Progress in Low Temperature Physics, Vol. 12

PROGRESS I N LOW TEMPERATURE PHYSICS

XI1

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PROGRESS IN LOW TEMPERATURE PHYSICS EDITED BY

D.F. BREWER Professor of Experimental Physics University of Sussex, Brighton

VOLUME XI1

1989

NORTH-HOLLAND AMSTERDAM. OXFORD. NEW YORK . TOKYO

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(Revised for vol. 12) Gorter, C.J. (Cornelius Jacobus) Progress in low temperature physics. (Series in Physics) Vol. 12 edited by D.F. Brewer. Vol. 12 has imprint: Amsterdam; New York: North-Holland; New York, N.Y.: Sole distributors for the U.S.A. and Canada: Elsevier Science Pub. Co. has imprint: Amsterdam, North-Holland Pub. CO.; New York, lnterscience Publishers. 1 . Low Temperatures. 1. Brewer, D.F. (Douglas Forbes) 11. Title. 111. Series. 536.56 55-14533 QC278.G6 PRINTED IN THE NETHERLANDS

PREFACE

It may seem eccentric to start the twelfth volume of Progress in Low Temperature Physics with a chapter on high-temperature superconductivity, however distinguished its author. When the discovery of high- T, superconductivity (defined by Ginzburg in Chapter 1 as Tc> 30-40 K) in rather complicated compounds was announced, many low-temperature physicists must have decided that its development was now in the hands of the expert materials scientists, and that their attention was better placed elsewhere. Nevertheless, if this were to mark a sharp decline of low-temperature physicists’ interest in superconductivity, it seemed appropriate to have some sort of obituary in the form of, as Ginzburg quotes from my letter to him, “an article of more general interest giving an historical perspective of the various speculations in the past on the possibility of such (HTS) superconductors, and the possible mechanisms for the superconductivity in the recently discovered materials.” Professor Ginzburg responded with la chapter of characteristic style - and with the short response time characteristic of Soviet authors to this series - which is well worth reading for more than its scientific interest. Other chapters illustrate the wide range of physics which are more usual low-temperature topics. Study of spin polarized quantum systems was much stimulated by a conference with this title held at Aussois in 1980. A considerable amount of work on ’He has been done in France, and the chapter on spin polarized ’He gas by Betts, Laloe and Leduc is a collaboration between authors from Paris and Sussex. It follows two other related articles, on spin polarized atomic hydrogen in Volume X, and on 3He-4He solutions in Volume XI. Chapter 3 is another article on helium, by Nakayama, dealing with Kapitza thermal boundary-resistance at, mainly, millikelvin temperatures. Although this is another characteristic, although not exclusively, low-temperature phenomenon, which is technically very important in the problem of thermal contact, it has not yet received attention previously in this series. The increasing availability of millikelvin temperatures emphasizes not only its practical importance but also the growing interest in the detailed mechanisms of the interaction of 3He quasiparticles with solid surfaces, a topic which was also treated in Volume XI. The subject of charge density waves discussed by Gruner in Chapter 4 is one of much current interest in metal physics. As such, it is one which many would assert is not really a topic for a low temperature physics review. The arid philosophical argument about what low temperature physics is, if V

vi

PREFACE

anything, has not deterred me from including the chapter here. It follows previous articles on metal physics in Volumes X and XI. The final chapter, from Helsinki, on multi-SQUID devices and their applications, represents somewhat of a departure in subject matter for this series. Previous volumes have concentrated almost exclusively on the physics of low temperatures, partly because application in the liquid-helium temperature range are not extensive, apart from superconducting magnets. The advent of the Josephson effect changed that, and the economic and social climate is inclined to favour applications: IUPAP in its sponsored conferences, for example, specifically encourages the inclusion of applications in the programme. To return to high-temperature superconductivity: as experimental information and theory have progressed, it seems to be emerging that the physical mechanism may be quite different from the usual BCS superconductivity, and that measurements at quite low temperatures (tens of millikelvins) may be very important in determining what it is. So there are still some interesting and rewarding things to be done by low-temperature experimental physicists. Not quite salue, perhaps, but at least not quite vale, and I think I can promise at least one more contribution to the series in the future. As usual, I am very grateful to colleagues for discussions about topics of current interest in the physics of low temperatures, to the authors who gave their valuable time to writing, and to the publishers, especially Peter de ChPtel and Anita de Waard, for their help. To future authors - if they are there - I should like to extend my anticipatory thanks, and to remind them that inopi benejcium bis dar, qui dot celerirer. Sussex, 1989

D.F. Brewer

CONTENTS VOLUME XI1 Preface ..................................................... Contents .................................................... Contents of previous volumes ...................................

Ch. 1 . High-temperature superconductivity: some remarks. V.L.Ginzburg..........................................

V

vii xi

1

1. Introduction ................................................... uestions of priority .................................. 2. Significance 3. Experimental studies of superconductivity . .. 4 . Microscopic theory of superconductivity ................................

5 . Critical temperature in the BCS theory

......

6. Superconductivity mechanism in the "neon" s 7. Superconductivity mechanism in the "nitrogen" superconductors . Ways for

raising T,

..............................................................

...................................... 8. Exciton mechanism of HTS . .. 9 . Nature of HTS in metal oxides ............................................ 10. Concluding remarks ..... References ........................................

................... .................

21 26 32 41 41

Ch. 2. Properties of strongly spin-polarized 'He gas. D.S. Befts. F.Laloeand M . Leduc .................................. Introductory remarks ........................................................ 1. The0ry .................................................................. 1.1. Introduction .............................................. I ............................................. 1.3. Characteristic lengths ................................................ 1.4. Transport properties ..................................... 1.5. Classical transport theory ............................................. 1.6. Quantum mechanical transport theory 1.7. Thermal conductivity ................................................ 1.8. Viscosity ..................................... 1.9. Spin diffusion ....................................................... 1.10. Equation of state .................................................... 2. Polarisation methods ...................................................... 2.1. Optical polarisation of 'He nuclei ..................................... 2.2. Relaxation processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Experiments on spin rotation effects and spin waves ..................... 3.2. Experiments on thermal conductivity . . . . . . vii

47 47 47 51 52 54 56 59 66 67 70 73 78 78 89 93 93 98

...

CONTENTS

VIII

3.3. Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Liquid-gas equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ... , _ . .. ...... ........

106 109 111

Ch. 3. Kapitza thermal boundary resistance and interactions of helium quasiparticles with surfaces, Tsuneyoshi Nakayama . . . . . . . . . . 115 1. lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Kapitza thermal boundary resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definition of the Kapitza resistance R , . . . . 2.2. General expression for the Kapitza resistanc 2.3. Anomalous behaviour of the observed Kapitza resistances 2.3.1. The liquid 'He-sintered powder interface at mK tem 2.3.2. The liquid He-bulk solid interface above about 3. Fenni liquid theory of the Kapitza resistance.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Heat transfer due to zero-sound excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Emission of zero-sound from a small particle into liquid 'He 3.1.2. Heat flux from a spherical particle at temperature 7 . .. . . . . . 3.1.2.1. Energy current into liquid ' H e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.2. Kapitza conductance t o zero-sound excitations . . ..... 3.2. Energy transfer due to inelastic scattering of single quasiparticles interface . ,.............. .................. ..... 4. Anomalous Kapitza resistance between sintered powder and liquid 'He at mK temperatures ......................... .................................. 4.1. Heat exchanger using submicrometer metal particles . . . . . 4.2. Soft phonon-modes in sintered powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Heat transfer due to the effect of soft phonon-modes . 5. The magnetic channel of heat transfer between sintered PO 5.1. Surface characteristics of submicron metal particles and surface magnetic .. impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 5.2. Heat transfer due to magnetic coupling at mK temperatures uid 'He-4He mixtures and sintered

............................... ............ .................................... 6.2. Magnetic channel . . . . .

....................................

117 118 118 120 124 124 126 128 128 128 130 130 132 135 140 140 141 144 149 149 152 158 159 162 165

7.1. Kapitza conductance k, and phonon transmission coefficient across the

................................................

165 168 171 175 176

.................................................

179 186 187 189

7.2. High-frequency phonon scattering at the liquid He-solid interface . . . . . . . . . . 7.2.1. Cause of diffuse scattering at the surface without liquid He . . . . . . . . . . 7.2.2. Specular versus diffuse scattering of bulk phonons . . . . . . . . . . . . . . . . . . 7.2.3. Diffuse signals in the time-of-flight reflection signals. . . . . . . . . . . . . . . . . 7.2.4. Reduction of the diffuse signal at the solid surface in contact with 8. S u m m a r y . . . ...................................................... ........... ... Appendix . . . . . . . . . . . . . . . . . . . . . References . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS

ix

Ch. 4. Current oscillations and interference efects in driven charge density wave condensates. G. Griiner ......................

195

1. Introduction ............................................................. 2. Basic notions and observations in charge density wave dynamics ............... 2.1. The charge density wave ground state and model compounds .............. 2.2. The dynamics of the collective mode .................................... 2.3. Frequency and field dependent transport ................................ 3 . Current oscillations ....................................................... 3.1. General features ............ ..................................... 3.2. Current-frequency relation ............................................. 3.3. Size effects and fluctuation phenomena .................................. 3.4. Broad band noise .................................................... 4 . Models of charge density wave dynamics .................................... 4.1. The classical particle model ............................................ 4.2. Models including the internal degrees of freedom ........................ 4.3. The tunneling model .................................................. 5. Interference phenomena ................................................... 5.1. Harmonic mode locking ............................................... .................................... 5.2. Subharmonic mode locking ... 5.3. Nonsinusoidal and pulse drives ........................................ 5.4. Fluctuations and coherence enhancement ................................ 6 . Conclusions .............................................. List of review papers ........................................................ References .................................................................

197 201 201 205 211 217 217 220 223 226 227 227 232 237 239 239 246 255 259 262 265 266

Ch. 5. Multi-SQUID devices and their applications. Risto Ilmoniemi andJuhha Knuutila .....................................

271

1. Introduction ............................................................. 2 . SQUIDs ................................................................. 2.1. Single-junction (rf) SQUIDs ........................................... 2.1.1. General ........................................................ 2.1.2. Rf SQUID in the hysteretic mode ................................. 2.1.3. Discussion ...................................................... 2.2. Double-junction (dc) SQUIDS ......................................... 2.2.1. Operation ...................................................... 2.2.2. Problems with practical devices ................................... 2.2.3. The state of the art .............................................. 2.3. Electronics ........ ........................................ 3. Applications: biomagnetism ................................................ 3.1. Measurement techniq .............................. 3.1.1. Magnetically shielded rooms ..................................... 3.1.2. Gradiometers ................................................... 3.2. Neuromagnetism ..................................................... 3.2.1. Origin of neuromagnetic fields .................................... 3.2.2. Spontaneous activity ............................................

273 273 274 274 276 279 280 280 284 286 289 292 293 293 295 296 297 298

X

CONTENTS

3.2.3. Evoked fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Clinical aspects of MEG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Cardiac studies . . . . . .................................... 3.4. Other biomagnetic ap .............................. 3.5. Multichannel neuromagnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Optimization of multichannel neuromagnetometers . . . . . . . . . . . . . . . . . . 3.5.2. Existing multichannel systems ....................... ........... 3.5.3. Planar gradiometer arrays . . . . ..................... 3.5.4. Use of multichannel magnetometers . . . . . 4 . Other multi-SQUID applications . . . . . . . . . . . . . . . . . . . . . . . .......... 4.1. Geomagnetism .................... 4.2. Physical experiments . . . . . . . . . . . . . . 4.2.1. Accelerometers and displacem 4.2.2. Monopole detectors ....................... ........... 5 . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 302 303 304 305 310 319 323 326 326 328 328 329 332 333

Author Index

................................................

341

Subject Index

................................................

361

CONTENTS OF PREVIOUS VOLUMES

Volumes I-VI, edited by C.J. Gorter Volume I (1955)

I.

The two fluid model for superconductors and helium 11, C.J. Gorter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16 Application of quantum mechanics to liquid helium, 11. R.P. Feynman ................................... 17-53 111. Rayleigh disks in liquid helium 11, J.R. Pellam.. . . . . . 54-63 IV. Oscillating disks and rotating cylinders in liquid helium 11, A.C. Hollis Hallett . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-77 V. The low temperature properties of helium three, E.F. Hammel ........................................ 78-107 VI. Liquid mixtures of helium three and four, J.M.Beenakker and K.W. Taconis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-137 The magnetic threshold curve of superconductors, B. VII. Serin ........................................... 138-1 50 VIII. The effect of pressure and of stress on superconductivity, 151-158 C.F. Squire ...................................... Kinetics of the phase transition in superconductors, T.E. IX. Faber and A.B. Pippard . . . . . . . . . . . . . . . . . . . . . . . . . . 159- 183 Heat conduction in superconductors, K. Mendelssohn 184-201 X. The electronic specific heat in metals, J.G.Daunt . . . . 202-223 XI. Paramagnetic crystals in use for low temperature XII. research, A.H. Cooke.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 224-244 XIII. Antiferromagnetic crystals, N.J. Poulis and C.J. Gorter 245-272 x IV. Adiabatic demagnetization, D. de Klerk and M.J. Steenland ............................................ 272-335 xv. Theoretical remarks on ferromagnetism at low temperatures, L. NCel . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 336-344 XVI. Experimental research on ferromagnetism at very low temperatures, L. Weil.. . . . . . . . . . . . .. . . . .. . . .. . . . . . 345-354 XVII. Velocity and absorption of sound in condensed gases, A. van Itterbeek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355-380 XVIII. Transport phenomena in gases at low temperatures, J. 381-406 de Boer ......................................... xi

nii

CONTENTS OF PREVIOUS VOLUMES

Volume I1 (1957) I. 11.

Ill. IV. V. VI. VII.

VIII. IX. X. XI. XII. XIII. XIV.

Quantum effects and exchange effects on the thermodynamic properties of liquid helium, J. de B o e r . . . . . . . . Liquid helium below 1"K, H.C. Kramers . . . . . . . . . . . . Transport phenomena of liquid helium I1 in slits and capillaries, P. Winkel and D.H.N. Wansink.. . . . . . . . . Helium films, K.R. Atkins . . . . . . . . . . . . . . . . . . . . . . . . . Superconductivity in the periodic system, B.T. Matthias Electron transport phenomena in metals, E.H. Sondheimer .......................................... Semiconductors at low temperatures, V.A. Johnson and K. Lark- Horovitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The De Haas-van Alphen effect, D. Shoenberg . . . . . . Paramagnetic relaxation, C.J. Gorter . . . . . . . . . . . . . . . Orientation of atomic nuclei at low temperatures, M.J. Steenland and H.A. Tolhoek . . . . . . . . . . . . . . . . . . . . . . Solid helium, C. Domb and J.S. Dugdale . . . . . . . . . . . Some physical properties of the rare earth metals, F.H. Spedding, S. Legvold, A.H. Daane and L.D. Jennings The representation of specific heat and thermal expansion data of simple solids, D. Bijl.. . . . . . . . . . . . The temperature scale in the liquid helium region, H. van Dijk and M. Durieux . . . . . . . . . . . . . . . . . . , . . . . . .

1-58 59-82 83-104 105-137 138-150 151- 186 187-225 226-265 266-291 292-337 338-367 368-394 395-430 43 1-464

Volume 111 (1961) 1. 11. 111.

IV. V. VI. VII.

VI11. 1x. X.

Vortex lines in liquid helium 11, W.F. Vinen.. . . . . . . . Helium ions in liquid helium 11, G. Careri . . . . . . . . . . The nature of the 301-transition in liquid helium, M.J Buckingham and W.M. Fairbank. . . . . . . . . . . . . . . . . . . Liquid and solid 'He, E.R. Grilly and E.F. Hammel 'He cryostats, K.W. Taconis . . . . . . . . . . . . . . . . . . . . . . . Recent developments in superconductivity, J. Bardeen and J.R. Schrieffer ............................... Electron resonances in metals, M.Ya. Azbel' and I.M. Lifshitz ......................................... Orientation of atomic nuclei at low temperatures IT, W.J. Huiskamp and H.A. Tolhoek . . . . . . . . . . . . . . . . . . . . . . Solid state masers, N. Bloembergen . . . . . . . . . . . . . . . . The equation of state and the transport properties of the hydrogenic molecules, J.J.M. Beenakker . . . . . . . . .

1-57 58-79 80-1 12 113-152 153-169 170-287 288-332 333-395 396-429 430-453

CONTENTS OF PREVIOUS VOLUMES

XI.

Some solid-gas equilibria at low temperatures, Z. Dokoupil .......................................

xiii

454-480

Volume IV (1964)

I. 11.

111.

IV.

V. VI . VII.

VIII.

IX.

X.

Critical velocities and vortices in superfluid helium, V.P. Peshkov ........................................ Equilibrium properties of liquid and solid mixtures of helium three and four, K.W. Taconis and R. de Bruyn Ouboter ........................................ The superconducting energy gap, D.H. Douglass, Jr. and L.M. Falicov ................................ Anomalies in dilute metallic solutions of transition elements, G.J. van den Berg ......................... Magnetic structures of heavy rare-earth metals, Kei Yosida ......................................... Magnetic transitions, C. Domb and A.R. Miedema ... The rare earth garnets, L. NCel, R. Pauthenet and B. Dreyfus ......................................... Dynamic polarization of nuclear targets, A. Abragam and M. Borghini ................................. Thermal expansion of solids, J.G. Collins and G.K. White .......................................... The 1962 3He scale of temperatures, T.R. Roberts, R.H. Sherman, S.G. Sydoriak and F.G. Brickwedde . . . . . . .

1-37

38-96 97- 193 194-264 265-295 296-343 344-383 384-449 450-479 480-514

Volume V (1967) I.

11.

111.

I v. V. VI.

The Josephson effect and quantum coherence measurements in superconductors and superfluids, P.W.Anderson ............................................ Dissipative and non-dissipative flow phenomena in superfluid helium, R. de Bruyn Ouboter, K.W. Taconis and W.M. van Alphen ............................ Rotation of helium 11, E.L. Andronikashvili and Yu.G. Mamaladze ..................................... Study of the superconductive mixed state by neutrondiffraction, D. Gribier, B. Jacrot, L. Madhav Rao and B. Farnoux ...................................... Radiofrequency size effects in metals, V.F. Gantmakher Magnetic breakdown in metals, R.W. Stark and L.M. Falicov .........................................

1-43

44-78

79- 160

161- 180 181-234 235-286

xiv

VII.

CONTENTS OF PREVIOUS VOLUMES

Thermodynamic properties of fluid mixtures, J.J.M. Beenakker and H.F.P. Knaap ......................

287-322

Volume VI (1970) I. 11. 111.

IV.

v. v1. VII. VIII. IX.

X.

Intrinsic critical velocities in superfluid helium, J.S. Langer and J.D. Reppy ........................... Third sound, K.R. Atkins and 1. Rudnick . . . . . . . . . . . Experimental properties of pure He3 and dilute solutions of He3 in superfluid He4 at very low temperatures. Application to dilution refrigeration, J.C. Wheatley . . . . . . . . Pressure effects in superconductors, R.I. Boughton, J.L. Olsen and C. P a l m y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superconductivity in semiconductors and semi-metals, J.K. Hulm, M. Ashkin, D.W. Deis and C.K. Jones . . . Superconducting point contacts weakly connecting two superconductors, R. de Bruyn Ouboter and A.Th.A.M. de Waele ....................................... Superconductivity above the transition temperature, R.E. Glover 111 .................................. Critical behaviour in magnetic crystals, R.F. Wielinga Diffusion and relaxation of nuclear spins in crystals containing paramagnetic impurities, G.R. Khutsishvili The international practical temperature scale of 1968, M. Durieux .....................................

1-35 37-76

77-161 163-203 205-242

243-290 291-332 333-373 375-404 405-425

CONTENTS OF PREVIOUS VOLUMES

xv

Volumes Vll-XI, edited by D.F. Brewer

Volume VII (1978) 1.

2.

3. 4. 5. 6.

Further experimental properties of superfluid 3He, J.C. Wheatley ....................................... Spin and orbital dynamics of superhid 3He, W.F. Brinkman and M.C. Cross ............................. Sound propagation and kinetic coefficients in superfluid 3He, P. Wolfle ................................... The free surface of liquid helium, D.O. Edwards and W.F. Saam ...................................... Two-dimensional physics, J.M. Kosterlitz and D.J. Thouless ........................................ First and second order phase transitions of moderately small superconductors in a magnetic field, H.J. Fink, D.S. McLachlan and B. Rothberg Bibby ............ Properties of the A-15 compounds and one-dimensionality, L.P. Gor'kov ............................... Low temperature properties of Kondo alloys, G. Griiner and A. Zawadowski .............................. Application of low temperature nuclear orientation to metals with magnetic impurities, J. Flouquet.. .......

1-103 105- 190 191-281 283-369 371-433

435-516 517-589 591-647 649-746

Volume VIII (1982) 1. 2. 3. 4.

Solitons in low temperature physics, K. Maki.. ...... 1-66 Quantum crystals, A.F. Andreev ................... 67-132 Superfluid turbulence, J.T. Tough . . . . . . . . . . . . . . . . . . 133-220 Recent progress in nuclear cooling, K. Andres and O.V. Lounasmaa ..................................... 221-288 Volume IX (1985)

1.

2. 3.

Structure, distributions and dynamics of vortices in helium 11, W.I. Glaberson and R.J. Donnelly ........ The hydrodynamics of superfluid 'He, H.E. Hall and J.R. Hook ....................................... Thermal and elastic anomalies in glasses at low temperatures, S. Hunklinger and A.K. Raychaudhuri ....

1-142 143-264 265-344

xvi

CONTENTS OF PREVIOUS VOLUMES

Volume X (1986) 1.

2. 3. 4.

Vortices in rotating superfluid 'He, A.L. Fetter.. . . . . . Charge motion in solid helium, A.J. Dahm .......... Spin-polarized atomic hydrogen, I.F. Silvera and J.T.M. Walraven ....................................... Principles of ab initio calculations of superconducting transition temperatures, D. Rainer . . . . . . . . . . . . . . . . .

1-72 73-137 139-370 37 1-424

Volume X I (1987) 1.

2. 3. 4.

5.

Spin-polarized 'He-4He solutions, A.E. Meyerovich Long mean free paths in quantum fluids, H. Smith . . . The surface of helium crystals, S.G. Lipson and E. Polturak ........................................ Neutron scattering by 4He and 3He, E.C. Svensson and V.F. Sears ....................................... Characteristic features of heavy-electron materials, H.R. Ott . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-73 75-125 127-1 88 189-2 14 2 15-289

CHAPTER 1

HIGH-TEMPERATURE SUPERCONDUCTIVITY: SOME REMARKS* BY

V.L. GINZBURG P. N. Lebedev Institute of Physics, USSR Academy of Sciences, Moscow, USSR

* Submitted

to “Progress in Low-Temperature Physics” in November 1987.

Progress in Low Temperature Physics, Volume X I 1 Edited by D.F. Brewer @ Elsevier Science Publishers 9.V., 1989

Contents I . Introduction . . . . . . . . . ................................ 2. Significance of HTS. Questions of priority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental studies o ............... 4. Microscopic theory of superconductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Critical temperature in the BCS theory . . . . . . . . ....... 6. Superconductivity mechanism in the ”neon” sup ................. 7 . Superconductivity mechanism in the “nitrogen” superconductors. Ways for raising

r, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8. Exciton mechanism of HTS . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 9. Nature of HTS in metal o x i d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . emarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.............................................

2

3 3 7 12 15 19

21 26 32

1. Introduction

Reproducible high-temperature superconductors were obtained in 1986-87, as late as 75 years after the discovery of the superconductivity effect. We apply the term “high-temperature superconductors” to those superconductors for which the critical temperature T, exceeds 30-40 K and, what is especially important, is higher than the boiling point of nitrogen under the = 77.4 K. normal pressure, TLIN2 Less than a year has passed (at the time of writing) since the discovery of the “nitrogen” superconductors with T c 380-90 K. So, it is quite natural that the situation in the field of high-temperature superconductivity (HTS) remains, on the whole, not entirely clear, particularly concerning the detailed mechanism of superconductivity. In these circumstances it would be too early to write a review of HTS. The editor of this series thought, however, that the discovery of HTS should be reflected in this volume and he suggested tFat I should write “an article of more general interest giving an historical perspective of the various speculations in the past on the possibility of such (HTS) superconductors, and the possible mechanisms for the superconductivity in the recently discovered materials”. It is precisely this task that I have set for myself. It seems reasonable here to give also some information well known to the specialists for the convenience of the nonspecialist readers. It proved to be rather difficult to write such an article in the style typically employed for scientific papers and reviews. At any rate, what I was writing proved to be some sort of an essay and therefore I decided not to avoid using the first person singular pronouns, and to make digressions concerned with matters of priority and so on. The suitability of such a style in the present publication can, of course, be questioned but in my opinion there are no reasons to try to unify the style of all reviews. As an illustration, note that the Annual Review of Astronomy and Astrophysics (Annual Rev. Inc. California, USA) publishes even quasi-biographical notes by scientists starting from 1975.

2. Significance of HTS. Questions of priority

The discovery of HTS produced an exceptional interest and attracted intense attention in the scientific community and the general public. This discovery is unquestionably highly important for physics and for solid-state physics 3

4

V.L. GINZBURG

specifically. It is also unquestionable that discoveries of no lesser and even greater scientific significance have been made in physics in the last two or three decades. They have not produced such a strong response, though. There is hardly any doubt that the extreme attention drawn by HTS is caused primarily by the understanding of the potential uses of HTS in technology, medicine and so on. Until now the ever-expanding use of superconductors necessarily involved cooling with liquid helium. But there is a relative shortage of helium and it is expensive. For instance, in the USSR a liter of liquid helium costs about 10 roubles. Helium is extracted from natural gas the resources of which are limited and it can be assumed that in a few decades the shortage of helium will become acute. On the other hand, nitrogen is a main component of the atmosphere and it is very easy to separate and liquefy. A liter of liquid nitrogen is cheaper than that of helium by a factor of several hundreds. In addition, nitrogen is a more effective coolant than helium since its heat of evaporation is larger by an order of magnitude. Finally, since helium is so expensive, in most installations the gaseous helium is not discharged into the atmosphere but is recovered, reliquefied and used again. This requirement, in addition to other complications, makes the cryogenic apparatus for helium much more complex and expensive than for nitrogen. Moreover, its reliability is therefore diminished. An accident involving just a discharge of helium into the atmosphere can be catastrophic or, at least, result in the stoppage of the installation involved in the accident. Nitrogen, of course, will be released into the atmosphere and therefore the installations with the nitrogen superconductors will be very much simpler to operate. Thus, the applications of superconductors cooled by liquid nitrogen will increase the economic and the technological benefits by a factor of hundreds if not thousands. At the same time, an improvement in the efficiency by a few percent or even less is regarded as a considerable advance in electrical and heat engineering. It must be added that, according to some indications, a critical temperature T, as high as 200-300 K can be achieved for some superconductors. Such materials are still unstable, though, and it is too early to discuss their application. If, however, materials with T, 200-300 K can be made stable and easy enough to obtain, then the range of HTS applications will be extended even further. Apan from the above reasons, which explain the exceptional attention drawn to the discovery of HTS,it must also be noted that the new materials proved to be quite easy to manufacture - metaphorically speaking they can be cooked in any kitchen. This is why the nitrogen superconductors are being manufactured by schoolchildren less than a year after their discovery (see, for instance, Grant 1987).

-

HIGH-TEMPERATURE SUPERCONDUCTIVITY

5

I m u l d like to note that I had been working on the HTS problem since 1964, always believed in the feasibility of producing HTS, and always promoted the studies in this field. It was precisely the ease with which they have been produced that proved to be unexpected even for me (I believed that after the first success it would take a relatively long time to reproduce the results and, so to say, to master the HTS effect). On the whole, the significance of HTS is clear enough, though I hope that the above remarks were not superfluous. It seems worthwhile to note also the following facts. Firstly, the practical application of the HTS is not something trivial and self-evident, it will require considerable efforts. A wide variety of potential uses may be named now, new ones will be suggested but, in any case, much work is to be done before new devices are designed and deployed. Secondly, it would be quite wrong to think that the helium superconductors (or, to be more precise, the use of the superconductors cooled with liquid helium) are becoming or will become redundant. In addition to purely physical experiments, the low temperatures are needed in various apparatus for reducing the thermal noise (in the classical domain, that is for hw < ksT, the thermal noise for instance in conductors is proportional to the temperature T; here w is the frequency, for instance, the frequency of the circuit being used). The discovery of the HTS, therefore, will by no means detract from the significance of the low-temperature physics though it will of course strongly influence this science as a whole. Since the discovery, analysis and future application of the HTS have a vast, though potential, economic and, hence social effect, it must affect the work style and general attitudes in physics and technology. For this and other reasons, that will be clear from the following discussion, I would like to make a few brief remarks on questions of priority. The questions of priority are an old concern of the scientific community. Suffice it to recall the considerable attention paid by Newton to this subject (see, for instance, Westfall 1982). The history of science tells us of the great efforts wasted in fruitless priority disputes and the pain they brought to their participants. In our time such disputes are, of course, different in form but hardly in substance. A good illustration is given by the endless discussion of the contributions made by Logentz, PoincarC and Einstein to the special relativity theory (I have even written a paper entitled “How and who did create the relativity theory?” which has appeared in several Russian publications and in which I also discuss the general problem of priority; see the collection of papers, Ginzburg 1985). A general discussion of the subject would be out of place here but I would like to tell about an incident that took place during the 18th International Conference of Low-Temperature Physics (LT-18). A “Special symposium on high T, superconductors” was held at a plenary session of the

6

V.L. GINZBURG

conference on August 21, 1987. Five reports were to be delivered at the symposium, but I was not expected to participate. A day before it one of the organizers of the LT-18 had suggested, quite unexpectedly for me, that 1 make a brief report, too. I can only guess at the reasons for this suggestion, but it was obviously well-intended. Therefore I agreed, though I was in a quandary - indeed, what could 1 say in a few minutes after the reports of which 1 did not have any prior knowledge. Bearing in mind the tremendous excitement generated in the HTS studies I thought it would be worthwhile and opportune to remind the audience about the need to be cautious in raising priority questions. With this purpose I demonstrated a transparency with the following statements: Priority questions are dirty business. Priority mania or supersensitivity is illness. I commented briefly on these statements whose meaning seemed quite clear to me - as a person with fifty years' experience of work in physics and astrophysics I recommended not to attach excessive significance to priority questions. Incidentally, I made similar remarks in my introductory report delivered on August 3, 1987 to the 20th International Cosmic Ray Conference held in Moscow. As far as I could see, my statements and comments were then recognized exactly as 1 expected - as a piece of advice presented in a half-joking manner. Unfortunately, the response to my address at the LT-18 was different. A few persons told me (and perhaps many thought but did not tell me!) that they understood it as a complaint or a reproach. Humans are, of course, complicated creatures. It is not rare that people say something while thinking just the opposite. Apparently, some people suspected that while calling for restraint in the questions of priority I was, in fact, pursuing some personal ambitions. If I had expected that kind of misunderstanding I would not, of course, have made this address at all or, at least, would have worded it differently. I think it would be shameful to speak at a conference about one's claims or complaints. I can only make the assurance that I was misunderstood and that I do not have any priority claims or complaints. My work has been generally recognized. In fact, I consider myself to be quite lucky in this respect, since most of my papers and books have been published only in Russian or in translation with small circulation. In addition, I could not attend many international conferences (for instance, apart from the LT-18 I could attend only the LT-10 which was held in Moscow in 1966 though my first paper on physics of low temperatures was published in 1944; see Ginzburg 1944). I hope that the above remarks are relevant and no further comments are needed. 1 would like only to emphasize that when I cite below my papers

HIGH-TEMPERATURE SUPERCONDUCTIVITY

7

and even quote from them I do it exclusively for the sake of information. Such information can prove useful to those interested in the history and sociology of physics and science in general. As for me, I would prefer not to resume discussion of such questions (and, perhaps, I shall have no more chances for doing that) so I decided to use the.occasion of writing this paper for this purpose.

3. Experimental studies of superconductivity*

Let me begin with a brief account of the history of experimental studies on superconductivity. The liquefaction of helium by Kamerlingh Onnes in 1908 was the first milestone in this direction. By the way, up to 1923 liquid helium was produced only at Leiden. In 1911, Kamerlingh Onnes discovered superconductivity in mercury ( T , = 4.1 K). In 1913, the superconductivity of lead ( T, = 7.2 K) was discovered. It was observed in 1914 that superconductivity was destroyed by a magnetic field, the values of the critical field H , ( T ) being not very large (for example, in the case of mercury even at T + 0, when H , has its maximum value, it is about 400 Oe). Naturally, the critical current I, was also found to be small. Thus, immediately after the discovery of superconductivity, two obstacles appeared in the path of its practical application (the first superconducting magnet was constructed by Kamerlingh Onnes in 1914). The first obstacle was the temperature barrier (low values of T,), and the second was the magnetic and current barriers (relatively weak fields H , and currents I,). The magnetic and current barriers were overcome by the discovery in the 1930s of alloys which are now called superconductors of the second type (see Berlincourt 1987). For such materials, superconductivity vanishes only in a field Hc2(T ) which can be enormously high. For example, Hc2(0) 200 kOe for Nb,Sn. As a result (as late as the 1960s) strong superconducting magnets were constructed. For high-temperature superconductors discovered recently, the field Hc2 is generally still stronger. As regards the critical current I,, for high values of Hc2 it is mainly determined by the way materials are treated, that is, by technology. Therefore, there are apparently no serious reasons to fear that there may be any difficulties in principle in overcoming the magnetic and current barriers for high-temperature superconductors. Table 1 shows how the critical temperature T, was raised over the years up to 1973. I would also like to draw attention to some values of boiling

-

* In this section and some other sections I have used some parts of my introductory to the

HTS conference in Trieste delivered on July 5, 1987 (see Ginzburg

1987a).

report

V.L. GlNZBURG

8

Table 1 Material

Year of discovery of its superconductivity

Critical temperature

r, (K) 4.1 1.2 9.2 18.1 20 to 21 20.3 23.2-23.9

191 1 1913 1930 1954 1966 191 1 1973

points and melting points under atmospheric pressure (see table 2). Of course, these facts are well known but they are relevant to our subject. Now let us discuss the attempts to overcome the temperature barrier made since 1973 when the superconductor Nb,Ge with T,= 24 K was discovered. In connection with the recent advances we must note the work on BaPb,_,Bi,03 ceramics carried out since 1974 (Sleight et al. 1975). The highest temperature T,= 13 K for such ceramics is obtained for x =0.25. These and other ceramics had attracted considerable attention even before the latest discoveries as evidenced by more than 200 entries in the reference list of the review by Gabovich and Moisseyev (1986). The reproducible superconducting materials with high temperatures T, discovered in 1986-87 belong precisely to this class of metal-oxide ceramics. We shall discuss the metal oxides below but here we must note that high-temperature superconductors were perhaps produced in 1978 and later though they were “nonreproducible”. These include CuCl (Brandt et al. 1978, Chu et al. 1978, Lekowitz et al. 1979, Vezzoli and Bera 1981), CdS (Brown et al. 1980), Nb-Si films (Ogushi et al. 1983) and some other materials (the latest results for CuCI and CdS are discussed by Lefkowitz

Table 2 Substance Boiling temperature, Tb( K ) under atmospheric pressure Melting temperature T,,,( K ) under atmospheric pressure

He

H,

Ne

N,

0,

H,O

4.2

20.3

27.2

17.4

90.2

373.16

14.0

24.5

63.3

54.1

213.16

HIGH-TEMPERATURE SUPERCONDUCTIVITY

9

et al. 1987, Homan and MacCrone 1987, and Collins 1987). It is often argued (at least by Soviet workers) that the above reports of strong diamag= netic anomalies at temperature above and even much higher than Tb.N2 77.4K are erroneous, that is, they involve no HTS. But the experiments were carried out by various laboratories, particularly for CuCI, and, as far as I know, no errors have been found. Obviously, if the superconducting phases had been observed they were unstable and the conditions required for producing specimens with ‘‘anomalies?’ remain unclear. There is no rigorous proof, therefore, that HTS was observed in CuCl and some other materials before 1986. It would be instructive to analyze the causes for the lack of clarity in this matter. As for me, I never saw any reasons to believe that all the reported diamagnetic anomalies were erroneous. The detection of a large (in magnitude) diamagnetic susceptibility by itself in the absence of a similar sharp anomaly of resistance does not rule out the existence of grains of the superconducting phase that do not form a continuous electric circuit. In principle, superdiamagnetism not associated with superconductivity can also exist (see Ginzburg et al. 1984 and the references cited there). It is more probable, however, that it was precisely the HTS that was observed in those experiments. In view of the latest results, one can suggest, for instance, that the CuCl specimens contained copper oxides. Incidentally, A.P. Rusakov told me that in his later (after 1978) experiments with CuCl he was especially careful about preventing oxidation and perhaps that was why no strong diamagnetic anomalies were then detected. In addition, a large number of experiments have been reported in which quite different (metal oxide) specimens exhibited “nonreproducible” superconductivity-type anomalies at temperatures as high as room temperature (see below). In short, I believe that, quite probably, HTS of the “nonreproducible” kind has been observed, at least after 1978. Incidentally, in materials science nonreproducibility is not a strange phenomenon; it has been known, for instance, for some semiconductors. It was only in 1986 that Bednon. and Muller (1986) reported reproducible superconductivity with T,- 30-40 K for the La-Ba-Cu-0 ceramics. This report was entitled, though, “Possible high T, superconductivity in the Ba-La-Cu-0 system” and, indeed, it lacked the evidence that the resistance R = 0. But, in fact, it was the first reproducible “neon” superconductivity, = 27.2 K. that is, for which T, was higher than the boiling point of neon, Tb,Ne This work was, undoubtedly, of a great importance. I am not acquainted with the details of the subsequent studies in the HTS field carried out in Switzerland, Japan and USA which can be found in the reviews that will be published in Part 3 of the LT-18 Proceedings (the reports at the abovementioned symposium). Further details can be found in the review by Golovashkin (1987). However, I shall mention some milestones on the road

I0

V.L. GINZBURG

to the HTS development. For instance, Cava et al. (1987) reported a fairly sharp and undoubtedly superconducting transition with T, = 36.2 K for the La, $r0 *CuO, ceramics. The “nitrogen” superconductivity with T, > Tb,N2 -- 77.4 K was later obtained by replacing ,,La with ,,Y and some other elements. For the system Y-Ba-Cu-0 Wu et al. (1987) produced superconductivity with T,,, = 93 K (under atmospheric pressure) and T,, = 80 K (at T,,, the resistivity R starts to decrease sharply and at T,, we have R = 0 within the accuracy of the measurements). Similar results were obtained soon afterwards by groups of experimenters in many countries (the first work performed in the USSR is reviewed by Golovashkin 1987). Interestingly, the “neon” superconducting ceramics had been manufactured several years before their superconductivity was discovered. For example, Shapligin et al. ( 1979) produced the above-mentioned La, $r0 ,CuO, ceramics. Shapligin et al. even studied the resistivity of these ceramics but not in liquid helium. Naturally, they did not discover their superconductivity ( I believe that these ceramics did not include any “nitrogen” superconductors). In recent tests these old specimens proved to be superconducting and therefore they experienced little or no ageing. According to my information, similar ceramics had been manufactured before 1986 in Japan and France though they had not been examined for superconductivity. One should learn a lesson from that. Clearly, all new materials with metallic or semiconductor properties must be “tested for superconductivity” in liquid helium, and in successful cases T, must be measured. A wide variety of the “nitrogen” superconductors are now known. A typical one is the YBa,Cu,O,-, system. It has T,=93 K, for instance, for y = 0.1. When y is higher, for example, y = 0.8, the system does not exhibit superconductivity. In this system yttrium can be replaced with a number of other elements without a significant reduction in T,. The “nitrogen” superconductors are generally rather easily obtainable (I mean ceramics rather than single crystals) and, as mentioned above, they have even been manufactured by school pupils (Grant 1987). An intense search of the HTS materials with even higher critical temperatures is going on. Various criteria for “true” or “genuine” superconductivity have been put forward. Under ideal conditions the resistivity R of a genuine superconductor must, of course, vanish and the full Meissner effect (the expulsion of the magnetic field from the interior of the superconductor) must occur in a weak magnetic field. The suggested requirement that the effect be stable in time does not seem necessary. This condition is, of course, significant (perhaps, decisive) from the viewpoint of applications. In terms of physics, however, the metastable phases are by no means inferior to the stable ones

HIGH-TEMPERATURE SUPERCONDUCTIVITY

11

and, in fact, the concept of stability itself is, to a certain extent, arbitrary (for instance, diamond can be regarded as a metastable phase). As for the requirement of reproducibility, it is highly significant for proving the very existence of superconductivity. Let us assume, for the sake of argument, however, that we have accidentally produced a stable material with R = 0 which exhibits the Meissner effect. Obviously, it is a superconductor even if we cannot manage to produce it once again for a long time. The suspicious cases thus involve a combination of instability with nonreproducibility. In any case, at the time of writing (October 1987) I know of no “genuinely” superconducting material whose critical temperature T, has been proved to be much higher than 100 K. Some data on such “potential superconductors” with T c 3 100-150 K were reported to the LT-18 Conference (see Proceedings LT-18). It should also be mentioned that dgushi et al. (1987) reported T,, = 315 K and T,, = 255 K for La-Sr-Nb-0 films. These specimens also exhibited some diamagnetism and a dependence of T, on the current. Huang et al. ( 1987) reported evidence suggesting that specimens of the EuBaZCu306+, system contained inclusions of superconducting phase with T,- 230 K; see also the review by Golovashkin (1987). (A sign of the exceptional importance of HTS is the appearance of the newsletter “High T, update” that is published fortnightly in the USA. The main item in the newsletter is a list of preprints which contained about a hundred titles in the issue No. 9 published on September 1, 1987. At least 2500 papers on HTS will be published a year if the publication rate is kept at this level, and perhaps more since I d o not know the scope of coverage by the newsletter. The newsletter contains also sections for news and other topics. The issue No. 9 reported the production of stable and reproducible specimens of the system Y( BaSr)-Cu-0 with T, = 240 K, although apparently it is not the temperature T,, . Another group of experimenters have also produced specimens with T, = 260 K.) From the viewpoint of the theory, at least at present, there are no limitations preventing the existence of superconductivity at room temperature and even at higher temperatures (see below). We cannot safely predict, however, what critical temperatures will be obtained, especially, .for practicable materials. An intense search is underway and we shall probably know soon the possible limits for raising T, for metal oxides. At any rate, I shall not be surprised at all if in the very near future easily obtainable “room-temperature’’ superconductors are manufactured. (In view of the recent advances it is instructive to recall the papers which stated quite firmly that the highest possible value for T, was 25-30K; Matthias 1970, 1971.)

I?

V.L. GINZBURG

To conclude this section, let us discuss briefly the relevant terminology. Before 1986 the term “high-temperature superconductor” was often applied even to the materials with Tc> 10 K, for instance, to Nb,Sn with T, = 18.1 K. In my papers the term HTS was applied only to superconductors with a critical temperature near the boiling point for the liquid air, that is, T,> 80-100 K (see, for instance, Ginzburg 1968,1970,1972). At present, ceramics with T,> 30 K are typically referred to as HTS materials. A term is, of course, a matter of convention and it does not have a special significance. In my opinion, however, it would be more appropriate to apply the term HTS to materials with T, > T,,.,: = 77.4 K. This suggestion is supported by the fact that the temperatures T,-20-40 K were long regarded as being obtainable even for the “conventional” (phonon) mechanism of superconductivity (see, for instance, Ginzburg 1968, Ginzburg and Kirzhnits 1982). In order to prevent confusion, however, in this paper I apply the term HTS to all new “neon” and “nitrogen” superconductors, that is, the materials = 27.2 K. This approach is supported by the fact that all with T, > Th.Ne such materials known now are metal oxides and clearly have something in common (apart from the composition, the common features are the layered structure of the new materials and the presence of oxygen in them).

4. Microscopic theory of superconductivity A microscopic theory of superconductivity, which can be called reliable or fairly comprehensive, was first put forward in 1957, i.e. 46 years after the discovery of superconductivity. Of course, I mean the theory formulated by Bardeen, Cooper and Schrieffer (Bardeen et al. 1957). However, this theory did not grow out of nothing: a lot of work had preceded it (see, for instance, Bardeen 1956). Incidentally, I have participated in discussions of the following question on a number of occasions: what prevents us from creating a new theory (in fact, the discussions concerned problems other than superconductivity)? Do we lack additional facts or information, or are we waiting for the advent of a genius, a “new Einstein”? It is impossible to provide an unequivocal answer to this rhetorical question, since all depends on the prevailing situation. But as far as superconductivity is concerned, the following can be stated unambiguously: both Einstein and Bohr, and many other leading scientists of a lower calibre, were interested in the problem of superconductivity and made a number of useful comments (Einstein 1922, Hoddoson et al. 1987). However, they were not able to formulate the microscopic theory of superconductivity. It seems to me that, in the first place, such a

HIGH-TEMPERATURE SUPERCON1)UCTIVITY

13

theory had to wait until the discovery of superfluidity of He 11, which was finalized only in 1938.* Then it became generally clear that superconductivity was the superfluidity of the electron liquid in metals. This concept, treated in the context of Landau's theory of superfluidity, helped the understanding of some aspects of the problem (Landau 1941, Ginzburg 1944, 1946) though not at once. The matter was complicated by the opinion (even Landau held it for a time) that superfluidity of helium 11 had no relation to the Bose-Einstein condensation whose possibility and even inevitability under certain conditions was suggested by Einstein (1925). The relation of superfiuidity to the Bose-Einstein condensation or, at least, to the Bose statistics of the 4He atoms became obvious only after 1948 when liquid 'He whose properties sharply differed from those of liquid 4He had been produced (liquid 'He is known to lack a lambda point in the temperature range T > 0.1 K while for 4He we have TA=2.17 K). Many scientists, however, had believed in such a relation even before liquid 'He was produced (London 1954). One would think that then efforts had to be made to understand how electrons in 3 metal could produce a Bose system, for instance, by binding electrons into pairs with zero spin. In fact; however, the concept of such pairing was by no means trivial. It was very hard to imagine two electrons forming a bound state because normally they repel each other. Perhaps I do not have a complete picture but I know only one paper published before 1954 that mentioned a relation between superconductivity with formation of pairs and their Bose-Einstein condensation. It was written by Ogg (1946) who believed he had discovered superconductivity in metal-amonia solutions. As far as I know he was mistaken though I do not know the details of the experiment. However, Ogg suggested that electrons were bound in pairs which underwent a Bose-Einstein condensation. In 1952 I analyzed the * In science, discoveries are made in different forms and in quite different ways. The discovery of superconductivity and superfluidity is a very good illustration of that. For instance, superconductivity was discovered in 1911 with confidence and practically by a stroke by Kamerlingh Onnes. In the same year Kamerlingh Onnes observed some strange phenomena in liquid helium at 7'-2.2 K (nonmonotonic dependence of the density on temperature) and in 1922 he discovered an anomalously fast flow of the helium films. In 1928 the concept of two phases -helium I and helium 11- was put forward by Keesom and Wolfke and in 1932 Keesom and Clausius obtained a clear lambda curve for the specific heat of liquid helium at the lambda transition. In 1936 W. Keesom and A. Keesom discovered superhigh thermal conductivity of helium I 1 and finally in 1938 Kapitza, and Allen and Misener independently discovered superfluidity of helium I I - its frictionless flow through narrow slits and capillaries. A complete enough understanding of superfluidity was achieved by 1941 in the works of Kapitza and Landau. Therefore, we can say that it took thirty years to discover the effect of superfluidity. To save space, no references are given here, especially as a full enough list was given by London (1954).

V.L. GINZBURG

14

properties of a charged Bose gas and noted that the Meisnner effect had to occur in it (Ginzburg 1952). In this paper, however, I never mentioned the possibility of the formation of such a Bose gas from electrons and apparently I did not care for the concept at the time. In fact, in the paper on the tY-theory of superconductivity (this is how 1 refer to the macroscopic theory put forward in 1950 by Landau and me (Ginzburg and Landau 1950); we wrote when replacing -ihV with (-ihV - ( e / c ) A ) that “ e is the charge for which there are no reasons to believe that it differs from the electron charge”. I had some doubts about that even at that time ( I believed it would be better to take some effective charge e * ) but 1 could not find any convincing arguments in favour of a difference between e and e*. Later, when I analyzed experimental data I saw that the agreement of the Y-theory and experimental data could be improved by introducing the effective charge e * = ( 2 - 3 ) e (Ginzburg 1955). But Landau objected against this suggestion (Ginzburg 1955) since he noted that such an effective charge could depend on the pressure, temperature, composition and so on, and therefore it had to be regarded as a function of coordinates. But then the gauge invariance is violated. I failed to overcome this difficulty and to introduce the charge e*. Now the solution of the problem seems quite clear - any universal charge e * = ne where R is an integer is quite admissible from the viewpoint of gauge invariance and, in fact, in superconductors including the newlydiscovered ones we have e* = 2e (Gough et al. 1987). Is it not curious (it seems to be sheer blindness now) that such a conclusion was not drawn by me, by Landau, or by anybody else? I have spoken about this problem at length also in order to draw attention to the contribution made by Schafroth who was killed in a plane crash in 1959. Like me, he started by considering a charged Bose gas (Schafroth 1954a) and then put forward the idea of pairing in the most clear and definite form, even suggesting that the electron-phonon interaction could be responsible for pairing (Schafroth 1954b). Schafroth et al. (1957) developed their theory but left the question about the nature and mechanism of pairing unanswered. The pair size was assumed to be small (of atomic dimensions) and the critical temperature T, was determined to a first approximation by the formula for the temperature of Bose-Einstein condensation of an ideal gas: T, =

3.31 h’n”’ m*k,

where m* is the mass of the pair, m =9.1 x 10 28 g is the mass of a free electron, and n is the number density of pairs with zero spin. In recent years, and especially now when new superconducting materials have been obtained, Schafroth’s model with “tiny” pairs is being widely

HIGH-TEMPERATURE SUPERCONDUCTIVITY

15

discussed, and we shall return to it later. Thus, it is obvious that the generally accepted belief that the introduction of pairing in the theory of superconductivity is due to Cooper (1956) is entirely unfounded. The merit of Cooper’s work and, in the first place, of the subsequent BCS theory (Bardeen et al. 1957) lies in the real demonstration of pair formation (“large”, or Cooper, pairs) and of their collective “condensation” even under a weak attraction between electrons near the Fermi surface of the electron gas. I must emphasize that the above brief discussion was somewhat one-sided (perhaps even biased). An essential difference between superconductivity and superfluidity is, of course, the occurrence of the Meissner effect in superconductors that was discovered in 1933. A similarity can, of course, be found between the effect of rotation on the helium I1 and the effect of the external magnetic field on a superconductor but it is a different matter. The significant aspect is that superconductivity does not reduce to infinite conductivity and at equilibrium the external magnetic field cannot penetrate into the interior of a superconductor (we are discussing, of course, Type I superconductors for H < H, or Type I1 superconductors for H < H c , ) . In this connection, a significant contribution to our understanding of superconductivity (apart from an even irrespective of the similarity to superfluidity) was made by the concept of the macroscopic quantum state and the rigidity 6f the wave function in this state (see London 1950). The concept of the energy gap in the excitation spectrum was also promoted in this way (see London 1950, Ginzburg 1944, 1946, Bardeen 1956,1963). In the BCS model the Bose-Einstein condensation is somewhat indefinite in character because the size of the pairs is much larger than the distance between them. Owing to the above factors, the similarity to superfluidity apparently made no contribution to the formulation and presentation of the BCS theory (Bardeen et al. 1957). But now we understand that the BCS model with weak coupling is one limiting case while the Schafroth model (with localized pairs) is another limiting case. To be more precise, this fact was realized long ago but somehow the prevailing feeling was that all real superconductors are described by the BCS theory. Now it is clear that there are reasons not to limit the discussion to the BCSscheme and to consider also the general case (see below).

5. Critical temperature in the BCS theory The above considerations do not, of course, detract in any way from the significance of the BCS model and this approach on the whole. In fact I believe it quite possible and even probable that new superconductors with

V.L. GINZBURG

16

T,> 30 K are described basically by the BCS model. We shall continue this discussion below and now we shall base our treatment precisely on the BCS theory. The well-known formula of the BCS theory T, = 8 exp( - 1/ Aerr)

(2)

and, what is more important, the entire BCS theory were strikingly successful. Indeed, according to this formula, the value of T, is determined by only two parameters, viz. k B 8 , which is the region of attraction between electrons near the Fermi surface, and Aefr = N ( 0 ) V, which is a dimensionless parameter characterizing the intensity of this attraction. In the original version of the BCS theory, herr=N ( 0 )V < 1, where N ( 0 ) is the density of states on the Fermi surface and V is the matrix element of the interaction energy. It is not surprising that the BCS theory is indeed the real microscopic theory of superconductivity for the chosen model. However, the fact that this model gives a fairly accurate description of real superconductors with a weak coupling (Ae,,-* 1) can by no means be considered obvious. As an example, we can consider the BCS formula connecting T, with the energy gap 2 4 ( 0 ) in the superconductor spectrum for T = O , which has been confirmed experimentally:

2A(O) =3.52kBT,.

(3)

This relation might not be satisfied for the simplest model. Of course, the BCS theory was developed and refined (in the USSR, the contributions made by N.N. Bogoliubov, L.P. Gor'kov and G.M. Eliashberg are worth mentioning). The reasons behind the astounding success of the BCS model are discussed in Ginzburg and Kirzhnits (1982). In fact, the success of the BCS model is largely explained by the fact that it is a mean field theory. On the other hand, such a theory is, to a certain extent (formally only in the vicinity of Tc),equivalent to the Landau theory of second-order phase transitions, that is, the V-theory of superconductivity in this case, which also includes only two parameters in the simplest case. Besides, it is precisely for the conventional superconductors that the mean field theory is generally suitable owing to the smallness of the fluctuations near T,. The smallness of the fluctuations in its turn is due to the large correlation length to (Ginzburg 1960, White and Geballe 1979). Let us now go over to the calculation of the critical temperature and its possible maximum value. For the phonon mechanism of superconductivity, i.e. when the attraction between electrons is due to their interaction with the lattice vibrations, or phonons, the temperature 8 0,) in formula (2) is the Debye temperature. For the normally-accepted values @,,s500 K

-

HIGH-TEMPERATURE SUPERCONDUCTIVITY

17

and Aerr=S 1/3, we obtain Tc=S500 e-3 = 25 K. In general, this explains the fact that T c s 10-30 K. In a more realistic theory, and for weak coupling (under the condition I ) , we have

where A and p are dimensionless coupling constants for the phonon (exciton) and Coulomb interaction respectively, and 0,= EF/k , is the temperature corresponding to the Fermi energy EF of the metal. Further, in the commonly used approximation (homogeneity, isotropy, weak coupling) we obtain p - A = 4re2N(O)

where ~ ( wk,) is the longitudinal permittivity of the material, and the angle brackets indicate averaging over the momenta hk. It has been suggested that the requirement of stability of a metal leads to the condition ~ ( 0 k, ) > 0. If this be so, then A s p in accordance with eq. ( 5 ) , and superconductivity would be possible only because p* < p. This would indicate a low upper limit for T, (Cohen and Anderson 1972). Such a conclusion, however, is incorrect since the stability requirement actually has the form l/c(O, k ) s 1 (where k # 0), the values of ~ ( 0 k, ) < 0, and hence A > p are admissible. This important result was obtained by Kirzhnits (1976, 1987) and is associated with the following. The response function obeying the dispersion relations is not E but rather I / a As a matter of fact, the electric field E = ( I / E ) D where , D is the induction satisfying the equation div D = 477pext.The external charge pextcan be controlled, and hence it is D that plays the role of the “cause”, while the field E is the “consequence”. We cannot arbitrarily change E, and hence the function E in the formula D = EE need not obey the dispersion relations. To be more precise, the function ~ ( 0k, + 0) for small values of k --* 0 is also a response function, and hence the necessary condition for the stability of the system is c(0, k -+ 0) a 0. But in the theory of superconductivity, we are interested in large values of the wave vector k Usually, k is of the order of k , , and hence we should impose only the requirement l/e(O, k ) s 1. As mentioned above, this is in agreement with the condition ~ ( 0 k, ) < 0. What is more, it is just the values ~ ( 0k, ) < O that are realized in most metals (Dolgov et al. 1981, Dolgov and Maksimov 1982). Usually, p* 0.1 (the parameter p =S OS), and in some cases A s 3 (for example, in the PbBi alloys, A s 2.6; see Dolgov and Maksimov 1982, Wong and Wu 1986).

-

V.L. GINZBURG

I8

For intermediate and strong coupling when her[> 1, formula (2) under condition ( 4 ) is obviously inapplicable. For this case, a number of formulas of type ( 2 ) have been proposed with, say, Acfr=(A - p * ) / ( l + A ) .

(6)

As an illustration, note that for p* = 0.1 and A = 1 or A = 3, the temperature T, = 0.1 10 and T, = 0 . 2 5 0 , respectively, in accordance with (2) and (6). In other words, we obtain the values T,= 30 K and T,==75 K for 0 = Or,==300 K.Many other similar formulas for T,, obtained on the basis of Eliashberg's equations (Ginzburg and Kirzhnits 1982, Dolgov and Maksimov 1982, Wong and Wu 1986) and having a semi-empirical or model character, lead to high values of T, for appropriately chosen parameters. For example, the following formula has been proposed by Wong and Wu (1986) (we present it for p * = O . l ) :

0 T, ->( 1 +0.53A ) exp[ - 1.25/( A -0.1 1 )], 5.42

(7)

where

A =2

I

dw

a 2 ( w )F ( w ) w

(see the above papers for notation). According to ( 7 ) , Tc=0.078,, and T, = 0.31 8,, for A = 1 and_ A = 3 respectively. For strong coupling, when A b 10, we have T,- &/A (here 6 = h G / k , , where (I, is a certain mean frequency in the phonon spectrum; see Allen and Dynes (1979, Ginzburg and Kirzhnits (1982, Chapter 4). In any case, there are no restrictions in principle imposed on the possible values of T, by the conventional theory of superconductivity. Thus, it is clear that large ("high-temperature") values of T,- 100 K can be obtained for A =s3 and 0 O,, 8 0,) if at the same time eD3 300 K. The only question is whether such values of O,, and A can be realized for the phonon mechanism of superconductivity. For each type of material, there exists a maximum value of T, since the parameters A and 0,) are not independent. Thus, an increase in the value of the electron-phonon coupling constant A leads to a decrease in the phonon frequencies (due to the screening by conduction electrons and other reasons). This effect can be approximately described by the formula (see Dolgov and Maksimov (1982) in this connect ion ) :

- - -

HIGH-TEMPERATURE SUPERCONDUCTIVITY

19

where woo is the bare - not renormalized - (“pseudoatomic”) phonon frequency and C is a constant. As a result, we obtain the maximum value of T, which is proportional to woo. Superconductors obtained before 1986 were characterized by parameters leading to T C s25 K. Hence, and in view of the rough estimates presented above, it was usually concluded that T, < 30-40 K for the phonon mechanism of superconductivity (see, for example, Ginzburg and Kirzhnits (1982, Chapter 1). It was always stipulated, however, that this conclusion is not general. For example, it was mentioned that O,, 3000 K for metallic hydrogen and some (though not all) calculations lead to the values T,100-300 K (see Ginzburg and Kirzhnits (1982), Dolgov and Maksimov (1982) and the literature cited therein). Many metals are known to have high values of OD(for example, O,(Al) = @,(Ti) = 430 K, O,(Os) = 500 K, OD(Mo) =450 K, and @,(Be) = 1400 K). The high value of 8, for Be even led to the suggestion that a temperature T,- 100 K can be attained for beryllium under certain conditions. The presence of light atoms H and C in organic superconductors raised hopes (Budzin and Bulaevskii 1984) that the high phonon frequencies in this case might result in large values of T,. So far, HTS of such types have not been produced, but the possibility cannot be ruled out.

-

6. Superconductivity mechanism in the “neon” superconductors

Naturally, it is especially important at present to explain the mechanism of superconductivity in recently discovered materials, viz. metal oxide ceramics. Let me start with the alloy La,.8Sro.,Cu04and similar materials with T,=36 K. According to Cava et al. (1987) and Kwok et al. (1987), superconductivity in this case may be caused by the usual phonon mechanism described above. The high value of T, is attributed to the high frequency wo of vibrations in the “subsystem” formed by a Cu ion surrounded by a distorted octahedron of oxygen ions. A comparison with the superconducting metal oxide ceramic BaPb,-,Bi,03 (for this alloy, T, = 13 K for x = 0.25) for nearly identical values of OD leads to T, = 36 K under the assumption that the density of states N ( 0 ) near the Fermi surface is 2-3 times higher for La,.8Sr0.ZC~04. Kwok et al. (1987) assumed that for the alloy La,.,,Sro,,,Cu04 the value of N ( 0 )is 3-5 times higher than for BaPb0.,Bio.,O, in accordance with the measurements of the field H c 2 . Taking A = 1.2 for Putting A = 3 by BaPbo.75Bio.z503, we get A =3.1-3.9 for Lal.85Sro.15Cu04. way of an example and using formula (7) with an identical value of OD,. we obtain Tc(La,.,,Sro.,5Cu04)= 3.2Tc(BaPb0.75Bi0.2503) = 42 K. The same conclusion is drawn by Weber (1987) from quantitative analysis based on

V.L. GINZBIJRG

20

calculations (Mattheiss 1987) of the electron structure for the tetragonal crystal LalCu04 (such computations have been carried out also by Mazin et al. (1987)).According to Mazin et al., the values Tc=30-35 K for A = 1.5 are quite natural for the system Laz ,(Ba, Sr),CuO,. The light oxygen atom vibrating at a high frequency wo plays an important role in this case. A distinguishing feature of the metal oxide ceramics is that they have high values of wgowing to the presence (associated with light oxygen atoms) of rigid optical modes of lattice vibrations (phonons) which are strongly coupled with conduction electrons. In turn, this is due to a significant contribution of ionic-covalent bond in these compounds. About 20 valence electrons per unit cell contribute to the formation of this bond (the metallic bond is formed by just 1 to 2 conduction electrons which exert a rather weak influence on the lattice). Thus, it can be assumed that there is a tendency towards the optimization of the contributions from ionic and metallic bonds in metal oxide ceramics (see Ginzburg and Kirzhnits (1982, p. 170). The above discussion does not mean, of course, that the nature of superconductivity in the La-Sr-Cu-0 system has been understood in full. The nature of superconductivity in BaPb, ,Bi,O, is also a matter of controversy (see Gabovich and Moisseyev 1986). As noted above, however, the La-Sr(Ba)-Cu-0 system may have the conventional phonon mechanism of superconductivity. This i; also suggested by the results on the isotopic effect in the La, 8sSr,,,5Cu0, ceramics (Batlogg et al. 1987b, Faltens et al. 1987). The observed isotopic effect (the change in T,) was caused by the replacement of I6O with '"0.The isotopic effect is usually described by the where M is the mass of the isotope. For a singlerelation T,- M component metal the parameter a if superconductivity is caused by electron-phonon coupling and anharmonicity is ignored (as noted above, in the case of the phonon superconductivity mechanism 0 in the BCS formula is the temperature 9,)which is proportional to the characteristic frequency wg of lattice vibrations that is proportional to M -I"). The parameter Q has a different value for a complex (multicomponent) system and when anharmonicity is taken into account. According to Batlogg et al. ( 1987b) for the La, ssSr, ,5Cu04ceramic a = 0.16k0.02. Faltens et al. (1987) reported values of Q varying from 0.1 to 0.37 for various specimens. When anharmonicity is ignored the theory yields values of a that exceed experimental results (apparently these values are considerably higher than 0.16). I t would be interesting to analyze the isotopic effect for BaPb,Bi,-,O,. At any rate, a clear manifestation of the isotope effect suggests that the phonon mechanism plays a significant role for the La-Sr-Cu-0 system. On the other hand, the smallness of the observed values of a in comparison with the predicted values can be explained both by anharmonicity and by the 'I,

=:

HIGH-TEMPERATURE SUPERCONDUCTIVITY

21

effect of another superconductivity mechanism, for instance, the exciton mechanism that will be discussed below.

7. Superconductivity mechanism in the “nitrogen” superconductors. Ways for raising T,

We must state at the beginning that the mechanism of superconductivity in the “nitrogen” superconductors (and, partially, in the “neon” superconductors as discussed above) is still unclear. This is explained by the lack of reliable experimental data, particularly for single crystals. In the light of the above estimates we can admit logically that the temperatures T, 100 K can be explained in the framework of the conventional phonon mechanism. As an illustration, note that (7) with A = 3.5 and T, = 250 K yields eD = 680 K and this value is by no means excessive. If the experimental result for Q D , the electronic component of the specific heat and other parameters confirmed such a possibility then we would obtain, roughly speaking, a “trivial solution” for the high-temperature superconductivity in the cases under consideration. Indeed, the limit of Tc=S40 K for the phonon mechanism has never been substantiated since it was derived in a rather arbitrary way from rough estimates and few available experimental results. Now we, from this point of view, have produced materials with such parameters OD and A as lead to high temperatures T,. This conclusion can be reliably refuted, probably, only when we obtain sufficiently detailed results on the phonon spectra of the “nitrogen” superconductors and other data needed for calculating T, (though we assume here that we are dealing with BCS superconductivity). The widely held opinion based on incomplete data, and intuition suggest, however, that the “nitrogen” superconductors are unlikely to have the phonon mechanism of superconductivity. Note, first of all, the lack of a marked isotope effect (see, however, below) in the “nitrogen” superconductors of the type of YBa2Cu307and EuBa,Cu,O, (Batlogg et a]. 1987a, Bourne et a]. 1987). The replacement of I6O with I8O was found to produce no noticeable change in T, (the measurements gave a = 0.0 f 0.027 and 0.0 f 0.02). As noted above, this result by itself does not rule out the phonon mechanism (superconductivity in Ru and Zr is assumed to be due to the phonon mechanism but for them a =O; for PdH and PdD we even have a
-

....

