beaglehole 1976; Kaiser and GTkrt 1976) suggest that serious deviations from the HF nonmagnetic picture occur
644
G. GRUNER AND A. ZAWADOWSKI
[CH. 8 , § 5
with the density of states having a sharp top and broad tails indicating that Up,,(.$ is not much smaller than one in these cases. With increasing temperature the bound state is progressively destroyed, the narrow central peak decreases and is smeared out by the thermal fluctuations; above T, only the single particle resonances appear and the impurity looks like a genuine magnetic moment. Room temperature impurity resistivities in copper and gold, shown in fig. 3, are characteristic of this situation. It is important to emphasize that the s-d model focuses on the central peak, the main features of which are well reproduced by calculations. The broad resonances appear only as a temperature independent potential scattering background through the Schrieffer-Wolff transformation. The density of states as depicted in fig. 20 is, moreover, in good overall agreement with the results of perturbation calculations valid for the symmetric Anderson model: the central peak and single particle tails are well resolved for strong Coulomb correlations (see fig. 15). Also, the dominant pole approximation of the model (Zlatic et al. 1974) leads to a density of states rather similar to that proposed here on experimental grounds. These approximations cannot however account for the fine details ofthe many-body resonance, in particular for the logarithmic temperature dependences above TK. The central peak aspect of the Kondo problem thus appears to be well confirmed, and a broad range of experimental information can be interpreted in terms of this picture. It is also in accordance with theoretical results, in certain limits of the Anderson model for which the calculations were performed and reliable theoretical solutions are available. 6. Conclusions
Recent experimental and theoretical developments of dilute alloys of 3d elements with simple metals have been reviewed with particular emphasis on the low temperature properties. It is well established by various types of experiments that a model, proposed by Anderson, in which the Coulomb correlations between the delectrons and the mixing of d- and s-host states is of importance, is capable of accounting for most of the experimental findings. Due to the low dimensionality of the problem, correlation effects are of crucial importance. These have mainly been discussed in the strongly magnetic limit where Coulomb effects are important. The Anderson model reduces to the so-called s-d model in this limit, where the well-defined spin of the impurity is weakly coupled to the host states.
CH. 81
PROPERTIES OF KONDO
ALLOYS
645
The gradual transition from a high temperature state -where the impurity looks magnetic-to a low temperature nonmagnetic state, is a central feature of dilute alloys. The high temperature properties can be adequately described by perturbational treatments while at low temperatures Fermi liquid theory is appropriate. The crossover between the two limits can be accounted for numerically by using Wilson’s renormalization group method. It seems reasonable to say that the basic aspects of the dilute alloy problem are solved but further efforts are needed to arrive at a complete agreement between theory and experiment. In particular the effect of orbital degeneracy has not been completely cleared up. This is also the case when the impurity has less or more than a half-filled d-shell. Only a phenomenological description can be given for these situations; even so this description incorporates both the single particle and many-body aspects of the problem. Although the Kondo effect itself is a phenomenon restricted to a limited class of solids, ideas developed in this field have been proved to be extremely useful in discussions concerning other phenomena. The relation of the Kondo problem to X-ray absorption in metals and to superconductivityhas already been mentioned. Recent developments in chemisorption, surface physics and in the one-dimensional metal problem are strongly based on approaches applied first to dilute alloys. The understanding of the many-body aspects of the dilute alloy problem will certainly influence further developments in the field of magnetism in metals and also in other fields of solid state physics.
Note added in proof Berman and So (1978, Phys. Rev. Lett. 40,53) performed careful tunnelling experiments on junctions doped with magnetic impurities, and obtained the scattering amplitude as a function of energy in a wide energy range. The experiments confirm the behaviour of the scattering amplitude discussed in $4 and 5. The Fermi liquid theory was generalized recently for degenerate orbitals by Blandin and Nozieres (to be published) and by Mihhly and Zawadowski (to be published). Yoshimori’s results were recovered only in the s-d limit for S = 5/2, the general results depend on five nonuniversal Fermi liquid parameters.
References Abrikosov, A.A., 1965, Physics2,5. Abrikosov, A.A. and A.A. Migdal, 1970, J. Low Temp. Phys.3,519. Alloul, H., 1975, Phys. Rev. Lett. 35,460. Alloul, H., 1977, Physica B86,449. K
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G. GRUNER AND A. ZAWADOWSKI
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Anderson, P.W., 1961,Phys. Rev. 124,41. Anderson, P.W., 1967,Phys. Rev. 164,352. Anderson, P.W., 1970,J. Phys.C: SolidSt. Phys.3,2346. Anderson, P.W., 1972,Comments Solid St. Phys. 5,72. Anderson, P.W. and W.L. McMillan, 1967,in: Marshal, W., ed., Theory of Magnetism in Transition Metals. International School of Physics Course XXXVII, Varenna, 1966 (Academic Press, New York, London) p. 62. Anderson, P.W. and G. Yuval, 1971,J. Phys. C:Solid St. Phys. 4,607. Anderson, P.W., G. Yuval and D.R. Hamann, 197Oa,Phys. Rev. B1,4664. Anderson, P.W., G. Yuval and D.R. Hamann, 1970b.Solid St. Commun. 8,1033. Babic, E. and G. Griiner, 1976,Physica 84B,31. Bassett, A. and D. Beaglehole, 1976,J. Phys. F 6,1211. Beaglehole,D. and H. Will. 1972,J. Phys. F 2,43. Bell, A.E. and A.D. Caplin, 1975,Contemp. Phys. 16,375. van den Berg, G.J., 1964,in: Progress in Low TemperaturePhysics, Vol. IV, ed. C.J. Goner, (North-Holland, Amsterdam) p. 194. Berthier, C. and M.Minier, 1973,J. Phys. F: Metal Phys. 3,1169. Berthier, C. and M. Minier, 1973,J. Phys. F: Metal Phys. 3,1268. Blandin, A., 1967,J. Appl. Phys. 39,1285. Blandin, A., 1973,in: Magnetism, Vol. V, ed. E. Suhl (Academic Press, New York) p. 58. Bloomfield, P.E. and D.R. Hamann, 1967,Phys. Rev. 164,856. Brenig, W. and W. G6m, 1968,Z.Phys. 217,188. Caplin, A.D. and C. Rizzuto, 1968,Phys. Rev. Lett. 21,746. Cooper J.R. and M. Miljak, 1976,J. Phys. F 6,ll. van Dam, J.E and G.J. van den Berg, 1970,Phys. St. Solidi 3,11, van Dam, J.E., P.C.M. Gubbensand G.J. van den Berg, 1972.Physica 62,389. Drcw, H.D. andR.E.Doezema, 1972,Phys. Rev. Lett. 28,1581. Fowler, M., 1972,Phys. Rev. B6,3422. Fowler, M. and A. Zawadowski, 1971,Solid St. Commun. 9,471. Friedel, J., 1956,Can. J. Phys. 34,1190. Gruner, G., 1972,Solid St. Commun. 10,1039. Griiner, G., 1974,Adv. Phys. 23,941. Graner, G. and A. Zawadowski, 1972,Solid St. Commun. 11,663. Griiner, G. and A. Zawadowski, 1974,Rep. Progr. Phys. 37,1497. Heeger, A.J., 1969, in: Solid State Physics Vol. 23, eds. F. Seitz, D. Turnbull and H. Ehrenreich(Academic Press, New York) p .283. Iche, G. and A. Zawadowsk, 1972,So lid St. Commun. 10,1001. Ishii, 1976,Progr.Theor. Phys. 55,1373. Kaiser, A.B. and P.N. Gilbard, 1976,J. Phys. F 6,2209. Kasuya, T., 1956,Progr. Theor. Phys. 16,45. Kedves, F.J., M. Hord6s and L. Gergely, 1972,Solid St. Commun. 11,1067. Kondo, J., 1964,Progr. Theor. Phys., Japan 32,37. Kondo, J., 1969, in: Solid State Physics Vol. 23, eds. F. Seitz, D. Turnbull and H. Ehrenreich(Academic Press, New York) p. 183. Krishnamurthy, H.R.,K.G. Wilsonand J.W. Wilkins, 1975,Phys. Rev. Lett. 35,1101. Kr06, N.and Z.Szentirmay,1972.Phys. Lett. MA, 173. Langer, J.S. and V. Ambegaokar, 1961,Phys. Rev. 121,1090. Langreith,D., 1966,Phys. Rev. 150,516. Loram. J.W., T.E. M a l l and P.J. Ford, 1970,Phys. Rev. B2,857. Loram, J.W., R.J. WhiteandA.D.C. Grassie, 1972,Phys-Rev. BS,3659.
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Menyhkd, N., 1972,Solid St. Commun. 11,423. Menyhhrd, N., 1973,Solid St. Commun. 12,215. Mezei, F. and G. Gruner, 1972,Phys. Rev. Lett. 29,1465. Mizuno, K., 1971,J. Phys. SOC.Japan 30,742. Muller-Hartmann, E., 1969,Z. Phys. 223,277. Myers, H.P., L. Wallden and A. Karlsson, 1968,Phil. Mag. 18,725. Nagaoka, Y., 1956,Phys. Rev. 138,A 111 2. Nagasawa, H.,1972,Solid State Comm. 10,33. Nozibres, P., 1974,J. Low Temp. Phys. 17,31. Nozibres, P., 1975, in: Proceedings of LT14, Vol. 5, eds. H. Krusius and H. VUOfiO (North-Holland, Amsterdam) p. 339. Nozibres, P.,1976,Low Temp. Phys. Conf. Rivier, N., 1976,Physica 84B,50. Rizzuto, C., 1974,Rep. Progr. Phys.37,147. Rudermann, M. A.and C. Kittel, 1954,Phys. Rev. 96,99. Sakurai, A. and A. Yoshimori, 1973,Progr.Theor. Phys., Japan49.1840. Schotte, K.D. and U. Schotte, 1971,Phys, Rev. B4,2228. Schrieffer, J.R. and D. Mattis, 1965,Phys. Rev. 140,1412. Schrieffer,J.R. and P.A. WOW,1966,Phys. Rev. 149,491. Sblyom, J. and A. Zawadowski, 1974,J. Phys. F:Metal Phys. 4,80. Souletie, J., 1972,J. Low Temp. Phys. 10,143. Star, W.M., 1971,Ph.D. Thesis, University of Leiden. Suhl, H., 1965,Physics 2,39. Tholence, J.L. and R. Tournier, 1970,Phys. Rev. Lett. 28,867. Triplett, B.B. and N.E. Philipps 1971,Phys. Rev. Lett. 27,1001. Wilson, K.G., 1975,Rev. Mod. Phys. 47,773. Wilson, K.G., 1973, in: Proceedings of the Twenty-Fourth Nobel Symposium, eds. Bengt Lundquist and Stig Lundquist, June 12-16, 1973 at Aspenosgarden, Lerum. Sweden (AcademicPress, New York and Amsterdam). Yamada, K., 1975a.Progr. Theor. Phys. 53,970. Yamada, K., 1975b,Progr. Theor. Phys. 54,316. Yoshida, K., 1957,Phys. Rev. 106,893. Yoshida, K. and K. Yamada, 1970,Progr. Theor. Phys. Suppl. No, 46,244. Yoshida, K. and A. Yoshimori, 1973, in: Magnetism, Vol. V, ed. H. Suhl (Academic Press, New York) p. 253. Yoshida, K. and K. Yamada, 1975,Progr. Theor. Phys. 53,1286. Yoshimori, A., 1976,Progr. Theor. Phys. 55,67. Yuval, G. and P.W. Anderson, 1970,Phys. Rev. B1,1522. Zawadowski, A. and G. Gruner, 1974,J. Phys. F:Metal Phys. 4,L202. Zittartz, J. and E. Muller-Hartmann, 1968,Z. Phys. 212,380. Zlatic, V., Gruner, G. and N. Rivier, 1974,Solid St. Commun. 14,639.
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CHAPTER 9
APPLICATION OF LOW TEMPERATURE NUCLEAR ORIENTATION TO METALS WITH MAGNETIC IMPURITIES BY
J. FLOUQUET Laboratoire de Physique des Solides, Orsay, France
Progress in Low Temperature Physics, Volume VIIB Edited by D.F. Brewer 0North-Holland Publishing Company, 1978
Contents Introduction 651 1. Comments on nuclear orientation 652 1.1. Principles of NO 652 1.2. Application to solid-state physics 657 1.3. Experimental conditions 664 2. Magnetism of an impurity: Kondo effect 669 2.1. Magnetism of an impurity in a nonmagnetic host 669 2.2. The Kondo effect 675 2.3. Influence of the hyperfine coupling 680 2.4. NO in the single impurity study 682 2.5. TKscale. Experimental possibilities of the different hyperfine methods 688 3. The hyperfine field. The origin of magnetism 689 3.1. Atomic structure and hyperfine fields 690 3.2. 3d hyperline fields 691 3.3. 4f hyperfine fields 698 4. Single impurity effects 698 4.1. Weak Kondo coupling 699 4.2. Strong Kondo coupling case 706 4.3. Intermediate coupling 709 4.4. Kondo coupling. Degeneracy of the ground state 710 5. Interaction effects 713 5.1. Magnetism of interaction impurities 713 5.2. Sensitivity of NO to interaction effects 719 5.3. Spin glass behaviour in a NO experiment 720 5.4. Appearance of magnetism due to interaction effects 726 5.5. Kondo lattice or magnetic ordering in cerium compounds 731 6. Other applications 731 6.1. Use of NO in low temperature physics 731 6.2. In nuclear physics 733 7. Conclusion 734 Appendix 737 ReEerences 741
Introduction
The aim of this article is to describe the study of dilute alloys by Nuclear Orientation (NO). Such investigations have experienced a burst of activity in recent years, particularly with the elucidation of phenomena related to the Kondo effect. Parallel to the Kondo problem, considerable attention has been given to: (i) the description of the impurity magnetism itself, e.g. the question of the existence of orbital magnetism in 3d materials, the role of crystalline field effects in rare earths; (ii) the coupling among the impurities when the impurity concentration increases; (iii) the link between magnetic impurities and decrease of the superconductivity. With NO experiments, one can study at almost OK: (i) the appearance and the behaviour of magnetism for an isolated impurity as a function of an applied field; (ii) the symmetry of the magnetic interaction when spin glass behaviour occurs; and (iii) the impurity magnetization even in the case of a strong magnetic response of a superconductinglattice. This article is written in a simple form in order to show: (i) firstly for the solid state physicist the possibilities and the limitation of NO methods in the study of condensed matter ($ 1); (ii) for the nuclear physicist the fundamental interest of the study of dilute alloys ($2). Section 3 describes how hyperfine measurements such as NO are direct tests of the magnetism origin. In $4, the discussion is concerned with single impurity effects, mainly the problem of the Kondo effect, while in $ 5 the different coupling among magnetic impurities is discussed; $ 6 gives examples of practical applications in low temperature and nuclear physics. Finally, my main aspiration has been to convince the reader that the NO method is a very nice and simple tool to measure the magnetic behaviour of an impurity and that the relative weakness of some results has mostly been due to the experimentalists. More precise information can certainly be extracted especially when NO experiments are combined with other techniques. It should be pointed out that the given examples are chosen for simplicity and to provide illustrations of well-defined magnetic behaviour. Most of the examples are chosen from work performed in Orsay or in Grenoble since
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I have had the opportunity to look in more detail at the experimental results. More general references are given in the various tables of results.
1. Comments on nuclear orientation The measurement of the anisotropy of y rays or particles emitted by radioactive nuclei of spin 1 touches on several different domains in physics: nuclear physics (determination of nuclear dipole and quadrupole moments, experiments on parity conservation, mixing between different transition multipolarities), atomic physics (existence of hyperfine coupling), solid state physics. It will be explained in the following why a well-defined quantum state of spin I with a 2 angular momentum 1, = M emits radiation with an angular distribution which is characteristic of this state and what are the mechanisms responsible for the selection of a pure quantum state. In order to give some practical support to this section, I shall describe briefly how an actual experiment is performed. The reader who wishes a more precise description should refer to the general review of Blin-Stoyle and Grace (1957), De Groot et al. (1965) or Shirley (1966) on NO and the book of Freeman and Frankel on hyperfine interactions (1967).
1.1.
PRINCIPLES OF NO
I .I .I. SimpIe considerations on nuclear orientation Nuclear orientation is the ordering of nuclear spins in space. For example such ordering may occur for a nuclear spin 1 subjected along the Z axis of quantization to an applied field H if the population of the state 1 I, 1, = M) is higher than that of the other 21 sublevels. Such ordering can be observed by various techniques like magnetization, NMR, polarized neutron scattering or by detection of the angular distribution of particules or y rays emitted by the nuclei. The nuclear ordering can be obtained dynamically or statically. The method called nuclear orientation which will be described here, is restricted to static nuclear ordering detected by the angular distribution of particles or y rays emitted by radioactive nuclei during their decay. The population of the 21+1 nuclear sublevels is governed by the Boltzmann factor aM = exp(-E,/k,T) which contains the ratio of the energy E M of
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the state 11, M ) and the thermal energy k,T. To obtain an appreciable nuclear ordering, the thermal energy must be comparable to or smaller than the splitting energy E,-E,-, between two nuclear sublevels. In order to give an order of magnitude, we note that for nuclei with dipole moments of one nuclear magneton in an applied field of 100 kOe, the Zeeman energy is equal to the thermal energy at 3 mK. In solids, as we will see later, such ordering may occur easily via the hyperfine coupling which produces an effective field Herron the nuclei of up to 1 MOe. The main point of the NO methods is that the angular distribution of the emitted a, 8, y rays is characteristic of the initial state. To illustrate this point, we take the simple case of a spin Z = 1 which decays to a ground state Z = 0 by a dipolar y transition (fig. 1). The angular distributions W r of the y rays emitted from different sublevels II, M) in a direction Z' which make an angle 0 with the Z axis of quantization are: W: =
w;~ = +a2[l+cos2e],
w: = u2 sin28.
At high temperatures, when all the sublevels are equally populated, the observed angular distribution, which is the sum of each component, is isotropic. In the opposite situation, when only one state is populated, the angular distribution is strongly anisotropic. For example a detector along the Z axis records no counts if only the I1,O) state is populated. The solid state physicist will explain why the 11,O) state is the lower level and what is the origin of the splitting between the different sublevels. As the transition from the different sublevels 11,M) cannot be observed directly due to the poor resolution of the detector, the observation of a pure IZ, M) state is obtained by lowering the temperature. It is worth pointing out at this stage the relation of NO to other nuclear techniques. In an angular correlation experiment the selection of a pure IZ, M) state is realized by the condition that a photon emitted from the intermediate IZ, M) nuclei must be in coincidence with the photon which has selected this intermediate state. In a Mossbauer experiment, the (Z,M) selection of the source nuclei (or absorber) is achieved by the Doppler shift of the absorber (or source). Finally we must remark that the equality W i = W;' in y ray measurements is due to parity conservation in a y transition. As a state 11, 1) cannot be resolved here from a state 11, - l), the sign of the effective hyperfine field cannot be determined but only its magnitude. More generally a y
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anisotropy measurement does not detect the first moment of the orientation of the nuclei B, which is proportional to the average value of M y
If 80
n
E’
FP ( 0 )
bu a dr up0 lor intcr ac t ion :1
72’ ( 8 )
0
Mognctic intkoetlm
+I
-Q -1
-
-
Fig. 1. Electric dipole radiation. The main point is that the photon has only angular momentum 1 or - 1 along its direction of motion.
+
but rather the alignment which is an even function of the M quantum number. The first even moment B, is proportional to the thermal average defined by : B2
N
(M2-+Z(Z+ 1)).
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On the other hand, /3 transitions which violate parity conservation are sensitive to the B, parameter. If the circular polarization is detected, the sign of the effective hypefine field can be deduced. As most NO experiments are performed by y anisotropy measurements we restrict now the following discussion to the observation of y transitions. 1.1.2. Mechanism of Orientation
In insulators, the different mechanism of nuclear orientation are well described by the following Hamiltonian (Abragam and Pryce 1951)
where Hx, Hy, H, represent the components of the applied field, Z and S are the nuclear and electronic spin of the observed atoms, A and B are the magnetic hyperfine coupling between Zand S, D is a possible anisotropy of S due to crystal field effects and P is the quadrupolar anisotropy of I due to the coupling between the nuclear quadrupole moment and the electric field gradient. The use of the first term leads to brute force polarization of the nuclei. To achieve appreciable orientation, large magnetic fields and very low temperatures must be employed, Such experiments can be useful principally for nuclear physics. By the combined effects of the Zeeman electronic term and the hyperfine magnetic coupling (A, B), nuclear orientation can be achieved with relative low applied fields since the electronic moment is polarized with a small applied field and the corresponding hypefine field ASz/gnpncan reach 1 MOe. This mechanism, proposed by Oorter (1948) and Rose (1949), describes the case of a free paramagnetic impurity. When a strong anisotropy exists (DS;),its conjunction with the hyperfine magnetic cciupling gives appreciable NO even in zero field (Bleaney 1951). This mechanism has been used extensively in the study of paramagnetic salts, the radioactive ions being either a constituent of the salt or a substituted ion. Pound (1949) suggested that the quadrupole term P leads to nuclear alignment even in zero applied field. Generally, the quadrupolar nuclear splitting is lower than the two previous Gorter, Rose and Bleaney cases.
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The Abragam and Price Hamiltonian neglects completely the effect of conduction electrons. Directly applied to metals, it leads for example to the conclusion that no hypertine coupling occurs for a diamagnetic impurity aside from the brute force term. In 1959, Samoilov et al. showed by NO that a strong orientation can be reached for diamagnetic impurities dissolved in ferromagnetic lattices. The essential role of conduction electrons was explained later by Daniel and Friedel (1963). With the case ofthe Kondo effect, we shall see later another example of the major influence of the conduction electrons. 1.1.3. Generalformulation
Generally, y decay is more complicatedthan in our first example: the observed y ray follows several or y transitions (see practical examples on fig. 2). The expression for the y distribution is simple if the following conditions are respected: (i) The parent nuclei 1 I, M ) are in thermal equilibrium with the lattice. (ii) The Hamiltonian of the parent nuclei or intermediate nuclei has a cylindrical axis 2 of quantization. (iii) The lifetime of the intermediate nuclei li is shorter than the nuclear relaxation time in order to ensure that no re-orientation occurs in the intermediate levels. All these conditions are easily obtained when the lifetime of the parent nuclei is long enough to be sure that the nuclei are in thermal equilibrium since the lifetime of the intermediate state is generally very short (fig. 2). Assuming these restrictions, the angular distribution of the y ray, emitted at angle 0 with respect to the 2 axis, is given by the general expression
w(e)= K
gKBK U,F,pK (cos 0)
where gK is a solid angle correction which characterizes the detector, U, is a nuclear parameter which describes unobserved nuclear transitions, F K is a nuclear parameter which describes the observed transition, pK(cos8) iS a Legendre polynomial and BK is a nuclear orientation parameter which is related to the population W(M)of the parent nuclear sublevel. The series is restricted to even values of K in order to respect the y parity conservation. We will assume here that in all the decays observed, the nuclear parameters g&#K are well known, The solid state information is given by the two terms B,P,(cos 0).