?-I

V.L. GINZRURG

Let us consider possible causes for an increase in T, starting with a BCS-type theory. Even the simplest formula ( 2 ) of the BCS theory, written in the form

allows us to identify the main factors which lead to an increase in T,. Apparently, these factors are the increases in the width of the characteristic interaction region 0,the density of electron states N ( 0 ) on the Fermi surface, and the interaction V responsible for pairing. All three types of mechanism involving these factors will be discussed below. Let us begin with the mechanism leading to an increase in the density of states N ( 0 ) .It is well known that the standard theory of superconductivity regards the density of states N ( E ) of electrons in a normal material as a smooth and slowly varying function of the electron energy E near the Fermi surface. Hence the final formulas contain only the density N ( O ) = N ( E = EF)at the Fermi boundary. The situation may change ifthe superconducting transition temperature T, is relatively close to the temperature T, of the strwtural phase transition, at which an energy gap 2A,(O) of a dielectric nature appears in the energy spectrum of the material. This gap may be due to the congruence of different parts of the Fermi surface of the material.* Near the gap boundaries, the density of states in the allowed region increases and vanes strongly according to the law N ( E ) E-"', where E is the distance from the gap boundary (physically, an increase in the density of states is due to their "expulsion" from the gap). Let us now suppose that the Fermi level of electrons in a doped semiconductor lies (before their superconducting pairing) in the allowed band just near the boundary of the dielectric gap (see fig. 1). Then the superconducting pairing of the electrons due to their phonon attraction (or due to the attraction of some other type, say an exciton type; see below) will be enhanced on account of the dependence of N on E. Accordingly, the critical temperature T, will be higher than in the case without a dielectric gap, and will be described, in view of the strong dependence of N ( E ) on E, by a non-exponential formula (see Kopaev and Rusinov 1987, Ginzburg and Kirzhnits 1982, Chapter 2, where the calculations are made in the weak coupling approximation).

-

* "The congruence" means that the electron energy spectrum on relevant parts of the Fermi surface with electron momenta p satisfies the condition E ( p ) = - E ( p + Q) for a certain set of vectors Q. Such a situation (or one similar to it) is called "nesting".

HIGH-TEMPERATURE SUPERCONDUCTIVITY

23

‘t

Fig. 1. Density of states N ( E ) near the dielectric gap.

Another factor, viz. the increase in the attraction V between electrons, is realized, in particular, in the bipolaron model of superconductivity. According to this model, a strong electron-phonon interaction contracts the rather wide conduction band to a narrow polaron band which corresponds to a fairly large mass of the carriers, that is polarons. Because of this interaction, pairs of polarons may combine to form a bound state, or bipolaron, with a large binding energy (roughly speaking, this corresponds to the confinement of both electrons in the same polarization well). The aggregate of bipolarons, which are Bose particles, may undergo BoseEinstein condensation whose temperature is described by formula ( 1 ) for a low number density n of bipolarons. In the model under consideration (see Alexandrov et al. (1986) and the literature cited therein), this temperature is just the critical temperature of superconductivity. For n 102’-1022cm-3 and rn* lo2 m, we obtain Tc-30-100 K. The formation of pairs with a small radius (local pairs) and their subsequent condensation form the basis of not only the bipolaron model, but also some other models. It should be stressed that the BCS model and the entire approach it involves are applicable to the conditions when the conduction band width (in the normal state) and, accordingly, the Fermi energy EFis considerably higher than the coupling energy k,O causing superconductivity (see (2)) and the gap width 24. Therefore, in formulas of the type of eqs. (2) and (6) we have an effective dimensionless coupling constant (see also below)

-

-

24

V.L. GINZBURG

(To prevent misunderstanding, note that in the BCS theory the coupling with A 2 I is called a strong coupling while the condition of weak coupling is written as A,,-,.< 1. The constant A,,,. in (9) is rather arbitrary in character and for A,,, 2 1 it is not identical to Aefr in the RCS theory.) This condition (9) is, of course, not quite rigorous and, for instance, we may consider the BCS approach even for A , , , < 3 . In fact, larger values of A C f , were not encountered in earlier studies. I f the interaction energy and hence the coupling energy in pairs is even higher then we obtain the other limiting case involving localized pairs and their subsequent Bose-Einstein condensation (this case may be referred to as the Schafroth model). The intermediate range between these two limiting cases (corresponding formally to A,.rr& 1 and A ? , , % 1 ) is especially difficult to analyze. It is quite possible, however, that it is precisely such “intermediate” coupling that occurs in the new superconductors. The above conclusions are presented, for instance, by Bulaevskii et al. (1984) in a more exact and detailed form. Though it is a fascinating and significant problem its further discussion would be unsuitable in this paper. The mechanism of attraction between pairs can be different both in the BCS case (large pairs) and in the Schafroth case (the localized or small pairs). For instance, as noted above, the bipolaron model includes the phonon mechanism of attraction but the pairs are localized. Then the penetration depth in a superconductor for a weak magnetic field is

where e* = 2e and n is the number density of the pairs. For instance, for m* = 100 m and n = 3 x 10” cm we have T, = 60 K according to ( 1 ) and fi = 5 x 10 ’ cm. I n general, we have 6 ( r n * ) ~ - ’ ” Tv4 , according to (1) and ( 10). We can see from the above illustration and formulas ( 1 ) and (10) that in the bipolaron model at T, 100 K the values of 6 are higher by an order of magnitude than the values typical of Type I superconductors (for instance, we have f i ( 0 ) = 5 x 10 -’ cm for Al, 6 ( 0 )= 3.9 x 10 * cm for Pb, and so on). Formula ( 3 ) is, of course, inapplicable to the models with localized pairs. The pair dissociation energy 2 A is considerably larger than kHT, but absorption (for instance, of electromagnetic waves) can occur not only with dissociation of pairs but also with excitation of the collective modes. I n this case the pairs exist even at T > 7,and this fact is, of course, reflected by the absorption spectra and so on. i n the frequently used Hubbard model (see, for instance, Bulaevskii et al. 1984, Zvezdin and Khomskii 1987) the part of the above parameter A,.,, (see formula ( 9 ) ) is played by the ratio U / t where the parameter t is of the order of the conduction band width and U describes the attraction

’

-

-

2s

HIGH-TEMPERATURE SUPERCONDUCTIVITY

between electrons (or holes). The entire range of the values of herr can be qualitatively analyzed with the use of this or similar concepts. The results are shown schematically in fig. 2 (details are reported by Bulaevskii et al. 1984).

I am not acquainted with any results of comparisons of the available experimental data and the predictions made in various theories of localized pairs and, in fact, these theories themselves are not quite clear to me. For instance, the bipolaron model, apparently, must exhibit a strong isotope effect but I do not have a clear picture of this issue. My understanding of other models with localized pairs is also insufficient. The same is true for the models with non-phonon interaction put forward by Anderson (1987) (the “resonating valence bond theory”), Eliashberg (1987), Zvezdin and Khomskii (1987), Schrieffer (1987) and other authors even irrespective of the pair sizes. Therefore, I shall make only some general comments here. The discovery of HTS has, so to say, stirred up the theoretical thinking and pushed into the limelight even some old concepts. Firstly, these are the models with localized pairs and generally the models that do not fit into the conventional BCS model (theory). Secondly, the models for pairing and, specifically, superconductivity that do not involve phonons are now

A

\ \

/‘ coherence length [ , = i ( O )

\\ \

\

/

\I

/

/

//

\

\

‘\

// /’

BCS limit

0

1

Fig. 2. The dependence of T,, the coupling energy 2 A ( O ) of the pair and the coherence length toon the dimensionless coupling energy Aerr-- U / I .

26

V.L. GINZBURG

widely discussed. The new (differing from the BCS theory) models are typically far from completion and often insufficiently clear or even largely unclear (not only to me since I received this impression after talking to a number of competent theorists and taking part in specialized discussions). On the other hand, it is quite obvious that it does not make sense any longer to continue working only within the framework of the BCS model and the phonon coupling mechanism. This conclusion is supported by the observation of superconductivity in materials with heavy fermions (“heavy electron superconductivity”). Though this phenomenon is far from being clear (Ott 1987, Pethick and Pines 1987) we have good reason to think that this superconductivity involves spin effects, rather than the phonon coupling. Pethick and Pines (1987) suggested a mechanism of the “attractive interaction between the heavy electrons which results from virtual exchange of antiferromagnetic f-electron moment fluctuations”. Apparently, we may just as well speak about the exchange of spin (of course, virtual) waves ( I must note that, as far as I know Achiezer and Pomeranchuk (1959) were the first to discuss the effect of the virtual spin wave exchange on T,). There is no doubt about some relation between HTS in new materials and antiferromagnetism, a n d it explains the ?tempts to relate HTS and, so to say, the spin effects (Emery 1987, Anderson 1987, Schrieffer 1987, Pethick and Pines 1987 and other authors). As far as I can see, however, no clear evidence supporting these concepts has been found yet. It is still at least admissible, therefore, to account for HTS within the framework of the BCS model (theory), though using the exciton coupling mechanism. Since I have worked only with this model myself and believe it to be not only possible but even probable (as far as 1 can judge with the available experimental data) I shall discuss the exciton mechanism of HTS at some length here.

8. Exciton mechanism of HTS At the end of section 7 I digressed from the logical outline of this review discussing possible mechanisms of superconductivity that d o not involve the BCS theory. Now I return to the main subject, that is, to finding feasible explanations for the increase in T, in the framework or, at least, with the use of the concepts of the BCS model. In this context, we have only the third possible mechanism to discuss, that is, the increase in T, owing to the growth of the pre-exponential factor 0 in an equation of the type of ( 2 ) in the BCS model.

HIGH-TEMPERATURE SUPERCONDUCTIVITY

27

The interaction between the conduction electrons that gives rise to superconductivity may be mediated by other (so to say, “bound”) electrons, apart from lattice vibrations (phonons). We could use the term the electron mechanism of superconductivity in this connection. But since superconductivity in a metal is an electronic phenomenon the superconductivity mechanism that differs from the phonon mechanism is usually referred to as the exciton mechanism in order to prevent a confusion in terms. Other terms have also been used, though. What is more important, when we are dealing with the exciton mechanism we do not include in it the spin waves and, generally, the excitations associated with the spin variables. We have been using the term exciton mechanism for a long time (Ginzburg 1968, 1970, 1972, Ginzburg and Kirzhnits 1982) precisely because it gives a vivid illustration of the process. The point is that in the phonon mechanism the interacting conduction electrons exchange phonons, while in the exciton mechanism they exchange excitons (excitations of the electron type). In “good” metals (as also in a free electron gas), there is only one type of electron excitation, viz. plasmons, for the energy E, = f i w , ~EF.To a first approximation (for the free electron gas, this is an exact result), the plasmon frequency w , = (47re2n/m)’/’ = 5 . 6 4 ~lo4 n1/2 (the numerical value corresponds to the charge and mass of a free electron). For n lo’’ cm up- 10l6 and fiw, = E,- 10 eV (for a degenerate electron gas, w , = (47re2n/m)’/’ = ( \ 6 ~ ) 1 / 2 w F ,a = e 2 / 7 r h U F ) . The plasmons being considered here are longitudinal vibrations in the system of conduction electrons and their exchange does not take into account the effect of other (“bound”) electrons. Hence, it is natural that the plasmons are found to contribute in the simplest case only towards the screening of the Coulomb interaction and influence the parameter p (see (4)). Some metals or “sandwich”-type systems, however, may contain electronic excitations other than plasmons (see, for example, Agranovich and Ginzburg 1984), having frequencies we < w F = EF/ fi. In any case, the frequencies we may considerably exceed w D = kBOI,/fi. For instance, for the exciton energies E, = hw, 0.1-1 eV the temperature Q,= E , / k , - lo3-lo“ K while for the typical metals we have OF= EF/k , 10’ K and OD< lo3 K. It is precisely the replacement of the temperature O,, with 8,in the BCS formula (2) and in other equations for the phonon mechanism that is the essential step in the concept of the exciton mechanism. The transition from the temperature ODto 8, is sometimes described with the use of the isotope effect owing to which the characteristic phonon frequencies up,,- M I / ’ where M is the ion mass. In the case of the exciton (electronic) mechanism the part of ions is played by the “bound” electrons with the mass m which is, generally, of the order of the mass of free electrons. Hence we have

-

-

-

’,

V.L. C I N Z B U R G

28

- ,/PO+,,, -

and 9, d - @ , , ~ 100 @,, . This approach does not give anything essentially new (there are no special reasons for repeatedly explaining t h e well-known fact that the temperature @, is considerably higher than

W,

@I,).

As far as I know, Little (1964) was the first to consider the HTS problem using a modern approach that, in fact, involved an exciton mechanism. Little considered quasi-one-dimensional conducting polymer chains with molecular “side branches”. Electron polarization of these side branches must, in principle, increase T, for the chain or, to be more exact, determine T, for the conduction electrons in the chain. In the same year I and Kirzhnits considered two-dimensional superconductivity, for instance, for the surface levels quite independently of the HTS problem (Ginzburg and Kirzhnits 1964). I still consider this subject to be of considerable interest but there is no space to discuss it here. After reading Little’s paper I immediately applied his idea to the above concept, that is, suggested the possibility of increasing T, in thin metallic layers or on a surface by coating it with a dielectric material (Ginzburg 1964). Similar systems include the insulatormetal-insulator sandwiches and various layered compounds. The underlying concept is that the electronic excitons which, in principle, can exist and easily propagate in insulators ( Agranovich and Ginzburg 1984) will be absorbed and emitted by the conduction electrons in the metal layers (fig. 3). The excitons in this model play the part of phonons in the phonon mechanism and therefore we may talk of the “exciton mechanism” of superconductivity. A highly significant concept here is also the possible penetration of the conduction electrons (that is, their wave functions) into the insulator (semiconductor) layers (Allender et al. 1973a). The question of fluctuations was not discussed in 1964. It soon became clear, however, that fluctuations are particularly dangerous in the quasi-one-dimensional case and that their significance in the two-dimensional case is considerably smaller though they are still harmful. In any case, the one-dimensional conducting chains are still to be manufactured and genuinely onedimensional superconductors have not yet been produced. In contrast, the “sandwiches” are easier to produce and, what is more important, there exist, or can be fabricated, numerous layered compounds that can be regarded as “stacks” of sandwiches. The superconducting layered compounds known until recently had only rather low critical temperatures, though. But the new (high-temperature) superconductors are precisely the layered systems. The exciton mechanism can, of course, be effective only if the increase in the pre-exponential factor 0 in (2) and other formulas is not accompanied with a decrease in the parameter Aerr. Is it possible? No restrictions can be

HIGH-TEMPERATU RE SUPERCON DUCT1 VlTY

29

,metal

Fig. 3. Insulator-metal-insulator sandwich (a schematic representation).

imposed on the value of A from the general considerations for the exciton mechanism (Ginzburg and Kirzhnits 1982). The exciton mechanism can be effective, however, only when cel(O, k) < 0 where e,,(w, k) is the component of the longitudinal permittivity due to electron polarization (we can write approximately for the total longitudinal permittivity e( w , k) = eph+ eel - 1 where eph- 1 is the contribution of the lattice). I t is, apparently, not easy to obtain eel(O,k ) < 0 since such permittivity values undermine the stability of the lattice (this can be seen from the simplest “jelly” model in which the squared ion frequency is written as the ratio between the squared ion plasma frequency and the permittivity q ( 0 , k ) and, hence, is negative for ecl(O,k ) < O ) . In any case, in theoretical calculations we must take into account the exchange-correlation interaction (that is, the difference between the mean electric field and the acting field) and move outside the approximation of weak coupling. N o consistent theory satisfying these requirements has been developed yet (Dolgov and Maksimov 1982, Sham 1985, Ginzburg 1987b). In general, the exciton mechanism has unclear prospects for being used in the framework of “high theory”. Under such circumstance one has to do with various heuristic considerations and estimates that are naturally not very reliable, especially when

V.L. GINZBURG

30

we are concerned with finding a numerical value of T,. A variety of such estimates has been reported (see Ginzburg and Kirzhnits 1982, and the references cited there). We can hardly expect that suitable exciton bands for the “bound” electrons with E,< El.can occur in a homogeneous three-dimensional metal with E , > 1-3 eV. This is why the exciton mechanism seemed to be an unlikely model for the typical three-dimensional materials (see, however, Geilikman 1973). As for the “sandwiches”, the model calculations (see Ginzburg and Kirzhnits 1982, Chapter 8), provide evidence that a significant increase in 7;. can be expected only for a metal film with a thickness of the order of the atomic size in a “sandwich”. Under such condition the “sandwiches” are hardly promising materials and we must look for superconductivity primarily in the layered materials. The first studies of such materials using intercalation (that is, introduction of intermediate “insulator” layers) were started about 1970 (Gamble et al. 1971). The results were quite interesting - for instance, intercalation led to a significant increase in T,. The “classic” layered materials of the type of TaS,, NbS2 and so on have not yet exhibited high values of T,, though (Friend and Yoffe 1987). An early analysis of the relatively weak influence of conventional intercalation on 7,was performed by Bulaevskii and Kukharenko (1971) and it is discussed in detail in Chapter 6 of the book by Ginzburg and Kirzhnits (1982 ). The explanation is that when large molecules with suitable levels ( E , 0.1-1 eV) are intercalated between the conducting layers they cannot provide for effective transfer of the momentum hk hk,. . In other words, such systems lack excitons that are needed for raising T,. The following statement has been made in this respect (Ginzburg and Kirzhnits 1982, p. 39; we have replaced here hR with E, and 9 with k ) : “Thus, for layered compounds with molecular layers, on account of the large molecule size ( a 3 10-20 A), which is necessary for obtaining low excited levels ( E , < 1 eV), the exciton mechanism turns out to be ineffective. To overcome this difficulty, it is necessary to try to introduce into the dichalcogenide compounds (or other layered compounds) layers of the semiconductor type instead of molecular layers. There can exist in such layers delocalized excitons having propitious values of E, and at the same time existing at k k F . Thus, further investigation of layered compounds of the most diverse types is absolutely necessary. In this case, for the elucidation of the mechanism of superconductivity, it would be interesting not only to vary as far as possible the thickness and composition of the conducting and poorly conducting (dielectric) layers, but also to measure the variation of T, with pressure or deformation (we have in mind the determination of the depen-

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HIGH-TEMPERATURE S U P E R C O N D U C T I V I T Y

31

dence of T, on the distance between the conducting layers for a given composition of the compound)”. I think it quite probable that (see below) HTS systems of the type of Y-Ba-Cu-0 (and, to some extent, of the type of La-Sr(Ba)-Cu-0) provide precisely the above conditions - they have a suitable exciton level (electronic excitation level) and a capacity for transferring a large momentum when v i p a l excitation of this level occurs. I think it is worthwhile to note here a fact that helps one to appreciate the significance of the difference between the acting (effective) field E,,, and the mean electric field E for obtaining negative values of the permittivity E making possible a strong attraction between electrons (when E is negative the attraction immediately follows from the equations V = e’/u or V (w, k ) = 47re2/ekZ).Consider the simplest model of isotropic point dipoles with polarizability a.The permittivity for a set of such dipoles with concentration N can be found from the following well-known relations:

E=l+----

4 r aN 1-faN’

Here P is the polarization and the factor f expresses the above-mentioned difference between the acting field and the mean field. For randomly distributed point dipoles (or for a cubic lattice of such dipoles) we have f = 4 r / 3 , and (1 1) leads to the well-known Clausius-Mossotti-LorentzLorenz formula ( ~ - 1 ) / ( ~ + 2 ) = 4 7 r a N / 3If. f a N + 1 then and for faN > 1 we may obtain E < O . But in the static case and for long waves ( k+ 0) we cannot have E < 0 since for E + 0 the system becomes unstable and undergoes restructuring, for instance, the ferroelectric transition (Ginzburg 1987d). As noted above, however, the permittivities E < 0 are admissible for k # 0. Though formulas (11) are, strictly speaking, inapplicable for k # 0 they indicate that in order to obtain negative permittivities E we must take into account the difference between E,, and E and, in addition, the conditionfaN > 1 must be satisfied. If other atoms are present in the vicinity of the conducting atomic plane (for instance, the plane of the atoms Cu and 0) then the product faN can be large only in the case of a close contact or mutual penetration of the electrons of the “polarizer” atoms and the conduction electrons (indeed, f=S 1 and a u3 where u is the atomic size; if N < f 3 then faN i1). This estimate is, of course, rather rough but it explains why theoretical HTS models must include overlapping

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V.L. GINZRURG

of the wave functions of the “polarizer” electrons and the conduction electrons.

9. Nature of HTS in metal oxides

All newly-discovered “neon” and “nitrogen” HTS systems are layered materials (the same is true for their “prototype” .-the superconductor BaPb, rBi,03). In these materials the C u - 0 layers with a high conductivity alternate with layers of La or Y and Ba and so on (see, for instance, David et al. 1987, Simonov et al. 1987). Such materials naturally exhibit a strong anisotropy which is manifested in the studies of single crystals. For example, the nitrogen superconductor YBa,Cu,O,-, exhibited anisotropy of the critical magnetic field and the critical current. The ratio H ; , / H ! , 10 for this material and a similar ratio was measured for the critical currents (Dinger et al. 1987). Here H:, and H ! , are the critical fields that are, respectively perpendicular and parallel to the conducting C u - 0 layers. Strong anisotropy, particularly of the field Ifc-,(T), was found also by some other workers (see, for instance, Worthington et al. 1987). The quasi-twodimensional nature of metal oxides is particularly vividly manifested by the fact that the replacement of the non-magnetic Y ion with magnetic rare-earth ions (for instance, the Eu ion) does not result in a significant decrease in the temperature T,. This result suggests that the electrons responsible for superconductivity are concentrated in the C u - 0 planes and their wave functions are very small in the planes containing the ions of Y, Eu and so on. As noted in section 8, the layered materials seem to be the most advantageous in terms of the efficiency of the exciton mechanism. The lack of the isotope effect in the nitrogen superconductors noted in section 7 in any case does not contradict the assumption about the exciton (non-phonon) HTS mechanism. (In fact, Leary et al. (1987) have found a small isotope effect in the Y-Ba-Cu-0 system which, apparently, does not refute the above conclusion since phonons must play some part even when we have the dominant exchon mechanism.) In my opinion, an especially striking manifestation of t h e exciton mechanism in the YBa2Cu,0,-,. superconductors is the strong exciton absorption found in a specimen with JJ = 0.1 ( T, = 93 K). The absorption peak corresponds to the energy 0.37 eV (0,4 x 10’ K ) . On the other hand, a non-superconducting specimen of this system with y =0.8 does not exhibit this exciton absorption band (see fig. 4; Kamaras et al. 1987). A purely longitudinal electronic oscillation (a wave or an exciton) does not, of course, manifest itself in the infrared absorption.

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0

5Ooo

33

loo00

Frequency (an-')

Fig. 4. Conductivity as a function of frequency in the superconducting material YBa,Cu,O, and the non-superconducting material YBa,Cu,O, (Kamaris et al. 1987).

But it must be remembered that in non-cubic crystals (and in cubic crystals with spatial dispersion) excitons are, generally, neither longitudinal nor transverse in character (for details see, for instance,, Agranovich and Ginzburg 1984).* The ratio 2 A ( 0 ) / k , T c and the penetration depth for the magnetic field S ( T ) are significant parameters describing a material. According to the

* S. Tajima, H. Ishii, H. Takagi and S. Tanaka in a preprint of a paper submitted to Phys. Rev. Lett. ( I received it on October 29, 1987) reported the result obtained by measuring the electromagnetic radiation reflected from the single crystals of metal oxides. They suggest that the absorption peak for a superconducting material that can be found from fig. 4 is an artefact produced by the treatment of the measurement results for polycrystalline specimens. If this suggestion is correct the above arguments supporting the exciton mechanism are no longer valid. In my opinion, however, the problem remains open since Tajima et al. studied only single crystals with a certain orientation with respect to the wave vector q of the incident light. I d o not quite understand how such measurements, even if we also take into account the data for polycrystals, can lead to the conclusion about the absence of the exciton absorption peak for all orientations of q with respect to the crystal axes. It must be added that, as mentioned previously, the excitons of importance for HTS have k = k,. So an absence of excitons with k = q = 2 r / A = los cm- corresponding to the optical domain (where A is the wavelength of light) does not prove the absence of excitons, or electronic excitations, having k == k,= 107-10Rcm-'. But, of course, the presence of an excitonic peak even for q lo5 could be used as an indication (no more) of the additional presence of excitons with k > 10' cm-l.

'

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V.L. GINZBURG

34

reported results, the ratio 2A/kT, varies between 0.6 and 10. Kamaras et al. (1987) have reported for a number of experiments 2 A / k,T, = 3.5 which fully agrees with the BCS model with weak coupling (see formula (3)) including the case of the exciton mechanism. According to some other > 3.5 and, specifically, 241 knT, = 4.7 1.2 was found by reports, ?A/ kHTL Collins et al. (1987). Gershenson et al. (1987) gave 2 A / k B T , = 0 . 6 . This result differs so strongly from other results that it would hardly be worth mentioning if the measurement technique used in this study did not seem, in principle, to be especially reliable. In this technique the heat absorbed in the specimen is measured, rather than the reflection or transmission of the electromagnetic waves for the superconducting film. On the whole, no completely reliable data on the ratio 2 A / k s T , have been reported by the time of writing (October, 1987). Harshman et al. (1987) have used the technique of muon spin relaxation for measuring the penetration depth 6 ( T ) of a field in the YBa,CuO,-, material (here y = 2.1 + 0.05). When T + 0 (specifically, at T = 6 K ) the measurements gave 6 = 1365 A and this result and the specific heat value yield the “carrier” concentration in the normal state n = 0.6 x 10” cm-3 and the effective carrier mass m* = 4 m (it means that there is one carrier per unit cell or of a carrier per Cu atom). The observed dependence of the penetration depth on the temperature T suggests an s-pairing mechanism, that is, the BCS-model superconductivity. Finally, note that Gurvitch and Fiory (1987) analyzed the results on the temperature dependence of the resistivity and concluded that superconductivity in the nitrogen and neon materials has a non-phonon mechanism. The available results, thus, indicate that the newly-discovered nitrogen superconductors are materials with a pronounced layered structure and the dominant exciton mechanism of the HTS described by the BCS model (see, however, the footnote on p. 33). The phonon mechanism plays a significant part in the neon superconductors (the La-Sr-Cu-0 system and so on). The exciton mechanism, though, also makes a contribution in these materials (this fact accounts for the relatively weak isotope effect). Of course, phonons come into play also in the nitrogen superconductors but their contribution is small (that is, their effect on T,). The temperatures T, 300 K and even higher can, apparently, be obtained for 0, 5 x 10’ K. Little (1987) seems to hold similar views (on the nature of HTS in newly-discovered materials). Indeed, Little ( 1987) presented specific model calculations and estimates making use of the “chemical” concepts (orbitals and so on). I have no experience with these concepts and “language“. Another possible approach here is to make use of the permittivity language and I have made some remarks in terms of this concept above.

*

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35

It is clear, however, that we are talking here about something that has not yet been proved, or, as it is now fashionable to say, about a possible scenario. We cannot say at this point whether a correct theory can be constructed with this scenario. We have a good reason therefore to consider other scenarios. In addition to the above opinion, I should also mention the scenario that is currently popular among the staff of our department (E. Maksimov, 0. Dolgov, L. Bulaevskii and others). For the new (neon and nitrogen) superconductors a significant role is played by the phonon mechanism for which 0 @,in a formula of the type (2) for strong coupling. On the other hand, the electronic excitons also make an important contribution which can be regarded as a kind of the total suppression of the Coulomb repulsion, that is, they make the parameter small (see formula (4)). In other words, in the case of weak coupling we would have Aerr==A. For coupling of an intermediate strength for which A 1 the effective parameter herr(see, for instance, formula (6)) also increases since p = 0. When 0, is large owing to the presence of oxygen (see section 5 ) we obtain a high T, even for A 1. As the purely “excitonic” scenario, this one seems to be highly promising in my opinion. The third scenario, that attracts considerable attention for a good reason, involves the inclusion of the spin effects (see the end of section 7). To conclude this section, note that in the HTS the coherence length is relatively small. Under such conditions the part played by fluctuations near T, is enhanced, that is, the critical region is expanded. It is an interesting problem but I lack the space to go into it in this paper. I hope to discuss it elsewhere in the near future. Therefore, 1 shall only cite some references on the subject (Ginzburg 1960, 1987c,d, Ginzburg and Sobyanin 1987, Bulaevskii et al. 1984, White and Geballe 1979, Gor’kov and JCrome 1985).

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10. Concluding remarks

Some results and considerations presented in this paper may become dated by the time it is published. It cannot be prevented since the developments in the HTS field are hardly predictable. This is why I suggest returning to the history of HTS studies since this subject cannot become dated. In fact, some basic events in HTS history have already been noted above but I think it worthwhile to add some material here especially since I would prefer not to return to this subject in the future. HTS studies, particularly those that concerned the exciton mechanism, attracted considerable attention after 1964. An important event of this first period was the International Conference on Organic Superconductors held in Honolulu on September 5-9, 1969 (Proceedings of this conference were

36

V.L.. GINZBURG

edited by Little, 1970). It should be emphasized that the organic superconductors and intercalated layered superconductors were still unknown at the time. I believe that it was undoubtedly the discussion of the possible exciton mechanism of superconductivity that stimulated the search for such superconductors and studies of them (Gamble et al. 1971, Jtrome and Schulz 1982, Gor’kov 1984). The intercalated layered materials and the organic superconductors (produced for the first time in 1980) are clearly interesting by themselves or, to be more precise, irrespective of the HTS problem. Their manufacture, however, did not raise T, to the range of nitrogen or even neon temperatures. I shauld like to add a few remarks on the theory. What are the parameters and conditions determining T,? What is the upper limit for T, and is it sufficiently high for producing HTS? Many workers, probably, have though about these questions. A systematic and extensive programme of studies in this field was attempted in the I.E. Tamm Department of Theoretical Physics (P.N. Lebedev Physical Institute of the USSR Academy of Sciences, Moscow). This work was summarized in the book “High Temperature Superconductivity” to which I have repeatedly referred here (Ginzburg and Kirzhnits 1982; various chapters were written by L. Bulaevskii, V. Ginzburg, D. Khomskii, D. Kirzhnits, Yu. Kopaev, E. Maksimov and G . Zharkov). The book had been published in Russian in 1977 but its English translations appeared much later for unknown reasons. We suggested making considerable additions and changes in the English translations to the publishers but were allowed only some slight changes since the publishers thought it was too much trouble to make any significant additions (see the Preface to the book). I mentioned this fact here just to illustrate the typical attitude to HTS studies that prevailed before 1987. Below I cite the last part (section 6 ) of the introductory chapter written by me for this book since it is simpler than to repeat again the conclusions presented in it (note that the text is exactly the same as in the original Russian edition published in 1977).

“On the basis of general theoretical considerations, we believe at present that the most reasonable estimate is Tc=s300 K (see ( I S l ) ) , this estimate being, of course, for materials and systems under more or less normal conditions (equilibrium or quasi-equilibrium metallic systems in the absence of pressure or under relatively low pressures, etc.). In this case, if we exclude from consideration metallic hydrogen and, perhaps, organic metals, as well as semimetals in states near the region of electronic phase transitions (Chapter S), then it is suggested that we should use the exciton mechanism of attraction between the conduction electrons.

HIGH-TEMPERATURE SUPERCONDUCTIVITY

37

In this scheme, the most promising - from the point of view of the possibility of raising T, - materials are, apparently, layered compounds (Chapter 6) and dielectric-metal-dielectric sandwiches (Chapter 8). However, the state of the theory, let alone of experiment, is still far from being such as to allow us to regard as closed other possible directions, in particular, the use of filamentary compounds. Furthermore, for the present state of the problem of high-temperature superconductivity, the soundest and most fruitful approach will be one that is not preconceived, in which attempts are made to move forward in the most diverse directions. The investigation of the problem of high-temperature superconductivity is entering into the second decade of its history (if we are talking about the conscious search for materials with T,B90 K with the use of the exciton and other mechanisms). Supposably, there begins at the same time a new phase of these investigations, which is characterized not only by greater scope and diversity, but also by a significantly deeper understanding of the problems that arise. There is still no guarantee whatsoever that the efforts being made will lead to significant success, but a number of new superconducting materials have already been produced and are being investigated. Therefore, it is, in any case, difficult to doubt that further investigations of the problem of high-temperature superconductivity will yield many interesting results for physics and technology, even if materials that will remain superconducting at room (or even liquid-nitrogen) temperatures will not be produced. Besides, as has been emphasized, this ultimate aim does not seem to us to have been discredited in any way. As may be inferred, the next decade will be crucial for the problem of high-temperature superconductivity.” Let us discuss now the studies conducted by other authors. As mentioned in section 5 , Cohen and Anderson (1972) suggested that owing to the stability condition for metals, ~ ( 0k,) 3 0 , the critical temperature T, has an upper limit of the order of 10 K. This report contained, though, some reservations that made it possible to question the general character of this result. At the end of my paper published in 1972 I also presented objections against it. Kirzhnits (1976) later demonstrated that the stability condition is I/e(O, k ) s 1 and, hence, the values ~(0, k ) < 0 are quite admissible. I have repeatedly mentioned this fact which leads to the conclusion about the absence of some general restrictions for the temperature T,, at least, imposed by permittivity considerations. Many papers treated the exciton mechanism and various restrictions which it involves (see, for instance, Allender et al. 1973a,b, Inkson and Anderson 1973, Bardeen 1978, VujiCiC 1979, Sham 1985). Owing to the significance of the paper by Allender et al. (1973a), note that, as mentioned by Allender et al. (1973b) and in Chapter 8 of

SX

V.L. GI NZB UR C

Ginzburg and Kirzhnits (1982). its criticism by Inkson and Anderson (1973) did not concern the models of exciton mechanism that are of real significance (excitons must make a contribution even for large wave vectors k). It is hardly worthwhile to discuss here other critical responses* t o various suggestions for raising the temperature T,. Until 1986-87 while the prospects of HTS remained unclear the attitudes to this problem varied widely. Quoted below is a paper by Bardeen (1978): “In view of the large number of experiments that have been done and the wide variety of materials tested, many, including Bernd Matthias, have been pessimistic about prospects for finding excitonic superconductivity. While these experiments d o show that the conditions for observing it must be very exacting, they d o not rule it out completely. Since the potential importance of high-temperature superconductivity is so great, I feel that the search should be pursued vigorously even though the prospects for success may be small”. Bardeen concluded his note in the following way: “The search for superconductivity from excitonic mechanisms has been fruitful in leading to new scientific insights and there is still hope that the dream of high-temperature superconductivity will become a reality. It is a challenging problem that will i,ntrigue material scientists for a long time to come.“ In 1978 when the above statements were made I generally agreed with Bardeen’s views though I did not think that a really large number of experiments had been done and, on the whole, I was more optimistic. For instance, my report to a session of the World Electrotechnical Congress held in Moscow in 1977 contained a table that is also given here (table 3). *T he same i \ true. of course, for the total rejection ol’ the possihility of producing HTS voiced by Matthias (1970, 1971 1. Since Matthias is no longer with us I mentioned this fact here only to explain the following to the readers. The editor of Physics Today, H.L. Davis. sent me the paper by Matthias published in 1971 suggesting that I write an answer to it. In my return letter of August 29, 1971. I wrote that the content and spirit of the statements made hy Matthias in the above papers were such that I did not wish to enter into open polemics with him. But I wrote that I had decided “to give some information to Physics Tod\ay or any other responsible organization or person in the A I P and help them to come to their own conclusions ahout some of B. Matthias’ claims and remarks”. I attached to the letter the concluding part of my paper that wa5 published later (1972). I wrote at the end of that letter: ”Finishing t h k long letter, 1 would like once more to stress the point that I definitely refuse to enter into polemics with B. Matthias and wrote o n l y to give some available information. I f such information is not needed to you or somebody else you can throw this letter in a basket!“ Unfortunately, as I later learned from letters sent by M. Strongin, K. Brueckner and H.L. Davis, my letter not only had not been thrown away but somebody had made numerous copies of i t and distributed them among physicists in the U S A . I had not visited the USA since 1969 and, of course, I had no idea who had distributed my letter and why. Under such circumstances I had to agree to publication of my letter but the editors of Physics Today tinally decided that it was not necessary. Since many physicists know about this deplorable incident I had to tell here all I know about it to prevent misunderstandings.

HIGH-TEMPERATURE SUPERCONDUCTIVITY

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Table 3 Material or system

Superconductivity mechanism (nature of the atrraction between the conduction electrons)

Maximum possible value of T,.,,, (rough estimates in K )

Metallic hydrogen or deuterium, alloys, based on them Organic metals, hydrogen-containing metals, and so on Materials with electronic structural phase transitions Three-dimensional metals with exciton bands, and so on Laykred compounds and dielectricmetal-dielectric sandwiches Filamentary (quasi-one-dimensional) compounds CuCl and CdS with impurities, obtained under pressure and as the result o f special processing Non-equilibrium superconductors

phonon

100 to 200

phonon

50 to loo(?)

phonon o r exdton

100 to 300 ( ? )

exciton

100 to 300 (?)

exciton

100 lo 300

exciton

100 to 300 (?)

?

300(?)

There are no clear limits, but in practice it is probable that K T,.,,”S

It is typical that only for the layered materials (apart from metallic hydrogen) I did not put a question mark concerning the predicted critical temperature. I must admit, however, that with passage of time I became increasingly prone to doubts, too. This is why in the Waynflete Lectures on Physics (Ginzburg 1982) I exactly reproduced table 3 but put a question mark for the layered materials too (although of course in 1977 there was no reference to CuCl or CdS). In my last publication on HTS (Ginzburg 1984) that had been written before 1987 for a popular science magazine I described the situation in the field in the following words. “It happened somehow that the research in the field of high-temperature superconductivity became unfashionable (there is a good reason to speak of fashion in this respect since fashion sometimes plays a significant part in the research work and the scientific community). It is hard to achieve anything by making admonitions. It is, typically, some obvious success (or reports of success though they may be erroneous) that can radically and rapidly reverse the attitudes. When they sense a “rich strike” the former doubters and even dedicated critics are capable of turning coat and becoming

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ardent supporters of the new work. But this is another subject that is rather a concern of the psychology and sociology of science and technology and I shall not dwell on it here. In short, the search of the high-temperature superconductivity can readily lead to unexpected results and discoveries, especially as the predictions of the existing theory are rather vague.” That is what happened in 1986-87. The prospects of physical studies in the HTS field are generally clear enough. It is quite possible and even probable that HTS materials that have been discovered belong to only one or, say, two types. Are there any reasons that prevent the existence of several types of HTS materials? In short, as before the search for HTS must be conducted without any prejudice on as wide a scale as possible. The wide application of HTS in technology is clearenough. In conclusion, I should like to note only the following aspect. There are reasons to expect the fabrication of highly efficient superconducting storage batteries, that is, the superconducting magnets storing the magnetic energy. If such storage batteries can be produced the thermonuclear fusion reactors may prove to be superfluous. Indeed, according to the current predictions, such reactors will be very expensive and extremely large ( I am speaking about the reactors with magnetic confinement of plasma), they will utilize radioactive tritium (the use of pure deuterium is projected for the distant future) and, finally, in some variants use “blankets” of fissionable materials. So in fact, it is an expensive, complicated and “dirty” power production process. On the other hand, the use of the renewable energy sources, such as solar radiation, wind and tide, is made increasingly efficient but encounters primarily difficulties in the energy storage and in energy transmission with minimal losses over large distances. If these difficulties can be removed by the superconducting devices and transmission lines then, as noted above, the fusion power may prove to be unattractive and unnecessary. The HTS studies thus are of the utmost importance for the future of the power production. Even pessimists, however, predict extensive and promising applications of HTS not only there, but generally in technology and in medicine.

Notes added in proof New high-temperature superconductors have been synthesized, including layered compounds with Cu (systems Bi-Ca-Sr-Cu-0, TI-Ca-Ba-Cu-0, etc.). The highest values for T,, and T,,, 195 and 162 K respectively, were obtained for a compound tentatively identified as TICa,+ Ba,Cu,O, (Liu et al. 1988). However, these figures are not confirmed yet, and Tee= 125 K

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41

is the highest established critical temperature at the time of this writing. Superconductivity without Cu with T,, = 30 K was discovered for cubic Ba,.,K,.,Bi03 (Cava et a1 1988). For materials without magnetic ions (like Cu), the theories which connect HTS with spin effects are, as far as 1 understand, excluded. For other known HTS materials, the nature and mechanism of superconductivity is open to question. Very popular was the RVB-theory by Anderson and others, but if some of the experimental results are correct, there are grounds to prefer BCS-type theories or, in any case, theories with pairing near the Fermi level of some Fermi liquid. Concretely for the Ba2CaSr2Cu20, system, and probably for other similar ones too, most likely pairing of 2p holes on 0 takes place (Takahashi et al. 1988). Let us also stress the absence of the linear term in the specific heat, expected in RVB-theory, for the compound Bi4(Sro.,Cao.4)6C~,0.y (see Sera et al. 1988). About the mechanism of pairing, we would like to mention the paper by Quader and Salamon (1988) where arguments are put forward which speak against a decisive role of the spin fluctuations for pairing. So, pairing due to an exitonic mechanism (also called mechanism of charge fluctuations) which is especially near to the heart of the author, still is, in any case, quite possible. According to G. M. Eliashberg (1988) there is the possibility of the phonon mechanism as well. A macroscopic theory for superconductors with small coherence lengths, mentioned at the end of section 9 of the present paper, was already developed (Ginzburg 1988 and, in detail, Bulaevskii et al. 1988). Acknowledgements

I am grateful for comments and advice (particularly, made after reading the manuscript of this paper) to L.N. Bulaevskii, P. Campbell, D.A. Kirzhnits and E.G. Maksimov. References Achiezer, A.I., and I.Ya. Pomeranchuk, 1959, Zh. Eksp. & Teor. Fiz. 36, 859. Agranovich, V.M., and V.L. Ginzburg, 1984, Crystal Optics with Spatial Dispersion and Excitons (Springer, Berlin). Alexandrov, A.S., J. Ranninger and S. Robaszkiewicz, 1986, Phys. Rev. B 33, 4526. Allen, P.B., and R.C. Dynes, 1975, Phys. Rev. B 12, 905. Allender, D., J. Bray and J. Bardeen, 1973a. Phys. Rev. R 7, 1020. Allender, D., J. Bray and J. Bardeen, 1973b. Phys. Rev. B 7,4433. Anderson, P.W., 1987, Science 235, 1195; see also Phys. Rev. Lett. 58, 2790. Bardeen, J., 1956, Handb. Phys. 15, 274.

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Bardeen, J., 1963, Phys. Today 16(1), 19. Bardeen, J.. 1978, J. Less-Common Met. 62, 447. Bardeen, J., L.N. Cooper and J.R. Schrieffer, 1957, Phys. Rev. 108, 1175. Batlogg. R., R.J. Cava, A. Jayaraman, R.B. van Dover, G. Kourouklis, S. Sunshine, D.W. Murphy, L.W. Rupp, H.S. Chen, A. White, K.T. Short, A.M. Mujsce and E.A. Rietman, 1987a, Phys. Rev. Lett. 58, 2333. Batlogg, B., G. Kourouklis, W. Weber, R.J. Cava, A. Jayaraman. A. White. K.T. Short, L.W. Rupp and E.A. Rietman. 1987b. Phys. Rev. Lett. 59, 912. Bednorz, J.G., and K.A. Miiller, 1986. 2. Phys. B 64,189. Berlincourt, T.G.. 1987, IEEE Trans. Magn. MAG-23, 403. Bourne, L.C., M.F. Crommie, A. Zettl, H.C. Loye, S.W. Keller, K.L. Leory, A.M. Stacy, K.I. Chang, M.L. Choen and D.E. Morris, 1987, Phys. Rev. Lett. 58, 2337. Brandt. N.B., B.V. Kuvshinnikov, A.P. Rusakov and V.M. Semenov, 1978, Pis'ma v Zh. Eksp. & Teor. Fiz. 27, 37 [1978. Sov. Phys.-JETP Lett. 27, 331. Brown, E., C.G. Homan and R.K. MacCrone, 1980, Phys. Rev. Lett. 45, 478. Bulaevskii, L.N., and Yu.A. Kukharenko, 1971, Zh. Eksp. & Teor. Fiz. 60, 1518. Rulaevskii. L.N.. A.A. Sobyanin and D.I. Khomskii, 1984. Zh. Eksp. & Teor. Fiz. 87, 1490. Bulaevskii, L.M., V.L. Ginzburg and A.A. Sobayin, 1988, Zh. Eksp. & Teor. Fir. 94, 355 [Sov. Phys.-JETP, to be published]. Buzdin, A.I., and L.N. Bulaevskii, 1984, Usp. Fiz. Nauk 144, 415. Cava, R.J., R.B. van Dover, 8 . Ratlogg and E.A. Rietman, 1987. Phys. Rev. Lett. 58, 408. C'ava, R.J., et al., 1988, Nature 332, 814. Chu, C.W., A.P. Rusakov, S. Huang, S. Early, T.H. Geballe and C.Y. Huang, 1978, Phys. Rev. B 18, 2118. Cohen, M.L.. and P.W. Anderson, 1972, Proc. AIP Conf. on d-Band and f-Band Superconductors, 1971, New York, p. 17. Collins. R.T.,Z. Schlesinger, R.H. Koch, R.R. Laibowitz. T.S. Plaskett, P. Freitas, W.J. Gallagher, R.L. Sandstrom and T.R. Dinger, 1987, Phys. Rev. Lett. 59, 704. Collins, T.C., 1987. Ferroelectrics 73,469. Cooper. L., 1956. Phys. Rev. 104. 1189. David. W.I.F.. W.T.A. Harrison, J.M.F. G u m , 0. Moze, A.K. Soper, P. Day, J.D. Jorgensen, D.G. Hinks, M.A. Beno, L. Soderholm. D.W. Capone, I.K. Schuller. C.U. Segre, K. Zhang and J.D. Grace, 1987, Nature 327, 310. Dinger. T.R.. I . K . Worthington. W.J. Gallagher and R.L. Sandstrom, 1987, Phys. Rev. Lett. 58. 2687. Dolgov, O.V., and E.G. Maksimov. 1982. Usp. Fiz. Nauk 138.95; see also, 1983, Trudy FlAN - Proc. P.N. Lebedev Phys. Inst. Acad. Sci. USSR 148, 3. Maksimov, 1981, Rev. Mod. Phys. 53, 81. Dolgov, O.V.. D.A. Kirzhnits and €6. Einstein, A., 1927, Gedenkboek Kamerling Onnes, Leiden, p 429. Einstein, A,, 1925, Sitzungsber. Preuss Akad. Wiss. Phys.-Math. KI. 3. Eliashberg. G.M., 1987, Pis'ma v Zh. Eksp. & Teor. Fiz. 46.94. Eliashberg. G.M.. 1988. JETP Lett. 48, 275. Emery, V.J., 1987, Nature 328. 756. Faltens,T.A., W.K. Ham, S.W. Keller, K.J. Leary, J.N. Michaels, A.M. Stacy. H.C. Loye. D.E. Morris, T.W. Bardee, L.C. Bourne, M.L. Cohen, S. Hoen and A. Zettl. 1987, Phys. Rev. Lett. 59. 915. Friend, R.H., and A.D. YoHe, 1987. Adv. Phys. 36, 1. Gahovich, A.W., and D.P. Moisseyev, 1986, Usp. Fiz. Nauk 150, 59?. Gamble, F.R., J.H. Osiecki, M. Cais, R.'Pisharody, F.J. Disalvo and T.H. Geballe. 1971. Science 174, 493. Geballe. T.H., and C.W. Chu, 1979, Comments Solid State Phys. 9, 115.

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Geilikman, B.J., 1973, Usp. Fiz. Nauk 109, 65. Gershenson, E.M., G.N. Holzman, B.S. Karasik and A.D. Semenov, 1987, Pis'ma v Zh. Eksp. & Teor. Fiz. 46, 186. Ginzburg, V.L., 1944, Zh. Eksp. & Teor. Fiz. 14, 134. Ginzburg, V.L., 1946, Superconductivity (USSR Academy of Sciences, Moscow) 204 PP. [in Russian]. Ginzburg, V.L., 1952, Usp. Fiz. Nauk 48,26 [German translation: 1953, Fortschr. Phys. 1, 1011. Ginzburg, V.L., 1955, Zh. Eksp. & Teor. Fiz. 29, 748 [1956. Sov. Phys.-JETP 2, 5891. Ginzburg, V.L., 1960, Fiz. Tverd. Tela 2, 2031 [1961, Sov. Phys.-Solid State 2, 18241. Ginzburg, V.L., 1964a. Zh. Eksp: & Teor. Fiz. 47. 2318. Ginzburg, V.L., 1964b. Phys. Lett. 13, 101. Ginzburg, V.L., 1968a. Usp. Fiz. Nauk 95, 91. Ginzburg, V.L., 1968b, Contemp. Phys. 9, 355. Ginzburg, V.L., 1970, Usp. Fiz. Nauk 101, 185 [1971, Sov. Phys.-Usp. 13, 3351. Ginzburg, V.L., 1972, Ann. Rev. Mater. Sci. 2, 663. Ginzburg, V.L., 1982, Waynflete Lectures on Physics (Pergamon Press, Oxford). Ginzburg, V.L., 1984, Energia 9, 2. Ginzburg, V.L., 1985, About Physics and Astrophysics, Collection of Papers (Nauka, Moscow). Ginzburg, V.L., 1987a. High-temperature superconductivity, in: Proc. Adriatic0 Res. Conf. on High Temperature Superconductivity, Italy, Triest, 5-8 July, 1987, p. 3. Ginzburg, V.L., ed., 1987b, Superconductivity, Superdiamagnetism, Superfluidity (Mir, Moscow). Ginzburg, V.L., 1987c, Proc. Int. Conf. LT-18, Part 3, p. 2046. Ginzburg, V.L., 1987d, Ferroelectrics 76, 3 (Trudy FlAN - Proc. P.N. Lebedev Phys. Inst. Acad. Sci. USSR 180, 3). Ginzburg, V.L.. 1988. Phvsica C 153-155, 1617. Ginzburg, V.L., and D.A. Kirzhnits, 1964, Zh. Eksp. & 'l'eor. Fiz. 46,397. Ginzburg, V.L., and D.A. Kirzhnits, eds, 1982, High Temperature Superconductivity (Consultants Bureau, New York). Ginzburg, V.L., and L.D. Landau, 1950, Zh. Eksp. & Teor. Fiz. 20, 1064. Ginzburg, V.L., and A.A. Sobyanin, 1987, Proc. Int. Conf. LT-18, Part 3, p. 1785. Ginzburg, V.L., A.A. Gorbatsevich, Yu.V. Kopaev and B.A. Volkov, 1984, Solid State Commun. 50,339. Golovashkin, A.I., 1987, Usp. Fiz. Nauk 152, 553. Gor'kov, L.P., 1984, Usp. Fiz. Nauk 144, 381. Gor'kov, L.P., and D. Jerome, 1985, J. Phys. (France) 46, L-643. Gough. C.E., M.S. Colclogh, E.M.Fogan, R.G. Jordan, M. Keene, C.M. Muirhead, A.I.M. Rae, N. Thomas, J.S. Abell and S. Sutton, 1987, Nature 326,855. Grant, P., 1987, New Sci. 115, N 1571, 36. Gurvitch, M., and A.T. Fiory, 1987, Phys. Rev. Lett. 59, 1337. Harshman, D.R., G. Aeppli, B. Batlogg, R.J. Cava. E.J. Ansaldo, J.H. Brewer, W. Hardy, S.R. Kreitzman, G.M. Lake, D.R. Noakes and M. Senba, 1987, Phys. Rev. Lett. . Hodduson, L., G. Baym and M. Eckert. 1987, Rev. Mod. Phys. 59,287. Homan, C.G., and R.K. MacCrone, 1987, Ferroelectrics 73, 481. Huang, C.Y.,L.1. Dries. P.H. Hor, R.L. Meng, C.W. Chu and R.B. Frankel, 1987, Nature 328, 403. Inksoa, J.C., and P.W. Anderson, 1973, Phys. Rev. B 8, 4429. Jkrorne, D., and H.J. Schulz, 1982, Adv. Phys. 31, 209. Kamarb, K., C.D. Porter, M.G.Doss, S.L. Herr, D.B. Tanner, D.A. Bonn, J.E. Greedan, A.H. O'Reilly, C.V.Stager and T. Timusk, 1987. Phys. Rev. Lett. 59, 919. Kirzhnits, D.A., 1976, Usp. Fiz. Nauk 119, 357.

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Kirzhnits, D.A., 1987, Usp. Fiz. Nauk 152, 399. Kopaev. Yu.V., and A.1. Rusinov, 1987, Phys. Lett. A 121, 300. Kwok. W.K., G.W. Crabtree, D.G. Hinks, D.W. Capone, J.D. Jorgensen and K. Zhang, 1987, Phys. Rev. B 35, 5343. Landau, L.D., 1941, J. Phys. (USSR) 5. 71. Leary, K.J., H.C. zur Loye, S.W. Keller, T.A. Faltens, W.K. Ham, J.N. Michaels and A.M. Stacy, 1987. Phys. Rev. Lett. 59, 1236. Lefkowitz, I., J.S. Manning and P.E. Bloomfield, 1979, Phys. Rev. B 20, 4506. Lefkowitz, I., W.E. Estes. W.E. Hatfield and P.E. Bloomfield, 1987. Ferroelectrics 73, 487. Little, W.A., 1964, Phys. Rev. A 134, 1416. Little, W.A., ed., 1970, Proc. Int. Conf. on Organic Superconductors (Interscience, New York). Little, W.A., 1987. A New Model High 7, Excitonic Superconductor (preprint). Liu, P.S.. et al., 1988. High-7, Update 2, 16. August IS. London. F., 1950. Superfluids, Vol. 1, Superconductivity (Wiley, New York). London, F., 1954, Superfluids, Vol. 2, Macroscopic Theory of Superfluid Helium (Wiley, New York ). Mattheiss, L.F., 1987, Phys. Rev. Lett. 58, 1028. Matthias, B.T., 1970, Comments Solid State Phys. 3, 93. Matthias, B.T., 1971, Phys. Today 24(8), 21. Mazin, 1.1.. E.G. Maksimov, S.N. Raksheev, S.Yu. Savrasov and Yu.A. Uspenskii, 1987, Pis'ma v Zh. Eksp. & Teor. Fiz. 46, 120. Ogg, R.A., 1946. Phys. Rev. 69. 243. Ogushi, T., K. Obara and T. Anayama. 1983, Jpn. J. Appl. Phys. 22, L 523. Ogushi, T., Y. Hakuraku, Y. Honjo, G.N. Suresha, S. Higo, Y. Ozono, 1.1. Kawano and T. Numata, 1987, La-Sr-Nb-0 Films as a Candidate for Superconductor with T, up to 255[?71]K (preprint); see also R o c . LT-18, p. 1141. Otc, H.R., 1987, in: Progress in Low Temperature Physics, Vol. 11, ed. D.F. Brewer (NorthHolland, Amsterdam) p. 215. Pethick, C.J., and D. Pines, 1987, in: Proc. Int. Workshop on Novel Mechanisms for Superconductivity, Berkeley, June 1987 (preprint). Quader and Salamon, 1988, Solid State Commun. 66,975. Schafroth. M.R., 1954a. Phys. Rev. 96, 114Y; see also 1955, Phys. Rev. 100, 4 6 3 . Schafroth. M.R.. 1954b, Phys. Rev. 96, 1442. Schafroth, M.R.. S. Butler and J. Blatt, 1957, Helv. Phys. Acta 30. 93. Schriefier, J.R.. 1987, Talk at Conf. LT-I8 and preprint. Sera, M., et al.. 1988, Solid State Commun. 66. 1101. Sham, L.J., 1985, Physica t3 135, 451. Shapligin, I S . . B.G. Kahan and V.B. Lasarev, 1979, Zh. Neorg. Khim. 24. 1478. Simonov, B.I.. V.N. Molchanov and B.K. Veinstein. 1987. Pis'ma v Zh. Eksp. & Teor. Fiz. 46. 199.

Sleight. A.W., J.L. Gillson and P.E. Bierstedt. 1975, Solid State Commun. 17, 27. Takahashi, T., el al., 1988, Nature 334, 814. Vezzoli, G C ' . , and J. Bera, 1981, Phys. Rev. I3 23, 3022. Vuji[?5S]ciC, G.M., 1979. J. Phys. C 12, 1699. Weber, W., 1987, Phys. Rev. Lett. 58. 1371. Westfall. R., 1982, Never At Rest: A Biography of lsaak Newton (Cambridge Univ. Press, Cambridge). White. R.M., and T.H. Geballe, 1979, Long Range Order in Solids (Academic Press, New York). Wilson. J.A., 1978, Philos. Mag. B 38, 427. Wong, Zheng-yu, and Hang-sheng Wu, 1986, J. Phys. C 19, 5459. Worthington, T.K., W.J. Gallagher and T.R. Dinger, 1987, Phys. Rev. Lett. 59, 1160. Wu, M.K., J.R. Ashburg, C.J. Torng, P.H. Hor, R.L. Meng. L. Gao, Z.J. Huang, Y.Q. Wang and C.W. Chu. 1987. Phys. Rev. Lett. 58, 908. Zvezdin, A.K., and D.I. Khomskii, 1987, Pis'ma v Zh. Eksp. & Teor. Fiz. 46, 102.

CHAPTER 2

PROPERTIES OF STRONGLY SPIN-POLARISED 'He GAS BY

D.S. BETTS University of Sussex, School of Mathematical and Physical Sciences, Brighton, East Sussex, BN1 9Qh, England

and

F. LALOE and M. LEDUC Universite' de Pans VI, Laboratoire de Spectroscopie Hertzienne de I' Ecole Nonnale Supirieure, 24 rue Lhornond, 75231 Paris Cedex 05, France

Progress in Low Temperature Physics, Volume X I 1 Edited by D.E Brewer @ Elsevier Science Publishers B. V., 1989 45

Contents .............................. .. Introductory remarks . . . . . . . . . . . . .............. 1. Theory., . . , . , , . , . . . . . .......................... ........................... 1 . 1 . Introduction . , , . . . . . . . . . . . . . . , . . . . . . . I.?. The interaction potential , , . . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Characteristic lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Classical transport theory. . . . . . . . . . . . . . . . . . . . . . . . ............... 1.6. Quantum mechanical transport theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Thermal conductivity . . . . . . . . . . . . . . . . . ................... 1.8. Viscosity . . ................................................ 1.9. Spin diffusion,. . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . 1.10. Equation of state . . . . . . . . . . . . . . . .......... .................... 2. Polarisation methods.. . . . . . , . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . 2.1. Optical polarisation of 'He nuclei . . . . . . . . . . . . . . . . . 2.2. Relaxation processes .................................. 3. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Experiments on spin rotation effects and spin waves . . . . . . . . . . . . . . . . . . . . . 3.2. Experiments on thermal conductivity , , . . . . . . . . . . . . . ... 3.3. Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Liquid-gas equilibrium . . . , . . . . . , , . . . , . . . , ......................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

47 47 47 51 52 54

56 59 66 67 70 13 78 78 89 93 93 98 106 109 111

Introductory remarks Earlier volumes in this series have included chapters on spin-polarised atomic hydrogen by Silvera and Walraven (1986) and on spin-polarised 3 He-4He solutions by Meyerovich (1987). Of the three hydrogen isotopes in atomic form, 'H (ordinary hydrogen) and 'H (or Tfor tritium) are bosons, whereas 'H (or D for deuterium) is a fermion as is 'He. Quantum degeneracy can be expected but there are substantial experimental challenges in trying to achieve the necessary combinations of sufficiently high densities and sufficiently low temperatures. 'He dissolved in superfluid liquid 4He-II is also a fermion system, and if the fraction of 'He in the solution is less than about 6% the solute behaves in many ways like a perfect Fermi gas which can be cooled towards the absolute zero of temperature without condensation occurring; Fermi degeneracy effects have been experimentally demonstrated in a number of ways. In this present chapter we concentrate on the real Fermi gas of 'He which cannot be cooled into the strongly degenerate regime because condensation into the liquid phase comes first. There is obviously some overlap between this topic and those of Silvera and Walraven (1986) and Meyerovich (1987) and we have tried where possible to avoid covering the same ground; perhaps most conspicuously we have not dealt in detail with the gas-like behaviour of 'He dissolved in superfluid liquid 4He-II. All these systems can be spin-polarised, though not all by the same techniques, and interest in their properties was greatly heightened by a conference on spin-polarised quantum systems in 1980; the proceedings (edited by Laloe (1980)) remain a valuable source of information.

1. Theory 1.1. INTRODUCTION

Recent work on spin-polarised 3He gas has been comprehensively described in a wide range of published articles, many of which have been used heavily in the preparation of the present chapter. Where these are reports of original research the references are given at the appropriate points in the text in the normal way. Some reviews have also appeared and are useful as source material; these are not always explicitly referred to in the text, but they 47

48

D.S. BETTS ET AL.

importantly include Lhuillier and Laloe (1979, 1980), Lhuillier and Leduc (1985). Laloe et al. (1985). and Retts and Leduc (1986). For many purposes, where we are concerned with 'He atoms with thermal kinetic energies, the atoms behave as single particles (rather than the composites which we know them to be). In the ground state each has an angular momentum characterised by a quantum number I = $ whose origin is entirely nuclear. In an ordinary gas of such atoms at, say, room temperature and atmospheric pressure, the two spin levels m, = f 5 are equally populated and the sample is described as having zero nuclear polarisation. The properties and supporting theories of such a gas are usefully reviewed by Keller (1969). One may imagine a different situation in which one of the two levels is preferentially occupied at the expense of the other. This would be described as a state of non-zero nuclear polarisation, and in the extreme case (all in one level, none in the other) it would be 100%. In general, if there were N , and N, atoms respectively in each level, the polarisation M is defined as

We shall reserve until later a discussion of the practical means of producing polarisation M >O, and of the processes of nuclear relaxation which tend to destroy such polarisation. Meanwhile we shall see that one may expect some spectacular effects in polarised 'He. The extreme case with M = 100% is often denoted as 'He: and we consider this extreme case first. 'He, with nuclear spin quantum number I =;, is a fermion and all the differences between the ordinary (non-polarised) fluid and the polarised fluid derive from the Pauli principle of antisymmetrisation. These are purely statistical eHects, by contrast with the effects of zero-point motion which arise from atomic mass alone. I t is immediately clear why the Pauli principle plays a particularly large role in 'He,: we have an assembly of atoms which are totally indistinguishable, whereas for 3He with M < 1, the direction of the nuclear spin may serve as a label which breaks the indistinguishability. Two consequences follow: ( i ) The density of states p ( E ) of 'He; ( M = 100%) is different from that of ordinary .'He. This can be immediately understood in the approximation of independent particles valid for a dilute gas. For 'HeT there are fewer spin states accessible by a factor of two. Because of this, the radius of the Fermi sphere at T = 0 must be multiplied by the Fermi energy by 2?'", etc. At finite temperatures, the gas of polarised atoms has a greater degree of degeneracy than that of the unpolarised gas. Clearly these effects have analogues in an interacting fluid, although these are more difficult to evaluate.

STRONGLY SPIN-POLARISED 'He GAS

49

(ii) The interactions between atoms act differently in 'He?, being apparently reduced. Let us again simplify the problem by thinking about the dilute gas, which permits us to ignore all except two-body collisions. These latter are governed by the interatomic potential represented in fig. 1.1. Effective collision cross sections can be calculated from the interaction potential 4 ( r ) by a standard partial wave analysis of scattering. In this analysis, each partial wave with a given I-value corresponds to particles of angular momentum % ( / ( I + l))''2. For elastic scattering, which is valid for the present application, there are as many outgoing particles of angular momentum lfi as are incident on the scattering centre with angular momentum Ih. Hence the normalisations of the outgoing and incoming parts of each partial wave must be the same. Only a phase factor is left undetermined; each outgoing partial wave has experienced a phase shift S1. As a result the coherent sum of the partial waves no longer gives a plane but a distorted wave, corresponding to the scattering that has taken place. The values of the phase shifts depend on 4 ( r ) and on the kinetic energy of the approaching atoms. At very low energy, when the de Broglie wavelength of the atoms greatly exceeds the range uoof the potential +( r), all phase shifts tend to vary as lka0)*'+'where k is the (small) wavenumber; in these circumstances So, though itself small, greatly exceeds all the other 8,which may therefore to a good approximation be neglected. This, but for condensation, would be the case in a gas of unpolarised 'He as T+O, where only collisions between atoms with zero relative orbital angular momentum would

AZlZ HELIUM POTENTIAL

8l

6

Y

. *.

-10,

-121 0

'

"

'

'

0.25

,

.,

,..

"

'

0.50 Separation in nm

"

"

' 0.75

Fig. 1.1. The interaction potential between two helium atoms. The depth of the well is about lo-' eV (about 10 K ) and the range of the potential is about 0.3 nm. Calculated from a formula given by Aziz et al. (1987).

50

D.S. BETTS ET AL

be important in the calculation of properties of the gas. For simplicity we mention here only the rather artificial limiting case in which there is no condensation and the gas is considered in the low temperature limit. It is at this point that the Pauli principle plays its crucial role. It dictates that if two atoms have their nuclear spins parallel, the orbital wavefunction associated with the two atoms must be antisymmetric on exchange. It follows easily that the only permitted values for the orbital angular momentum correspond to odd values of 1. Thus if the gas is completely polarised ( M = 100°/o), collisions with I = 0 , 2 , 4 , . . . , are ruled out. At very low temperatures, therefore, the only important phase shift a0 contributing to the collision cross section would be pre-empted by this symmetry consideration, and the result would be that the atoms became effectively transparent to each other and so would not interact. See fig. 1.2. In a real gas at finite temperatures, even if it could be fully polarised, this stark result would of course be softened by the fact that the odd-I phase shifts a,, 6,, as,. . . , could no longer be neglected so the interaction would not disappear completely. The arguments given so far apply to collisions in which the nuclear spins are parallel. In a gas with M = 1, all collisions are of this nature. Otherwise, with M < 1, some collisions will be between atoms whose nuclear spins are

-

de Broglie wavelength

Fig. 1.2. Wavepackets associated with two ‘He atoms in the same spin state (parallel nuclear spins) undergoing a collision. The direct influence of interactions is ignored for the sake of simplicity, and a one-dimensional model is taken. ( a ) Wavepackets just before the collision. ( b ) During the collision the wavepackets overlap and, due to interference effects, the amplitude or the wavefunction is negligible close to x = 0. There is an “exchange hole” of a size comparable to the d e Broglie wavelength Ad”. (From Lhuillier and Leduc (1985).)

STRONGLY SPIN-POLARISED 'He GAS

51

not parallel and these are therefore distinguishable. In such collisions, the interference effects shown in fig. 1.2 will not occur and the interaction can come fully into play. For a general value of M,how many collisions will there be in each category? We may combine equation ( 1 . 1 ) with the simple conservation equation, Nr+ N J = N,

(1.2)

to obtain the fractions ft = N,/ N and f r = N 1 / N respectively: ft =$( 1 + M ) ;

f i = $(1 - M).

(1.3)

If we now choose a pair of atoms at random, the probability Pp that they will have parallel nuclear spins (tt or 5.5.) is, Pp=ff+f:=a(1+M)2+:(1

-M)2=f(l+M2),

while the probability Pa that they will have antiparallel spins

Pa= 1 - P, = f(1 - M 2 ) .

(1.4)

( t i or Jf)

is,

(1.5)

1.2. THEINTERACTION POTENTIAL

The interaction potential is quite well known. A simple Lennard-Jones form has often been used:

4(r) =44(a/r)'2-(~/r)61.

(1.6)

with constants €/kg= 10.22 K and a = 0.2556 nm as originally given by De Boer and Michels (1939). Aziz et al. (1979) have given a more precise form used in some calculations, and the best current version is probably that due later to Aziz et al. (1987). Both Aziz versions may be expressed as follows:

4 ( r ) = €4*(X),

(1.7)

where x = r/rm

and

4*(x) = A * exp(/3*x2- a * x ) - F ( x )

[:.d+-ji+p :".I ,

(1.9)

with either F ( ~ ) = e x p [ - ( ( D / x ) - l ) ~ ] for x < 0,

(1.10)

F ( x )= 1

( 1 . 1 1)

or for x 3 D.

D.S. BETTS ET AL.

52

The numerical values are shown below in table 1 , and it should be noted that all but e / k B and rm are dimensionless. Not all figures displayed are significant. Displayed digits are given specifically for the avoidance of round-off errors. 1.3. CHARACTERISTIC LENGTHS

There are three characteristic lengths which could be expected to play parts in determining the properties of a gas of ’He, whether or not it is polarised. These are as follows. ( 1 ) The mean distance n - ’ 1 3 between atoms, where n is the number density. For some purposes it is convenient to express this as ( k B T / p ) + ” ’ and numerically this is equal to 4.7( T / p ) ’ / ’nm where T is in K and p in torr ( 1 Torr= 133.3 Pa). Table 2 shows how the magnitude of this quantity varies as a function of temperature below the critical temperature T, ( = 3.32 K) for (i) a constant pressure of 1 Torr and (ii) the vapour pressure. ( 2 ) The range a, of the interatomic potential. This is to some extent a matter of definition but it is obviously of the order of u in the Lennard-Jones interaction (eq. (1.6)) or of rm in either of the Aziz potentials. We take a, = 0.3 nm for present purposes. (3) The de Broglie thermal wavelength AT=h(2.rrmkBT)-’/*. This is a measure of the size of the wave packet surrounding an atom in the gas, and numerically A T = 1.01 nm. Values are given in the table below. All three quantities, ( l ) , (2) and (3), are given numerical values in table 2. Two new effects come from quantum mechanics, and these may be thought of as “diffraction” and “symmetry” effects. The distinction between them is related to the magnitudes of the length parameters as follows. For reasons which will become clear, we shall be concerned in this article mainly with a gas at temperatures between 1 and 4 K and at pressures typically of order 1 Torr. This is a dilute gas, that is, a gas of ’He for which n-’/’ + a,. Table 1 Aziz et al. (1979) F/k,,

= +10.8

K

r, = +0.29673 nm

A* = +5.448504 x 10‘ a* = +13.353384

p* = 0.0

+

c6 = 1.3732412 c,, = +0.4253785 c,,,= +0.178100

0=+1.241314

Aziz et al. (1987)

E l k B = +10.948 K rm= +0.2963 nm A* = +1.8443101 x lo5 a* = +10.43329537 p* = -2.27965105 c6= +1.36745214 cg = f0.42123807 c I 0 = +0.17473318 0=+1.4826

53

STRONGLY SPIN-POLARISED ’He GAS Table 2

3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 I .O 0.8 0.6

6.93 6.78 6.62 6.46 6.29 6.1 1 5.92 5.72 5.50 5.26 4.99 4.70 4.36

-

0.76 0.80 0.84 0.89 0.95 1.03 1.1 1 1.22 1.37 1.56 1.84 2.28 3.08 4.90

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

0.56 0.58 0.60 0.63 0.65 0.68 0.71 0.75 0.80 0.85 0.92 1.01 1.13 1.30

In these circumstances, when two atoms collide, the deflection of the relative velocity resulting from the encounter is much the same as if, on the classical theory, each atom were supposed surrounded by a wave field whose linear extension is of the order of AT. This field gives rise to diffraction effects. Thus for example rigid spheres are deflected by each other’s wave field even when no actual contact occurs, so that in this case the effective diameter is enlarged. In general, depending upon the form of the interatomic interaction, there may be an enlargement or a diminution of the effective diameter. This is the “diff faction” effect. At thermal equilibrium the “symmetry” effectbecomes noticeable when the temperature is lowered further until AT rises towards equality with n-”3. There is then a state of congestion in the gas, and this can be related to the average occupation of accessible singleparticle states. In the case of a gas of 3He atoms, further reduction of temperature would lead steadily towards a state of Fermi degeneracy but this is virtually impossible to observe. For example, a gas with a number density corresponding to a pressure of 1 Ton at a temperature of 1 K has a Fermi degeneracy temperature of 35 mK, which is well below its condensation temperature of 250mK and therefore inaccessible. In order to see strong effects of symmetry in ’He, it is necessary to deal with the gas-like system of ’He dissolved in liquid 4He-II rather than the real gas itself; see Meyerovich (1987). Two further remarks should be made at this point. Firstly, significant degrees of polarisation are attainable only by putting the gas into a metastable state. Equilibrium polansation due to an applied magnetic field B is

D.S. BEITS ET AL.

54

never large for realistic fields and temperatures, being given by M(equi1ibrium) = tanh( h y B / 2 k , T ) ,

(1.12)

with h y / 2 k B= 7.78 x K per tesla. For example, with T = 2 K and B = 20 T, M is only 0.78%, and even 100 T would only produce M = 3.9%.We return to this matter in greater detail at a later stage, but it must be borne in mind that substantial polarisation cannot be a permanent state in gaseous ‘He; after a certain relaxation time, shorter or longer depending on a number of parameters, the nuclear moments of the atoms tend to return to a true thermodynamic equilibrium in which the moments are distributed in a nearly random fashion in space. The second remark concerns the small magnitude of the nuclear magnetic moment of ‘He. I n most of what follows (except in the evaluations of relaxation times) we shall ignore the dipole-dipole coupling between nuclear spins. This approximation is fully justified except at extremely low temperatures: at a distance of the order of a,, the magnetic field due to the nucleus is of the order of T and the corresponding magnetic interaction energy is of the order of 0.1 pK. One may also observe that solid 3He at the melting pressure (34 atm) becomes antiferromagnetically ordered below about 1 mK (Halperin et al. (1978)). 1.4. TRANSPORT PROPERTIES

Consider the regime where

for example in the gas at a pressure of about 1 Torr and a temperature in the range 1-4 K according t o table 2 above. For n-”3>AT one cannot observe at thermal equilibrium any effects of nuclear polarisation. The equilibrium properties of the gas are well described within the framework of classical statistical mechanics irrespectively of polarisation (see section 1.10). On the other hand, for A,&a, it is essential to take account of the statistical correlations brought by the Pauli principle into the atomic collisions which govern the transport properties of the gas. For a gas with M = 100% (fully polarised) the effective interaction is reduced as described earlier, and this brings about a corresponding increase in the mean free path of the atoms. It follows that the viscosity and thermal conductivity of the gas at low temperatures may be expected to be increased by polarisation and we shall see that more precise calculations confirm this expectation. In order to calculate transport properties of dilute gases it is necessary to solve the Boltzmann equation to an appropriate approximation by considering the detailed dynamics of binary collisions and statistical expressions

STRONGLY SPIN-POLARISED 'He GAS

55

for the fluxes. The original key papers in which the methods were established (by Hilbert, by Chapman and by Enskog) in the years 1912-22 have been conveniently collected, in English translations where necessary, by Brush (1972) who includes a commentary and a thorough bibliography. Some authoritative later expositions of the classical and quantum mechanical approaches have been given by Grad (1958), Bernstein (1966), Hirschfelder et al. (1954), and Chapman and Cowling (1970). We do not propose in this article to recall the theory of transport in gases in its entirety. It is necessary however to give some of the essential stages in order to be precise about the influence of symmetry on the results. The starting point is the Boltzmann integro-differential equation which relates the fluxes of the dynamical variables (e.g. mass flux, momentum flux, and energy flux) of the particles to the external forces acting on the particles. Symbolically, we may write this equation (1.14)

which has as solutions the single-particle distribution functions f(r, p, r ) describing the nanequilibrium system in terms of the time and the momentum and position vectors of the particles. The time rate of change o f f is given as the sum of two terms, the first of which is the streaming term. The second is the collision term, and, as denoted by the appearance of J(ff), this depends on the dynamics of the particles undergoing binary collisions only. The work of Hilbert, Chapman and Enskog (referred to above) resulted in a special solution for f as a power series: f = f o + a f , + a 2 f 2 + **

* .

(1.15)

Here the term fo is the local Maxwell-Boltzmann distribution and leads to Euler's equations of hydrodynamics for ideal fluids. To first order, when ( f o + a f l )is taken, the Navier-Stokes equations result. We shall not go beyond this approximation in this article. The power series solution is valid provided that the ratio (expressed through the coefficient a ) of the mean free path between collisions to the size of the gas container is much less than unity. The function fl contains the gradients of such quantities as the temperature, the velocity, and M which is also a hydrodynamical variable here, and these must not vary significantly over a mean free path. Furthermore,fl contains the effects of the interparticle potential and is independent of density. There is obviously a difficulty in applying this approach to the extreme case of 3He gas with M = 100% because then the effective collision cross section tends to zero as T - , 0. In such a case one would observe that below a certain temperature, momentum and heat would be carried in a ballistic manner.

56

D.S. BE'ITS ET AL.

The hydrodynamic regime is governed by the Navier-Stokes equations. In this approximation, the fluxes of heat and momentum are proportional to first-order spatial derivatives of the macroscopic properties of the gas density, mean speed, temperature. The coefficients of proportionality, which are the transport coefficients which we seek, are themselves expressed in One obtains also the coefficients terms of collision integrals denoted by L?"*". of viscosity 77 and of thermal conductivity K in the forms of rapidly converging series of which classically the first terms are (1.16)

where c , / m is constant-volume heat capacity per unit mass (3kR/2m for present purposes). The collision integrals are easily calculated from (1.17)

where mr = m / 2 is the reduced mass, g the relative velocity of approach, and y a reduced velocity defined by (1.18)

The characteristics of the interaction are taken into account in the angleaveraged cross sections Q"'( g ) defined by (1.19)

where a ( g , 0 ) is the probability of scattering of two atoms with relative velocity g through an angle 8. 1.5. CLASSICAL TRANSPORT

THEORY

Classically the evaluation of a ( g , 0 ) is a simple matter of following the possible trajectories with the result that in the integral a ( g , 8 ) sin 0 dB is replaced by g6 d b where 6 is the impact parameter (see fig. 1.3) and B(g, 6) is given in terms of the interaction potential +( R ) by ( 1.20)

where R , denotes the value of R at closest approach, as shown in the figure. In the simplest case, where atoms are treated as rigid elastic spheres

STRONGLY SPIN-POLARISED 'He GAS

57

Fig. 1.3. Dynamics of a collision between two 'He atoms A and B. BB' represents the path of B relative to A. The impact parameter b is the distance from A to the initial direction of approach, R , is the distance of closest approach, and 0 is the angular deflection of B.

of diameter u without a field of force, it is trivial to follow the recipe to obtain cos f O ( g , b) = b / u ,

(1.21)

Q"'(g) = (2.rr/3)u2,

( 1.22)

(1.23)

so that from eqs. (1.16), ( 1.24)

It is important to recall that we are still expounding the classical theory with the simplest hard-sphere interaction. This is not adequate for a full description of some of the transport phenomena to be discussed later, but

D.S. BETTS ET AL.

58

it has the advantage of producing results which it is easier to visualise. We

will take it only one step further. We refer to expressions from Chapman and Cowling (1970) for the first approximations to the viscosity ( C & C equation (12.4, 1)) and thermal conductivity (C & C equation (13.4, 1)) of a monatomic gaseous binary mixture in which both atomic types are elastic spheres without fields of force. The molar fractions, masses, and diameters are taken as ( x , , m , , a , )and (x2, m 2 , a2)respectively, where (x, +x,) = 1. In such a gas there are three types of collision, namely, 1-1 with a , ,= a, , 2-2 with u 2 , = a 2 and , 1-2 with u , , = ~ ( a , + u 2We ) . can adapt the expressions to our needs as follows. Firstly, we take the two types of atom as t and 1instead of 1 and 2, using eq. (1.3) to relate x, and x2to polarisation M: x , = .ft = ;( 1

+ M ),

xa =fi =

I( 1

-

M ).

(1.25)

Secondly we take all spheres to have the same mass m = m , = m , . Thirdly, since there are only two categories of collision, those for which the appropriate diameter is up(moments parallel) and those for which the appropriate diameter is a, (moments antiparallel), we substitute (1.26)

u,,=u?2=Wp

for collisions involving parallel moments and

+

(1.27)

! ( w , , a??) = a,

for those involving antiparallel moments. After some tedious algebra we arrive at (1

+ M 2 ) + ( 1- M 2 ) (1 +3r)/4

)

(l+M2)r+(l-M')(5+3r2)/8 '

(1.28)

(1+M2)+(1- M2) (27+32r)/59 ( 1 + M 2 ) r+ ( 1 - M2)(43+ 16r2)/59 ( 1.29)

(1.30)

is the ratio of the'two cross sections (parallel and antiparallel) involved. For M = 1 , all collisions will be of type tt and eqs. (1.28) and (1.29) reduce to the forms of eqs. (1.24) with upfor a. Equations (1.24) are also recovered for r = 1 whatever the value of M since all collisions then have the same cross section. The case M = O also gives a result like eq. (1.24) but with :[a:+ a:] in place of uf.This all makes for a reasonably consistent description and even the terms ( 1 + M 2 ) and ( 1 - M 2 ) can be recognised in that $(1 + M 2 ) and :(I - M 2 )are, respectively, the fractions of all collisions with parallel and with antiparallel moments, as shown above.

STRONGLY SPIN-POLARISED 'He GAS

59

Another way of writing these equations, which has a bearing on the way in which some experiments have actually been performed, is as follows:

8 M 2 (1 - r ) [T(M)I, -[T(o)ll (1 + r)(5+3r) - M2(1 - r)(5-3r)' [T(O)Il

(1.31)

for viscosity and [K(M)II - [ K ( O ) l l - [K(O)ll

59M2(1 - r ) (1 + r)(43 + 16r) - M 2 (1 - r)(43 - 16r)'

(1.32)

for thermal conductivity. For r = 1 these expressions are zero (that is, [q(M)l1 and [ K ( M ) ]are ~ independent of M); for r t l they vary as M 2 / (1 - M 2 )and if in addition M is small the expressions are approximately proportional to M 2 . Somewhat similar results can be obtained by more simple-minded kinetic theory arguments. Transport properties are proportional to an appropriate mean free path A ( M ) which must allow for the existence of two different collision cross sections. Bearing in mind the relative probabilities of the two types of collision (see eqs. (1.4) and (1.5)) we may suppose that A - ' varies as [f(1+ M 2 ) a i $(1- M 2 ) a i ] .It then follows readily that

+

(1.33) ( 1.34)

This expression is in fact very similar to eqs. (1.31) and (1.32). For r = 1 it is zero (that is, v ( M ) and K ( M )are independent of M); for r < l it is approximately M2/(1- M 2 ) and if in addition M is small, the expression is approximately proportional to M'. 1.6. QUANTUMM E CHANICAL TRANSPORT

THEORY

The classical approach discussed above is simply inadequate for the complete description of the richness observed in the behaviour of spin-polarised 'He gas. Lhuillier and Laloe (1982a,b) have provided the necessary theory in the form of solutions to the proper quantum mechanical version of the Boltzmann equation. There is also a considerable literature relating to the gas-like system of 'He dissolved in superfluid liquid 4He-II.There are some strong similarities, and in addition these solutions can be experimentally investigated in the Fermi degenerate state because condensation does not occur when the concentration of 'He is less than about 6% even as T+O.

D.S. BETTS ET AL.

60

Moreover these solutions can be polarised significantly simply by the application of a magnetic field. This chapter is concerned with the gas phase only, since the previous volume in this series contains a chapter by Meyerovich ( 1987) entitled “Spin-Polarised 3He-4He Solutions” which is complementary. See also the chapter entitled “Spin-Polarised Phases of ’He” by Meyerovich (1989) from which many references to the work of Bashkin and Meyerovich can be obtained; some of these will be mentioned later. Transport in gases, properly considered, is a matter of some technical complexity and we shall attempt only a summary of the results here. Lhuillier and Laloe (1982a,b) were able to produce results which should be valid for arbitrary values of M.The starting point was a detailed study of a binary collision, making use of a preceding article by Pinard and Laloe (1980) and importantly including the effects introduced by particle indistinguishability. The authors write a Boltzmann equation for a spin operator p s ( r , p) which replaces the usual Boltzmann distribution function f ( r, p) which appears in eq. (1.14). This follows the work of Waldmann (1958), and also calculations concerning quantum fluids (see for example Silin (1957)). There is a restriction to the study of dilute gases, and the Boltzmann equation obtained does not include any degeneracy effects. Nevertheless the structure of the (binary) collision term is more complex than in its classical counterpart. Instead of only one collision cross section, there are a number of cross sections which depend in different ways on the interatomic potential. Depending on the physical situation of interest (the macroscopic gradients of velocity, temperature, orientation, etc.) and of the value of the polarisation M, different linear combinations of the independent cross sections play the dominant role. First of all there are the four real quantities, u k ( 0 ) ,uY(O), T ; ‘ ( O ) , and &; each of these has dimensions of area (but note that, confusingly, the constant u in the Lennard-Jones interaction equation ( 1.6) is a length) and the first three (but not the fourth) depend on 8. These are all related to the interatomic potential through the phase shifts a1as follows u, = k - 2 C exp[i(bl- 6,.)](21+ 1) I. I’

x (21’+ 1) sin

sin 61~Pl(cos O)Pl,(cos O), ~ y = k - ’ C( - 1 ) ‘ ~ 0 ~ ( 6 , - 6 1 . ) ( 2 1 + 1 )

(1.35)

1. I’

x (21’+ 1) sin 6, sin 61~Pl(cos O)Pl.(cosO),

(1.36)

T ; I = k - ’ C (-1)1sin(Sl-61,)(21+1) I. I ‘

x (21’+ 1) sin

a1sin 61~Pl(cos O)Pl.(cos O),

( 1.37)

(1.38)

STRONGLY SPIN-POLARISED 'He GAS

61

It was mentioned earlier that the wavefunction after scattering is phase shifted by relative to the case with zero scattering potential. Calculation of SI for a given potential is in general a complicated business involving numerical integration of the radial wave equation for the relative motion of the two atoms, going outwards from the centre of the potential and proceeding to large distances to find the asymptotic behaviour and hence the phase shifts. This equation is (1.39)

The asymptotic form of the solution of this equation, which is an acceptable function at the origin, is r-' sin(kr - : l ~ + 6 ~ ) .

( 1.40)

Standard texts which discuss the mathematics of this include Schiff (1955) and Mandl (1957). Earlier papers reporting work involving the evaluation of 6, for particular potentials include Massey and Buckingham (1938,1939), De Boer and Michels (1939), Buckingham et al. (1941), De Boer et a]. (1949, 1950), De Boer and Cohen (1951), Kilpatrick et a]. (1954), Keller (1957), Monchick et al. (1965), and Bernstein (1966). The results shown in fig. 1.4 for 8, as a function of k for 'He are from Lhuillier (1983).

ta

Fig. 1.4. Collisional phase shifts 6,for 'He as a function of the reduced wavenumber k* = b (where u is the length parameter 0.2556 nm which appears in eq. (1.6)). k* is proportional to the square root of the kinetic energy, and k* = 1 corresponds to a kinetic energy of 2.46 K. (From Lhuillier (1983).)

D.S. BETTS ET AL.

62

Associated with each of a k ( 0 ) ,a;"(@), and T ' , " ( e ) there are "angleaveraged cross sections" defined like the Q") in eq. (1.19). These are Qi"[uA]'2n Q " ' [ a y ]= 2~

Q'"[T;']

= 2~

I:

( 1 -cos'e)a,(e) sin

I,: 1:

e de,

(1.41) ( 1.42)

( 1 -cost O)a',"( 0 ) sin 0 do,

( 1 - cos'

e)T;"(

e) sin e de.

( 1.43)

The k-dependences of some of these, Q"'[T;"], Q"'[ak], Q("[ae,"], and are shown in figs. 1.5 and 1.6. For calculations of viscosity, thermal conductivity, longitudinal spin diffusion and their couplings, in the first approximations developed by Lhuillier and Laloe (1982a,b), only three of these are needed, namely Q ' " [ u k ] ,Q'2'[a,],and Q'"[at"], for which the integrals reduce to the following forms Q'''[al,]=4nk ' C ( I + l ) sin2(6,-61+,),

(1.44)

Q'"[ak] = 4 ~ k - ~( I C + 1)(1+2)(21+3)~'sin2(61-6,+2),

( 1.45)

I

I

Q'"[crz"] = 4 ~ k - ( -~1 ) ' ( I + 1 ) ( I + 2)(2I+3)

' sin'(6, - 61+2). (1.46)

I

In addition to the more familiar transport properties, there are "identical spin rotation" effects. These are not peculiar to 'He, and are observable

Fig. 1.5. 'He cross sections Q ' " [ a Aand ] Q'"[u;L]from eqs. (1.45) and (1.46). (From Lhuillier (1983).)

STRONGLY SPIN-FQLARISED 'He GAS

Fig. 1.6. 'He cross sections Q"'[ry] from eqs. (1.37) and (1.43). and (From Lhuillier (1983).)

63

T;&

from eq. (1.38).

for example in the scattering of a polarised electron beam by a polarised target. In such scattering, the electrons are not subject to explicit spindependent forces yet, in accordance with Pauli's exclusion principle, the scattering amplitudes are different in the case when incident and target electrons have parallel spins and when they have antiparallel spins. The result (Byrne and Farago (1971)) is a rotation analogous to the Faraday rotation in optics. The axis of the polarisation rotation is defined in this case by the target polarisation and is independent of the direction of the incident electron beam. Byrne and Farago (1971) give an explicit expression for the angle of rotation. The identical spin rotation effect in 'He is discussed by Lhuillier and Laloe (1982a,b), Lhuillier (1983), and Lhuillier and Leduc (1985). So far we have had in mind a gas of 'He in which the atoms have nuclear spins which are either parallel or antiparallel. But new effects arise in the case of a gas having not only longitudinal orientation but also transverse orientation. Let us again begin by considering a binary collision, but now one in which the nuclear spins are at some angle other than 0 or T.It is impossible to find a reference frame in which the two atoms are in a pure spin state. Let us arbitrarily choose as an axis the spin direction of one of the atoms, say atom 1. In spin space, the two atoms may be symbolically represented as 11) = I+),

( 1.47)

la=al+)+PI-).

( 1.48)

D.S. B E T l S ET AL.

64

In the absence of interaction, this situation would not evolve with time. With an interaction which is independent of spin, it is tempting to draw the same conclusion, but this would be false because of the presence of exchange terms. Because of the symmetry constraints referred to above, the two components of atom 2 (eq. (1.48)) interacting with atom 1 d o not sample the interaction potential in the same way and the corresponding phase shifts are different. This supplementary phase shift appears as a rotation of the spin of atom 2 as if there were an effective magnetic field. This simplified description obscures a fundamental aspect of the process by building into eqs. (1.47) and (1.48) an asymmetry of view regarding the respective roles of the two atoms. In fact it is clear that the spin of each atom turns during the collision and that, the total spin being a conserved quantity, the rotation takes place about the resultant of the two spins. The situation is represented in fig. 1.7 from Lhuillier and Leduc (1985). The axis Oz is directed along the resultant of the initial spin directions M!,')and M ? ) of a pair of interacting atoms. Before the collision, the spin of each atom has longitudinal and transverse components M zand M I (fig. 1.7a). During the collision (fig. 1.7b), the spins of both atoms turn around their resultant Oz. After the collision the components M , of both atoms are unchanged but their transverse components have evolved (see figs. 1.7a and 1 . 7 ~ )in that M:" and Mi" differ from MI" and M:".It is as if a virtual magnetic field were acting. The idea of exchange fields is familiar in dealing with magnetism in condensed phases; it can now be seen to play a role in the interaction of two isolated atoms. We shall return to these matters in discussing experiments on spin waves. For the present, we return to the topic of collision cross sections and the fact that two of them are required to describe the identical spin rotation effects. These are 7;:d (eq. (1.38)) which measures the identical spin rotation effect in the transmitted beam and, evaluating eq. (1.43) with t = 1,

Q "' [ rY ]= 87r-'

Z(-I)'(]+

1 ) sin(& -

sin 6' sin a,+,,

( 1.49)

/

which measures the identical spin rotation effect in the scattered beam. Finally, eight of the collision integrals we actually need are from eq. (1.17) with appropriate values of ( 1 , s):

fl".'" Ty],

fp.ll[

R(2.2"ak3,

a".')[ay3.

7;1],

( 1S O )

n'l'"[ T?],

A further three come from similar averages of

7;;d

using the definition (1.51)

STRONGLY SPIN-POLARISED ‘He GAS

65

Fig. 1.7. A schematic representation of the “identical spin rotation effect” during a collision between,two ’He atoms at low temperature. (a) Before the collision, atoms 1 and 2 carry the and Mt), respectively. (b) During the collision, the magnetisations initial magnetisations MI’’ M ,and M, rotate around their vector sum directed along Oz. (c) After the collision, the initial and MY)have been shifted to MI” and Mi”,respectively. (From Lhuillier magnetisations Mi’’ and Leduc ( 1 9 8 9 . )

where T& is the cross section evaluated from eq. (1.38) with k = (2mrkBT)’”y/h, and these are =(I)[

T fex wd

]

9

-;;(a[

T fex wd

3

3

T ( 3 ) [ T fex wd

1.

(1.52)

All the quantities in eqs. (1.50) and (1.52) have the dimensions of swept volume per unit time and this may be thought of as the product of a mean thermal velocity 2[kBT/mnr]’” and an area. A common practice is to scale would have unit all the l2“*” and E“)in such a way that f2*(‘.’)and 8*(s) value for classical rigid spheres of diameter cr and then arbitrarily take

66

D.S. BETTS

ET AL.

u = 0.2556 nm for 3He (this choice comes of course from the Lennard-Jones

interaction equation (1.6) but is nevertheless arbitrary in the sense that the end products of the calculation, for example viscosity or thermal conductivity, are independent of the choice). The conversions are as follows: 2 0 ( 1.J ' ( 1.53) (s + 1)![ 1 - ( 1+ (- 1)')/(2 + 2t)]7Tu2' (1.54) Some of these results occur for example in the first approximations for the thermal conductivity [ K ( M ) ] and ~ the viscosity [77(M)II. CONDUCTIVITY 1.7. THERMAL

(1.55)

(1.56)

(1.58) It is interesting to look at some limiting cases. Lhuillier and Laloe (1982a) show that as T+O all the fl*('." in the thermal conductivity expressions approach equality so that tr-*-16/43 and & + + 1 as T - 0 . ( 1.59) Thus from eq. (1.55), [ K ( M ) ]diverges ~ as M + 1. At sufficiently low temperatures, therefore, a high nuclear polarisation would increase the thermal conductivity dramatically, though there is very little chance of ever observing this in experiments on the gas. At experimentally accessible temperatures the collision integrals must be numerically evaluated. For M = 0 the result (1.56) applies; for M = 1 some algebraic manipulation of eq. (1.55) gives ( 1.60)

STRONGLY SPIN-POLARISED 'He GAS

67

Now eqs. (1.56) and (1.60) differ only in the factor 2 multiplying R * ( 2 * 2 ) [ a ~ ] in the denominator. Lhuillier and Laloe (1982a) point out that one might have expected to be able to interpolate between (1.56) and (1.60) using (1.4) and (1.5) to obtain a denominator equal to (1 + M 2 ) X { ~ * ( 2 ' 2 ) [ a k ] - ~ * ( 2 ' 2 ) [ a e k x ] } + ( 1 - M 2 ) X ~ * ( 2 ' 2Iakl. )

Unfortunately this only coincides with the correct interpolation (1.55) in the limits M = O and M = 1. For general M it does not contain the three other quantities R*('*')[ak], n * ( l ' z ) [ a k ] , and n*"'"[ak]. Physically the failure of this seemingly intuitive interpolation is because it totally ignores the correlations between the velocities and the spin orientations of the atoms. 1.8. VISCOSITY

The viscosity can be written in the form (1.61)

where (1.62)

( 1.63)

(1.64)

Again it is interesting to look at some limiting cases. As mentioned above, Lhuillier and Laloe (1982a) show that as T + 0 all the R*"*" in the viscosity expressions approach equality so that CI-.-3/5

and

&++1

as

T+O.

( 1.65)

Thus from eq. (1.61)' [ v ( M ) l 1 diverges as M + 1. At sufficiently low temperatures, therefore, a high nuclear polarisation would increase the viscosity dramatically, though there is very little chance of ever observing this in experiments on the gas. At experimentally accessible temperatures the collision integrals must be numerically evaluated. For M = 0 the result (1.62) applies; for M = 1 some algebraic manipulation of eq. (1.61) gives (1.66)

Table 3 . ...

T(K)

,]*I

I,I 1

fi*l'.')[u,

[Vh]

fl*".?'[ a k ] L?*Il.3) [ah 1 --

fl*12,21[uy]

51

+

K

+

&

51

+

9

+

52

(6- 6,) ( 5 2 - 1 , )

0.6 0.8

1.3966 1.3550

1.3115 1.3303

1.3383 1.3330

1.3235 1.2993

-0.3509 -0.1 196

+0.0630 +0.0235

-0.0467 -0.0195

+0.0877 +0.0325

-0.0191 -0.0100

-0.1097 -0.0430

-0.1068 -0.0425

1.o 1.2 1.4 1.6 1.8

1.3417 1.3298 1.3163 1.3019 1.2875

1.3227 1.3044 1.2835 1.2633 1.2447

1.3040 1.2738 1.2470 1.2244 1.2050

1.3204 1.3459 1.3653 1.3773 1.3829

+0.1638 +0.3994 +0.5660 +0.6686 +0.7200

-0.0347 -0.0898 -0.1331 -0.1623 -0.1784

+0.0337 +0.1001 +0.1632 +0.2102 +0.2363

-0.0483 -0.1263 -0.1891 -0.2323 -0.2563

+0.0210 +0.0700 +0.1218 +0.1627 +0.1858

+0.0684 +0.1899 +0.2963 +0.3725 +0.4147

+0m93 +O. 1963 +0.3109 +0.3950 +0.4421

2.0 2.2 2.4 2.6 2.8

I .2735 1.2602 1.2477 1.2360 1.2250

1.2280 1.2129 1.1993 1.1869 1.1756

1.1883 1.1735 1.1603 1.1484 1.1376

1.3838 1.3815 1.3770 1.3710 1.3642

+0.7333 +0.7199 +0.6889 +0.6471 +0.5994

-0.1839 -0.1815 -0.1739 -0.1631 -0.1506

+0.2429 +0.2348 +0.2177 +0.1962 +0.1733

-0.2644 -0.2607 -0.2489 -0.2324 -0.2136

+0.1913 +0.1835 +0.1677 +0.1482 +0.1281

+0.4268 +0.4163 +0.3916 +0.3593 +0.3239

+0.4557 +0.4442 +0.4166 +0.3806 +0.3417

+0.5493 +0.4993 +0.4510 +0.4053 +0.3628

-0.1374 -0.1244 -0.1118 -0.1001 -0.0893

+0.1513 +0.1309 +0.1128 +0.0970 +0.0833

-0.1940 -0.1747 -0.1 564 -0.1394 -0.1238

+0.1091 +0.0920 +0.0773 +OM47 +0.0542

+0.2887 +0.3031 +0.2553 +0.2667 +0.2246 +0.2337 +0.1971 +0.2041 +O. 1726 +0.1780

+0.3237

-0.0794

f0.0716

-0.1098

+0.0455

+O. 1510

3.0 3.2 3.4 3.6 3.8

1.2148 1.2052 1.1961 1.1877 1.1797

1.1651 1.1555 1.1466 1.1383 1.1305

1.1277 1.1185 1.1100 1.1021 1.w46

1.3568 1.3492 1.3415 1.3338 1.3262

4.0

1.1721

1.1232

1.0876

1.3188

+0.1553

The entries are not accurate to the number of figures quoted. According to Lhuillier the numerical uncertainty is believed to be of the order of a few parts in 1000.

p

g

3 -I

P

r

STRONGLY SPIN-POLARISED ’He GAS

69

Now eqs. (1.62) and (1.66) differ, as for thermal conductivity, only in the in the denominator. Similarly the interpolafactor 2 multiplying R*(2’2)[aekx] tion between M = 0 and M = 1 does not take the form which intuition might suggest; the full expression (1.61) must be used. Table 3 was calculated by Lhuillier for the potential due to Aziz et a]. (1979) with the length parameter u taken as 0.2556 nm (this same choice must therefore be taken when the tables are used to evaluate [ K ( M ) ] ,and [77(M)ll).With thesevalues we can plot [ K ( M ) ] and , [ q ( M ) ] ,as functions of T and M. Their behaviour is similar but not identical except in the limiting cases M = 0 and M = 1. In fig. 1.8 we show the predicted behaviour of [ K ( M ) ] ,versus T. The “bump”, which is particularly pronounced in the region of 2 K can be traced to a phase-shift cancellation in the p(/ = 19 and f ( l = 3) waves for k* = 1.75 corresponding to T = 3.25 K (see fig. 1.4). Such a cancellation is a pure dynamical consequence of the balance between attractive and repulsive parts of the interaction. Its observation is made possible by the elimination, through the Pauli principle, of the even-l channels of interaction. In fig. 1.9 we seek to compare eqs. (1.55) and (1.61) by plotting the quantity [ X ( M )- X ( 0 ) ] / [ M 2 X ( O )where ] X represents [ K ] , from eq. (1.55) or [TI, from eq. (1.61). To be precise we have plotted the limiting forms as M + 0, namely (r2- 6,) and ( Cz - 5,)from table 3. In fact these forms are accurate in the temperature range of interest to better

(K)

."

-.

-.

-.

".

I".

Fig. 1.8. The thermal conductivity of gaseous 'He calculated from eqs. (1.55)-(1.58) as a M = 0.4 function of temperature for five different polarisations as follows: M = 0.0 (-), (--. . .-), M = 0 . 6 (- . - . - . -), M =0.8 (- - . - -), M = 1.0 ( - - - - - - - - - ) . (From Lhuillier (1983).)

D.S. BETTS ET AL.

70

I)

.4

.3

.2

.1

: with X=q(viscosity)

or K (thermal conductivity)

.a

1

2

3

4

TIK Fig. 1.9. An aspect of the temperature dependence o f [77(M)], and [ K ( M ) ] ,The . plots are of ( [ X ( M j ] , - [ X ( O ) ] , j / M ’ [ X ( O ) ] ,versus temperature for low values o f M less than about 2 5 % , where X is viscosity r) or conductivity K . The equations used are (1.55). (1.56). (1.61) and ( 1.62). with numerical values from table 3.

than 2 % provided that M C 25%. It is interesting to note that [ K ] I and [ 711 have exactly the same temperature dependence for M = 0 and M = 1 but not for general M. 1.9.

S P I N DIFFUSION

Lhuillier and Laloe (1982b) have shown that the spin diffusion in a partially polarised system is not isotropic, the diffusion of the longitudinal component of the polarisation being quite different to that of the transverse component. Emery (1964) first pointed out that, for an unpolarised gas, the particle indistinguishability effects in collisions give a negligible contribution to the spin diffusion coefficient; his paper explained why there had been an apparent discrepancy between Chapman-Enskog theory and spin diffusion measurements on (for example) unpolarised gaseous 3He by Luszczynski

STRONGLY SPIN-POLARISED ‘He GAS

71

et al. (1962). This was confirmed by Lhuillier and Laloe (1982b) who also showed that the presence of a significant nuclear polarisation can radically change the situation: in fact the spin diffusion phenomenon is then dominated by identical spin rotation effects, and the corresponding hydrodynamic equations become highly non-linear and anisotropic. Instead of remaining a purely dissipative process, the spin diffusion phenomenon also acquires an oscillatory character giving rise to spin waves. Such properties are rather unusual in a gas, and their origin is quantum interference effects during collisions between identical atoms. The existence of spin waves in gases was first predicted by Bashkin (l981,1984a,b, 1986). Spin waves in degenerate Fermi liquids are now quite well known following the theory of Silin (1957) who showed that in Landau’s theory of a Fermi liquid, spin waves are a consequence of the anisotropic part of the molecular field. It has also been shown both theoretically (Leggett 1970, Leggett and Rice 1968) and experimentally (Corruccini et al. 1971, 1972) how spin waves affect spin diffusion in a liquid in spin-echo experiments. Spin waves in liquids have also been studied by Bashkin and Meyerovich (1981) and Meyerovich (1983). They have been detected in both normal and supertluid liquid 3He by Masuhara et al. (1984). And there have been investigations in dilute solutions of 3He in liquid 4He-II which in many respects (see for example Meyerovich ( 1987)) are quite well described by gas models; measurements have been reported by Owers-Bradley et al. (1983), Gully and Mullin (1984), and Ishimoto et al. (1987). The microscopic “identical spin rotation effect” has been described earlier. It is not manifested macroscopically in a sample with spatially homogeneous polarisation; but it can be detected where the orientation of the polarisation varies across the sample so that two atoms from points rl and r,, each carrying the corresponding spin orientations, collide at r. Consider for example a situation in which the polarisation can be described as the sum of a large component M, constant across the sample, and a small inhomogeneous transverse component M+ = M, + iMy set to zero at the centre of the cell. Firstly, the random thermal motion of the atoms as they cross the cell will tend to remove any inhomogeneities. This will contribute to a current of transverse orientation opposing the original cause and proportional to -VM+; this is the usual Fick law. But secondly, superposed on this, there is the identical spin rotation effect which has quite different rules, and although it is associated with the chance nature of collisions it does not have a fully stochastic character. At each binary collision the rotation effect is partially cumulative around the z-axis and it pulls the transverse component of the polarisation current into rotation about that axis. This gives another contribution to the current of transverse orientation proportional to iM,. Lhuillier and Laloe (1982b) have performed

D.S. B E n S ET AL.

72

the detailed gas kinetic theory which can be used to describe this case as follows: The diffusion of the longitudinal component of the spin magnetisation is described to a good first approximation by a familiar Fick law (1.67)

J ( M,) = - DOV Mz, where

( 1.68)

This result coincides with the classical result and has no dependence on M . A more refined approximation by Lhuillier and Laloe (1982b) does bring in a weak dependence which can be written out in an explicit form and evaluated; we will not pursue this matter any further here since the correction is for practical purposes negligible. It is when one considers the diffusion of transverse components of the spin magnetisation that the new phenomena appear. The transverse polarisation current is described to a first approximation by

J ( M,) = -

Dl3

( 1 + ipM,)VM+,

1+p2M1

( 1.69)

where (1.70) The dimensionless coefficient p is thus the ratio of the sum of “identical spin rotation cross sections” to the cross section for collisions between distinguishable particles. The conservation of total spin during a binary collision leads to an equation of continuity aM+ -+div at

J ( M + )= 0.

(1.71)

In the case we are considering here (IM+I< Mz), eqs. (1.69) and (1.71) can be combined to give the equation of motion (1.72) This equation, written in a form due to Nacher et al. (1984) shows that a perturbation in transverse orientation can be propagated (as indicated by

STRONGLY SPIN-POLARISED ’He GAS

73

the imaginary term), giving rise to a new mode of oscillation for which the dispersion law for a wavenumber k can be written (1.73) where, as usual the imaginary term on the right represents damping. The product p M gives the relative importance of the particle indistinguishability effects in spin diffusion; for example, for a cell where the lowest diffusion mode has a diffusion time rD,the frequency of the transverse spin wave is given (in the Larmor rotating frame, and provided that p M < 1) by ( 1.74)

Sw = p M / r D .

In other words, p M is a spin oscillation “quality factor”, independent of the gas density n (whereas 6w itself is inversely proportional to n). Table 4 gives numerical values of the three collision integrals, including L?*”,”[a,]which was also given in table 3, and of the coefficient p. 1.10. EQUATIONOF

STATE

Consider first the unpolarised gas ( M = 0), with a number density in the range listed in table 2. The conventional way of expressing deviations from Table 4

0.6 0.8 1.o

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

+0.0168 +0.0314 +0.0739 +0.1147 +0.1426 +0.1552 +0.1538 +0.1412 +o. 1202 +0.0936 +OM37 +0.0324 +0.0010

-0.0293 -0.0577 -0.0839 -0.1075 -0.1283

-2.0469 -2.1723 -2.0186 -1.7844 -1.5394 - 1.3085 - 1.0998 -0.9147 -0.7522 -0.6102 -0.4865 -0.3791 -0.2861 -0.2056 -0.1364 -0.0770 -0.0263 +0.0167

1.3966 1.3550 1.3417 1.3298 1.3163 1.3019 1.2875 1.2735 1.2602 1.2477 1.2360 1.2250 1.2148 1.2052 1.1961 1.1877 1.1797 1.1721

- 1.4536 - 1.5800 - 1.4494 -1.2556 - 1.0612 -0.8859 -0.7348 -0.6074 -0.5015

-0.4140 -0.3421 -0.2630 -0.2347 -0.1949 -0.1623 -0.1355 -0.1 134 -0.0952

D.S. B E T S ET AL.

74

the ideal gas equation of state pV=NAkRTis in the form of a virial expansion:

B C PV - I ++ + , * N,kBT V V

~-

. .,

(1.75)

where NA is Avogadro's number, V is the molar volume, B is the second C the third, etc. We shall consider virial coefficient with units m'mol only the second, B, here since even that will not be easy to observe in the temperature range of interest below about 2 K. There are good expositions of the business of calculating second virial coefficients for helium, for example, by Hirschfelder et al. (1954) and Keller (1969). For the purposes of calculatihg second virial coefficients for arbitrary polarisation M, the most convenient source for 'He treated as a Lennard-Jones gas (see eq. ( I .6)) is by Boyd et al. (1969), based on earlier work by Boyd et al. (1966) (for a hard-sphere gas) and Kilpatrick et al. (1954). The calculation involves the evaluation of the following expressions which leave out terms involving bound pairs, as is appropriate for 'He, R( M = O ) = B , i r e c l + B e x c h n n g e r

( 1.76)

where ( 1.77)

where, as before, A-r is the de Broglie thermal wavelength and (1.78)

where ( 1.79) x

G - ( k ) = C (-l)I(2l+1)61(&),

( 1.80)

I -0

where the 6, are the phase shifts calculated from the interaction as before. The contribution Bexchange in eq. (1.78) can be seen to have two parts, one of which is the perfect Ferrni gas term containing the factor in. As the temperature rises above about 2 K the two parts tend to cancel each other, leaving BdirCc, as the dominant contribution. Lhuillier and Laloe (1979)

STRONGLY SPIN-POLARISED 'He GAS

75

adapt the above formalism to the case of full polarisation ( M = 1 ) by taking the combination

B ( M = I ) = Bdirect+2Bexchange.

(1.81)

The sums in eqs. (1.79) and (1.80) are over all integer values of 1, but when eqs. (1.77) and (1.78) are combined into eq. (1.81) there is cancellation of the terms with even I; physically this is because with M = l all nuclear spins are parallel so that the Pauli principle allows only odd 1. In general the expression for B ( M ) is B ( M ) = Bdirect+( l+Mz)Bexchmge.

(1.82)

In table 5 we reproduce (with some rounding off) some of the values of Bdirecland BCxchange calculated by Boyd et al. (1969) together with corresponding predictions for B ( 0 ) from eq. (1.76) and B(1) from eq. (1.81). Using these values, one can calculate B( M) from eq. (1.82). For the special case of M=O (unpolarised gas), one can refer to Matacotta et al. (1987). The calculated values of B(0) and B ( l ) are plotted in fig. 1.10. There are as yet no measurements on the polarised gas in the temperature range of interest although the unpolarised case has been carefully analysed recently by Matacotta et al. (1987). The lower temperature range below about 1.4 K will be difficult for experiments aimed at studying the effect of polarisation because of short relaxation times (see below), and it is unfortunate that it is only in that range that B ( 0 ) and B( 1) are expected to become significantly different. Data for B ( 0 ) in the liquid helium range of temperatures have been reported by Keller (1955) (reanalysed by Roberts et al. (1964)), Grimsrud and Werntz (1967), Cameron and Seidel (1989, and Matacotta et al. (1987). The latter authors find good agreement between theory (based on the most recent interaction potential of Aziz et al. (1987)) and experiment down to 1.5 K. The experimental data were fitted to a polynomial, B(O)= 16.69-336.98/ T+91.04/ T2-13.82/ T', which is believed to be more reliable than the B ( 0 ) column in table 5 . If virial coefficients higher than the second are ignored, which is certainly justified for the conditions under which gas polarisation effects can be studied experimentally, then the equation of state (1.75) takes the simple form

(1.83) where B(T) is given by the calculation described above. It is now only a matter of standard thermodynamics to calculate the deviations of thermodynamic quantities from their perfect-gas values. This can easily be done for

Table 5

979.70 290.70 126.60 64.8 1

- 1354.70 -639.70 -420.46 -3 14.17

-375.00 -349.00 -293.86 -249.36

+604.70 -58.30 -167.26 - 184.55

I .2 1.4 1.6 1.8

108.1047 82.23801 65.26080 53.41515 44.76470

36.37 21.68 13.49 8.68 5.73

-251.01 -208.91 - 178.74 -155.97 -138.16

-2 14.64 - 187.23 -165.25 -147.29 - 132.43

-178.27 -165.55 - 151.76 -138.61 - 126.70

17 12

2.0 2.2 2.4 2.6 2.8

38.22077 33.12916 29.07553 25.78603 23.07318

3.86 2.65 1.84 1.30 0.93

- 123.80 - 11 1.98 - 102.06

- 1 16.08 - 106.68

-93.61 -86.33

-119.94 - 109.33 -100.22 -92.3 I -85.40

3 2 2 I 1

3 .O 3.2 3.4 3.6 3.8

20.80475 18.8851 1 17.24353 15.82671 14.59382

0.67 0.49 0.36 0.26 0.20

-79.97 -74.38 -69.42 -64.99 -61.00

-79.26 -73.89 -69.06 -64.73 -60.80

-78.59 -73.40 -68.70 -64.47 -60.60

4.0

13.51308

0.15

-57.39

-57.24

-57.09

0.2 0.4 0.6 0.8 1 .O

1208.647 427.3212 232.6042 151.0809

-98.38 -91.01 -84.47

83 43 26 U

@

m

"

6 4

1 1

I

r, >

r

STRONGLY SPIN-POLARISED 'He GAS

77

polarised Fig. 1.10. The calculated second virial coefficients for unpolarised ( = 0) an ( M = 1 ) 'He as functions of temperature (from Lhuillier and Laloe (1979)). For general M, one can use eq. (1.82) to give the interpolation formula B ( M ) = ( 1 - M2)B(0)+M2B(1). ,I

internal energy, enthalpy, entropy, heat capacities, Joule-Thomson coefficient, velocity of sound, etc. (see for example Hirschfelder et al. 1954). We illustrate this here only for the constant-volume heat capacity C,. This is given by the formula ( 1.84)

(1.85)

Use of the polynomial of Matacotta et al. (1987) gives an explicit form for the bracketed quantity in eq. (1.85) and if one takes the example of unpolarised gas at a pressure of 1 Tom at 1.5 K as a representative combination, the correction to the molar heat capacity due to non-ideality is only 0.1%. The difference between the polarised and unpolarised cases is then of little more than academic interest and the classical value Cv = $NAk, can be used with confidence independently of polarisation M.

D.S. BETTS ET AL.

78

2. Polarisation methods

2.1.

O P T I C A L POLARISATION OF

'HE

NUCLtl

Unfortunately, as shown in section 1.3, the polarisations achievable merely by the application of large magnetic fields are too small to allow the experimental testing of many of the predictions referred to in section 1. There is however a practical technique based on optical methods originally demonstrated by Colegrove et al. (1963). This has been fully reviewed by Laloe et al. (1985), and more recent developments with L N A lasers have been described by Schearer et al. (1986) and by Daniels et al. (1987). The present section is no more than an edited version of those authoritative papers. First, a brief summary of the steps may be helpful to readers unfamiliar with the techniques of atomic physics. The method involves optical pumping and does nor produce equilibrium polarisations. The steps are as follows. - A glass vessel, containing 'He gas at a pressure in the region of 1 Torr, is placed in a homogeneous magnetic field of order 1 mT. - A weak electrodeless discharge is struck in the gas raising some atoms from the 1 'So ground state to the metastable state 2 'S,. - Circularly polarised laser radiation with wavelength A = 1.08 pm (corresponding to the 2 'SI + 2 'Po transition) is directed along the field axis into the electrically excited gas. - The laser beam carries angular momentum and the 2 'Sl metastable atoms, in being resonantly bounced up to 2 'Po and relaxing back, acquire electronic polarisation. Their nuclei also become polarised through hyperfine coupling. - Finally the nuclear polarisation of the ground state ( 1 'So) atoms builds up to that of the metastable atom population through spin-exchange collisions. -The polarisation can be measured by the N M R response of the 1 'So atoms, or optically by recording the intensity of the 1.08 pm fluorescent light scattered by the 2 'S, metastables at right angles to the laser beam. We now look at these stages in more detail. The 'He energy levels used are shown i n fig. 2.1. Since the spin of the 'He nucleus is I there are two hyperfine sublevels in the 2 ' S , state, with quantum numbers F = 1 and F = l;in the 2 'P state there are five sublevels: 2 'Po (with F = i), 2 'PI (with F = f and F = i) and 2 3Pz(with F = and F = i). A radiative transition between the 'Soground state (para-helium) and the 2 'SI state (ortho-helium) is strongly forbidden, and the lifetime of ortho-helium, determined by diffusion to the wall, is a few milliseconds. For this reason the 2 'S, state is metastable. Optical transitions can occur between the 2 'Sl and 2 'P states;

=:

STRONGLY SPIN-POLARISED 'He GAS

l

'

s

u

79

I

Fig. 2.1. Energy-level diagram of 'He atoms. The relative intensities of the hyperfine components C , to C, are shown in fig. 2.2.

they correspond to a resonant line of ortho-helium with a wavelength A = 1083 nm. Figure 2.2 shows the components of the hyperfine structure of this line. The rf glow discharge produces metastable ortho-helium atoms, and since these have a relatively long lifetime, it is possible to accumulate a significant number coexisting with a large number of para-helium atoms. As can be seen from fig. 2.1, the level diagram of ortho-helium is of such a nature that it is possible to achieve optical pumping by means of the resonant line at A = 1083 nm. The ortho/para gas mixture is exposed to a weak static magnetic field Bo of order 1 mT (larger than the Earth's field so that the direction of polarisation is controlled, but not so large as to spread out the line structure shown in fig. 2.2) which imposes a quantisation axis, and illuminated with circularly polarised resonant light in the direction parallel to B,. To illustrate the optical pumping cycle we consider a pair of levels of ortho-helium, say 2 'S, ( F = $ ) and 2 3P0( F = $ ) which are coupled by the hyperfine component C, in fig. 2.1. Useful references on this matter include

D.S. BETTS ET AL.

80

Fig. 2 2. Components of the hyperfine structure of the 'He emission line A = 1083 nm.

Cohen-Tannoudji and Kastler (1966) and Happer (1972). In the magnetic field &, both these levels split into two Zeeman sublevels, with quantum numbers mF = f !. For the electric dipole transitions associated with the absorption of a circularly polarised resonant photon (for definiteness, we choose the w' polarisation) of the C , component, there are selection rules which state that an optical transition can occur only from the Zeeman sublevels 2 'SI ( F = mF = -:) to the sublevel 2 'Po ( F= 4, m F = +$).Atoms in the 2 ' S , ( F = +,mF = + 4) sublevel cannot absorb light in this component of the hyperfine structure at all. Since the inverse transition, 2 'PO (F = l) + 2 ' S , ( F = i), accompanied by the spontaneous emission of a photon, is allowed to both Zeeman sublevels of the 2 IS, ( F= l)state, ortho-helium atoms accumulate in the 2 'SI ( F= $, mF = + $) sublevel, so that an orientation of the total angular momentum ( F ) of the metastable atoms arises in the system. The weak rf discharge and the optical pumping of the ortho-helium atoms by circularly polarised resonance light create a situation in which two spin systems coexist in the cell: atoms of para-helium ( 1 'So ground state, with an electronic angular momentum J = 0 and a nuclear spin I = 5) and the atoms of ortho-helium, oriented by the optical pumping (2'SI metastable state with F = i, where F = J + I). The orientation of the total angular momentum in an optical pumping cycle gives rise to both an

:,

:,

STRONGLY SPIN-POLARISED 'He GAS

81

electronic polarisation ( J ) and a nuclear polarisation (I)of a system of atoms. The physical reason for this process is the hyperfine interaction ale J, which occurs in both the 2 3P excited state and the 2 ' S , metastable state. The nuclear polarisation (I)which arises in the 23S, state can be transferred to the 1 ' S o state during binary collisions of 3He atoms 'He42 'S1)+'He(l 'So)+3He(2'S1)+'Het(l 'So), which are called "collisions with an exchange of metastability". The interaction between the colliding atoms is electrostatic and does not directly affect the nuclear spin I. If the spin states of the nuclei of the two colliding atoms are identical, such a collision cannot be distinguished from ordinary elastic scattering, since there is no change of any sort in the expectation value of either the electronic or nuclear angular momentum in the system. If, on the other hand, the colliding atoms have different nuclear spin states, the result is an exchange of the projections of the nuclear spins. Dupont-Roc et al. (1971) and Leduc (1972) report a study of the theory of the exchange of metastability in collisions of 3He atoms and have measured the effective cross section for this process. The cross section is (7.6 *0.4) x lo-': cm2 at T 300 K and the corresponding interaction time s. It follows that f i r 4 1, where fi describes the in the collision is TZeeman and hyperfine interactions of the nuclear spin I, and under this condition the nuclear polarisation (I) which exists in the 23S1 state is transferred entirely to the 1 ' S o ground state. This state thus forms a reservoir in which the 3He atoms, which accumulate with a nuclear polarisation, are preserved for a long time. In early experiments the light source used for optical pumping was a gas-discharge helium lamp with an rf discharge, which emits many lines of the fine and hyperfine structure in the spectrum. Such light sources have a low power, typically less than 10 mW in the band of the resonant line at A = 1083 nm. Daniels. and Timsit (1971) and Timsit and Daniels (1971) reported polarisations of 15-25% at room temperature with a gas pressure of 1 Torr, and this is probably the best that can be done in this way. More recently, progress has been achieved by replacing lamps with lasers. Mollenauer (1980) reported the use of a laser with an active medium consisting of (Fl)* colour centres in a sodium fluoride (NaF) crystal and certain modifications by Leduc et al. (1980) and TrCnec et al. (1982) made it possible to improve the stability and reduce the spectral width of its output while maintaining a high power level. Figure 2.3 is a block diagram of the laser. The (F:)* colour centres in NaF, which have an absorption peak at A = 870 nm, are excited by a dye laser, which is in turn pumped by the light from a krypton ion laser emitting the lines A l = 645 nm and A2 = 690 nm. Divalent impurities (Mg2+or Ca")

-

D.S. B E T S ET AL.

82

645 nm

1

2

Fig. 2.3. Block diagram of a laser arrangement yielding a wavelength A = 1083 nm.( 1 ) Krypton ion laser. (2) Dye laser. ( 3 ) Laser using (F;)* colour centres in NaF.

in a concentration of 10-4-10-3in the NaF crystal serve as electron capture centres. The resonator shown in fig. 2.4 was used to operate the (F:)* laser in a single mode. The NaF single crystal is placed in a vacuum flask, cooled with liquid nitrogen over half the distance between spherical mirrors M, and M,, with a radius of curvature 75 mm. The beam from the dye laser (A = 890 nm) enters the resonator through mirror M , , and 95% of the intensity of this beam is reflected to the single crystal from mirror Mz.There are further mirrors in the ring as indicated in the diagram. In addition there are in the resonator two Fabry-Perot etalons FP, and FP2which are required to determine the lasing mode and its scanning. Etalon FP, is an air-filled etalon, formed by two prisms with Brewster angles of incidence; one of the prisms can be moved by means of a piezoelectric ceramic allowing electronic

vv

r------

B

LF

4

-_

“1

Fig. 2.4. Optical resonator of the colour-centre laser. M, to M, are mirrors. FP, and FP, are Fabry-Perot etalons. LF is a Lyot filter. P, and P, are rotating prisms. 0, and O2 are optical windows. VV is a vacuum volume. NaF is a sodium fluoride crystal. B, is a Faraday cell. (From Trtnec et al. (1982).)

STRONGLY SPIN-POLARISED 3He GAS

83

self-tuning of the frequency of the mode generated by the laser. Etalon FP2 was a quartz plate 0.5 mm thick whose rotation, together with that of plates P, and Pz made it possible continuously to tune the frequency. With a transmittance of 35% of the output mirror M,, this resonator could provide an output power of about 300 mW at the required wavelength A = 1083 nm. The output frequency could be tuned continuously over a range of 50 GHz and was stable to within 5 MHz. Figure 2.5 shows the arrangement used by the Paris group to produce nuclear polarisation of 'He at room temperature. The 1083 nm beam is expanded and passed through a quarter-wave (A/4) plate producing circular polarisation. The beam then passed through the gas held at a pressure of about 0.3 Torr in a cylindrical cell of diameter 5cm. The cell was at the centre of a Helmholtz coil which gave a static uniform magnetic field lBol 0.5 mT directed along the optical axis. The uniformity of this field over the volume of the cell is most important since even small gradients of the order of lo-' T cm-' sharply reduce the relaxation time of the nuclear magnetisation (to -1 min) and lower the maximum degree of polarisation. We shall have more to say about relaxation processes in section 2.2. The polarisation of the nuclei may be detected and measured using standard NMR methods, applying an rf magnetic field to the sample (see, for example, Abragam (1961) or Slichter (1980)). There is also an optical method (see PavloviE and Laloe (1970), Pinard and van der Linde (1974), and Leduc et al. (1984)) which involves measuring the degree of circular polarisation with a wavelength of 668 nm which is emitted by a discharge in the direction of the field Bo. The components required for this optical method are shown in fig. 2.5, including a A/4 plate, a linear analyser A, and an interference filter I F which singles out the line at 668nm. These

-

I

2

1

5

-

3

--

60

IF A h f 4 3

-

p

-

L

b

I

hf4

-

t

Fig. 2.5. Block diagram of the apparatus used in Pans to polanse 'He nuclei optically. ( I ) Krypton ion laser. (2) Dye laser. (3) Colour-centre laser. (4) Cell holding 'He. (5) Photomultiplier. (A/4) Quarter-wave plate. (A) Linear analyser. (IF) Interference filter for the wavelength 668 nm. (From La102 et al. (1989.)

84

D.S. BETIS ET AL.

components are positioned on the source side of a photomultiplier as shown in the figure. When the h / 4 plate is rotated at a frequency w , the photocurrent is proportional to the intensity of the light of alternately left-hand and right-hand circular polarisations, so that the photocurrent becomes modulated at a frequency of 20. This modulation makes it possible to use synchronous detection, after which the output signal is directly proportional to the degree of nuclear polarisation of the 'He. The dependence of the achievable polarisation on various experimental parameters has been the subject of an investigation by Nacher and Leduc (1985). Important parameters include the laser power, the pressure of the gas, and the particular one of the nine hyperfine components of the line at 1083 nm being used. Figure 2.6 illustrates the pattern of the results of this calculation. It turns out that in order to maximise the polarisation it is better to use a laser in three-mode, rather than single-mode, operation. In the three-mode case, a very slight change in the construction of the resonator allows lasing on three longitudinal modes, separated from each other by about 150 MHz, near the C, component of the hyperfine structure, that is, 2 ' S ,( F =5 ) +2 ' P 0 ( F = 4 ) . Nacher et al. (1982) reported results showing that theory and experiment agree very well on this. The degree of polarisation of the 'He nuclei achieved at room temperature in these experiments was about 70% at a 'He pressure of about 0.3Torr in the cell and using a laser-beam power of 300mW. Quite similar results can now be obtained with neodymium-doped solid-state lasers, such as LNA (Daniels et al.

Fig. 2.6. Results of calculations by Nacher and Leduc (198s) giving the nuclear polarisation of an optically pumped gas of 'He as a function of the laser power. Each curve is for a different frequency of the laser, tuned to a different component of the line at 1.08 )rm. ( a ) Calculations assuming no collisional disorientation of the 2 'P state (low pressure). (b) Same calculations assuming a total depolarisation of the 2 'P state (high pressure).

STRONGLY SPIN-POLARISED 'He GAS

85

(1987)), which have the advantage over colour centre lasers of operating at room temperature and without deterioration in time. The most interesting experiments on the polarised gas have to be done at helium temperatures. Now it has been demonstrated (see Nacher et al. (1982), Himbert et al. (1983) and Leduc et al. (1983)) that the techniques described above can be adapted for use with a cell maintained at low temperatures. It is essential for the glass to be coated with something which prevents the wall relaxation which would otherwise occur due to magnetic impurities. This is not a problem at room temperature because the dwell time of 3He atoms is then very short, whereas the dwell time rises as the temperature falls. The crucial quantity which governs the wall relaxation of nuclear spins at low temperatures is the adsorption energy A W of an atom of 'He on the wall. Since the bulk relaxation times in dilute gaseous 3 He are very long (of order lo5h), nearly all the spin depolarisation occurs while the atoms are adsorbed on the inner surface of the container where they experience the effect of local magnetic fields created by paramagnetic impurities. The relaxation time varies as usual as exp( -qA W/ k g T ) ,where q is in the range 1 to 2 depending on some details of the interaction, and Lefkvre-Seguin et al. (1985) report that the optical pumping process is completely quenched by the wall relaxation whenever k,T< q A W/15. Consequently only walls with 'He adsorption energies below about 10 K can be used for optical pumping below 1 K. Now since the adsorption energy of one 'He atom on a glass surface is of the order of 200 K, uncoated glass walls are clearly completely inappropriate. According to LefkvreSeguin et al. (1985), an efficient way to reduce A W is to coat the walls of the cell with several layers of solid H2 which reduces A W to 12*3 K, allowing experiments to be performed down to 1-2 K. Further reduction requires a further layer of condensed 4He on top of the H2 coating. The value of A W for an atom of 'He on the surface of condensed 4He seems not to be very precisely known, but it seems probable (see Al-Shibani (1977) and Edwards and Saam (1978)) that it does not exceed 5 K, so that experiments below 1 K are made possible. Figure 2.7 shows the cryogenic arrangement reported by Himbert et al. (1983). The cell C contained a mixture of 'He, 4He and H2 with densities corresponding to room-temperature filling pressures of 1 Torr, 2 Torr and 1.5 Torr, respectively. The gas mixture was designed to coat the inner walls of the cell in the following way. As the cell is cooled towards helium temperatures, the hydrogen condenses first and provides a preliminary coating of hydrogen followed, on cooling below 1 K, by a condensed layer of helium-4. The results of Himbert et al. (1983) demonstrated that the optical pumping of 3He works down to about 0.5 K (the lowest temperature available) where the nuclear relaxation time could still be longer than 10min. This is the good news; the bad news is that

D.S. BETIS ET AL.

86

r.f.