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METALS WITH MAGNETIC IMPURITIES
52u" 3,
p'33% E C 56%
-37 €2 835
2'
54~r
"cr
5% 7.1 d
5*
0.7pr 1,332 0' Fc5a
6o Ni
Fig. 2. Radioactive decay of "Mn, 5AMn,"Co, 6oCo.The broad arrows correspond to the NO observedyrays. The angulardistributionsof these y raysare drawn on figs.3 and 6.
1.2. APPLICATION TO SOLID-STATEPHYSICS
1.2.I . Hyperfne coupling-nuclear orientation parameter The BR terms depend on: (i) nuclear constants such as nuclear dipole or quadrupole moment; (ii) solid state parameters which describe the NO mechanism; (iii) the temperature. Knowledge of two of these items will determine the third. The method can be used for nuclear, solid state physics or thermometry at low temperatures. We restrict the discussion here to the solid state applications. The general BK expression is given by B&) =
1 (2K+ 1)' M
+
C[IKI, MO] W ( M ) ,
(1.1)
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where C[IKZ, MO] is the well-known Clebsch-Gordan coefficient. The information given by NO experiments results from the comparison of the experimental value W(0) with the theoretical value computed for different hyperfine couplings. Table 1 indicates the B,, B, terms for three simple Hamiltonians in the high temperature limit for the special case of a nuclear spin I = 1 and electronic spin S = 3 (Blin-Stoyle and Grace 1957, p. 564). Table 1 The first nuclear orientation parameters for three simple nuclear orientation mechanisms (r= 1, s = +). Harniltonian
B1
Bz
Wc see from this table that y-NO experiments cannot detect the sign of He,, but only that of P - that the separation between a quadrupole and a magnetic coupling can in principle be obtained at high temperature where the respective B, terms show quite a different temperature dependence. All the above discussions assume that all the observed nuclei have the same axis of quantization which means practically that a macroscopic axis of symmetry can be defined which coincides with the local axis. As we will show in $ 5 , we must check whether or not this condition is fulfilled in a given case. We have drawn on figs. 3-6 the y anisotropy of well-known 3d NO isotopes "Mn, s4Mn, 58C0,6oCowhich correspond to the y decay of fig. 2. All the nuclei are subject to an effective hyperfine field equal to +200 kOe. We must remark: (i) that the low temperature sensitivity depends on the relative sign of the U,F,, U4F4 terms. This effect is shown by comparing the 58C0, fig. 5, with the 6oCo, 'Mn, figs. 3,4 and 6 and noting that the "Co U4F4 term has an opposite sign to the U,F, term. While the 52Mn,54Mnand 6oCo have their maximum sensitivity in the axial direction, the 58C0loses its sensitivity completely along this direction. (ii) that the sensitivity of the
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nuclear probe to any misalignment between the real 2 axis and the experimental Z axis (defined generally by the applied field) depends on the strength of the ratio U4F4/U,Fz when both terms have the same sign (see fig. 3 for "Mn and fig. 4 for 54Mn).(iii) that with a relatively low counting rate the accuracy in the determination of an effective field can approach one percent. 1.2.2. Illustrating cases Before 1964, most of the nuclear orientation experiments were performed with paramagnetic salts which were cooled down by demagnetization of the salt itself. Now, most of the NO experiments are performed in metallic materials which are cooled indirectly using an auxiliary demagnetization salt or dilution refrigerator; the main experimental difference is that the production of the low temperature is independent of the sample preparation. In the actual studies three different fields can be distinguished: (i) measurement of hyperfine fields in ferromagnetic materials, (ii) measurement of hyperfine interactions in the presence of radiation damage, (iii) study of dilute magnetic alloys. The first two subjects are often studied using nuclear resonance destruction of the y ray anisotropy (NMR/NO). We do not describe here the details of this method (destruction of y anisotropy by a radiofrequency field at the saturation conditions for the nuclear resonance) since generally the NMR detection is not easy in the case of a magnetic impurity dissolved in nonmagnetic metals due to the fast nuclear relaxation times of the impurity (Waltstedt and Narath 1972). In this case, the NMR/NO detection adds the supplementary disadvantage of a radiofrequency power limitation to the maintenance of low temperatures (Spanjaard et al. 1971). The recent review by Stone (1976) gives interesting applications of NMR/NO. In order to illustrate the NO impact in solid state physics, we have shown in table 2 different experiments which have been performed. Four examples are related to metal physics, two examples to insulators. The first y anisotropy signal was observed in 1951 at Oxford and at Leiden by the detection of the 6oCo anisotropy emitted from Co nuclei, dissolved in a single crystal of CuSO4RbZSO46H,O(tutton salt) (Daniels et al. 1951; Gorter et al. 1951). The more famous NO experiment is the observation of the B decay nonparity conservation (Wu et al. 1957).
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m
d .LL
58
60
co
co
Fig. 5. Fig. 6. Figs. 3-6. The figures represent the y ray distribution W of the respective "Mn, 54Mn, "Co, 60Coprobes, submitted to a hyperfine field of 200 kOe, for five different temperatures T.The nuclear parameters defined in eq. (1.1) are: zMn 54Mn WO 60Co
Z g&nl@rJ
uzF2
uz4
6 3.07 -0.3741 -0.1591
3 3.30 -0.4949 -0.4467
2
3.99 -0.2988 +0.7127
5 3.75 -0.4206 -0.2428
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Table 2 Examples of NO experiments. Experiments' CO6OCO 98Au
&iS4Mn E60Co 137mCein CMN "Mn in (CL)MN
Comments
References
First observation of a hyperhe field in a ferromagnetic metal First observation of a hyperfine field on a diamagnetic impurity in a ferromagnetic metal First experiment performed in dilute alloy Detection of the NMR by NO Determination of the absolute temperature scale of CMN Rotational cooling
Grace et al. (1955) Khutsishvili (1955) SamoYlov et al. (1959)
Cameron et al. (1964) Matthias et al. (1966) Huntziger et al. (1970) Lubbers et al. (1967)
'CMN, (CL)MN are abbreviations for cerium magnesium nitrate and cerium lanthanum magnesium nitrate, respectively.
1.2.3. Limitation of the method. Comparison with other methods In order to emphasize the interest of such measurements, we describe the advantages of the technique. The main advantage is that NO observations are single counting experiments which allow a low content of radioactive nuclei, typically 10' atoms; the corresponding concentrations are generally less than I ppm. Other advantages are: (i) large range of isotopes available, notably of 3d elements (V, CryMn, Co) and 4f elements (Ce, Eu, Tb, Yb); (ii) high energy of the observed y rays (see fig. 2), typically 500 keV, which allows working with bulk materials easily characterized by other techniques (resistivity-magnetization) ; (iii) its detection entails no resonant condition. The main disadvantage of the method is its integral character which does not allow the direct decomposition into different hyperfine couplings (see 0 5). The interpretation of the results will be unambiguous if the physical situation is simple, for example a simple magnetic or quadrupolar coupling. The interpretation becomes more difficult in the presence of a mixed hyperfine coupling or of different sites in the lattice. Experimentally, the assumption of an unique hyperfine coupling must be checked over a large range of temperature. Unfortunately, this condition is often not realized in many experiments which have been reported up to now.
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NO gives the same information as ordinary magnetization experiments (M) with somewhat more sophisticated response and no background signal due to the lattice. The main difference is, as described in figs. 3-6, its higher sensitivity to any local breakdown of the macroscopic symmetry defined by the external field (see 0 5). The other differencesare: (i) its specificity (observation of a well defined nucleus); (ii) its hyperfine character which is more sensitive to the different magnetic terms since for example spin and orbital contributions give quite different hyperfine response (see 0 3). Hyperfine measurements such as differential perturbed angular correlation, Mossbauer effect or NMR which are spectroscopic methods are more powerful techniques. Unfortunately conditions on the hyperhe coupling and on the electronic or nuclear relaxation time limit their practical range of investigations. NO is often more easy to perform and can be a good tool to continue studies which cannot be carried on by NMR; an example will be given in 5 5 for the AuCo alloys. Another possibilityfor detecting a static thermal NO effect is offered by determining the intensity of the Mossbauer lines which depend on the parent or stable nuclear populations if the source or absorber is at a low temperature. If a source like PtS7Cois cooled to sufficiently low temperature so that the 7C0 parent nuclear sublevels become unequally populated, the intensity of the Mossbauer lines depends both on the population of the '7C0 parent nuclei which is a function of the "Co effective field, and on the transition probability between the different 57Fe Mossbauer sublevels. Such NO is observed at higher temperatures than in an ordinary experiment since the variation of the corresponding intensity depends on the first order parameter B,. In comparison with ordinary NO the disadvantage is generally the higher impurity concentration related to the low energy of the Mossbauer y rays; the advantages are: (i) only intensity ratios in a single spectrum are necessary. This leads to less possibility of systematic error than NO where warm counts must be obtained; (ii) no external magnetic field is necessary to provide a quantization axis.* For an absorber such measurements are of rather low utility despite the fact that the same phenomenon is able to give directly the temperature and the hyperfine coupling of the sample nuclei through their Mossbauer line splitting and intensity strength (Kalvius et al. 1970). The nuclear specific heat anomaly C, is directly linked to the hyperfine coupling. The advantage here is that the magnetic coupling can be determined
'
*Gonzales Jimenez et al. (1974) have observed the orientation of a paramagnetic state at low temperatures in zero field for the first time for a cubic symmetry impurity.
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in zero field and eventually in an applied field. The disadvantage is that the impurity concentration must be high enough to overtake signals arising from other contributions. Its comparison with NO or M is powerful for investigating the presence of different sites since C, has a rather specific response for magnetic or non-magnetic sites. Its efficiency will be shown in 0 5. 1.3. EXPERIMENTAL CONDITIONS In a nuclear orientation experiment, three different parts can be distinguished (i) the attainment of low temperatures, (ii) the detection of y rays, (iii) the preparation of samples. 1.3.1. Refrigeration and y ray detection
The necessary low temperatures can be achieved by adiabatic demagnetization of a paramagnetic salt or by dilution of 3He in 4He. By demagnetization of cerium magnesium nitrate (CMN), temperatures down to 3 mK can easily be reached on a sample thermalized at the end of a thermal link in contact with the salt (Williams 1968); with commercial dilution refrigerators, most of the experiments reach temperatures near 10mK. In the future more fruitful solutions for NO seem to be: (i) demagnetization of enhanced nuclear moment compounds (PrNi, , PrCu,) where temperatures down to 1 or 2 mK, respectively, can be reached (Andres 1976); (ii) attainment of a continuous temperature down to 3 mK by dilution refrigerators using either high surface heat exchangers (Frossati et al. 1976a*) or multimixingchamber systems (De Waele et al. 1976). Whatever the method of attaining the low temperature, the experiment must be performed in a wide range of temperature. At the present time, the best results seem to be obtained with the old technique (demagnetization) where the temperature is continuously varied during subsequent heating of the salt. The counting system is an ordinary single counting arrangement. Two types of detector can be used: (i) ordinary NaI(T1) scintillators which have a poor resolution but a high efficiency; (ii) Ge(Li) detectors which have a high resolution (-2 keV) but a lower efficiency. If all the nuclear parameters *Frossati has recently reached a continuous temperature of 2.7 mK.
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are well known in the y expression of the y ray distribution only one detector need be used. The experimental parameter, measured usually in the direction of the applied field, the so-called axial anisotropy E(O), is defined as the differencebetween the warm and cold counts normalized by the warm counts E(0) = ""(0)
-"(O)]/N"(O).
The NO results are reported often in terms of an effective hyperfine field, Herr, which corresponds to the experimental value E(0) calculated from a brute force-type Hamiltonian:
A more satisfactory solution is to use two counters parallel and perpendicular to the macroscopic axis in order to have two self-consistent measurements E(0) and E(3n) at the same time. Each experimental point is obtained by the comparison of two y ray measurements: (i) the y anisotropy E,.(O) of an isotope A with a known hyperfine coupling; (ii) the y anisotropy of an isotope B with an unknown hyperfine coupling. E(T) will define the B temperature. Figure 7 shows the general set-up used at Orsay. A CMN salt is demagnetized and temperatures down 3 mK are obtained on the sample which can be polarized in fields up to 40 kOe. The electronic pulses received from the detectors are amplified and analyzed in amplitude in a multichannel analyzer following a 10-minute cycle. Each point is stored on a paper tape which is then analyzed by a computer for an automatic research of the y ray photopics. This procedure minimizes the eventual electronic drifts. Simultaneously, direct integral measurements allow a crude analysis of the running experiment. Using simple electronic devices, a NO apparatus can be almost automatic and must be regarded as the equivalent of a NMR or Mossbauer spectrometer (Benoit 1976). A more important point is the sample preparation, which we now make some comments on.
I .3.2. Metallurgy The activity of a NO sample is typically around 10 pCi which corresponds to 10" nuclei with a lifetime T of one month. The local impurity concentration will depend on the modes of preparation which can be classified as follows :
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(i) ordinary metallurgy, (ii) direct activation, (iii) implantation. The last of these can be su,divided into direct implantation after a nuclear reaction, or indirect implantation using an isotope separator. Timer
Fig. 7. NO set up of Orsay (Benoit 1976). The main point is that the exchange gas, which ensures thermalization of the paramagnetic salt with the 1.5 K fixed point, is scaled at 4.2 K.This NO spectrometer can carry out automatically two full runs including cold and warm counts. h is the 1.5 K fixed point, i and j are respectively the demagnetization and polarizing magnet, k is the paramagnetic salt, I a 50 Hz heating, m the Ge(Li) detector, A the NO thermometer (for example e 6 0 C o ) ,B the NO sample (for example G s 4 M n ) , MA the multichannel analyzer, CS the current power supply, S the shunt which measures the current in the polarizing magnet, V two voltmeters. The spectrum represents a typical NO measurement in a G S 4 M n , e 6 0 C o experiments. The clocks 1, 2, 3, 4 d e h e respectively the time of (1) the demagnetization salt, (2) the cold counts, (3) the 50 Hz reheating of the salt, (4) the warm count normalization. A resistor bridge R gives continuously the temperature of the paramagnetic salt. For each 10 min top given by the analyzer, a serializcr picks up successively the information received from the different measurements.
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A radioactive sample can be prepared in a classical way by diffusing the radioactive isotope in the metallic sample or melting together both materials. As a practical example, we describe a C U ~ Alloy ~ Mpreparation. ~ 54Mn can be bought in a carrier-free chloride solution: 'carrier-free' means that there are no substances other than HCl, H,O, Mn. In order to eliminate parasitic ions like iron, the best method is to clean this solution using an ether or a resin separation. A small part of this purified activity is electropolated or directly painted on the Cu sample. In the first case, the active sample is generally melted using a high-frequency induction furnace; in the second case, the chloride is decomposed and diffused during annealing under a hydrogen atmosphere; further homogenization can be obtained by melting the sample. (If the chloride is decomposed during the melting the first step can be bypassed directly). Finally, the activity is thus diluted homogeneously in the sample which can be characterized by other techniques; the important point is the low concentration of radioactive nuclei (< 1 ppm). Direct activation can be induced in stable alloys by different types of particles (neutron, proton, deuteron). Generally as the concentration of the stable impurity must be rather high (> lo00 ppm), two problems may occur: (i) magnetic interactions among the impurities; (ii) disturbance of the lattice symmetry. One example is given by the preparation of the La137mCealloys which can be directly obtained by bombarding a La foil byaproton beam. We obtain a high dilution of the magnetic impurities, but also the disadvantage of a lattice disturbance produced by the incoming particles and by the recoil of the created radioactive ions. Around 1967, implantation techniques seemed sufficiently powerful to prepare any type of alloy using an isotope separator plus a post acceleration. To illustrate the principle of the method, we describe the preparation of an AuYb alloy. Two radioactive isotopes seem available for a NO study: "'Yb and 169Yb.As a first step, the Yb203oxide is irradiated in a neutron reactor in order to get the desired activity. This powder is then set in an isotope separator where two different beams of and la9Yb can be extracted separately. Using a post acceleration of 100 keV, these incoming ions are selectively implanted in a gold lattice with a penetration depth ranging to about 100 A. This leads to a local concentration around 100 ppm. If we compare the implantation to a direct activation its advantages are the selection of a well-specified isotope and a greater dilution. However in all this picture we have completely neglected the defects produced during implantation: each incoming ion produces around lo00 defects before it stops completely. The problem is to know what alloys we are studying. A
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'
simple comparison of an Au' 5Yb implanted sample with the same sample melted after implantation(Ben0it et al. 1974) clearly shows that they are not equivalent (fig. 8). In implanted samples, the Yb ions are far from the idealized situation of substituted ions in an unperturbed lattice. As our task is not to describe the physicsof radiationdamage, weconclude that the separator implantation is a very nice tool to select the activity and only a first step in the preparation of a dilute alloy. Classical metallurgy must then be performed in order to eliminate the radiation damage and to obtain finally a simple alloy.
-
20 (TI
Yb
-1s Eb
-l a
-!
/:+', '
of 283 KeV rbrultr for
H 3 10 kOe .2'5Eamelted after implantation .'~Ebimplanted, aarnple
I
25
50
l5
+K-'
Fig. 8. Anisotropy of the 283 keV y ray of the '"Yb probe implanted in a gold lattice (P) or melted after implantation (E.). (Benoit et al. 1974a.)
When a heavy nuclei of mass M is submitted to a nuclear reaction by a light particle of mass m and of energy Eo ,the produced radioactive nucleus has a recoil energy around:
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As charged light particles (a, p, d) must have a rather high energy in order to lead to appreciable nuclear cross section a high recoil energy (typically 500 keV) is obtained. This energy can be used to perform a direct implantation in another lattice. A typical arrangement consists of first a thin foil, where the nuclear reaction occurs, followed by a second foil, where the ions recoil. Such a procedure was realized previously in Berkeley (Brewer 1969) in order to prepare a Cu5*Mn sample. The "Mn activity is produced by a (a, 3n) reaction on avanadium foil with 33 MeV incoming a particles. The average penetration depth is typically around lOOOA. Eska et al. (1976) have recently used the same procedure for a AuMn alloy using the supplementary property that the influence of the other magnetic impurities can be minimized by a previous oxidation, which does not oxidize the incoming ions which are further implanted at room temperature. As we have underlined before, the main problem remains the annealing of defects. Careful investigations must be made in order to be confident of its valid application to the study of dilute alloys. In conclusion, I would like to emphasize that (i) ordinary metallurgy can be performed with radioactive samples; the main difficulty, coming from contamination troubles, is easily solved as the usual activity is very low. (ii) special techniques like implantation can be employed successfully for qualitative investigations (is an impurity magnetic or not in such materials?) but certainly not for more accurate quantitative studies. 2. Magnetism of an impurity: Kondo effect
2.1. MAGNETISM OF AN IMPURITY IN A NON-MAGNETIC HOST 2.I . I . The Friedel approach (Blandin 1973)
The problem of a magnetic impurity in a metal was first treated with the exchange Hamiltonian (Zener 1951)
which describes the coupling of the spin S of a local moment interacting with the conduction electrons of spin s. Since the existence of a localized spin is assumed in this Hamiltonian from the beginning, it is not appropriate to ask why 3d impurities have or have not a local moment.
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An important advance in such understanding was initiated by Friedel (1956) who used a resonant scattering theory within an unrestricted H F approximation. The presence of an impurity produces a local spherical potential V(r) which scatters the free electrons of the metal (Friedel 1952). The important point is that the extra-ionic charge 2 associated with a foreigh atom is screened out over a distance of an atomic radius (charge neutrality) apart for long-range oscillations of the electronic density p(r), p(r) = -(u/2nr’) cos (2k,r
+$),
k,, r being respectively the Fermi wave number and the distance from the impurity, and u, 4 are parameters depending on the scattering by the impurity.* When the potential V ( r ) is nearly strong enough to create a bound state with quantum number I, resonance scattering occurs. Such a situation is well known in atomic and nuclear physics. The virtual bound state, which resonates with the I spherical component of the plane waves, is characterized by a finite broadened width (A) and a position ( E ) near the Fermi level. As, in this virtual bound state, the electrons are well localized, local exchange and correlation forces ( V ) must be taken into account; they favour spin alignment according to the Hund’s rule. Using these arguments, Blandin and Friedel (1959) predicted the occurrence of 3d impurity magnetism as a function of the electronic parameters of the lattice (influence on the A width) and of the impurity (influence of the screened charge and the correlation forces). Similarly, they showed (Blandin 1961) that when the spin degeneracy is lifted in a virtual bound state, the oscillating charge density in the matrix is associated with an oscillating spin density of the same wavelength;? this leads to a Rudermann-Kittel-like oscillation (1954) and to a spin glass coupling among the impurities which can be characterized by an isotropic distribution of the blocked local moments (Blandin and Friedel 1959). 2.1.2. Anderson Hamiltonian
In 1961, Anderson derived a simple Hamiltonian which is equivalent to the Friedel approach (Daniel and Friedel 1966). The Anderson Hamiltonian is *This phenomenon is basically the diffraction effect known in optics. The impurity can be regarded as a hole which disturbs the homogeneous crystal. tThe scattering by a spin dependent potential in normal metals was discussed first by Friedel (1952) for the problem of a proton with its bound electron dissolved in a simple Fcrmi gas.
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widely used for theoretical investigations using the HF approximation. It is given by
The first term is the band Hamiltonian, CG being the creation operator in the conduction band. The second and third terms describe the local d state of the impurity (the creation operator Cde of a d electron of spin a). The d state with one electron has energy Ed, the d state with two electrons 2Ed+U.The interesting case will be when Ed c EF and Ed+U > Ep (fig. 9). The last term Vkd describes the resonant scattering and leads to the virtual bound state picture (fig. 9). The mixing vk, gives rise to a broadening A of the localized state,
I
PO
I
P-0
Fig. 9. The d bound states of energy Ed and 2Ed+ U are shifted and broadened by the
resonant scattering with the conduction electrons.
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( p being the density of states of the conduction electron at the Fermi level) and to a shift of d-a virtual levels with respect to the Fermi level (energy shift). In the HF approximation, the Coulomb interaction inside the d level is approximated by the relation
where (nd-,) describes the average occupancy of the d-a level; fluctuations between electrons of opposite spin a -u are completely neglected. With this restricted approximation, for the above model of one d electron, the occurrence of magnetism necessitates the following condition
A non-magnetic impurity is described by the picture of two virtual d levels at the Fermi level. In real cases (orbital degeneracy), the general magnetism condition is given by a Stoner-like condition between the pd density of states of the virtual bound state at the Fermi level and Uerr, an effective interaction :
Figure 10 represents the 3d magnetism occurrence along the 3d elements as a function of the strength of the ratio A / U ; the sharp line shows the transition in HF theory between a non-magnetic regime where (n,,) = (nd-,) and a magnetic regime where (n,,) # (n,,). While it has long been evident that the sharp demarcation between magnetic and non-magnetic impurities is an artifact related to the Hartree-Fock approximation, because there must be no sharp phase transition for one impurity, it is only recently that a unified view has been put forward in the theory of the Kondo state. In the limit where the positions of the virtual bound states with opposite spins are well above and below the Fermi level as pictured in fig. 9. Schrieffer and Wolf have shown that the Anderson Hamiltonian can be reduced to an exchange part and a classical potential V,, HR = +JS*s+V,.