113

p

K 4He

___)

60

Fig. 2.7. Schematic diagram of an apparatus reported by Himbert et al. (1983) for polarising 'He at helium temperatures. (Cryogenics) B, is a "He bath, C a T'F'yrex cell containing the optically pumped 'He atoms, E a copper heat exchanger, 1, the liquid 'He injection line, I, the filling tube for B,,I, the vapour pressure line, Pthe 'He pumping line, R the 'He refrigerator, and V the vacuum pumping line. (Optics) A is an analyser, Bo a 7G magnetic field, L, a discharge fluorescence light, L, a laser beam providing the 1.08 pm pumping light, Q a rotating quarter wave plate, and R F is 10 MHz radiofrequency producing a weak discharge in the cell.

metastability exchange rates in the gas decrease seriously as the temperature falls, with the result that the maximum achievable polarisation is significantly lower than at room temperature. Leduc et al. (1983) quote a maximum achieved value of M = 25% which compares. disappointingly with room temperature values reaching almost 70% in some cases. There is another method involving the use of a double cell, which has been used in two important recent experiments described below, originally designed by McAdams and Walters (1967). This can produce polarisations up to about 50% at helium temperatures with hydrogen coating only and might be expected to do better with a second layer of helium-4. Figure 2.8 shows the double-cell arrangement as described by for example Leduc et al. (1983), Nacher et al. (1984) and La102 et al. (1985). The upper chamber,

STRONGLY SPIN-POLARISED 'He GAS

87

-v

CIRCULARLY POLARIZED LASER BEAM

MIRROR

r-l

-

--M-

5.4cm

HIGH TEMP. THERMAL SHIELD

FOAM INSULATION-

-

71 cm

LOW TEMP. THERMAL SHIELD NMR

coi Ls

3 cm

LIQUID Fig. 2.8. Double cell used in Pans (Leduc et al. (1983)) for optical polansation of 'He gas at low temperatures. The upper cell at 300 K is where the optical pumping is done. The nuclear

polarisation is transferred by spin exchange to the lower cell immersed in a liquid helium bath.

-

with volume V, 100 cm3 at room temperature T ,, communicates through a capillary with inner diameter about 2 mm with the lower chamber, with volume V,- 10 cm3 at helium temperature T,.The optical pumping takes place in the upper chamber, with the laser beam directed parallel to the vertically directed small static magnetic field Do. This field Do is produced by several current loops whose axis is that of the laser beam. The relative positions of the loops and the currents in them are adjusted to ensure uniformity along the connecting capillary between V, and V,. The gas density is another important parameter contributing to the achievement of polarisation as demonstrated by Nacher et al. (1982) in single-cell room-temperature experiments, the optimum filling pressure being close to half a torr (see fig. 2.9) corresponding to a number densitv of about 1.6 x 10l6atoms ~ m - With ~ . lower pressures, metastables can diffuse faster to the walls where their lifetime ends; with higher pressures the lifetime of the metastables is shortened by the greater collision frequency

D.S.BETTS ET

88

0

1

AL.

2

p (torr)

Fig. 2.9. Variation with pressure o f the polarisation produced by laser optical pumping of 'He at ambient temperature. The most favourable pressures are between 0.3 and 3 Torr. (From Nacher et al. (1982).)

within the gas. There is quite a narrow practical range and one normally works with room-temperature number densities within the range 10l6 to lo" atoms cm-3. In a double cell, the helium-temperature number density is enhanced according to the simple ideal gas calculation (ignoring the capillary volume) n,kBT,=pl=p,=n2kBT2, ( n , VI + nzV2)kBTI = PAId v,+ V A where pfillis the filling pressure at T I .Hence n2ln1 = TI1 T2,

(2.1) (2.2) (2.3)

These equations enable one to calculate the necessary pAl,in order to fix n , at the required optimum and then to deduce n , . Note that for typical conditions n , is approximately proportional to T2 (because V,T, s V,T2 in the denominator of eq. (2.4)) and that therefore n, is fairly insensitive to T 2 . Taking as an example T2= 3 K, V2= 10 cm3, T I= 300 K, V , = 100 cm3, and p s l l= 5 Torr one finds n , = 1.6 x 10l6atoms cm.~' and n, = 1.6 x 10" atoms cm-3. in order to increase the nuclear relaxation time in the cold chamber, a small amount of molecular hydrogen is mixed with the 'He on filling. When V, is cooled to helium temperatures, the hydrogen forms a solid film which serves as a counter-relaxation coating. This is because the relaxation time of 'He nuclei on solid hydrogen can be several days (Barb6 et al. (1975)

STRONGLY SPIN-POLARISED 'He GAS

89

and Lefkvre-Seguin et al. (1985); also see below). The temperature distribution along the capillary connecting V , and V2 is an important matter if relaxation during passage from V , to V, is to be minimised. The relaxation rate is tolerable at T, (hot) because although there is no hydrogen coating, the dwell time of 'He atoms on glass is small and any magnetic impurities have only a small chance to cause depolarisation. And the relaxation rate at T2(cold) is tolerable because of the hydrogen counter-relaxation coating. What has to be minimised is the area within the capillary which is neither cold enough to retain a hydrogen coating nor hot enough to discourage visiting 'He atoms from dwelling too long and so being relaxed by magnetic impurities. The practical solution to these requirements is to protect the connecting capillary with two copper heat shields. One keeps the temperature of the lower part of the capillary near T2,while the other keeps the upper temperature near 100 K. The small distance between these shields minimises the region with intermediate temperatures where the polarised 3He atoms could relax rapidly through collisions with the uncoated wall. This solution causes some cryogenic problems due to heat transfer and consequent helium boil-off rates, but it has been shown to fulfil the purpose of transferring polarisation by diffusion from V , to V, with high efficiency. Theory by Leduc et al. (1984) indicates that the polarisation in V, should approach 96% of that in V , and experiments by the same authors, in which detection was by NMR at T2 and by optical means at TI, verified that it was close to 100% of it. 2.2. RELAXATIONPROCESSES

A full discussion of several of the mechanisms of relaxation has been given by Timsit et al. (1971). The relaxation time 7, of a polarised sample of 'He gas may result from several mechanisms and we can write 1 1 _1 --_1 +-+-+-, 71

7d

78

Tad

1

(2.5)

Tab

where 7 d relates to the magnetic dipole coupling between 'He, nuclei; T~ relates to coupling between 'He nuclei and field gradients; T , ~relates to coupling between adsorbed 3He nuclei and ferromagnetic impurities on or near the container walls; and Tab relates to coupling between absorbed 'He nuclei and ferromagnetic impurities embedded in the material of the container walls. We now deal briefly with these in turn, giving key references only from which the broader literature can be accessed. A good general reference is that of Timsit et al. (1971) although it is concerned primarily with room temperature effects and care must be taken when extrapolating to low temperatures. The expressions given below are generally valid.

D.S. BElTS ET AL

90

Chapman (1975) has shown how well theory and experiment are in agreement for the value of 7 d as shown in fig. 2.10. An estimate can be obtained by considering the effects of binary collisions and this leads to the expression (Abragam (1961))

where y is the gyromagnetic ratio, ( V2>the thermal mean square velocity, (J the distance of closest approach, and rcOl1 the mean time between collisions equal to 6.2 x lO-’’/pT’/’ s ( p in g cm-’; T in kelvin). Numerical substitution leads to the approximate result 74 = 115 T ’ l 2 / ps ( p in g cm-’; T in kelvin). Figure 2.10 shows the results of more precise theory compared with experimental data and we see that for example at T = 3 K and n = loi8atoms (corresponding to p = 5 x g ~ m - ~rd ) ,is about 4 x lo7s which is over one year. We conclude that this mechanism can safely be neglected for present purposes. Schearer and Walters ( 1965) have considered the effects of field gradients, and made measurements on ’He gas at T = 3 0 0 K, and more recently a

I

I

I

2

I

I

I

I

10 12 14 16 18 20 T IK Fig. 2.10. The quantity p ~ versus , temperature, where p is the density of the polarised gaseous >He and T~ is the relaxation time due to the magnetic dipole coupling between the nuclei (from Chapman (1975)). According to eq. (2.6) this would vary as T”’,but a more thorough treatment (shown as a dashed line) gives results which agree well with experiment and confirm that T, is very high. 4

6

I

8

STRONGLY SPIN-POLARISED ’He GAS

91

generally applicable theory has been given by Lefkvre-Seguin et al. (1982). Gradient effects are important because any field inhomogeneity encountered by a randomly moving atom may be pictured as a randomly varying field applied to the same but motionless atom. A Fourier decomposition of this fluctuating field will in general yield a component lying in a direction perpendicular to that of the main field Bo and oscillating at the nuclear Larmor frequency w o . This field component will then induce transitions between the two magnetic levels *4 of the nucleus and contribute directly to the nuclear relaxation rate. In general this is a rather complicated matter, with T~ depending on Bo, the field gradient G, and at least three characteristic ~ to 3/[8r112(’.’’[~~]] times including the time between collisions T , ~ (equal where n is the number density), the diffusion time across the cell TD (approximately equal to 0.62r2nm3R“”’[Uk]/keT for a sphere of radius r), and l / w o where wo (the Larmor frequency) is equal to yBo with y being the nuclear gyromagnetic ratio. Lefhre-Seguin et al. (1982) have addressed the theory of this and obtained a general formula for T ~ .In one limit, W O T Ds= 1 , this reduces to a form given earlier by Schearer and Walters (1965). Although the formula in this limit is not appropriate for experiments of interest in this chapter, we mention it briefly here:

where G is the magnetic field gradient (part of which will be due to the magnet itself and therefore proportional to Bo while the other part may be an ambient gradient in the laboratory), ( U 2 )is the thermal mean square velocity proportional to T. The numerical form of (2.7) becomes “=

B~ ~ ; + 6 . 6 3x 1 0 1 5 p 2 ~ 2.28 x 101’pT3’2 ’

(5)

where Bo is in gauss and G is in gauss cm-’. Work by Schearer and Walters ( 1965) showed satisfactory agreement between this theory and experiment ~, Bo=lOG, and G = at 300K. If we take p = 5 ~ 1 0 - ~ g c r n -T=300K, 5x G cm-’ as representative values, eq. (2.8) gives T~ = 3 x lo* s (about ten years!). Equation (2.8) is not appropriate to the conditions in all of the experiments discussed in this chapter, but it is generally true to say that suitable design can ensure that field gradients are less destructive of polarisation than other mechanisms discussed below. Timsit et al. (1971) have given a helpful discussion of the physics of adsorption (for 7J. and absorption processes (for T,,,). In practice these are the most important mechanisms causing depolarisation and Timsit et al. (1971) identify the dipolar coupling between a ’He nucleus and a flipping magnetic dipole in the glass as the most important of all.

D.S. BEITS ET AL.

92

Their arguments lead, subject to some model assumptions, to the following expressions. We have omitted their arguments here and simplified their notation.

where N and N a d are the numbers of free and adsorbed ’He atoms, respectively, in the container (taken to be a sphere of diameter d ) , fFe is the atom fraction of iron in the glass (iron being the commonest paramagnetic impurity in glass), and T ; ~is the intrinsic relaxation time of a ’He nucleus near a paramagnetic site on the wall surface. The number Nad depends on temperature according to (2.10) where, in addition to quantities already defined, u:d is the lattice vibration frequency of an adsorbed atom, and Qadis the (positive) binding energy of an adsorbed atom. The thermal mean speed ( U )is, of course, proportional to TI”. Hence, from (2.9) and (2.10), (2.11) For Tab, the arguments of Timsit et a[. (1971) lead to expressions which parallel eqs. (2.9) and (2.10), (2.12) where /3 is the number of nearest neighbours for a ’He atom absorbed in the glass, v,h is the jump frequency of a ’He atom diffusing through the glass, and N a b is the number of ’He atoms dissolved in a surface layer of thickness (Ar) equal to the mean diffusion jump distance. This number is given by =

6 NkBT(A r ) S d

(2.13)

where S is the (temperature-dependent) solubility of ’He in glass, and (2.14)

STRONGLY SPIN-POLARISED ’He GAS

93

where Doexp(-Qab/k,T) is the diffusion coefficient of 3He in glass expressed in terms of the constant Do and the (positive) activation energy Qab.Hence from (2.11), (2.12) and (2.13), (2.15) The forms of Tad and T~~ are not particularly transparent but it is of interest to note that they have different signs in the exponentials. Thus at sufficiently high temperatures Tab will be the smaller, whereas at sufficiently low temperatures Tad will be the smaller. Substitution of suitable numerical values (given by Timsit et al. (1971)) suggests that at room temperature Tad and Tab are of comparable magnitude, depending of course on the type of glass used. It is sensible therefore to assume that Tad is the more important of the two in the helium range of temperatures. Substituting values into eq. (2.11 ) yields for a sphere of 6 cm diameter, (2.16) Let us assume that fFe is 1 in lo4. Then at 300 K this expression is equal to 7.8 x lo6 s. As temperatures fall from, say, 20 K to 4 K, Tad falls sharply s. The lesson is that, even from 1.4 x 10’ s (a day and a half) to 2.5 x allowing for approximations and uncertainties, it would be impossible to retain polarisation for enough time to do an experiment were it not for the expedient of coating the glass with some material which acts as a barrier between the spins and any paramagnetic impurities. A very thorough study of the beneficial effects of hydrogen and some other coatings has been made by Lefhe-Seguin et al. (1985) and it was found for example that in a spherical cell of diameter 3 cm, the relaxation time was as high as 1000 min at about 3.7 K, falling to 10 min at about 2.3 K. Times of this order allow a range of experiments to be performed, although at the lower temperatures some haste is necessary.

3. Experiments 3.1. EXPERIMENTS ON

SPIN ROTATION EFFECTS A N D S P I N WAVES

The theory of these effects has been discussed in sections 1.6 and 1.9 and experiments designed to measure the quantity p given by eq. (1.70) have been performed by Nacher et al. (1984) and Tastevin et al. (1985). Nacher et al. (1984) first used a method in which spin waves were excited by applying a magnetic field gradient over the sample, conditions being such

D.S. BETIS ET AL.

94

that the atoms are able to explore all the volume of the container before depolarisation is brought about by the field gradient (the condition for motional averaging is discussed by Abragam (1961) and by Slichter (1980)). This situation is only possible when the walls of the container have a very weak relaxation effect. The effects of spin waves can then be simply and completely described (see Lefkvre-Seguin et al. (1982)) in terms of the longitudinal relaxation time T,, the transverse relaxation time T,, and a frequency shift Aw of the precession of M, the transverse component of M. The origin of the shift can be understood from the following simple argument which is valid when (M,l is small compared to the longitudinal component Mo which is closely equal to M. Now the shift depends on the field gradient in the sample, and would be strictly zero if M were uniform (because of angular momentum conservation). The role of the variation FBo of the longitudinal component of the magnetic field across the sample is precisely to create a gradient of spin polarisation, that is, to couple the total transverse magnetisation M, of the sample to a spin wave. But this wave oscillates with a slightly different frequency and is also damped at a rate l / r D . The situation is then simply that of two coupled oscillators with different frequencies and damping, and one can show from an elementary theory of coupled oscillators that the equation of motion for M , acquires a damping rate 1/T2 and a frequency shift Aw given by

(3.2) where for the purposes of the experiments carried out by Nacher et al. (1984) it was unnecessary to know the magnitude of the coefficient a which occurs in both these equations. These experiments made use of the ratio of (3.2) to (3.1): T2Aw = @ M,

(3.3)

or, equivalently, A 4 = PM,

(3.4)

where A 4 was the phase shift accumulated during a time T 2 . Nacher et al. ( 1984) set out to measure p = Aq5/ M as a function of temperature. A general sketch of the experimental arrangement is shown in fig. 3.1 and many of the features of polarised 'He gas experiments discussed in various sections above are in evidence here. The 'He gas is contained in a sealed 'Pyrex cell, consisting of two different parts connected by a 3 mm inner diameter tube: a 5 cm inner diameter by 5 cm long cylindrical container where the atoms are submitted to a discharge and optical pumping at room

STRONGLY SPIN-POLARISED 'He GAS

I

95

kb C

d II

BO

Fig. 3.1. Double cell used by Nacher et al. (1984) for the detection of spin rotation effects and spin waves in polarised 'He. (a) Circularly polarised laser beam. (b) Upper 'He container submitted to optical pumping at 300 K. (c) Thermal shielding at high temperature. (d) Insulating foam. (e) Helium bath at 4.2 K. (f) Cylinder of brass coated with superconducting lead. (g) Thermal shielding at low temperature. (h) NMR pick-up coils (other sets of coils for tilting the spins and varying the field gradients are not shown). (i) 'He sample. (j) Pumped liquid 4He bath at adjustable temperature. (P) Exit to vacuum pump.

temperature, and a 13 mm inner diameter spherical bulb where the NMR experiments are done at low temperatures. Gaseous diffusion inside the connecting tube transfers polarisation from the upper cylinder to the lower bulb. With this experimental arrangement, optical pumping can be done at a temperature where it is much more effective than at a few kelvin. The source of light for optical pumping is a colour centre laser operating at A = 1.08 p,m. The sealed cell contains a mixture of 3He and molecular hydrogen, which freezes on the cold parts of the inside walls of the cell. This provides a coating which, as explained above, strongly reduces the nuclear relaxation rate. The pressure of the 3He gas was p = 0.3 Torr corresponding to (eq. (2.4)) n , = 1OI6 atoms cmP3 and n, = 10" atoms at high and low temperatures respectively. Thermal shields are used to control the temperature gradient along the tube. Their purpose was to restrict the thermal gradient to within a small region, in order to reduce the area of

96

D.S. BET73 ET AL.

surfaces at intermediate temperatures (cold enough to relax 'He nuclear spins but not cold enough to be covered by a few protective layers of solid hydrogen). Nuclear polarisations of 30% or more in the lower bulb can be obtained with a laser power of about 100 mW. As shown in fig. 3.1, the measurement bulb was inside a small isolated inner cryostat of 5 cm inner diameter placed inside the main helium cryostat. In this way, the temperature of the bulb could conveniently be varied between 1.5 and 8 K. In some cases it was found useful or necessary to heat this cell to about 8 K for a few minutes in order to improve the quality of the cryogenic coating and allow polarisation transfer. The NMR system had two separate and orthogonal pairs of induction and detection coils. The latter coils had a diameter of 26 mm, and were wound with 40 turns each. The N M R frequency was about 12 kHz, which corresponded to a (vertical) magnetic field of about 0.4mT. The stability of this field was greatly improved (to about 10 I ' T) by using a vertical superconducting hollow cylinder to trap a fixed flux. This cylinder was a 1 mm thick layer of lead on the inside surface of a brass former of length 22 cm and a diameter of 9.2 cm. The shielding factor obtained in this way was about lo4 in good agreement with theory (Thomasson and Ginsberg (1976), Bardotti et al. (1964)). The field gradient was of the order of 2 x lo-' Tm-', depending on how the superconducting cylinder was cooled in the 4 x T vertical field produced by three horizontal coils of 1 m diameter. By adjusting the currents in three sets of compensation coils it was possible further to reduce gradients at the cell by a factor of about ten. The quantities measured in these experiments were the precession frequency and the transverse relaxation time T2 of a small transverse component of the nuclear polarisation at various positive and negative values of the longitudinal polarisation M. Reversals of M were produced by N M R n pulses, and 7r/20 pulses were used to sample M in an almost non-destructive way. The polarisations were deduced from the amplitudes of the responses to 7r/20 pulses by calibrating the detection system with the signal from a small circular coil. Accuracy was limited by the relatively high signal-to-noise and was in the region of 5-10% in M. For the particular cell geometry employed, the time constant for build-up of nuclear polarisation at 4.2 K in the lower cell was about 20 min. This time was effectively limited by a narrowed section of capillary in the connecting tube; a 5 cm length of 1 mm inner diameter included a 1 cm part whose inner diameter was only 0.3 mm just above the lower cell. Although the capillary increased somewhat the time required for M to rise to a steady limiting value, it also increased the confinement time in the lower cell and strongly reduced spurious frequency shifts due to spin diffusion within the connecting tube.

STRONGLY SPIN-POLARISED ’He GAS

91

After an acceptable value of M was obtained (20-30%), a series of NMR measurements was started. The procedure was to alternate TI10 pulses to measure T2 and the spin precession frequency, and T pulses to reverse M. Repeating this operation 5 or 10 times reduced M by a factor of about five, so that measurements of the spin frequency for a whole set of positive and negative values of M was obtained. T2was found to be practically independent of M. Typical values of T2 were in the range 0.5 to 5 s, and of A w / ~ T were in the range 1 to 10 mHz. The authors point out that great care had to be taken to avoid, or allow for, spurious frequency shifts arising in a variety of ways. Some types of shift were independent of M and these were eliminated by a suitable choice of experimental procedure. Other types, which were well understood, depended on M but fortunately not on 6Bo or T 2 .The procedure adopted at each temperature for dealing with these was to measure values of A 4 l M for various T2 (i.e., 6Bo) and then to extrapolate to zero T2 (i.e., infinite 6Bo). This is illustrated in fig. 3.2. The results for p are shown in fig. 3.3. The agreement with theory discussed above in section 1.9, and represented by the full curve in the diagram is very satisfactory. Later experiments were performed by Tastevin et al. (1985) using a so-called “direct” detection method, in which no static magnetic field gradient is necessary. The NMR coils were connected in opposition so that they created a pure rf field gradient with zero spatial average and excited (or detected) spin waves. Figure 3.4 shows the experimental arrangement in which two pairs of coils, each in opposition, are mounted around a

Fig. 3.2. Measurements reported by Nacher et al. (1984) at 2.5 K. The horizontal scale is the relaxation time T2of the transverse nuclear polarisation. 1/ Tz is proportional to the square of the field gradients over the sample. The vertical axis gives the phase shift A& (measured during a time T2)divided by the spin magnetisation M.

98

D.S. B E I T S ET AL.

Fig. 3.3. Measurements of the quality factor p of the spin waves as a function of temperature. The full line is theoretical (see section 1.9).The data are from Tastevin et al. (1985); see also Lhuillier and Leduc (1985).

spherical sample. The first pair was used for generating rf pulses creating an initial space-dependenttilt of the nuclear magnetisation, while the second picked up the induced signal. In ordinary NMR, the axes of the induction and pick-up coils are orthogonal to minimise their direct coupling; but in the arrangement shown in fig. 3.4 the cross-talk is minimised when the axes are at 45". The method offered some advantage in precision, particularly at temperatures above about 2 K. Below 2 K there were some unresolved discrepancies, as discussed by Tastevin et al. (1989, one possibility being the presence of a 'He monolayer. Spin waves have also been studied in similar work on spin-polarised hydrogen by Johnson et al. (1984) and on spin-polarised 3He dissolved in superfluid liquid 4He-II by Gully and Mullin (1984), and by Ishimoto et al. (1987).

3.2. EXPERIMENTS ON

THERMAL CONDUCTIVITY

This section is based primarily on the work described by Leduc et al. (1986), and Leduc et al. (1987). The theory of the thermal conductivity has been discussed in section 1.7. Figure 3.5 shows a general sketch of the experiment. The nuclear polarisation was obtained by optical pumping using the methods described above with a recently-developed LNA laser (Schearer et al. (1986)

STRONGLY SPIN-POLARISED 'He GAS

99

i ~ i Fig. 3.4. The induction coils (a) are connected in opposition and create an rf magnetic field with zero average value over the spherical sample, as shown by the magnetic field lines (dashed). The static field B, and the initial magnetisation are perpendicular to the plane of the figure. Just after a short resonant rf pulse, the direction of the transverse magnetisation varies over the sample as indicated by the large open arrows; it can be described by I = 1 diffusion modes of the transverse magnetisation. The detection coils (b), also in opposition and with their axes at 45" to the induction coils, receive a flux which is maximum at this initial time. For coils in opposition, the induction coupling in the absence of spins is minimised for a 45" angle, instead of 90" for coils in parallel. (From Tastevin et al. (1985).)

and Daniels et al. (1987)), capable of delivering 200-300 mW at the necessary wavelength of 1.083 p n . In the double-cell format, the two 'MPyrex glass cells were connected by a tube of 75 cm length and sealed after initial cleaning and filling with 'He (2.2Torr at room temperature) and H2 (0.62 Torr at room temperature). When the lower cell was cooled, the 'He pressure dropped to about 0.5 Torr, and the hydrogen formed a solid coating on the inner cold walls of the glass. The purpose of this procedure was to protect the polarised 'He against depolarisation as described in section 2.2. The same motive led to a cell design avoiding the use of any metal inside, and taking care that nearby wires were nonmagnetic and/or reasonably distant from the 'He. A cell optimised for the measurement of unpolarised 3He gas would have been quite different (see, for example, Betts and Marshall (1969)), and compromises had to be made. Figure 3.5 also shows an enlargement of the conductivity cell. It was a right circular cylinder whose end plates were 2 mm thick and 3 cm in diameter, separated by 1 cm, and with thick external silver layers intended to provide the best approximation to isothermal planes needed for sensible analysis of the data. There

I00

D.S. BETTS ET AL.

Fig. 3.5. Apparatus used by Leduc et al. (1987) to measure thermal conductivity. The 'He gas is contained in a double glass cell, in a homogeneous magnetic field B,. A weak gas discharge is maintained in the upper cell at ambient temperature where the atoms' nuclei are polarised by the method described in section 2.1. Spin diffusion downwards brings polarisation into the lower cell, and this is monitored by NMR. The lower cell is also shown on a larger scale, with carbon resistor thermometers R, and Rz.The upper and lower external surfaces are silvered and the lower is covered with a thin heating pad.

was a small orifice in the centre of the upper plate to allow passage of gases and polarisation. The glass side walls were about 0.5 mm thick; ideally they would have been much thinner than that in order to reduce the conduction of heat through the walls relative to that through the 'He to an insignificant magnitude, but in practice the primary constraint was mechanical. The temperature 0,of the upper plate was effectively thermally anchored to the pumped liquid 4He inner bath (see fig. 3.6) and could be varied at will

STRONGLY SPIN-POLARISED 'He GAS

101

Fig. 3.6. Sketched (not to scale) plan and side elevation of the conductivity cell showing the geometrical quantities S, S', I and I', together with the positions of the resistance thermometers measuring 8,and e,. Numerical values of these are used in eqs. (3.5) to (3.9).

in the range 1.3-4.2 K by adjusting the pumping speed. The temperature OPof the lower plate could be raised above 0, by means of a non-inductive winding of manganin resistive wire glued to the silvered surface with GE-7031 varnish, known for its favourable thermal conductivity (McTaggart and Slack (1969)). Heater powers were varied to achieve suitable temperature differences and were typically of the order of 1OOp.W for temperature differences in the region of 10-100 mK. 0, and 0,were measured using Allen-Bradley nominally 1 0 0 R resistors (originally investigated by Clement and Quinnell (1952)) which rise to about 1 kR, 2 kR and 20 kR at 4.2, 3 and 1.3 K respectively, offering suitable sensitivities in the range 0.2 R mK-' to 2 0 R mK-'. The pair of resistors were chosen to be wellmatched and were calibrated by reference to the vapour pressure of liquid 4He in the inner bath using the EFT-76 temperature scale (Durieux and Rusby (1983)). Resistances were measured with a bridge having sufficient sensitivity for use at the low powers necessary at low temperatures.

102

D.S. B E T S ET AL.

The upper plate of the conductivity cell was thermally anchored to the inner bath as described above. The rest of the cell was thermally insulated by the vacuum space which could be pumped to 10 6Torr, so that the heating power 0 supplied to the lower plate had to pass upwards through the glass and 'He to the inner bath. Raw data could be obtained in the form of 0, 8, and et at various T and M. A first approximation for the thermal conductivity K might be obtained then by using the basic relation

where S is the cross-sectional area of the contained 'He and 1 its length. This formula would be exact for an insulated solid cylinder with upper and lower surfaces taken as isothermal planes at 0, and OP, respectively. Problems of analysis always arise however with fluids because these have to be contained, and the containing walls themselves have the ability to conduct heat. In fact in the experiments being described here, a large fraction of the total heat conducted by the filled cell passed through the side walls. If those glass walls could be taken as having the perfect geometry of a cylindrical tube, and if in addition the two end plates could be taken as having infinite thermal conductivity, then a theorem due to Lazarus (1963) could be invoked to prove the exact relation with which to replace eq. (3.5). This would be (3.6)

where K , , ~is~ the thermal conductivity of the '"Pyrex walls, and S' the cross-sectional area of glass in the tube ( S as before is the area of the contained cylinder of 'He). Clearly if either K~~~ or S' were zero, then eq. (3.5) would be recovered. Otherwise K ~ ~has ~ toSbe' determined in a separate experiment in which the cell is evacuated. Unfortunately eq. (3.6) is exact only when the conditions mentioned above are exactly fulfilled. In the real cell the geometry is not perfect and the thermal conductivity of the end plates is finite. A further modification of eq. (3.6) can be made in a plausible though not exact way, by assuming not only that the upper and lower plane surfaces are isotherms (at 8,and @( respectively) but also that the inside upper and lower plane surfaces are isotherms (at, say, 0:and 0;respectively). It is then a straightforward exercise, bearing in mind Lazarus' theorem, to arrive at (3.7)

STRONGLY SPIN-POLARISED ’He GAS

103

where I’ is the combined thickness of the upper and lower end plates (S and 1 as before are the area and length of the contained cylinder of ’He, and S’ the cross-sectional area of glass in the tube). This modification would be strictly true for the geometry shown in fig. 3.6 and has the virtues that it reverts to the form of eq. (3.6) if l’=O with S’ finite and to the form of eq. (3.5) if I’ = 0 and S’ = 0. There remained a problem as to the correct ~ possibility ~ . would be to use published values procedure for finding K ~ One (e.g. Zeller and Pohl (1971)). Another would be to deduce it from measurements made on the evacuated cell, using eq. (3.7) with ~ 3 =“ 0;~ this however would be of dubious validity in the extreme case in which all the heat must flow through the tubular walls. Moreover, the geometry of the glass was unavoidably imperfect so that not only was K~~~ needed but also some way of choosing appropriately-averaged values of S’ and I’ (S and 1 may be taken as reasonably well known). The following procedure was adopted. Firstly the cell was evacuated so that the dominant thermal resistance was in the walls; the term containing I’ was therefore neglected in eq. (3.6), which then resolves to give the “measurable” result

Secondly the cell was filled with superfluid liquid 4He-II which effectively has almost infinite thermal conductivity relative to the glass so that the dominant thermal resistance was in the end plates; the term containing S‘ was therefore neglected in eq. (3.6), which then resolves to give the “measurable’’ result (3.9) These subsidiary “measurements” of S ‘ K p y , from (3.8) and K ~ I‘ from ~ ~ (3.9) / were then used in eq. (3.7) to obtain ~ 3 ” ~In. the case of unpolarised gas, the results could be compared with the accepted results of earlier authors (Betts and Marshall (1969) and Keller and Kemsk (1969)). The comparison reveals a small systematic discrepancy which is almost certainly due to the geometrical problems associated with the use of glass, and the resulting need to make approximations as described above. It was not a major concern in that the object of the experiment was to study diferences between ~ ( 0 ) and K ( M as ) discussed below. The magnetisation M of the gas was measured in the usual way with a set of crossed coils located in the evacuated space around the cell; free induction decay signals at 10 kHz could be detected after small tilting angle pulses in the induction coils. The separate calibration of the N M R signals

D.S. BETTS ET AL.

104

was obtained by comparison with optical measurements of circular polarisation of lines emitted by the discharge of the nuclear polarisation in the upper cell as described earlier (PavloviE and Laloe (1970)). The main measurements were concentrated on changes occurring in the thermal conductivity when M was deliberately destroyed. First the gas was polarised using the whole paraphernalia of laser and discharge. When an acceptable level of polarisation was achieved (typically 20-30%), the laser and discharge were switched off to avoid spurious heat inputs to the cell. Once this had been done, the time scale for action became restricted by the relaxation time (typically 30 min at 2.5 K, falling to 3 min at 1.8 K), and it was virtually impossible to take data below about 1.5 K. The temperature 9, was held as constant as possible manually (typically within 1 mK during the course of the measurement sequence) and Oe was continuously monitored under the influence of a steady heat flux (typically about 100 bW) and displayed on a chart recorder. The procedure was evolved so as to minimise dependence on measuring absolute values of conductivity. It involved destroying the polarisation M by a 7r/2 N M R pulse and observing the effect on as exemplified by fig. 3.7; SO( was typically 1 or 2 mK for 20-30% polarisation. The corresponding decrease SK in thermal conductivity was then finally obtained using eqs. (3.7) and (3.9).

3.5&0

1

depolarked

He 7112 NMR pulse

polarized ’He

3.535 t0

time

Fig. 3.7. A typical recorder trace of the resistance of RL (converted into temperature 6 , )as a function of time. The power 0 in this case was 107 p W and the temperature of the upper plate 8,was kept approximately constant at about 3.430 K. The initial nuclear polarisation, M = ? S % . was suddenly destroyed at time f,by an N M R pulse. The subsequent rise in temperature, about 2 m K in this case, provided raw data from which the conductivity change, K ( M )- ~ ( 0 )was . deduced. (From Leduc et al. (1987).)

STRONGLY SPIN-POLARISED 'He GAS

105

It was important to carry out three checks for consistency. Firstly, at a fixed temperature SK is predicted (see eq. (1.55)) to be closely proportional to M 2 for the conditions achievable in these experiments, and this was borne out in practice. Secondly, it was demonstrated that with a zero heating rate Q, SO, was also found to be zero, and this eliminates possible worries about spurious heat generated by the 1r/2 NMR pulse and/or the consequent depolarisation. Thirdly, there was the important and interesting matter of whether convection could safely be assumed to be absent, bearing in mind the fact that the fluid was being heated from below. Experimentally its absence was demonstrated by showing that the temperature difference across the cell was proportional to the heat current Q; non-proportionality would have been a clear signal that convection was present. We shall consider this matter in more detail in the following section, partly because of its intrinsic importance to the correct analysis of thermal conductivity and partly because a deliberate study of the onset of convection would be a feasible and interesting extension to the work described here. The experimental results obtained by Leduc et al. (1987) and in later work by the same group, yet to be published, are shown in fig. 3.8. Generally

Fig. 3.8. The thermal conductivity data of Leduc et al. (1987) in the convenient form ( K ( M ) M 2versus temperature, appropriate for low M. The circles and triangles represent data taken in November/Decembcr 1986 and June/July 1987, respectively, with the dotted area intended simply as a guide for the eye. For comparison, the predictions of Lhuillier (1983) are also shown with a dashed line, corresponding to the Lennard-Jones potential, and a dashed and dotted line to the Aziz potential. K(O))/

D.S. B E T S ET AL.

106

speaking, there is good qualitative agreement between theory and experiment. There are however significant differences, in particular the position of the maximum, which may point to the need to extend the theory to higher order. The most recent experimental work, not yet published, has confirmed that K ( M, T ) becomes independent of M at about 1.5 K as shown in fig. 3.8. It is clearly of interest to extend the measurements as far down in temperature as possible; the limitation is that the relaxation time becomes uncomfortably short below 1.5 K. 3.3. CONVECTION The thermal expansion coefficient a of 'He gas is positive, whether or not it is polarised, and may be assumed to be closely equal to the perfect gas value of 1/ T deg-'. Nevertheless heating from below, though it is a necessary condition for convective motion, is not a sufficient condition. For a given cell, filled with a given fluid under specified circumstances, the applied temperature difference has to exceed a certain well-defined minimum value before convection becomes the favoured mode. There is thus an onset condition, first studied by Benard and Rayleigh (see Saltzman (1962) for a collection of historic papers and references). There is a large literature on Rayleigh-Binard convection but suitable key references should include Saltzman (1962), Threlfall (1975), Ahlers and Behringer (1978), Busse (1978), Libchaber and Maurer (1978), Maurer and Libchaber (1979), and Drazin and Reid (1981). The experimental work of Threlfall (1975) on helium is particularly relevant to the present discussion. Data on the onset of convection are best shown in a plot of Nusselt number (Nu) versus Rayleigh number (Ra), these two dimensionless parameters being defined as follows. The Nusselt number is the ratio of the actual heat flow to that which would occur if conduction were the only mechanism. Thus, for a cylinder of cross section S,with axial heat flow, (3.10) where AT/Ax is the temperature gradient. The Rayleigh number is a more complicated combination of fluid properties including the isobaric thermal expansion coefficient (a),the density ( p ) , the isobaric specific heat (C,), the thermal conductivity ( K ) , and the viscosity (v),together with the external parameters g (the acceleration due to gravity), AT (the temperature difference between the upper and lower plates, and I (the distance between the plates). The dimensionless Rayleigh number is given by Ra =

'

gap C,,A 77' KT

(3.11)

STRONGLY SPIN-POLARISED 'He GAS

107

For sufficiently low values of AT, Nu = 1 independently of Ra, corresponding to the transmission of heat entirely by conduction. If AT is raised, there is observed to be a critical value of Ra, denoted by Ra,, beyond which Nu rises above unity, that is, the fluid begins to transmit heat by convection. The value of Ra, has been calculated for simple geometries and boundary conditions and is generally in the region of 1700. Threlfall (1975), in a systematic experimental study of the effect in gaseous helium in a cylindrical cell having a diameter of 5 cm and length 2 cm, found Ra, = 1630 as shown in fig. 3.9. In principle, therefore, it is possible to use eq. (3.10) with low AT so that Nu = 1 to find K, and then, by raising AT until the onset of convection is observed, to use eq. (3.11) with an appropriate value of Ra, to deduce 7.The idea is attractive because it allows a measurement of both K and 7 in a single experimental run. In practice it should also be possible in polarised 'He gas although the required numerical parameters are not ideally favourable. The main difficulty is that the optimum density, from the point of view of producing high polarisations, gives lowish values of

1630

Ra Fig. 3.9. Experimental data for gaseous 4He obtained by Threlfall (1975). The definitions of the Nusselt number Nu and the Rayleigh number Ra are given in eqs. (3.10) and (3.11), respectively. The cell geometry (diameter 5 cm; length 2 cm) was similar to, but not identical to, that of Leduc et al. (1987). The transition to convection is clearly seen, with a critical Rayleigh number of 1630.

D.S. BEITS ET AL.

1 OR

Ra for sensible values of AT. However, as an example of what might be possible, if the pressure and temperature of the unpolarised 3He in a cylindrical cell of 5cm diameter and 2cm length were 2Torr and 2 K respectively, then raising AT from 5 to 100 mK could be expected to sweep Ra/Ra, from about 0.19 through 1.0 (onset of convection) to about 3.8. Table 6 shows some other estimated possibilities at different temperatures, again for unpoiarised 'He at 2Torr in the same cell. The two columns correspond to the fairly arbitrary choices of AT = 5 mK (a suitable lower limit if measurement sensitivity is to be maintained) and A T = 50T mK ( T in K), a suitable upper limit if A T / T is not to amount to more than 5 % . Bouchaud and Lhuillier ( 1986) have given theoretical consideration to the Rayieigh-BCnard instability discussed above, and have also pointed out that there is a magnetic equivalent in which magnetisation is substituted for temperature and the Stern-Gerlach force for the gravity force. Such an equivalence arises from the formal similarity of the equations describing the two situations. Moreover the two effects can be superposed so that magnetic effects could for example be set to act either with or against the thermal effects. The equations governing this mode coupling were originally given by Lhuillier and Laloe (1982b). For 'He they take the forms J T = - K ( M , T ) V T + L l , ( M ,T ) 7 T M ,

(3.12)

+ LZl(M , T ) V In T,

(3.13)

JM

= -DO( T)VM

where JTand J M are the heat and polarisation current densities. We have already referred in some detail to the quantities K and Do, and Lhuillier (1983) gives plots of L,? and L,, which, like K and 7, show strong M dependence in the region of 2 K. These equations suggest some fascinating

Table 6

T

1.4

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3 .O

Ral Ra, for AT = 5 m K

0.75 0.45 0.29 0.19 0.13 0.10 0.070 0.053

0.040

Ra/ Ra, for AT = 5OTmK ( T i n K)

10.6 1.2 5.2 3.8 2.9 2.3 1.8 1.5 1.2

STRONGLY SPIN-POLARISED 3He GAS

109

experimental possibilities for investigating hydrodynamic instabilities in spin-polarised 'He gas, but these have yet to be attempted.

3.4.

L I Q U I D - G A S EQUILIBRIUM

It would be straying too far from the purpose of this chapter to discuss the properties of spin-polarised 'He in liquid form, interesting though these are. It is necessary however to remark that the production of polarised liquid is not altogether easy since the application even of very large magnetic fields only causes small polarisation. One method which has received attention makes use of effects at the solid-liquid transition, as originally proposed by Castaing and Nozikres (1979). An alternative, recently demonstrated by Tastevin et a]. (1988) and relevant to the topic of this chapter is based on the rapid liquefaction of spin-polarised 3He gas. The apparatus used is shown in fig. 3.10. The idea is first to polarise 'He gas using the methods described in section 2.1 and then to reduce the temperature quickly so that the initial gas pressure exceeds the vapour pressure and condensation occurs. Tastevin et al. (1988) have demonstrated that this method can lead to nuclear polarisations in excess of 40% in the condensed liquid at about 400 mK where the vapour pressure is very low. The relaxation time of the polarisation in the liquid is a few minutes. There are expected to be effects

- -

-

\\ II II

r r I I I I -

-

Fig. 3.10. The experimental arrangement used by Tastevin et al. (1988) for producing polarised liquid 'He. The container of the 'He sample is made of three different cells, connected by a long tube. The upper cell A is at ambient temperature, where the nuclear polarisation is produced by laser optical pumping. The intermediate cell B is at about 4 K and acts as a polarised gas reservoir. The lower cell C is immersed in the liquid 'He bath whose temperature can be varied by pumping; it is surrounded by the NMR pick-up coils.

110

D.S. BETTS ET AL.

02

04

06

08

1.0

M Fig. 3.1 1. The predicted phase equilibrium diagram for partially polarised ’He at T = 0.3 K. with a liquid density assumed to be independent of M,and with mechanical energies per atom assumed to be E = -2.5 K and E, = -2.2 K for unpolarised and fully polarised liquid respectively. With these parameters the liquid always has a smaller polarisation than the vapour with which it is in equilibrium but at T = 1.8 K, for example, this is reversed. Such diagrams depend very sensitively on 7, E and E , . (From Stringari et al. (1987).)

of polarisation on the phase diagram. These were first discussed by Lhuillier and Laloe (1979), and have recently been considered phenomenologically in some detail by Stringari et al. (1987). They have drawn some interesting conclusions about possible behaviours, although these depend strongly on the validity of the phenomenological model for free energy and on the currently unknown difference between the energy per atom in the unpolarised and fully polarised liquid states. As an example, fig. 3.1 1 shows the predicted interrelation between vapour pressure and polarisation at T = 0 . 3 K when particular assumptions are made: the liquid phase has a smaller polarisation than the vapour phase, a situation which can be reversed by raising the temperature. This example is relatively simple but more complicated “metamagnetic” effects might arise in which the liquid phase separates, with the two liquid phases having different polarisations. Such effects can also be represented in phase diagrams, of which Stringari et al. (1987) give a number of examples. Clearly, experimental data are needed.

STRONGLY SPIN-POLARISED ’He GAS

111

References Abragam, A., 1961, The Principles of Nuclear Magnetism (Oxford). Ahlers, G., and R.P. Behringer, 1978, Phys. Rev. Lett. 40,712. AI-Shibani, K., 1977, Doctoral Thesis (University of Sussex, U.K.). Aziz, R.A., V.P.S. Nain, J.S. Carley, W.L. Taylor and G.T. McConville, 1979, J. Chem. Phys. 70, 4330.

Aziz, R.A., F.R.W. McCourt and C.C.K. Wong, 1987, Mol. Phys. 61, 1487. BarbC, R., F. Laloe and J. Brossel, 1975, Phys. Rev. Lett. 34. 1488. Bardotti, G., B. Bertotti and L. Gianolio, 1964, J. Math. Phys. 5, 1387. Bashkin, E.P., 1981, Pis’ma v Zh. Eksp. & Teor. Fiz. 33, 11 [Sov. Phys.-JETP Lett. 33, 81. Bashkin, E.P., 1984a. Pis’ma v Zh. Eksp. & Teor. Fiz. 40, 383 [Sov. Phys.-JETP Lett. 40, 11971.

Bashkin, E.P., 1984b, Zh. Eksp. & Teor. Fiz. 87, 1948 [Sov. Phys.-JETP 60, 11221. Bashkin, E.P., 1986, Usp. Fiz. Nauk 148, 433 [Sov. Phys.-Usp. 29, 2381. Bashkin, E.P., and A.E. Meyerovich, 1981, Adv. Phys. 30, 1. Bernstein, R.B., 1966, in: Advances in Chemical Physics, Vol. X,Molecular Beams, ed. J. Ross (Wiley-Interscience, New York) ch. 3, p. 75. Betts, D.S., and M. Leduc, 1986, Ann. Phys. (France) 11, 267. Betts, D.S., and R. Marshall, 1969, J. Low Temp. Phys. 1, 595. Bouchaud, J.P., and C. Lhuillier, 1986, Phys. Lett. A 116, 99. Boyd, M.E., S.Y. Larsen and J.E. Kilpatrick, 1966, J. Chem. Phys. 45, 499. Boyd, M.E., S.Y. Larsen and J.E. Kilpatrick. 1969, J. Chem. Phys. 50, 4034. Brush, S.G., 1972, in: Kinetic Theory, Vol. 3, The Chapman-Enskog Solution of the Transport Equation for Moderately Dense Gases (Pergamon Press, Oxford). Buckingham, R.A., J. Hamilton and H.S.W. Massey, 1941, Proc. R. SOC.London Ser. A 179, 103.

Busse, F.H., 1978, Rep. Prog. Phys. 41, 1929. Byrne, J., and P.S. Farago, 1971, J. Phys. B 4, 954. Cameron, J.A., and G.M. Seidel, 1985, J. Chem. Phys. 83, 3621. Castaing, B., and P. Nozitres, 1979, J. Phys. (France) 40,257. Chapman, R., 1975. Phys. Rev. A 12, 2333. Chapman, S., and T.G.Cowling, 1970, The Mathematical Theory of Non-Uniform Gases, 3rd Ed. (Cambridge). Clement, J.R., and E.H.Quinnell, 1952, Rev. Sci. Instrum. 23, 213. Cohen-Tannoudji, C., and A. Kastler, 1966, in: Progress in Optics, VoI. V, ed. E. Wolf ( North-Holland, Amsterdam). Colegrove, F.D., L.D. Schearer and G.K.Walters, 1963, Phys. Rev. 132, 2561. Corruccini, L.R., D.D. Osheroff, D.M.Lee and R.C. Richardson, 1971, Phys. Rev. Lett. 27,650. Corruccini, L.R., D.D. Osheroff, D.M. Lee and R.C. Richardson, 1972, J. Low Temp. Phys. 8, 229.

Daniels, J.M., and R.S. Timsit, 1971. Can. J. Phys. 49, 525. Daniels, J.M., L.D. Shearer, M. Leduc and P.-J. Nacher, 1987, J. Opt. SOC.Am. B 4, 1133. De Boer, J., and E.G.D. Cohen, 1951, Physica 17, 993. De Boer, J., and A. Michels, 1939, Physica 8, 409. De Boer, J., J. van Kranendonk and K.Compaan, 1949, Phys. Rev. 76.998, 1728. De Boer, J., J. van Kranendonk and K. Compaan, 1950, Physica 16, 545. Drazin, P.G., and W.H. Reid, 1981, Hydrodynamic Stability (Cambridge). Dupont-Roc, J., M. Leduc and F. Laloe, 1971, Phys. Rev. Lett. 27, 467. Durieux, M., and R.L. Rusby. 1983, Metrologia 19, 67.

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Edwards, D.O.. and W.F. Saam, 1978, The free surface of liquid helium, in: Progress in Low Temperature Physics, Volume Vlla, ed. D.F. Brewer (North-Holland, Amsterdam), p. 283.

Emery, V.J., 1964, Phys. Rev. A 133, 661. Grad, H., 1958, Handb. Phys. 12, 205. Grimsrud. D.T., and J.H. Werntz Jr, 1967, Phys. Rev. 157, 181. Gully, W.J., and W.J. Mullin, 1984. Phys. Rev. Lett. 52. 1810. Halperin, W.P., F.B. Rasmussen, C.N. Archie and R.C. Richardson, 1978, J. Low Temp. Phys. 31, 617. Happer, W., 1972, Rev. Mod. Phys. 44, 169. Himbert, M., V. Lefevre-Seguin, P.-J. Nacher, J. Dupont-Roc, M. Leduc and F. Laloe, 1983, J. Phys. Lett. (France) 44, L523. Hirschfelder, J.O., C.F. Curtiss and R.B. Bird, 1954, Molecular Theory of Gases and Liquids (Wiley, New York) [reprinted 19641. Ishimoto. H., H. Fukuyama, N. Nishida, Y. Miura, Y. Takano, T. Fukuda, T. Tazaki and S. Ogawa, 1987, Phys. Rev. Lett. 59, 904. Johnson. B.R., J.S. Denker, N. Bigelow, L.P. Levy, J.H. Freed and D.M. Lee, 1984, Phys. Rev. Lett. 52, 1508. Keller, W.E., 1955, Phys. Rev. 98, 1571. Keller, W.E., 1957, Phys. Rev. IM,41. Keller, W.E., 1969, in: Helium-3 and Helium-4 (Plenum Press, New York) ch. 3, p. 61. Keller. W.E., and J.F. Kerrisk, 1969, Phys. Rev. 177. 341. Kilpatrick, J.E.. W.E. Keller, E.F. Hammel and N. Metropolis, 1954, Phys. Rev. 94, 1103. Laloe, F., ed., 1980, Proc. Conf. on Spin Polarised Quantum Systems, 21-26 April, 1980, Aussois, France, J. Phys. (France) Colloq. 0 ,suppl. 7, 1980. Laloe, F., M. Leduc, P.-J. Nacher, L.N. Novikov and G. Tastevin, 1985, Usp. Fiz. Nauk 147, 433 [Sov. Phys.-Usp. 28, 9411. Lazarus, R.B.. 1963, Rev. Sci. Instrum. 34, 1218. Leduc, M.. 1972, Thkse d'Etat (Ecole Normale Supirieure, University of Paris, France). Leduc, M.,G. Trinec and F. Laloe. 1980, J . Phys. (France) 41, C7-75. Leduc, M., P.-J. Nacher, S.B. Crampton and F. Laloe, 1983, in: Proc. AIP Conf. on Quantum Fluids and Solids,, 11-15 April. 1983, Sanibel Island, FL, U.S.A., eds E.D. Adams and G.G. lhas (AIP, New York) p. 179. Leduc, M., S.B. Crampton, P.-J. Nacher and F. Laloe. 1984, Nucl. Sci. Appl. 2, 1. Leduc, M., F. Laloe, P.-J. Nacher, G. Tastevin, J.M. Daniels and D.S. Betts, 1986, in: Proc. 14th Int. ConP. on Quantum Electronics, 9-13 June, 1986, San Francisco, U.S.A. (American Physical Society) p. 138. Leduc, M., P . J . Nacher, D.S. Betts. J.M. Daniels, G. Tastevin and F. Laloe, 1987, Europhys. Lett. 4. 59; see also Proc. 18th Int. Conf. on Low Temperature Physics, 20-26 August, 1987, Kyoto, Japan, Jpn. J. Appl. Phys. 26, suppl. 26-3, 213. Lefevre-Seguin, V., P.-J. Nacher and F. Laloe, 1982, J. Phys. (France) 43, 737. Lefkvre-Seguin, V., P.-J. Nacher, J. Brossel, W.N. Hardy and F. Laloe, 1985, J. Phys. (France) 46, 1145. Leggett, A.J.. 1970, J. Phys. C 3, 448. Leggett. A.J., and M.J. Rice, 1968, Phys. Rev. Lett. 20, 586. Lhuillier, C., 1983, Transport properties in a spin polarized gas, 111. J. Phys. (France) 44, 1. Lhuillier, C., and F. Laloe. 1979, J. Phys. (France) 40. 239. Lhuillier. C., and F. Laloe, 1980, in: Proc. of C N R S Colloq. on Spin Polarised Quantum Systems, 21-26 April. 1980, Aussois, France, ed. F. Laloe, J. Phys. {France) Colloq. C-7, suppl. 7, C7-51.

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Lhuillier, C., and F. Laloe, 1982a, Transport properties in a spin polarized gas, 1. J. Phys. (France) 43, 197. Lhuillier. C., and F. Laloe, 1982b, Transport properties in a spin polarized gas, 11. J. Phys. (France) 43, 225. Lhuillier, C., and M. Leduc, 1985, Ann. Phys. (France) 10, 859. Libchaber, A., and J. Maurer. 1978, J. Phys. Lett. 39, L369. Luszczynski, K., R.E. Norberg and J.E. Opfer, 1962, Phys. Rev. 128, 186. Mandl, F., 1957, Quantum Mechanics, 2nd Ed. (Butterworths, London) sect. 38. Massey, H.S.W., and R.A. Buckingham, 1938, Proc. R SOC.London Ser. A 168, 378. Massey, H.S.W., and R.A. Buckingham, 1939, Proc. R. SOC.London Ser. A 169, 205. Masuhara, N., D. Candela, D.O. Edwards, R.F. Hoyt, H.N. Scholz, D.S. Sherrill and R. Combescot, 1984, Phys. Rev. Lett. 53, 1168. Matacotta, F.C., G.T. McConville, P.P.M. Steur and M. Durieux, 1987, Metrologia 24, 61; briefer presentation, in: Roc. 18th Int. Conf. on Low Temperature Physics, 20-26 August, 1987, Kyoto, Japan, ed. Y. Nagaoka, Jpn. J. Appl. Phys. 26, suppl. 26-3, 1679. Maurer, J., and A. Libchaber, 1979, J. Phys. Lett. (France) 40, L419. McAdams, H.H., and G.K. Walters, 1967. Phys. Rev. Lett. 18, 436. McTaggart, J.H., and G.A. Slack, 1969. Cryogenics 9, 384. Meyerovich, A.E., 1978, Phys. Lett. A 69, 279. Meyerovich, A.E., 1983, J. Low Temp. Phys. 53, 487. Meyerovich, A.E., 1987, Spin-polarised 3He-4He solutions, in: Progress in Low Temperature Physics, Vol. XI, ed. D.F. Brewer (North-Holland, Amsterdam) p. 1. Meyerovich, A.E., 1989, Spin-polarised phases of 'He, in: Anomalous Phases of 'He, eds W.P. Halperin and L.P. Pitaevskii ( North-Holland, Amsterdam) to be published. Mollenauer, L.F., 1980, Opt. Lett. 5. 188. Monchick, L., E.A. Mason, R.J. Munn and F.J. Smith, 1965, Phys. Rev. 139, A1076. Nacher, P.-J., and M. Leduc, 1985, J. Phys. (France) 46,2057. Nacher, P.-J., M. Leduc, G. Trenec and F. Laloe, 1982, J. Phys. Lett. (France) 43, L525. Nacher, P.-J., G. Tastevin, M. Leduc, S.B. Crampton and F. Laloe, 1984, J. Phys. Lett. (France) 45, L441. Owers-Bradley, J., H. Chocholacs, R.M. Mueller, C. Buchal, M. Kubota and F. Pobell, 1983, Phys. Rev. Lett. 51, 2120. PavloviE, M., and F. Laloe, 1970. J. Phys. (France) 31, 173. Pinard, M., and F. Laloe, 1980, J. Phys. (France) 41, 799. Pinard, M., and J. van der Linde, 1974, Can. J. Phys. 52, 1615. Roberts, T.R., R.H. Sherman and S.G. Sydoriak, 1964, J. Res. Natl. Bur. Stand. Sect. A 68,567. Saltzman, B., ed., 1962, Selected Papers on the Theory of Thermal Convection (Dover, New York). Schearer, L.D., and G.K. Walters, 1965, Phys. Rev. A 139, 1398. Schearer, L.D., M. Leduc, D. Vivien, A.-M. Lejus and J. Thiry, 1986, IEEE J. Quantum Electron. QE-22, 713. Schiff, L.I., 1955, in: Quantum Mechanics (McGraw-Hill, New York) ch. 5. Silin, V.P., 1957, Zh. Eksp. & Teor. Fiz. 33, 495, 1227 [1958, Sov. Phys.-JETP6, 387, 9451. Silvera, I.F., and J.T.M. Walraven, 1986, Spin-polarised atomic hydrogen, in: Progress in Low Temperature Physics, Vol. X, ed D.F. Brewer (North-Holland, Amsterdam) ch. 3, p. 139. Slichter, C.P., 1980, The Principles of Magnetic Resonance, 2nd Ed. (Springer, Berlin). Stringari, S., M. Barranco, A. Polls, P.-J. Nacher and F. Laloe, 1987, J. Phys. (France) 48, 1337. Tastevin. G., P.-J. Nacher, M. Leduc and F. Laloe, 1985, J. Phys. Lett. (France) 46,L249. Tastevin, G., P.-J. Nacher, L. Wiesenfeld, M. Leduc and F. Lal0.5, 1988, J. Phys. (France) 49, 1

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Thomasson, J.W., and D.M. Ginsberg, 1976, Rev. Sci. Instrum. 47, 387. Threlfall, D.C., 1975, J . Fluid Mech. 67, 17. Timsit, R.S., and J.M. Daniels, 1971, Can. J . Phys. 49. 545. Timsit, R.S., J.M. Daniels and A.D. May, 1971, Can. J . Phys. 49, 560. Trenec. G., P.-J. Nacher and M. Leduc, 1982, Opt. Comrnun. 43, 37. Waldmann, L., 1958, Z. Naturforsch. a 13. 609. Zeller. R.C., and R.O. Pohl. 1971, Phys. Rev. B 4, 2029.

CHAPTER 3

KAPITZA THERMAL BOUNDARY RESISTANCE AND INTERACTIONS OF HELIUM QUASIPARTICLES WITH SURFACES BY

TSUNEYOSHI NAKAYAMA Department of Applied Physics, Hokkaido University, Sapporo 060, Japan

Progress in Low Temperature Physics, Volume XII Edited by D.E Brewer @ Elsevier Science hblishers B. V., 1989 115

Contents I . Introduction . . . . . . . ........................................... 2. Kapitza thermal bou nce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definition of the Kapitza resistance R, ................................. 2.2. General expression for the Kapitza resistance R, . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Anomalous behaviour of the observed Kapitza resistances . . . . . . . . . . . . . . . . . 2.3.1. The liquid 'He-sintered powder interface at mK temperatures . . . . . . . . 2.3.2. The liquid He-bulk solid interface above about 1 K . . . . . . . . . . . . . . . . . 3 . Fermi liquid theory of the Kapitza resistance ................................. 3.1. Heat transfer due to zero-sound excitations . . . 3.1.1. Emission of zero-sound from a small pa 3.1.2. Heat flux from a spherical particle at temperature 7 3.1.2.1. Energy current into liquid 'He ............................ 3.1.2.2. Kapitza conductance h, due to zero-sound excitations 1.2. Energy transfer due to inelastic scattering of single quasiparticles at th

....................... tween sintered powde ................... 4 . I . Heat exchanger using submicrometer metal particles ...................... 4.2. Soft phonon-modes in sintered powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Heat transfer due to the effect of soft phonon-modes ...................... 5 . The magnetic channel of heat transfer between sintered powder and liquid 'He . . 5.1. Surface characteristics of submicron metal particles and surface magnetic impurities . . . . . . ............... ................ 5.2. Heat transfer due to magnetic coupling at mK temperatures . . . . . . . . . . . . . . . 6 . Thermal boundary resistance between liquid 'He-4He mixtures and sintered powder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Acoustic channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Magnetic channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Anomalous Kapitza resistance between liquid He and a bulk solid above about 1 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Kapitza conductance h , and phonon transmission coefficient across the interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. High-frequency phonon scattering at the liquid He-solid interface . . . . . . . . . . 7.2.1. Cause of diffuse scattering at the surface without liquid He . . . . . . . . . . 7.2.2. Specular versus diffuse scattering of bulk phonons . . . . . . . . . . . . . . . . . . 7.2.3. Diffuse signals in the time-of-flight reflection signals . . . . . . . . . . . . . . . . . 7.2.4. Reduction of the diffuse signal at the solid surface in contact with liquid He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . Summary . . . . . . . .................................... Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . ................................................

I16

117 118 118 120 124 124 126 128 128 128 130 130 132 135 140 140 141 144 149 149 152 158 159 162 165 165 168 171 175 176 179 186 187 189

1. Introduction

Heat exchange between dissimilar media is an important subject for experiments at low temperatures. When heat is transferred from a solid into liquid He or vice versa, a temperature discontinuity appears at the interface. Although this phenomenon was first discovered almost half a century ago, by Kapitza (1941) during his experimental investigation of the heat transport in superfluid 4He, it is only in recent years that low temperature physicists have studied this problem in detail, in connection with condensed matter physics. Indeed, various subfields of condensed matter physics such as surfaces, magnetism, disordered systems, etc., are involved in this. Initially, there was a suggestion (Gorter et al. 1951) that the phenomenon was related to the superfluid phase of liquid 4He (He 11). However, a similar temperature discontinuity also appears at the interface of normal liquid 'He and a solid (Lee and Fairbank 1959), and it has been established that the thermal boundary resistance is not peculiar to He 11. In fact, thermal boundary resistances have by now been reported for many interfaces between dissimilar media. These include the interfaces between nonmagnetic or magnetic solids and quantum media such as solid 'He or 4He, 3He-4He mixtures, superfluid 3He. In principle, a thermal boundary resistance or so-called Kapitza resistance occurs between any dissimilar media at low temperatures, even for the interface between two quantum media like the solid 4He-liquid 4He or liquid 4He-hydrogen gas interface (Huber and Mans 1982, Maris and Huber 1982, Peach et al. 1982, Castaing and Papoular 1983, Graf et al. 1984, 1985, Kagan et al. 1984, Goldman 1986). Khalatnikov (1952, 1965) presented a theory for the liquid 4He-solid interface based on the idea that heat is carried by phonons, but these phonons have a high probability of reflection at the interface due to the large acoustic mismatch between a solid and liquid 4He. This so-called acoustic mismatch (AM) theory predicts that the Kapitza resistance R K , whose definition will be given in section 2.1, is proportional to the inverse third power of the temperature ( T - 3 ) . However, the Kapitza resistances observed at the liquid 4He (or 'He)-solid interfaces are, in general, some orders of magnitudes smaller than the value of the AM theory in addition to disagreement with the predicted T - 3 temperature dependence. This anomalous behaviour, often called the anomalous Kapitza resistance, is well established experimentally in two cases: the liquid 'He-sintered powder interfaces at millikelvin (mK) temperatures and the liquid 4He (or 'He)-bulk 117

118

T. NAKAYAMA

solid interfaces above about 1 K. These have been long-standing unsolved problems in low temperature physics. To understand the underlying mechanisms for creating the anomalous heat conductance is of great importance for low temperature physicists. As far as the former problem at mK temperatures is concerned, the investigation seems to be entering a new stage in the past few years due to recent experimental and theoretical efforts, in comparison with the time when the reviews on this problem were written by Harrison (1979) and Nakayama ( 1984a). The present article treats mainly the investigations on the anomalous Kapitza thermal boundary resistances between liquid 'He and metal particles observed at mK temperatures. The problem of the Kapitza resistance above about 1 K will be surveyed in section 7. Section 2 begins with the definition of the Kapitza resistance R K . The AM theory is explained in a different manner from the conventional procedure based on phonon transmission and reflection at the solid-liquid He interface. This enables us to treat in a simple way the Kapitza resistance between a Jinite-size system such as small particles and liquid 'He, which is the main subject in section 4. In section 2, the experimental data for R K both at mK temperatures and above about 1 K are compared with the predictions of the A M theory. The Fermi liquid theory for the Kapitza resistance R K is explained in section 3, where the emphasis is laid on the theoretical treatment of the heat transfer mechanisms due to both zero-sound excitations and single quasiparticle excitations. Section 4 describes the effect of the elastic softening of sintered powders on the anomalous behaviour of the Kapitza resistance observed at the liquid 'He-sintered powder interfaces. Section 5 deals with another heat transfer mechanism originating from the magnetic coupling in connection with the surface characteristics of sintered metal particles. Section 6 discusses the problem of the Kapitza resistance between 'He-4He mixture and sintered powder. This is of great importance with respect to the possible discovery of the superfluid transition in the dissolved 'He in a dilute 'He-4He solution. The problem of the anomalous Kapitza resistance above about 1 K is described briefly in section 7 with emphasis on recent investigations using the new techniques of high-frequency phonon generation and detection.

2. Kapitza thermal boundary resistance 2.1. DEFINITION OF THE

KAPlTZA RESISTANCE R K

Consider the heat transfer from liquid He at temperature T, into a solid at T2 (fig. 2.1). The net heat flux across the interface Is given by the difference

KAPITZA THERMAL BOUNDARY RESISTANCE

Temp.

T2

119

T1

Fig. 2.1. Temperature discontinuity at the liquid He-solid interface. AO(T) is the net heat flux from liquid He at TI = Tz+AT into solid at T z .

of the heat flux per unit area AQ = OL( 7-11 - OS( T2)

(2.1)

I

os(

where Q,( Tl) is the heat flux from liquid He into the solid, and T2)the flux from the solid into the liquid, respectively. The net heat flux has to vanish when the temperatures are equal ( Tl = T2),namely QL( TI)= & ( T I ) . If the difference in temperature ( A T = Tl - T2)is small, eq. (2.1) becomes

where the heat conductance across the interface is defined as

The Kapitza resistance RK is defined as the inverse of the Kapitza conductance RK = l / h K . RK is usually expressed in units of m2K/W. In the above definition of RK, a temperature discontinuity ( A T ) is presumed at the interface. The validity of this assumption can be understood as follows: Provided that phonons are incident from liquid He into the solid, the critical angle of incidence, at which phonons transmit into the solid, becomes 0; = sin-'(cJoS) according to Snell's law, where cL and us are the velocities of phonons in liquid He and the solid. Hereafter, the subscripts 'S' and 'L' indicate the solid and the liquid, respectively. Since cL= 2.4 x lo4 cm/s for He I1 below 1 K and us is typically 5 x loscm/s, one 0: are totally reflected. The finds that 0',-3". Phonons incident at phonon transmission coefficient tLS from the liquid into the solid is determined from the ratio between the incident (A,) and reflected ( A , ) amplitude of a phonon as t,=

~-IA~/A,,~~.

(2.4)

T. N A K A Y A M A

120

The transmission coefficient tLs can be obtained easily in the case of normal incidence at O L = O , in terms of the acoustic impedance Z L = p L c Land Z s = psus, where p, and pLare the mass densities of the solid and the liquid, respectively. The result becomes

where the condition Z, > Z L is taken into account. The transmission coefficient tLs is equal to rsL in the case of the normal incidence, where tsL means the coefficient from the solid into the liquid. Since ps -- 10 g/cm3 and pL= 0.142 g/cm3, we have a very small transmission with tLs= 2 x Thus the large acoustic mismatch, in addition to the small critical angle (Of = 3”), severely limits the effective heat exchange between these two media. This is the reason why the temperature discontinuity appears at the interface. 2.2. GENERAL EXPRESSION

FOR T H E

KAPITZARESISTANCE RK

In deriving the expression for the Kapitza resistance RK in the acoustic mismatch (AM) theory (Khalatnikov 1965), a part of the lengthy calculation is taken up with estimating the angular integral with respect to the transmission coefficient fLS(0). Here we try to formulate R K from a different point of view (Nakayama and Nishiguchi 1981) from that of the AM theory (Khalatnikov 1965). The merit of this approach lies in being able to obtain the Kapitza resistance R K between finite-size system like a submicron metal particle and liquid He without much difficulty. The conventional derivation of RK will be dealt with in section 7, where the problem of the Kapitza resistance above about 1 K will be described on the basis of this formulation. Emission of sound waues from a small particle Consider the case in which a small particle with fixed center of mass is vibrating elastically in liquid He 11. This vibration causes a periodic compression and rarefaction of the density of liquid 4He near it. These propagate in the form of sound waves. The mean rate of emitted energy from the small particle is given in terms of the square of the fluid velocity u ( r ) (Landau and Lifshitz 1982), E = ipLcL Iu(r)I’ d S ,

(2.6)

where pL and cL are the mass density and the sound velocity of liquid 4He, respectively. The factor appears as a result of averaging eq. (2.6) over the period of the vibrations. The integral is performed over a closed surface surrounding the small particle. The energy carried away by sound (phonons)

KAPITZA THERMAL BOUNDARY RESISTANCE

121

is supplied by the kinetic energy of surface motion of the small particle. The velocity potential in liquid He is taken as the scalar function 4 ( r ) defined by u ( r ) =grad # ( r ) .

(2.7)

Sufficiently far from the spherical particle, having its center at the origin, the velocity potential may be written simply in the form

R being the radius of the particle. The functionf,(O, 4) is determined from the boundary conditions at the surface of the particle. By taking the gradient of eq. (2.8) in spherical coordinates, one obtains Dqb)=

exp(iqr - iqR) r2

(2.9)

The expression for the squared velocity at a distance r in terms of the surface velocity u ( R ) of the particle becomes (2.10)

Substituting eq. (2.10) into eq. (2.6) and taking the closed surface of the integration to be a sphere of radius r, the energy flux divided by 4rR2 is found to be (2.11)

where dl2 =sin 8 d e d4. At the surface of the small particle ( r = R), the fluid velocity u ( R , e , 4 ) must be equal to that of the surface motion of the small particle. Provided that the surface displacement of the small particle is defined as u ( R ) ,its time derivative is simply the velocity of the surface motion as uq(R)= Uq(R).

(2.12)

In the limit R + 00, eq. (2.1 1 ) reduces to the rate of emission of sound waves per unit area from a plane surface. As a result, the formula for the energy flux between a bulk solid and He I1 is obtained by replacing the radial component of the displacement u, by that of the z-component of Cartesian coordinates (see fig. 2.2). For deriving the formula of heat transfer, it is convenient to write down eq. (2.11) in terms of the quantized displacement (Nakayama and Nishiguchi 1981). If u,(r) represents the eigenmode

T. N A K A Y A M A

122

/Fig. 2.2. Definition of Cartesian coordinates in the limiting case of R --fa.

of elastic waves belonging to the eigenfrequency w J , the elastic waves can be quantized by replacing the amplitudes in u,(r) by the boson operators a, and a:, where J stands for a set of possible quantum numbers specifying the eigenmodes. Consider an isotropic solid with a plane surface in contact with He 11. Since He I1 has a small mass density ( pL= 0.142 g ~ r n - ~compared ) with that of a solid, the boundary conditions for the displacement field may be taken as those for a stress-free surface. Thus the displacement-field operator of a longitudinal phonon is written as u ( r )=

c eq(")Ps",

112

(a, + a:,) exp(iqll* x) cos q z z ,

(2.13)

Q

where e, is the polarization vector for a longitudinal phonon. Putting v i ( z = 0)= U i ( z = 0) into eq. (2.1 1) and taking the thermal average of the squared velocity at a finite temperature, the heat flux density & ( T ) from a bulk solid with a plane surface into He I1 is given by

O s ( ~=)I TP LTCrL [ e x p ( - P W

Ui(0)

U;(O)+I,

(2.14)

P

where H = I h o p f a , and P = 11kBT. The eigenfrequency wq in the solid equals to that of the longitudinal waves in He 11, i.e., wq = cLq, originating from the energy conservation law. Substituting eq. (2.13) into (2.14), and ) ~the Bose-Einstein distribution replacing the thermal average ( a ; ~ , by function n ( w , ) , eq. (2.14) is expressed as (2.15)

After the integration, we obtain Qs( T )=

r 2 p L c L k ;T4F 30ps v i h ' '

(2.16)

KAPITZA THERMAL BOUNDARY RESISTANCE

123

where us is the velocity of phonons in the solid and F takes the value of 2 in the present simplified case. For the eigenmodes in a half space, see the description given in section 7.2.1. Taking into account the contribution from transverse waves with polarization perpendicular to the surface, as well as surface waves localized in the vicinity of the solid surface, Khalatnikov (1965) performed a more rigorous calculation of &( T) and obtained n2pLcLk;T4F

QdT ) = 30psv:fi3

'

(2.17)

where vT is the velocity of transoerse phonons in the solid and F is a function of mass densities and phonon velocities of the solid and the liquid, respectively. For example, F takes the value of 1.57 m the case of the copper-liquid He11 interface. The numerical values of F for various materials are given in table 1 by Challis and Cheeke (1968). By differentiating eq. (2.17) with respect to T according to eq. (2.3), the explicit form of the Kapitza resistance R K is given by

RK

=

15fi3p,v: 2a2pLcLk;T3F '

(2.18)

The theory of the Kapitza resistance between normal liquid 'He and a solid was proposed by Bekarevich and Khalatnikov (1960), and Gavoret (1965), where sound energy from the solid is considered to be transferred at the interface to the collective zero-sound modes in normal liquid 'He. Taking into account the contribution from the transverse zero-sound (Fomin 1968), R K takes the form

RK=

15fi3p,v: 2n2pLk;T3(a,cL0+~ 2 c m ) F '

(2.19)

where cLoand cm are the velocities of longitudinal and transverse zero-sound in normal liquid 'He. The factors a , and a2take values of the order of unity and are expressed in terms of Landau parameters Fo and F,, whose expressions will be given in section 3.1.2. The form of eq. (2.19) implies that the physical origin of the heat transfer is identical with that of the solid-He I1 interface (eq. (2.18)). A detailed discussion of this point will be given in section 3.1.1. The Kapitza resistances R K obtained in eqs. (2.18) and (2.19) are expressed only by material constants, and are proportional to the inverse third power of temperature. In addition, it should be noted that, for bulk solids, the actual value of R K for liquid 'He becomes of the same magnitude as that of R K for He 11, since a, = 1.54, a2=0.64, cLo= 3 . 4 6 ~and ~ c w = uF, where uF= 1.8 (0atm)-3.9 (27 atm) x lo3 cm/s.

124

2.3. ANOMALOUS BEHAVIOUR

T. NAKAYAMA OF T H E OBSERVED KAPITZA RESISTANCES

2.3.1. The liquid 'He-sinrered powder interface at mK temperatures As described in the previous section 2.2, the Kapitza resistance increases with decreasing temperature. If the Kapitza resistance R K is proportional to T-' down to 1 mK as shown by eq. (2.19), the efficiency of heat exchange per unit area at 1 mK is lo9 times worse than that at 1 K. In order to overcome the difficulty of heat exchange at mK temperatures, it is necessary to make the contact area between a solid and liquid 'He large. For this reason, small metal particles with micron size are used in the heat exchangers of dilution refrigerators (Wheatley et al. 1968, Radebaugh et al. 1974, Frossati et al. 1977). which have come into wide use in low temperature laboratories. For example, powder (nominal particle size 700 A) compressed at room temperature at about 200 bar to a filling factor of 36% and then sintered at 180°C in vacuum for 30 min gives 1.3 m2/g (Rogacki et al. 1985). However, if the Kapitza resistance R K shows the T - j dependence as predicted by eq. (2.191, it is apparent that there still remains a difficulty of heat exchange at mK temperatures even using small metal particles of submicron size. Fortunately, this predicted T-' temperature law is not realized at mK temperatures as first pointed out by Avenel et al. (1973). They showed that the Kapitza resistance R K between various metals (Cu, Au, Pd) and liquid 'He observed at mK temperatures deviates considerably from the theoretical prediction of eq. (2.19) in both magnitude and ternperature dependence. This anomalous behaviour has been established by many research groups. Experimental work before 1979 is described in the excellent review by Harrison (1979), in which the data not only for various metal particles but also for magnetic insulators (CMN) are described in detail. Early data on R K did not indicate a magnetic field dependence up to several 10 mT (Andres and Sprenger 1975, Ahonen et al. 1976, Ahonen et al. 1978). However, Perry et al. (1982) reported the first instance of a magnetic field dependence (-0.8 T) of R K for a system of Pt powder packed in an equal volume of graphite powder, in contrast to the acoustic mechanism which predicts no field dependence. Subsequently, Osheroff and Richardson (1985) investigated in detail not only the magnetic field dependence but also the pressure dependence of R K between liquid 'He and compressed silver powder. They observed a strong increase in the thermal boundary resistance with the application of magnetic fields up to 9.4 T. The observed R K rises continuously with magnetic fields up to -0.2 T, but does not show a field dependence above 0.2T. This result implies that the data on R K above 0.2 T are attributed to the nonmagnetic channel. It should also be

KAPITZA THERMAL BOUNDARY RESISTANCE

125

noted that Osheroff and Richardson (1985) observed the Kapitza resistance RK to increase somewhat with applied pressure. These features will be discussed in detail in sections 4 and 5. The typical Kapitza resistances R, measured are shown in fig. 2.3, in which the data for silver particles are compared with the prediction for a bulk solid given by eq. (2.19). The common characteristic features of the data are the following: (i) Both the temperature dependence and the observed magnitude are in fair agreement with the theory (eq. (2.19)) above about 10 mK. (ii) As the temperature is decreased from 10 mK to 1 mK, the data deviate considerably from the theory for both the temperature dependence and the magnitude. Figure 2.4 shows the magnetic field dependence of RK observed by Osheroff and Richardson (1985). It should be emphasized that even the data at 0.385 T in fig. 2.4 deviate strongly from the theory for a bulk solid. It is evident that some unknown processes transferring heat effectively are present. This has been a long-standing unsolved problem in low temperature physics since its first observation by Avenel et al. (1973). The theoretical description on this subiect will be dealt with in detail in sections 4 and 5.

T(rnK1 Fig. 2.3. Thermal boundary resistance R K observed between silver particles and liquid 'He as a function of temperature. The symbols are taken from Andres and Sprenger (1975): (O), Ahonen et al. (1978): (0).The solid line represents the prediction from the AM theory for bulk silver (eq. (2.19)).

T. NAKAYAMA

I26

lo'

I

1

10

.

. . . 100

T(mI0

Fig. 2.4. Magnetic-field dependence of the Kapitza resistance R, between silver particles and liquid 'He at saturated vapour pressure. The filled circles (0)are the data of R, at zero magnetic field and the open circles (0) are R, at 0.385 T. The solid line denotes the theoretical prediction for the bulk silver-liquid 'He interface. These are taken from Osheroff and Richardson (1985).

2.3.2. The liquid He-bulk solid interface above about 1 K There is another interesting finding on the problem of the Kapitza resistance, which is quite different from the phenomenon described in the previous section 2.3.1. These are observed above about 1 K at the interfaces between liquid (or solid) He and a bulk solid. To show the deviation of the experimental data from the T-' law of eq. (2.18), the product R,T3 observed is plotted against temperature T in fig. 2.5, in which the results for solid or liquid 'He and 4He are taken from the work of the Illinois group (Anderson et al. 1964, 1966, Anderson and Johnson 1972, Folinsbee and Anderson 1973, Reynolds and Anderson 1976). The discrepancy with the AM value (eq. (2.18)) is evident above 1 K. In addition, the observed values are much less than those of the AM theory even below 0.1 K. One should note that the magnitudes of the observed R K above about 1 K are essentially the same among liquid 4He (or 'He) and solid 4He (or 'He), while below 0.1 K the magnitudes of R K for these several forms of He are quite different (see fig. 2.5). These interesting observations are of 1960-1976 and this so-called Kapitza problem above about 1 K has attracted much interest among low temperature physicists for many years.

KAPITZA THERMAL BOUNDARY RESISTANCE

Lqud

-

'He

I27

-

10'

T(K)

Fig. 2.5. Thermal boundary resistance R, multiplied by T3between liquid 3He or 4He and copper under various pressures as a function of temperature. The symbols are for liquid 'He (O), solid 3He (0).liquid 4He (A), and solid 4He (A).Curves A, B and C are for liquid 'He, liquid 4He, and solid 'He, respectively. The data are taken from the work of the Illinois group (Anderson et al. 1964, Folinsbee and Anderson 1973, Reynolds and Anderson 1976). The arrows at the right side indicate the prediction from the acoustic mismatch theory (eq. (2.18)) for an ideal Cu surface.

Although the history of this problem is longer than that found at mK temperatures (Avenel et al. 1973), the underlying mechanisms are not yet fully understood. Many good reviews on this subject have been published (Pollack 1969, Challis 1974, Anderson 1976, Wyatt 1981, Zinov'eva 1985), which report the status of investigations when these articles were written. The problem is essentially how to understand the unexpectedly large transmission of high-frequency phonons ( 3100 GHz) across the interface between the He system and a bulk solid, where the He system is liquid or solid 'He and 4He. In ordinary experiments on heat transfer, information on the polarization, frequency and propagation direction of phonons are averaged out. As tunable sources of monochromatic phonons have become available, with which phonons with frequencies in the terahertz regime can be generated (see the book edited by Bron 1985), phonon spectroscopy with a high resolution has become another promising tool for the study of phonon scattering at the liquid He-solid interface. In section 7, recent efforts to understand the problem will be surveyed with emphasis on the application of high-frequency phonon generation and detection techniques to this problem.

T. NAKAYAMA

128

3. Fermi liquid theory of the Kapitza resistance 3.1. HEATTRANSFER

D U E TO Z E R O - S O U N D EXClTATlONS

3.1.1. Emission of zero-sound from a small particle into liquid -3He Liquid 'He below about 0.1 K behaves as a Fermi liquid. The theory of a Fermi liquid, developed by Landau (1957, 1958), postulates that the classification of energy levels remains unchanged when the interaction between 'He atoms is gradually switched on. It is well known, of course, that liquid 'He undergoes the phase transition into superfluid 'He at around 1 mK as first found by Osheroff et al. (1972). The thermodynamic quantities of liquid 'He can be calculated by taking account only of weakly excited energy levels lying close to the ground state. In liquid 'He these are single-particle excitations and zero-sound excitations. Bekarevich and Khalatnikov (1960) and Gavoret (1965) applied Landau theory to the problem of the Kapitza resistance. Toombs et al. (1980) have justified this phenomenological theory from the microscopic viewpoint by taking into account both zero-sound and single-quasiparticle excitations. This section deals with the heat transfer between a solid and liquid 'He due to zero-sound excitations in terms of the Landau Fermi liquid theory, using as illustration a small particle with radius R as mentioned in section 2.2. This treatment is not only useful to discuss the heat transferring mechanism between sintered powder and liquid 'He, which is the main subject matter of this article, but also to derive simply the Kapitza resistance R K at the liquid 'He-bulk solid interface as a limiting case of a large particle with radius R + co (fig. 2.2). The quasiparticle distribution function n ( p , x, 1 ) of liquid 'He in the nonequilibrium state is a function of time r and position x (we define x as the position vector from the center of a particle). Hereafter, the spin index u will be suppressed using the definition of the momentum p = ( p , (T). Consider the situation in which the distribution function differs by a very R being small amount 6 n ( p, x, t ) from the equilibrium one n o ( p ) at 1x1> IR(, the instantaneous position of the surface of the particle. In terms of 6 n ( p, x, t ) the kinetic equation is written as a w p , x, at

1)

+ 0;

a m p , X, ar

1)

= l[n(p')I,

(3.1)

where I [ n ( p ' ) ] is the collision integral and Sri(p, x, 1 ) is defined as an0(p) 6€p.

6ii( p , x, t ) = 6 n ( p, x, t ) --

a€P

(3.2)

KAPITZA THERMAL BOUNDARY RESISTANCE

129

Here 6ep is the variation of energy for a small deviation 6 n ( p ' ) . Defining up by the following equation: (3.3) 6ep can be expanded in terms of the Landau parameters keeping the first two terms of the expansion 6ep = 1f,' 6np' = P'

c ( f i + f ;P^

'

P^') vp'8( p'-pF)/uF

P'

-6,

(3.4) where p^ = p/IpI, and F i and Fi correspond to a spherical change of volume of the Fermi sphere and distortion of the sphere proportional to cos Op, respectively. The density of quasiparticle states at the Fermi surface N ( 0 ) is defined as N(O)=rn*pF/(.rr2h3).The parameters F i and Fi take the values of 10.07 and 6.04 at zero pressure. The moments vo and v 1are defined by the following integral with respect to solid angle Op = N(0)(fGvo+ffp^*v l ) = FivO+F;v1

(3.5)

The kinetic equation (3.1) is rewritten as

a VP -+up at

- grad( up+ h e p )= I [ v,]

At low temperatures, the mean free time of collisions 7 between quasiparticles is given by 7

-- lo-''/ T2s.

When the condition 1 holds, the collision integral can be omitted in eq. (3.6). This is satisfied in the temperature regime TC0.1 K, where there exists the zero-sound collective mode. It should be noted here that the mean free time is associated with inelastic scattering between quasiparticles and is not due to elastic scattering by the boundary. This becomes important when discussing zero-sound excitations in liquid 'He immersed in sintered powder. Consider a spherical elastic particle in liquid 'He whose surface is executing vibrations (Nishiguchi and Nakayama 1982). These vibrations act as a periodic perturbation to liquid 'He and cause a change in the distribution function as

T. N A K A Y A M A

I30

where R is the radius of the particle. f ( 0 , d ) is a dimensionless function depending on the polar( 0) and azimuthal(4) angle. Substituting eqs. (3.4) and (3.7) into eq. (3.6), one has the kinetic equation at a distance far from the origin ( r = 1x1Z= R )

+

+

( p - S ) up ( F~uO F3 vI * 6 )= 0,

-

(3.8)

where p = p^ 4 = cos 0,, s is the dimensionless velocity of a collective mode defined by s = w / upq, and vFis the Fermi velocity of the quasiparticle. The direction of q is taken to be parallel to that of x due to the spherical change of the distribution function. From eq. (3.8), up becomes

By taking the direction of ulr to be parallel to 9, the vectors written in terms of the row-vector representation

6 and v, are (3.10)

and v1 = ( h r .

Vlflr

hd).

(3.11)

The result is that the change of the distribution function due to the surface vibrations of a spherical particle gives rise to

3.1.2. Heat flux from a spherical particle at temperature T 3.1.2.1. Energy current into liquid ' H e The vibrating surface of a spherical particle produces excitations in liquid 'He as described in section 2.2, and the excitations carry away energy from the small particle. The energy current takes the following form according to the Landau theory

(3.13)

where we define d7=-

V

d'p .

KAPITZA THERMAL BOUNDARY RESISTANCE

Using the relations eP = uF(p -pF) +6ep, - S ( p - p , ) / v , , eq. (3.13) yields (vp6cp

J ( x ) = t]FN(O)

up = uFi,

and

+ 6e2) -.d o p pp4v A

131

ani/aep = (3.14)

Substituting eq. (3.4) into eq. (3.14) and performing the integration, the expression for J ( x ) in terms of moments vo and v1 is given by, J ( x ) = oFN(O)[(1 +$F;)F&vl+F;vl

*

(3.15)

[vz]],

where the second-order moment [ vzlVis defined by the relation (3.16)

For spherical zero-sound, it is sufficient to take into account the radial component of energy current J ( x ) , J,(x)=uFN(o)

[

F ; ( ~ + $ F ; ) ~ ~ ~ , ,1+ F~ ;l i -r. 8, d

1

.

i ~ 2 "

Using the number- and momentum-conservation law derived from the kinetic equation (3.8), one has Jr(x) = u F N ( O ) [ F ~ s L v ~ + F I S L v : , + F ? s T ( Y:g)]. (3.17) The fluid velocity u and the number density fluctuation 6 n are related to the moments vo and v, as follows: U=/

and 6n =

Up6?lp d T = N ( O ) O F ( 1 + f F s ) V l ,

I

6np d r = N ( 0 ) v o .

(3.18)

(3.19)

Using eqs. (3.18) and (3.19), the heat current J , ( x ) can be rewritten as

(3.20)

where the velocities of longitudinal and transverse zero-sound are given by cL= SLUF and cT=sTuF, respectively. By integrating eq. (3.20) with respect to a closed surface surrounding a small particle, the averaged energy flux E becomes E. = y /1 J r ( x ) r 2 d 0 4wR

T. NAKAYAMA

132

The energy currents J at the respective positions x and R are related to each other through the equation

Using this relation the averaged energy flux E becomes

E=-

'I

47r

J,(R)dO.

(3.21)

Because of the presence of viscosity in normal liquid 'He, the appropriate boundary condition is the nonslip condition, i.e. all components of fluid velocity near the surface equal those of the surface velocity of the spherical particle. It should be mentioned here that Ritchie et al. (1987) reported the measurements of both the real and the imaginary parts of the transverse acoustic impedance of an oscillating surface in contact with dilute solutions of 'He in 4He, as well as with pure 'He with and without 4He surface boundary layer. In particular, they emphasized the effects of 4He-rich surface layer on the boundary conditions. One can put the condition that the fluid near the moving surface oscillates in phase with the surface motion of the spherical particle. The condition is expressed as na,u

= U,

where a,. = (tir, tio, ti4) is the time derivative of the surface displacement of a spherical particle. u is the fluid velocity near the surface and n the number density of liquid 'He which is taken to be the global average value. From eqs. (3.18), (3.20) and (3.21) we obtain the energy flux E taking the time average, (3.22)

Here the prefactors a , and a2 are defined by (3.23)

where p L is the mass density of liquid 'He: p I . = nmHerand cL and cT are the velocities of longitudinal and transverse zero-sound, respectively. 3.1.2.2. Kapitza conductance hK due to zero-sound excitations The energy flux Q,( T) at a finite temperature T can be obtained by taking the thermal average of eq. (3.23) and summing over all the vibrational

KAPITZA THERMAL BOUNDARY RESISTANCE

133

modes of a spherical particle as

.

.

fJ=e.4

where the angular brackets denote the thermal average at T. The suffixes ‘s’ and ‘t’ denote the spheroidal and toroidal modes, respectively. J specifies the set of quantum number (I, m,0 ) .The nature of these modes and their eigenvalue equations are described in the Appendix. The thermally averaged squared-displacement can be written in terms of the Bose-Einstein distribution function nB(w,T ) as

Substitution of the expression for uJ,[and u ~into , ~eq. (3.24) leads us to the equation

1 2

J

x [a,CL(Ai)2G~~(R)+a2cT€(As) GsB9(R)+(A:)2G:(R)II.

(3.25)

The factors A: and G; are defined in the paper by Nishiguchi and Nakayama (1982). The prefactor (21+ 1) in eq. (3.25) denotes the degeneracy due to the sum over the quantum number m. Let us consider the heat flow from a small particle at temperature T + AT into liquid ’He at T. The Kapitza conductance hK is defined by

From eq. (3.25), the expression for the conductance becomes

I 2

J

1 2

J

+a,cT€(A,) Gso4(R)+(At 1 Gt ( R ) l l ,

(3.26)

where the numerical factors a, and a2 and the velocities of zero-sound cL and C, can be calculated from the Landau parameters Foand F;. Quoting the Landau parameters from Wheatley (1979, one can estimate a, = 1.54, a2= 0.64, cL= 3.456uF, and c,= vF, respectively.

T. NAKAYAMA

134

The solid curve in fig. 3.1 shows the calculated results (eq. (3.26)) of the Kapitza resistance R K between liquid 'He and one silver particle with R =0.5 p m in radius, where the eigenvalues up to 11 = 100 are taken into account, see Appendix. One sees from fig. 3.1 that the resistance exhibits a T - dependence above about 10 mK for R = 0.5 pm and the magnitude coincides with the bulk limit. The physical meaning of this result is evident because the dominant phonons contributing to the heat transfer in this temperature range have much shorter wavelengths than the size of a panicle. As a result, the shape of the particle is irrelevant and R K approaches the bulk limit. At low temperatures, the resistance R K increases exponentially with decreasing temperature. This exponential behaviour is caused by the size effect of small particles, i.e., considering the lowest eigenvalue from eq. (3.26) the asymptotic form of R K sufficiently below 10 mK becomes

where w o is the lowest angular frequency corresponding to the spheroidal mode with spherical symmetry. The Kapitza resistance observed for silver particles with R = 0.5 pm is plotted in fig. 3.1. Above roughly IOmK, the calculated resistance for

t Fig. 3.1. Calculated boundary resistance R , as a function of temperature between liquid 'He and a silver particle with 0.5 pm radius. The observed data from Andres and Sprenger ( 1975) are given for dirty silver particles with R = 0 . 4 p n in radius ( A ) and for clean silver particle with R = 0.55 pm (0).respectively. After Nishiguchi and Nakayama (1983).

KAPITZA THERMAL BOUNDARY RESISTANCE

135

R =0.5 pm agrees well with the observed data in both magnitude and temperature dependence. At temperatures less than 10 mK, there is a large discrepancy between experiment and the calculated value. This, of course, suggests the inadequacy of the isolated particle model to explain the experimental data. 3.2. ENERGYTRANSFER

DUE TO INELASTIC SCATTERING OF S I N G L E

QUASIPARTICLES AT THE INTERFACE

In addition to the excitations of collective zero-sound modes described in the previous section 3.1, single-particle excitations play a role in energy transfer across the solid-liquid 'He interface. In this section, the energytransfer mechanism due to inelastic scattering of single particles at the interface is described, following Toombs et al. (1980). They treated this problem from the microscopic viewpoint in terms of quantum mechanical perturbation theory, and justified the conclusions of the semiclassical treatment (excitation of zero-sound is the dominant heat-transfer channel) of Bekarevich and Khalatnikov (1960) afid Gavoret (1965). Consider the case in which a quasiparticle with momentum hk is approaching or leaving the periodically vibrating surface. This process gives rise to energy transfer across the interface due to the inelastic scattering of a quasiparticle. This situation is illustrated in fig. 3.2. As a first step for clarifying the physical basis of this process, the Kapitza conductance h , between a 'gas' of 'He atoms and a solid is treated in this section. This procedure will be generalized to the case of 'liquid' 'He taking into account the interaction between 'He atoms. If the solid surface is displaced by an amount u, along the z-axis, the energy change SE per unit volume of 'He gas is expressed by (3.27)

Fig. 3.2. Schematic diagram of inelastic scattering of a 'He quasiparticle from state k to k + p by a vibrating wall. q indicates the wave number of a phonon in a solid.

136

T. NAKAYAMA

where V = L3 is the volume of 'He gas. The stress tensor IZE is the energy density as understood by putting u, = L in eq. (3.27). The second quantized form of the stress tensor HS', can be obtained by combining the z-component of kinetic energy of an 'He atom with the following field operator (3.28)

Here the boundary condition at the interface is not considered, for simplicity. The explicit form of US', is written down by taking into account the conservation law of the particle number (Zubarev 1974), (3.29)

The interaction Hamiltonian is expressed in quantized form using eq. (3.27)

(3.30) where the displacement u, is given by the z-component of eq. (2.13). Using eqs. (3.28)-(3.30),the second quantized form of the interaction Hamiltonian becomes (3.31)

Here

where qll and pIIare the parallel components of the momenta at the interface. The Kronecker delta represents the momentum conservation law parallel to the interface as a consequence of the assumption of a plane interface. A is the contact area of the interface. The z-compoment of the polarization vector of the displacement expressed by eq. (2.13) is given by eqz. The k, p ) is represented by element Mo( (3.32)

The interaction Hamiltonian Hso yields the energy-transfer process in which the absorption or emission of a phonon with momentum hq in the solid is accompanied by the inelastic scattering of a 'He atom from state k to k + p (see fig. 3.2).

KAPITZA THERMAL BOUNDARY RESISTANCE

The heat flux density gas at T is written QSG=

1

137

oScfrom the solid at temperature T + AT into ’He

AoqW(k+k+p),

(3.33)

4p.q

where the transition probability W (k -* k + p ) is given by

-&+,(I

-h ) ( I + “q 116 ( E k + p - E k - hoq )

*

Here no( T + AT) is the Bose-Einstein distribution function and fk( T ) the Fermi distribution function at temperature T. Expanding &( T + AT) with respect to a small temperature difference AT, the net heat flux can be expressed by the relation = hKAT.

Using this, the Kapitza conductance hK is obtained as (3.34)

where

(3.35)

The above eq. (3.35) can be calculated by putting the appropriate condition q < k , because the phonon energy hoq is small enough compared with the Fermi energy EF at temperatures considered here. The result is o k4 RG( oq) = lF.

(3.36)

41r2

From eqs. (3.34) and (3.36), the Kapitza resistance and a solid takes the form

RK =

1 5psv2h’ lr2pLvFkiT’ .

RK

between 3He gas

(3.37)

The numerical estimate of eq. (3.37) becomes of comparable order with the conductance hK (or resistance RK) of eq. (2.19) which attributes it to excitations of the collective zero-sound modes. It should be emphasized, however, that this is not actually the case for ‘liquid’ ’He, in which the interaction between quasiparticles plays a key role. If this interaction is

T. NAKAYAMA

138

properly taken into account, i.e., the effect of a back-flow current of excited quasiparticles, the magnitude of the Kapitza conductance hK due to the inelastic scattering of single quasiparticles is severely reduced and becomes negligibly small compared with the conductance due to collective zero-sound excitations. The physical background of this conclusion is described in the following. Considering the mutual interaction of 'He atoms, the proper stress tensor of liquid 'He consists of the sum of eq. (3.29) and the following term, 1711(X) =;uG+(x)G+(x)G(x)G(x), (3.38) where a is the strength of the interaction potential defined by V ( x )= a 6 ( x ) . The sum of eqs. (3.29) and (3.38) becomes, in terms of the Hartree-Fock approximation,

(3.39)

where n, is the number density of 'He atoms in the liquid. The factor Fo (Landau parameter) is defined as Fo= i u N ( O ) , where N ( 0 ) is the quasiparticle density of states at the Fermi surface expressed by

where m* is the effective mass of a quasiparticle. From eqs. (3.31), (3.32) and (3.39), the unscreened matrix element taking into account the mutual interaction becomes h2 + fk:Fo] . (3.40) m The matrix element M(&,p ) refers to the scattering of quasiparticles with definite momenta. In our case, the vibrating solid surface produces a quasiparticle-quasihole pair which propagates into the system undergoing multiple scattering of other quasiparticle-quasihole pairs. This dynamical screening of the liquid-solid interaction is described in fig. 3.3, where the A4 ( k, p ) = -[( kz + :p, )'

--

&+ qk A

-

Fig. 3.3. Vertex correction incorporating the screening. The angular frequency w means that o f phonons in a solid, k and k + p indicates 'He quasiparticles approaching and leaving the \,lid surface, respectively. The definition of M ( k ) ,X o ( k ) , and A(&',k ) are given in the text. After Toombs et al. (1980).

KAPITZA THERMAL BOUNDARY RESISTANCE

139

black circle is the screened matrix element fi(k, p ) and the open circle the unscreened matrix element M(&,p ) of eq. (3.40). Introducing the scattering amplitude for a quasiparticle A(&,k', p ) , the screened matrix element can be expressed as &&, P)= M ( S P) -1M ( k ' , P)Xo(k',P M k , k',PI,

(3.41)

h'

where xo(k,p) is the particle-hole propagator of the noninteracting system expressed as

The scattering amplitude A(k,k') represents a process in which two quasiparticles with momenta hk and hk' exchange momentum hp and energy hw,. The scattering amplitude takes a simple form in the case of only one-nonzero Landau parameter Fo as follows (3.42)

where A = w/(uFp) and A+l g ( A ) = 1 - f A log-. A-1

After some manipulation one obtains the screened matrix element A?( k, p ) ,

where = i - i .The effect of mutual interaction is included in the screened matrix fi(&, p). Then the proper expression of R K due to single-quasiparticle excitations can be obtained by replacing Mi(&,p) for noninteracting 'He atoms in eq. (3.35) by A?(&, p ) . Figure 3.4 shows the calculated conductances where the two contributions due to single quasiparticles hip and collective zero-sound excitations h r are compared. The abscissa is the ratio between the velocities of zero-sound cLoand the Fenni velocity of quasiparticles uF as defined by s=cL0/vF.Because the actual value of s is about 3.5, we conclude that the contribution to the heat transfer from the inelastic scattering of a single quasiparticle is negligibly small. The physical origin of this result is due to the fact that the interactions between quasiparticles are very strong, and the back-flow current of excited quasiparticles modifies the effective heat transfer.

140

T. NAKAYAMA

S

Fig. 3.4. The conductance h',P from single particles. h z from zero sound. The solid curve represents the sum of these contributions: h: = h;'+ hF. The abscissa is the reduced zero-sound velocity s = cLo/ur. After Toombs et al. (1980).

4. Anomalous Kapitza resistance between sintered powder and liquid 'He at mK temperatures 4.1. HEATEXCHANGER

USING SUBMICROMETER METAL PARTICLES

The previous section 3 has dealt with the Fermi liquid theory of the Kapitza resistance R K between an isolated particle and liquid 'He. Noticeably the anomalous behaviours of the Kapitza resistance at mK temperatures are observed for sintered (or pressed) powders. Sintered powder constitutes an important part in the dilution refrigerators, which are now widely used in the laboratories. Thus the problem of the Kapitza resistance between sintered powder and liquid 'He is of interest not only in its own right but also from practical implications. The principle of the dilution refrigerator, which was originally suggested by London (1951) and London et al. (1962), is described briefly in the following. The development of the dilution refrigerator allows us to perform experiments at mK temperatures (see the reviews by Lounasmaa 1974, Betts 1976, Andres and Lounasmaa 1982, Richardson and Smith 1987). The first successful cryostats of this type were built by Neganov et al. (1966) and Hall et al. (1966).Cooling is produced in the mixing chamber by causing 'He atoms from the upper 'He-rich phase to move across the phase-separation boundary to the lower phase, which has 6.4% of 'He dissolved in superfluid He 11. This cooling process can be understood in analogy with evaporation (quantum mechanical): namely, the upper phase corresponds to liquid and the lower phase to vapor. The process can be made continuous by circulating 'He in the system with a pump at room temperature.

KAPITZA THERMAL BOUNDARY RESISTANCE

141

A heat exchanger is a vital part of the design of a dilution refrigerator, where the incoming liquid 3He is gradually cooled by the outgoing colder liquid. As described in previous sections, the Kapitza resistance is a severe problem in the design of a successful heat exchanger, making it important to maximize the heat-exchange area. Copper sinter was originally used in a heat exchanger by Wheatley et al. (1971) in their dilution refrigerator. Radebaugh et al. (1974) measured the Kapitza resistance of various metals, and introduced 'submicron silver powder', so-called Japanese ultrafine powder, to build more effective heat exchangers. Concerning the development of research on ultrafine particles in Japan, see a review by Hayashi (1987). The advantage of using submicron metal particles lies in making the contact surface area large. For example, the surface area of 10 g silver powders with size 700 A becomes almost 13 m2 (Rogacki et al. 1985). Frossati et al. (1977) constructed a dilution refrigerator using silver powder to reach 2 mK in the continuous mode. Submicron metal particles are used for thermal contact with the 'He system down to temperatures of 0.1 mK (Guenault et al. 1983, Owers-Bradley et al. 1983). However, it should be emphasized that if the T 3law of the Kapitza resistance of eq. (2.19) is valid at mK temperatures, the use of metal particles cannot be the final solution for designing an effective heat exchanger. The history of low temperature physics tells us that it is not necessarily legitimate to apply an argument based on the extrapolation of an experimental law measured at higher temperatures (Richardson 1981). The actual case is shown in fig. 2.3, where the temperature dependence of the Kapitza resistance R K around 10 mK changes dramatically from the predicted T-' law, i.e., at temperatures less than around 10 mK the resistance RK becomes roughly proportional to T ' .Thus it is evident that the use of submicron particles leads not only to a large contact area, but also to the appearance of an anomalous temperature dependence of the Kapitza resistance. This anomalous temperature dependence, roughly proportional to T ' ,is the key element for constructing a successful heat exchanger. 4.2. SOFTPHONON-MODES

I N SINTERED POWDERS

As shown in section 3.1, the internal vibrations of a particle of submicron size are irrelevant to effective heat transfer below about 10 mK. This is due

to the finite-size effect as understood from the following simplified arguments. The lowest frequency vo of 8 particle with diameter d is of the order of uS/2d, where us is the sound velocity in the particle determined from the elastic constants. For instance, the lowest vo becomes about 2 GHz for a particle of 1 pm in diameter. If heat exchange occurs from the excitation or absorption of zero-sound due to surface vibrations of a particle, the

142

T. N A K A Y A M A

resistance will increase exponentially at a temperature lower than that corresponding to vo. This argument has been given in section 3.1.2.2. At temperatures higher than that corresponding to the finite lowest eigenfrequency, the calculated resistance exhibits a T 3dependence and the magnitude is in agreement with the bulk limit. This is because the dominant phonons contributing to the heat transfer in this temperature regime have much shorter wavelengths than the size of the particle, i.e., the particle shape is irrelevant and the resistance approaches the bulk limit. The above argument was demonstrated quantitatively in the case of isolated particles for the normal liquid 3He as shown in fig. 3.1. However, when the particles are bridged in the sintering process, new low-energy vibrational modes associated with the characteristic property of sintered metal-powder should appear (Harrison and McColl 1977). It is expected that this might play a role in effective heat transfer. Sintered powders are characterized by particle size (or mass), bridges, pores, and an oxide layer (fig. 4.1). The various properties of sintered powder have been studied by some experimental groups (Iwama and Hayakawa 1981, Rogacki et al. 1985, Hayashi et al. 1986). Rogacki et al. (1985) made a systematic study of the influence of temperature, time, pressure and atmosphere during pretreatment and sintering on the surface area, packing factor, structure, hardness, and electrical conductivity of sinter produced from 900 8, Cu powder. Iwama and Hayakawa (1981) made a detailed study on the neck growth stage in the sintering of powders of Au, Ag, A1 and Cu. They observed that neck growth is severely affected in an atmosphere of 0, in all the materials. Hayashi et al. (1986) performed Mossbauer spectroscopy to investigate the dynamical properties of the sinter. Sintered particles of Au as well as Cu-Au and Ag-Au were employed in Mossbauer spectroscopy, Au being used in order to investigate the low frequency modes in sinters. The average particle diameter was 5008, for the Au and Cu-Au sinters, and 2500 A for the Ag-Au sinters. From the analysis of the width of Mossbauer spectra, they obtained a characteristic frequency (temperature) of the sinter of -25 mK. Frisken et al. (198 1 ) suggested a characteristic frequency corresponding to the Debye frequency of about 25 mK from the measurements of elastic constants. It should be noted that these two different experiments lead to the same magnitude of characteristic frequency. The low-energy vibrational modes are produced from the elastic deformation of the bridges connecting the particles. Kingery and Berg (1955) investigated the relation among the size of the bridges, annealing temperature and heating time in the sintering process. One can deduce the size of the bridge for the sintered particles from electron microscope photographs. Taking into account the sintering conditions used in the heat-transfer

KAPlTZA THERMAL BOUNDARY RESISTANCE

143

Fig. 4.1. T h e electron microscope picture of sintered Cu powder at ( A ) 80°C and ( B ) 130°C. Characteristic necks between grains grow after sintering at 130°C. By courtesy of Rogacki et al. (1985).

T. NAKAYAMA

144

experiments the radius R and length 1 are estimated to be of the order of 1/10 of the particle radius R. The necks could be modeled by rods with cross section ma2 and length I, and the maximum frequency corresponding to the Debye cutoff frequency can be estimated approximately from the elastic deformation of a rod. As suggested by Nishiguchi and Nakayama (1983), the maximum angular frequency can be expressed from the balance condition of forces as (4.1)

where M and E denote the mass of the particle and Young’s modulus of the rod, respectively. The frequency is estimated to be of order wE/ k B = 10 mK 20 mK for a silver particle 1 pm in diameter with Young’s modulus E = 8.27 x 10” dyne/cm2 and mass density ps = 10.49 g/cm. It should be emphasized that the value estimated above from the simple argument is of the same order as the frequency obtained by Frisken et al. (1981) and Hayashi et al. (1986). Thus the particles execute periodic vibrations around their equilibrium positions and these constitute a perturbation to the liquid ‘He in this temperature region. Although the detailed structure of the vibrational density of states (DOS) of the sinter is not an important factor in discussing the Kapitza resistance at mK temperatures, one should mention the characteristics of the DOS of sintered powders, which form a three-dimensional percolating network. It is anticipated (Deptuck et al. 1985, Maliepaard et al. 1985, Page and McCulloch 1986) that this system is fractal (in a statistical sense) at shortlength scales. The nature of vibrational modes in a percolating cluster has received considerable attention in recent years (see the review by Orbach 1986). From the scaling argument, Alexander and Orbach (1982) have conjectured that the DOS’s of percolating clusters in all Euclidean dimensions obey the universal law g ( w ) - w ” ’ in the regime above the characteristic frequency w,. Vibrational modes in this regime are called fracton. There exists also the phonon regime, where the DOS follows the conventional Debye law g ( w ) - w 2 below w,. The detailed description of the DOS for sintered powders will be given in section 6.1, where the Kapitza resistance between ’He-4He mixtures and sintered powder at subrnillikeluin temperatures is dealt with in connection with the characteristics of the DOS of sintered powders.

-

4.3. HEAT TRANSFER

DUE TO THE EFFECT OF SOFT PHONON-MODES

Let us consider the energy emission due to low-energy thermal vibrations of particles in sintered powder immersed in liquid ’He. The expression for

KAPITZA THERMAL BOUNDARY RESISTANCE

the energy flux eq. (3.22)

fi

I45

from one particle is written down in the same form as

[ u e ( R ) 2 + u , ( R ) 2 dR. ]

877

(4.2) Here the prefactors a, and a2 are defined as (4.3) and (4.4) Here pL is the mass density of liquid 'He defined by pL= nmHe, where mHe and n are the bare mass of a 'He atom and the number density of liquid 'He, respectively. cLand cT are the velocities of longitudinal and transverse zero-sound. When small particles in sinter vibrate without volume change, the surface displacement u in eq. (4.2) can be replaced by the particle displacement q from its equilibrium position using the identities u:( 0 ) = qz cos

e,

u;( 0 ) = qz sin

e,

u;( 0 ) = 0.

Here q is the small displacement of a particle from the equilibrium position (see fig. 4.2). The angular integral in eq. (4.2) can be performed in the following forms,

I I

lu:l2 do1477 = fq:,

Iuil' do1477 =$&,

@

Fig. 4.2. The definition of surface displacements. See the text for details.

T. N A K A Y A M A

I46

Substituting these relations into eq. (4.2), and when all three degrees of freedom of a particle are taken into account,

E =:,pL.(a,c,+2a2cT)141'. Taking into account the contribution from all particles, we obtain 1

E =PL(u,cL+ 2a,c,) c ( Q J * A , , 6s

(4.6)

I

where A, denotes t h e surface area of the ith particle with radius R,, and S is the total exposed area of the assembly of particles immersed in liquid 'He. The heat current OF(T )from sintered powder into liquid 'He is defined by the thermal average of eq. (4.6), OE(

T)

=z 1

~L(aicL+2a,c,) C ( 4 3 ~ 1 .

(4.7)

I

Recalling the fact that the Einstein model can describe well the phonon specific heat down to one-tenth of the Debye frequency (in our case this is wE), we can take a thermal average of (4,)2 of eq. (4.7) in terms of the Einstein model to describe the acoustic channel at rnK temperatures where we are interested in. The result is

where M, is the mass of the small particle on the ith site. The final result for OF(T) becomes

where use is made of

EL=-=---A 1 8

3 MIS Mo 4nR'ps'

Here M , is the average mass of particles in the sinter. The temperature derivative of eq. (4.9) yields the Kapitza conductance h: as (4.10)

where 8 = h w E / k Band R is the averaged particle radius. The total conductance h , between sintered powder and liquid 'He is given by the sum of two contributions hK = h E + h z ,

(4.1 1)

KAPITZA THERMAL B O U N D A R Y RESISTANCE

147

where h', denotes the Kapitza conductance due to the internal vibrations of particles given by eq. (3.26). Figure 4.3 shows the comparison between the experimental data and the calculated resistance R K = 1/ hK of sintered silver powder of radius R = 0.5 p m with characteristic temperature 0 = 15 mK. The total resistance exhibits a T-' variation above about 10 mK, and at temperatures less than 10 mK the calculated one shows a rather moderate temperature dependence. The essential point is that the particles in sinter are well described by the Einstein model, because we are seeing rather 'high' temperature properties of sintered powders even in the mK temperature region, originating from the relatively large mass of particles and weak couplings among particles. Let us consider the heat transfer mechanism due to single-quasiparticle excitations taking into account the effect of pores in the sinter. When powder particles in the sinter vibrate thermally at a finite T, quasiparticles are scattered inelastically by the vibrating wall and this scattering contributes to the energy exchange. Rutherford et al. (1984) described the pores in the sinter as a set of boxes of finite sizes, each of which contains 'He quasiparticles. Each box has a mass M and is free to vibrate as a three-dimensional oscillator, where energy is transferred to the 'He quasiparticles via the 10'

-

lo5

P

Y

z lo4 Y

10'

10' 1

10 T(mK)

100

Fig. 4.3. Contribution from the acoustic channel due to the coupling between zero sound and soft modes. The solid curve shows the calculated resistance with tl = IS mK. The dashed line represents the resistance from one small particle with R = 0.5 pm calculated in section 3.1.2. The observed data are from Andres and Sprenger (1975). After Nishiguchi and Nakayama ( 1983).

T. N A K A Y A M A

148

shaking of the box. They assumed that the frequency spectrum of vibrational modes of the sinter are distributed with a constant density of states over the frequency range 0 to wD, where w D corresponds to the Debye cutoff frequency of the sinter. The expression for h , obtained (Rutherford et al. 1984) is expressed as hK

=

5.6Am*’E:k;TD , Mn2h4

(4.12)

where D and A are the density of states of shaking box modes and the interface area of the box, respectively. They postulated the constant density of states of shaking box modes as (4.13) where N is the number of powder particles of diameter d in the sinter and uD is the bulk Debye velocity. The treatment is entirely analogous with that of Toombs et al. (1980) for ’He gas, in which the Debye density of states is used for describing their formula given by eq. (3.37), i.e., the result of eq. (4.12) can be rewritten by using the Debye density of states w 2V D ( w )= 27r2v;.

(4.14)

By replacing w with the dimensionless variable k g T / h w , one has

5.6Ak:kiT’I - 8h2(M / V ) n 4 v ; ’

h -

(4.15)

where the factor I means the following integral *

x’dx 7r4 -_ exp(x)-1 1 5 ’

(4.16)

Dividing by the interface area A and using the relation k: = 37r2n, the conductance h , is expressed by (4.17)

where a is a number of order unity. This is the same as eq. (3.37) in both magnitude and temperature dependence. Especially, it should be mentioned that eq. (3.37) was derived for the case of noninteracting ’He atoms (gas). Thus the physical ongin of the T-linear dependence of hK of eq. (4.12) comes from the assumption of a constant density of states for sintered powder. As stated in section 4.2, there are two frequency regimes characterizing the DOS of the sinter: the phonon regime ( w z ) and the fracton regime ( w ” ’ ) . For the sinters considered here the crossover frequency o, becomes

KAPITZA THERMAL BOUNDARY RESISTANCE

149

of the order of 10 mK as will be estimated in section 6.1, where the Kapitza resistance of 'He-4He mixtures will be discussed. The result of eq. (4.12) should be modified at mK temperatures. In addition, it should be noted that, if the multiple scattering of quasiparticle-hole excitations in liquid 'He is taken into account, the magnitude of the Kapitza conductance expressed by eq. (4.12) is much reduced as described in section 3.2, where the dominant heat transfer channel is shown to be through the zero-sound excitations rather than through single-particle excitations (see fig. 3.4).

5. The magnetic channel of heat transfer between sintered powder and liquid 'He 5.1. SURFACE CHARACTERISTICS

OF SUBMICRON METAL PARTICLES A N D

SURFACE MAGNETIC IMPURITIES

Metal particles used in experiments, especially Ag or Cu, would be covered with an oxide layer or other adsorbed gases such as O2and H 2 0with about 100 A thickness. There is some evidence, using various techniques, that oxygen with a paramagnetic moment is strongly adsorbed in a variety of forms: 0, ,0;.and 0-on sintered silver powder (Kummer 1959, Kobayashi et al. 1972, Ido and Hoshino 1974). Nishiguchi and Nakayama (1983) have pointed out that the oxygen impurities located at or near the metal surface play a crucial role in a magnetic coupling to the 'He nuclear spins. However, due at that time to a lack of experimental data on the dependence of R K on magnetic field up to -1 T, the estimate of the areal density of the adsorbed oxygens was not reasonable. In the case of the magnetic insulator (CMN in particular), Peshkov (1964) and Wheatley (1968) suggested the possibility of direct energy transfer from 'He quasiparticles to paramagnetic atoms in CMN or vice versa. In this connection, Potter (1976) and Nakayama (1984b) anticipated that adsorbed oxygens on the surface could form a magnetic adsorbed layer, and that magnetic coupling could occur between oxygens and 'He nuclear spins. There is no doubt that studies of the magnetic properties of 'He atoms adsorbed on metals are important in connection with the magnetic contribution to heat transfer across the sintered powder-liquid 'He boundary. Saito et al. (1985) reported the results of cw and pulsed NMR experiments for 3 He atoms adsorbed on small copper particles with an average diameter of 5 0 0 w . The linewidth and line shift were measured for both the 'He and 63 Cu nuclei. Figure 5.1 shows the first derivative of the absorption signal of 'He at 0.18 K and 9.4 MHz. The sharp lines are due to the liquid 3He marker. The

I so

T. NAKAYAMA

Fig. 5.1. First derivatives of N M R absorption for 'He adsorbed on Cu particles (Cu-A and Cu-B) and the marker as unshifted 'He. The inset shows an experimental cell with a capillary side cell for 'He as an unshifted mark. After Saito et al. (1985).

upper curve is for 2.7 layers of 'He adsorbed on sample Cu-A (oxygen content 0.25*0.05 at wt%) and the lower curve is for about 1.5 layers on Cu-B. The layers were thermally annealed in the region from 2.2 to 4.2 K for half a day. The scale for the external field sweep is 0.166 mT per division for the upper curve, and 0.503 mT for the lower. As the homogeneity of the in a sphere of 1 cm diameter, the shift electromagnet is better than 5 x ( d H / H ) is detectable within an experimental accuracy of 1 x In fig. 5.1, dH,/H (from the resonance field of liquid 'He) is evaluated to be -(3.2+ 1 . 0 ) ~ and dHB/H = -(1.2*0.3)x which are listed in table 5.1. The negarioe signs are of interest. Measurements of the Knight shift (Knight 1956) and linewidth for "Cu in samples Cu-A and Cu-B were done in parallel. The observed values are also listed in table 5.1, where the shift is defined as the same as that of the Knight shift. We see that the Knight shift for decreases with increasing oxygen content, while the absolute values of the line shift and linewidth for the 3He resonance grow larger with larger oxygen content in the Cu samples. The widths become broader with increasing oxygen content for the Cu samples. A small diamagnetic shift is observed for the adsorbed 'He. The experimental results

KAPITZA THERMAL BOUNDARY RESISTANCE

151

Table 5.1 Line shift and linewidth of NMR for "'Cu and 'He adsorbed on Cu samples, named Cu-A and CU-B,respectively. (*): As reference field, the resonance of liquid 'He was used. ~~

b3cu Knight shift

Sample, CU-A (0:0.2s wt%) Sample, Cu-B (0:0.45 wt%)

Adsorbed 'He

dH,-,(0.2 K ) (x T)

Shift (*)

( x 10-3)

2.26

6.2*0.1

-(0.032* 0.01)

2.10

6.7 *O.l

-( 0.120

(x

* 0.03)

dHp.,(0.2 K ) (X IO-~T) 0.45 (2.7 layers) 0.80 (1.5 layers)

concerning the linewidth and line shift of the adsorbed 3He are summarized as follows: (i) a larger width for larger substrate oxygen content, (ii) a larger shift for larger substrate oxygen content, (iii) a negative sign of the shift. Oxygens chemisorbed or adsorbed on the substrate play an important role in these features. For feature (i), it is possible to estimate the areal density no of chemisorbed oxygen from the linewidth data in a manner identical to the work of Nakayama (1984b). Assuming a dipole interaction between the 'He nuclear spin and the electron spin of the chemisorbed oxygen, the linewidth becomes of the order of d H = zpe/f3,where z is a number of order unity representing the effective coordination number and f is the mean distance from any localized spin to adsorbed 'He. The mean distance T is related to the areal density no of localized spins by I/f2 = no. Using the observed width d H = 0.08 mT from table 5.1, the areal density no becomes of the order of l O I 4 This agrees with the number density of surface oxygen calculated with the chemical analysis shown in table 5.1, 0.45 wt%, assuming that most of the oxygen is near the surface. For features (ii) and (iii), the observed shifts are small and their signs are inverted compared with the usual Knight shift. A reasonable hypothesis for explaining (ii) and (iii) is the existence of a small demagnetization field at the position of the adsorbed 3He due to the local magnetic moments of the chemisorbed oxygen. Perry et al. (1982) have reported that the thermal resistance between liquid 'He and Pt particles depends on the applied magnetic field at 0.8 T. The theoretical analysis using the Fermi contact coupling between conduction electrons and 'He nuclear spins was made by Perry et al. (1982) by changing the roles of spins in the formulation of the magnetic Kapitza resistance described by Leggett and Vuorio (1970), i.e., the localized electron

T. N A K A Y A M A

15'

spins and the nuclear spins of 'He quasiparticles are replaced by the localized nuclear spins of 3He adsorbed on Pt and the spins of conduction electrons in metals, respectively. The heat transfer between the adsorbed 3He and quasiparticles in the bulk 'He occurs through the exchange interaction. However, Nishiguchi and Nakayama (1983), Nakayama (1984a), and Hood et al. (1987) claimed theoretically that such a strong coupling can never be derived for actual systems. The theory proposed by Perry et al. (1982) requires a srrong magnetic coupling of the Fermi contact type between 'He nuclear spins and the conduction electron spins in Pt as large as lo2 to lo' times the value of the dipole coupling between the corresponding nuclear spins. The theory to explain the observation suggested, if such a strong coupling occurs, that the magnetic coupling could produce a Knight shift for the adsorbed 'He with a magnitude of d H / H 5 x 10 -'. Since the anomalous behaviour of Kapitza resistances below 20 mK has been established not only for Pt particles but also for other metal particles, such a large Knight shift could be observable as well as for 'He adsorbed on Cu or Ag particles, if the mechanism explaining the anomalous resistances below 20 mK is the same. Paying attention to this point, Saito et al. (1985) concluded that the Fermi-contact-type coupling suggested by Perry et al. ( 1982) is absent for the 'He-Cu system. Such a large Knight shift as suggested by Perry et al. (1982) was also not observed in the measurements on 'He adsorbed on small silver particles. This is consistent with the results of the Kapitza resistance between liquid 'He and Ag particles presented by Osheroff and Richardson (1985), who observed that the magnetic field and pressure dependence for the liquid "He-silver system was in conflict with the experimental results for the Pt-'He system (Perry et al. 1982). Finally, it should be mentioned that the theoretical work on the magnetic Kapitza resistance by Leggett and Vuorio (1970), which tried to explain the experiment for CMN (Abel et al. 1966), is the prototype for many subsequent works (Guyer 1973, Mills and Btal-Monod 1974a,b, Challis 1975, Nakajima 1978, Nakayama 1984b). Especially, Mills and Btal-Monod (1974b) investigated in detail the magnetic coupling between randomly distributed electron spins in a dilute magnetic alloy and 'He nuclear spins, in which they predicted that the Kapitza resistance R K should be proportional to T at very low temperatures.

-

'

5.2. HEAT TRANSFER

D U E TO M A G N E T I C C O U P L I N G A T

mK

TEMPERATURES

Osheroff and Richardson (1985) have found a novel magnetic field dependence of the Kapitza resistance between silver particles and liquid 'He in the temperature range 1-5 mK. The main features of their results are:

KAPITZA THERMAL BOUNDARY RESISTANCE

153

(i) the thermal boundary resistance RK observed increases with increasing pressure, (ii) the observed RK rises continuously with magnetic fields up to about 0.2 T, (iii) the observed RK does not show a magnetic field dependence above 0.2 T. It should be noted that all of these characteristic features are in contrast with the experimental results of the thermal resistances for the platinum particles-liquid 'He interface (Perry et al. 1982). The following is an analysis, according to Nakayama (1986b), of the origin of the magnetic field dependence of RK between sintered silver and liquid 'He observed by Osheroff and Richardson (1985). It is natural to consider, from the experimental evidence by Saito et al. (1985) and Osheroff and Richardson (1985) and the theoretical prediction (Nakayama 1984b), that the localized magnetic impurities (oxygens in a variety of forms: 02,O;, 0, 0-, etc.) play a crucial role for magnetic coupling with 'He nuclear spins. Since the thermal boundary resistance RK observed doubles from its zero-field value at about 200 G, this characteristic field should be a measure of the broadening of the energy splitting due to randomly distributed localized spins. The spacing between localized magnetic impurities estimated from the characteristic field is about P = 5 A (Osheroff and Richardson 1985). Above 2 kG, the Kapitza resistance RK observed does not vary with magnetic field, which implies that the magnetic coupling is suppressed due to the locking of localized spins and the thermal boundary resistance above 2 kG comes from nonmagnetic mechanisms. On the other hand, the RK observed at zero field is attributable to two conduction mechanisms (magnetic and nonmagnetic) since these two mechanisms provide independent paths for the heat transfer across the boundary. As a result, one can separate the thermal conductances (the inverse of RK) at zero field as follows: hK(total)= hK(mag)+ h,(nonmag),

(5.1)

where hK(total)denotes the conductance at zero field. Using this relation, the purely magnetic conductance hK(mag) is obtained by subtracting the observed values hK(nonmag) at 3.85 kG from hK(total) observed at zero field. Figure 5.2 shows the thermal boundary resistances: RK(nonmag), RK(mag) and RK(total), which are obtained from the data at saturated vapor pressure (SVP)observed by Osheroff and Richardson (1985). It should be noted, as seen from fig. 5.2, that the &(mag) values plotted by black triangles have a dip at about 2.5mK. We understand also that at temperatures above about 4 mK the nonmagnetic mechanism is important for effective heat transfer and below 4 mK the magnetic mechanism is dominant.

I54

T. NAKAYAMA

0 0 0

D Fig. 5.2. Thermal boundary resistance (R,) between silver particles and liquid 'He at saturated vapour pressure (Osheroff a n d Richardson 1985). The black circles ( 0 )are the data for are R,(nonmag) at 0.385 T. The R,(total) at zero magnetic field a n d the open circles (0) black triangles (A)denote the magnetic contribution of R,(mag). The solid curve is the theoretical result obtained by eq. (5.9). After Nakayama (1986b).

First, let us discuss the results of R,(mag) plotted by black triangles in fig. 5.2. Hereafter, the arguments are concentrated on the case of normal liquid 'He by treating the data at SVP above 1 mK, and the superfluid phase of liquid 'He is not considered here. Provided that the 'He quasiparticle with momentum hk approaches the interface and is scattered by flipping the 'He nuclear spin due to magnetic interaction with localized magnetic impurities near the surface (fig. 5.3), the Kapitza conductance h,(mag) = I/R,(mag) is expressed as

Fig. 5.3. Schematic illustration of inelastic scattering of 'He quasiparticles by the magnetic coupling with thy magnetic impurities.

KAPITZA THERMAL BOUNDARY RESISTANCE

I55

where f ( k ) is the Fermi distribution function for the 3He quasiparticles and W,, is the transition rate of a 3He atom from an occupied state k to an empty state k’. In eq. (5.2),the localized spins contributing to the transition are expressed by the two-level system with an energy splitting A with distribution n(A). The expression for n ( A ) is important in the present analysis and we shall discuss it later. Now the transition rate W f i .from the state k is written down as 21T W,. = - / ( M + A

1, k‘lH’lk, M ) 1 2 6 ( ~ k ,Q - + A),

(5.3)

H’ being the interaction Hamiltonian of the two-level system. The magnetic coupling between 3He nuclear spins and the electronic spins is taken to be dipolar type expressed as V=

cB I d x I d y d a B ( x - y ) s , ( x ) s B ( y ) ,

0.

a,~=x,y,z,

(5.4)

where duB denotes the dipole interacting between the nuclear magnetic moment pn of 3He atom and pe of localized electron spin, which is (5.5) daa (1 r 1) = [ pepn(r26aa - 3 rarB )I/ r 5 . In eq. (5.4), s(x) and S ( y ) are the spin densities for 3He nuclear spin and localized electron spin. By substituting the relation S p ( y )= 1,S,(y -I?”) and the second quantized form of s,(x), the dominant interaction Hamiltonian becomes,

where a: and ak are the creation and annihilation operators for 3He atoms with the momentum trk and spin c.The factor V is the volume of a half space occupied by liquid ’He. The symbol S: = S ; +is’, expresses the raising operator for the spin state characterizing a localized state at site R, in the effective mean near the surface. The factor J(k, k‘) is the Fourier transform of dipole interactions in a half space. It has been shown (Mills and BCalMonod 1974a) that the dipole interaction behaves like an effective contact interaction when the heat exchange is dominated by scattering with momentum transfer of the order of the ’He Fermi momentum p -pF. Therefore one can take J ( k , k‘) to be the contact type by setting the dipole interaction erg cm3. d ( x ) = J 6 ( x ) , where J takes the value of 0.99 x Substituting eq. (5.3)into eq. (5.2), one obtains the Kapitza conductance hK as (Nakayama 1984b, 1986b), n(A)A3dA hK = (5.7) exp(A/ k B T )- exp( -A/ k B T )’

T. NAKAYAMA

156

where use is made of

The exchange enhancement effect KeR of quasiparticles (Stoner enhancement) is included through eq. (5.7), which increases the conductance h K = RK'by one order of magnitude for pure 'He. Now let us describe the nature of the distribution function n ( A ) in eq. (5.7) according to Nakayama (1984b). The important point in the random spin system (dipole spin glass, for example) is the potential energy as a function of the simultaneously specified orientations of all of the spins. In this connection, the specific heat of the random spin system is well described by assuming the two-level system as shown by Villain (1979), which implies that the transition involves the simultaneous rearrangement of a small number of spins. The width of the energy distribution function n ( A ) of the above-mentioned two-level system can be estimated as 5 = z& i', where z is a number of order unity representing the effective coordination number and i is the mean distance from any localized spin to the nearest one. This estimate of is reasonable since the effective field acting on any given spin is dominated by the few spins which happen to be the closest (fig. 5.3). Following the discussion by Villain ( 1979), the non-vanishing distribution function at A = O is assumed. The energy distribution function n ( A ) in eq. (5.7) is taken as a Gaussian such as

a

n( A ) =

__ A J ~ no

exp( -A2/ A ' ) ,

(5.8)

where no is the areal density of the two-level system near the surface in a projected mean. Combining the density no and the width L,we can write the distribution function n ( A ) as a function of one variable from down A = zp:ny2. By replacing the variable A in the integral of eq. (5.7) with a dimensionless one x = A / k , T , one can find the characteristic feature of the temperature dependence of the resistance. Using a dimensionless variable x, eq. (5.7) becomes

The dimensionless width A / k , T in the distribution function in eq. (5.9) varies with the temperature, namely, the width shrinks with increasing temperature. Combining the temperature dependence of the integral with

KAPITZA THERMAL BOUNDARY RESISTANCE

157

T 2 in the prefactor in eq. (5.7), one can obtain numerically that RK has a minimum around T, = 6/2.5ks for the Gaussian distribution. By using the areal density of magnetic impurities estimated as no= 1/(5 A)’, we can evaluate the temperature T, as 2 3 mK where the resistance RK(mag)has a minimum. Here the relation 6 = zpczn;” is used. It should be emphasized that this T, is in good agreement with the temperature where the observed RK(mag)has a minimum as seen from fig. 5.2. At higher temperatures than T,, we have a simple expression for the resistance from eq. (5.9) as

-

(5.10) erg cm3, m* = By taking suitable values for the factors J = 0.99 x 3.01mHe, kF=7.86x lo7 cm-’, K:,= 12, and n0=4.6x 1014cm-2,we find RK(mag)Z0.47X 1O8T(m2K/W).