The essential point is that the resonant mixing Vkd leads to a positive antiferromagnetic J constant which can be defined as a function of the d virtual bound state by the relation
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The positive sign of J is thus a characteristic of a resonant mixing as the usual direct atomic exchange would give a negative J value. Practically, the exchange Hamiltonian describes only the situation of the Mn impurity in a normal lattice. Although the Anderson Hamiltonian is more suitable than the exchange Hamiltonian for describing the variety of dilute alloys, the latter is more simple and important results have been obtained from it. In the following sections, the normal potential is neglected;
Ti
Mn
NI
T
H.F picture
Fig. 10. Schematic representation of the condition for formation of a local moment in the Friedel-Anderson model (Hartree-Fock approximation). Impurities outside the curve are non-magnetic (NM); the diagram is for 3d impurities in a simple metal host.
only the exchange part is taken into account since we discuss the magnetic properties of strongly magnetic HI; impurities. It must be emphasized that the spin value S attributed to an impurity is not only specific to the initial unperturbed d state but partly given by the HF treatment of the conduction electron mixing ( Vkd). A quantitative description of the magnetic phenomena can be given when the conduction band is more complicated than a free electron structure. Phase shift analyses have been performed by Friedel et al. (1966) for narrow bands and applied to Pd, Pt lattices by Campbell (1967). This problem has been extensively studied by Moriya (1967). 2.1.3. Response of an electron gas to the impurity magnetism
In order to avoid a complex discussion, we present here the crude picture given by Anderson (1967) and consider the simple form of the exchange
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Hamiltonian. Reducing the S vector to an ordinary number S, its action on the electron gas can be summarized by linear response theory. The polarized center S at the site ro perturbs the free electron gas by the local presence of an effective field H(ro) SJ. The electronic response at site r , rn(r), depends on the spatial susceptibility of the lattice x(r-ro), N
If the interaction among the conduction electrons can be omitted (case of a normal metal like Au), the spatial susceptibility has the well-known shape of the RKKY oscillations given by the function Fo(r)equal to [Ruderman and Kittel 1954; Yosida 19571 : Fo(r)
- -i3-
r:x-cos
x],
x = 2k,r.
This magnetic polarization carries the magnetic information and will be responsible for magnetic interactions among the impurities. The energy of interaction between impurities localized on sites r l , rj is given by
As (i) F(r) oscillates far from the impurity site and (ii) S has 2S+ 1 degrees of freedom, in low concentration ranges the magnetic ordering among impurities corresponds to magnetic moments isotropically coupled without any preferred direction. (Caroli 1967 has computed the resonant interaction process using directly the Anderson Hamiltonian.) In some lattices like Pd, Pt which have an enhanced Pauli susceptibility, the interaction among the conduction electrons cannot be neglected. The main result is that the spatial susceptibility has an enhanced response with a well-definedsign over a range d which covers many atomic distances (d 8 A for the Pd lattice). This enhanced response leads (i) to an abnormal polarization of the lattice around the impurities. This gives rise to the giant moment phenomenon (fig. 11); the magnetism of the impurity is not only localized on its sites but is accompanied by the electronic response magnetism which is carried by the surrounding atoms up to a distance equal to d (the giant moment of iron in palladium is near 1 2 ~ ~(ii) ) ; to a fcrromagnetic coupling among the impurities inside the ferromagnetic polarization clouds.
-
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2.2. THEKONDOEFFECT 2.2.1. Simple considerations
In 1933, experimentalists had already observed that the resistivity Of metal samples thought to be very pure has a minimum at low temperatures. As the resistivity of pure metals decreases as the temperature decreases, it
I I
\C
\
\‘*\El
\
\ /
lst
ncighbour
Fig. 11. The full line A describes the spin polarization of a normal host as a function of the distance r from the impurity site. The dashed curves B and C correspond to the schematic spin polarization of an enhanced lattice; the curve B describes the case of an induced moment, located on the matrix sites, weaker than the moment which can be localized according to the d band structure, the curve C describes the extreme saturation process. The ECo, PdMn alloys CORespOIId to the B case, the pdFe, PJCo are in an intermediate regime. In an enhanced lattice, the charge screening of the impurity (located mainly in the impurity cell) has a quite different radial dependence than the spin polarization which is built up at long distances by electron hole excitation.
was recognized experimentally that this minimum must be attributed to the existence of 3d magnetic impurities. This behaviour has been a theoretical puzzle as an exchange model leads in second order in J to a resistivity contribution R independent of the temperature,
R,
-
J z S [ S + 11.
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The theoretical explanation was given in 1964 by Kondo who showed that a logT contribution occurs if we take into account the third power of J in the perturbation :
The resistivity minimum appears as a characteristic of an antiferromagnetic coupling (J > 0) which leads to a logarithmic increase of the resistivity at low temperature. Abrikosov (1965) has shown that higher order in the perturbation must be taken into account when the temperature decreases. Below a characteristic temperature defined by
the coupling between the conduction electrons and the localized moment becomes so strong that a perturbation calculation breaks down. The impurity seems to be described by two regimes: (i) a weak coupling regime (T > TK)where the impurity carries a welldefined moment; (ii) a strong coupling regime (T < TK) where the roles of the conduction electrons cannot be described by a HF approximation. In the weak coupling region, the principal role of the resonant coupling J is to give rise to dynamic effects in which the conduction electrons are scattered from the impurity without a strong correlation effect. Thus, locally, on the impurity site, the conduction electrons are limited to providing a reservoir which produces relaxation of the spin S whereas the occupation of the state I S, S,) is practically determined by their unperturbed energies. For the case of S = 4 (the only case studied theoretically below TK), the ensemble (impurity-conductionelectrons) is well described by the degenerate non-perturbed state,
where xa represents the two states of S, and *v the Fermi sea without electron-hole excitation. In the strong coupling region, by contrast, we no longer have a simple relaxation process, but rather a correlation between the impurity spin and that of the scattered electrons which means that the scattering of a second electron depends on the spin of the first. The low temperature singlet state
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theory indicates that as T + 0 the combined state @ tends toward a singlet of the type (Yosida and Yoshimori 1973)
where ll/u schematically represents the combination of electrons of spin o with a hole of spin-o. The fluctuation between the two states +$ is no longer controlled by the thermal energy kBT but by the coupling energy kBT'. Experimentally if the susceptibility can be fitted by a Curie-Weiss law in a certain range of the ratio TIT,,
it is well known that the 6 value depends on the experimental range of temperature. We will discuss briefly the present theoretical situation and describe the temperature dependence of the susceptibility. As the limit of the weak coupling is well described, the two main points are to obtain an exact solution (i) for the strong coupling case, and (ii) in the cross-over regime.
2.2.2. Low temperature solution
Using scaling theory, Wilson (1974) has shown that two stable points in J space describe at OK the solution of the ferromagnetic and antiferromagnetic cases: these fixed points are respectively J = 0 and J = + co. The fundamental states of both cases (J 0) are completely different: for J c 0 the impurity is strongly magnetic and its susceptibility diverges at 0 K, while for J > 0 the ground state must be a singlet (J -+ + 00) and the susceptibility is constant at 0 K. Wilson reached an exact zero-temperature solution and made finite temperature corrections. Nozi6res (1974) has shown that, in the strong coupling limit (T c TK), the low temperature properties can be derived phenomenologically in the same spirit as in the usual Landau theory of Fermi liquids. Because the singlet remains somewhat polarizable an indirect repulsive interaction between electrons of antiparallel spin occurs. This interaction leads to a zerotemperature susceptibility
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As in the Pauli susceptibility of a metal (TF-+ TK),the low temperature excitations give only a quadratic deviation in T. Recently Yamada (1975) and Yosida and Yamada (1975), investigating the Anderson Hamiltonian with electron-hole symmetry (eq. (2.1) : Ed = -3U+EF, the so-called symmetrical Anderson model, E, = -3U) by a perturbation expansion in a power series of the Coulomb energy U have: (i) clearly demonstrated the presence of the H F and Kondo mechanism for the occurrence of magnetism; (ii) proved the Nozi6res Fermi liquid theory based on the s-d exchange model. Figure 12 represents the density
1.0
0.5
0 0
5
10
Fig. 12. The density of states for the d electron calculated by Yamada for the symmetrical Anderson Harniltonian (E,, = -)U) using a perturbation expansion in a power series of the electron correlation U. This calculation shows the reasonable structure in the whole range of E/A including the large U limit. In the HF magnetic case (U/nA = 2 or 5). the broad peaks appear near E = +)U. (The F e d energy is taken as the zero energy, right-hand side are only shown.)
of d states, p d , obtained by Yamada as a function of the relative energy E/A for various values of q = U/nA, the HF parameter. For strongly H F magnetic impurities (q > l), three peaks occur: the two located at E, = &+U describe the HF regime, the third located at the Fermi level represents the Kondo coupling. The striking result is that for q < 1 only one peak occurs almost independent of the strength of q. The paramagnetic impurity has a zero moment. For > 1, the paramagnetic impurity moment is completely quenched below TKby the Kondo correlation; above TK, the paramagnetic impurity carries a well-defined moment since the thermal energy averages
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the sharp peak centered at the Fermi level. The HF transition (q = 1) shows a broadened Kondo anomaly, the entropy of the magnetic moment shifts the local moment regime to the U < d region (Friedel 1976). Krishnamurthy et al. (1975) have schematically represented the different regimes of the impurity as a function of the three variables T, U,A (fig. 13). The drawn surfaces describe respectively the planes T = xA, T = U and the cross-over surfaces T = T,(A, U).As is noted by the authors, in contrast to the drawing,
t T.
T (Temperaturo)
nA
plane
T=T,
A ( d level width)
(A#)
U.mA
Fig. 13. Schematic sketch of the various schemes arising from the susceptibility results
for the symmetric Anderson model. The high temperature free orbital regime has no physical meaning as the strong coupling limit for U = 0.
the separation between the various regimes is quite fuzzy; notably the definition of an infinite Kondo temperature for the zero-moment paramagnetic impurities has no physical meaning. 2.2.3. Finite temperature solution
Finite temperature calculations were made numerically in the cross-over region by Wilson (1974) using an S = exchange Hamiltonian. Figure 14 shows the variation of k,TX1g2& as a function of logT/TK. At high temperature
+
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which corresponds to the pure Curie law limit. At zero temperature
If an empirical Curie-Weiss relation is used near OK, one can see that e =2 . 5 ~ ~ . Recently numerical calculations performed by Krishnamurthy et al. (1975) using a symmetrical Anderson model led to the same dependence. 0.25
I
I
I
I
I
I
I
Fig. 14. Universal plot of kBTX(T)/(gPB)' versus logT/TK,where z(T) is the impurity susceptibilityfor the spin 3 exchange Kondo Hanultonian (Khrishnamurthy et al. 1975).
2.3. INFLUENCEOF THE HYPERFINE COUPLING This discussion neglects the fact that, a t low temperature, the localized moment may have a hyperfine coupling with its nucleus,
Two different cases occur, according t o whether the hyperfine coupling is greater or less than the Kondo coupling. For k,TK > A, the local moment is strongly coupled to the conduction electrons, and the appearance of the induced moment on the impurity is described by the pure electronic term
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In the full Hamiltonian, the hyperfine term may be regarded as aperturbation; the spinsIandSare thus uncoupled. The hyperfine coupling can be reduced to
For kBTK< A, the situation is completely different. In low field, the local moment is strongly coupled to the nuclear spin. We must take the s u m P = I + S . If the hyperfine coupling is such that a well-isolated ground state I: can be defined, the Kondo Hamiltonian is only a perturbing term. Its action on the ground state can be reduced to
where a is defined in table 3 as a function of A (Benoit and Flouquet 1976). The striking result is that a Kondo coupling may appear or disappear due only to the action of the hyperfine coupling if I and S are antiferromagnetically coupled. If the Kondo temperature is defined as a function of the degeneracyn of the angular momentum (Cornut and Coqblin 1972) by TK = TFexp+ l / J p n ,
we remark that TKchanges even when Z and S are ferromagneticallycoupled. Table 3 Variation of the a coefficient of the exchange Hamiltonian in the electron-nuclear ground state as a function of the hyperfine coupling and the electronic and nuclear spin. AcO
S > I s<1
S=&
s< I
S -
s+ I
S -
S+I
I 2z+ 1
-
A>O
s+ 1
s-r+l --
I-
S
s+ 1
-- 1
21+ 1
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In this case, the Kondo temperature as a function of I is for Z = 0,
TK = TF exp [+2/Jp(2S+ l)],
for I # 0.
TK = TF exp [2/Jp(2S+S/F)].
As F = I+S > S, the main results are
-
Using Jp(2S+ 1) = 0,068, a typical number for an alloy like AuMn, we get in the F + 00 limit
When the ground state F is a singlet electron nuclear state (A > 0,
I = S = +), no exchange coupling may occur as long as this ground state is well isolated. A good example will be given in $ 4 by the AuYb study of the Grenoble Group.
2.4.
NO IN THE SINGLE IMPURITY STUDY
We again emphasize that NO experiments have the advantages of permitting magnetic measurements to be made at low impurity concentrations, including the observation of the local moment appearance at 0 K without any resonance condition. The first point is experimentally important if we want to study a well isolated impurity without any magnetic interaction with the other impurities. As the NO signal can be detected only if the impurity is sufficiently polarized, we are interested in the cases where TK is lower than 10 K. The low magnetic field study will give the strength of the Kondo coupling beside the hyperfine coupling, the high magnetic field study will restore the local moment completely decoupled from the conduction electrons and from the nuclei so long as H > kBTK/gpB, H > AI/gpB. The limits A > kBTK,A < kBTKwill be called respectively the weak and strong Kondo coupling regime. This definition is almost the same as that used by Wilson and NoziBres, since in NO, A has often the same order of magnitude as the thermal energy in order to allow an appreciable y ray
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anisotropy. The main difference occurs, as was remarked in the last section, at very low temperatures (Flouquet 1971a; Flouquet and Brewer 1975; Hartmann 1976).
2.4.1. Weak Kondo coupling regime kBTK6 A
In a first approximation, the main coupling is described by the following hyperfine Hamiltonian
where gpBHSz , g & f I , are respectively the electronic and the nuclear Zeeman terms. The NO response, for example the axial anisotropy E(O), can be computed knowing the electron-nuclear wave function and energy levels of this full Hamiltonian. In zero applied field, the different eigenstates are defined by the total angular momentum F,and their energies are related to A by EF = +A{F(F+ l)-S(S+l)-l(Z+
1)).
When the Zeeman terms appear, the component F, is an eigenstate of the Hamiltonian which can be diagonalized inside each F, value. For high magnetic fields (gp,H 6 A), the hyperfine coupling can be treated as a perturbation on the S, electronic states which are well decoupled. Such a situation is well known in atomic physics when the spin-orbit coupling is comparable with the applied field; the low and high field limits correspond to the Zeeman and Paschen Bach effects respectively. As the two parameters g,p, and Z are assumed to be well known the theoretical function E:(O) seems to depend on the three parameters A, S, g. In fact, when the electronic magnetization is completely saturated, the impurity of spin I sees an effective field equal to
+
Heff(sat)= H H,,(sat), where &,(sat) is related to A S by the relation
684
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[CH. 9, $ 2
As this value is easily obtained in a high field measurement, two paa-r meters S,g can be chosen. E: is the analogue in NO of the Brillouin function B: in magnetization experiments. It will be computed for different values of the applied field and of the temperature. If one is in the habit of representing NO results in the form of an effective hyperfine field, one calculates the hyperfine field H: which corresponds to the theoretical angular distribution E@). Figure 15 compares H i and B; as a
Fig. 15. Comparison of an NO and M response to the free spin Hamiltonian YP,, [eq. (2.311.
function of the applied field at 10 mK for the special case of a 54Mn probe which corresponds to the values I = 3, S = 2.5, g = 2, &(sat) = -400 kOe. For H = 0, the relative splittings between each successive F level are 35, 29 22, 16, 10 (in mK). The smooth variation of H i expresses the weak mixing by the applied field between each Flevel ; the electronic response reaches rapidly the Brillouin limit since (i) the Zeeman electronic term is three orders of magnitude greater than the corresponding nuclear one; (ii) the degeneracy of the electron nuclear state is rather high (12 for the ground state F = 5.5 at H = 0). Figure 16 shows the H i temperature dependence for different values of the applied field. Contrary to the behaviour of the Brillouin function, H; is almost independent of 1/Tfor a constant applied field.
CH. 9, 8 21
METALS WITH MAGNETIC IMPURITIES
Au Mn
- H i (kOe)
t
S . 512
685
Hhf.-400k0e
g.2
H in kOa 40
400
1
200 0,1
4
100 0,03
c
0
I
I
I
1
*
The sensitivityof these NO functions to the S determination is rather good as is shown by table 4 which gives at 10 mK for three different applied fields -E Sof the axial function of the previous the relative variation AE = s4Mn probe. As the anisotropy is measured with an accuracy better than AE = 0.5%, the choice between the two values S, S + $ can be made without any ambiguity in the weak Kondo coupling regime. fi:, like the Brillouin function, is sensitive almost solely to the product Sg. Table 4 NO sensitivity to the electronic spin determination assuming a g factor equal to 2. At 10 mK, AE(”Mn) = E$+’/’-E2s for &,(sat) = -400 kOe.
150
9.44
600
14.4
1200
9.5
8
6
8 3.3
5
1.9
4.6 3.3 1.2
3.6 2.7 0.79
[CH. 9, 8 2
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686
Finally we mention that this function is defined for a given value of the nuclear spin and changes appreciably when another isotope with a different spin is observed: the effective field picture, in the weak Kondo coupling limit, is nuclear spin-dependent as indicated by table 5 which represents at 10 mK the H i values of the 52Mn,"Mn probe. The first step in the NO analysis is thus to compare the measured effective field with the calculated curves H@, T ) . Agreement with a particular curie indicates that the system is indeed in the weak Kondo coupling region. Disagreement may indicate either a wrong choice of the constants S and g , or that the system is in fact entering the strong coupling region (A 4 kBTK). Table 5 In the case of S = 9, g = 2, free spin function HOsfor 3ZMnand 54Mn nuclei submitted to a saturation hyperfine of -400 kOe. H (W 52Mn
20 45
50 106
100 178
200 253
500 326
40
93
158
230
310
H25'z (kOe) 54Mn
2.4.2. Strong Kondo coupling limit kaTK B A In this case the NO response is described by an effective hyperfine field proportional to the magnetization and, of course, independent of the probe nuclear spin [see 9 2.3, eq. (2.3)]. The fundamental difference between the strong and weak coupling is that, in one region, He,, is proportional to the magnetization and, in the other, it is not. Usually, the Kondo coupling is evaluated by the expression (aHeff/aH)H-+O = R &/3kBONO
[see eq. (2*3)i,
where R is the hyperfine ratio between the saturated values AS/g,,pn and the magnetization. 2.4.3. Cross-over region A
N
kBTK
The NO response, like the magnetization response, is quite difficult to calculate in the cross-over region as no decoupling between I, S, and s can be
CH.9, 5 21
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687
performed. In order to visualize this region, Benoit et al. (1976) have used a very crude model for the singlet formation. Instead of the N-body Kondo electronic coupling, they have chosen a simple antiferromagnetic interaction between two spins S = 3 and K = 3, 2: = 2kBToS* K.
The effective spin K represents the Kondo behaviour of the conduction electrons and the Zeeman term is restricted to the S spin in order to simulate the low polarizability of the conduction electron gas. Here, the magnetization (m) and NO responses can be computed exactly for a well-defined isotope. The main result is that, if the low field responses are described in terms of Curie-Weiss laws with Om, ONO parameters,
-
then eNO is always greater than 8, for kBTo A and tends to Om only in the strong coupling limit (kBT,,> A). In the cross-over region, the proportionality hypothesis between He,, and m (often made in NO) leads to an overestimate of T K . 2.4.4. Remarks on the decoupling between Zand S
In a spectroscopic method like the MBssbauer effect (ME) or NMR the decoupling of I, S is attained through the rapid fluctuation of S with respect to the nuclear Larmor period 7, (typically h/A). For an electronic relaxation time T~ < z,, the time evolution of Z is determined by relaxation processes and the nucleus is submitted to a hyperfine field proportional to the impurity magnetization
When zS $ T,, the electronic and nuclear spins are strongly coupled. It is well known that in zero applied field it is possible to observe in a ME experiment the free precession of the nucleus and to detect the T~ variation in the region zS 7, by the line broadening of the MSssbauer spectrum. N
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[CH. 9, 8 2
Similar dynamic considerations were previously invoked to describe the decoupling of I and S in NO. These arguments are conceptually wrong as a NO experiment is a static measurement. In most cases, the electronic relaxation effects do not change the nuclear eigenstates and hence the density matrix which enters in the y anisotropy calculation. In dilute alloys, whatever the electronic relaxation time of the local moment, in the absence of any Kondo coupling, the experimental data must fit a free spin function H:. For a ferromagnetic exchange, whatever the strength of T&, the NO response must fit a free function H: even in the case of infinite coupling (T& 4 1 at experimental temperature), our response is described by the fixed point J = 0. It is a property of the Kondo state that the decoupling between Z and S occurs for A/kBTK << 1 (limit J --t + 00).
2.5. TKSCALE. EXPERIMENTAL POSSIBILITIESOF THE DIFFERENTHYPERFINE METHODS
We comment here on the possibility of observing directly the impurity on its sites by NMR, ME, NO and compare them as a function of T K . NMR experiments are easy to perform in high Kondo temperature alloys (TK > 100 K) since the relaxation time is rather long. Two pieces of information are usually extracted: the Knight shift and the relaxation time. One of the interests of such studies is to discuss the consistency between these two observables and thus the limitation of the theoretical models in describing a true physical case (Narath 1973a; see 0 3). As TK decreases, the NMR line width and its relaxation time increase as the magnitude squared of the coupling; the impurity nuclear resonances are difficult to observe except for the special cases of abnormally low hyperfine coupling. At the present time only the Yb resonance in gold has been observed at low temperatures using a very strong magnetic field (Folstaedt et al. (1975). Before 1974, the only possibility of observing directly the low TK impurities seemed to be provided by radioactive methods which necessitate different criteria for observation. The basic NMR experiment of Boyce and Slichter (1974) on CuFe, who resolved five satellite lines from the main Cu host resonance,has offered the possibility of making specific translated impurity resonance through the satellites. These satellites correspond to a defined position in the lattice with a weaker electronic coupling p&) than those of the impurity itself. The application of this method to the CuFe alloys (TK 27 K) has given rise to the nice experiment of SkaletoF(1975) on N
CH. 9, 8 31
METALS WITH MAGNETIC IMPURITIES
689
single crystals and to the first dynamical studies through the Kondo crossover (Alloul 1975, 1976). ME leads (i) only to static information if .r& 4 1: this is always the situation found for the well-known isotope of 57Fe;(ii) to both dynamic and static information if we can cross over the region .rS/z, 1. The first abnormal Kondo departure of .rS from the well-known Korringa law was seen in a ME experiment performed by Gonzales and Imbert (1973) on a AuYb alloy. ESR experiments give theoretical information on the hyperfine structure and on the impurity relaxation. For 3d elements in normal metals, the hyperfine structure has never been resolved and direct measurement of the relaxation time is masked by the bottleneck problem (Hasegawa 1959). For 4f elements both types of information have been observed since (i) the J (4f) value is lower than the J (3d) value; (ii) the bottleneck is open for 4f ions in a non-S state. In most cases, J has a negative sign and a constant value near 0.1 eV; the main mechanism is the direct atomic exchange (see Orbach 1975; Orbach et al. 1974). NO, as ME, cannot give any information in high TKsystems as the experimental signal is too low. We describe in $0 3 and 5 an intermediate situation (AuCo) where both NO and NMR were performed. In parallel to these general considerations, practical criteria dictate that only one technique offers the opportunity for looking at a definite nucleus. For example, the stable atoms of Ce, having even numbers of nucleons, carry no nuclear moment; on the other hand, the 137mCeisotope is an excellent NO probe. We restrict this discussion here, neglecting other methods like muon precession and angular correlation.
-
3. The hyperfine field. The origin of magnetism In high magnetic fields when the magnetic impurity is completely polarized, NO will give the magnitude of the saturation hyperfine field (&(sat)), The sign of Hhfcan be obtained, (i) in some cases by following the relative competition between the applied and the hyperfine fields: when the electronic moment becomes completely saturated, the sole variable is the applied field; (ii) by performing anisotropy measurements or low temperature ME; (iii) by detecting the degree of circular polarization of the y rays. For example, the positive sign of Hhf at a Co impurity dissolved in palladium was measured by all four methods. M
J. FLOUQUET
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[CH. 9, 5 3
I will show that measurements of the saturation hyperfine field and possibly its sign are relevant to the question of the origin of magnetism. The discussion is not restricted to the NO values as a better understanding is obtained by a complete compilation of the data.
3.1, ATOM~C STRUCTURE AND HYPERFINE FIELDS
The hyperfine interaction between an electron of spin and orbital angular momenta s, 1 with a nuclear spin I is equal to
The first term, called the contact term, exists only for an s electron as the delta function indicates. The second and third terms describe respectively the orbital and dipolar effects. If this Hamiltonian is summed over all the electronic shells n with a restricted HF approximation (same radial wave function for eachn shell), an ion like M n Z +in an s state will have no hyperfine coupling. Using a spin-dependentradial wave function for the n shell a core polarization effect in the inner shells leads to a hyperfine field proportional to the spin S of the unfilled shell,
being the wave function of the different shells n at the impurity site. For 3d ions, H:p is intrinsically negative with a 3d free ion value Ro = H,d,/gpBSof - 120 kOe per Bohr magneton which is approximately independent of atomic number (Freeman and Watson 1965). The orbital part, Rorb, depends on the atomic number and increases gradually from a value of + 180 kOe/pBfor the TiZ+ions to a value of +650 kOe/p, for the NiZ+ions. For a 3d element, the orbital magnetism term has often a bigger hyperfine response than the corresponding spin term. For the 4f shell, more localized than the 3d shell, the hyperfine contribution is essentially orbital except for the s state ions E t + , G d 3 + where the only contribution is the 4f core polarization. Table 6 lists the hyperfine value of the 4f free ions (Kondo 1961). $"JO)
N
CH. 9, 6 31
METALS WITH MAGNETIC IMPURITIES
691
Table 6 4f free ion hyperfine field. 4f
4f
I 1
Ce3+
Pr3+
NdZ+ Pm3+
Sm3+
Tb3+ 3.14
3.33
4.33
3.93
3.43
Dy3+
Ho3+
Er3+
Tm3+
Yb3+
5.98
1.46
1.64
4.13
4.12
3.2. 3d HYPERFINEFIELDS 3.2.1. Magnetic impurity. Limitation of the models Table 7 represents the NO results obtained for different alloys; the results are summarized by: (i) the Sg determination if a fit to a H: function can be reached; (ii) the strength of the Kondo coupling given by the Weiss parameter 8; (iii) the values of the saturation hyperfine field and magnetization; (iv) the direct ratio R of these two values in the case of a normal metal host or the ratio R of the saturation hyperfine field by the local magnetization on the impurity site for the case of an enhanced matrix like Pd, Pt (see 44). Table 8 describes the ME results obtained for a "Fe probe, and table 9 the NMR results on the impurity site; the magnetic impurity examples are those of MoCo and WCo alloys. Finally, table 10 gives: (i) the NMR results obtained for M F Fe, Co, Ni ions dissolved in a dilute ferromagnetic alloy of PdFe 2% (Le Dangh Koi et al. 1976); (ii) the g value of the ferromagnetic resGance detected by Bagguley et al. (1974). As tables 7 and 8 show, R generally is negative with an average value of 70 kOe/pB, that (i) the main magnetic part is the spin one; (ii) the resonant mixing is really an I = 2 coupling between a d localized state and an electron of well-specifiedpartial wave function I = 2 [an I = 0 electron mixing should give a strong positive hyperfine contribution (R, MOe)]. Both conclusions agree with the Anderson picture that the spin magnetism is more easily fulfilled than the orbital one. If r describes the exchange 3d coupling between two electrons in a different m = I, orbital and U the Coulomb interaction in the symmetrical Anderson Hamiltonian, the two conditions for magnetism are given (see Coqblin and Blandin 1968) by N
(U+If)/nd > 1
and
(U-r)/nA > 1
Q W N
Table 7a NO results on 3d impurities in normal host.
M Ref. 1K
CuCr
1 2 H 2 25 mK 60 mK 25 mK
-
AgMn LuMn
4omK
> I1201 - 153 - 238 -242 - 280
- 328 - 328
- 51 -74 -76 - 80
-400
4.5
- 88
120 mK 280 mK
- 170
2.5 3.2
- 68
- 188
300 mK 20 mK
-180
4.8
- 37.5
2 2
-400 2K &Co
3.0 3.2 3.2 3.5
K = +29%
- 58
Williams (1970) Brewer (1974) Williams (1970) Williams (1970) Campbell (1967) Pratt (1969) Flouquet (1971b) Cameron (1966) Flouquet (1970b) Williams (1968) Lagendijk (1969) Flouquet (1971a) Marsh (1972) Marsh (1970) Marsh (1972) Taurian (1974) Thomson (1976) Taurian (1974) Holiday (1970) Boysen (1973)
Daybell (1969) Vochten (1976)
Lutes (1964) Hirschkoff (1971a) Manhes (1971)
I-'
Mahhes (1971) Hanson (1976) Manhes (1971) Caplin (1972) Newrock et al. (1971) Li et al. (1974) Collings (1962)
Boucai (1971) Narath (1973b)
8 \o
cQ1
W
CH. 9,
0 31
METALS WITH MAGNETIC IMPURITIES
693
Table 7b NO results on 3d impurities in an enhanced host.
aMn
- 360
$ 2
-400 PAC0
$ 3 1 2
- 370
l0OmK 1.6K 1.6K 70mK
PtCo QMn
+ 210 +200 -200 -210 -360
11 3.6 5.2
- 80
Gallop" (1970) Thomson (1975) (ml = 5pB) Flouquet (1977b) Gallop. (1970) Flouquet (1977a) - 100 Gallop (1970) (ml=2pB) Ali (1974) -70 Gallop (1970) N
7.5
-
(ml=
Star (1975) Bozorth (1961) Crangle (1965) Tissier (1972) Tholence (1976)
SPB)
a Gallop neglects completely the giant moment occurrence as indicated by the H : fit of the ZCo results; the E M n experiment was performed in a narrower range of temperature than that of Flouquet et al. (1976). The apparent discrepancy between the H," functions, found by Gallop and by Flouquet et al. for the B M n alloy, is certainly due to a difference in the purity of the sample (see Flouquet et al. 1976b).
Table 8 ME results on Fe" in different non-magnetic host (Kitchens and Taylor 1974).
Cu Ag
Au Mo W Pd
Pt
27 1.6 0.45 1
-90 -36 -190 -112
2.3 -2.2 2.2 2.4
0.43
-75.5 -311
2.4
-301
6
N
10
Steiner(1973a) Steiner(1975) Steiner(l973b) Maley(1970) Perez Ramirez (1977) Kitchens(l965) -31.4 -103 Craig(1972) (ml = 3 ~ 8 ) - 100 Kitchens (1965) (mi = 3pB) -47 -16.4 -86 -46.5
Tholence (1970) Hanson (1976) Tholence (1971) WPP (1968) Amamou (1976) Bozorth (1961) Crangle (1965)
"The ME experiments of Ericson et al. (1970,1971), performed at low temperature, show clearly that the 'CO impurity is less magnetic than the s7Feimpurity in Pd and Pt matrices {TK(Co)> T,(Fe)}. In each alloy, the hyperhe field of the Co impurity, which is deduced in zero field from the ME line intensity, decreases with the impurity concentration, while the hyperfine field of which is deduced from the line splitting (8 1.2.3), is almost concentration independent.
694
[CH. 9, 5 3
J. FLOUQUET Table 9
NMR results on impurity sites. [When the Knight shift K is decomposed into Kd and Kerb, the results agree with the Dworin and Narath analysis (1970I.l
AjV A_o
290
guco CuNi
gV &Cr &lMn SCo wco -
45 1.8
-1.5 29 5.2 1.28 +0.30 -0.38 -2.01 -6.8
+ + +
-3.1
I .6
-0.3 -0.4 -1.4 - 3.5
1.6 0.1
Narath (1973)
1 .o 1.5 - 26 -1.4
Narath (1 976)
Table 10 NMR and ESR results of 3d impurities dissolved in a dilute ferromagnetic alloy of palladium.
Mn Fe Co Ni
-400
0.01
+230 175
0.15 0.25
- 300
+
2.12 2.17 2.33 2.45
-
+ +
'Hhf, dHhrlHhrdescribe respectively the frequency and the relative linewidth of the NMR observed on the impurity dissolved in a ferromagnetic alloy of pdFel% (Le Dangh Khoi 1976). 'g represents the g values derived from ferromagnetic resonance in the different alloys pdMn, pdFe, pdCo, PdNi with a content of impurity near 1 % (Bagguley and Robertson 1974; Alquid et al. 1976). 'The last column describes the cases where skew scattering effects were detected (Senoussi et al. 1976).
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695
As the spin magnetic condition is often just fulfilled, the orbital magnetic condition is not usually reached for 3d impurities in the Friedel-Anderson approach. We do not discuss the possible origins of a core polarization attenuation (for more details the reader can refer to Narath 1973a). But the appearance of an abnormal ratio for AgFe, MoCo, WCo and abnormal hyperfine sign for PdCo, PdNi must be explained, In insulators, theCo2+on is known as an ion which often carries an orbital moment; even if the initial splitting C of the crystal field gives a singlet orbital ground state, the spin orbit coupling (AJ5.S)leads to an orbital part in the ground state which appears rigidly coupled with the initial spin part; an induced Van Vleck contribution independent of the temperature is provided by the applied field mixing among crystal field levels.
-
1.6 1.4
';;; 1.2
g I Q
1.0 0.8
I
0.6
0.4 0.2 r
0 0
I
l l l 20 40
l
l 60
l
l l l l l l 80 100 120 140 H (kOe)
Fig. 17. Magnetic field dependence of low temperature 59C0resonance shifts in 0.1 at % WCo. The solid lines are smooth crystal field fits to the data (Narath 1976).
Narath (1976) has interpreted the NMR results of Ws9C0 (fig. 17) in a pure ionic model with a T z orbital singlet ground s&e and crystal field excited levels. The small hyperfine field which varies in temperature is attributed to a spin-orbit induced orbital contribution in the ground state; the temperature-independent Knight shifts, which appear in high magnetic fields, are attributed to the local moment Van Vleck susceptibility. Assuming a spin-orbit equal to 0.02 eV, from these two hyperfine values a consistent crystal field splitting, C = 0.5 eV is obtained. The same analysis for the
[CH. 9, 8 3
J. FLOUQUET
696
hyperfine field of Co in Pd can be made assuming a T z spin contribution of 3 pB and an orbital contribution, given by
with Rorb= 500 kOe/p,, Rspin= - 100 kOe/pB, equal to 1 p B . Careful studies on the CuFe NMR satellites by Boyce and Slichter (1975) are interpreted taking into account the crystal field and the energy level parameters. The values of the crystal field splitting C derived for these three examples are:
-
C(eV)
WCo 0.5
PdCo CuFe 0.2
0.5
-
Although these values have the same order of magnitude as the width A of the virtual bound state (Asd 0.5 eV), the observation of the crystal field structure does not contradict the Friedel approach. C and A correspond to two different mechanisms: (i) C describes an excitation inside the d level; (ii) A a transfer of a d electron to the Fermi sea. Clearly, d correlation factors must be incorporated in the Friedel-Anderson treatment which neglects intra-atomic correlations. Hirst (1971, 1974) has derived a simple criterion for the observation of a crystal field structure in the extreme case of strong intra-atomic correlations with respect to the k-d mixing, as in this situation the impurity must be stabilized in a well-defined configuration 3d" with n fixed: the splitting of an intra-configuration is seen if its energy (for example C ) is lower than the energy of excitation, E,,, to an upper nf 1 interconfiguration. The main problem for the 3d impurities is that the criterion of applicability, defined as E,, > A , is generally not fulfilled due to the large value of A ( A 0.5 eV). This difficulty appears when Hirst (1971) attempts to fit the 3d hyperfine field in a pure ionic approach because such fitting requires the assumption of two somewhat dubious hypotheses: an abnormally high strength of the direct coupling and an attenuation of the orbital coupling. A correct description of the 3d magnetic impurities lies between the ionic and the Friedel-Anderson model.
-
3.2.2. Non-magnetic impurity Table 9 which represents the NMR results, mainly for non-magnetic impurities, gives the Knight shift values decomposed into an orbital and spin
CH. 9,
METALS WITH MAGNETIC IMF'URITIES
31
697
term according to the Dworin and Narath (1970) calculation which has shown that, in the non-magnetic regime, the orbital susceptibility has an important contribution given by the following expressions 4&21+ Xorb = nkBTsfl
1)
,
2&(21+ 1) Xspin
= nkBTsf2
'
where the two characteristic temperatures T,, are linked to the magnetism conditions by
These expressions show: (i) that for r = 0, xorb- 2xspin;(ii) that Tsfl is always lower than Tsfl.If the T,, are understood to be fluctuation temperatures, these expressions suggest that the spin and orbital parts fluctuate with two different parameters. Spin-orbit coupling may mix the different modes. The striking results of table 9 is that the Dw6rin and Narath analysis can be made only for the impurities far from the HF magnetic transition. Near the HF instability, the AuCo studies show (i) an important orbital contribution (K k0.3); (ii)he impossibility of treating the orbital and spin magnetism independently. For the less magnetic AuNi case, Dwarin and Narath (1970) have interpreted the results by the supplementary assumption of a crystal field splitting C = 0.2eV. The hypefine properties reflect the continuity which occurs at the HF instability. The actual studies performed on 3d impurities show that the crystal field and the spin-orbit coupling must be taken into account even for a 3d magnetic impurity near the HF instability. Good evidence of this is given not only by hyperfine measurements but also by transport properties like skew scattering (see for example the recent experiment of Senoussi et al. 1976 on PdFe, PdCo, PdNi alloys referred to in table 10). We remark finally that the orbital effect must lead to an orbital ESR g shift. Although the results of the ferromagnetic resonance cannot be directly extrapolated to the local state of the impurity, table 10 shows that the g value seems strongly linked to the orbital contribution.*
=
*In ECo, E F e alloys, ferromagnetic ESR leads to g factors respectivelyequal to 2.29, 2.15 (Bagguley et al. 1974). The appearance of an abnormal behaviour for the Co impurity in platinum seems confirmed by the recent high field magnetizationexperiments of Tissier (1977 )which show a very high Van Vleck susceptibility proportional to the impurity concentration. New high field NO experiments in the ferromagnetic alloys would check the origin of the Van Vleck contribution.
698
[CH. 9, 5 4
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3.3. 4f HYPERFINE FIELDS For 4f impurities, by contrast, excellent agreement is found between insulator and alloy values if the crystal field effects are taken into account. All the hyperfine couplings determined by ESR, ME, NO are in excellent agreement with the ionic model. Deviations may occur near a valence change (Lid 1). Table 11 lists the hyperfine fields measured by NO for different alloys, and the corresponding crystal field ground state is indicated; well resolved crystal field structures were observed. The Hirst criterion seems well proved since, for the 4f impurities, C A 0.02 eV (Coqblin and Blandin 1968) and E,, I eV 3 C.
-
N
N
Table 11 NO results on 4f impurities.
eNO
Hr'
r7 > 10K
AgYb
&ce
r7
&Eu
&uTb &uY b
kce g2&ce
Hdsat) (kW
rb
Ref.
+ 560 + 560
r7
- 134
S
Flouquet (1971b) Flouquet (1974) Flouquet (1970a) Barclay (1975a)
r7
state non-magnetic doublet r3with fust excited level r7at a splitting of 1 K r, pcndetermination r7 100 mK 560 r, 4oomK r,
+
Barclay (1975b) Benoit (1974) Flouquet (1971b) Benoit (1973)
As the nuclear moment of 17sYbis unknown and the hyperfine coupling of Yb in Au is known from the ESR experiment of Tao et al. (1971), the NO experiment in a high magnetic field leads to the nuclear moment determination (Krane 1972; Spanjaard 1971;Benoit 1974). 'Hr is a fit to a weak coupling function H,: given by a well-isolated ground state r of the crystal field levels. bTis the ground state determination made by a Hr fit or by the value of Hdsat).
4. Single impurity effects
The examples to be given in this section illustrate the different couplings defined in 9 2. They were chosen particularly in order to show: (i) the capability of NO measurements; (ii) the interest of comparative NO and M
CH.9, 8 41
METALS WITH MAGNETIC IMPUFUTIES
699
studies (while the former lead to a local response derived from the value of the hyperfine coupling and a non-local response from the possible enhancement of the Zeeman electronic term, the M experiments always give a global response); (iii) the possible extension of the NO results to more interesting M experiments; (iv) the interconnection between different measurements which can elucidate the magnetic behaviour. Points (ii) and (iv) will be used in the study of giant moments (PdMn, PtCo, PdCo). Point (iii) appears clearly in the recent magnetizationand &tivG measurements performed in Grenoble on AuYb alloys, point (iv) in the connection between magnetism and superconductivity. We shall not discuss experimental details, but we point out that: (i) for the PdMn, PdCo, PtCo alloys the specific nature and the dilution of NO expexments2low the observation of an impurity in very dilute alloys where the contribution of the residual parasitic impurities dominates; (ii) in Lace alloys, the magnetic behaviour of the Ce impurities can be observed with a lattice in a superconducting state so long as most of the lattice parts are in a normal state; (iii) the NO experiments can be performed on bulk materials usable for other techniques. For example, for the PdMn, PdCo and AuMn experiments described here, in order to avoid anydoubtabout the purity of the samples, magnetization and neutron analysis experiments were performed. Finally we point out that the discrepancy between M and NO measurementsperformed on the CuMn, AgMn alloys are not due to spurious effects but have a well-defined interpretation. Contrary to the widespread view, the NO advantage of a greater dilution probably gives the best value of the Kondo coupling. 4. I . WEAKKONDOCOUPLING 4.1.1. Asimpleexample: AuYb -
The AuYb alloy is an excellent example of the weak Kondo coupling limit ( A BTBTK).The existence of a predominant antiferromagnetic coupling is clearly demonstrated by: (i) a negative ESR g shift (Tao et al. 1971); (ii) a resistivity minimum (Murani 1973); (iii) a deviation of the electronic relaxation time T~ from the well-known Korringa law. Figure 18 shows the Mtissbauer results of Gonzales and Imbert (1973). With a pure Korringa law, the product T T should ~ be a constant at low temperature. NO experiments (Benoit 1974a) have shown that the 175Ybimpurities dissolved in gold are well described by the spin function H: using an effective
700
[CH.9, 0 4
J. FLOUQUET
ii, 160
140 130 120 110 -
preliminary Mossbauer results
100 -
80
90
I
I
I
0.6 0.81
I
2
I
I
3 4
I
l
l
6 8 10
I
---+ Kelvin
I
20 30
T
Fig. 18. Ionic relaxation rates for AjYb, measured via the MBssbauer line broadening in the 170Yb spectrum (Gonzales Jimenez and Imbert 1973). Relaxation rates obtained by ESR are also shown (Tao et al. 1971). The relaxation does not follow a Korringa law but increases logarithmically at low temperatures as expected from a Kondo anomaly.
a melted after implantation He,,
Yb
-function H r7 spin free 1 -70 K-' T
I
I
0
1
'
2
HinkOe
Fig. 19. The NO resulis for dilute "'Yb in Au host, showing a fit to the free spin function P 7 .
CH. 9, 0 41
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spin S = 3, with g and A corresponding to the r7crystal field ground state. This fit shows that the Kondo coupling is much lower than the hyperfme coupling (fig. 19). This behaviour must also be observable in an ordinary magnetization experiment. The element Yb has a number of radioisotopes some of which lack dipole nuclear moments; the experimental situation is particularly favourable in that the isotopes 1 7 ' Y b and "jYb have moments of opposite sign, and furthermore, the former isotope fulfills the conditions for electron-nuclear singlet coupling (A > 0, S = I = 3). Table 12 lists for 171Yb,173Yb,and 174Ybthe nuclear spin value I, the hyperfme coupling A and the splitting energy C, between the zero field electron nuclear ground Table 12
Variation of the nuclear spin, the hyperfine coupling and the splitting between the electronuclearground state and the first excited state for different stable isotopes of Yb.
"IYb I
3
A (a) 127.8 CF (a)127.8
173Yb
I -35.1 105.3
114yb
0 0 0
state F and the first excited level. Figure 20 represents the hyperfine level of 71Ybas a function of the ratio gpgH/A, fig. 21 the corresponding susceptibility. The residual slope below 20 mK is due to the presence of Yb isotopes other than 71Yb(Frossati et al. 1976); this observation is no more than the well-known Breit-Rabi effect in atomic physics. Recently, resistivity measurements made at low temperature (T < 200 mK) by Mignot et al. (1976) show the breakdown of the Kondo scattering below the formation of the Yb electron-nuclear singlet (fig. 22).
-
4.1.2. Observation of an enhanced electronic Zeeman term: The PdMn giant
moment
Star et a]. (1975) have shown by a magnetizationexperiment that amanganese impurity dissolved in a Pd lattice carries a giant moment of 7.5 pB. As in PdFe and PdCo alloys, an important fraction of the magnetization is due to moments induced on the palladium.