(5.11)

This agrees with the empirical relation above 2.5 mK ( R K= 0.45 x lo8 T(m2K/W)) plotted in fig. 5.2 by black triangles. At temperatures much smaller than T,, RK(mag) is very sensitive to the shape of the energy distribution function close to A =O. If a simple Gaussian distribution function is taken, one has RK(mag) T P 2 .The theoretical curve obtained from eq. (5.9) is shown by a solid line in fig. 5.2, where the values of parameters noted below eq. (5.10) are used. In view of our simplified model the agreement should be regarded as satisfactory. It is concluded here that the characteristic behaviour of the magnetic resistance R,(mag) observed is well described by the dipole coupling between randomly distributed magnetic impurities and ’He nuclear spins. As remarked earlier, the nonmagnetic contribution to the thermal conductances h,(nonmag) should be identified as the observed conductances above 2 kG because there is no sign of field dependence there. It should be emphasized that the observed data still deviate appreciably from the theory for bulk solids in both magnitude (about 10-lo2 times smaller) and temperature dependence (see the data at 3.85 kG and zero field are shown in fig. 2.4). This discrepancy could be explained from the coupling between the low-lying vibrational modes in the silver-particle system and the collective excitation in liquid 3He as described in section 4.3 (see fig. 4.3). The validity of the soft-mode picture for our particle system has been confirmed by recent investigations (Maliepaard et al. 1985, Deptuck et al. 1985, Page and McCulloch 1986, Lambert 1985, Burton and Lambert 1986). It should be noted that Osheroff and Richardson (1985) observed the pressure dependence of RK for high magnetic fields near 1 mK (see fig. 3 in their paper): they found that RK at 29 bar is somewhat larger than the

-

158

T. N A K A Y A M A

observed value at SVP. This experimental finding seems to reflect the existence of the superfluid phase of liquid 'He, since at such high magnetic fields the acoustic coupling is dominant and this coupling is weakened by the superfluidity (slip boundary condition). To summarize, the anomalous thermal resistance between metal particles and liquid 'He observed in the mK region are attributed to two compatible magnetic and nonmagnetic mechanisms, i.e., the origin of unexpectedly small thermal resistances is not unique but comes from both magnetic and nonmagnetic mechanisms in the mK region. In particular, it should be emphasized that the experiments by Osheroff and Richardson (1985) have revealed for the first time the magnetic and nonmagnetic contribution of the anomalous thermal conductance between small particles and liquid 'He at mK temperatures. Finally, we should give a comment on the Kapitza resistance between sintered powder and solid 'He. In this case both of the magnetic and the acoustic coupling are important as well as the case of liquid 'He. At high magnetic fields, the magnetic coupling is frozen and the acoustic coupling becomes dominant. Because the velocity of acoustic phonons in solid 'He (which transfer heat) is close to that of zero sound in liquid 'He, the same temperature dependence and magnitude of R K as that for liquid 'He are expected for solid 'He at high magnetic fields. In fact, these tendencies were observed by Mamiya et al. (1983) and Greywall and Busch (1987). For the magnetic channel, heat is transferred by spin diffusion in solid 'He (Guyer 1973, Morii et al. 1979).

6. Thermal boundary resistance between liquid ' H c + ~ H ~mixtures and sintered powder

The cooling of a dilute 'He-4He solution down to a few tens of FK has been of great interest associated with the possible discovery of the superfluid transition in the dissolved 'He. This cooling is, however, very difficult due to the problem of the Kapitza resistance between the refrigerant and the helium sample. To date, the lowest temperature of a mixture achieved using sintered Ag powder is close to 200gK (Chocholacs et al. 1984, Ishimoto et al. 1987). The Kapitza resistance R , between the mixture and Ag sinter at mK temperatures has been observed by many researchers (Frossati 1978, Osheroff and Cotruccini 1981, Ritchie et al. 1984, Chocholacs et al. 1984). The results indicate that the observed R K between dilute solutions of 'He in 4He and sintered Ag powder are proportional to T-I. The 3He-4He mixture represents a unique Fermi liquid, whose density, and hence degeneracy temperature, can be varied at will. Noticing this point, it has

KAPITZA THERMAL BOUNDARY RESISTANCE

159

been pointed out (Ritchie et al. 1984, Chocholacs et al. 1984) that the observed R K between dilute solutions of 'He in 4He and sintered Ag powder vary as T;' (see the data given in figs. 6.2 and 6.3 in section 6.2). It is natural to consider two independent channels of heat transfer for the system. These are due to acoustic and magnetic mechanisms, which will be discussed in the following subsections (Nakayama 1988). 6.1. Acousric

CHANNEL

In the case of normal liquid 3He, the collective mode (zero-sound) has played a role in opening up the acoustic channel, as shown in section 4.3. For 3He-4He mixtures, one should notice that the collective modes (phonons) in a 3He-4He solution do not couple with the vibrational modes of sintered powder, since at such low temperatures the metal particles of micrometer size in the sinter vibrate without volume change of individual particles (due to the size effect), i.e., the low-energy vibrational modes of the sinter do not excite phonons into the 'He-4He solution due to the slip condition at the boundary (in the case of liquid 'He, the nonslip condition leads to the excitation of zero-sound). The plausible acoustic channel is via the direct interaction between 3He quasiparticles and the low-energy vibrational modes in the sinter, which has been suggested by Rutherford et al. (1984).

-

Sintered powders with packing fraction of about 0.4 0.5 are normally used for facilitating the cooling of the dilute 3He-4He mixture into the mK and sub-mK temperature range, where the sinter forms a three-dimensional percolating network and takes a fractal structure at shorter length scale than some characteristic length 6 (in a statistical sense). It is suggested that the sintered powder is fractal at short-length scales (Deptuck et al. 1985, Maliepaard et al. 1985, Page and McCullough 1986). Alexander and Orbach (1982) have conjectured from the scaling argument that the density of states (DOS) of percolation clusters for all Euclidean dimensions exhibits universally the w dependence in the regime above the characteristic frequency w,. This regime is called thefracton regime (localized mode) or short-lengthscale regime. There is also the regime called the phonon regime or longlength-scale regime, where the DOS obeys the conventional law D ( w ) w 2 sufficiently below w, (fig. 6.1). This conjecture has been confirmed by computer simulation (Grest and Webman 1984, Yakubo and Nakayama 1987). Lambert (1985) has made a computer calculation of the DOS for sinters with varying percolation density p. The effective medium theory (Demda et al. 1984) predicts that the DOS is proportional to w 2 below some characteristic frequency w , and becomes constant above w , , which is

"'

-

T. NAKAYAMA

I60

* U C

OD

Fig. 6.1. The expected density of states (DOS) of percolating cluster. w, is the crossover frequency between the Debye density of states ( w ’ ) and the ‘fracton’ density of states ( w ” ~ ) .

expressed as w<wc,

D ( w ) = ( : ) * 3 w D9N - 2w,’.

D ( w )=

9N 3 w D - 2w,

;

W>Wcr

where the frequency oDis the cutoff frequency corresponding to the mode with wavelength equal to an inter-particle distance and N is the number of metal particles per unit volume, respectively. Putting w, = 0, eq. (6.2) yields the same density of states as eq. (4.13). From the above arguments, one may conclude that the characteristic frequency w , of the sintered powder is an important factor for predicting the power of the temperature dependence of the Kapitza resistance. It should be emphasized also that the percolation clusters become elastically soft as a function of the percolation density p. This is involved in the sound velocity vSinof the sintered particles, i.e., the magnitude of the sound velocity becomes as small as one-quarter or less compared with the corresponding bulk values (Robertson et al. 1983). One must note, from the experiments on ultrasonic propagation (Robertson et al. 1983, Maliepaard et al. 1985, Page and McCulloch 1986), that sound with frequencies of several 10 MHz can be propagated through the sinter. This implies, at least, that the characteristic frequency w , is larger than several 10 MHz, which corresponds to Rw,/k,> 1 mK. Let us estimate the characteristic frequency w , for a sintered powder with packing fraction f = 0.5, which is assumed to be composed of spheres (1000 A in diameter). This choice of the numerical values is quite reasonable because the sintered powders used in the experiments are made from silver particles with the nominal size 700 A. Provided that the system takes a

-

KAPITZA THERMAL BOUNDARY RESISTANCE

161

simple cubic composed of spheres (1000 A in diameter), this has the filling factor fo and packing fraction f is given by

f = nu?J= f o p ,

(6.3)

where u and n are the volume of one particle and the occupied number of particles in unit volume. The percolation density is defined by p. From eq. (6.3), p is estimated to be 0.96 for the sinter off = 0.5. Using this value of p and taking into account the dimensionality of the system, the average distance between vacant sites becomes -3a (this was estimated from the relation a ( l -p)-"'- 3a). As a result, the approximately estimated characteristic length 5, becomes of the order of 5000 A. This length is related to the characteristic frequency Y, by v,= u s / & . Since the sound velocity of the sinter with packing fraction f 0.5 is us = los cm/s (Robertson et al. 1983), we have Y,- 2 GHz, which corresponds to a few tens of mK in a temperature scale. We should note here that the value of T, = 15 mK recovers well the acoustic channel of the Kapitza resistance for pure 'He at mK temperatures as shown in fig. 4.3. Thus, at subrnillikeluin temperatures, the Debye-phonon picture eq. (6.1) is valid for sintered powders composed of the particle size of around 1000 A. The dissolved 3He atoms confined in pores diffuse into the adjacent pores through narrow connecting channels. In this process, the sizes of the pores and channels become an important factor for the diffusion of 'He atoms. The following is a discussion of the energy spectrum of 3He atoms in pores and the diffusion of 'He atoms through channels; special attention is paid to the size of pores and channels (Nakayama and Yakubo 1987). Pores in a sinter have irregular shapes and constitute a random network as seen from electron microscope pictures (fig. 4.1). Consider, first, the special case where 'He atoms are confined in small pores of l 0 0 A in diameter which are connected by cylindrical channels of 7 A in diameter and 10 A in length. The zero-point energy Eo of 'He quasiparticles in the channel becomes much higher than that in a pore, namely, E,= 1.41 K for d = 7 A. Since the Fermi energy EF of a 5 % 'He-4He mixture becomes 0.33 K, 'He atoms must tunnel through the potential barrier with a height of I K in order to diffuse into the adjacent pore. The tunneling probability 1/r is estimated to be lo5s-' in this case (Nakayama and Yakubo 1987). The condition for the localization of 3He quasiparticles w r > 1 in the pore holds for the temperature region around 1 mK, where w is the dominant angular frequency of thermal vibration of the sinter at temperature T. In the actual sintered powder used in the experiments, the average pore size D is of the order of 1000 or more, and the size of channels connecting pores is of the order of one-tenth of 0, i.e., 100 A. For this situation, 'He atoms can move freely through the narrow channels in contrast with the

-

-

-

T. N A K A Y A M A

162

above case. Thus the dissolved ‘He atoms in the sintered powder can be treated as free particles with a continuous spectrum. The formula for the Kapitza conductance hK obtained in eq. (3.34) can be applied to our system. Since we are interested in the millikelvin or submillikelvin temperature range, the phonon density of states of eq. (6.1), which is proportional to w ’ , should be taken into account. In this case, the conductance becomes, using the formula for the Fermi ‘gas’ derived in eq. (3.37),

h K=

cktktT’

(6.4)

15h2p,o:,,’

where c is a number of order unity and ps is the mass density of the sinter, respectively. The Kapitza resistance ( h K = 1/ R K ) is proportional to T-’ and TL’ ( - kE4) in contradiction to the experimental results. The magnitude of h , at 1 mK is also too small to explain the data. It should be noted here that us,, is the velocity of sound in the sinter, and eq. (6.4) is identical with eq. (4.17) when the mass density pL in eq. (4.17) is replaced by that of dissolved ’He atoms. 6.2. MAGNETICC H A N N E L

It has been reported that the Kapitza resistance R , of the interface between dilute ’He-4He mixtures and sintered powder varies as R K T in the millikelvin temperature region (Radebaugh et al. 1974, Frossati 1978, Osheroff and Corrucini 1981, Ritchie et al. 1984, Chocholacs et al. 1984). In addition to the above temperature dependence, Ritchie et al. (1984) and Chocholacs et al. (1984) have investigated the effect of varying the ‘He concentration in the solution, that is, the Fermi temperature TF, on R K . The results indicate that R K is proportional to T i ’ (fig. 6.2). These T-* and T i ’ dependences are not recovered from the acoustic channel mechanism in the submillikelvin temperature region as presented in section 6.1. In this section, let us consider another mechanism of heat transfer between ’He dissolved in He I1 and sintered powders. This is due to the magnetic coupling between ’He quasiparticles and magnetic impurities (i.e., 0 2 ,O ? , 0, 0-, etc.) in the vicinity of the surface of sintered powder (Nakayama 1984b), which has played an important role in the case of pure ‘He-sintered powder interfaces (see section 5 ) . One might suppose, at first sight, that magnetic coupling is irrelevant to the heat transfer due to the fact that ‘He atoms are preferentially adsorbed at the surface of the sinter, and interrupt the effective coupling. This is not always true for the following reasons. For example, even for the pure ‘He-sintered powder interfaces, the first few ’He adsorbed layer are localized at the interface and they are not important for direct coupling (dipole

-

KAPITZA THERMAL BOUNDARY RESISTANCE

I 10

20

I

I

50 100 200

500 1000 2000

163

I

TF(~K) Fig. 6.2. The Fermi temperature dependence of observed R, between a ' H e 4 H e mixture and silver particles. After Chocholacs et al. (1984). The data are taken from Ritchie et al. (1984): (A),Osheroff and Corruccini (1981): ( O ) ,Frossati (1978): ( O ) ,Chocholacs et al. (1984): ( A ) .

coupling) between 'He quasiparticles in bulk liquid 3He and localized magnetic impurities. That is, although adsorbed 'He atoms play a role as a second-order process in the perturbation theory, they do not contribute to the direct energy exchange (the first order) from the sinter into liquid 'He or vice versa. In addition, the observation of the TFdependence of R K for 3He-4He solutions is the clear evidence of the irrelevance of the first few adsorbed 4He layer to the heat transfer. This result is reasonable because the magnetic dipole coupling is proportional to the distance F 3and is not a short-range one. From the above argument one can apply the formula R K of eq. (5.9) to the case of the heat t.ransfer between dissolved 'He atoms and the sinter as well. The Kapitza resistance R K due to the magnetic coupling between dissolved 3He nuclear spins and magnetic impurities is expressed, using formula (5.9), as

In the above formula, the Stoner enhancement factor K c f l in eq. (5.9) is omitted because the exchange interaction between 3He quasiparticles in 4He is negligibly small. The energy distribution function n ( A ) in eq. (6.5) should take the same form used in section 5, which is expressed as

n0 n ( A) =7 exp( - A 2 / Z 2 ) , A&

164

T. N A K A Y A M A

where the width d of the distribution function n ( A ) can be estimated in a similar manner to that in eq. (5.8),

A= &if3,

(6.7)

where z is a number of order unity representing the effective coordination number and i is the mean distance from any localized spin to the nearest one. The width of the distribution function d / k , T in the integral of eq. (6.5)varies as the temperature. Combining n ( A ) with the factor x3/(ex- e-") &) in eq. (6.5),the integral can be easily obtained as r ~ ~ 7 ~ ~ / ( 4atdsufficiently low temperature T < d/k, where the width d takes a value of 5 mK as estimated in section 5. As a result, the Kapitza resistance'due to the magnetic coupling between dissolved 3He atoms and the sinter in the submillikelvin temperature region is expressed as

-

It should be emphasized that eq. (6.8) varies as T P 2and T,' (- k F 2 ) whose , dependences are in agreement with the experimental features reported by Ritchie et al. (1984) and Chocholacs et al. (1984). By taking suitable values for the factors for a 5% solution, J = 0.99 x erg cm3, m* = 2.46 m 3 , loa

5E

Y 106 l o 7 r

v h

cr'

Osherof f - Corruint 8 0 % Frossati et al 6 L */. Ritchie el d 0 3 o/. Rild7ie et d 1 2 .I.

0

A

%!I

'

'

"""' 1

'

'

"'.'.I

10

'

'

h , , d

100

T (mK) Fig. 6.3. Thermal boundary resistance R , between a 3He-4He mixture and Ag-sinters as a function of temperature (Frossati 1978, Osheroff and Corruccini 1981, Chocholacs et al. 1984, Ritchie et al. 1984). Straight lines are theoretically calculated by eq. (6.8) (Nakayama 1988).

KAPITZA THERMAL BOUNDARY RESISTANCE

165

EF= 4.57 x lo-’’ erg, and no = 4.0 x loi4cm-2, we find R K = 32 T-2 (m’ K/W) in the submillikelvin temperature region. This result is plotted by the straight line (below) as well as the experimental data in fig. 6.3. The upper line is for a 1.3% solution: RK=80 T-’(m2 K/W). It is remarkable that the observed T-’ and T;’ dependences of RK are recovered by the magnetic coupling model. If the magnetic coupling is a dominant channel between 3He-4He mixtures and sintered powders, the similar magnetic field dependence on RK with that for pure 3He will be observed. Although it is reported by Ritchie et al. (1984) that the observed R K for 3He-4He mixtures is insensitive to the magnetic field (-3 T) above 5 mK, it seems that this point requires further experimental investigation; especially the data at submillikelvin region are interesting.

7. Anomalous Kapitza resistance between liquid He and a bulk solid above about 1 K

7.1.

KAPITZA CONDUCTANCE

hK

A N D PHONON TRANSMISSION

COEFFICIENT ACROSS THE INTERFACE

The problem of the anomalous Kapitza resistance observed in the temperature range 1 K-2 K has a long history compared with that described in previous sections (Beenakker et al. 1952, White et al. 1953, Fairbank and Wilks 1955, Dransfeld and Wilks 1958, Challis et al. 1961, Kuang Wey-Yen 1962). Experiments on RK between liquid (or solid) helium and copper are described in section 2.3.2 and some typical results have been shown in fig. 2.5. The observed data deviate significantly from the acoustic mismatch prediction (RKT3= const.) above about 1 K. Thus, the anomalous Kapitza resistance observed above about 1 K presents an interesting problem as well as that in the mK temperature region. In section 2.2, the theory of Kapitza resistance has been dealt with from a different point of view from the conventional treatment by Khalatnikov (1952, 1965) and Little (1959), which incorporates the transmission coefficient t ( 0 ) of a phonon incident at angle 8 across the interface. The quantity t ( 0 ) is determined by applying the boundary condition at the interface, where the effects of the total or internal reflection of bulk (B) phonons as well as the surface waves propagating along the interface are involved. How the transmission coefficient t ( 0 ) is introduced in the AM theory is shown briefly in the following. When a phonon with energy h o is incident from liquid helium into a solid, an energy h o f L S ( 0is) transmitted to the solid. As a result, the heat

T. N A K A Y A M A

I66

flux

o,

from liquid He into a solid is defined by

where the angular integral ( 0 ) should be performed in the half space. The Bose distribution function for phonons is expressed as nB( f i w / k B T )in eq. ( 7 . 1 ) , and cL is the velocity of phonons in liquid He. Assuming the simple Debye density of states for phonons, which is valid at temperatures considered here, the integral of eq. (7.1) yields 0L.S

=

7r'pLkicLT' FLS, 30psh3V:

(7.2)

where the factor FLsis given bv

F,

=

ps 2Pl.

('> ' jo'rLS(

0 ) cos OL d(cos 0,).

(7.3)

CI.

If the transmission coefficient r ( 0 ) is not a function of frequency w , the heat flux will be proportional to p,and the product R K T 3 has no adjustable parameters, and hence is constant. From the data plotted in fig. 2.5, one sees that there are two distinct temperature regions in which the physical properties of R, are quite different, namely, T s 0 . 1 K and T a 1 K, i.e., there is a decrease by one order of magnitude in the quantity R,T3 as the temperature is increased from 0.1 K to 1 K. Characteristic features are as follows: ( i ) Above 1 K, RK is essentially the same for liquid 'He or 'He, solid 'He or 'He, while below 0.1 K the magnitude of RK for those several forms of He are quite different (see fig. 2.5). ( i i ) RK is very sensitive to surface treatment (Challis et al. 1961, Kuang Wey-Yen 1962, Johnson and Little 1963, Anderson and Johnson 1972, Folinsbee and Anderson 1973, Opsal and Pollack 1974, Synder 1976, Rawling and van der Sluijs 1978, 1979). As seen from eqs. (7.1)-(7.3), the Kapitza conductance h,, defined by h K = aOLs/aT, involves information on the transmission coefficient f ( 0) through the factor FLs.Defining the average coefficient by the equation

oLs

f1.s

=2

1,)'

~ L S ( ~ Lcos )

0L 4 c o s 0 d ,

(7.4)

the conductance h , can be expressed approximately as

hK/ 7'

= aiLs,

where a takes a constant value determined by material constants.

(7.5)

KAPITZA THERMAL B O U N D A R Y RESISTANCE

167

In fig. 7.1, the Kapitza conductances h K divided by T 3 (eq. (7.5)) are plotted from the data of the Illinois group (Anderson et al. 1964, 1966, Anderson and Johnson 1972, Folinsbee and Anderson 1973, Reynolds and Anderson 1976). The arrows indicate the theoretical values predicted by the AM theory for various phases and isotopes of helium. From the same figure, one can conclude that the transmission coefficient fLs depends strongly on temperature T above -0.1 K. This implies that the average transmission coefficient fLsdepends on the frequency w of incident phonons, in contrast with the AM theory. It is generally believed that, below 0.1 K, the modified AM theory incorporating phonon attenuation is .valid; in particular, the attenuation of Rayleigh waves plays a role in modifying the AM theory for ideal surfaces (Andreev 1962, Peterson and Anderson 1972, Haug and Weiss 1972). This point has been confirmed experimentally by Zinov'eva (1978,1980) and Zinov'eva and Sitnikova (1983). Since a in eq. (7.5) is proportional to pLcL, it is expected, if the acoustic mismatch picture is valid, that the data for h K / T 3depend on whether the He is solid or liquid. However, experimental data above 1 K show that the value of h K / T 3becomes almost the same for solid He and liquid He. These experimental facts suggest that some unknown mechanisms of effective heat transfer are present in this temperature range.

:?:.*:-i

Liqrid 4Hc--c

,

'O-", 01

1.0

T(K)

Fig. 7.1. The Kapitza conductance h , divided by T3as a function of temperature. The definition of symbols is the same as that in fig. 2.5. The arrows at the right side indicate the prediction from the acoustic mismatch theory.

T. N A K A Y A M A

168

-

At around 1 K 2 K, where the Kapitza resistance (or conductance) shows anomalous behaviour, excited phonons have a frequency range of 80- 160 GHz because their frequency is related to T through hw,,, = 3.8kBT. The average transmission coefficient above about 1 K, i.e., above 80 GHz, can be estimated from fig. 7.1 as

i= 0.1.

(7.6)

This value is two orders of magnitudes larger than the value from eq. (2.5) obtained by applying the acoustic mismatch boundary conditions. Thus, the Kapitza problem around 1 K-2 K is linked closely with the explanation of the anomalous features of high-frequency phonon transmission across the liquid He-solid interface.

7.2. HIGH-FREQUENCY PHONON HE-SOLIDI NT E R F AC E

SCAITERING

AT THE L I Q U I D

In the preceding section 7.1, it has been shown that the Kapitza conductance h , involves information about the transmission coefficient t ( 0 ) averaged over the polarization of phonons, propagation direction, and frequency w. In order to understand the origin of the anomalous Kapitza resistance, it is more desirable to observe directly the transmission (or reflection) coefficient t ( 8, w ) of high-frequency phonons with definite polarization, propagation direction, and frequency. In recent years, with the advent of high-frequency-phonon generation and detection techniques, it has become possible to study the scattering of phonons of known polarization, frequency and propagation direction at the interface. The generation of high-frequency phonons utilizes various techniques such as heat pulse, tunneling junction, and thermal conduction etc. (see the book edited by Bron 1985). Using these techniques, a great deal of effort has been devoted to the study of the phonon transmission mechanism at the liquid He-solid interface. Early attempts were made by Trumpp et al. (1972), Guo and Maris (1972,1974), Sherlock et al. (1972), Ishiguro and Fjeldly (1973), Swanenburg and Wolter (1973), and Kinder and Dietsche (1974). The heat pulse technique has the advantage that, through the time of flight analysis of a heat pulse, one can identify which modes and frequencies play a role for effective phonon transmission. Using the heat pulse techniques, Taborek and Goodstein (1979, 1980) succeeded in distinguishing clearly the phonon modes when reflecting at sapphire surfaces in contact with liquid 4He (fig. 7.2). Heat pulse experiments have confirmed that diffuse signals in phonons reflected from a surface are severely affected by placing liquid He (rather than vacuum) at the interface (Horstman and Wolter 1977, Weber et al. 1978, Folinsbee and Harrison

KAPITZA THERMAL BOUNDARY RESISTANCE

169

TimeIpsl

Fig. 7.2. Time-of-flight signals due to backscattering of phonons at a sapphire surface. The dotted line indicates the reduction of diffuse signals at the surface in contact with liquid 4He. The numbers 1.2 and 3 correspond to longitudinal (L),fast transverse (FT),and slow transverse (ST) phonon, respectively. After Taborek and Goodstein (1980).

1978, Taborek and Goodstein 1980, Marx and Eisenmenger 1981, 1982, Basso et al. 1984, Eisenmenger 1986). Important experimental facts obtained using these new techniques are summarized as follows: (i) Transmitted phonons into liquid He obey a cosine (Lambertian) law (Wyatt et al. 1974, Mills et al. 1975, Sherlock et al. 1975, Wyatt et al. 1976, Wyatt and Page 1978). See fig. 7.3. (ii) Surface irregularities are necessary for effective energy transfer (Weber et al. 1978, Basso et al. 1984, Mok et al. 1986). See fig. 7.4. (iii) Anomalous transmission into liquid He occurs above about 80 GHz (Sabisky and Anderson 1975) and there seems to exist an isotope effect between 3He and 4He (Koblinger et al. 1983, Heim et al. 1983). See fig. 7.5. (iv) Diffuse signals in time-of-flight reflection experiments: signals are severely affected by placing liquid He instead of vacuum at the interface. See fig. 7.2. As a first step, one must clarify the origin of diffuse signals observed in phonon reflection experiments at surfaces nor in contact with liquid He (Nakayama 1985, 1986a). Klitsner and Pohl (1986, 1987) have made some extensive experimental studies on the cause of diffuse scattering at Si-crystal surfaces by deposited thin films (without liquid He), through measurements of the thermal conductivity in the boundary scattering regime. Their results will be discussed at the end of this section along with the theoretical analysis of the cause of diffuse scattering.

T. NAKAYAMA

Fig. 7.3. The angular dependence of phonon emission from the cleaved surface of NaF to liquid ‘He at a heater temperature of 2.1 K. The measurements were made at 24 bar. After Wyatt and Page (1978).

Fig. 7.4. Phonon pulses reflected from LiF surfaces: dashed-dotted line, before cleaving in vacuum; solid line, freshly cleaved at I K; dotted line, with helium. After Weber et al. (1978).

KAPITZA THERMAL BOUNDARY RESISTANCE

171

Phonon Frequency [ G H t I 50

100

1

I

150

200

250

,

I

1

I

85 GHz

R

1 L

0 02

1

OC

T=lK I

I

06

I

I

I

08

1

10

I

12

Phonon Energy CmeV1 Fig. 7.5. Effective reflection coefficient R / R , as a function of phonon frequency. R is normalized to unity at 75 GHz ( R J . A strong breakdown of the phonon reflectivity at the ~ o l i d / ~ H e boundary occurs when the phonon frequency exceeds 85 GHz. The experimental situation is illustrated in the inset. After Koblinger et al. (1983).

7.2.1. Cause of di#iuse scattering at the surface without liquid He

The cause of diffuse scattering of the phonons actually lies in the surface irregularities which violate translational invariance parallel to the surface. Various surface irregularities can be considered: a rough surface, imperfections like dislocations in the vicinity of the surface, and surfaces covered by chemisorbed or physisorbed impurities. Hereafter, our arguments are concentrated on the case of rough surfaces, since the complicated surface state does not lend itself to simple understanding of the problem. Our task in physics seems to be to understand phenomena in as simple a way as possible. Provided that an isotropic elastic continuum occupies the half space z 2 0 with a stress-free boundary at z = 0, the displacement vector at a point x = (r, z ) and time t can be expanded in terms of eigenmodes (Ezawa 1971),

[ a J u J ( z exp(ik. ) r-iwt)+H.c.],

(7.7)

where ps is the mass density of the medium and J = (k,c, m ) labels a set of quantum numbers which specifies the eigenmodes of phonons, k is a two-dimensional wave vector, c the velocity of a waue front traversing the surface, and m specifies the mode. The sum over J in eq. (7.7) is defined as

(7.8)

T. N A K A Y A M A

172

where D, denotes the spectral range of the velocity c. R represents the Rayleigh mode (representative surface mode) whose amplitude decreases exponentially with the distance z from the surface. In eq. (7.7), u, and its Hermitian conjugate a: are the annihilation and creation operator of the ]-mode phonon, respectively. There arefive eigenmodes specified by 1 (Ezawa 1971). For the transverse (T) phonons, there exist two kinds of eigenmodes which have a velocity (of a wave front traversing the surface) spectrum c a cT, where C, is the velocity of T phonons. The first mode is the TH mode polarized parallel to of incidence and reflection is the surface [see fig. 7.6(a)]. The angle related to c by cot' 6 H = p 2 ( c )= (c/cT)' - 1 and the range of the velocity c is from cT (OH = ~ / 2 )to infinity ( & = 0). Note that c is related to the incident angle OH. The other T mode (referred to as the TV mode) consists of T phonons polarized in the sagital plane followed by evanescent pseudosurface-waves [see fig. 7.6(b)J. The velocity of a wave front c of this mode is confined in the finite range c T s c s cL, where cL is the velocity of longitudinal (L) phonons. Another mode consists of mixed longitudinal (L) and transverse (TV) waves with vertical polarization, which interact with each other through the surface. Let us consider a rough surface whose height from the plane ( z = 0) is given by a function f ( r ) , where r is the two-dimensional position vector. The bumps of roughness can be described as a mass density fluctuation. The spatial dependence of the mass density fluctuation p ( x ) is expressed by combining the roughness function f(I ) and the Heaviside step function as g(x) = goo(z +f(r ) ) , where go is the mean value of the mass density. For f ( r ) small compared with the wavelength of phonons, one can expand the step function as e(z+f(r)) = e(z)-f(r)Wd.

(7.9)

Thus, the random part of the mass density is separated as Ap( r ) = pof( r ) , where Ap( r ) has dimensions of g cm-'. The perturbed Hamiltonian due to

Surface

"

TH

I

\

TH

TV

I

!

TV

b

Fig. 7.6. ( a ) Transverse mode polarized parallel to the surface. This mode is denoted as the TH mode. (b) Transverse mode polarized in the vertical plane (TVmode). The longitudinal part i s localized in the surface.

KAPITZA THERMAL BOUNDARY RESISTANCE

173

the mass density fluctuation (Nakayama 1976, 1985, 1986a) is represented by

(7.10)

[a:(t)--J(f)l[a:'(f)--J'(f)l

where F ( k + k') is a two-dimensional Fourier transform of the roughness function f(r ) and k is the two-dimensional wave vector, respectively. The transition rate of the J-mode phonon into the J'-mode phonon is obtained as

X

1:

U j ( Z ) . U j f ( Z ) 6 ( Z ) d2

l2.

(7.11)

In eq. (7.11), the ensemble-averaged Fourier transform of the roughness function is defined by

( F ( k+ k')') = S

J

-

d r exp[i(k+ k') r](f(r)f(O)),

where S is the normalized area introduced by integrating over r. Following the formulation of scattering theory in quantum mechanics, the differential cross section of J-mode phonons is expressed in terms of (7.11) as, da(J+J')=

flw,y(J+ J ' ) 9

QJ

being the incident energy flux of J phonons with the velocity c,, given

QJ

- 12 P S / U J / 2 w 2 c J s -

(7.12)

Let us illustrate the case in which the transverse phonons polarized parallel to the surface (hereafter referred to as TH phonons) are incident at angle OH on a rough surface and are scattered into bulk and surface phonons (hereafter referred to as B phonons and R phonons, respectively). The displacement vector of TH phonons is written as (7.13)

T. N A K A Y A M A

174

where

UfH(Z)

=o.

Here wA is the angular frequency, k the two-dimensional wave vector parallel to the surface, and P’ = ( c/ c, ) ?- 1. The energy flux of TH phonons can be given from eq. (7.12) by

The cross section of T H phonons into TH phonons is obtained as (7.14) where 4 is the angle between the two-dimensional vector k and k‘, and the factor W is obtained by assuming the white noise for the correlation function as W=

(aG)’w4 87~’~:

’

where a is the characteristic length parameter of the surface roughness, and -

Ap is the average amplitude of surface density fluctuation. In a manner

identical to that used in deriving eq. (7.14), the cross sections of the other processes ( J + roughness -* J ‘ ) contributing to the time-of-flight spectra are obtained ( Nakayama 1986a). For the other important process converting TH phonons into R’ phonons (TH +roughness+ R’), one obtains d(r(TH + R’) =

TW

sinZ4ff d 4 2c,ci K

(7.15)

Here the numerical factors f,and K are written as fi

= 1 -2y77/(1+

v2L

K = ( Y - 7 I( Y - II + ~ Y T ~ ) / ~ Y T ’ , where y z = 1 - (cR/cT)’ and v 2= 1 - ( cR/cL)’. Although the cross section expressed by eq. (7.15) is not observable directly in the time-of-flight reflection experiment due to the position of the detector, R phonons converted at the rough surface should be rescattered into bulk (9) phonons because of the roughness, and constitute the diffuse signal in the time-of-flight experiments. The details will be given in section 7.2.3.

KAPITZA THERMAL B O U N D A R Y RESISTANCE

175

7.2.2. Speculur versus difluse scattering of bulk phonons

High-resolution time-of-flight phonon-reflection experiments have revealed that the reflected signals are composed of both specular and diffuse parts (Taborek and Goodstein 1980, Marx and Eisenmenger 1981, 1982, Burger et al. 1985). The scattering probability of a TH phonon incident at an arbitrary angle OH with angular frequency w into the diffuse part is expressed by hw T(TH + J’) hoT(TH + R) (7.16) tdiff =

x,,

+

1

QH

OH

where T(TH + J‘)is integrated over the scattered angle: T(TH + J’)=

II

y(TH -+ J ’ ) dx d4.

The first term of eq. (7.16) can be explicitly rewritten as

Here [JB] is the set of all J’s except R. The factor F, = I, + I , + I3 is the numerical constant which has the value 3.5, where the first term (II= 7r) corresponds to the process TH + TH, and 1, and I , correspond to the decay processes into TV phonons and L phonons. The transition rate of a TH phonon into a R phonon becomes

Hence, the first term of eq. (7.16) becomes t,(w,T H +

B) =

(a 6 ) 2 w 4 F 1

8 7 r 2 p ~ c ’~

(7.17)

and the second term is obtained as (7.18)

It is clear from eqs. (7.17) and (7.18) that the component of ‘the diffuse scattering increases with increasing frequency proportional to the fourth power of frequency. The component of the speculur reflection coefficient is obtained by extracting the part of the diffuse scattering from unity as

r,= l - ( t , + t z ) .

(7.19)

T. NAKAYAMA

I76

The sapphire surfaces (mildly anisotropic) used in experiments (Taborek and Goodstein 1980, Northrop and Wolfe 1984) have a roughness scale of the order of 6 = 100 A. This indicates that the surface can be treated with mean variation in depth and width of 6 = 100 with the areal density w = 0.56 -'. The characteristic length of roughness should correspond to the length scale a given in eqs. (7.17) and (7.18), and one can replace (ohp)' by w ( A M ) ? ,where the averaged mass of a bump is estimated as A4 = psa3= 3.99 x lo-'"g for a = 100 A. The resultant probability of diffuse scattering for TH phonons of frequency v in GHz yields for sapphire crystal, rd,ff = 1 0 - ' ~ ~ ~ ,

(7.20)

where the following values for sapphire are used: cL= 1 1 x lo5 cm/s and c T = 6 x l o 5 cm/s. Note that tdim is proportional to the fourth power of frequency. Equation (7.20) indicates that bulk phonons with frequency around 100GHz are scattered dominantly into a difuse part and, for sufficient low frequencies ( < 100 GHz), most of the incident phonons are specularly reflected. It should be emphasized that the probability of diffuse scattering rdirf cannot exceed unity. From this condition one can estimate the frequency regime where the present analysis is valid. In the case of a roughness parameter 6 = 100 A, one has the condition Y < 200 GHz. The value is reasonable because the corresponding wavelength of 200 GHz phonons becomes about 300 A. This implies that the present theoretical model is valid for phonons with larger wavelength than the roughness scale a. If the wavelength A of phonons becomes much shorter than the roughness scale a, the geometrical scattering by roughness becomes much more relevant. For A a one expects a crossover, which would correspond to the transition between specular and diffuse scattering.

-

7.2.3. Difuse signals in the time-ofiflight reflecrion signals

Let us consider the case in which the heater and bolometer are very small and close together. Figure 7.7 shows the geometry of our system with the definition of the thickness of crystal h and the polar coordinate r. Each element of the area d A = r d r d 4 on the top surface is irradiated by B phonons emitted from the heater of the Lambertian source and the element dA re-radiates the phonons. Since the heater and bolometer are assumed to be very small and close together, the bolometer detects only the phonons backscattered with the same angle as that of incident phonons, hence the element dA can be considered as a new source. Defining S ( r ) as the heat flux emitted by the heater and taking into account the time delay of arrival at the bolometer t = 2 d / c , , where d 2 = r 2 + h Z , the fraction of the reflected

KAPITZA THERMAL BOUNDARY RESISTANCE

177

rough surface

/

Source 0, Detector Fig. 7.7. Geometric arrangement of the system. The heater and bolometer are assumed to be close together.

intensity is expressed by dR, =

d a ( J + J ’ , c, 4) coS2 8 S ( t - 2 d / C ~ ~ ! ) ( r 2 + h2)2

(7.21)

+

Here ~ j j is, defined as ~ j j=, 2cJcJ‘/(cJ c J , ) .The cross section d a ( J + J ’ ) is obtained by eqs: (7.14)-(7.15). The diffuse signal R,(t) as a function of time is obtained by integrating eq. (7.21) over r (over the irradiated surface) and assuming the heat pulse to be described by a delta function

S ( t ) = S( t&. This is valid for a crystal of about 1 cm in thickness because heat pulses used in the experiments were of 10-100 ns duration. The relation between r in eq. (7.21) and x in eqs. (7.14)-(7.15) is obtained from

+

x 2= COt28 1 = (h/ T)’

+ 1,

x = C/ cJ.

Using this relation, the diffuse signal, which is a function of time, is represented by R, ( t ) = So

C J. J ’

jO2= 1r d4

dr

d o * ( J + J ’ ) COS28S ( ~ - ~ ~ / C J J , ) . (7.22) (r2+ h2)2

Here the definition of the cross section is d a * ( J + J’)= d a ( J + J‘)/dx d 4 . Component of Rphonons in difuse signals. B phonons scattered at the surface have a high probability of mode conversion into R phonons. To see the effect of mode-converted R phonons on the diffuse signals, we consider the transition rate of R phonons by roughness. The inverse of the lifetime of R phonons into B phonons can be obtained by using eq. (7.11) as well:

T(R+ B) =

2 Wf:o’F,

c:cRK

(7.23a) ’

T. N A K A Y A M A

I 7x

This transition rate is identical to that obtained by Maradudin and Mills (1976) except for the numerical factor. For the process ( R + roughness + R), the transition rate is obtained as (7.23b) From the ratio of eqs. (7.23a) and (7.23b), one can conclude that the transition rate of R phonons into R phonons is 3.21 times as large as that of R phonons into B phonons: f ' ( R + R) = 3.21/*(R+ B). The effective lifetime of R phonons taking into account eqs. (7.23a) and (7.23b) is 7R

=

s,

(7.24)

where v is in the GHz range. In deriving eq. (7.24) the values for sapphire are used. By taking v = 50-100 GHz, T~ becomes 10 ns. These results indicate that, when R phonons propagate along the surface, R phonons should be backscattered into B phonons with the lifetime 7R. Figure 7.8 shows the calculated shape of reflection signals. The thicknesses of crystals are taken to be h ;= 0.5 cm. For the calculation of the

-

0

1

2

3

Time(psec1 Fig. 7.8. Calculated results o f reflection signal taking into account all phonon modes ('urve A comes from mode conversion to B phonons. Curve B represents the component of the mode-convened R phonons. The thickness of the crystal is taken to be h = 0.5 cm. The peaks ( 1-61 correspond to the processes L -, L, L -,TV. L + TH, T V -,TV, TV + TH and TH + TH. respectively. After Nakayama ( 1986a).

KAPITZA THERMAL B O U N D A R Y RESISTANCE

179

time-of-flight reflection spectra, the parameters for sapphire are used by identifying the velocity of slow transverse phonons (ST) as that of TH phonons and the fast transverse phonons (FT)as TV phonons; cST= 6.0 x 10’ cm/s and cFT= 6.5 x los cm/s, and cL= 11.0 x lo5 cm/s. Curves 6, 5 and 4 correspond to the T phonons TH + TH, TH + TV, and TV-,TV. Curves 3, 2 and 1 are due to the L phonons L-, TH, L + TV and L-, L. Curve A comes from the mode conversion from B phonons to B phonons, and curve B shows the component of the mode-converted R phonons. The ratio of height between the curve A( B + B’) and B( B -,R-, B’) is defined by PJ-J*=

da(J +R+ J’) d a ( J + J‘) *

(7.25)

The ratio of height of curves 6A and 6B is PTH-TH = 0.2 and of 2A and 2B is ~ ~ ~ - ~ = (Nakayama 0 . 0 7 1986a). These are calculated from eq. (7.25). In fig. 7.8 curves 2 and 3 are rounded in comparison with the other processes. This indicates the absence of the forward scattering in the processes L + TH and L-,TV. This is attributed to the fact that the roughness has been simplified as the mass defects in the present calculation. 7.2.4. Reduction of the diffuse signal at the solid surface in contact with

liquid He It has become clear in the preceding section that the diffuse signals are composed of high-frequency phonons with the frequency above about 100 GHz. This is consistent with the experimental results of heat transfer, in which the anomalous heat conduction across the interface is observed above about 1 K (see fig. 7.1.). Burger et al. (1985) have observed only specular reflection and no He effect for low frequency phonons. For high frequency phonons a remarkable change of the diffusely scattered component was found when the Si surface was in contact with liquid He. Thus it is necessary to explain the reason why high-frequency phonons contribute to the effective heat transfer, by incorporating the surface irregularities. This subject has been studied theoretically in two different points of view: the modification of acoustic mismatch (AM) theory by incorporating surface irregularities (Little 1961, Adamenko and Fucks 1970, Haug and Weiss 1972, Peterson and Anderson 1972, Khalatnikov and Adamenko 1972, Sheard and Toombs 1974, Shiren 1981, Shen et al. 1981), and the quantum mechanical extension taking into account the interaction between phonons and the He system or adsorbed impurities (Toombs and Challis 1971, Anderson and Johnson 1972, Rice and Toombs 1972, Sheard et al. 1973, Cheeke and Ettinger 1976, 1979, Nakayama 1977, 1985, 1986a, Maris 1979, Kinder 1981, Kinder and Weiss 1986).

I80

T. NAKAYAMA

Let us consider the latter viewpoint, in which the quantum mechanical interaction between He system and phonons in a solid is taken into account. Apart from poorly defined surfaces such as metal surfaces (hard to handle theoretically), one can consider well-Characterized surfaces such as those of sapphire used in the phonon reflection experiments. It has been well accepted for these cases that the first few adsorbed layers of He are immobile at sufficiently low temperatures with a density similar to that in bulk solid He at a pressure of about 100atm (Brewer et at. 1965). At a location far from the range of the attractive substrate potential, the liquid He should maintain its bulk properties (superfluid). The He atoms between the first two adsorbed layers and bulk liquid are bound weakly to the substrate and their motion is quite restricted. In fact, how to describe this state is still open to question (e.g., Lauter et al. 1983). One possibility is to regard it as a dense fluid with no long-range order at temperatures around 1 K (Nakayama 1977, 1985) where most experiments of phonon reflection and transmission have been performed. An important task in understanding the phonon transmission quantummechanically is to determine the type and strength of coupling between phonons and the He system (Mans 1979, Nakayama 1985, 1986a, Haug et at. 1987). Under circumstances in which phonons are incident into the surface in contact with liquid He, the interaction has the effects of scattering and energy absorption. The two types of interaction considered for our system are the displacemenr-type coupling and the deformation coupling.

Displacement-rype coupling. The displacement-type coupling between adatom and substrate phonons has been studied for various problems since the work of Lennard-Jones and Strachen (1935). Here we consider a TH phonon incident at angle 0 to the surface (see fig. 7.6a). The interaction Hamiltonian can be represented by the quadratic form with respect to the relative displacement between the He atom and the substrate (7.26) where a is the coordinate of the He atom measured from the equilibrium position, and f is the coupling constant between the He atom and the substrate. The one-phonon (THmode) absorption probability is defined by

where is the incident energy flux defined by eq. (7.12) and hwd is the energy difference between the ground state and the first excited state of the He atom bound in the attractive potential from the substrate. Using first-order perturbation theory, the transition rate (s-') for normal incidence

KAPITZA THERMAL BOUNDARY RESISTANCE

of the TH mode phonon

r D

181

due to the coupling eq. (7.26) becomes, (7.27)

where N / A is the number of He atoms per unit area. In deriving eq. (7.27) the relation w i =f/mHe is used. As a result, one has the one-phonon absorption probability (7.28)

It should be noted that this absorption probability is expressed by the given physical parameters, and the sum rules presented by Mans (1979) can also be obtained from eq. (7.28). Using the sum rules, Maris (1979) has discussed in detail the phonon absorption by adsorbed He atoms. The two-level tunneling state (TLS) model was introduced for liquid 3He or 4He by Andreev (1978), Andreev and Kosevich (1978) in explaining the observed T-linear specific heats above the quantum degenerate temperature (no long-range order). The concept of the TLS’s for the He system is quite analogous in many respects to the TLS model in glasses originally proposed by Anderson et al. (1972) and Phillips (1972). The TLS’s are responsible for the universal low-temperature properties shared by all configurationally disordered systems. There are two essential differences in the TLS’s between liquid He and glasses. One of them is due to the high tunneling probability of the He atoms because of the large overlap of wave functions. The second one is that the density of states per unit energy n ( E ) of the TLS’s is larger than that of glasses by a factor of the order of lo4 as will be shown in the following. The TLS model has also been introduced independently by Nakayama (1977, 1985) for the He system close to the interface, where the positions of He atoms should be quite irregularly distributed (no long-range order). This is a particularly attractive possibility because it leads to lowenergy states, and it appears that a phonon should couple well to these states (Nakayama 1977, 1985, 1986a). The maximum energy difference Em can be estimated to be 1 0 0 K from the binding energy of the van der Waals potential to the substrate. The magnitude of the level density Y( E ) per one He atom becomes z / Em,where z is the number of neighbouring vacant positions. By taking z - 5 and Em 100 K, the density of states per atom can be estimated as v ( E ) = 3 x lOI4 erg-’. The number density of He atoms in the first few adsorbed layers is N 1015cm-’, so that the density of states per unit area no becomes

-

-

-

no= 3 x loz9erg-’ cm-’

.

T. N A K A Y A M A

182

This is quite large compared with the density of tunneling states i n glasses erg-' cm-'. per unit area: no= The absorption rate of a TH phonon by the TLS with a broad distribution of energy difference can be obtained by replacing the function S ( f i w k h w , ) N / A in eq. (7.28) by no. Taking account of the temperature dependence, one has (7.29) By using the explicit value of no= 3 x lo2' erg ' cm-*, the absorption probability I,, (corresponding to the transmission coefficient)becomes

I , ,= 6.9 x l 0 - ' v 2 , where v is expressed in GHz. If one considers a phonon of 100 GHz incident at the surface, the rate becomes ID= 6.9 x This is too small to transfer energy effectively in the frequency regime considered here, as concluded by Maris (1979). Dejonnorion coupling. The other important interaction arises from the coupling proportional to the strain called deformation coupling. We should bear in mind that the physical nature of He atoms close to the boundary includes the contribution from the He atoms and the substrate surrounding the He atoms. When a TH phonon is incident at the rough surface, the substrate surface atoms will be deformed by an incident phonon, i.e., the phonon works as a deformation coupling proportional to the strain enp.The He atoms close to the surface should change the states by rearranging the atomic configuration quantum-mechanically from this coupling. As a consequence, the He system close to the intertace has a new energy state E,-, which differs from the initial one E,. Because the spread of the wave packet of He atoms is small with respect to the spatial change of the strain, one can estimate the atomic energy difference as

(7.30)

The strength of deformation coupling constant g, = aE/ae,,p for the TLS can be estimated from eq. (7.30) by postulating the complete deformation of eOp= 1. Due to this deformation the change of the binding energy of the He atom should be of the order of the van der Waals potential so that the deformation coupling constant g becomes about 100 K. Thus, the interaction Hamiltonian between a phonon with the wave vector k and the He system

KAPITZA THERMAL BOUNDARY RESISTANCE

183

is expressed in the second quantized form, (7.31)

Hk = 8 k T k * x ,

where the operator a, is the Pauli matrix and 7, the strain of a TH photon. Using eq. (7.31) and assuming a wide distribution of the energy difference, the absorption probability of TH phonons becomes (Nakayama 1986a) fs = 27rg2nowtanh[ Psc:

21.

(7.32)

If one takes the values for the mass density of a solid and the velocity of a TH phonon for sapphire as ps = 3.99 g ~ m - ~c, ,= 6.0 x lo’ cm s-’, and g = 100 K, the absorption probability for frequency v in GHz is ts -- 2.56 x 10-6v.

(7.33)

For the incident phonon of v = 100 GHz, the absorption probability becomes t, -- 2.56 x This is too small to explain the experiments (-lo-’) as well as the case of the displacement coupling obtained in eq. (7.29). Apart from phonon absorption by the adsorbed He system, there is an interesting possibility that adsorbed air molecules constitute two-level tunneling states identical to those of glasses (Kinder 1981, Schubert et al. 1982, Basso et al. 1984, Kinder et al. 1985). The phonon absorption rate for this case is obtained as well by using eq. (7.32), which is the same as that obtained by Kinder (1981). From the density of states per unit volume obtained for fused silica, no= cm-3 erg-’, as a typical value, one can estimate the density of states per unit area as no= lo2’ cm-* erg-’. The deformation coupling constant is known to be of the order of g = 1 eV for glasses. Using these values, the scattering rate for frequency v in GHz yields t,=1.12x10-6v.

(7.34)

This is also too small to explain the phonon-reflection experiments as well as eq. (7.33). In eqs. (7.29), (7.33) and (7.34), it has been shown that the direct interaction process of phonon absorption (B phonons to He system) is negligible. These theoretical results are natural in some sense for the following reason. The cause of interaction (for example, TLS’s) postulated is present in the two-dimensional sheet. When bulk (B) phonons traverse the sheet, the number of TLS’s interacting with B phonons is so small that these states do not yield observable effects on the transmission or reflection coefficient of bulk phonons. This conclusion is consistent with the recent reports by Klitsner and Pohl (1986, 1987), who made a systematic study of phonon scattering at polished silicon surfaces by deposited thin films in the temperature range 0.05-2.0 K corresponding to dominant frequencies

T. N A K A Y A M A

184

from 5 to 180GHz. They prepared ex- or in siru-deposited thin films of metals, nonmetals, and condensed gas with average thickness from 2 8, to lo4 8, as well as a clean and a rough surface. Depositions and measurements are done in truly clean (ultra-vacuum) conditions using a remarkable method. In the course of these experiments, they found that diffuse scattering is associated with the islands of discontinuous thin film, or microscopic structural irregularities. There was no evidence of scattering by some kind of unknown interface state or individual atoms or molecules at the interface. Let us remind ourselves of the experimental evidence that diffusely scattered phonons'play a key role in the effective transfer of energy. It has been clarified that diffuse signals are due to two causes. One is the direct scattering of B phonons at irregular surfaces (B+ roughness -* B), and the other is due to the mode-converted R phonons ( B+ roughness .+ R -,B). Thus, another possibility for effective energy transfer occurs through the interaction between R phonons and the He system: The R phonons converted from B phonons at the rough surface interact with the He system, and the energy of R phonons is absorbed by the He system (see fig. 7.9). Note that R phonons propagate along the interface, and the scattering process is quite different from the case of B phonons mentioned earlier. I f the lifetime of mode-converted R phonons due to interaction with the He system ( T H ) is shorter than that of R phonons due to scattering by roughness ( T ~ ) , the phonon energy converted into R phonons should be transferred into the He system. As a consequence, the component of R phonons of diffuse signals (-20%) should vanish when liquid He is present. This process has been discussed in detail by Nakayama (1985, 1986a). in which both the displacement-type and the deformation coupling between R phonons and TLS's have been taken into account. The lifetime due to the displacernenttype interaction between R phonons and the He system is calculated in a straightforward way by replacing the surface displacement uTH(0) in eq. (7.26) by that of R phonons. Taking the numerical values for sapphire, we have the lifetime of R phonons with frequency i n GHz under the condition L

8

L

4

8

Fig 7.9. Two possible channels o f energy transfer. Process ( 1 ) represents the direct procesb, and ( 2 ) represents the R phonon mediated process of energy transfer.

KAPITZA THERMAL BOUNDARY RESISTANCE

A w ~> 2 k s r

185

*

7 H . d a 0.88 x i o - 2 V - 3 s. (7.35) The lifetime of R phonons due to the deformation interaction can be obtained by using eq. (7.31) and the wave function of R phonons. The numerical result for R phonons with frequency in GHz for sapphire gives

7H,s = 1.02 x 1 0 - ~y - 2 s. (7.36) It should be noted that the frequency dependence of eqs. (7.35) and (7.36) is different from that of bulk phonons. This is because of the fact that the energy density of R phonons is localized in the vicinity of the surface depending on its wavelength, i.e., the energy density is frequency dependent. By comparing the lifetimes of R phonons due to scattering by roughness and the defects (TLS), one can conclude that the mode-converted R phonons can be absorbed effectively by the He system for a frequency of about 100 GHz. If this is the case, the diffuse tail arising from the mode-converted R phonons vanishes when the surface is in contact with liquid He at around 100 GHz of phonon energy, i.e., the diffuse signal to the mode-converted R phonons (curve B in fig. 7.8) vanishes. This picture is also valid for the process postulating the presence of LTS in adsorbed air molecules. Finally, the possibility should be mentioned that adsorbed air molecules and He atoms combine together to form defect states (e.g., TLS’s), because it may be wrong to think of the exposed surface and He system as having a definite boundary. This means that adsorbed air molecules are not closely packed, and He atoms can penetrate between them. This possibility has been pointed out by Vuorio (1972), although the description is made in a quite different manner. In particular, if the wavelength of incident bulk (B) phonons is comparable with the thickness of these layers (matching layer), the interaction between B phonons and the He system must be affected, and this would constitute a highly absorbing layer for phonons. In any case, the modeconverted surface phonons play a key role in anomalous phonon transmission. It is worth mentioning the situation where a solid surface (characterized by roughness scale a ) in contact with liquid He receives phonons with much shorter wavelength than the roughness scale a (A < a ) . In this limiting case, the phonon transmission coefficient should equal, if only the acoustic mismatch boundary condition is taken, the value predicted by the acoustic mismatch theory for bulk solid (see the discussion in section 2.2). The calculation incorporating only surface roughness must recover this limiting conjecture. This section has reviewed the current understanding of the puzzling phenomenon of the Kapitza resistance above about 1 K, by enlightening the application of new techniques of generation and detection of highfrequency phonons. Although the problem has become clearer in recent

I86

T. NAKAYAMA

years due to the accumulation of experiments and theories, we have to admit that some detailed work has yet to be done to clarify the mechanism of high-frequency phonon transmission.

8. Summary The important interesting features of the Kapitza thermal boundary resistance have been reviewed. Section 2 was devoted to the introduction of the Kapitza thermal boundary resistance, and has cleared up the subject by comparing the existing data with the acoustic mismatch theory. In section 2.3, the anomalous Kapitza thermal boundary resistance observed at the liquid 'He-sintered powder interfaces at mK temperatures, reported for the first time by Avenel et al. (1973), has been discussed together with the anomalous Kapitza resistance observed at the liquid He-bulk solid interface above about 1 K. Section 3 has outlined the Fermi liquid theory for the Kapitza resistance, by illustrating energy transfer from a small particle into liquid 3He. In this section, the contribution from both the zero-sound excitation and the inelastic scattering of single quasiparticles have been explained theoretically. Sections 4 and 5 dealt with the main topics of this article: the Kapitza resistance between liquid 3He and sintered powder at mK temperatures. The acoustic channel of heat transfer has been described in section 4 by incorporating the characteristic features of elastic properties of sintered powders. In this section, it has been pointed out that the soft-phonon modes peculiar to sintered powder play an important part in the effective heat transfer. The contribution of the magnetic coupling to the effective heat transfer in the case of the liquid 3He-Ag powder interfaces was discussed in section 5 . Here, it was shown that magnetic coupling plays a relevant role in the anomalous heat transfer as well as the acoustic channel discussed in section 4. Especially, it is important to point out that magnetic impurities (i.e., 0 2 , 0;. 0, 0-, etc.) at the silver surface play a key role in the anomalous heat transfer. In section 6, the Kapitza resistance between liquid 'He-4He mixtures and sintered powder was discussed, illuminating the observations ( R K E ;' and T-') at millikelvin temperatures. Here, both the acoustic and the magnetic channels have been discussed in detail, and it has been suggested that the same magnetic coupling in the case of the pure 'He-Ag powder interface is important. In particular, it is remarkable that the observed Kapitza resistances for liquid 'He and 3He-4He mixtures are understood in a consistent way.

-

KAPITZA THERMAL BOUNDARY RESISTANCE

187

Section 7 reviewed the current aspects of investigations on the longstanding problem of the Kapitza resistance (probably from the time of the 1950's) about about 1 K, by paying attention to the application of new techniques of high-frequency phonon-generation and detection to this problem. There are two interesting problems on the Kapitza resistance, which have been omitted in this article. These are the Kapitza resistance between magnetic materials and liquid 3He, and the Kapitza resistance between two quantum media like the solid He-liquid He or liquid He-hydrogen gas interfaces. Experimental reports up to 1979 on the former problem are described in detail in the review by Harrison (1979). Recent experiments on this subject have been reported by Saito (1981) for CTS, Jutzler and Mota (1984), Batey and Main (1984) and Marek et al. (1986) for CMN, Fujii and Shigi (1987) for CPS, Jacquinot et al. (1986) for HoVO,, Mizutani et al. (1987) for TmVO,, and Ytterboe et al. (1987) for PrNiS. Although these topics are not yet fully understood, the rapid developments in the past several years encourage experiments and theories.

Appendix: Surface vibrations of a small particle

In order to calculate the energy flux given by eq. (3.22) from a small particle into liquid 3He, one must obtain the surface displacement u ( R ) from eigenmodes of phonons in a particle. The displacement vector u ( r , t ) is expressed through a scalar potential 4 , and vector potentials 42and 4', using the equation u( r, 1 ) = grad

4 , + curl 42+ curl 4'.

(A.1)

The first term represents the longitudinal mode with dilation, and the others are two transverse modes. Without loss of generality, the vector potential is taken as

4, = ( r , 0,O)4i,

i = 2,3.

(A.2)

The above definition of the vector potential gives us a good perspective for determining the eigenmodes in the spherical body. Consider a small particle in contact with liquid 'He. Since liquid 'He compared with that of the has a small mass density ( p L = 0.0815 g small particle, the appropriate boundary conditions determining the eigenmodes may be taken as those for a stress-free surface, which are expressed in the following forms u a P ( R )= O ;

a,p = r, 8, 4.

(A.3)

T. N A K A Y A M A

18R

This condition allows us two types of oscillations in a spherical particle: One is the spheroidal mode and the other is the toroidal mode. The spheroidal mode is expressed by the sum u J .= ~ grad

4 I + (YJ curl curl 43,

(A.4)

and the toroidal mode is UJ,, = curl 4 2 ,

(A.5)

where the lower suffices ‘s’ and ‘t’ mean the spheroidal and toroidal mode, respectively. The J acts for a set of quantum numbers specified by (1, rn, w), where 1 and rn give the order of spherical harmonics and w is the angular frequency. The factor aJ in eq. (A.4) indicates the ratio of the second to the first term. The potential functions are expanded by the associated Legendre polynomial P ;I(cos 0) and the spherical Bessel function j r ( x ) as

4,(r, r ) = C A:”’j,(k,r)P;I(cos0 ) cos rn4 1. m

x exp( -iwkr),

(rn s 1, j = 1 , 2 , 3 ) .

(A.6)

Here the suffix j means the longitudinal ( j = 1) and the transverse ( j = 2 , 3 ) modes, respectively. A;“‘ are the expansion coefficients. The wave number k, is defined by the relation @k = k,cj. From the boundary conditions expressed by eq. (A.3),the eigenvalue equations are obtained as

(A.7)

where $, = k, R and q = k,R ( i = 2 . 3 ) . The above equation is solved numeri2 k:/k: = cally with respect to the single variable using the relation v 2 / t = (2 P + A )IP. The displacement u obtained from the vector potential & has no radial component of oscillation as understood from the definition of eq. (A.2) expressed as ul(r,t)=curl$,=

(

0,-sl,:O:?~

-a? ):

KAPITZA THERMAL BOUNDARY RESISTANCE

I89

For instance, taking I = 1 and m = 0 in the potential eq. (Ah), we have

42= A:.oj,(k2r)P,(cos

0) e-'"'.

(A.9)

Substituting eq. (A.9) into eq. (A.8), the displacement u becomes u , ( l = 1, m =0) = (O,O, -A:"j,(k2r)

sin 0) e-i"'f.

(A.lO)

This yields the lowest torsional oscillation, which is called a roroidal mode. are called the The modes determined from the potentials 4, and spheroidal modes, which have the radial component of surface oscillation. The eigenvalue equation (A.7) has been solved numerically by Nakayama and Nishiguchi (1981), and Nishiguchi and Nakayama (1982) for silver and copper particle.

Acknowledgements I am grateful to my colleagues, K. Yakubo and N. Nishiguchi, for many valuable comments and discussions. I have benefitted greatly from conversations and correspondence with L.J. Challis, J.D.N. Cheeke, K. Dransfeld, W. Eisenmenger, G. Frossati, Y. Fujii, H. Fukuyama, J.P. Harrison, H. Kinder, M. Kubota, H. Ishimoto, H.J. Maris, F. Pobell, K. Rogacki, S. Saito, T. Shigi, F.W. Sheard, and A.F.G. Wyatt. I would like to thank K. Kimura and K. Yakubo for producing the typescript and figures. This research was granted in part by the Iwatani Naoji Foundation's Research Grant, the Suhara Memorial Foundation, and a Grant-In-Aid from the Ministry of Education, Science and Culture, Japan.

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Rutherford, A.R., J.P. Harrison and M.J. Scott, 1984, J. Low Temp. Phys. 55, 157. Sabisky, E.S., and C.H. Anderson, 1975, Solid State Commun. 17, 1095. Saito, S., 1981. Phys. Rev. 24, 459. Saito, S., T. Nakayama and H.Ebisawa, 1985, Phys. Rev. B 31, 7475. Schubert, H.. P. Leiderer and H. Kinder, 1982, Phys. Rev. B 26, 2317. Sheard, F.W., and G.A. Toombs, 1974, J. Phys. C 4.61. Sheard, F.W., R.M. Bowley and G.A. Toombs, 1973, Phys. Rev. A 8, 3135. Shen, T.J., D. Castiel and A.A. Maradudin, 1981, J. Phys. (France) 42, C6-819. Sherlock, R.A., A.F.G. Wyatt, N.G. Mills and N.A. Lockerbie, 1972, Phys. Rev. Lett. 29, 1299. Sherlock, R.A., N.G. Mills and A.F.G. Wyatt, 1975, J. Phys. C 8, 300. Shiren, N.S., 1981, Phys. Rev. Lett. 47, 1466. Swanenburg, T.J.B., and J. Wolter, 1973, Phys. Rev. Lett. 31, 693. Synder, N.S., 1976, J. Low Temp. Phys. 22, 257. Taborek, P., and D. Goodstein, 1979, J. Phys. C 12, 4737.