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T
1 0
111 Yb: A I
.127,8 m"K I 1
b
I
x=- g C'BH A ;
Fig. 20. Hypefine level scheme of '"Yb. The dashed lines are the asymptotes of the mF = 0 levels when x = g p B H / A+ 00.
90.6 */. cnr I c he d
Our results
"Ir'
- Calculated
1I-l 0
V
1
50
Fig. 21. Susceptibilityof &"'Yb
I
100
versus l/T(Frossati et al. 1976).
1/T ( K - 1 ) .
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As was emphasized, NO experiments will give: (i) the value of the local moment on the impurity site by the detection of the saturation hyperfine field; (ii) the value of the induced electronic polarization which does not contribute to the hyperfine constant, as it is located outside the impurity site but contributes to the electronic Zeeman term by an enhancement factor (1 +a) of the local term.
[ nR .cm
10
50 100
500 1000
5000 T (mK)
Fig. 22. The squares and triangles refer to the &'71Yb resistivity of a 415 at ppm alloy; circles, 0,to the &17"Yb resistivity of a 396 at ppm alloy (Hebral et al. 1976). The different logarithmic slopes reflect only the presence of parasitic impurities of iron (CFe Sppm). The important point is the occurrence of a maximum in the Kondo scattering near the temperature where the singlet electron-nuclear state becomes isolated from the triplet excited level (fig. 20).
-
As the saturation hyperfine field &(sat) = -370f 10 kOe is close to that of -400 kOe measured for the Mn nuclei dissolved in gold, the local moment m, of the manganese in palladium is close to the value attributed to Mn impurities in AuMn alloys, which is described as an S state with S = and g = 2. Locally, the Mn impurity in the enhanced palladium matrix has the same behaviour as in a normal matrix. Knowing the saturation hyperfine field, comparison of the experimental anisotropy with an unenhanced free spin function E:o (S = 5,go = 2) shows an enhanced electronic term (fig. 23). The experimental value fits, down to
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10 mK, the enhanced function Ef with S = f,g = 3; g is here only a phenomenological parameter which reflects the shift of the local go value (go 2) by the enhancement factor 1 +a, g = go(l +a)*
Below 10 m K a gradually increasing deviation occurs due to the spin-glass interaction between the Mn impurities and the parasitic iron impurities (at a concentration near 3 ppm) (Flouquet et al. 1976).
Fig. 23. 835 keV y ray axial anisotropy of 54Mn dissolved in palladium and gold as a function of l/Tfor H = 50, 100,200 Oe. The curves represent the correspondingfree spin value Ej5I2 which does not take into account any possible magnetic enhancement (Flouquet et al. 1976).
Since with the rather high value of I = 3, the composition of I and S leads to a free spin function H: which is only sensitive to the product S g , the choice of S and g is quite arbitrary using such a fitting. The fixed value S = $ is in agreement with the linear response theory developed later for the - results, and the ESR g shift reported now. PtCo In the common description of the giant moment, the induced moment M i is rigidly coupled to the local moment M d . The fundamental question is how to understand the dynamic behaviour. Alquie et a]. (1976) have PdMn alloy; their recently performed accurate ESR experiments on a three main observations were: (i) an unusually large g factor (g 2.1 in the bottleneck regime); (ii) an even more unusual g shift (dg 0.6) when
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the bottleneck becomes broken, i.e. when the impurity ESR resonance becomes decoupled from that of the conduction electron; (iii) a strong Korringa term for the impurity relaxation time (Tzs lo-” sec at 1 K) (fig. 24). If this result confirms qualitatively the existence of a giant moment, we must not forget that the derived values are given using the Hasegawa equations which are well established for a normal lattice but must be reconsidered in the presence of a giant moment; the dynamics of the giant moment is an open problem. N
c C J
2 2
f
4200
3700
3200
2700
H i n 0.
2200
Fig. 24. ESR experiments at 1.6 K of Alquie et al. (1976) on a C M n alloy containing 500 ppm of Mn. The (1) signal corresponds to a sample annealed for 10 h at 1300 K; the ESRg factor is equal to 2.146 (point A). The (2) curves describe the ESR of the same sample after a scratched surface process; the ESR g factor increases up to 2.58 (point B) while the linewidth increases from 303 Oe to 653 Oe. The (3) curve describes the ESR signal obtained after an annealing of the sample (2) for 3 h a t 600 K; the ESR signal reaches its bottleneck limit g = 2.145 and the linewidth decreases down to 305 Oe. The C point represents a parasitic cavity signal.
If unsuitable dynamical arguments are used for the decoupling between 1 and S i n NO experiments, one sees that the Mn impurity fulfills the condition for the fast relaxation regime ( 0 2.4) as at 10 mK, zs lo-’ sec is much shorter than a typical Larmor period zn = 2 x lo-* sec. A NO fitting to a Brillouin function B;’’ would lead to a considerably stronger signal than the observed H i t 2 , which is characteristic of weak Kondo coupling.
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4.1.3. Other examples Figure 23 shows that the AuMn alloys fit the free spin function E;'' above 10 mK. Below 10 mK, the deviation will be interpreted as spin glass behaviour presumably among the Mn impurities. Such free spin behaviour has been previously observed by Williams (1968) and by Lagendijk (1969). The weak coupling situation occurs also for the AuCr alloys (Williams 1970) and for AuCe, AgCe (Flouquet and Marsh 1 9 7 ~ F l o u q u e1971). t In these examples the l o c z e d moments are assumed to be in a well-isolated ground state; when the splitting C between crystal field levels is comparable to the Zeeman electronic term, the applied field mixing between the levels must be taken into account. Such a situation was observed by Barclay et al. (1975) for the 16'Tb isotope dissolved in gold. 4.2. STRONG KONDO COUPLING CASE 4.2.1. Proportionality between H,,and M
As nodirect comparison has been performed for magnetic impurities in normal lattices, we can only establish in table 13 that for T, 2 0.3 K good agreement occurs between the Curie-Weiss Kondo constant measured by NO and M experiments. For example, the accurate NO experiment performed by Brewer (1974) on CuCr allows verification of the proportionality law. In the c& of AuCo, where the orbital and spin magnetic contributions have almost the same weight, a comparison cannot be made with the susceptibility measurement since each different hyperfine contribution must be taken into account. The excellent agreement between NMR and NO experiments demonstrates the point that the fast relaxation regime and the strong Kondo coupling limit give the same result (tables 7-9).
-
Table 13 Comparison for strong Kondo coupling of the Curie-Weiss temperature measured by NO and magnetization results.
CuCr ZnMn PtCo AJzbCe
1 0.28 1.6 0.4rt0.1
1 0.35 rt 10 1.6 0.3
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4.2.2. The appearance of a giant moment
Older susceptibility measurements show that, in the exchange enhanced Pt lattice, the Co impurity carries a giant moment of 3 . 6 which ~ ~ can be decomposed into equal induced (mi) and local parts (md)(Crangle and Scott 1965). NO experiments by Gallop and Campbell (1968) and extrapolated susceptibility measurements of Tissier and Tournier (1972) show that the Co impurity goes through a Kondo transition near 1.6 K. A direct comparison between a magnetization experiment, m = m i + m d , and a pure local hyperfine measurement (Herr md) will be re!evant to the problem of the formation of the induced moment on the matrix sites as a function of the appearance of the local moment. Ali et al. (1974) have extended the Gallop NO experiments in fields up to 70 kOe in order to cover the magnetic field range used by Tissier and Tournier. The comparison of NO and M is described in fig. 25. The hyperfine effective field appears to be proportional to the global magnetization up to the complete development of the local magnetization, which is known to be proportional to the hyperfine effective field in the strong Kondo coupling limit: the induced moment on the matrix is then proportional to the local magnetization. The particular advantage of PtCo is to extend the demonstration over a very large range of applied fields. The existence of a Kondo-like state for the larger giant moment of PdCo ( M = 12 pB, TK = 100 mK) (Flouquet et al. 1976a) makes questionabFthe classical description of a giant moment. Usually these moments are described by a ferromagnetic exchange coupling between the spins S, s of the impurity and the conduction electrons. Taking into account the NO results obtained on PtCo and PdCo alloys and following Morya’s analysis for the existence of a local moment (1967), the appearance of magnetism on the Co sites can be summarized in two steps. Firstly, thelocalmagnetism M,(r,)appears onthe site ro oftheimpurity ; its HF existence depends on the narrow band d, its Kondo coupling* is characterized by k,Ti. Secondly we must take into account the long-range reaction of the lattice. The high density of states of the narrow band d leads:
-
-
*The origin of the Kondo-like coupling for the giant moment is an open problem. The resistivity of ptCo and PdCo alloys shows a logarithmic decrease. Loram et al. (1971) have discussedthe different possible mechanisms.They have ruled out a sole Kondo scattering of s electrons from local Co moments; the temperature dependent resistivity may result from s electron scattering from enhanced spin ffuctuations within the impurity cell (see Lederer and Mills 1968).
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-1V
-
I"
Magnetization deduction
of M, (HI N.0 experiments
-50
tia!
-!I
Sample 1
2 3
Results h
0 0
H (kOe)
Fig. 25. NO results for very dilute E C o (Ali et al. 1974) compared to the extrapolated magnetized curve of Tissier and Tournier (1972). The magnetization curve is scaled by the factor 60 kOe/pB,
(i) to a strong local response of the electron gas described by the action of a fictitious local field h,(ro) which produces a strong spatial polarization M(r) linked to the lattice susceptibility ~ ( r - r , , )by the relation
(ii) to a decrease of the Kondo temperature T i . The ferromagnetic description only takes (i) into account. This crude description undervalues the feed-back process between the Kondo temperature and the strongly correlated electron gas. The observed linear response between the induced and local moment for the PtCo alloy suggests that more generally, the electronic polarization responds linearly to the local magnetism whatever the strength of the field. Analogous agreement has been found experimentally by Boyce and Slichter
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(1974) and Alloul (1975) on CuFe alloys well below and well above T K , where no extra Kondo polarization occurs in the Kondo state (see Ishii 1976). (It must be noted that a linear response theory may break down for an enhanced lattice like Pd in the case of a strong giant moment.) Finally, as the magnetic moment on the Co sites is slowly restored by the applied field, an attempt can be made to compare the experimental data with theoretical expressions which described the local moment induced by the applied field. Ishii (1970), using a singlet Kondo approach, has given at 0 K the simple expression
which approaches the saturation following a (TK/H)’ law. Gallop (1969) Pt-Co data are fitted well by the has pointed out experimentally that the simple function
which approaches saturation with a (TK/H) law. The Gallop picture corresponds to a simple model in which, in the symmetrical Anderson Hamiltonian, the spin up and down virtual bound states are shifted by the applied field symmetrically with respect to the Fermi level by an amount 6 E = d(gpBH/2kBT’. For the AgFe alloy, Steiner et al. (1975) have found also that the experimentalresultzaturate more slowly than the Ishii formula; as for PtCo, a satisfactory agreement is found with a resonance level model (Schot; and Schotte 1975). Gotze and Schlottman (1974) have derived an ‘exact’ temperature dependence of the magnetization for the special case of a spin S = +,a comparison with this model is made by Steiner and Hulner (1975) for the AgFe alloy. If the low field behaviour of the Kondo systems Fe in Cu, A g s u , Mo is mainly determined by their Kondo temperature, the magnetization in high fields cannot be scaled by an universal behaviour (Perez Ramirez and Steiner 1977). For example, if the resonance level model gives a reasonable fit to the AgFe and CuFe alloys, it cannot fit the data of AuFe and MoFe alloys; s p z a l propezes must be taken into account. 4.3. JNTERMEDIATE COUPLING The results of Campbell (1967) and Flouquet (1971b) yield a value
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8 N o = 25mK on CuMn and
8NO
[CH. 9, 9 4
= 40mK on AgMn. Magnetization
experiments made by Hirschkoff et al. (1971a) on G M n and Doran and Symko (1975) on AgMn yield respectively 8, = l G K and 8, = OmK. We note agreementwith the Benoit (1976) statement that, with intermediate coupling (A kBTK), ONO is larger than 0,. ( 5 2). But the main problem is to know the significance of 8,. Three mechanisms contribute to the apparent change of the behaviour of a single impurity in a magnetization experiment performed at very low temperature. (i) The conduction electrons, which cooperate in the Kondo coupling, are in a narrow band kBTKnear the Fermi level (fig. 12); if the concentration of impurities is such that the number of electrons cannot realize a complete Kondo compensation for each impurity, the Kondo coupling appears to be reduced (see Gruner and Zawadowski 1974). This consideration defines a critical concentration, C , , for the strong coupling regime (C, TK/TF). For a CuMn or an AgMn alloy, the critical concentration, C, 5 ppm, is just around the IowG value used by Hirschkoff et al. and Doran and Symko. (ii) It is well known that the coupling with the other impurities changes the behaviour of the single impurity and leads to a distribution of the Kondo coupling (Souletie and Tournier 1971; Tournier 1974). (iii) Finally, the reduction of TKby hyperfine coupling is more noticeable in the low temperature susceptibility measurements than in the NO experiments which are performed in higher fields than in susceptibility experiments. All these three mechanisms are minimized for the reported NO experiments since the concentration of the impurity is a t least one order of magnitude lower than in M experiment. We emphasize here that the agreement between NO and M experiments is better if the high temperature susceptibility data only are interpreted following the numerical analysis of Krishnamurthy. For example, for an AgMn alloy, the susceptibility measurement of Manhes (1971) and Hanson (1976) leads to an extrapolated value 8, = 60mK at OK which is in an excellent agreement with the NO data. New measurements would seem interesting in order to understand the discrepancy between magnetization experiments. In the next section, we will see a particular example of a strong TK variation as the temperature decreases. N
N
N
-
4.4. KONDOCOUPLING. DEGENERACY OF THE GROUND STATE
-
A more fundamental contradiction seems to occur for the Lace alloys
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where the NO results give a 8 value near 100 mK and the high temperature M experiments yield 8 near 27 K (Flouquet 1971; Edelstein 1971). By incorporating the crystal field in an Anderson model corrected for the 4f impurities, Cornut and Coqblin (1972) have explained this apparent conflict. The exchange constant J and the degeneracy n of the occupied levels change when the different excited states of the crystal field become depopulated; the variation of J is related t o the different initial energy Ed between the crystal field levels. This effect is basically the same as that of the variation of the Kondo temperature due to hyperfine coupling (42.3); the main difference is that in the latter case the product Jn has only a smooth variation. The Lace alloy is, on the contrary, a spectacular example where the parametezn and consequently the Kondo temperature varies appreciably with the crystal field occupancy. Such an anomaly was also explained by De Gennaro and Borchi (1973) by adding the crystal field effect to the exchange Hamiltonian. Benoit (1976) has shown that their procedure requires the definition of an arbitrary sign for the exchange constant. It is now well established that the Coqblin and Schrieffer (1969) Hamiltonian is the best description of the kf mixing. In this Hamiltonian, the mixing is not restricted, as for the exchange Hamiltonian, to the transition M-M’ = f 1 between M (theZcomponent of the 4f angular momentum) and M‘ (the corresponding k angular momentum), but is allowed for all the M-M‘ transitions. The alloy Lace offers the advantage that the strength of the J coupling may be changed by varying the experimental conditions under pressure. The pressure variation of the hyperfine field of cerium in lanthanum is shown in fig. 26. At low field, the coupling J increases under pressure as reflected in the decreasing initial slope 6Heff/6H.At high applied field, the curves show the same &(sat). These results illustrate a situation in which the impurity well described by the ionic model, shows a strong sensitivity to the resonant coupling. Benoit et a]. (1974b) deduce, within the framework of a bound 4f level, the pressure variation of the relative position of the 4f state: SE/A 1 as well as the minimum initial position of the bound state E4, > -34. One often hears the term ‘chemical pressure’ as it seems possible to change the Kondo temperature by alloying two lattices in which the impurity has quite different values of TK.Since the Ce impurity is magnetic in lanthanum and non-magnetic in thorium, the alloy La,Th, - - -,Ce fulfills this condition. Pena et al. (1977) have performed NO and M experiments on such an alloy which appears as a very complicated system where the magnetic behaviour must be interpreted taking into account the local environment of each impurity. N
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The supplementary interest of the NO experiment is to give here direct information on a magnetic impurity in a superconductinglattice. The problem of the coexistence of magnetism and superconductivity has undergone a burst of activity since it leads to elucidation of the competition between Cooper pair formation, responsible for superconductivity, and the pair breaking which occurs by spin-flip scattering at an impurity site. The depression of superconductivity by a magnetic impurity has been studied by Abrikosov and Gorkov (1960) using second order perturbation theory.
P (kbar) - 2 --o--
-.+,-
10 2 (after cycling)
L -110 K” T
20
H(kOe)
Fig. 26. NO curves for @Ce at various’appliedpressures (Benoit et al. 1974). The solid and dash-dotted curves were taken at 2 kbar before and after cycling to 10 kbar respectively. All measurements were made on the same sample at high and low pressure.
More recently Zittartz and Miiller-Hartmann (1970) have shown that pair breakinghas its maximum amplitude when the Kondo temperature approaches the critical temperature. Striking re-entrant curves of the superconductive transition T&) as a function of the impurity concentration n may occur. The magnetic information was obtained mostly by the use of theoretical expressions relating the magnetism to transport properties of the alloy. The confirmation by NO of the indirect information obtained by these last methods supports the validity of the theories. Experimentally the amusing fact is that the alloy La,Th,-,Ce, which is a complicated case for an NO and M experiment (Pena et al. 1976), shows a very nice re-entrant superconductive depression (Huber et al. 1974).
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5. Interaction effects
Interaction effects among magnetic impuritieswere often regarded as parasitic effects which mask the single impurity behaviour. We shall emphasize here that the NO sensitivity to the axial alignment of the nuclear spin makes it possible to distinguish between purely paramagnetic, ferromagnetic and more complex ordering situations. Up to now the interactions among the impurities have not been systematically studied by NO. We present: (i) for the NO specialist the importance of reversible and irreversible effects; (ii) the possibilities offered by NO experiments. A further possibility offered by NO is the observation of magnetism due to the presence of other impurities. Two specific examples are given: the PtCo alloy where ferromagnetic pairs form at a distance up to the range ofthe giant moment and the AuCo where only first neighbour coupling can modify the single impurity behaviour leading to pair and triplet groups. Finally, NO experiments on Ce compounds are reported and the occurrence of a Kondo lattice is discussed. 5.1. MAGNETISM OF INTERACTIONIMPURITIES 5.1.1. Spin glass
We consider the coupling between impurities which carry a well-defined moment; this assumes that the Kondo coupling (kBTK) is much weaker than the exchange interaction coupling kBT, of the localized moment with the other impurities which is linked to the J constant by the relation
In a normal lattice, the magnetic interaction among the impurities is characterized by (i) a spatial disorder of the impurities (like in a glass), (ii) a spin disorder of the directioh of the magnetic moment. As the analogy with atomic motion in a glass is found mostly in the rearrangement of the spin direction (the spatial disorder is quenched), the magnetic ordering below T, is called a spin glass. The isotropic distribution of the magnetic moment is due to the oscillating Ruddermann-Kittel-like polarization of the electronic gas; the r - dependence of the polarization gives rise to some kind of corresponding state at low concentration c: the specific heat C, the magnetization M and the ordering temperature T, follow simple functions
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of the reduced parameters T/c, H/c (Blandin 1961; Souletie 1968; Souletie and Tournier 1969). The occurrence of an interaction coupling is characterized by (i) a change in the low field magnetization slope aM/aT at T, (Schmitt and Jacobs 1957; Kouvel 1963; Tholence and Tournier 1974), (ii) a broadened anomaly in the specific heat, ( 5 ) a remanant magnetization 6, below T, (Owen et al. 1957), (iv) the appearance of well-resolved ME lines (Violet and Borg 1966) and of an antiferromagnetic-like ESR signal (Owen et al. 1957). Until 1971, the molecular field models (Blandin and Friedel 1959; Marshall 1960; Klein and Brout 1963) seemed to explain reasonably well the magnetic behaviour below the transition T, although the quantitative agreement with low temperature results (x xo, C T)was explained by a non-physical Ising model. In the Klein and Brout models, all the spins are assumed to be parallel and antiparallel to a given direction; the exchange coupling of one spin with the others can be restricted inside a correlation sphere of radius R, , which corresponds to a coupling with n, = 2.27 magnetic impurities. The theoretical interest for this subject reappeared when Cannella (1971) discovered a sharp cusp in the low frequency initial susceptibility. (The magnetic anomaly at T, is strongly decreased in higher fields by the occurrence of remanent effects.) As the sharp cusp in the susceptibility seems to characterize a well-defined transition, Edwards and Anderson (197s) have developed a new theoretical model where there is a ground state with the spin aligned in a defined direction even if each 2 direction appears to be random. An order parameter q defined as the average of (S;) gives a nonzero value below T, . Two different ‘schools’ of thought appear among the experimentalists. One points out that the occurrence of irreversible effects explains most of the spin glass properties; the other will prove the occurrence of a phase transition at T, and regards the irreversible effects as after-effects which can be related to the existence of a great multiplicity of low-lying metastable states or to some additional anisotropic interaction neglected in the calculation of Edwards and Anderson. For example, previously Kouvel (1963) argued that the slope deviation aM/aT is not related to a sudden phase transition but to the existence of irreversible effects due to anisotropic coupling among the spins below T,. Tournier (1965) has analyzed the irreversible properties with a model described by Nee1 (1961) in order to explain the properties of fine antiferromagnetic grains. In this picture, no paramagnetic spins are coupled in a superparamagnetic cloud which has a random statistical possibility of carrying an N
-
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uncompensated moment Mualigned along an anisotropic field in zero applied field; the volume of the cloud is defined by the physical dimension of the grain. The main point is that a potential barrier,
leads to a strong temperature variation of the transition probability per unit time ( T i ' ) of an uncompensated moment along one of its direction H a , T, = To exp E,/k,T.
For a temperature Tb, called the blocking temperature, the lifetime of M,(T,) becomes so long compared with the characteristic time of the measurement that this uncompensated moment appears as blocked and thus does not contribute to the low field susceptibility(the presence of a cusp and of a change in slope BMpT in higher magnetization experiment). Taking into account the different orders of magnitude, Tb is approximately, in a M experiment, equal to E, = 20kBTb.
Above T,, the grain behaves like a superparamagnetic group which follows a Curie law. The T, transition corresponds to the higher temperature Tbin the Ntel-like approach. In dilute alloys, the observations of (i) the susceptibility cusp, (ii) the remanent magnetization with a logarithmic time decrease support strongly a NCel-like picture. Holtzberg et a]. (1976) have recently shown using a crude molecular field picture that the anisotropic dipolar energy, 3p2(s1W2* r
W,
may explain the cloud formation. The size of the cloud results from the competition between (i) the internal dipolar energy of the cloud, (ii) the exchange energy between the no inner spins and the spins outside the cloud. Experimental results lead (i) to no 400, independent of the concentration for one specific system, (ii) to an anisotropy energy equal to
-
E,
M;AdlgLW
where A , represents the width of the distribution of the anisotropy fields which are produced by the dipolar coupling. The quadratic term Mi represents the coherent mechanism which creates the macroscopic anisotropy
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inside the cloud. As the dipolar and exchange coupling follow a r - dependence, scaling laws are observed also for the remanent field. The presence of irreversible effects can be detected in a magnetization experiment in two different ways. The thermoremanent magnetization (TRM) corresponds to a procedure where the sample is initially magnetized in a field above its ordering temperature then cooled through the transition in the applied field. The field is switched off later. The isothermal remanent magnetization (IRM) is obtained when H is applied below T, and is then reduced to zero. According to the Ntel model, in dilute alloys, (i) the IRM begins by varying as H 2and the TRM as H, (ii) the IRM and TRM saturation a,,(T) are the same at each temperature. The remanent saturation o,(T) follows the simple scaling law exp( -aT/C). During the magnetization of a spin glass, two different regimes may be obtained: (i) For H = HR, the TRM, or,achieves its complete saturation, ars;each impurity is poorly polarized as the ratio between the applied field, H and the molecular field, HM= kBTc/gpB,produced by the other interacting impurities, is low at HR (HR/HM 20). (ii) For H $ H,, each localized moment becomes slowly decoupled from the random molecular field; the saturation magnetization os of each impurity is ,reached with a linear law in H-' (as (1 -H,/H)). (See theoretically, Larkin 1970; Matho 1976.) Table 14 represents the magnetic observables measured by Tholence (1973) in an AuFe alloy containing 0.5 % of Fe. Crude application of scaling laws will givzhe equivalent values for another impurity dilution. N
N
Table 14 Tholence results on A_e,.oo, (1973).
0.3
360
42
70
1500
arH(0)represents the TRM at zero Kelvin in a magnetic field H, a,,(O) the corresponding saturation value which is achieved for H 2 H,,. os(0) represents the saturation magnetization (m) of each impurity. are the mean values of ars(0),oris given per magnetic atom, E a n d the N k l model. no is independent of the concentration. T,, HM, As, E, HEfollow the scaling linear concentration law.
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If the 'grain' behaviour is now well established the occurrence of a welldefined transition at T, is an open question and a great puzzle for the theoreticians (Fischer 1977). We underline that the first properties do not reject the second one but on the contrary can occur if an interaction coupling occurs inside the grain. A naive picture seems to be that (i) just above T,, the alloy is an arrangement of some local ordered moment in a superparamagnetic cloud together with some free paramagnetic impurities; (ii) below T, (assumed the real transition of the alloy), the coherence length Iz among the spins increases with the ratio T,/T ( A + 00 for T = 0). The observation of irreversible effects at T = T, is just the manifestation of the fact that coherence between the spin occurs at T,: the reversible and irreversible effect appear at the same temperature (the important point (Blandin 1976) is that no n,, the Brout and Klein percolation number). At 0 K, all the spins are frozen and the motion of one local spin cannot be independent of the others (collective excitations).* 5.1.2. Influence of the hyperjne structures
According to the discussion of i2.3 and the results of table 3, the strength of the magnetic interactions must also change appreciably when the electronnuclear ground state F is well isolated from the other excited F states (A > kBT) and when the interaction occurs well below the characteristic hyperfine coupling (A > kBT,). Without hyperfine coupling the ordering temperature, T:, is related to the degeneracy of the localized moment by the factor S(S+ I) [eq. (5.1)]. For a well-isolated electron-nuclear ground state, the ordering temperature T: must be evaluated taking into account the F degeneracy and the change of the apparent exchange constant (table 3). For a ferromagnetic coupling (A < 0),T,' is related to T," by the expression Tf T,"
-=-
S I+S
X-
Z+S+1 S+1
< 1.
For S = +, this ratio reaches the limit 0.33 for large value of I. *In parallel to a microscopic observation of the dipolar influence, important studies are the dynamic observations just around and far below To. The results of Murnick et al. (1975), McLaughlin and Alloul (1977) and Levitt and Waldstedt (1977), seem to confirm: (i) a strong variation of the coherencelength below T,; (ii) a complete freezing of the spins at 0 K (these conclusions are in excellent agreement with the Massbauer experiment of Violet and Borg 1966).
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For an antiferromagnetic coupling, the two situation S >< Z lead to a decrease of T,. For S = +,the interaction vanishes for I = 3 and the ratio reaches the limit 0.33 for Z 4 00.
5.1.3. Appearance of magnetism by the presence of other impurities
In the cases discussed in the last section, the impurities carry their full moment. If an isolated impurity appears as non-magnetic (T < TK),the presence of other impurities may modify the behaviour of the single impurity. Caroli (1967) has shown, using an Anderson model, that two impurities at a distance r modify their energy position in the Fermi sea and the width of their virtual bound states according to a change 8pd(EF)of the d density of st at es,
(#1is a virtual bound state parameter). As this quantity will be positive or negative, it will be expected for long distances that there will be an equal probability of increasing or decreasing the impurity magnetism (Tournier 1974). Magnetization experiments have clearly shown the existence of ferromagnetic pairs, antiferromagnetic pairs are not directly detected as the pairs appear as non-magnetic. Good examples are given in CuFe, AuFe - by a Cz magnetic contribution which can be detected down to temperatures as low as TK/3000.(Tholence and Tournier 1970; Hirschkoff et al. 1971b; Frossati et al. 1976c.) This discussion neglects the influence of the spin polarization. At relatively high concentrations it is well known that the molecular field HM created by the surroundings can induce a magnetic moment initially blocked by a Kondo coupling. For a strongly enhanced matrix like Pd or Pt, the interaction of pair impurities will be restricted mainly to the ferromagnetic range A of their spin polarization since (i) the giant moment can be regarded as a large scattering centre of radius r = A with a low electronic and spin polarization outside their giant moment range L; (ii) the complex Fermi surface leads to destructive interference of the spin and electronic polarization. Finally in a first neighbour coupling depending on the sign of the electronic response, ferromagnetic pairs or triplets (AuCo) - or antiferromagnetic pairs (AuV) may occur.
-
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5.2. SENSITIVITY OF NO TO INTERACTION EFFECTS 5.2.1. High sensitivity of the NO experiments to any local misalignment
In order to show the NO sensitivity to any misalignment (2,fi) between the local axis 2 of quantization and the macroscopic i? axis defined by the applied field, we have drawn for different temperatures in fig. 27 the axial response H,,,(O) of a 6oCo nucleus submitted to a hyperfine coupling Hhf = - 200 kOe along a 2 axis which forms an angle 0 with the axis of the counter. Although the corresponding magnetization response along H given by M~
=
s cOS
e,
s
H,,~,
is independent of the temperature, the NO response depends strongly on the temperature as suggested by figs. 3-6. Assuming a constant angle of mis-
Fig. 27. Influence of a misalignment angle. He,, describes at different temperatures the 6oCoNO response to a misalignment 0 between the macroscopic axis, defined by the appliedfield, and the local axis 2 of quantizationof the Co nuclei, submitted to a hyperfine coupling of Hbl= 200 kOe (Taurian 1974).
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alignment (defined by a vertical line), we observe that the effective field response leads to a decrease of Hefras the temperature decreases (fig. 27). We have already pointed out that for a single impurity behaviour, the two regimes A 2 keTKare described by an effective field which is independent of the temperature. The above arguments show that when the symmetry of the external field breaks down locally, striking results appear in the temperature dependence of the effective field. NO is thus an excellent test of an eventual local breakdown of the macroscopic symmetry defined by H.
5.2.2. Low sensitivity to a site decomposition In a concentrated alloy, different types of sites may occur; their resolution is poor by NO which is an integral method. For example, if 6oCo nuclei in equal proportion are respectively submitted to a hyperfine coupling of -200 kOe and - 120 kOe along the same 2 axis, without special caution, the experimentalist associates an unique effective field He,, to the measured anisotropy which is the average of that produced by each class of impurity. Between 20 and 10 mK, the relative variation of 6Heff/He,,is only equal to 4 %; very low temperatures must be achieved to obtain a better resolution. The discussion can be summarized by underlining that compared with a magnetization experiment: (i) NO is more sensitive to any misalignment (2,I?) than to an attenuation of the average magnetization; (ii) NO has almost the same bad resolution in the decomposition of different sites. Generally in a concentrated alloy only an analysis of concentration effects allows the extraction of the magnetic behaviour of different sites.
5.3. SPINGLASS BEHAVIOUR IN A NO EXPERIMENT 5.3.1. The models
The NO experiment performed by Compton (1968) on a A u ~ sample, ~ M ~ shows the characteristic spin glass decrease of Heff(T)(fig-28). As at that time (1965), the sample was not prepared with special caution, a content C,,, of the most probable iron parasitic impurity was accidentally dissolved during the sample preparation. In order to interpret these experiments, Compton developed a crude model with the following hypotheses: (i) the magnetic coupling is described by an isotropic molecular field H,,
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"ef f in kOe 350
300
250
200
Fig. 28. NO Compton results (1968) on a &uS4Mn alloy.
(ii) the electronic spin is blocked along the vector sum H,of the applied field Hand the molecular field HM.(Its magnetization is completely saturated along H, .) (iii) the nuclear local axis of quantization coincides with the vector axis H,(the nuclei are submitted to their saturation hyperfine field). The axial anisotropy along H is only an average of the anisotropy of each nucleus in agreement with its local axis of quantization H,:
where
is given by eq. (1.1). Performing the integration over all the 8 angles, EH can be simply written
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where a and A,(a) are respectively the ratio H H / Hand the corresponding attenuation coefficient. The first two even values are 5 - 3a2 3(1 -a2)’ A , @ ) = -8 16a
A&)
=
{30(1-~~)’-35(1 -a’)’} 64a
105~~-190~~+81 96
Figure 29 describes (the full lines), for a Mn nucleus submitted to a hyperfine field of -400 kOe the NO response as a function of different values of the a parameter in terms of an effective field. As was emphasized for the H i function, this response will be characteristic of the NO parameters. The main objections to this model are that: (i) for low values of a(-0), the hypothesis of an isotropic molecular field HM seems incorrect; (ii) for a 1, for a certain range of solid angles, H , is too low to achieve the blocking of S ; (iii) in high magnetic fields the direct nuclear Zeeman term N
I
100 I
200
300
I
I
f
inK” t
Fig. 29. Theoretical models for a 54Mnprobe dissolved in a spin glass alloy and submitted to a hyperfine coupling of -200 kOe. The solid lines describe the effective field, computed in the crude Compton model, for different ratios of H M / H ;the dashed lines represent the corrected results taking into account the Zeeman nuclear alignment (b = Hhr/H).The new point is the abnormal decrease of Heti as the temperature decreases.
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must be taken into account, which leads to a nuclear orientation axis different from the electronic axis and consequently to a lower reduction of the effective field. Taking into account all these remarks, numerical calculations can be performed using a maximum angle of disalignment O,,(O < Omax < in) and a supplementary ratio b equal to H,JH (Taurian 1974). Figure 29 represents (dashed lines) the NO response for b = - 10, H = 40 kOe and a random distribution of 8. A more sophisticated isotropic molecular distribution can be used. Table 15 describes the A,, A4 coefficients respectively for an unique molecular field parameter HM = aH, square wave probability and a Lorentzian distribution of width A, = aH. Table 15 The A 2 , A , attenuation coefficients for an unique molecular field HM = aH,a square wave probability up to HM= aH and a Lorentzian distribution of width A M = aH. Unique molecular field
Square probability
Lorentzian probability
a
A2
A4
A2
A4
A2
A4
0.1 0.3 0.5 0.7 0.9 1.I 1.3 1.5 1.7 1.9
0.9900 0.91 16 0.7630 0.5621 0.3434 0.1941 0.1311 0.0956 0.0731 0.0578
0.9670 0.7239 0.3449 0.0142 -0.0849 -0.0209 -0.0083 -0.0042 -0.0023 -0.0014
0.9967 0.9703 0.9192 0.8468 0.7591 0.6678 0.5896 0.5259 0.4738 0.4308
0.9889 0.9048 0.7584 0.5894 0.4452 0.3557 0.2989 0.2583 0.2275 0.2034
0.9742 0.8217 0.6840 0.5807 0.5021 0.4409 0.3923 0.3529 0.3206 0.2934
0.9200 0.6388 0.4702 0.3650 0.2953 0.2466 0.21 1I 0.1843 0.1632 0.1466
As A2(a) and A4(a) follow quite different dependences, the use of two counters perpendicular and parallel to the applied field will be fruitful in the study of the molecular field distribution. (The A, and A4 coefficients are simply related to the axial and equatorial anisotropy by the expressions
The B,U,F, , B4U4F4are known from the paramagnetic behaviour.
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5.3.2. Spin glass behaviour Spin glass behaviour has been previously observed in many NO experiments as a parasitic effect; we have seen examples for very dilute alloys of PdMn and AuMn. Similar effects were observed for PdMn and PdMnH,zloys by TGmson et al. (1975, 1976) and for AuMn alloys by many people who have attempted to prepare a NO thermGeter (Konter 1976; Hiraki and Ono 1974). The transition from single impurity effect to spin glass behaviour is shown on figs. 30 and 31 in the case of very dilute alloys of PdFeX6'Co, Hhf in KOP
FPCO C,(pprn)
0.1 KQ
b
A!&-;, I
?
v 7.8 +' 6 4 021.6
.+
.
I
Fig. 30. Variation of the hypefine field, Hhl = Heff-H, of 6oCo versus T-*for four different applied fields (40 000/3000/500/100 Oe). The different lines represent the corresponding mean values (Flouquet et al. 1977).
-=
< 1 ppm, x < 100ppm). For the more dilute alloys (x 20 ppm), above 500 Oe, an independent value of Herrwith the temperature and the concentration describes a single impurity effect from which the Kondo coupling can be extracted; on the contrary for the alloy of 72 ppm, a spin glass coupling appears clearly. In this range of concentration, the magnetic coupling between the impurities is principally located outside the ferromagnetic polarization carried by the giant moment. Gallop (1968) had previously mentioned that such a behaviour occurs up to a critical concentration, C,= 1000 ppm, which is the concentration where two impurities begin to interact insidetheir ferromagneticpolarization. The same critical concentration appears in the ordering temperature T, of the PdFe alloys (Chouteau and Tournier 1971) for C < C,,T, Cz,and for C, C,,T, C. Similar (E
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conclusions were recently reached by Nagamine et al. (1976) in order to interpret muon precession experiments. Recently, Sanchez et al. (1977) have observed the transition from single impurity to spin glass effects for AgMn alloys and have checked over a large range of concentration (14 to 3000 ppm) the spin glass behaviour. Their main observations are that (i) the NO results can be described by the scaling parameter h = H / c (ii) the best fit is obtained with a Lorentzian distribution centered at H , = 0 with a width A,/C = 480 kOe%.
W'CO
I,
1
m .
GRpprn
+
+ in K-'
+
200
Fig. 31. The temperature dependence of the hyperfine field of the 6oCo diluted in a 72 ppm iron sample shows a characteristic spin glass behaviour. The horizontal lines are the mean values obtained on more dilute samples, fig. 30 (Flouquet et al. 1977).
Similar experiments performed on PtMn* by Benoit et al. (1977a) have shown that (i) the spin glass behaviouroccurs for a concentration up to 1 % and the fit by a Lorentzian distribution is obtained with a width A,/C = 40 kOe% which has an order of magnitude lower than that found for the AgMn alloys (see 5 5.1.3), (ii) above a concentration of 1%, the G is not respected and the behaviour of the alloy is mostly a scaling 1 characteristic of antiferromagnetic short-range interactions between the Mn *For E M n alloys, the NO characteristic of the single impurity shows a temperature dependence of Hell(Benoit et al. 1977; Thompson et al. 1977). The transition between the regimes of the single impurity and the spin glass does not appear directly on the results. This unusual behaviour for a cubic lattice can be interpreted as either a variation of the hyperfine coupling through the Kondo condensation (T. 10 mK) or misalignment effects due to preferential axis. Further experiments on single crystals are being made.
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impurities. Similar conclusions were given by Tholence and Wassermann (1976) performing magnetization experiments; in the spin glass regime, the width A, deduced from the NO results is in excellent agreement with that estimated by Sacli et al. (1974) using their specific heat measurements. Remanent effects were studied on AgMn and PtMn, PtCo alloys by Sanchez et al. (1977) and Benoit et aL(1977a). Qualitatively, the thermoremanent anisotropy was found to be rather low in good agreement with a Heisenberg coupling between the spins. Quantitatively, its magnitude depends strongly on the sample preparation; for example with cold working procedure, the sign of the gamma ray anisotropy was found to reverse below a certain critical value of the applied field, this critical value increasing with increasing Mn concentration. The reversed sign of the anisotropy can be understood as the indication that locally the electronic spins 'prefer to be' perpendicular to the applied field as it occurs for an antiferromagnetic coupling. This behaviour must be further investigated and correlated with the ESR results which were generally interpreted in a simple antiferromagnetic scheme (Owen et al. 1957). On the contrary, for the ferromagnetic alloy of PtCo strong remanent anisotropy was detected with no reversed sign (Ben% et al. 1977a). 5.4. APPEARANCE OF MAGNETISMDUE TO INTERACTIONEFFECTS 5.4.1. A giant moment: PtCo
Figure 32 represents the effective field seen by 6oCo nuclei dissolved in different alloys of PtCo,. For a given magnetic field, the main points are (i) the increase ofx,,, with concentration, and (ii) the constant value of He,, as a function of T(Taurian 1974). They suggest that only ferromagnetic coupling occurs. The Co impurity initially isolated with a TK 1.6 K is coupled with the other impurities which interact through their positive ferromagnetic polarization. As was shown by Tissier and Tournier (1972) and Costa Ribeiro et al. (1974), at low concentration, pairs within the critical ferromagnetic range (r 5 A) of the polarization are more magnetic than the single impurity. A simple picture of the decreasing Kondo tempsrature of the pairs (7'') can be given in a linear response model. So long as T! > T, the Co moment is an induced moment aligned along H. Assuming a constant positive value of the polarization in its ferromagnetic range (fig. a), the amplification m,/mo of the one impurity magnetism, mo = x0H, can be regarded as the
-
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response to (i) the applied field, and (ii) the molecular field H , produced by the other impurities located inside the ferromagnetic cloud with a concentration Ci,, . The following relations can be written
since the polarization is proportional to the impurity xi and lattice xPr susceptibility. At low concentration, the Kondo temperature follows a C2 dependence.
200
I /...
...... . * . . . ..
100
0
Y
H (k10
20
30
L
40
Fig. 32. Hyperline fields points for more concentrated ptCo samples compared to the magnetization curve determined by Tissier and Tournier (1972) for the same samples. The magnetization data were normalized by the same factor as in the fig. 25.
5.4.2. First neighbour interactions: AuCo - alloys
In the last example, the hyperfme coupling defined by H,,,(sat) has the same limit whatever value the Kondo temperature has. This phenomenon corresponds to an impurity which is fully magnetic in the HF sense. An interesting case is the AuCo alloy which is on the borderline of the HF transition. AlthoughTKfor the single impurity is high (-700 K), the low-temperature Knight shift is so large (+29 %) that it can be detected in NO and NMR experiments (Holliday et al. 1971 ; Boysen et al. 1973; Narath and Barharn
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1973). With such a high Kondo coupling, in a normal host, the magnetism of the impurity can be modified only by first neighbour interactions which are well known to be ferromagnetic for the Co impurity (see Moriya 1967). In order to explain their specific heat and susceptibility results, Boucai et al. (1971) proposed a model where pair and triplet impurities are submitted respectively to a Kondo coupling 7’’’ = 20 K and T z ’ = 0 K. With these strong differences in TK, it seems particularly interesting to have a direct check of the Friedel-Anderson prediction which suggests a spin magnetism below the H F instability. The NO method offers the possibility of such studies which are difficult to perform here by NMR due to the extreme line broadening for the paramagnetic impurities and to the possible presence of a quadrupolar broadening when Co neighbours are present. For the first neighbour antiferromagnetic coupling of AuV, Narath and Gossard (1969) managed to observe both pair and single impurity signals by NMR (K“’ = - 1.5%, K(’) = 0.6%). Table 16 Characteristics of the &Co alloys and their corresponding betaparticle asymmetries. Sample preparation
Co conc. (at %) 0.958 1.901 5.184 11.335 11.335
Sample No. 1 2 3 4 5
Relative concentrations of Co species CN,
CN2
89.09 79.43 52.79 23.61 23.61
9.67 16.14 23.86 15.60 15.60
cNj
1.10 3.46 11.58 11.53 11.53
%+ 0.14 0.96 11.76 49.26 49.26
Beta-particle asymmetries from concentrated CoAu alloys oriented at low temperature (constant geometry). C o s (Ifh< negative) is shown for comparison. Sample Co conc. (at %)
Fraction of maw. atoms
11.33
60.8 %
5.18
23.3%
1.901
4.42% 0
100 ppm 3
3
~
0
~
Np (cold)
N )-
0.9948 f 0.0011 0.9850+0.0040 1.0037+0.00?4 1.0006f0.0037 1.0200f0.0017 1 .W62f 0.0020 1.0091+0.O004 0.8267+_0.0012
H
(kW 40.0 20.0 40.0 20.0 40.0 20.0 71.9 10.0
sign (Heft)
-
+ f
+ + +-
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Table 16 represents the results of the fl anisotropy experiment performed by Boysen et al. (1975) on different alloys. The sign of the hyperfine coupling, given by that of the B anisotropy, changes from positive to negative when the number of magnetic triplets increases. Further, by separating the various contributions of the y anisotropy according to the Boucal analysis a net negative Knight shift must be attributed to the Co pairs. This constitutes a direct demonstration in a single alloy system of the increasing dominance of spin magnetism when TKbecomes increasingly low.
1
co
100
0
10
20
90.