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Taborek, P., and D. Goodstein, 1980. Phys. Rev. B 22. 1550. Toombs, G.A., and L.J. Challis, 1971, J. Phys. C 4, 1085. Toombs. G.A.. F.W. Sheard a n d M.J. Rice, 1980, J. Low Temp. Phys. 39, 273. Trumpp, H.J., K. Lassmann a n d W. Eisenmenger, 1972, Phys. Lett. A 41,431. Villain, J., 1979, in: Ill-Condensed Matter, eds R. Balian, R. Maynard and G . Toulouse (North-Holland, Amsterdam) p. 522. Vuorio, M., 1972, J. Low Temp. Phys. 5, 1216. Weber. J.. W. Sandmann, W. Dietsche and H. Kinder, 1978, Phys. Rev. Lett. 40, 1469. Wheatley. J.C., 1968, Phys. Rev. 165, 304. Wheatley, J.C., 1975, Rev. Mod. Phys. 47, 415. Wheatley, J.C., O.E. Vilches a n d W.R. Abel. 1968, Physica 4, 1. Wheatley, J.C.. R.E. Rapp and R.T. Johnson, 1971, J. Low Temp. Phys. 4, 1. White, D.. O.D. Gonzales and H.L. Johnston, 1953, Phys. Rev. 89, 593. Wyatt, A.F.G., 1981, in: Nonequilibrium Superconductivity, Phonons, a n d Kapitza Boundaries (Plenum Press. New York) p. 31. Wyatt, A.F.G., and G.J. Page, 1978, J. Phys. C I t , 4927. Wyatt, A.F.G., N.A. Lockerbie a n d R.A. Sherlock, 1974, Phys. Rev. Lett. 33, 1425. Wyatt, A.F.G., G.J. Page and R.A. Sherlock, 1976, Phys. Rev. Lett. 36, 1184. Yakubo, K., and T. Nakayama. 1987a. in: Proc. 18th Int. Conf. on Low Temperature Physics, Jpn. J. Appl. Phys. 26, suppl. 26-3. 883. Yakubo, K., and T. Nakayama, 1987b, Phys. Rev. B 36, 8933. Ytterboe, S.N., P.D. Saundry, L.J. Friedman. C.N. Gould and H.M. Bozler, 1987, in: Proc. 18th Int. Conf. on Low Temperature Physics, Jpn. J. Appl. Phys. 26, suppl. 26-3, 379. Zinor'eva. K.N.. 1978, Pis'rna v Zh. Eksp. & Teor. Fiz. 28, 294 [Sov. Phys.-JETP Lett. 28,2691. Zinov'eva. K.N.. 1980, Zh. Eksp. & Teor. Fiz. 79, 1973 [Sov. Phys.-JETP 52. 9961. Zinov'eva. K.N.. 1985, in: Low Temperature Physics (Mir, Moscow) p. 78. Zinov'eva. K.N.. and V.I. Sitnikova. 1983, Zh. Eksp. & Teor. Fiz. 57, 576 [Sov. Phys.-JETP 57. 3321. Zubarev, D.N., 1974. Nonequilibrium Statistical Thermodynamics (Plenum Press, New York) p. 248.

CHAPTER 4

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS IN DRIVEN CHARGE DENSITY WAVE CONDENSATES BY

G. GRUNER Department of Physics and Solid State Science Center, University of Calgornia Los Angeles 90024, USA

Progress in Low Temperature Physics, Volume X I 1 Edited by D.F. Brewer @ Elseoier Science Publishers B. V., 1989 195

Contents

.................. I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Basic notions and observations in charge density wave dynamics . . . . . . . . . . . . . . . 2.1. The charge density wave ground state and model compounds 2.2. The dynamics of the collective mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Frequency and field dependent transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 3. Current oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . 1 . General features . ............................................ 3.2. Current-frequency ............................................ 3.3. Size effects and fluctuation phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Broad band noise . . . . . . . . . . . . . . . . . . . . 4 . Models of charge density wave dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . I . The classical particle model . . . . . . . . . . . ........................ 4.2. Models including the internal degrees of 4.3. The tunneling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Interference phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Harmonic mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Subharmonic mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Nonsinusoidal and pulse drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Fluctuations and coherence enhancement . . . . . . . . 6 . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of review papers . . . ... .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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197 201 20 1 205 211 217 217 220 223 226 227 227 232 237 239 239 246 255 259 262 265 266

1. Introduction

In highly anisotropic metals, electron-phonon interactions may lead to a new type of ground state at low temperatures with features both similar to and different from the well-known superconducting ground state. In contrast to superconductors, the condensate is formed of electron-hole pairs with a total momentum q = 2kF where kF is the Fermi wavevector, reflecting the 2 k , singularity of the Linhard response function in one dimension. The electron density has a spatial dependence, and in one dimension it is given by Ap = p o + p I cos(2kFx+4),

(1.1)

where po is the unperturbed electron density, p , and 4 are the amplitude and phase of the spatially oscillating charge density, called the charge density wave (CDW). The period of the charge density wave is determined by the Fermi wavevector and for a partially filled electron band, the CDW is incommensurate with the underlying lattice. Like the superconducting ground state, the charge density wave state also develops below a certain transition temperature (called the Peierls transition temperature Tp, Peierls 1955), and the mean field treatment of the thermodynamics of the phase transition is essentially the same as that of a superconductor. The central feature of both ground states is the development of a single particle gap A, which has a well-defined relation to the transition temperature, and of a collective mode which also determines the low frequency electrodynamic response. Charge density waves, such as given by eq. (1.1) have been observed in several inorganic and organic, linear chain compounds where the anisotropic crystal structure leads to highly anisotropic bands. The transition temperatures are, in general, somewhat below room temperature, and the appearance of single-particle gaps at *kF turns the metals into semiconductors, or semimetals with the gaps also observed by optical studies. The periodic modulation of the charge density leads also to periodic modulation of the atomic displacements, and therefore, the period can be determined by structural studies. The scattering intensity is proportional to A ( T), and available experiments are in fair agreement with a BCS-like temperature dependence of the order parameter (Fleming 1981). The collective mode can couple to the applied external electric field, and this for a dc electric field of sufficient magnitude, may lead to the translational notion of the condensate as envisioned by Frohlich (1954). For an 197

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198

incommensurate C D W vanishingly small elctric fields would, in principle, be able to displace the collective mode. However, impurities, lattice imperfections, etc., pin the C D W to the underlying lattice, and this leads to a so-called threshold field E,, below which the condensate is pinned, and above which the current-carrying state, with nonlinear current-voltage characteristics is observed. For small impurity concentrations the threshold field is small, of the order of 10 mV/cm in various materials. The restoring force due to imperfections also shifts the oscillator strength associated with the ac response to finite frequencies, and the response can, in general, be adequately described by a simple expression, the same as that of a harmonic oscillator except at low frequencies, where additional contributions to a(@) occur, either due to the tunneling process or due to the internal degrees of freedom of the condensate. The pinning frequency oois much smaller than d / h , and the response is in general, overdamped, U ~ T >1. The effective mass m* is large, because in the dynamical response the kinetic energy of the electrons and ions has to be included, and values of m * / m , - lo3 with mb the band mass are typical. Because of the total momentum 2kF associated with the electron-hole condensate, the charge density wave ground state has a fundamental periodicity with the wavelength A = .rr/k,. The translational motion of the CDW, with a drift velocity Ud, leads to a characteristic frequency fo = v d / h ,which corresponds to the frequency related to the displacement of the condensate by one wavelength. The dc current is given by I = nevd with n the number of electrons in the collective mode. For a one-dimensional metal at T = 0, n = 2kF/ 71; and consequently the relation between current density per chain j andf;, is given by j/.f0 = 2.

These oscillations have first been observed by Fleming and Grimes (1979) and the linear relation, eq. (1.2) was later demonstrated by Monceau and coworkers (Monceau et al. 1980). The nonlinear and frequency dependent conductivity, together with the current oscillations, can be described by phenomenological equations of motion. Among the models, which assume that quantum effects can be neglected, the simplest, called the classical particle model (Griiner et al. 1981) is formally identical to that of the resistively shunted Josephson junction. Although the model does not account for all the details of the w and E dependent response it reproduces several of the main observations, including the current oscillations with the relation given by eq. (1.2). It is not surprising therefore, that a broad variety of observations can be made on materials with a C D W ground state, which are similar to those made on Josephson junctions. The combined application of ac and dc fields, for

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example, leads to various ac interference phenomena, some of which, in reference to the close relation to findings in Josephson junctions, are in general referred to as Shapiro steps. These have been investigated recently in detail, and observations on charge density waves have also been used to test various theories of nonlinear dynamical behavior, such as questions concerning transitions to chaos in deterministic systems. An alternative description of the nonlinear current-voltage characteristics is based on the concept of tunneling processes (Bardeen (1986, 1987) and references cited therein). The model gives a nonlinear current-voltage characteristic somewhat different from that of the classical particle model, and a periodic potential which is nonsinusoidal. The frequency dependence is accounted for by a formalism which has been applied earlier to superconductor-insulator-superconductor (SIS) junctions. With the form of the potential and of the I - V curve given, the main features of the current oscillations and of the various interference effects can be described without further assumptions. The experiments which study the dynamics of the collective mode can be classified as follows. 1. Frequency and electricfield dependent conductivity where the time average current ( j (1 ) ) is measured in the presence of dc or small amplitude ac drives. The dc conductivity is defined as a d , = (j(t ) ) / Edcwhere Edcis the applied electric field, and the small amplitude ac response is characterized by uaac = Re u ( w )+ i Im a ( w ) where Re a ( w ) and Im a ( w ) are the real and imaginary components of the conductivity. Here it is assumed, that linear response theory is appropriate, and the response to a sinusoidal ac drive E,, = E,, sin w t is also sinusoidal, with no higher harmonics. Because the dc response is nonlinear, it is obvious that this is not valid for arbitrary ac fields, and nonlinear ac effects may occur for finite fields. These experiments serve as tests of the various models which lead to predictions concerning the dc and ac response of charge density waves, and the ac response can also be used to evaluate the fundamental parameters of the problem, such as the effective mass m*,the pinning frequency wo and damping constant I / T. Frequency dependent conductivity studies have been performed over a rather broad frequency range, from audio to submillimeter wave frequencies, and by now the w dependent response of the pinned mode is fully characterized in many materials. Similarly, the dc conductivity has also been explored in detail, and compared with various theories. Experiments performed at finite fields and frequencies have also been conducted, and the nonlinear ac response has also been examined in detail in various materials. Such studies are supplemented by the measurements of rectification and harmonic mixing, both performed over a broad range of

200

G . GRUNER

frequencies and applied dc and ac fields. These studies will not be summarized here; it should be mentioned, however, that (perhaps not surprisingly) models which account for the dc and small amplitude ac response are successful in describing the response in finite fields and frequencies, and also in the presence of joint ac and dc excitations. 2 . Sepctral response in the current carrying stare. In the nonlinear conductivity region, the spectral features of the current include a large amplitude

broad band noise and current oscillations (often called narrow band noise N B N ) , with a fundamental given by eq. (1.2) and several harmonics f n = nso with slowly decaying intensity. The current oscillations have a finite spectral width, and also display temporal fluctuations which can be studied by examining the time dependence of the Fourier transformed current. The origin of the current oscillations has been studied by using imaginative lead configurations for current and voltage contacts and also by employing thermal gradients to break up the coherence which leads to current oscillations. The observation of current oscillations is suggestive for a significant coherence throughout the specimens, and it is expected that the range of current-current correlations (j(f , 0), j ( t, r ) )is comparable to the dimensions of the samples. The finite widths associated with the current oscillations (when viewed by employing a spectrum analyzer which detects the Fourier transformed current) and the broad band noise cannot be explained by external noise terms, and represent the dynamics of the internal degrees of freedom within the condensate. 3 . Interference experiments where both dc and a c field are applied, and where either the dc or the ac response is measured. Many of the observations are similar to those made on Josephson junctions, but the relevant frequencies for CDW dynamics are in the radio frequency, instead of in the microwave range. Features of mode locking between the intrinsic current oscillations and the externally applied ac field are closely related to the features of the current oscillations, but additional effects, such as synchronization and coherence enhancement by external drives can also be investigated by the joint application of dc and ac drives. Also, in contrast to microwave signals, waveforms different from sinusoidal can be applied at radio frequencies, and these can be used to examine phenomena such as transitions from the pinned to the current carrying state in detail.

This review focuses on the current oscillations and interference phenomena in an attempt to give an overview of the progress which has been made in the field, and to summarize the open questions, both from the experimental and from the theoretical point of view. The paper is

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201

organized as follows. First, the basic notions and experiments on charge density wave dynamics are summarized, with emphasis on the frequency and field dependent response. This is followed by the summary of experiments on current oscillations, and by the discussion of attempts made to establish the origin of oscillating response in the nonlinear conductivity region. A short description of the various models is given, followed by a detailed discussion of the various interference effects which occur in the current-carrying CDW state driven by both dc and ac electric fields. The status of the field and a summary of open questions concludes this review. Several reviews on the various aspects of charge density waves have appeared recently and these are listed at the end of the paper.

2. Basic notions and observations in charge density wave dynamics 2.1. THECHARGE

DENSITY WAVE GROUND STATE AND

MODEL COMPOUNDS

The condensate which is the focus of this review has many features which are similar to superconductivity. Simple arguments which show the essential features of the ground state are as follows. Consider a one-dimensional ( 1 D ) metal. In the absence of an interaction with the lattice, the ground state is as shown in fig. l(a), where the electron states are filled up to the Fermi level and the underlying lattice is that of a periodic array of atoms with lattice constant a. As first pointed out by Peierls (1955) this state is not stable for a coupled electron-phonon system. In the presence of an interaction between the electron and the lattice, it is energetically more favorable to distort the lattice periodically with period A related to the Fermi wavevector kF, h = 7T/kF.

(2.1)

A lattice distortion with this period opens up a gap at the Fermi level, as shown in fig. l(b) where the situation appropriate for a half-filled band is drawn. As states only up to *kF are occupied, the opening of the gap leads to the lowering of the electronic energy. In contrast, the lattice distortion leads to an increase of the elastic energy, but in one dimension the total energy (electronic+lattice) is lower than that of the undistorted metal (this is the consequence of the divergent Linhard function at q = 2kF in 1 D). Consequently a distorted state is stable at T = 0 K. The gap opening also leads to the modification of the electron density, much in the same way as in the nearly free electron theory of metals. The electron density p = /+I2 will be a periodic function of the position x with the period given

G. GRUNER

202

-

0

0

0

0

0

0

0 'atoms

a

-1

mcta I

p(rl

00

00 >

20

00

OO\

atoms

insulator Fig. I . Peierls distortion in a one-dimensional metal with a half filled band. ( a ) Undistorted metal, ( b ) Peierls insulator.

by (2.1), and determined by the band filling. Thus, for an arbitrary band filling, the period of this modulated charge density (and the accompanying periodic lattice distortion, see fig. I(b)) will be incommensurate with the underlying lattice. At finite temperatures, normal electrons excited across the single-particle gap A screen the electron-phonon interaction. This in turn leads to a reduction of the gap as the temperature increases and eventually to a second-order phase transition, as in the case of a superconductor. The above features of this so-called Peierls transition and of the collective mode can be described by the mean field treatment of the one-dimensional electron-phonon Hamiltonian

H

=

trw:b:b,

c;<,ck,,+ A. m

k.
where c; ( c k ) , 6 : ( b , ) are the electron and phonon creation (annihilation)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

203

operators with momenta k and q, u denotes the spin, &k and u: are the electron and phonon dispersions and g( k ) is the electron-phonon coupling constant. With a complex order parameter

A e-’ = g(2kF)( b 2 k , + b+ZkF), (2.3) where A and 4 are real, the displacement field of the ions is given by

The electronic part of the Hamiltonian can be diagonalized by setting up a self-consistent equation for A by using a mean field approximation replacing bZkFby ( b 2 k F ) . Using a linear dispersion relation to describe the electron band near the Fermi energy sF &k = UF(lkl-kF), (2.5) where uF is the Fermi velocity, the relation between the gap A, and the dimensionless electron-phonon coupling constant A ’ = g2(2kF) ( w:kF cF)-’ leads to the BCS gap equation at T = 0,

A ( T = 0) = 2 0 exp( - l / A ’ ) , (2.6) where the cutoff energy D in the gap equation is the one-dimensional bandwidth. The temperature dependence of A also has the characteristic form which is the same as that of the superconducting gap and vanishes at the transition temperature T p .The latter is related to the zero temperature gap by T p= A ( T = 0)/ 1.76. (2.7) The temperature dependent carrier concentration in the condensate n,( T) is also related to A ( T ) , and

where 5(3) is a third-order zeta function. At temperatures T < T p ,ncDw( T ) is given by (2.9) while at T = 0, ncnw( T ) = 1. The spatially dependent electron density can also be evaluated, and at T = 0

G . GRUNER

204

where po is the electron density in the absence of electron-phonon interaction. The appearance of a gap in the single-particle excitation spectrum, together with the collective mode, which is described by a complex order parameter (see eq. (2.3)), is a feature reminiscent of superconductivity. One group of materials, where such types of ground state have been observed are composed of chains, and have highly anisotropic electronic properties. There is a large overlap of the wave functions parallel to the chains, and the single-particle bandwidth Dllis approximately -1-2 eV in this direction. The overlap is much weaker in the other directions, and consequently D,,the bandwidth perpendicular to the chains, is one or two orders of magnitude smaller. The first compound for which the dynamical behavior of the CDW ground state has been examined (Monceau et al. 1976) is the linear chain compound NbSe,. The material has two transitions, one at 149 K and the other at 59 K. These correspond to CDW formation on different chains in this structurally fairly complicated material. Together with NbSe3 several members of the so-called trichalcogenide ( M X 3 ) family undergo second-order metal-semiconductor transitions below room temperature. Other examples include halogen-tetrachalcogens (MX4), h,, where h = I, Br or CI and bronzes such as the called blue bronze &.3Mo03. The dc conductivity of three representative compounds is shown in fig. 2. In all cases a metal-insulator transition, indicated by arrows occurs somewhat

\

IToSe41p I

I

I

0.01

0.02 l / T (K-')

Fig. 2. Temperature dependence of the electrical conductivity in o-TaS,, in (TaSe,),l and in K,, ,MOO,. The arrows represent the Peierls transition temperatures, they are evident by examining the temperature derivatives d R / d Z

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

205

below room temperature and the single-particle gap evaluated from the temperature dependence of the conductivity below Tp is (when lowdimensional fluctuations are also taken into account) in broad agreement with a BCS expression. Other transport properties and the magnetic susceptibility also clearly establish the development of the single-particle gap below the phase transition. That this is associated with the development of incommensurate CDWs is evidenced by diffraction studies which establish also the wavevector q which characterizes the distortion. In all cases, the incommensurate CDW develops along the chains and typically A 4a with a the lattice constant along the chains. High resolution X-ray scattering experiments indicate substantial phase coherence both parallel and perpendicular to the chain direction. The significance of this observation will be discussed later. The temperature dependence of the scattering intensity I gives directly the temperature dependent gap as 1 - A 2 which, in general is close to the BCS form (Griiner 1988). The structural and electronic characteristics of the materials, and the main features of the phase transitions are summarized in the various reviews listed at the end of the paper, and will not be discussed here.

-

2.2. THE D Y N A M I C S O F

THE COLLECTIVE MODE

The dynamics of the CDW mode is described in terms of the time dependent order parameter. As the order parameter has two degrees of freedom (amplitude and phase), both amplitude and phase fluctuations will occur. These can be accounted for (Lee et al. 1974) by assuming that A(x, r ) is given by (2.11)

A(x, r ) = ( A + 6 ) eim,

where A is the equilibrium order parameter, 8 and dJ are the fluctuations from the equilibrium value. To lowest order in 6 and dJ the amplitude mode corresponds to the excitation AZk,+ A-ZkF= 2 4 + 28, the phase mode corresponds to A2kF+ A-2,+ = 2iA4. The dispersion relation of these modes have been evaluated, using the Hamiltonian (2.2). The electron-phonon interaction transforms the phonons near 2kF into an optical and an acoustic branch with the dispersion relations m 3m

0:= A ’ w : k F + y ( u F q ) 2 , optical mode, acoustic mode,

(2.12) (2.13)

G . GRUNER

206

where uF is the Fermi velocity and the effective mass of the condensate is given by m* 43 _ - 1 +-.

m

A&,

(2.14)

The optical mode has a gap, while the acoustic mode is gapless. It is therefore, in general, assumed that the fluctuations of the amplitude mode can be neglected at low temperatures with most of the dynamical properties due to the dynamics of the phase of the condensate, only. The total kinetic energy of the mode is due to the kinetic energy of both the electrons and of the oscillating ions and consequently, the effective mass is significantly larger than the single-particle mass. Treating the phase as a classical field, the Lagrangian density associated with d ( x , 1 ) is given by (2.15)

where n is the electron density. The first term is the kinetic energy of a line of mass density m * n per unit length. The potential energy is the second term, with a phenomenological elastic constant K characterizing the distortions of the CDW ground state. The dispersion relation corresponding to wave-like excitations of the form exp( -iwr - kx) is given by a’= ( ~ / m * ) ( 2 k , q ) ’ ,

(2.16)

and the microscopic theory (Lee et al. 1974) leads to the following expression for the elastic constant K

= m ( i h u t / k,)’.

(2.17)

The excitations, which correspond to local modulation of the phase 4 with a period 9 are called phasons. Similarly to a superconductor, the phase b(x, 1 ) of the condensate plays an important role in the dynamics of the collective mode. A rigid displacement of the CDW leads to an electrical current. With j = = ne(dx/dr) (where n is the carrier density and ud the drift velocity), and $I = 2k,.x one obtains e d d per chain. j = -ST

dt

(2.18)

A compression of the wave leads to the change of the electronic density,

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

207

and therefore 1 d4 - per chain. dx

n =-

(2.19)

IT

Equations (2.18) and (2.19) are different from those which relate d(x, t ) to the current and chemical potential in a superconductor, for which j = -6 d 4 / d x and p = 6 d4/dr. Equations (2.18) and (2.19) can also be derived by noting that the Fermi surface is tied to a moving CDW or to a compressed CDW and evaluating the total current or electron density using a band picture. In that case, it is assumed that, as in a superconductor for a slowly varying +(x, t ) (on the time scale much less than A J h ) , the Fermi surface is tied to the position and time dependent condensate. Consider first a slowly varying +(x), for which ( l / k F ) ( d 4 / d x ) < 1. This can be described as a change in the wavevector 2kF+d4/dx, and consequently the single-particlegap is shifted from f k F t o f(kF+i(d4/dx)).Thesituation is shown in fig. 3. If the electrons do not change their density then the gap is removed from the Fermi surface. This leads to raising electrons above the gap, and to a large increase of the total electronic energy. A change of the electron density, by tying the gap to the Fermi surface involves less

I I I

J\ I I

I I

I b

!

-

+-. 1 kF

2wFh

, I

Fig. 3. Displaced Fermi surface for a position (a) and for a time dependent (b) phase. The states are filled up to the Fermi level, *(k,+idd(x)/dx) in fig. 3a, and up to i ( k b i (1/2u,h) x (d&/dr)) in fig. 3b. The dotted lines refer to the original Fermi surface at *kr..

G . GRUNER

208

energy. As p ( x ) = 2kF(x)/27r,the gap is at the Fermi level if (2.20) the same relation as eq. (2.18). For a time dependent phase 4( I ) the inversion symmetry is broken and the dispersion relation becomes asymmetric. The electronic energies are given by c+k = [ v F kf

5 d 4 l d t + d']"'

(2.21)

near the gap. The energies corresponding to the displaced Fermi surface are given by E,&

=

-A

zti d 4 / d t ,

(2.22)

and the total current (2.23) the same as eq. (2.19). This relation, and also eq. (2.21), are expected to break down for strong local distortions of the condensate. In such cases both the amplitude and phase perturbations have to be included. This however has not been considered in detail. In the presence of an applied electric field E the equation of motion (2.24) and the frequency dependent conductivity (2.25) with the real part Re d

nem

w )= y S ( w ) .

(2.26)

Re a ( w ) then has a S ( w ) peak at zero frequency, with an oscillator strength .f = m e ' / m * , a feature reminiscent of superconductivity, The optical conductivity, including both the zero frequency collective mode, and contributions from carrier excitations across the single-particle gap is displayed in fig. 4a. The fact that the phason mode is gapless has important consequences as far as the effect of impurities and other lattice imperfections is concerned.

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

209

W

WO

W

Fig. 4. Frequency dependent response of the collective mode (a) wit..mt pinning, and (b) with pinning and damping. The response at frequencies w > 2A/8 is due to single particle excitations.

Fukuyama and Lee (1978) and Efetov and Larkin (1977) considered the problem of impurity pinning in detail. The Hamiltonian in one dimension (2.27) where the first term represents the elastic energy of the deformable CDW, and the second term describes the interaction between the CDW and the impurities distributed at random at positions i assuming that the impurity potentials have the form of V(x) = Vo6(x). In less than four dimensions, eq. (2.27) leads to a finite phase-phase correlation length and also to a finite pinning energy in the thermodynamic limit. The first term in (2.27) favors a uniform phase, the second favors local distortions around impurities. A dimensionless parameter E=-

VOP,

(2.28)

huFni'

*

1 (strong pinning) the tells us which of these is more important. For phase of the CDW is fully adjusted to every impurity site to obtain a

G . CRUNER

210

maximum potential energy gain, the cost in elastic energy for doing this is negligible. For < 1 (weak pinning) the phase cannot be adjusted fully at every impurity site but only over a length scale Lo longer than l / n , the average distance between the impurities. For strong impurity pinning, where the elastic energy of the CDW is neglected, the pinning energy is trivially given by

(2.29) where n, is the number of impurities per unit volume. In this limit the pinning energy is proportional to the impurity concentration. As the phase is fully adjusted at every impurity site, the phase-phase correlation length EPIN

= AEPOT = - vpP,

9

Lo- l / n l .

(2.30)

with the impurity concentration c, = 10 ppm, Lo is approximately 10 cm along the chain direction. The case of weak impurity pinning is more interesting. In three dimensions ( 3 D ) , assuming a volume Li over which the phase is constant but is adjusted to the impurity fluctuations, one obtains a potential energy gain A E w n = - VnPi(nt/G)”’,

(2.31)

the elastic energy, from (2.27)

EkL= : n f i u F /Li. (2.32) Minimizing the total energy with respect to Lo leads to a finite phase-phase correlation length (2.33) and

(2.34) i.e., the pinning energy is proportional to the square of the impurity concentration. With typical values of uF = lo7cm/s, Vo = lo-’ eV, p I = 0.1 po and n,= 10ppm the coherence length Lo-lOPm not much smaller than the typical length of the specimens being investigated. The coherence lengths perpendicular to the chains are given by L , = LIIuFL/ uFII, and are reduced from LII by the ratio of the bandwidths in the directions. Consequently, L A is of the order of I - I O - ’ pm comparable to the transverse directions of the specimens. The above estimates suggest, that, depending on the dimensions of the specimens, impurity concentrations, etc., the range of phase coherence can be either smaller or larger than the typical dimensions of the specimens. A coherent response is expected in the latter case, while in the former, internal degrees of freedom may play an important role in

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

21 1

the dynamics of the collective mode. For the same parameters, the pinning eV, much smaller than the single particle gap A, which energy EPIN is of the order of lo-’ eV in various materials. While these concepts have been confirmed by a variety of experiments on alloys and also on materials where pinning centers have been created by irradiation, the question whether weak impurity pinning has been observed remains controversial (see for example, the articles in “Charge Density Waves in Solids”, eds Gy. Hutiray and J. Solyom, 1985). Extended defects, such as grain boundaries have also been shown to be important, in particular, in materials where the residual impurity concentration is low and consequently, the overall pinning potential is small. In these cases, the phase-phase correlation length can also be comparable to the dimensions of the specimens leading to pinning by surfaces (Borodin et al. 1986, Gill 1985) or contacts which are provided to measure the transport properties. It is expected that the length scale Lo plays an important role both in the statics and in the dynamics of the collective mode. Broadly speaking, if Lo > 1, the length of the specimen, the response is expected to be coherent, with disorder effects due to random impurity distributions playing only a secondary role. For 1 > Lo, on the other hand, the role of internal degrees of freedom may become important, and these in turn may dominate the nonlinear and ac response, and also the various features of the time dependent current.

-

2.3. FREQUENCY AND

F I E L D DEPENDENT TRANSPORT

The interaction between lattice imperfections or boundaries, and the collective mode has several important consequences. Pinning leads to a preferred average phase of the condensate (e( r ) ) = 0, where ( ) refers to spatial average. For small displacement from the equilibrium position X,, the potential is given by V(x) = - l a 2 ,

(2.35)

and V(x = ; A ) = EplNas given by eqs. (2.28) and (2.33). Neglecting effects related to random impurity distributions, or possible local distortions of the collective mode, the equation of motion in the presence of an applied ac field can be written as -dx2 + f - + +dxi x = y e eE df2 dt m

iwt

,

(2.36)

where f = 1 / r is a damping constant (introduced phenomenologically in the above equation), and w , = J k / m . The restoring force leads to an overall shift of the oscillator strength from zero to finite frequency, while the finite damping disorder effects result in a finite width, with the expected a(w ) as

G . GRUNER

212

displayed in fig. 5. (Sridhar et al. 1986, Reagor et al. 1986, Reagor and Gruner 1986). The single-particle gaps, in the region of approximately 10'3-10'4Hz are well documented by optical studies, and, in the gap region the strong resonances correspond to the collective mode oscillating about the equilibrium positions, with the center frequency wo in the millimeter wave spectral region. Equation (2.36) leads to a frequency dependent conductivity with real and imaginary parts given by Rea(w)=-

ne'r

w'/ r 2

m* ( w t - w2)'

ne2r Ima(w)=-

(2.37a)

+w 2 / r 2 '

(wt-w*)w/r

(2.37b)

m* ( w i - w 2 ) ' + w2/r2'

and the above equations give a good overall account of the observed w

NbSc,

'i

0

c

KCP

L

)

FREQUENCY IHzI

Fig. 5. Frequency dependent conductivities, Re m(o)of various materials in their CDW state. The full lines represent contributions from transitions across the single-particle gaps, the dotted lines show the contributions due to the pinned collective mode.

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

213

dependent response, such as shown for TaS3 in fig. 6. The full line in the figure is a fit to the harmonic oscillator equation with parameters w0/27r = 5 x lo9 Hz, I / ? = 125 x lo9 Hz and m * / m = 940. Detailed studies on alloys of TaS3 (Reagor and Griiner 1986) and (TaSe,),I (Reagor et al. 1985, Kim and Griiner 1988) are in broad agreement with the concept of impurity pinning as wo increases with increasing impurity concentration. In contrast, the spectral width and intensity (the latter related, through the oscillator sum rule to the effective mass) are independent of impurity concentration for small amount of impurities. The description of the ac response in terms of eq. (2.37) neglects the important role of disorder and contributions to a ( w ) coming from the dynamics of the internal deformations of the collective mode, as well as quantum effects. These will be discussed in section 4. The small pinning energy associated with the collective mode suggest that for small d c electric fields, the collective mode can be depinned and

I

5

I

1

I

I

I

-

-0.4-

- 0.8 10'

-

I

1

I

1

10'

10'0 W/2H

1

I

lo'*

(HZ)

Fig. 6. Measurements of Re a(o) and Im u(o)in TaS, at T = 160 K. The full line is a fit to the expression eq. (2.37a), with parameters given in the text. The dotted line follows from a model where a distribution of the characteristic frequencies is included. (After Sridhar et al. 1985.)

G . GRUNER

214

driven into a current-carrying state with possibly nonlinear currents-voltage characteristics. The condition for this is approximately given by EPIN < eEA, where E is the applied dc field and A the period of the CDW. In the above relation the right-hand side is the energy gain associated with the displacement of the collective mode by one wavelength in the presence of the electric field. Indeed, nonlinear conductivity has been observed at moderate applied fields, in all the materials discussed before, with a sharp onset field, called the threshold ET for the nonlinear conductivity. The detailed form of the nonlinear conduction varies depending on external factors such as temperature, impurity concentration, or macroscopic inhomogenities (such as grain boundaries) in the specimen. The dc conductivity, defined as ( j (t ) ) / E where ( j (t ) } is the time averaged total current density and E is the applied electric field is shown in orthorhombic TaS, in fig.7. Below a threshold field ET, approximately 300 mV/cm, the conductivity obeys Ohm's law, and the temperature dependence of this component reflects the exponential freezing out of the electrons excited across the single-particle gap. In contrast, the strongly nonlinear conductivity for E > ET has a much weaker temperature dependence. The onset of nonlinear conduction is smooth as evidenced by the current-voltage characteristics also displayed in the figure. A behavior, similar to that displayed in fig. 7, is found in general for the other compounds, and the existence of the sharp threshold field ET is well established. ET is closely related to the pinning frequency w,; both increase with increasing impurity

1

s

* * ...*I

*

Iv/Cm

- .. *

~ ~ ~ m a *u l I

9

3

#)Vlcm

E( V k mI Fig. 7. Electric field dependent conductivity u(E ) in o-TaS,. The data are normalized to the room temperature conductivity. The inset shows typical dc I - V characteristics on the same material.

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

215

concentration in alloys, and also in materials where wo is larger, also a larger threshold field ET has been found (Griiner 1988). An empirical relation between the low frequency dielectric constant E ( o -* 0) ET = const x e, found for a broad range of materials, can be explained using general arguments about the pinning energy of the collective mode (Wu et al. 1984). The behavior in the figure can be described in terms of a two-fluid model involving electrons excited across the gap and electrons in the collective mode. The former has an ohmic contribution, the other, a nonlinear response which appears as E > ET. The total current can be written as (2.38)

where the subscripts n and CDW refer to current carried by the uncondensed and condensed electrons. The validity of such a two-fluid description has been clearly demonstrated recently by generating nonlinear current-voltage characteristics in an open circuit configuration ( It,,,I= 0) where the voltage was generated through the thermoelectric effect (Beyermann et al. 1986). The current-voltage characteristics are also often explored by measuring the differential resistance d V/dI, employing low frequency lock-in techniques. While the method cannot be used at high fields because heating effects, it is advantageous to study the behavior near threshold. The first such measurements (Fleming and Grimes 1979), where a well-defined ET was evident is shown in fig. 8 (Fleming 1981). The various models, which account for the nonlinear conductivity will be discussed later. The models lead to various predictions on the overall current-voltage characteristics, and on the detailed behavior for large currents and also close to threshold. The experimental state of affairs remains highly controversial, and is, most probably mainly due to the differences in the sample quality, extent of phase coherence (with respect to the dimensions of the specimens), temperature and other factors. Broadly speaking, highly coherent specimens lead to sharp current-voltage characteristics, which may also display negative differential resistances, while in materials where other evidences (such as long time relaxation effects in the pinned state (Mihily and Mihaly 1984) point to disorder effects, the nonlinear behavior is more gradual. This difference most probably also reflects the differences in the pinning mechanism. Experiments on ET versus the dimensions of the specimens (Gill 1982, Zettl and Gruner 1984, Monceau et al. 1986, Borodin et al. 1986) suggest, that for small samples, surface or contact pinning are important, with randomly pinned impurities playing a less important role. In addition to the above differences, the onset of nonlinear conduction is often rather dramatic and is accompanied by switching and hysteresis

G . GRUNER

216

0 4 F ;

;o

&

I; E

25 '

30 I

.LP

& i

(mV/cm)

Fig. 8. Normalized differential resistance as a function of dc bias in NbSe,. A threshold field for the onset of nonlinear conduction is clearly observed. The field dependence of the current is also shown for the low temperature CDW state. The solid lines correspond to eq. (4.21) using parameters indicated on the figure. (After Fleming 1981.)

effects. These have been observed in NbSe, (Zettl and Griiner 1982), TaS, (Mihaly and Gruner 1984), and &.,Moo3 (Maeda et al. 1986), such effects usually become more pronounced with decreasing temperature. The behavior most probably is due to extended pinning centers in the materials, and this has been demonstrated directly by locating (by using moving contacts) the exact positions in the specimen where such behavior is generated (Brown and Mihaly 1985). At very low temperatures T < Tp still a different type of nonlinearity, a rather steep current-voltage characteristic is observed with a threshold field one or two orders of magnitude larger than ETwhere the smooth nonlinear conduction such as displayed in figs. 7 and 8 occurs (Mihily and Tessema 1986). This low temperature behavior, while most likely also associated with CDW dynamics, will not be discussed further in this review. Also, I will not extensively discuss experiments which focus on switching effects.

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

217

3. Current oscillations

3.1. GENERAL FEATURES In Fig. 7, I refers to the time average dc current, measured in the presence of an applied constant voltage or to the applied d c current - depending on whether constant current or constant voltage configuration is used to measure the current-voltage characteristics. (In principle, the currentvoltage characteristics can be different, depending on whether a constant current or constant voltage configuration is employed. This difference, however, has not been studied in detail.) In the nonlinear conductivity region, the current has, in addition to the dc, an oscillating component, most conveniently studied by monitoring the Fourier transformed current. Whether an oscillating current or oscillating voltage is measured depends on the external conditions, and both have been observed. In the following, however, I refer to the phenomenon as current oscillation or “narrow band noise” (NBN) the latter is the traditional notation used. The first observation by Fleming and Grimes (1979) in NbSe3 is displayed in fig. 9. A substantial broad band noise (BBN) accompanies the nonlinear conduction, and superimposed a narrow band “noise”, with a fundamental and several harmonics is also observed. The spectrum moves to higher frequencies with increasing applied voltage. Similar observations have subsequently been made on other materials where nonlinear conduction has been observed, namely in TaS, (both orthorhombic and monoclinic phase), (TaSe,),I, ( NbSe4),.,I and &,,Moo3 or Rbo,,Mo03,and by now the phenomenon can be regarded as one of the fundamental observations on charge density wave dynamics. The overall features of the observed current oscillations depends sensitively on crystal quality and in general on the coherence associated with the current carried by the collective mode. It has been established, (Ong et al. 1984a,b, Ong and Maki 1985) that current or voltage contacts can introduce excess peaks in the Fourier transformed spectrum, and similarly mechanical damage to the specimens also leads to complicated “noise” patterns. In carefully manipulated specimens the current oscillations have a rather high quality factor and also display many harmonics (Weger et al. 1980). An example is displayed in fig. 10, again showing experimental results on NbSe, . The coherence associated with the current oscillations can be characterized by the quality factor Q of the peaks in the Fourier transformed current. The quality factor Q = A,/fo where A, refers to the width of the fundamental, was found to approach lo’, in contrast to Q - 4 observed in fig. 9. As a general rule, materials where the electronic structure is more anisotropic and consequently the coherence perpendicular to the chain

G. G R U N E R

218

i

- ro

-SOL' 0

I

04

'

'

08

"

( 2

"

16

'

1

2 0

FREOUENCY (YHZ)

Fig. 9. Fourier transform of the time dependent current in NbSe, for various applied currents. Narrow band "noise" results if the current exceeds the threshold value for nonlinear conduction. Currents and dc voltages are: (a) I = 270 FA, V = 5.81 mV; (b) I = 219 FA, V = 5.05 mV; (c) 1 = 154 FA, V = 4.07 mV; ( d ) I = 123 PA, V = 3.40 mV; (e) I = V = 0. T h e sample crosssectional area A = 136 Fm2. (After Fleming and Grimes 1979.)

direction is less well defined, show broader current oscillation peaks, which are also less prominent for specimens with larger cross sections (Mozurkewich et al. 1983). These effects however, have not been investigated in detail. There is also a broadly defined correlation between the width of the narrow band noise and the amplitude of the broad band noise: in materials, and also in specimens of the same material with smaller broad band noise amplitude the Q of the Fourier transformed current peaks is larger. At one end of this spectrum, NbSe, has rather small broad band noise and high Q for the current oscillations in both CDW states while &.,MOO, and (TaSe,),I have both large broad band noise at current oscillations with small quality factors. The material TaS, represents an intermediate case (Griiner 1988). The quality of the crystals used and of

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

-40

NbSe, Sample ii

I

-50

219

- II

T=44 K

603

-

a.

-70

-

-80

-

-90

-

1

-loot-,

I

0

, 2

,

, 4

,

,

,

6

, 8

,

, 10

Frequency IMHz) Fig. 10. Spectrum of the voltage oscillations measured in response to an applied dc current of a 2.0 mm long NbSe, sample at 44 K. A single fundamental and at least 23 harmonics are visible. (After Thorne et al. 1987c.)

the voltage and current contacts can also have an important influence (Beauchene et al. 1986) as macroscopic defects can lead to spurious velocity distributions in the specimens, while irregular contacts can lead to inhomogeneous current distributions. Current oscillations have also been studied in the time domain by recording the time dependence of the current or voltage following a pulse excitation. Such experiments have also first been performed in NbSe, (Fleming 1983, Bardeen et al. 1982), and current oscillations have been subsequently observed also in TaS, and &.,Moo3 (Fleming et al. 1985). The observation of oscillations in the time domain differs from studies employing Fourier transforms in several aspects. First, for a finite Q of the oscillations a dephasing is expected, with a time dependent amplitude Aj(f) given by

&(f)

-

- 4 1 -eXP(-t/T)l,

(3.1)

with T Q-'. Such gradual decrease of the oscillation amplitude has indeed been observed in all materials, with a more dramatic decrease for oscillations with smaller Q values (Parilla and Zettl (1985) and references cited therein, Fleming et al. (1985), Thorne et al. (1987~)).Because of the fluctuating oscillation frequency, the time dependence of the oscillation amplitude most probably represents the dephasing of the various CDW domains,

G . GRUNER

220

which are phase locked at the start of the applied pulse. Transient oscillations, induced by the sudden transition from the pinned to the current carrying state can also accompany the leading edge of the applied pulse. These transients have the fundamental oscillation frequency, their amplitude however, is greatly enhanced over the steady state oscillation amplitude which is observed in the Fourier transform current under dc current or dc voltage drive conditions. Pulses of finite length of the order of few oscillation periods can also lead to mode locking, to be discussed at length in section 5 , and the observations can then be regarded as a form of interference and locking phenomena which are the consequence of competing periodicities, here the inverse frequency of the oscillations and the duration of the applied pulse. 3.2.

C U R R E N T - F R E Q U E N C Y RELATION

The frequency of the oscillations is approximately proportional to the current carried by the CDW (Monceau et al. 1980), and subsequently it has also been shown that the ratio of the current to the frequency varies with temperature, approximately as the number of condensed electrons nCDW (Bardeen et al. 1982). Figure 11 displays the linear relation between ZcL,w and fo in a wide frequency range for a pure NbSe, specimen (Bardeen et

N

0

50

100

I50

F-ig. 1 1 . Relation between the CDW current and fundamental oscillation frequency in NbSe,. The inset shows Icow/fo versus temperature. (After Bardeen et al. 1982.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

221

al. 1982), with the ratio &DW/fO as the function of T in the insert. Normalizing to CDW current for one chain, the above observations can be described as (3.2)

where jcDw refers to the current per chain camed by the condensate. While the precise value of the constant c in eq. (3.2) is still debated it is between 1 and 2 for a broad range of materials. For NbSe,, TaS, (both orthorhombic and monoclinic form and for (TaSe&I early experiments (Monceau et al. 1983) were suggestive of ZcDw/fo = e indicating that the fundamental periodicity is related to a displacement of the collective mode by a half wavelength A /2. There are, however, several ambiguities in the analysis. In NbSe, only part of the conduction electrons are condensed into the collective mode, and the relation ZcDw/fo depends on the assumed number of condensed electrons. Similar problems arise in monoclinic TaS,. The definition of the fundamental is complicated in (TaSe,),I where occasionally the second harmonic has a larger intensity than the fundamental. In orthorhombic TaS, , the current-frequency relation is in general nonlinear due to the distribution of currents caused by the filamentary nature of the specimens. Using a mode locking technique to improve the coherence, the measured current frequency relation leads, in orthorhombic TaS, to (Brown and Griiner 1985) JCDW

- (2*0.l)e,

fo

(3.3)

and the above value was subsequently confirmed in an independent study, also employing mode locking (Latyshev et al. 1986). Careful NMR experiments on &.,MOO, specimens (Segransan et al. 1986) utilizing motional narrowing effects also lead to a relation IcDw/fo=(1.9*0.l)e in close agreement with the findings on orthorhombic TaS, . The above relation between the current and oscillation frequency is related to the fundamental periodicity associated with the phase 4 (Monceau et al. 1980). The CDW current, described in terms of drift velocity ud of the (rigidly moving) condensate is given by jcDw = n@d. Associating the fundamental frequency with a CDW displacement by one period, fo= ud/A, this then leads to j C D W l f 0 = nc(

T)eA,

(3.4)

with A = ?r/ kF and n, = 2kF/7r, eq. (3.2) with a constant c = 2 is recovered. Alternatively, the fundamental frequency can be related to the energy

G . GRUNER

222

difference A E between the two sides of the displaced Fermi surface shown in fig. 4. With (3.5)

AE = hfo and jcDw = n,evd also leads to eq. (3.2) with c = 2. The strictly linear relation between j c D w and fo as displayed in fig. 11, also suggests that the CDW velocity is constant throughout the specimen. Indeed j,,, is proportional to fo over a broad range of currents and frequencies only in high quality materials. In materials where other, independent evidence suggest that disorder plays an important role, deviations from the linear relation between j,,, and fo are observed (likely related to finite velocityvelocity correlation lengths) (Brown and Griiner 1985). The amplitude of the oscillating current increases with increasing collective mode current and appears to saturate in the high electric field limit (Weger et al. 1982, Mozurkewich and Griiner 1983). This has been studied in detail in NbSe,, and the amplitude of the oscillating current is displayed in fig. 12 as a function of oscillation frequency. In experiments where the oscillation amplitude was measured up to fo = 700 MHz (Weger et al. 1982, Thorne et al. 1986a,b) the oscillation amplitude was found to saturate in the large velocity limit.

-

.

I

!

'

3

/

80-

6'o

20

,' 0

40

-6

0 O

E '

0

60 FREOUENCY

80

100

I20

fMc(i1

Fig. 12. Amplitude of the fundamental oscillation in NbSe, at 42.5 K, versus oscillation frequency. The amplitude is plotted both as a voltage AV, (solid symbols, left-hand scale) and as an equivalent current density Aj, (open symbols, right-hand scale). The inset shows a typical spectrum. The dotted line is eq. (6.25). (After Mozurkewich and Griiner 1983.)

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

3.3.

223

SIZE EFFECTS A N D FLUCTUATION PHENOMENA

The current oscillations are suggestive of translational motion of the condensate in a periodic potential. Considerable effort has been made to gain information on the origin of the pinning by measuring the oscillation amplitude as a function of sample volume, and also under nonequilibrium conditions such as provided by temperature gradients applied to the specimens. Although all experiments indicate that the current oscillations disappear in the thermodynamic limit (Mozurkewich and Griiner 1983) several questions, mainly centered around the detailed mechanisms of pinning and the mechanism of oscillating current generation remain. Experiments performed both in NbSe, (Mozurkewich and Griiner 1983) and in (TaSe,),I (Mozurkewich et al. 1983) specimens with different length and cross section are in agreement with a volume dependence of the relative oscillation amplitude AjcDW/jCDW=

const n-”’,

(3.6)

where R is the volume of the specimens. The results can be understood by postulating that the specimens are composed of domains of size L3 where the current is coherent, but the currents in different domains oscillate with random phase. In this case, the oscillation amplitude is given b y AJcDw/jcDw = A ~ ~ ( L ~ / R ) ’ / ’ , (3.7) with Ajo the oscillation amplitude expected for a coherent (single domain) response. The latter can be estimated by assuming simple models (to be discussed later) and then L3-0.2 pm3 as an appropriate lower limit of the domain volume is obtained in NbSe, and in (TaSe,),I. The macroscopic domain size is most probably related to the Fukuyama-Lee-Rice (see, Lee and Rice 1979) coherence length discussed before. As discussed earlier eq. (2.33) leads, with reasonable values for the impurity potential and charge density wave amplitude, impurity concentration and Fermi velocity to Lo = 100 pm along the chain direction. For an anisotropic band structure, the coherence length is also anisotropic (Lee and Rice 1979), and in the directions perpendicular to the chains the phase phase correlation length L , is given by

-

-

A typical anisotropy of f+ll/vFI 30 leads to L , 0.3 pm and to a domain size V = LoL: of about 1 pm3, in broad agreement with the experimental results. A different set of experiments (Ong and Maki 1985, Ong et al. 1984a,b) performed in NbSe, however, indicates that for long specimens, the oscillation amplitude is independent of I, the length of the specimens,

224

G . GRUNER

and the observation was used to argue that the oscillating current is generated at the contacts and not at the bulk (see also Jing and Ong 1986). Also experiments (Jing and Ong 1986) where a length dependent oscillation amplitude was recovered, were used to establish the range of phase coherence, and in NbSe, the experimental results are suggestive of Lo 200 pm. The differencebetween the two sets of experiments may reflect the difference in contact configurations, impurity concentrations, etc. The issue was further investigated by Brown and Mihily (1985) who employed a sliding, nonperturbative contact configuration to measure the noise amplitude in different sections of the specimens investigated. The observations are consistent with independent domains oscillating with the same frequency, but with random phase, as suggested by eq. (3.6). In several specimens evidence was also found for sudden jumps in the oscillation amplitude when the contacts were moved along the specimens, indicating oscillating current generation by macroscopic defects such as grain boundaries within the specimens. Several studies have been made to distinguish between proposals where the oscillating current is generated in the bulk or at the contacts by applying a temperature difference between the two ends of the specimens. The argument for conducting such experiments is simple: with the two contacts at different local temperatures TI and T I ,they expect to generate oscillations at different frequencies, each corresponding to the local temperatures. Narrow band noise, generated in the bulk, however, may be smeared by a temperature gradient, or alternatively be split into several frequencies each corresponding to a velocity coherent domain. The first experiments (Zettl et al. 1985) on small NbSe3 specimens did not reveal any splitting of the fundamental oscillation, which was also determined by the average temperature ;( TI+ T2) of the specimen. The result is indicative of velocity coherence, comparable to the length 1 1 mm of the specimen. In subsequent experiments (Ong et al. 1985, Verma and Ong 1984) however, a splitting was found and this was taken as evidence for current generation at the contacts. It was shown later, however, by several groups (Brown et al. 1985b, Lyding et al. 1986) that by increasing the temperature gradient, further splittings can be induced, demonstrating that the various oscillation frequencies cannot be associated with contacts alone. In fig. 13, the observations made by Lyding et al. (1986) are displayed. A fundamental observed at f = 1.1 M H z for AT = 0 is split into several peaks with increasing temperature gradient. The width of the peaks is approximately the same as that of the peak for AT = 0, suggesting that the sample is broken up by the application of a temperature gradient into velocity coherent regions. The minimum velocity coherence estimated from the number of peaks and of the sample length is of the order of lOOpm, in qualitative agreement with the conclusions of earlier studies.

-

-

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

225

NbSe,

r, 1400

1

E

-

-

115.7 K

165

N/cr I

-

-

-

0.1

0.5

0.9

1.3

1.7

18.9

22.2 28.0

32.0 K

2.1

f (MHzJ

Fig. 13. Splitting of the fundamental NBN peak for the upper Peierls transition of NbSe, as a function of applied thermal gradient. The gradient is opened around an average temperature of 115.7 K and the sample length is 0.89 mm. (After Lyding et al. 1986.)

The current oscillations, such as displayed in figs. 10 and 11, have large temporal fluctuations both in position and in amplitude. The time scale of the fluctuations, is significantly larger than the inverse fundamental frequency, and is typically on the scale of microseconds. The phenomenon was first studied by Brown et al. (1985a,b) and investigated in detail by Link and Mozurkewich (1988) and by Bhattacharya et al. (1987). The probability densities of the noise voltages follow a Gaussian distribution, suggestingthat the current oscillations have many of the characteristics of Gaussian noise, and cannot be regarded as a coherent signal over a substantial period of time (Link and Mozurkewich 1988). This is demonstrated in fig. 14 where the probability distribution of the NBN voltages obtained for repeated scales of approximately 1 ms is displayed. The full line is a Gaussian distribution with a width u = 3.3 pV. The distribution reflects the underlying probability density in the recorded oscillation amplitudes which are far enough removed from each other that they are uncorreiated. This is appropriate for long sampling periods with strongly correlated oscillations expected for sampling with short time differences. A crossover from such coherent to Gaussian noise was also studied, and the typical time for the crossover is, for NbSe,, of the order of 10 ps. Similar experiments have not been performed in other materials, although temporal fluctuations of the oscillation amplitudes have been widely observed.

G. GR~JNER

226

n d

2m 0.04 0

a

n

0.02 0 VOLTAGE !JLV)

Fig. 14. Histograms of voltages of narrow band noise in NbSe,. The solid line represents the best fit Gaussian with u = 3.3 pV. (After Link and Mozurkewich 1988.)

3.4. BROAD B A N D

NOlSE

The Fourier transformed spectra, such as displayed in fig. 9 have also a significant broad band component, with an effective noise temperature corresponding to several thousands degrees K (Richard et al. 1982, Zettl and Griiner 1983). The amplitude of the noise is inversely proportional to the square root volume O-”* (Richard et al. 1982), and the observation is suggestive of broad band noise generation in the bulk. In pure NbSe, specimens, however (Thorne et al. 1986a,b), the noise amplitude could clearly be related to the presence of macroscopic inhomogeneities (such as breaks in the specimens) and this suggests that the broad band noise is generated at macroscopic boundaries such as grain boundary or breaks within the specimens. The issue here is similar to that raised in connection to the current oscillations, with the most probable situation that in general both macroscopic defects and impurities contribute to sources of the broad band noise as both lead to the disruption of the phase of the condensate. Clearly detailed experiments on alloys with different impurity concentrations would be required to clarify this point. The spectral dependence can be well described with a characteristic w - ~ behavior with a < l (Bhattacharya et al. 1985, Maeda et al. 1983, 1985) reminiscent of l/f noise widely observed under different circumstances in various current-carrying conditions. The observed field, frequency and temperature dependence of the broad band noise is inconsistent with a generalized form of the Nyquist noise (Maeda et al. 1983) and the following model has been proposed (Bhattacharya et al. 1985). For constant current, fluctuations of the threshold voltage VT within the specimens (assuming that E T depends on the position) leads to fluctuations of the cordial resistance R = V / I and the noise voltage is given by (6 v’(0))= 12(6R’( w )) = I * ( 8 R / 8 v’)’ (6 v+(W )). (3.9)

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

227

It is also assumed that SR/SVT is proportional to S V / 6 V T , i.e., R is a function of V- VT only, and that both quantities are independent of the frequency. (The latter valid well below the pinning frequency w,,.) Detailed measurements performed on TaS, indeed are in agreement with eq. (3.9). The volume dependence of the noise indicates that it is an incoherent addition of fluctuations generated in coherent volumes, given by L3. If the fluctuation of the effective pinning force 6ET is proportional to E T , then (3.10)

with S ( o) the spectral weight function, with S ( w ) dw = 1, and A the cross section of the specimens. Equation (3.10) is the consequence of the assumption that the number of independent entities generating the broad band noise is given by R / L 3 . An analysis of the experimental results leads to L i - 1 pm3 in TaS,, in broad agreement with coherent volumes estimated from the volume dependence of the current oscillations in NbSe, and in (TaSe.&I (Mozurkewich and Griiner 1983, Mozurkewich et al. 1983).

4. Models of charge density wave dynamics

The current oscillations which often have a high quality factor are also suggestive of a highly coherent response with distribution of CDW velocities (which would lead to a spread of the oscillation frequencies) throughout the specimens playing only a minor role. Consequently, the first attempts to account for the frequency and field dependent response have been in terms of the dynamics of a single coordinate, the spatially averaged phase ( 4 ( r ) ) .As discussed earlier, such descriptions are expected to be most appropriate to small specimens of pure NbSe3. On the other end of the spectrum, in (TaSe,)J and in K,,,Mo03, the significant amount of broad band noise and the weak current oscillations, suggest that the dynamics of the internal modes and the effect of randomly positioned impurities are important, and the treatment of the problem as having many degrees of freedom - close to the thermodynamic limit - is appropriate. Situations intermediate between these two limits have also been considered. 4.1. THE CLASSICAL

PARTICLE MODEL

The model, which in general, is called the classical particle model (Griiner et al. 1981, Monceau et al. 1982) assumes that the condensate can be represented by a single degree of freedom classical object, which moves under the influence of the electric field E ( t ) . The inherent periodicity of

G.

228

GRUNER

the condensate is transformed into a periodic potential V(x). The equation of motion for the average coordinate, which characterizes the position of the condensate, is given by dx2 1 dx -+-+ dt’ r d t

V(x)=-

eE(f)

(4.1)

’

m*

where l / r is a phenomenological damping constant. Assumption of a sinusoidal pinning potential V(x) = ( w i / q ) sin qx, where q = 27r/A, leads to dx2 1 dx -+--+-sinqx=~. dt2 r dt q

eE

(4.2)

m

The model is shown in fig. 15. The small amplitude ac response is that of a classical harmonic oscillator with the real and imaginary part of the conductivity given by eq. (2.37). As discussed earlier, eq. (2.37) leads to a fair agreement with experiments conducted in various materials. Differences ) on the basis of the single harmonic between Re U ( W ) and Im ~ ( wcalculated oscillator response and the experimentally found behavior will be discussed later. The model leads to a sharp threshold field ET where nonlinear conduction sets in, and in terms of the parameters of the problem A m*wi

E’=g e

(4.3)

dc conduction :

ac conduction: \

I

\ \ \

I

/

Fig. IS. Classical particle model of charge density wave transport. The model also describes the behavior of the resistively shunted Josephson junction, as indicated on the figure.

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

229

for a sinusoidal pinning potential. The velocity for E > ET dx _ -

dr

+

;T( eaE/ m* oi/Q) a + a c Z - ' [ c o s 2 ( o t ) - s i n 2 ( o t ) ] - -aa-2)1/2sin(or) 2(1 cos(or)'

(4.4)

where

and a = E / E , . Equation (4.3) predicts a fundamental oscillation frequency e2E2T2

f(E)=m*2( E - E : ) I / ~ ,

(4.5)

with harmonics in the CDW current at

-

I,, ( a 2 -1)'/2[a - ( a 2 -1)"*]" cos n [ o r + ~ / 2 + s i n - ' a ] ,

(4.6)

with n = 1,2,3, . . . . The time-averaged CDW current density for the entire sample is given by kDW/fO

= n A T ) eA.

(4.7)

where n, is the density of electrons condensed in the CDW state. Equation (4.6) leads to a simple relation between the CDW current density and the fundamental oscillation frequency JCDWlfO

(4.8)

= nc( T ) eA.

With A = r / k F and n, ( T = 0 ) = k F , eq. (1.2) is recovered. The model also leads to a rich harmonic content of the current oscillation and the higher harmonics decrease approximately as I,, K 2 ,for electric fields close to threshold. With increasing electric fields, the harmonic content decreases (Lindelof 1981). The substitution 8 = qpx leads to the dimensionless form

-

d2e/dt2+rde/dt+sin

e = E/E,,

(4.9)

where r=(W ,,T)-* , ET=(A/2a)(m*wi/e), and time t is measured in units of 00'. Equation (4.9) is formally identical to the equation which describes the behavior of resistively shunted Josephson junctions. -+G*+sin d24 dr2 dr

Q=

I/I,,

(4.10)

where CPis the phase difference across the junction, I is the current through the junction, and G = (RC",)-', where R and C are the resistance and

G . GRUNER

230

capacitance of the junction, and W , = 2el,/ Ch. I, is the dc critical Josephson current, and time is measured in units of w ; ’ . This formal analogy suggests a close formal correspondence between phenomena observed in Josephson junctions and in materials displaying CDW transport. For example, the current oscillations described by eq. (1.2) correspond to the ac Josephson effect while the threshold field ET corresponds to the critical current I , . This formal analogy will be used later to analyze a variety of interference experiments in the presence of combined dc and ac electric fields. Equation (4.9) is also analogous to the equation describing other systems, for example a damped pendulum in a gravitational field or a phase-locked loop configuration of a voltage controlled oscillator. The pendulum analogy, in the context of Josephson junctions, was first pointed out by Anderson (1967). In table 1 the parameters which enter in the classical description of CDW transport, the parameters which describe the response of Josephson junctions, and the parameters appropriate to the mechanical pendulum analog are collected, together with the respective equations of motion. A very simple model using electronic circuit analogs has also been proposed to describe the experimental findings in CDW systems (Weger et al. 1982). The low-field ac response can be described by a simple circuit composed of resistances and capacitances such as shown in fig. 16 which leads to Re a ( w ) and Im a(o)which is equivalent to that of a harmonic oscillator. To account for the nonlinear dc conductivity, a circuit element is included which starts to conduct if the applied voltage exceeds V,, the breakdown voltage. V, is a cutoff voltage at which the element ceases to conduct. The nonlinear dc element may be, for example, a discharge tube or silicon-controlled rectifier (SCR). The circuit as displayed in fig. 16 is a Table 1 Correspondence between classical particle model of CDW transport, the Josephson equation and the motion of pendulum under gravitation. CDW

applied field E mass m* damping coeff. I / s threshold field E , periodic potential Qx d2x I dx 0,; . -+- -+-sin Ox =-;eE dr2 r dr O m

Josephson junction dc current I,, capacitance C conductance I / R critical current I, phase difference @

Pendulum applied torque r, moment of inertia M , damping coeff. 0, torque due to gravity mgl angle from vertical O

h d2@ h I d@ -C,+- -+ I sin @ = Id=; 2e dr 2e R dr

d’tJ do M,-+ 0,-+ mgl sin 8 = r, dt2 dt

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

23 1

vdc

'b "C

time t Fig. 16. An electrical analog which describes certain aspects of CDW conduction: the relaxation oscillator. (After Weger et al. 1980.)

relaxation oscillator. The response to small amplitude ac driving fields ( V < V,) is that of an overdamped oscillator (the inertial terms, which would be represented by an inductance L, are not included). Dc current does not initially flow if the external voltage v d , < V, . As v d , exceeds V,,, the relaxation oscillator discharges, and then charging of the capacitance starts again. This leads to oscillations as shown in fig. 16. The frequency of oscillations is (4.11)

while the time average current is ( 1 )= (Vm- V C ) G

(4.12)

for vd, > V, . (I)= 0 for V < V,,, and is a strongly nonlinear function of V,, . It is also evident that (Z)-h as observed experimentally. The Fourier transform of the time dependent voltage is rich in harmonics, explaining the slow decay of the intensity of the harmonics in the CDW noise spectrum. While in this model there is no divergence in the ac dielectric constant below threshold, the differential conductivity d I / d V diverges as V + V, from above, a feature also obtained in the classical rigid-particle model.

G . GRUNER

232

The relaxation oscillator is the simplest nonlinear electronic circuit which describes qualitatively the observed field and frequency dependent response at low frequencies and also the current oscillations. An inductive term has also to be included in order to describe the full frequency dependent response observed. More complicated nonlinear feedback circuits, which are able to reproduce the sinusoidal potential, have also been extensively used to model the behavior of Josephson junctions (Fack and Kose 1971, Beasley and Hubeman 1982). Incoherent effects have been treated within the framework of the classical particle model by introducing a noise term r(I ) (Wonneberger and Breymayer 1984, Wonneberger 1983, Breymayer et al. 1982, Kajanto and Salomaa 1985). Assuming a Gaussian noise for r Focker-Planck treatment of the equation of motion, eq. (4.2) is possible. With the time correlation function

(r(t)r(1 ) ) = 2 L-a~),

(4.13)

where the parameter T,, is given by (4.14)

Tdf = E d ,

with Edcthe applied dc field, P the depinning probability, the model leads to hysteretic current-voltage characteristics, and, as expected, a finite width for the current oscillation. Also, higher order steps tend to be suppressed by noise. Other assumptions lead to rounding of the I- V characteristics, with the removal of sharp threshold (Arani and Ambegaokar 1982, Ambegaokar and Halperin 1969). 4.2. MODELS I N C L U D I N G

T H E I N T E R N A L D E C R E E S OF F R E E D O M

The observation of fluctuation effects associated with the current oscillations, broad band noise and the disappearance of the current oscillations in the thermodynamic limit (together with other evidences for disorder effects in the pinned CDW state) suggest that a single degree of freedom dynamics of the collective mode has to be extended to treat the dynamics of the internal deformations. The treatment of the pinning by Fukuyama, Lee and Rice (Fukuyama 1978, Lee and Rice 1979) which accounts for pinning effects and for macroscopic length scales associated with the phasephase correlation length has been used extensively as a starting point to describe the dynamics of the collective mode, and both hydrodynamic treatments, computer simulations and mean field treatments of the dynamical problem with many degrees of freedom are available. As discussed earlier, specimens with dimensions significantly smaller than the static phase-phase correlation length Lo, given by eq. (2.30) and (2.33) in the strong and weak impurity pinning limit the single degree of

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

233

freedom description of the dynamics is expected to be appropriate. Deviations from this behavior can be treated perturbatively, for specimens where L: n. The approach by Klemm and Schrieffer (1983, 1984) (see also Robbins and Klemm (1986)) treats the spatial fluctuations of the spatial and time dependent phase c$(r, t ) perturbatively with the equation of motion

-

aZc$ -+at2

1 ac$ --V2t$

7

- VoS,(r) sin(2kFr+ t$(r,t ) ) = eE(r, t ) ,

at

(4.15)

where S,(r) is the fluctuation in the impurity density. The dynamics of the collective coordinate c$(l) = (c$k =

I

cl

c$ dr3.