40
50
60
70
H,(kOe)
Fig. 33. Effective (net) hyperfine fields He,, at the nuclei of various Co sites in Co& T = 15 mK. Errors represent the scatter among the data points. The dashed line labeled Nlis from Holliday and Weyhmann (1970) and Boysen et al. (1973). These values were obtained on the assumption of a unique, concentration independent, and uniaxial Herr for each Co species. The lines have no theoretical significance and merely indicate linear approximations to the data (Boysen et al. 1975). (see table 15) derived from pray anisotropy data near
Figure 33 represents the effective field attributed to each class of impurities (see table 15). The main points are: (i) of the triplet and quartet atoms does not reach a saturation value although the Grenoble group claims that the corresponding magnetization is well saturated in a field of 20 kOe. (ii) H:f> of the pair atoms does not seem to reach a zero value in zero applied field as would be expected for an induced moment (TE' = 20K). The
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temperature dependence of H,$.4 with 1/T shows that some misalignment occurs in the magnetic triplet (see Boysen et al. 1975; Boysen 1976). Using the correctedCompton model (§5.3.1), Boysen et al. have derived a molecular field strength of 30 kOe for the triplet. Figure 33 indicates that the same strength of internal field parallel to H must be applied to pair atoms in order to understand the possible occurrence of a spontaneous moment. For Boysen et a]. (1975), the random molecular field HM characterizes an interaction coupling among the magnetic triplets which occur below their ordering temperature. We mention here that the dipolar energy ( E r ’ 0.5 K) inside the pair and triplet clusters may giveaninner mechanism of misalignment. This hypothesis is supported by different striking results in the Boucai (1971) and Costa Ribeiro et al. (1971) experiments. The remanent value ar,(0)per magnetic atom of the triplet and quartet atoms is appreciably higher than that of 0.05 pB previously reported for a dilute alloy of AuFe. If the Holtzberg et al. (1976) analysis is applied to the triplet and quartet results, the number no of atoms inside one cloud, defined by
-
is near the number of atoms which build up each triplet and quartet (no 4). The fundamental question not yet resolved is whether the triplet and quartet form a well-defined quantum object, with an angular momentum indicating the low value of no 4 (no n,, the Klein-Brout number). According to the recent observation by Holtzberg et al. (1976) of a remanent magnetization for the pair of europium atoms which occur in a dilute sample of Eu2Sr,,S,,,, the dipolar energy barrier may exist well above the triplet or quartet ordering temperature. The occurrence of an anisotropic dipolar potential will lead to an internal spin configuration which minimizes the exchange and dipolar coupling; the three moments carried by the triplet are not perfectly parallel to each other but form an angle which minimizes the total energy of the triplet quantum object. As in zero field, a pair contribution seems to appear in the nuclear specific heat experiment of Costa Ribeiro et al. (1971), which can detect only the occurrence of a spontaneous moment, the quantum pair object itself is perhaps no. described by a single Kondo parameter Tg’ = 20 K since this value forbids the presence of a spontaneous moment in zero field; some other mechanism may be present which leads to some coherence among the Co atoms and formation of pairs. N
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5.5. KONDOLATTICE OR MAGNETICORDERING IN CERIUM COMPOUNDS The cerium intermetallic compounds present generally a Kondo behaviour at high temperature or a valence instability (Parks 1977). The first cases are characterized by a high coefficient y of the linear temperature specific heat. For example, AI,Ce has a y coefficient, equal to 1800mJ/T2, two orders of magnitude higher than that of Pd. The open question is if at low temperatures the Ce moments undergo a Kondo lattice or a magnetic ordering (Jullien et al. 1977). Benoit et al. (1977b) have recently performed NO experiments on Ce nuclei of the intermetallic compounds AI,Ce, M2Ce, In,Ce which have y coefficients equal to 1800, 180, 140 mJ/T2, respectively. In low field, the gamma ray anisotropy has the sign predicted by a magnetic coupling along the applied field for the N,Ce compound but a reversed sign, characteristic of an antiferromagnetic-like behaviour, for the other alloys. A1,Ce appears to be a Kondo lattice (see Andres et al. 1975) as the effective field is linearly induced by the applied field; the other compounds are ordered but, for A1,Ce lattices, the average magnetic moment of cerium atoms does not reach its full ionic value within the crystal field but reaches an intermediate value defined by the competition of the Kondo effect and of the exchange field. NO experiments performed on the intermediate valence compound of Sn2Ce show that the magnitude and the sign of the gamma ray anisotropy correspond to a brute force NO mechanism along the applied field (Benoit et al. 1977b): at low temperatures, the Ce atoms appear non-magnetic as it occurs for a high Kondo temperature. N
6. Other applications
6.1, USEOF NO IN LOW
TEMPERATURE PHYSICS
6.1.1. Thermomerry
Below 40mK, NO thermometers are very useful and simple to set up. Primary thermometry can be performed if (i) all the hyperfine and nuclear parameters of the probe are well known, and (ii) the chosen alloy is prepared without parasitic effects. In a ferromagnetic host like Fe, Ni, Co, these conditions are fulfilled for the 54Mnand 6oConuclei, the most commonly used as thermometers; the hyperfine fields of the probes have been determined by conventional NMR
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or NMR/NO methods. Generally, relatively high magnetic fields must be used in order to be sure that a single axis of alignment can be defined. Even for single crystals oriented along their easy magnetization axis, fields above 3 and 5 kOe must be applied respectively for the cubic NiMn and FeMn thermometer. A Co single hexagonal crystal provides the possibility o f e r o field thermometry (see Lounasmaa 1974 for general references). The conditions (i) and (ii) cannot be assured for non-magnetic hosts which are sensitive to long-range disturbances (see $5 1.3.2 and 5). The main advantage of dilute alloys is their adjustable sensitivity through a chosen applied field value (Pratt et al. 1971). The alloy must be chosen mostly to minimize the interactions among the magnetic impurities of the probe (54Mn)and of the parasitic Fe atoms. Two alloys are suitable: PtMn which has a very low ordering temperature (T, 0.4% of Mn) (Tho%nce and Wassermann 1976); and CuMn where the Fe atoms have a high Kondo temperature and so a small i z e n c e on the 54Mn probe. On the other hand thermometry carried out with an Aus4Mn sample was found to provide variable results (0 5.3); below 15 m z each sample must be checked with another thermometer. N
6.I .2. Magnetic Kapitza resistance
One of the difficultiesin achieving good thermal contact between a liquid and a solid is the poor thermal conductance at the interface. At low temperature, the thermal impedance, defined by
where is the power applied to the warm part, S the interface area, 6T the temperature difference between the two materials, varies generally with a T - law which is predicted for a phonon conductance. In 1966, the discovery by Abel et al. that the impedance between the cerium magnesium nitrate salt and the liquid 3He is an order of magnitude lower than predicted by the T - 3 formula, led to the suggestion that the magnetic coupling between a localized moment (the Ce paramagnetic centre) and the nuclear spin of the liquid gives a more efficient mechanism than the phonon conductance (Leggett and Vuorio 1970). In metals such effects were observed later by Avenel et al. (1973) and analyzed carefully by Mills and Beal Monod (1975). The magnetic conductivity increases as T-' at low temperature for pure paramagnetic impurities;
CH. 9,
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its influence decreases below the ordering temperature T,among the localized moments since such a coupling blocks its degree of freedom. In order to increase the thermal contact between 3He and a dilute alloy, the important parameters to consider are: (i) a low T,per fraction of impurity which may allow a rather high content of impurities; (ii) a high value of the magnetic moment. Some of the most convenient alloys are those with a rare earth impurity like Gd3+ dissolved in a pure metal (Pd, Au, Ag, Pt) since these alloys fulfill both conditions. A 3d giant moment fulfills the second condition but leads unfortunately to stronger interaction effects; the only exception is given by the PtMn - alloys, but unfortunately the existence of a Kondo coupling (TK 10 mK) certainly reduces the magneticcontribution below TK. Recently Klein (1977) has developed pulsed eddy current heating of a radioactive thin metal foil in contact with 3He/4Hein order to measure the nuclear spin lattice relaxation of dilute 6oCo in Fe and J4Mn in Au. This method seems important for the study of Kapitza resistance.
-
6.2. INNUCLEAR PHYSICS
An excellent example of the interconnection between different fields is provided by the determination of the 7mCenuclear moment. Its initial value (p = 0.69 p,) was estimated by adjusting the high temperature value of the entropy S(T) of the cerium magnesium salt measured in calorimetric experiments (Hudson and Kaeser 1967, Mess et al. 1969) to the NO value (Huntziger 1970). With this value, the saturation hyperfine field (Hhf= 560 kOe), which is measured for Ce nuclei dissolved in a lanthanum host, does not correspond to the theoretical value of a r7ground state (630 kOe) by the factorf = 1.12. This discrepancy was resolved by the accurate calorimetric experiment of Fisher et al. (1973) which attributed at high temperature a lower entropy than that previously reported. At low temperature, the correction factor f = 1.17, which must be applied to the NO temperature scale defined by Huntziger, accounts almost exactly for the discrepancy observed in the Lace alloy. The NO study of radioactive impurities was performed in different nonmagnetic hosts in order to obtain pure nuclear information: typical examples are given by the measurement of the multipole mixing ratio of different transitions in 56Gd(Uluer et al. 1975) dissolved in a gold lattice. Generally, one of the nuclear transitions gives directly the parent nuclear parameter BK,
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the study of the other decays gives the nuclear information on the U,F, coefficient. Up t o now, we have considered a simple decay scheme without any disturbance produced in the intermediate levels. The study of Au' 69Yb alloys has led to spurious effects since (i) thermal reorientation m a y occur in the intermediate levels, with 36 and 660 nanoseconds lifetime, and (ii) a supplementary attenuation factor may arise in the 36 nanoseconds intermediate state if that state is not a pure 11, M ) state (Steenberg, 1963). Table 17 gives the results obtained in three laboratories in terms of the first even integral attenuation G2 observed for both mechanisms in applied fields close to 5 kOe (G, is defined by E(0) = G2g,CJ,F,B2). Table 17 Attenuation of the anisotropy of the 198 keV gamma ray emitted by the lS9Ybnuclei as a function of the Yb concentration.
10 OOO 50 <1
0.11 0.20 0.44
Krane et al. (1972) Pernuk (1976) Benoit (1974a)
As this attentuation depmds strongly on the impurity concentration (difference between the samples of 10 000 and 1 ppm) or on the metallurgical preparation (difference between the samples of 50 and 1 ppm), important effects seem to arise from the local symmetry of the surroundings. Table 17 shows clearly the advantage of a great dilution; solid state information was obtained by Benoit et al. (1974a) using the '"Yb probe which avoids the existence of auxiliary attenuations.
7. Conclusion NO studies of dilute alloys have shown that the Kondo coupling occurs as a general phenomenon for quite different impurities (3d, 4f) and lattices (normal and enhanced matrix). The interesting nature of the NO study of localized magnetic states is that the local properties are due partly to nonlocalized electron states.
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735
Spin glass studies can be performed by NO. In parallel with magnetization experiments, fruitful results could be obtained. The NO limitation to very low temperatures does not allow the study of the temperature dependence of the Kondo coupling. Such studies have been extensively performed by other methods like magnetization, Massbauer effect, NMR, resistivity, and specific heat. For example, the low temperature excitation, with a T z term in CuFe, was firstly detected by the specific heat experiment of Triplet and P h i F p s (1971) using the Maxwell relation to link the field variation of C to the susceptibility. Such a dependence suggests that the excitation spectrum of the impurity in a Kondo singlet is like the Fermi liquid excitation as shown by Nozi6res (1974) and Yoshida and Yamada (1975). [Such behaviour was previously noticed in NMR results (Narath 1971).] The Kondo problem seems now resolved for the non realistic case of a spin one half. The crudeness of such a description appears clearly when an attempt is made to correlate numerically basic properties like the specific heat and the susceptibility. One of the important aspects of the Kondo effect is that it has led experimentalists to consider fundamental questions concerning the significance of observables. This has, in turn, led to progress in the understanding of the connection between the measured quantities and the parameters of physical importance to the theoretician, a connection which earlier often appeared to be rather tenuous. Similarly, from the purely experimental point of view, the attempt to study isolated impurities has necessitated a detailed analysis of impurity interactions and of metal physics problems, and to the critical evaluation of the possibilities of various experimental methods. For example, the NMR experimentalist, who attempts to observe a well-defined paramagnetic impurity through the matrix resonance, is now able to separate several satellite lines in a CuMn alloy. Figure 34 represents the twelve satellite lines observed by AllGl(1976)ina CuMn - alloy: an unfamiliar picture for a NO experimentalist who can only observe an integral curve with his technique. Another important aspect is that stimulation from the Kondo effect work has led to the performance of a large variety of experiments. The AuYb alloy is an excellent example. The initial EPR result of Hirst et al. (19%) and the susceptibility experiments of Gainon et al. (1967) suggested that the AuYb alloy may correspond to an antiferromagnetic exchange coupling. Accurate results were then reported: the EPR shift measurement of Tao et al. (1971), which proved the negative sign of J, stimulated the Massbauer experiment of Gonzales and Imbert (1972) which they were later (1973) able to under-
736
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stand (by the occurrence of a deviation to the ordinary Korringa relaxation time) following the observation of a resistivity Kondo minimum by Murani (1973). In order to check the strength of the Kondo coupling, Benoit et al. (1974a) performed new NO experiments (following Spanjaard et al. 1971) which have shown (i) the influence of the defects produced during implantation, and (ii) the weakness of the Kondo coupling. This last effect has
Fig. 34. Satellite resonance in @Mn alloys (Alloul and Hippert 1976).
tempted Frossati et al. (1976b) to perform magnetization experiments using different isotopes, and Hebral et al. (1976) to observe the resistivity. With all this physical information, Fert and Friedrich (1974) performed skew scattering experiments, Follstaedt et al. (1975) observed for the first time the NMR resonance of a strongly magnetic impurity and Lasjaunias et al. (1976) have recently carried out nuclear specific heat measurements. Finally one of the difficulties among physicists is to find a language which is understood by a non-specialist. It is best for the NO people, in order to
CH.91
METALS WITH MAGNETIC IMPURITIES
737
open their ‘private club’ to show clearly what they measure by (i) drawing the y ray angular distribution of the observed nuclei, and (ii) quoting the sensitivity of detection of the different effects (single impurity-interactionmixing of a magnetic and quadrupolar interaction). Acknowledgements
I wish especially to thank Dr W. Brewer, Dr I. A. Campbell, Prof. J. Friedel, Dr K. Ishii, Dr M. Ribault and Dr J. Souletie for stimulating discussions particularly during the course of writing this paper. Particularly thanks are also due to Dr A. Benoit, Dr J. Boysen, Dr J. Sanchez and Dr 0. Taurian for their collaboration. I have especially benefited from the initial work performed in Oxford by Dr J. Gallop and Dr I. R. Williams who gave me the first NO ideas in the Clarendon Laboratory. It is a pleasure to thank Dr K. Andres, H. Alloul, Prof. A. Blandin, Dr M.Chapellier, Dr B. Coqblin, Dr P. Monod, Dr J. L. Tholence, Dr D. Thoulouze and Dr R. Tournier for numerous discussions. I am greatly indebted to Dr Alquie, Dr Alloul and Dr Mignot for permission to use their unpublished results. Appendix Free spin function H i of 54Mn computed for S = 4, g = 2 &(sat) = -400 kOe. The successive columns give the values of B2,B4,E(O), Herr, 1/T,<&>, for seven values of H ( g 2 = 0.9800 and g4 = 0.9300).
1.44
1.44 1.43 1.40 1.36 1.30 1.24 1.18 1.12 1.05 0.99 0.93 0.87 0.82 0.77 0.72 0
0.639 0.638 0.623 0.584 0.528 0.464 0.401 0.342 0.296 0.244
0.205 0.173 0.146 0.123 0.104 0.089
0.965 0.964 0.953 0.924 0.880 0.827 0.772 0.717 0.664 0.614 0.568 0.525 0.486 0.450 0.418 0.388
358 358 358 358 358 358 358 358 358 358 358 358 358 358 358 358
1Ooo.o 500.0
333.3 250.0 200.0 166.7 142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4 66.7 62.5
-2.50 -2.50 -2.50 -2.50 -2.50 2.50 2.50 -2.50 - 1.50 -2.50 -2.50 -2.50 -2.50 -2.50 -2.50 -2.50
-
- 3.00 -2.99 -2.99 -2.97 -2.94 -2.90 -2.85 -2.80 -2.75 -2.69 -2.63 -2.57 -2.51 -2.45 -2.39 -2.33
[CH.9
J. FLOUQUET
738
Appendix-Continued 0.67 0.63 0.59 0.56
0.076 0.065 0.055 0.048
0.361 0.336 0.313 0.293
358 358 358 358
58.8 55.6 52.6 50.0
-2.50 -2.50 -2.50 -2.50
-2.27 -2.22 -2.16 -2.11
0.639 0.638 0.626 0.594 0.544 0.484 0.423 0.365 0.313 0.266 0.226 0.192 0.163 0.138 0.118 0.101 0.086 0.074 0.063 0.055
0.965 0.965 0.956 0.931 0.892 0.844 0.793 0.740 0.689 0.641 0.595 0.563 0.514 0.479 04 6 0.416 0.388 0.362 0.339 0.318
377 377 377 377 377 377 378 378 378 378 379 379 379 379 380 380 380 380 380 381
1Ooo.o 500.0 333.3 250.0 200.0 166.7 142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4 66.7 62.5 58.8 55.6 52.6 50.0
-2.50 -2.50 -2.50 -2.50 -2.49 -2.49 -2.49 -2.49 2.49 -2.49 -2.99 -2.49 -2.49 -2.49 -2.49 -2.49 -2.49 -2.49 -2.49 -2.49
-3.00 -2.99 -2.99 -2.97 -2.95 -2.91 -2.87 -2.82 -2.78 -2.72 -2.67 -2.62 - 2.56 -2.51 -2.45 -2.40 -2.34 -2.29 -2.24 -2.18
0.639 0.634 0.602 0.542 0.468 0.393 0.326 0.267 0.218 0.177 0.144 0.118 0.096 0.079 0.064 0.053
0.965 0.962 0.938 0.891 0.831 0.768 0.706 0.647 0.593 0.544 0.500 0.460 0.423 0.391 0.361 0.334 0.310 0.287 0.267 0.248
298 298 298 300 302 303 305 307 308 310 311 313 314 315 316 317 317 318 318 318
1Ooo.o
-2.50
-3.00 -2.99 -2.98 -2.95 -2.90 -2.85 -2.80 -2.74 -2.68 -2.62 -2.56 -2.50 -2.44 -2.38 -2.32 -2.26 -2.21 -2.15 -2.10 -2.04
H - 3000kOe
1.44 1.44 1.43 1.41 1.37 1.32 1.27 1.21 1.15 1.09 1.03 0.97 0.92 0.86 0.81 0.77 0.72 0.68 0.64 0.60
-
H = 0.500 kOe 1.43 1.43 1.41 1.37 1.31 1.24 1.17 1.10 1.03 0.97 0.90 0.84 0.79 0.73
0.68 0.64 0.60 0.56 0.52 0.49
0.044
0.036 0.030 0.025
500.0 333.3 250.0 200.0 166.7 142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4 66.7 62.5 58.8 55.6 52.6 50.0
-2.49 -2.49 -2.49 -2.49 -2.49 -2.48 -2.85 -2.48 -2.47 -2.47 -2.47 -2.46 -2.46 -2.45 -2.45 -2.45 -2.44 -2.43 2.43
-
CH.91
METALS WITH MAGNETIC IMPURITIES
739
Appendix-Continued
H = 0.200kOc 1.44 1.42 1.35 1.27 1.17 1.08 0.99 0.91 0.83 0.76 0.69 0.63 0.57 0.52 0.47 0.43 0.39 0.36 0.33 0.30
0.639 0.607 0.521 0.418 0.325 0.248
0.187 0.141 0.106 0.080 0.061 0.046 0.035 0.026 0.020 0.015 0.012 0.009 0.007 0.005
0.965 0.941 0.875 0.790 0.706 0.629 0.560 0.500 0.448 0.402 0.362 0.326 0.294 0.266 0.241 0.218 0.198 0.179 0.163 0.148
205 205 210 214 218 221 224 227 229 230 232 233 233 233 233 233 232 231 229 228
10oo.o
0.957 0.868 0.743 0.628 0.531 0.451 0.386 0.331 0.285 0.246 0.214 0.185 0.161 0.141 0.123 0.108 0.095 0.08 0.07 0.06
137 137 142 147 151 153 155 157 158 158 158 157 156 155 153 151 149 147 145 142
10oo.o 500.0 333.3 250.0 200.0 166.7 142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4 66.7 62.5 58.8 55.6 52.6 50.0
500.0 333.3 250.0 200.0 166.7 142.9 125.0 111.1 100.0 90.9 83.3 76.9
71.4 66.7 62.5 58.8 55.6 52.6 50.0
-2.49 -2.49 -2.48 -2.41 -2.45 -2.43 -2.41 -2.40 -2.37 -2.35 -2.33 -2.31 -2.28 -2.26 -2.23 -2.21 -2.18 -2.15 -2.127 -2.098
-2.99 -2.98 -2.93 -2.87 -2.80 -2.72 -2.64 -2.57 -2.49 -2.41 -2.34 -2.26 -2.19 -2.12 - 2.05 -1.98 -1.92 -1.85 - 1.79 -1.72
H = 0.100kOc
1.43 1.35 1.21 1.08 0.95 0.83 0.73 0.64 0.56 0.48 0.42 0.37 0.32 0.28 0.25 0.22 0.19 0.17 0.15 0.13
0.628 0.513 0.365 0.248 0.165 0.110 0.073 0.049 0.033 0.022 0.015 0.010 0.007 0.005 0.003 0.002 0.002 0.001 0.001 0.Ooo
-2.49 -2.47 -2.43 -2.38 -2.33 -2.27 -2.22 -2.16 -2.10 -2.05 -1.99 -1.93 - 1.87 -1.82 -1.76 -1.71 -1.65 -1.60 - 1.56 - 1.51
-2.99
-2.93 -2.83
-2.72 -2.61
-2.49 -2.38 -2.21 -2.16 -2.06 - 1.96 -1.86 -1.77 - 1.68 -1.60 - 1.52
-1.44 -1.37 - 1.30 -1.23
J. FLOUQUET
740
[CH. 9
Appendix-Continued
H = 0.050 kOe 1.38 1.16 0.94 0.75 0.60 0.48 0.39 0.31 0.25 0.21 0.17 0.14 0.12 0.10
0.08
0.553 0.315 0.163 0.085 0.045
0.025 0.014 0.008 0.005 0.003 0.002 0.001 0.Ooo O.OO0 O.OO0
0.899 0.695 0.525 0.402 0.312 0.245 0.195 0.157 0.127 0.104 0.086 0.071 0.060 0.050
0.043
78
10oo.o
85
500.0
89 92 93 94 94 94 93 92 91 90 89 87 85
333.3 250.0 200.0 166.7 142.9 125.0 111.1 100.0 90.0 83.3 76.9 71.4
38 41 42 42 42 42 41 40 40 39 38 37 37 36 35 34 34 33 32 31
10oo.o
-2.47 -2.37 -2.25 -2.13 -2.00 -1.88 - 1.78 -1.66
-1.30 -1.23 -1.16 -1.11
-2.95 -2.79 -2.59 -2.40 -2.21 -2.03 - 1.87 -1.72 -1.58 - 1.46 -1.35 - 1.25 -1.15 -1.07 -1.00
-2.31 -1.98 - 1.66 -1.41 -1.21 - 1.05 -0.93 -0.83 -0.75 -0.69 -0.63 -0.59 -0.55 -0.51 -0.48 -0.45 -0.43 -0.40 -0.39 -0.37
-2.14 -2.30 -1.89 -1.57 -1.32 -1.13 -0.98 -0.86 -0.77 -0.69 -0.62 -0.56 -0.51 -0.47 -0.43 -0.40 -0.33 -0.37 -0.24 -0.30
- 1.55 - 1.46 - 1.38
H = 0.020 kOe 1.10 0.67 0.41 0.26 0.18 0.12 0.09 0.07 0.05
0.04 0.03 0.02 0.02 0.01
0.01 0.01 0.01
0.00 0.00 0.00
0.270 0.066 0.019 0.006 0.002 0.001 O.OO0
O.OO0 O.OO0 O.OO0 O.OO0 O.OO0 O.OO0 0.Ooo 0.Ooo
0.o00 O.OO0 0.Ooo
0.648
0.356 0.210 0.133 0.089 0.063 0.046 0.034 0.026 0.021 0.016 0.013 0.011 0.009
0.007 0.006 0.005
0.o00
0.004 0.004
0.Ooo
0.003
500.0
333.3 250.0 200.0 166.7 142.9 125.0
111.1 100.0 90.9 83.3 76.9 71.4 66.7 62.5 58.8 55.6 52.6 50.0
CH. 91
METALS WITH MAGNETIC IMPURITIES
741
References Abel, W.R., A.C. Anderson, W.C. Black and J.C. Wheatley, 1966, Phys. Rev. Lett. 16,273. Abragam, A. and M.H.L. Pryce, 1951,Roc.Roy. Soc.London AZOS, 105. Abrikosov, A.A., 1965, Physics 2,21. Abrikosov, A.A.and L.P. Gorkov, 1961, Sov. Phys. JETP12,lUl. Ali, M., W.D. Brewer, E. Klein, A. Benoit, J. Flouquet, 0. Taurian and J.C. Gallop, 1974,Phys. Rev. 10,4659. Alloul, H., 1975, AIP Conference29,300; Phys. Rev. Lett. 35,540. Alloul, H., 1976,Physica 86-88B,449. Alloul, H. and F. Hippert, 1976, to be published. Alquie, G., A. Kreissler and J.P. Burger, 1976, Journal Less Common Metals 49, 97 and to be published. Amamou, A., R. Caudron, P. Costa, J.M. Friedt, F. Gautier and B. Loegel, 1976, J. Phys. F. to be published. Anderson, P.W., 1961, Phys. Rev. 124,41. Anderson, P.W., 1967, Many-body Physics, Les Houches (Gordon and Breach, New York) p. 229. Andres, K., 1976,Physica 8688B, 1071. Andres, K., J.E. Graebner and H.R. Ott, 1975,Phys. Rev. Lett. 35,1779. Avenel, O., M.P. Berglund, R.G. Gylling, N.E. Philipps, A. Vetleseter and M. Vuono, 1973, Phys. Rev. Lett. 32,76. Bagguley, D.M.S. and J.A. Robertson, 1974, J. Phys. F4,2282. Barclay, J.A. and B. Perczuk, 1975a,Hyp. Interactions 1,15. Barclay, J.A. and B. Perczuk, 1975b,Sol. State Comm. 17,565. Benoit, A., J. Flouquet and J. Sanchez, 1973, Sol. State Comm. 13,1581. Benoit, A., J. Flouquet and J. Sanchez, 1974a,Phys. Rev. B1,4213. Benoit, A., R. Delaplace and J. Flouquet, 1974b,Phys. Rev. Lett. 32,222. Benoit, A., 1976,Thesis Orsay, unpublished. Benoit, A. and J. Flouquet, 1976, Physica8688B, 519. Benoit, A., J. Flouquet, J.L. Tholence and E.F. Wassermann, 1977a,to be published. Benoit, A., J. Flouquet, M. Ribault and M. Chapellier, 1977b, to be published. Blandin, A. and J. Friedel, 1959,J. de Physique 20,160. Blandin, A., 1961, Thesis Orsay, unpublished. Blandin, A., 1973, Magnetism V,eds. Rado and Suhl (Academic Press, New York) p. 58. Blandin, A., 1976, private communication. Bleaney, B., 1951, Proc.Phys.Soc.A64,315. Blin Stoyle, R.J. and M.A. Grace, 1957, Handbuch der Physik 41,555. Bouca’l, F., B. Lecoanet, J.L. Tholenceand R. Tournier, 1971, Phys. Rev. B3,3834. Boyce, J.B. and C.P. Slichter, 1974, Phys. Rev. Lett. 32,61. Boyce, J.B. and C.P. Slichter, 1975, AIP Conference29,335. Boysen, J., W.D. Brewer and J. Flouquet, 1973, Sol. StateComm. 12,1095. Boysen, J., W.D. Brewer and E. Klein, 1975, Hyp. Interactions 1,55. Boysen, J., 1976, Diplomarbeit Berlin, unpublished. Bozorth, R.M., Wolff, P.A., Davis, D.D., V.B. Compton and J.H. Wernick, 1961, Phys. Rev. 122,1157. Brewer, W.D., 1969,private communication. Brewer, W.D., 1974, Phys. Lett. 49A, 397. Cameron, J.A., LA. Campbell, J.P. Compton, M.F. Grant, R.W. Hill and R.A.G. Lines 1964,Proc. LT9 (Plenum Press,New York) p. 1033.
742
J. FLOUQUET
[CH. 9
Campbell,LA.. 1968,J. Phys. C. 1,687. Campbell, I.A., J.P. Compton, I.R. Williams and G.V.H.Wilson, 1967, Phys. Rev. Lett. 19,1319. Compton,J.P., R.A.G. Linesand G.V.H. Wilson, 1966,Phys. Lett.20,569. Cannella, V., J.A. Mydosh and J.I. Budnick, 1971, J. Appl. Phys. 42,1689. Caplin, 1972, unpublished,see Rizutto 1974, Rep. Prog. Phys. 37,147. Caroli, B., 1967,J. Phys. Chem. Solids28,1427. Chouteau, G. and R. Tournier, 1971,J. Phys. C32,1002. Collings,E.W.,F.T. HedgcockandY. Muto, 1964,Phys.Rev. 134,A1521. Compton, J.P., I.R. Williams and G.V.H. Wilson, 1968, Hyperfine Structure and Nuclear Radiations(North-Holland, Amsterdam) p. 793. Coqblin,B. and A. Blandin, 1968, Ad. Phys. 17,281. Coqblin, B. and J.R. Schrieffer, 1969,Phys. Rev. 185,847. Cornut, B. and B. Coqblin, 1972, Phys. Rev. B, 4541. Costa Ribeiro, P., J. Souletieand D. Thoulouze, 1970, Phys. Rev. Lett. 24,900. Costa Ribeiro, P., M. Saint Paul, D. Thoulouze and R. Tournier, 1974, Proccedings LT12, eds. Timmerhaus,O'Sullivan and Hammel (Plenum Press, New York) p. 520. Craig, P.P.,D.E.Nagle, W.A. Steyert andR.D. Taylor, 1972, Phys. Rev. Lett.9,12. Crangle, J. and W.R. Scott, 1965, J. Appl. Phys. 26,921. Daniel. E. and J. Friedel, 1963,J. Phys. Chem. Solids 24,1601. Daniel, E. and J. Friedel, 1965, LT9,933. Daniels, J.M., M.A. Grace and F.N.H. Robinson, 1951,Nature 168,780. Daybell. M.D., W.P. Pratt and W.A.Steyert, 1969, Phys. Rev. Lett. 22,401. Doran, J.C. and Q.G. Symko, 1974, Sol. State Comm. 14,719. DwSrin, L.and A. Narath, 1970, Phys. Rev. Lett. 23,1287. Edwards, S.F. and P.W. Anderson, 1975,J. Phys. F. 5,965. Ericson, T., M.T.Hiwonen, T.E. Katila and V.K. Typpi, 1970, Sol. State Comrn. 8,765. Ericson, T., M.T. Hiwonen, T.E. Katila and P. Reivari, 1971, LT12 (Academic Press of Japan) p. 766. Eska, G., E. Hagn and K. Andres, 1976, to be published. Felsch, W., K. Winzer and G.V. Minugerode, 1975, Z. Phys. B.21,151. Fert, A., A. Friedrich, 1974,AIP Conference24,466. Fischer, K.H., 1976. Physica, ProceedingsICM 76,II, 813. Fisher, R.A., E.W. Hornung, G.E. Brodale and W.F. Giauque, 1973, J. Chern. Phys. 58,5584. Flouquet. J. and D. Marsh, 1970a, Phys. Lett. 32A, 501. Flouquet, J., 1970b, Phys. Rev. Lett. 25,288. Flouquet, J., 1971,J. Phys. F 1,87. Flouquet, J., 1971a,Thesis Orsay. unpublished. Flouquet, J., 1971b, Phys. Rev. Lett. 27,515. Flouquet, J., 1973,Ann. Phys. 8,5. Flouquet, J. and J. Sanchez, 1974, Proceedings LT12, eds. Tirnmerhaus, O'Sullivan and Hammel (Plenum Press, NewYork) p. 520. Flouquet, J. and W.D. Brewer, 1975, Physica Scripta 11,199. Flouquet, J., 0.Taurian, J. Sanchez, M. Chappellier and J.L. Tholence, 1977a, Phys. Rev.Lett.38,81. Flouquet, J., M. Ribault, 0. Taurian, J. Sanchez and J.L. Tholence, 1977b, Phys. Rev., to be published. Follstacdt, D.M., W.J. Meyer, D.C. Barham and A. Narath, 1975, AIP Conference29,354. Freeman,A.J. andR.B. Frankel, 1967, HyperEne Interactions(AcademicPress, New York).
CH. 91
METALS WITH MAGNETIC IMPURITIES
743
Freeman, A.J. and R.E. Watson, 1965, Magnetism 11, eds. Rado and Suhl (Academic Press, New York) p. 167. Friedel, J., 1952, J. Phil. Mag. 43,153. Friedel, J., 1956, Can. J. Phys. 34,1190. Friedel, J., F. Gautier, A.A. Gomes and P. Lenglart, 1966, Quantum Theory of Atoms, Molecules and the Solid State, eds. Liiwdin (Academic Press, New York). Friedel,J., 1976, private communication. Frossati, G., J.M. Mignot, D. Thoulouze and R.Tournier, 1976b, Phys. Rev. Lett. 36,203. Frossati, G. and D. Thoulouze, 1976a, ICEC6(I.P.C.). Frossati, G., J.L. Tholence,D. Thoulouze and D. Tournier, 1976c, Physica B84,33. Gainon, D., P. Donzeand J. Sierro, 1967, Sol. State Comm.5,151. Gallop, J.C. and I.A. Campbell, 1968, Sol. State Comm. 6,831. Gallop, J.C., 1970, Thesis Oxford, unpublished. GonzalezJimenez, F. and P. Imbert, 1972, Solid State Comm. 11.861. Gonzalez Jimenez, F. and P. Imbert, 1973, Solid State Comm, 13,85. GonzalezJimenez, F., F. Hartmannand P. Imbert, 1974, Phys. Rev. B10,2122. Gorter, C.J., 1948, Physica 14,504. Gorter, C.J., J.O. Poppema, M.J. Steedand and J.A. Beun, 1951, Physica 17,1050. Gotze, W. and P. Schlottmann, 1974, J. Low Temp. Phys. 16,87. Grace, M.A., C.E. Johnson, N. Kurti, R.G. Serwlock and R.T. Taylor, 1955, Comm. Conf. Phys. Basses Temp. Paris 261. De Groot, S.R.,H.A. Tolhoeck and W.J. Huiskamp, 1965, Alpha, beta and gamma spectroscopy, ed. K. Siegbahn (North-Holland,Amsterdam). Gruner, G. and A. Zawadowski, 1974, Rep. Prog. Phys. 37,1497. Hanson, M., 1976, Thesis Gtiteborg, unpublished. Hartmann-Boutron,F., 1975, AM. Phys. 9,285. Hasegawa, H., 1969, Prog. Theoret. Phys. 21,483. Hebral, B., K. Matho and J.M. Mignot, 1976, to be published. Hiraki, T. and K. Ono, 1974, J. Phys. Soc.Japan 36,1205. Hirschkoff,E.C., O.G. Symko and J.C. Wheatley, 1971, J. Low Temp. Phys. 5,155. Hirschkoff, E.C., M.R. Shanabarger, O.G. Symko and J.C. Wheatley, 1971, J. Low Temp. Phys. 5,545. Hirst, L.L., G. Williams, D. Griffiths and B.R. Coles, 1968,J. Appl. Phys. 39,844. Hirst, L.L., 1970, Phys. Kondens Materie 11,225. Hirst, L.L., 1971, Z. Physik245,378. Hirst. L.L., 1974, AIP Conference 24,ll. Holliday, R.J. and W. Weyhmann, 1970, Phys. Rev. Lett. 25,243. Holtzberg, F., J.L. Tholence and R.Tournier, 1976, Proc. Amorphous Magnetism (Troy) (Pergamon Press) to be published. Huber, J.G., W.A. Fertig and M.B. Maple, 1974, Solid State Comm. 15,453. Hudson, R.P. and R.S. Kaeser, 1967, Physics 3,95. Huntziger,J.J. and D.A. Shirley, 1970, Phys. Rev. 2,4420. Ishii, H., 1970, Prog. Theor. Phys. 40,578. Ishii, H., 1976, Prog. Theor. Phys. 55,1373. Jullien, R.,J. Fields and S Doniach, Phys. Rev. Lett. 38,1500(1977). Kalvius, G.M., T.E. Katila and O.V. Lounasmaa, 1970, MBssbauer Effect Methodology 5,531 (Plenum Press, New York). Kitchens, T.A., W.A. Steyert and R.D. Taylor, 1965, Phys. Rev. 138,467. Kitchens, T.A. and R.D. Taylor, 1974, Phys. Rev. B9,344. Khutsishvili,G.R., 1955, Zh.Eksperim i Teor. Fiz. 29,894.
744
J. FLOUQUET
[CH. 9
Klein, E., 1977, Hyper. Inter. to be published. Klein, M.W. and R.Brout, 1963, Phys. Rev. 132,2412. Kondo, J., 1961,J. Phys. Soc.Jap. 16,1690. Kondo, J., 1964, Prog. Theor. Phys. 32,37. Konter, J.A., 1976, Thesis Leiden, unpublished. Kouvel, J.S., 1963, J. Phys. Chem. Solids24,795. Krane, K.S., C.E. Obsen and W.A. Steyert, 1972, Nuclear Physics A197,352. Krishnamurthy, H.R.,K.G. Wilsonand J.W. Wilkins, 1975, Phys. Rev. Lett. 35,1101. Lagendijk,I., L. Niesen and W.J. Huiskamp, 1969, Phys. Letters MA,326. Larkin, A.I. and De Khmelnitzkii, 1970, Sov. Physics JETP31,958. Lasjaunias,J.C., J.M. Mignot and D. Thoulouze, 1976, to bepublished. Le Dang Khoi, Veilletp and LA. Campbell, 1976, J. Phys. F. 6, L197. Lederer, P. and D.L. Mills, 1968, Phys. Rev. 165,837. Legeett, A.J. and M. Vuorio, 1970,J. Low Temp. Physics 3,359. Levitt, D.A. and R.E.Walstedt, 1977, Phys. Rev. Lett.38,178. Li, P.L. and W.R. Muir, 1974, Proceedings LT13, p. 495. Loram, J.W., G. WilliamsandG.A.Swallow, 1971, Phys. Rev. B3,3060. Lounasmaa, O., 1974, Experimental Principles and Methods Below 1 K (Academic Press, New York) p. 1974. Lubbers, J. and W.J. Huiskamp, 1967, Physica34,193. Lutes, O.S. and J.L. Schmidt, 1964,Phys. Rev. 134A, 676. Maley, M.P. and R.D. Taylor, 1970, Phys. Rev. 1,4213. MacLaughlin, D.F. and H. Alloul, 1977, Phys. Rev. Lett. 38,181. Manhes, B., 1971,Thesis, Grenoble. unpublished. Marsh, J.D., 1970, Phys. Lett. 33A,207 Marsh, J.D., 1972,Thesis Oxford, unpublished. Marshall, W., 1960,Phys. Rev. 118,1519. Maple, M.B., 1976, Appl. Phys. 9,179. Matthias, E. and R.J. Holliday, 1966,Phys.Rev. Lett. 17,897. Matho, K., 1976, to be published. Mills, D.L. and M.T. Beal Monod, 1975, AIP Conference29,6. Moriya, 1967, Marshalled. Prw. Varenna School 1966(AcademicPress, New York) p. 206. Murani, A.P., 1973, Solid State Comm. 12,295. Murnick, D.E., J.A. Fiory and W.J. Kossler, 1976, Phys. Rev. Lett. 36,100. Nagamine, K., N. Nishida, R.S.Hayano and Y.Yamazaki, 1976, Physica 86-88B, 489. Narath,A., 1971, Proc. LT12, ed.E. Kanda, p. 675. Narath, A. and A.C. Gossard, 1969, Phys. Rev. 183,391. Narath, A,, 1973a, Magnetism Vol. V, ed. Rado and Suhl (Academic Press, New York). Narath, A. and B. Barham, 1973b, Phys. Rev. B7,2195. Narath, A., 1976, Phys. Rev. B13,3724. NM, L., 1961, Physique des Basses Temphtures, Les Houches (Gordon and Breach, New York). Newrock, R.S., B. Serin, J. Vig and G. Boato, 1971, J. Low Temp. Phys. 5,701. Nozi&res,P., 1974,J. Low Temp. Phys. 17,31. Orbach, R.,1975, P r m d i n g s LT14, Vol. V, ed. Krusius and Vuorio (North-Holland, Amsterdam)p. 375. Orbach, R,. M. Peter and D. Shaltrel, 1974, Proceedings Conference of Haute Neulaz, eds. G. Cohen and B. Giovannini. Arch. Sc. Genhve 27,241. Owen, J., M.E.Browne, V. Arp and A.P. Kip, 1957,J. Phys. Chem. Solids 2,85. Perez-Ramires, J.G. and P. Steiner, 1977, J. Phys. F, to be published.
CH. 91
METALS WITH MAGNETIC IMPURITM
745
Perez-Ramires, J.G., L.K. Thomas and P. Steiner, 1977, J. Low Temp. Phys. 26,84. Parks, R.D., 1977, ed. Valence Instabilities and Related Narrow band Phenomena (PlenumPress,New York). Pena, R. Tournier, A. Benoit and J. Flouquet, 1977, Sol. State Comm. 21,971. Perczuk, B., 1975, Thesis Monash University, unpublished. Pound, R.V., 1949, Phys. Rev. 76,1410. Pratt, W.P., R.I. Shermerand W.A. Steyert, 1969, J. Low Temp. Phys. 1,469. Rose, M.E., 1949, Phys.Rev. 75,213. Ruddermann, M.A. and C. Kittel, 1954, Phys. Rev. %,99. Sacli, O.A., D.J. Emerson and D.F. Brewer, 1974, J. Low Temp. Phys. 17,425. Samoilov, B.N., V.V. Sklijarevskii and E.P. Stepanov, 1959, Soviet Phys. JETP36,1383. Sanchez, J., J. Flouquet and W.D. Brewer, 1977, Proceedings IV International Conference on Hyperline Interactions (North-Holland,Amsterdam)to be published. Schmidt, R.W. and I.S. Jacobs, 1957, J. Phys. Chem. Solids 3,324. Schrieffer, J.R. and P.A. Wolff, 1966, Phys. Rev. 149,491. Senoussi, S., A. Fert and LA. Campbell, 1977, Solid State Comm. 21,269. Shirley, D.A., 1966, Annual Rev. of Nucl. Science 16,89. Souletie, J., 1968, Thesis Grenoble, unpublished. Souletie, J. and R. Tournier, 1969, J. Low Temp. Phys. 1,95. Souletie, J. and R. Tournier, 1971, J. de Physique C1,172. Spanjaard,D. and F. Hartmann, 1973, J. Phys. F3,1178. Spaqiaard, D., R.A. Fox, J.D. Marsh and N.J. Stone, 1971, In: Hyperfke Interactions and Excited Nuclei, eds. Goldring and Kalish. p. 113. Stakelon, T., 1974, Thesis University of Illinois, unpublished. Steiner, P. W. Zdrojewski, D. Gumprecht and S. Hufner, 1973a, Phys. Rev. Lett. 31, 355 (1973). Steiner, P., G.N. Beloerskij, D. Gumprecht, W.G. Zdrojewski and S. Hufner, 1973b. Sol. State Comm. 13,1507. Steiner, P. and S. Hufner, 1975, Phys. Rev. 12,842. Stemberg, N.R., 1963, Phys. Rev.95,982. Stone, N., 1976, Hyp. Interact. 2,45. Tao, L.J., D. Davidov, R. Orbach and E.P. Ch.wk, 1971, Phys. Rev. B4,5. Taurian, O., 1974, Thesis Orsay, unpublished. Tholence, J.L. and R. Tournier, 1971. J. Physique 32C, 211. Tholence, J.L. and R. Tournier, 1970, Phys. Rev. Lett. 25,867. Tholence, J.L. and R. Tournier, 1974, J. Physique 35, (3,229. Tholence, J.L., 1973, Thesis Grenoble, unpublished. Tholence, J.L. and Wassermann, 1976, Physica 86-88B,875. Thomson, J.O. and J.R. Thomson, 1975, A I P Conference 29,342. Thomson, J.O. and J.R. Thomson, 1976, AIP Conference, to be published. Thomson,J.O., J.R. Thomson, P.G. Huray, S. Nave and T.L. Nichols, 1977, to be published. Tissier, B., R. Tournier, 1972, Sol. StateComm. 11,895. Tissier,B., 1977, Thesis Grenoble, unpublished. Tournier, R., 1965, Thesis Grenoble, unpublished. Tournier, R., 1974, Proceedings LT12 (1973) Vol. 2, ed. Timmerhaus, O'Sullivan and Hammel (Plenum Press, New York) p. 257. Triplett, B.B. and N.E. Philipps, 1971, Phys. Rev. Lett. 27,1001. Uluer, I., C.A. Kalfas, W.D. Hamilton, R.A. Fox, D.D. Warner, M. Finger and Do Kim Chung, 1975, J. Phys. G 1,476. Violet, C.E. and R.J. Borg, 1966, Phys, Rev. 149,540.
746
J. FLOUQUET
[CH.9
Vochten. M., M. Labro and S.Vynchier, 1976,Physica 8688B, 76. Waldstedt, R.E. and A. Narath. 1972,Phys. Rev. B6,4.118. De Waele, A.Th.A.M., A.B. Reckers and H.M. Gijsman, 1976,Physica 81B,323. Williams, I.R., 1968,Thesis Oxford, unpublished. Williams, I.R., LA. Campbell, S. Sanctuary and G.V.H. Wilson, 1970,Sol. State Comm. 8,125. Wilson. K., 1974, Proceedings of the Nobel Symposium XXIV, ed. B. Lunqvist and L.Lunqvist (Academic Press,New York) p. 68. Wu. C.S.. E. Ambler, R.W. Hayward, D.D. Hoppes and R.P. Hudson, 1957,Phys. Rev. 105,1413. Yamada, K., 1975,Prog. Theor. Phys. 53,970. Yosida, K. and K. Yamada, 1975,h o g . Theor. Phys. 53,1286. Yosida, K., 1957,Phys. Rev. 106,893. Yosida, K. and Yoshimori, 1973. Magnetism V, eds. Rado and Suhl (Academic Press, New York) p. 253. Zittartz, J. and E. Muller-Hartmann, 1970,Z. Physik 232,ll. Zener, C.,1951,Phys. Rev. 81,440.