(4.16)

where the spatial average taken over a coherent volume of Li is evaluated using perturbation theory in the adiabatic approximation. The picture which emerges from such treatment is that the collective mode, in the presence of random impurity distributions, has many metastable configurations, with many local threshold fields ET,the largest of which has to be exceeded by the applied field to obtain a current carrying CDW state. The calculated w dependent resppnse is similar to that given by the single degree of freedom dynamics, with corrections in the high frequency, (but w < w o ) limit (Robbins and Klemm 1986). The current-voltage characteristics are similar to that obtained by the classical particle model, with the differential resistance d I / d V diverging as E,is approached from above. For large CDW velocities, the calculated I - V curve is qualitatively different from that given by eq. (4.7), and d I / d V increases with increasing applied electric fields. The perturbation treatment in terms of fluctuations in the impurity density breaks down when the phase coherent volume L: becomes significantly smaller than the volume of the specimens. In this situation the internal degree of freedom can be taken into account by assuming a position ( r ) dependent phase, and the equation of motion is given by 1 d@(r) --=-7

dt

6% k,eE

+- m * '

tic$

(4.17)

where X i s given by eq. (2.27). The above equation assumes that relaxational dynamics is important and inertia effects are neglected. A hydrodynamic treatment, valid in the high velocity limit has been proposed by Sneddon et al. 1982 (see also Sneddon 1984a,b). The time-average current has a characteristic I,,, = u,E - C a dependence on the applied field, where C is a constant which depends on the overall pinning energy and the elastic constant of the collective mode. Whether this high field behavior has been

G. GRUNER

234

observed or not is a matter of acute controversy at present: and experimental results which are in agreement (Coppersmith and Littlewood 1986) and in clear disagreement (Thorne et al. 1987a,b,c) with the above theoretical result have been reported. The origin of the disagreement between the various experimental results is not clear at present, but heating effects and uncertainties associated with the exact sources of the collective mode pinning may lead to different results. The above treatments of the CDW dynamics break down in the small velocity limit (near threshold) and here a mean field approach has been suggested (Fisher 1983, 1984, 1985). The Fukuyama-Lee Hamiltonian on a discrete lattice can be written as

k’(r,-r,)(4,-4,)2-E V, c0~(4,--P,), 1.1

(4.18)

I

where the first term represents the elastic energy and the second the interaction of the collective mode with the impurities k’ is the renormalized elastic constant, and V, the impurity potential. The phase p, is a random variable between 0 and T. The relaxational dynamics, both for infinite (Fisher 1983) and for short-range (Fisher 1985) interactions lead to various critical exponents near ET. These are fundamentally different from those which are the consequence of the single-particle dynamics. The time average velocity, given by ( V t [I)

- ( E - ETY,

t =;

(4.19)

is in significantly better overall agreement with the experiments. Nevertheless, close to threshold, analysis of the various experimental results in terms of the above equations does not lead to an exponent = (Monceau 1985). Current oscillations are not recovered in this treatment, and in the thermodynamic limit for finite size specimens, heuristic arguments lead to an oscillation amplitude for finite volume Aj/j

- const( L : / f l ) ’ ’ * ,

(4.20)

with the constant depending on the applied electric field close to threshold. The above equation is in agreement with the observations made in NbSe, and in (TaSe,)?l (Mozurkewich and Gruner 1983, Mozurkewich et al. 1983). A variety of numerical simulations have also been employed to study both various aspects of pinning by randomly positioned impurities and of the field and frequency dependent response, again within the framework of relaxational dynamics. Both one-dimensional (Matsukawa and Takayama 1984, 19861, Teranishi and Kubo 1979, Sokoloff 1981, Pietronero and Strassler 1983 and three-dimensional (Tua and Zawadowski 1984, Tua and Ruvalds 1984) simulations are available; they all give the same overall characteristics of the frequency dependent and nonlinear response. In the

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

235

pinned state, the most prominent feature is the presence of many metastable states, which are close in energy, but far in configuration space. This leads, in general, to a glassy behavior, and to frequency and long-time relaxational phenomena, similar to those observed in glasses, spin glasses and other random systems. Low frequency tails, at frequencies w 4 w o , and nonexponential time decays in response to pulse excitations are indeed widely observed in various CDW materials, together with hysteresis and nonlinear ac response phenomena. These are all explained, at least qualitatively, by computer simulations (see, for example, Littlewood 1986 and references cited therein). As an example of the improved agreement between theory and experiment, Re a ( w ) and Im a ( w ) measured in TaS, are compared with calculations based on the Fukuyama--Lee model in fig. 17. The w dependent response is not modified at high frequencies w > w o , but the enhanced low frequency oscillator strength is properly recovered by the calculation. Similar improved agreement between theory and experiment has been obtained by using the Klemm-Schrieffer approach (Robbins and Klemm 1986) and by assuming strong impurity pinning (Tucker et al. 1986). The apparent good agreement between theory and experiment is not too surprising, as a simple approach, assuming a broad distribution of oscillator frequencies, also gives an excellent overall description of the experimental findings (Sridhar et al. 1985), and models which include the internal degrees R e o ( w )/omax I

-

harm

,

, , , I

I

. . , I

I

I I I I

I

n

m

L

.

osc 6=8

w/wo

Fig. 17. Real part of ac conductivity, normalized to its maximum urns. versus frequency u in units of pinning frequency w,,. The solid line shows the theory of weak impurity pinning with a scaled pinning energy y and damping constant S = ~ / o J ~ TThe . broken line gives the Fukuyama-Lee result for strong pinning and the dotted line is the harmonic oscillator response. Crosses are experimental data on NbSe, at T = 42 K, taken from D. Reagor et al. (After Bleher 1987.)

G . GRUNER

236

of freedom lead also to a distribution of characteristic energies with a wide distribution function. Whether the various models describe also the very low frequency response phenomena, such as nonexponential time decays remains to be seen. The various models which take the internal degrees of freedom of the condensate into account also lead to nonlinear current-voltage characteristics which is in improved agreement with experiments. This is demonstrated in fig. 18, where the calculated nonlinear conductivity u = I/ V curves are collected. The dotted line refers to eq. (4.7), the prediction of the classical particle model, the dash-dotted line and open circles refer to numerical simulations with somewhat different assumptions on the coupling between the various phase coherent regions which build up the collective mode. The overall feature of all of these numerical simulations is a rounded I - V characteristic, with broad agreement with the hydrodynamic treatment in the large velocity limit (Sneddon et al. 1982) and with the critical dynamics approach of Fisher (1983, 1985) near threshold. In all of these models, current oscillations, with the frequency-current relation as given by eq. (1.1) are the consequence of finite-size effects, and they disappear in the thermodynamic limit. The volume dependence is in agreement with eq. (4.22) and, as discussed subsequently, also with the experimental findings. The spectral features of the current oscillations have also been calculated (Pietronero and Strassler 1983, Matsukawa and Takayama 1984, 1986). As

1.0-

~

c c _ _ _ _ _ _ _

+ _ - - -

----

Clossicol Porlicle

- - ---Tw ond Zawodowski

(1984)

Molsukowa ond

I

0

10

20

F/F, Fig. IS. Electric field dependent conductivity which follows from the “classical particle model”, from the tunneling model, and from calculations which take the dynamics of internal degrees of freedom explicitly taken into account. The experimental results follow the full line closely.

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

237

expected, the Fourier transformed current leads to both broad band and narrow band noise components, again in qualitative agreement with the observations. Rather sharp NBN peaks, with quality factors comparable to those measured in NbSe, have, however, not been recovered by such calculations, which are based on 1D models. Whether this is the artifact of one-dimensionality or signals a more serious breakdown of the assumptions which have been made, remains to be seen. 4.3. THETUNNELING

MODEL

A rather different description has been proposed by Bardeen (1979, 1980, 1984, 1986) by recognizing that in the nonlinear conduction region, the observed current can be well described by the empirical expression (Fleming and Grimes 1979) b w =

AE(1 - W E ) exp(-EolE),

(4.21)

where A is constant, and Eo somewhat larger than E T . The above equation provides an excellent fit over a broad electric field range in different materials (Bardeen 1988). The quality of agreement between experiment and eq. (4.21) has been discussed at length by Thorne et al. (1987a,b,c). The exponential form is suggestive of tunneling processes, and the model is usually referred to as the "tunneling model". It has been discussed extensively in the literature (Bardeen 1984, 1985, 1986) and here I summarize only those aspects which are important in the context of interference experiments. It is assumed that pinning due to impurities can be represented as an effective periodic pinning potential of the general form

v(e)=

-cos 8 for - 1 ~ 1 2< 8 < 1 ~ 1 2 , mod IT, cos 8 for ~ / c2 8 c 3 4 2 mod IT.

(4.22)

The Fourier components of the potential are given by 4

Fq =-(-1)""/(4nZn

1).

(4.23)

The advancement of the average phase is due to coherent tunneling of electrons across the pinning potential. An equation of motion has not been derived and therefore a first-principles calculation of the current-voltage characteristics, and the ac response, are not available. Phenomenologically, the current is written as I ( t )=

Veff exP(- Vol VefJ

(4.24)

where the effective potential is the difference between the applied potential V(f) and the pinning potential, Verr=V ( f ) - vMv'(8). Here VM is the

G . GRUNER

238

maximum magnitude of the pinning force and I V'( 8)l = 1. The current is related to the time dependent phase 8 as (4.25)

The time dependence of 8 can be solved by using eqs. (4.22), (4.24) and (4.25) with an assumed value of V , . The small amplitude ac response has been derived by using the photon assisted tunneling theory originally applied to describe the behavior of superconductor-insulator-superconductor (SIS) junctions (Tucker 1979). It leads to a scaling between the electric field and frequency dependent response, in the form of

Consequently, the low frequency ac response is given by

with wo a characteristic frequency related to the characteristic field Eo which enters into eq. (4.21). The smooth increase of a ( w ) with increasing w describes reasonably well the low frequency behavior of the measured a(o) (Wu et al., 1984, Bardeen 1986), and the scaling relation, eq. (4.26) also accounts for the experimental results over a wide range of frequencies and electric field values. Because of the fundamental periodicity associated with the phase variable 8, the current carried by the condensate oscillates in time, with the frequency current relation given by

i l i o = e,

(4.27)

differing by a factor of two from the prediction of the classical particle model. The harmonic content of the current oscillations has been calculated (Thorne et al. 1986a) by assuming that V, = V,, and with parameters V, and A obtained by fitting the calculated w and V dependent response to the experimental results obtained in NbSe,. The ratio of the amplitude of the higher harmonics to the fundamental decays approximately as l / n , and furthermore, it is independent of the applied voltage up to rather high applied electric fields. This is in contrast to the prediction of the classical particle model where the higher harmonics decrease with respect to the fundamental, because as in the high velocity limit the effective pinning potential is expected to be close to sinusoidal (Coppersmith and Littlewood 1986).

C U R R E N T OSCILLATIONS A N D I N T E R F E R E N C E EFFECTS

239

5. Interference phenomena

The nonlinear current-voltage characteristics and the observation of intrinsic current oscillations are suggestive of a variety of interference effects which may arise in the presence of combined ac and dc fields. In case of Josephsonjunctions the phenomenon was used by Shapiro (1963) to demonstrate the existence of the ac Josephson effect and consequently many of the (formally similar) observations in driven charge density wave systems are also called “Shapiro phenomena”. Harmonic mode locking was first observed by Monceau et a]. (1980), and the observation was used to explore the relation between the oscillation frequency and time-average current. The experiments were extended to explore the formal analog between CDW dynamics and observations made on Josephson junctions, including the characteristic Bessel function dependence of the interference features on the drive amplitudes (Zettl and Griiner 1984). Subharmonic mode locking (Brown et al. 1984) has also been explored in detail together with incoherent effects, coherence enhancement (Sherwin and Zettl 1985) and spatio-temporal effects (Bhattacharya et al. 1987). In contrast to Josephson junctions where the characteristic frequency is in the microwave spectral range, here rf frequencies are important, and consequently, wave forms different from a sinusoidal drive can be employed (Brown et al. 1986a,b). 5.1. HARMONIC MODE

LOCKING

Interference effects were first detected by Monceau and coworkers (1980) in NbSe, subjected to dc and ac applied fields. The differential resistance d V / d l measured in the nonlinear conductivity region with an ac field of varying frequency is displayed in fig. 19, for various current levels. Steps in the direct I - V curves correspond to peaks in the derivative, they occur when the fundamental frequency of the current oscillations (or its harmonics) coincides with the applied frequency. Although the spectrum of the interference peaks is complicated because of spurious effects (introduced most probably in homogeneous carrier injection at electrical contacts) a fundamental and several harmonics, such as those labeled by 4F2, 8F2 and 12Fz in the upper part of the figure, are evident. Spectra such as those displayed in the figure were used to establish the linear current-frequency relation discussed earlier. In carefully selected pure specimens, interference effects are also evident in direct I - V curves, such as shown for NbSe, in fig. 20 (Zettl and Gruner 1984). The excitation applied to the sample was of the form V = Vdc+ V,,COS(W~) with w / 2 7 r = 100 MHz. For V,,=O, a smooth, nonlinear

G . GRUNER

240

c."

110 yA

200 1

I

I

1

3

5

7

9

9 IrIHz)

Fig. 19. Differential resistance d V/dl in NbSe, observed by sweeping the frequency of the rf current, for a constant dc current. The various sets of interference peaks are labelled by F, , F2, F, and F4 dc current. (After Monceau et al. 1980.)

I- V curve is observed with a well-defined threshold voltage V, where the conductivity starts to be nonlinear. At higher values o f Vat, well-defined steps, which we shall index with an integer n, appear in the nonlinear region. The step height SV, as defined in the figure, in general first increases with increasing VaCand then decreases. The position of the n = 1 step (identified in the figure) corresponds to a dc current I,, which, in the absence of ac fields, yields an intrinsic oscillation of frequencyf, = 100 MHz = u/27r.This

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

- 100

24 1

0 100 dc current (PA)

Fig. 20. Shapiro steps in the dc I- V traces for NbSe, when rf field is applied at frequency w / 2 n = 100 MHz and of amplitude Vat. The step height 6 V is defined in the figure. No Shapiro steps are observed for V,, = 0, while the maximum step height is at approximately V,, = 100 mV. The arrow indicates the dc current which yields a fundamental noise frequency j,= 100 MHz. The step index is n. (After Zettl and Griiner 1984.)

has been established by measuring the oscillation frequency directly. One should also note the presence of harmonic steps corresponding to n = 2 (wheref, = 200 MHz) and smaller subharmonic steps corresponding to n = (where f,=50MHz). The steps are thus clearly an interference effect between the intrinsic current oscillation and the externally applied rf excitation. Similar experiments have been performed in TaS, (Brown and Griiner 1985) and also in &.,MOO3 (Fleming and Schneemeyer 1986). For both materials the current oscillations are in general smaller and have a significantly larger spectral width. These features are also evident in the interference patterns, shown for TaS, in fig. 21. The definition of the step height is straightforward, and is indicated in fig. 20, and its dependence on the ac amplitude is displayed in fig. 22. These can be analyzed by employing the formal correspondence between the equation of motion of the classical particle model and that of the resistively shunted Josephson junction. For the latter the step height 81 for the nth step can be obtained by explicitly solving eq. (4.20). In the high frequency

G. G R U N E R

242

275v rf

01

- 200

1

1

- 100 1

1

0 100 sample voltoge (mV)

a

200

I

Fig. 21. Differential resistance vs. dc sample voltage ( a ) with no rf and (b) in the presence of 2.75 V rf at 15 MHz. The peaks in the derivative correspond to steps in the direct I - V curve. (After Brown and Griiner 1985.)

limit ( w P 2eZJR/ f i ) , computer simulations (see for example, Lindelof 1981) and analytic approximations, Clark and Lindelof (1976), Fack and Kose (1971) show that the height of the nth step is ~ ( o- 2)~ , ( w =o)IJ,(I,(w)/wGI~(w = 0))l. (5.1) where I , is the Bessel function of order n. The critical current I , depends on the applied ac current as = O ) =(JO(zl(w)/wGIJ(w =O))l. (5.2) The corresponding equations for a CDW system are, by direct analogy to eqs. (5.1) and (5.2) lJ(w)/lJ(o

6v =2 v A ~ = 0)l Jn( Vefr)l

(5.3) (5.4)

The above equations are appropriate in the high frequency limit for a strongly damped system where the capacitive term (for the Josephson

CURRENT OSCILLATIONS A N D INTERFERENCE EFFECTS

243

junction case) or the inertial terms (for the CDW case) can be neglected. Equation (5.3) predicts that the maximum step height 6 V,,, depends only on the maximum value of Jn and is thus independent of frequency w. In the low-frequency limit, still neglecting the junction capacitance or CDW inertia, computer calculations (Lindelof 1981) lead to solutions for the step heights in the I- V characteristics which closely resemble Bessel functions. In this low-frequency limit the maximum step height SV,,, is a strong function of frequency w. Equations (5.3)-(5.5) are appropriate to a coherent response in the current-carrying state. If we assume that (due to either a distribution of the parameters which represent the CDW response or inhomogeneities such as phase boundaries, etc.) only a fraction CY of the sample responds collectively to the external field, then phenomenologically eq. (5.3) becomes 6 v = 2a VA w = 0)IJn ( Veff) I

(5.6)

9

while eq. (5.4) remains unchanged. In general, CY can be both frequency and voltage dependent, reflecting a different amount of coherence under different circumstances. The parameters wo and 7 which determine V,, can be derived from the low frequency ac response, or alternatively, a fit to the data such as shown in fig. 22, can be used to calculate these parameters. The full line in fig. 22, I

I

I

I

I

I

oc otylilude V, ImV)

Fig. 22. Step height 6 V versus ac amplitude V,, for Shapiro steps in NbSe,. The rf frequency is 210 MHz and the step index is n = 1 . The solid line is the prediction of the classical model, eq. (5.62). with parameters w;r/277 = 80 MHz. V,-= 24 mV, a = 0.17. (After Zettl and Griiner 1984.)

G. GRUNER

244

is eq. (5.5) with W ~ =T 503 MHz and a = 0.17. The former agrees well with W ~ T calculated directly from the frequency dependent response and the value of a indicates a highly coherent response (Zettl and Griiner 1984). The formalism also leads to a reduction of the threshold field which follows an oscillatory behavior (Lindelof 19811, but this has not been observed in early experiments. Subsequent measurements on extremely well-defined specimens (Latyshev 1987, Thorne et al. 1987b), however, clearly recovered the oscillatory behavior of V,, as shown in fig. 23. Interference effects are also evident in the ac response measured in the nonlinear conductivity region. Figure 24 shows as a function of applied dc bias, the real and imaginary parts of the complex conductivity, Re a,,(w) and i m uar(w). measured at w/27r = 3.2 MHz. It is evident that neither the real nor the imaginary part of the frequency dependent response is strongly affected by an applied d c voltage as long as V,, < V,. However, for Vdc> V,, Re a,, measured at 3.2 MHz strongly increases, and the ac dielectric constant VT,rnV

VT,rnV

10

0

50

-

I

O:

200

400

Vn,rnV

Fig. 23. dc threshold voltage versus ac amplitude for several applied ac frequencies. The period of the oscillations with ac amplitude is roughly proportional to the ac frequency. T h e solid lines are curves calculated by using eq. (5.4). (After Latyshev et al. 1987.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

Nb9, T:42K

-

245

80,

oooo 0O0

dc boos volloge (mV1

Fig. 24. Real and imaginary part of the ac conductivity Re u ( w ) and dielectric constant c ( w ) and imaginary measured at w / 2 7 r = 3.2 MHz as a function of applied dc bias voltage. T h e threshold field is indicated by an arrow. (After Zettl and Griiner 1984.) 4

related to the ac conductivity by € ( l o )=

4 n Im a , , ( w ) 9

w

(5.7)

strongly decreases for increasing V,, . The dielectric constant approaches zero for large dc drive showing that there is no appreciable out-of-phase

G . GRUNER

246

component Re uac( o),on the other hand, approaches the high-frequency limit obtained from the frequency dependence of the small amplitude ac response (observed for the pinned charge density wave). In addition to the overall behaviors described above, figs. 24a and b also show that for NbSe3 both Reu,,(w) and E ( W ) have sharp anomalies for well-defined values of Vd, in the nonlinear conductivity region. Specifically, Re uacshows "steps" to higher conductivity values at V,, = 1.6, 2.3 and 3.3 mV. At these same values of vd,, shows well-defined inductive dips (Zettl and Griiner 1984, Fleming et al. 1985). The ac response has not been calculated using the Josephson equation, but arguments similar to those used in nonlinear circuit theory can be applied to account for the observations (Zettl and Griiner 1984). 5.2. SUBHARMONIC MODE

LOCKING

Interference phenomena have also been observed whenever pfex,= qf o for q integer, but not equal to one (Brown et al. 1985a). These are called the subharmonic Shapiro steps, and a rich array of such steps is displayed in fig. 25, again observed in NbSe3. A few of the p / q values are identified in

- 20

10

0

10

x)

sompk voltoqe V ImV) Fig. 25. Differential resistance of NbSe, with and without an applied rf voltage V r , . 7he

numbers indicate the various subharmonic steps. (After Brown et al. 1984.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

241

the figure; these were made by plotting I,,, versus we,[ and checking that the steps were p / q times that for the fundamental. In general, features with small p and q are more conspicuous and are both taller and wider. A complete mode locking would correspond to a plateau in d V / d I at the same level as the differential resistance below threshold. This can be understood as follows. The steps are regions of locking between the internal and applied frequencies when pfex,- q j n t is sufficiently small. If the locking within such regions is complete, the CDW velocity becomes fixed by fCxt and does not respond to changes in the applied dc voltage. Hence d V/d I rises to the linear resistance R o , which is due to the uncondensed electrons alone. This situation holds if the CDW velocity is coherent throughout the specimen. In reality the velocity coherence length must be finite, and Ant may vary spatially. If the variation of A, is larger than the width of the region over which locking can occur, locking will be incomplete, and d V/dI will rise to a level less than Ro. Observations under such circumstances are called interference “features”. Therefore, the height of d V/d I is expected to correlate with the degree of synchronization across the sample. The completeness of synchronization depends on various factors such as sample dimensions, ac amplitude and frequency, and in fig. 26, subharmonic peaks, several of them displaying complete mode locking, are shown (Hall and Zettl 1984). While complete mode locking is observed only for a few subharmonic interference peaks, in fig. 26, in subsequent experiments,

1

NbSe3

I

-16

I

I

0

15

bias current ( IJA 1 Fig. 26. Mode locked Shapiro steps in NbSe,. Over the mode locked region, d V / d I is independent of dc bias. The inset shows the subharmonic structure in detail, with corresponding p / s values. (After Hall and Zettl 1984.)

G. G R U N E R

248

performed in carefully treated NbSe, specimens up to 150 subharmonic steps between the threshold field and the 1 / 1 harmonic step have been observed (Thorne et al. 1988). The detailed dependence of the subharmonic steps on the ac amplitude and frequency was also investigated, and characteristic experimental results are displayed in fig. 29. Similarly to the Shapiro step corresponding to the fundamental (see fig. 22), subharmonic Shapiro steps also display a characteristic Bessel function behavior, and have in general an amplitude which is reduced when compared to the fundamental Shapiro step. Although it is anticipated that an analysis in terms of eq. (4.2) provides an adequate explanation for the experimental results displayed in fig. 29, such analysis has not been performed to date. The appearance of subharmonic mode locking has generated wide interest, mainly because within the framework of the classical particle model such as described by eq. (4.9) no subharmonics are expected for an overdamped response (Waldram and Wu 1982, Renne and Poulder 1974). Several, rather different explanations have been advanced to account for the subharmonic mode locking, emphasizing the importance of internal degrees of freedom, assuming a nonsinusoidal potential or making the ad hoc assumption that inertia effects are important in the current carrying state. In this latter case the time development of the phase 8 is described by a return map (Bak el al. 1984, Alstrom et a]. 1984) 8,-

I

= f ( o " ,0 ) .

(5.8)

where I / 0 is a "strobing rate" at which 8 is measured (R= fl/fdr,,,, is a convenient choice for ac driven systems). The reduction of the relevant coordinates provides the motivation for the study of simple maps and their relation to complicated dynamical systems. This idea has often been applied successfully for Rayleigh-Bernard instabilities in fluids. The subharmonic step widths, shown in fig. 25, can be analyzed analogously to that done for the circle map, where universal scaling properties have been conjectured for the quasiperiodic route to chaos (Bak 1983, Bak et al. 1984, Azbel and Bak 1984). The circle map is a one-dimensional map with the following rule for the phase

en+,= / ( 6 , ) = 6 , , + n - ( K / 2 n ) s i n ( 2 n B n ) ,

(5.9)

and i s considered a classic problem of competing periodicities, one coming from the phase space variable (0, is defined modulo 1) and the other from 0. For any value of K I the iterations of the map may converge to a limit cycle such that 8 , + , = p + O , for a nonzero interval R (indicating a p / 4 subharmonic). When K = 1 every rational R will result in this kind of trajectory, and the plateaus form a staircase structure such as shown in fig.

CURRENT OSCILLATIONS A N D INTERFERENCE EFFECTS

,

1 1 . .

m

249

I

s

I

NbSeg f = 25MHz

.#

*-

2'

- ....

.--

..-

0

1. Circle map k= 1

112

-

00

1.o

0.5

Fig. 27. Widths of interference peaks for a NbSe, sample and for the circle map, with p / 9 marked on the vertical axis. (After Brown e l al. 1986.)

27. The staircase is said to be complete if the number of steps in an interval

I of width greater than a discrimination level I-S(r) N(r)=----r

rf0,

D < 1,

r

obeys (5.10)

with S(r) the sum of the step widths that are greater than r. The exponent D defines the (fractal) dimension of the staircase. For K < 1 the staircase is not complete and D = 1. At criticality ( K = l), D = 0.87 was calculated (Bak et al. 1984) and a similar value was found by analog simulations (Yeh et al. 1984). while for K > 1 the staircase structure breaks down and the motion is chaotic. A staircase, derived from differential resistance curves similar to that presented in fig. 24, is also shown in fig. 27. For both the circle map and NbSe, system, the smallest steps have been left out of the figure, but the two staircase structures are qualitatively similar, and in both cases the steps are, in general, larger for smaller q values.

G . GRUNER

250

The experimental results displayed in fig. 24 have been used to evaluate the fractal dimension D (Brown et al. 1985a). The construction N ( r ) versus 1/ r is displayed in figure 28, leading to D = 0.91 0.03 in surprisingly good agreement with values calculated, and obtained from analog simulations. There are, however, several problems with such analysis. First, a complete staircase, D < 1 is expected only if the parameters of the system are measured at the phase boundary of the chaotic regime, i.e., for a well-defined V,, value. Experimental evaluations of D can also be limited by finite (instrumental or intrinsic) noise levels, smearing out the smaller steps. Also, computer studies of the sine circle map (Yeh et al. 1984), indicate that D-0.9 can be obtained over a finite range of the parameter K 1 if the smaller steps are not considered in the evaluation. This may occur for the CDW system as well. The good agreement between the fractal dimension, observed experimentally, and calculated on the basis of simple equations of motion, is most probably fortuitous, and cannot be regarded as evidence for a transition to chaos in this driven nonlinear system. An alternative, conceptually simple explanation for the subharmonic steps has been advanced by Bardeen (see Thorne et al. 1986a,b). The argument is based on the potential, given by eq. (4.14) and employs arguments used originally by Shapiro (1963) to account for the steps in the Josephson junctions irradiated by microwave fields. The theorem that no subharmonics are expected in the overdamped response holds only for a sinusoidal potential which does not have higher harmonic Fourier components. Any other potential gives both harmonic and subharmonic locking,

*

F"'"'

I

I

I

1 1 1 1 1 l

I

/

1

2

..

(Ilr)

Fig. 28. Experimental determination of the dimensionality D (see eq. (4))

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

25 1

and consequently the observations are not surprising, given V ( q ) as described by eq. (4.14). The approach which is used to calculate the magnitude of the interference regions is the following (Thorne et al. 1986, 1987a). The current carried by the collective mode is given by eq. (4.16), and the time dependence of the average phase is, in the presence of applied ac and dc fields d8/dt

=z - O d

-tA COS( W , , t t ) ,

(5.11)

where wd is the drift frequency wd = u,/ A and A is proportional to the amplitude of the applied ac current. The time dependence of the phase then is e(t)=w,t+(A/w)

sin(w,,,t)+Oo.

(5.12)

The pinning potential can be Fourier expanded, (5.13) where

(Y

= 1 for a periodicity given by the wavelength A. After some algebra m

m

V ( 8 ) = f a o +1

1

q = l p=-m

(

3

aqJ, qa-

~~~[(po-q~~)t+qaO (5.14) ~],

where Zp is the Bessel function of order p. The conditions of phase locking depend on the equations of motion for the collective mode, but in general, phase locking occurs if the time average pinning energy in the locked state is smaller than in the unlocked state. Without locking the time average pinning energy ( V ( 8 ) ) ,=fao,and for an additional polarization energy pwexr= (5.15) is obtained. The polarisation energy is less than zero for some range of O0, -8, < eo< 8,, and 0, determines the width of the step. V ( 0 ) as given by eq. (4.14) has been used to evaluate 0, for different p and q values. As expected, both the harmonics and the subharmonics display a characteristic Bessel function behavior in broad agreement with the experimental results. The period of the oscillations with ac amplitude A varies linearly with frequency and inversely with q also in agreement with the observations, as indicated by fig. 29. In general, the agreement between experiment and theory is satisfactory for a broad range of applied ac frequencies and

G . GRUNER

252

0.30

NbSe, Sample #1

T=12l K

5 MHz \

p/q: f

l/f

x

1/2

o

1/3

f

1/4

x 1/5

0.15

0.10

0.05

0.00

0

5

10

15

20

Fig. 29. Widths of selected steps in the dc I - V characteristic versus peak ac amplitude for applied ac frequencies of (a) 5 MHz and (b) 10 MHz. The period of the oscillations with ac amplitude vanes as I / q for the p / q step. The solid lines are guides to the eye. (After Thorne et al. 1987b.)

amplitudes (Thorne et al. 1987b). A similar agreement would, however, be obtained by any potential which has the same periodicity as that of eq. (4.14) and has smooth minima separated by cusps. A rather different concept has been developed by Tua and Ruvalds (1985) by Littlewood (1986), and by Matsukawa (1987) by extending the models which incorporate the internal degrees of freedom and which have been discussed in the previous section to account for the interference phenomena observed. Both conclude that internal degrees of freedom lead to subharmonic mode locking even under circumstances where single degree of freedom dynamics lead only to locking corresponding to harmonic frequencies. The classical dynamics of coupled domains has been used by Tua and Zawadowski (1984) to account for the current-voltage characteristics and for the finite size effects associated with the current oscillations. The extension of the model to combined dc and ac drives (Tua and Ruvalds 1985) leads to subharmonic steps, and to an apparent complete Devil's staircase. The overall amplitudes of the subharmonics agree well with the experiments, and the calculations also lead to a subharmonic structure which depends only slightly on the drive conditions, with no critical values dividing complete and incomplete staircase behavior, in striking agreement

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

253

with the experimental observations. The amplitude of the steps was found to decrease with the increasing number of domains, and the steps disappear in the thermodynamic limit. This feature of the results is in clear contrast with the hydrodynamic treatment of Sneddon et al. (1982) which predicts that (while the current oscillations are a finite size effect) interference peaks are recovered in the thermodynamic limit. The origin of this disagreement is not fully understood. The model where the internal degrees of freedom are incorporated following the Fukuyama-Lee-Rice model (Fukuyama and Lee 1978, Lee and Rice 1979) has also been used to study the response of charge density waves to a combination of dc and ac drives (Coppersmith and Littlewood 1986). As discussed earlier, second-order perturbation treatment in the hydrodynamic limit (Sneddon et al. 1982) leads to interference effects involving the fundamental and higher harmonics. Higher order perturbation theory leads to interference features (Coppersmith and Littlewood 1985a,b), i.e. sharp peaks in the d l / d V curves, but not full mode locking with plateaus as displayed in fig. 26. Complete mode locking is recovered by numerical simulations on finite size systems where the advance of the average phase ( 4 ) is monitored. The results of the calculations, which display advances A4/27r, which correspond to various mode lockings p / q , are shown in fig. 30. The detailed waveform near mode locking has also been calculated by Coppersmith and Littlewood. The overall tendency of the experimental results, which are suggestive so-called interference “features” (i.e. not complete mode locking) at high frequencies, and complete; mode locking with welldefined plateaus in d V/d l curves at low frequencies (Brown and Griiner, unpublished) is well reproduced by calculations which take the dynamics of internal degrees of freedom explicitly into account. Interference curves, calculated on the basis of single degree of freedom classical dynamics (called GZC) model, see eq. (4.2) and on the basis of the Fukuyama-LeeRice model are displayed in fig. 31, together with experimental curves, shown as the full line. The absence of well-defined “wings” in experiment and in numerical simulations based on the FLR model are taken as evidence of the dynamics of the internal mode. Both harmonic and subharmonic mode locking is recovered also by perturbational analysis of the FukuyamaLee-Rice model in the presence of combined ac and dc electric fields (Matsukawa 1987). The model leads to clear anomalies in the currentvoltage characteristics; whenever nq = mp, complete mode locking, such as shown for example in fig. 26, however, has not been recovered. The reason for this is not clear at present, it may be related to the deficiencies of the model itself (Bardeen 1988), or due to the breakdown of the perturbation theory. The conclusion has been sharply criticized recently (Thorne et al. 1987a,b,c), who argued that complete mode locking can also be obtained

G. G R U N E R

254

h

0

om m a 3

Om

5 -

- 2

O A 0

0 0 2

3

007

04 3

u =4 F = 16

OBBBOOI

A 55 IMPURITIES 0

25 IMPURITIES

0 10 IMPURITIES

n

0 0 71 1

O3 4

I

I

0.5

1

-

ton Fig. 30. Number of wavelengths moved per pulse (A4)/27r versus the pulse duration I,,, with a fixed Eon= 16 for two systems with U = 4, one with 55 degrees of freedom and the other with 10. Locking is demonstrated because (A4)/27r is always a rational fraction. Inset: Magnified portion of the plot, demonstrating that increasing the number of degrees of freedom causes the appearance of high order subharmonics. (After Coppersmith and Littlewood 1986.)

at high frequencies, in apparent disagreement with the classical descriptions of charge density wave dynamics. The disagreement between the various experimental results most probably reflects the difference in the degree of coherence in the materials which have been investigated. As a rule, in small and pure specimens, where the phase-phase coherence length may exceed the dimensions of the samples, a highly coherent response, with pronounced complete mode locking is observed even at high frequencies. In contrast, samples with less coherence

iJ F A

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

1. O r

Oa8-

E

H

U

5 U

0.6 -

I

I

\

0.4 -

I

'

.[ ..-..I

\ I

;

\

f

//

I

I

I

-........

/-

'\I

I

082 -

1

I

' \

I

I...* /

-

Experiment

-

' ........ --- FLR model

I OL

I

I

I

bc Y

r--

r--1

I

255

I

GZC model

~

4

l

l

l

I

6

l

l

l

voltage (mV)

l

l

2

4

6

voltage ( mV)

Fig. 31. Differential resistance d V / d l plotted versus dc voltage V, near the first harmonic feature Q. u wAc for NbSe, with ac frequency w,, = 25 MHz and ac voltage amplitude V, 50 and 75 mV. The sample's threshold voltage V,-2 mV. Also plotted are theoretical fits using the FLR deformable model (dotted line) and the GZC one degree of freedom result (dashed line). T h e tops of the peaks are not calculated for FLR because the perturbation theory breaks down when the change in d V/dl is large. (After Coppersmith and Littlewood 1986.)

-

-

display complete mode locking only at low frequencies. This difference, and the crossover between' the two behaviors have, however, not yet been investigated in detail. 5.3. NONSINUSOIDAL AND PULSE

DRIVES

In the majority of cases, interference effects have been investigated by applying a sinusoidal ac field E ( o )= Eo sin( weXtt) of various amplitude and frequency. Most of the relevant theories have also been worked out for this particular case, mainly because all experiments on Josephson junctions involving microwave fields have been conducted under such circumstances (Lindelof 1971). In the radiofrequency spectral range various

G . GRUNER

356

periodic waveforms can be applied, and this has been used (Brown et al. 1986a). to further investigate the various aspects of mode locking. For a periodic square wave drive which oscillates between two values, El and E 2 , the former for a time interval i,, the latter for i2, the fundamental period is i = i, + I,. t , , I , , and E l , E2 can independently be varied leading to a variety of conditions under which the dynamics of the collective mode can be investigated. The difference between a sinusoidal drive and pulse drive, the latter with I , = r2, is shown in fig. 32. The sharp spikes in the upper part of the figure correspond to the fundamental interference peaks, with the p / q = i subharmonics also evident on the figure. The amplitude of the interference peaks is approximately constant for the sinusoidal drive for small dc voltages, and starts to decrease when IEdc- Ea,I<(ETI,i.e., when the system is not driven back to the pinned configuration during the experiment. In contrast, the steps are suppressed near zero dc bias for the square wave drive, in the interval where the ac drive amplitude is large enough that the applied field drives the system from above threshold to below threshold (with opposite polarity) without allowing it to relax to the pinned configuration. As with the sinusoidal drive, the interference peaks are again suppressed large Edc, for which l E d c - Ea,I < ET. The above difference between the results for a sinusoidal and a square wave drive is clearly due to the fact that a finite time in the pinned state is required to observe substantial interference effects. While the overall features of fig. 32 can be reproduced by computer simulations using only single degree of freedom dynamics (Brown et al. 1986a), experiments can also be conducted by varying i,, the characteristic time spent below ET,with E2> ET,applied for time r2 varied. Such experiments point to the importance of internal degrees of freedom in the dynamics of the collective mode. The measured step height is displayed in fig. 33 as a function of the time 1, spent below threshold. As expected, the step height approaches zero as i, (called i- in the figure) gets shorter and saturates the long time ( 1 , > CO) limit. For short times the step height 6 V can be described by the expression

hV-

vT(tI/TO)

(5.16)

where T~ is the characteristic relaxation time of the system. For an overdamped classical particle T~ = &T. In the long i2 limit simple arguments, (Brown et al. 1986a,b) based on the classical particle model lead to a step height

Computer simulations, with 1, = 0.25 ps, obtained by fitting the observed w dependent response to eq. (5.16), lead to the dotted line in fig. 33. The difference between the observed step height and that which follows from

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

257

1

-2vT

NbSe, T = 48K

loot 01

I

I

- 10

-5

5

0

10

somple voltoge (mV1 Fig. 32. Differential resistance curves for (a) sinusoidal ac drive and (b) square wave drive for the same amplitude. (After Brown et al. 1986a,b.)

the classical particle model is significant, and is most probably the consequence of the dynamics of the internal degrees of freedom (Brown et al. 1986a,b). A different type of interference effect is called, in general, “pulse duration memory”. The notion of pulse duration memory effect refers to the observation which for &,,Moo3 is displayed in fig. 34 (Fleming and Schneemeyer 1986). Square wave voltage pulses which drive the system from the pinned 0.4

,

I

1

1

r----------

->E A

I

‘

*., 5 .- 0.2 -

al

I

0 tl

I

.c

I

A

A

I

---- a- ,A- - = - r - -- - - A A

A A A A A

I

---- Tt=2.0w Simulation, ~ 0 . 2 5 ~ s A

8 A

‘ A

0 - I

I

Fig. 33. Step height amplitude versus waiting time 1. The dotted line is a fit to the simulation based on the classical particle model. (After Brown et al. 1986a,b.)

G . GRUNER

258

1

I

I

1 0

I

I

I

I

I

I

I

I

1

I

I

I

1

I

I

I 400

I

200

1

1

Fig. 34. (a) Current oscillations in response to a square-wave driving field of about lOE, (inset) in &,,MOO, at 45 K. We data was obtained in a current driven configuration and has been inverted; however the current oscillations are also clearly observed in a voltage-driven configuration. (After Fleming and Schneemeyer 1986.)

to the current carrying state lead to transient current oscillations with frequencies given by eq. (1.2). The amplitude of the oscillations gradually decreases with increasing time and for t + 00, the magnitude of the oscillations correspond to those which are measured under dc conditions by detecting the Fourier transformed current. This can be thought of as the consequence of gradual dephasing of the current oscillations which start with the same phase but which have slightly different frequencies depending on the local currents within the specimen. At the end of the pulse, there is a sharp upward cusp in the observed current, and this occurs for a broad

C U R R E N T OSCILLATIONS A N D INTERFERENCE EFFECTS

259

range of pulse durations t o . The cusp suggests that the system adjusts itself to the pulse duration in such a way that its velocity is always rising as the pulse ends. Apparently, the system “remembers” the length of preceding pulses, hence the name of “pulse duration memory effect”. The upward cusp, as displayed in fig. 34, is not observed for single pulses, although the current oscillations are evident. Clearly a single-particle description does not account for this finding as the current is determined by the derivative of the potential at the position of the particle at the end of each pulse, and this can be both negative and positive, depending on t o . Consequently, the current at the end of each pulse is expected to be a sensitive function of the pulse duration. It has been proposed (Coppersmith and Littlewood 1985a),that the pulse duration memory effect is the consequence of the internal degrees of freedom, where a large number of metastable states and a negative feedback mechanism play a crucial role. A simplified one-dimensional version of the equation of motion, eq. (4.18), has been examined numerically where a sequence of square wave pulses of length to is applied, with the time tombetween pulses long enough that the system relaxes to its appropriate metastable states between the applied pulses. The same model leads also to the subharmonic mode locking discussed earlier. The calculated response wave forms resemble closely those observed by experiment and this is taken as evidence for the importance of internal degrees of freedom in the dynamics. A further elaboration of the model suggests that when the system with infinite degrees of freedom is subjected to identical repeated pulses, it attempts to reach a state for which further pulses to not induce further changes. This fixed point is the least stable, and the pulse duration memory effect is the system’s signature of being on the verge of its region of stability.

5.4. FLUCTUATIONS AND

COHERENCE ENHANCEMENT

Fluctuation effects associated with the current oscillations, discussed in section 3, are suggestive of coupled domains of size L: which oscillate at approximately the same frequency fo =j / 2 e with random phase. The Gaussian distribution of the voltages of the narrow band noise displayed in fig. 14, is suggestive for independent phase dynamics for the various domains. The mode-locking phenomena discussed earlier can be understood on the basis of the dynamics of a single degree of freedom system; however, the volume dependence of the current oscillation amplitudes suggest that internal degrees of freedom, and finite size effects are important

G . GRUNER

260

(Mozurkewich and Griiner 1983). Also, interference experiments, performed for relatively small ac drives give mode locking, smaller than that expected for a single degree of freedom dynamics, and this has been interpreted as partial mode locking (Zettl and Griiner 1984). The reason for this is again most probably the absence of complete phase coherence throughout the specimens. It is then expected that phase synchronization by the applied ac field leads also to the reduction of incoherent phenomena, such as the broad band noise, or the narrow band noise fluctuations observed in the current carrying state. These have been investigated recently in detail (Bhattacharya et al. 1987, Sherwin and Zettl 1985). The broad band noise spectrum is significantly reduced during mode locking, with a total noise power reduced comparably to that observed in the pinned CDW state (Sherwin and Zettl 1985). This most probably is related to the freeze-out of the internal degrees of freedom, and may signal the enhancement of the dynamic coherence length which characterizes the current-current correlations. Such an effect has been recovered by a perturbational analysis of the FukuyamaLee-Rice model in the presence of combined d c and ac drives (Matsukawa 1987). The reduction is most probably the consequence of phase synchronization between the various domains. This reduces the dynamics to that of a single domain with no fluctuations and consequently no broad band noise. The noise reduction depends on the applied ac amplitude, as shown in fig. 35, where the total measured noise power is plotted as a function of ac current in this current-driven experimental arrangement. The gradual reduction of the noise power is suggestive for a gradual phase homogenization, and a complete phase homogenization is achieved for ac drive amplitudes

1.0

-?

--

0

0.8 -

-' 0.6 >= 0.4 -

*

i

NbSes T=48K

1

a,,/en=eN H Z

0 0

T

>*

0

0.2 -

I

j -

1

1

1

0

I

I l l l L

Fig. 35. Total broad band noise amplitude versus amplitude oran applied rrcurrent in NbSe,. Intense rf field suppresses the noise by homogenizing the CDW phase. (After Sherwin and Zettl 1985.)

CURRENT OSCILLATIONS AND INTERFERENCE EFFECTS

261

which are typically about one order of magnitude larger than those which are required for complete mode locking. Fluctuations of the current oscillations are also influenced by the mode locking, and both the amplitude and frequency fluctuations are reduced. The system synchronized at the p / q = subharmonic steps displays reduced amplitude fluctuation reductions at short time scales, but may fluctuate between varous mode locked states (characterized by different oscillation amplitudes) over a large time interval. N o amplitude fluctuation reduction has been observed in a more detailed experiment (Bhattacharya et al. 1987), but the frequency distribution of the current oscillations was eliminated by mode locking. The phenomenon is shown in fig. 36. The upper part of the figure shows the fluctuations of the N B N in position and amplitude without the application of ac drive. When synchronization occurs, the N B N peak frequency does not fluctuate, but the amplitude fluctuation is unchanged. The absence of frequency distribution implies that at mode locking all

1.4

1.5 I .6 PEAK FREOUENCY (MHz)

1.7

PEAK AMPLITUDE (arb.units) Fig. 36. Hystogram of the temporal fluctuations in frequency (a) and amplitude (b) of a “bare” NBN ( V,, = 0) and a locked NBN p / q = 1/2 at o = 3 MHz). Results represent 500 scans of each case. In (b) the two hystograms are shifted arbitrarily for clarity. (After Bhattacharya et al. 1987.)

262

G. GRUNER

temporal fluctuations of the oscillation frequency are quenched, i.e., the velocity degrees of freedom are eliminated. Coherence enhancement and fluctuation suppression, such as observed in driven charge density wave systems, represent probably the most interesting consequences of nonlinear dynamics of many degree of freedom systems. Some aspects of this phenomenon, such as subharmonic mode locking, under circumstances where a single degree of freedom system would not lead to subharmonic locking, have been explored, and it remains to be seen whether the other aspects of the observations can be accounted for.

6. Conclusions

The current oscillation phenomena which occur in driven charge density wave systems in the nonlinear conductivity region and associated interference effects in the presence of dc and ac drives are clear manifestations of a new type of collective transport phenomenon, carried by an electron-hole condensate. The linear relation between the time average current and oscillation frequency reflect the fundamental 2kF periodicity associated with the electron-hole condensate, and the mere existence of current oscillations in macroscopic specimens is suggestive of macroscopic length scales involved. The collective mode is characterized by an amplitude and phase, and the length scales are related to the phase-phase correlation length, determined by the impurity concentration and the parameters of the collective mode, and the highly coherent reponse clearly demonstrates that they are comparable to the dimensions of the specimens investigated. In spite of a broad variety of experiments performed both in the time and in the frequency domain and conducted by employing under various combinations of ac and dc drive amplitudes, frequencies and waveforms, several unresolved questions remain. The linear relation between the time average currents and oscillation and oscillation frequency is by now well confirmed in all materials which display CDW transport phenomena; there is however, considerable uncertainty concerning the numerical factors involved. Experiments on TaS, suggest that simple arguments advanced in the Introduction are correct and the oscillation is related to the advancement of the phase by 27r, corresponding to the displacement of the collective mode by one wavelength. Careful experiments in other, structurally simple model compounds, such as (TaSe,)*I and &.,MOO, would be highly desirable. The relation, given by eq. (1.2) holds also only at T = 0, and experiments on NbSe, indicate that the ratio j / f o is proportional to the number of condensed electrons. In contrast, in TaS, the ratio was found to be only weakly temperature dependent (Brown and Griiner (1985). Further experi-

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263

ments, in particular the transition temperature near Tp, are needed to clarify whether the proposed relation

is appropriate. Considerable controversy exists concerning the spectral width of the current oscillations, and the related broad band noise. Broadly speaking, larger spectral widths are accompanied by larger broad band noise amplitudes suggesting that the two types of incoherent effects are strongly related, and both care determined by the relative magnitudes of the phase-phase correlation length and dimensions of the specimens. One particular model (Bhattacharya et al. 1987) related the broad band noise amplitude to the spatial and temporal fluctuations of the CDW current, through the relation ( S V 2 ) = ( 6 I 2 ) R Nwhere R N is the resistance due to the normal electrons. Fluctuations in CDW velocity lead also to fluctuations in the oscillation frequency explaining the above overall correlation. Studies of the broad band noise amplitude performed in TaS, and NbSe, as a function of the dimensions of the specimens indicate that it is a bulk effect and that noise generation is due to randomly positioned impurities. In contrast, studies on rather pure NbSe, specimens indicate that macroscopic defects, such as grain boundaries, are the main source of broad band noise. Macroscopic defects, and also random impurities, have been suggested also as the source of the current oscillations. The most probable explanation for the variety of findings is that in general, both impurities and extended boundaries (such as grain boundaries, contacts, etc.) contribute the noise generation (both broad band and narrow band); but more detailed studies in alloys, or in irradiated specimens where the concentration of pinning centers is systematically varied, are called for to settle this issue. Fluctuation effects, and clear evidences that the current oscillations disappear in the thermodynamic, infinite volume limit demonstrate that description of CDW dynamics within the framework of single degree of freedom models is not appropriate. Various experiments however, suggest that the dynamics of the collective mode is characterized by macroscopic length scales. The static phase-phase correlation length (called the Fukuyama-Lee length) Lo is, for typical impurity concentrations and CDW parameters of the order of 10-100 pm, along the chain directions and one or two orders of magnitude less perpendicular to the chains. This leads to a typical “domain” size of 10-2-1 pm in broad agreement with estimates of the phase-phase correlation length on the basis of length dependent threshold field (Monceau et al. 1986, Gill 1982, Zettl and Griiner 1984, Borodin et al. 1986) current oscillations amplitudes (Mozurkewich and

264

G . GRUNER

Griiner 1983) and broad band noise (Bhattacharya et al. 1985, Richard et al. 1982) implies that the static and dynamic aspects of the problem are characterized by the same length scale (except perhaps near threshold, see Fisher ( 1985)). As Lo is expected to be inversely proportional to the impurity concentration, similar experiments on alloys would be of great importance. Interference effects confirm many of the conclusions which have been reached on the basis of the current oscillation studies alone, but also emphasize the formal similarity between the Josephson phenomena and nonlinear CDW transport. While for small amplitude ac drives fluctuation effects, volume dependences for interference effects and current oscillations go hand in hand, large amplitude ac drives lead also the phase homogenization, and to interference effects which survive the passage to the thermodynamic limit. Whether this corresponds to the conclusions which have been arrived at on the basis of the hydrodynamic treatment of the CDW dynamics (Sneddon et al. 1982) remains to be seen. Considerable theoretical activity, generated by the observation of subharmonic interference peaks, led to several fundamentally different proposals on the origin of subharmonic locking. Early analysis indicated that the concept of Devil’s staircase behavior, tied to inertial effects may be appropriate. By now this possibility is unlikely, with two remaining suggestions: nonsinusoidal pinning potential and internal degrees of freedom being able to explain a broad variety of interference features and mode locking. Most probably, both are important as there is no a priori reason why the pinning potential should be periodic, and also there is broad variety of independent experimental evidence for the importance of the dynamics of the internal modes of the condensate. The former effect may be more dominant for small specimens where pinning by boundaries such as the surface of the specimens and contacts are important, while the latter is more important for larger and impure materials where pinning is due to randomly distributed impurities. While the experiments led to several interesting theoretical questions, such as why internal degrees of freedom dynamics mimics many effects which are also the consequence of inertial dynamics, or the concept of minimally stable states - the hope that driven charge density waves can be appropriate model systems for general questions concerning nonlinear dynamics now appears to be remote. This is mainly because both randomly distributed pinning centers (which lead to a broadly defined localization problem) and extended pinning centers (leading to essentially a boundary problem) are in general equally important and cannot be easily separated. Several issues related to the dynamics of charge density waves have not been discussed in this rewiew. In particular, the charge density wave dynamics of specimens which display a so-called “switching” behavior (Zettl and Griiner 1982, Hall et al. 1986) has not been mentioned. The

CURRENT OSCILLATIONS A N D INTERFERENCE EFFECTS

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phenomenon arises as the consequence of macroscopic defects which leads to dynamics similar to that observed in coupled Josephson junctions (Inoui and Doniach 1987). Several observations concerning transitions to chaos have also not been discussed and I refer to a recent review (Zettl and Griiner 1986) which covers this type of experiments. Mode locking phenomena also occur in the elastic properties (Bourne et al. 1986), a not too surprising observation given the fact that the CDW can be regarded as a coupled electron-phonon system, and the earlier observations on nonlinear elastic properties. Several models, with interesting dynamical properties have also not been discussed. Among them, the dynamics of Frenkel-Kontorova type models have been studied in detail (Sneddon 1984a, Coppersmith and Littlewood 1985a,b), and have been shown to display many of the features which are the consequence of models where random impurity pinning is explicitly taken into account, such models also lead to a broad variety of interference effects in the presence of combined dc and ac electric fields, discussed before.

Acknowledgements

I am grateful to John Bardeen, Sue Coppersmith, Peter Littlewood and Laszlo MihPly for many discussions, and to Stuart Brown for reading the manuscript. Support by the National Science Foundation Grant DMR 86-20340, is acknowledged.

List of review papers

Several review papers have appeared recently which cover the various aspects of charge density wave dynamics, see for example: Fleming, R.M., 1981, in: Springer-Verlag in Solid State Sciences, Vol. 23, Physics in One-Dimension, eds. J. Bernasconi and T. Schneider (Springer, New York). Griiner, G., 1983, Comments in Solid State Physics 10, 183. Griiner, G., 1983, Physica D 8, 1. Griiner, G., 1988, Rev. Mod. Phys. 60, 1129. Griiner, G., and A. Zettl, 1985, Phys. Rep. 119 117. Hutiray, Gy., and J. Solyom, eds., 1985, Lecture Notes in Physics, Vol. 217, Charge Density Waves in Solids (Springer, Berlin). Jerome, D., and L.G. Caron, eds., 1987 (Plenum Press, New York).

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Monceau, P., 1985, in: Electronic Properties of Inorganic Quasi-OneDimensional Materials, ed. P. Monceau (Reidel, Dordrecht). Ong, N. P., 1982, Can. J. Phys. 60,757. Tanaka, S., and K. Hchinokura, eds., 1986, Physica B 143 (Yamada Conf. XV, Physics and Chemistry of Quasi-One-Dimensional Conductors). Zettl, A., and G. Griiner, 1986, Comments in Condensed Matter Physics 12, 265.

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Verma, G., and N.P. Ong, 1984, Phys. Rev. B 30,2928. Waldram, J.R., and R.H. Wu, 1982, J. Low Temp. Phys. 47, 363. Weger, M.,G. Griiner and W.G. Clark, 1980, Solid State Commun. 35, 243. Weger, M.,G. Griiner and W.G. Clark, 1982, Solid State Commun. 44, 1179. Wonneberger, W., 1983, Z. Phys. B 53, 167. Wonneberger, W., 1985, Solid State Commun. 54, 317. Wonneberger, W., and J.-J. Breymayer, 1984, Z. Phys. 56, 241. Wu, Wei-Yu, A. Jhnossy and G. Griiner, 1984, Solid State Commun. 49, 1013. Wu, Wei-Yu, L. MihBly, G. Mozurkewich and G . Griiner, 1986, Phys. Rev. B 33, 2444. Yeh, W.J., Da-Run He and Y.H. Kao, 1984, Phys. Rev. Lett. 52, 480. Zettl, A., and G. Griiner, 1982, Phys. Rev. B 26, 2298. Zettl, A., and G . Griiner, 1983, Solid State Commun. 46, 501. Zettl, A., and G. Griiner, 1984, Phys. Rev. B 29, 755. Zettl, A., and G . Griiner, 1986, Comments Cond. Matter Phys. 12, 265. Zettl, A., M.B. Kaiser and G. Griiner, 1985, Solid State Commun. 53, 649.

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CHAPTER 5

MULTI-SQUID DEVICES AND THEIR APPLICATIONS BY

Risto ILMONIEMI and Jukka KNUUTILA Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland

and

Tapani RYHANEN and Heikki SEPPA Electrical Engineering Laboratory, Technical Research Centre of Finland and Laboratory of Metrology, Helsinki University of Technology, 02150 Espoo, Finland

Progress in Low Temperature Physics, Volume X I 1 Edited by D.F. Brewer @ Elsevier Science Publishers B. V., 1989 27 1

Contents I . introduction . . . . . . . . . . . . . . . . . . . . . ................................... ..... 2 . SQUIDS . . . . . . . . . .......................... ............................. 2.1. Single-junction (rf) SQUlDs . . . . . . . . . . 2.1.1. General . . . ........................................... ......... 2.1.2. Rf SQUID in the hysteretic mode . . . . . . . . . . . . . . . . . 2.1.3. Discussion . . . . . . . . . 2.2. Double-junction (dc) SQUlDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Operation . . . . . . . . . . . . . . . ................................. 2.2.2. Problems with practical devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. The state of the art . . ................................... 2.3. Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications: biomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1 .1. Magnetically shielded rooms . . . . . . . 3.1.2. Gradiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Neuromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Origin of neuromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Spontaneous activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Evoked fields . . . . . . . . . . . . . . . . . . . . . . . ...................... 3.2.4. Clinical aspects of MEG . . . . . . . . . . . . . ...................... 3.3. Cardiac studies ........................................ 3.4. Other hiomagnet ........................................ 3.5. Multichannel neuromagnetorneters . . . . . . . . . . ..................... 3.5.1. Optimization of multichannel neuromagnetometers . . . . . . . . . . . . . . . . . . 3.5.2. Existing multichannel systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Planar gradiometer arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4. Use of multichannel magnetometers . . . . . . . . . . . . . . . 4 . Other multi-SQUID applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Geomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Physical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Accelerometers and displacement sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Monopole detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

213 213 214 214 216 219 280 280 284 286 289 292 293 293 295 296 291 298 299 300 302 303 304 305 310 319 323 326 326 328 329 329 332 333

1. Introduction

The Superconducting Quantum Interference Device (SQUID) offers unrivalled sensitivity for the measurement of low-frequency magnetic fields. Recent developments in the fabrication technology and electronics have made it possible to construct reliable low-noise SQUIDs. Consequently, magnetometers with many SQUIDs have become feasible in many applications and the number of SQUIDs is no longer limited by the difficulty of their use. Further reduction of SQUID noise is no more necessarily needed, since the noise limit in the state-of-the-art multi-SQUIDS seems to be determined by dewar materials, environmental low-frrequency noise, and other sources external to the SQUIDs. Additional recent interest in SQUID applications is caused by the possibility of fabricating them from the new high- T, materials. In this review', we start with an overview of the operation of rf and dc SQUIDs, stressing, in particular, the theoretical understanding of complete SQUID circuits. This is necessary for the design of practical devices. Main attention is focused to the operation of the dc SQUID and to recent progress in realizing practical dc-SQUID structures with flux-coupling circuits. A more thorough discussion of SQUID circuits is presented in a companion paper (Ryhanen et al. 1989). So far the largest field of application for SQUID arrays is biomagnetism, the study of magnetic fields originating in biological organisms. The subfield currently attracting most interest is neuromagnetism, where these techniques are applied to investigations of the central nervous system. A potentially important clinical use of SQUIDs in the future is in magnetocardiography (MCG), the recording of heart activity via magnetic measurements. We will give a brief overview of biomagnetic measurement techniques, illustrating them with several examples. Principles of SQUID magnetometers and their application to biomagnetism are discussed. Existing biomagnetic multiSQUID systems and some plans for future instruments are described. In addition, applications of SQUIDs in geomagnetism and in some physical experiments are briefly discussed.

2. SQUIDs

This chapter is a brief review of Superconducting Quantum Interference Devices (SQUIDs), which are formed by interrupting a superconducting 273

R. I L M O N l E M l ET AL

274

ring by one or two Josephson junctions (Lounasmaa 1974, Tinkham 1975, Barone and Paterno 1982, Likharev 1986, van Duzer and Turner 1981). SQUI Ds have been studied intensively both theoretically and experimentally since the introduction of the double-junction interferometer (dc SQUID) by Jaklevic et al. (1964, 1965) and, in particular, since the invention of the rf SQUID by Silver and Zimmerman (1967). Both rf and dc SQUIDs are common in multichannel applications. In spite of the outstanding properties of SQUIDs as magnetic flux sensors, their impact outside research laboratories has remained modest. Imprudent operation of SQUIDs often leads to unexpected problems that discourage their use. Evidently, the complicated dynamics, caused by the strong nonlinearity and the lack of natural damping, is one of the main reasons for difficulties in practical applications. However, as this paper will show, these problems can be overcome. When discussing the suitability of a particular SQUID in a multichannel system, all the necessary circuits should be included in the analysis. In other words, the contributions to noise of preamplifiers, coupling circuits, postdetection filters, etc. should be as important objects of theoretical and experimental study as the SQUID itself. For example, theory predicts a flux noise of about Q,/& for a typical rf SQUID and about lo-"@,,/& for a dc SQUID; @,=2.07x10 "Wb is the magnetic flux quantum. However, in a practical measurement setup such figures are seldom reached. Excess noise is usually caused by the preamplifier in rf SQUIDs and by parasitic elements in the input circuits of dc SQUIDs. 2.1. SINGLE-JUNCTION (RF) SQUIDS 2.1.1. General

In an rf SQUID the low-frequency external flux in the SQUID ring is read out by superposing on it a high-frequency bias flux and monitoring the amplitude of the rf voltage by a preamplifier matched to the SQUID with a resonant tank circuit (fig. 1). The Josephson junction is typically described by the resistively-shunted-junction (RSJ) model, which consists of an ideal Josephson junction, a resistance R, and a capacitance C. The supercurrent passing through the ideal junction is related to the quantum phase difference across the junction cp by I, = I, sin cp where I , is the critical current of the junction. In a single-junction superconducting loop, cp = -27r( @+ n @ J / @,). If an external flux @a is applied to a SQUID loop with inductance L, the total flux Q = Qa+ LI, in the loop obeys the equation 2 ~ @Q,/

+ pL sin( 2 n Q l Q,,) = 2a@,/

Q, ,

(2.1)

k---Jq, MULTI-SQUID DEVICES

Tank circuit

275

SQUID

x

Josephson

junction 1,sinlp

Fig. 1 . A single-junction SQUID coupled to a resonant tank circuit. If BL= ZTLI,/#~>1 the SQUID is hysteretic, if BL< 1 it is nonhysteretic and called an inductive SQUID; I, is the critical current of the junction, Go is the magnetic flux quantum 2.07 x lo-'' Wb.

where p L = 2rLI,/ @, is the normalized inductance. The dynamics of the single-junction SQUID depends fundamentally on p L ; its influence on the @-@a characteristics is illustrated in fig. 2. When I , is not high enough to screen the ring, i.e., when p L < 1, @(@a) is single-valued; the SQUID is then nonhysteretic. Otherwise, .@( @a) is multivalued and the SQUID is hysteretic. The SQUID in the regime p L< 1 is called a nonhysteretic SQUID, an inductive SQUID, or simply an L-SQUID; its characteristics depend strongly on the parameters of the SQUID ring and the tank circuit. Consequently, the L-SQUID is rarely seen in practical applications. In contrast, the dynamics of the hysteretic SQUID is much less sensitive to parameter

Fig. 2. The total flux @ in the SQUID ring as a function of the applied flux @a for BL< 1 and for pL> 1. When BL> 1. the SQUID is hysteretic and transitions occur at @ = @== *((n + f ) @,, + LIJ.

R. ILMONIEMI ET AL.

216

variations; therefore, it is preferred in most applications, including multichannel magnetometers.

2.1.2. Rf SQUID in the hysteretic mode Operation of the $SQUID in the dissipative regime A consequence of high B L is that the total flux through the SQUID loop becomes a multivalued function of the applied flux. In this mode, the flux may jump by about one flux quantum as depicted by arrows in fig. 2 . Sinusoidal flux excitation of sufficient amplitude causes the SQUID to traverse hysteresis loops; the flux transitions involve dissipation of energy that is proportional to the area of the hysteresis loop. Rf SQUIDS are thoroughly discussed in the literature (Zimmerman et al. 1970, Mercereau 1970, Nisenoff 1970, Giffard et al. 1972, Clarke 1973, Jackel and Buhrman 1975, chapter 3 of the companion paper: Ryhanen et al. 1989); only a brief review will be presented here. The peak voltage across the tank circuit depends on the amount of flux needed to excite flux jumps. The points for flux transitions can be approxiLI,) (see fig. 2 ) . Before a transition, depending mated by @,= * ( ( n on the branch where the system is, the amplitude of the tank circuit voltage approaches one of two critical values:

+a)@,,+

eT

where up is the pump frequency; the mutual inductance M between the SQUID ring and the tank circuit coil L, is proportional to the coupling Because of energy transferto the SQUID, the voltage constant k = M/&. in the tank circ?it periodically drops, rising up again until the next transition takes place at V,. For fixed rf excitation, a peak detector draws a triangular pattern with period @, as a function of dja as shown in fig. 3 . The energy AE absorbed in one complete cycle is approximately the area of the loop in the @- Q a plane divided by the loop inductance (Zimmerman et al. 1970, Jackel and Buhrman 1975); for high P L ,

c,,

ec2

as seen from fig. 2. Although and depend on the dc flux threading the SQUID loop (eqs. ( 2 . 2 ) and ( 2 . 3 ) ) , A E appears insensitive to the point of operation. Consequently, the effective impedance of the tank circuit must depend on the point of operation; the rf SQUID acts as a flux-dependent

MULTI-SQUID DEVICES

1

I

-2 a0

I

I

I

277

1

I

I

1

I

- a0

0

a0

200

Fig. 3. Ideal triangular flux-voltage characteristic of the hysteretic rf SQUID with periodicity of one flux quantum.

resistor. Suppose @a = @0/2: as long as the energy fed into the t?nk circuit during each rf cycle does not exceed A,: :he peak voltage V, remains unchanged, resulting in a plateau in the V,I, characteristics. An increase of power creates another plateau until an energy 2AE is fed into the system during every rf cycle. Correspondingly, at Qa = Go the first plateau ends when the energy 2AE is exceeded. Realistic staircase patterns, corresponding to the cases @ . = ( n + f ) @ , , and Q a = nQ0, are plotted in fig. 4. When k2QT> r / 2 , where QT is the quality factor of the tank circuit (Giffard et a]. 1972, Jackel and Buhrman 1975), the plateaus overlap and proper adjustment of the rf bias current implies a perfect triangular pattern as in fig. 3. Noise in the hysteretic rfSQUID Thermal noise causes flyctuations in the points of flux transitions, tilting the plateaus of the jdVT characteristics (see fig. 4) and increasing the equivalent flux noise. As shown by Kurkijarvi (1972), the uncertainty in the flux jumps decreases when the frequency of the sinusoidal excitation is increased. Kurkijarvi and Webb (1972) derived an expression for the equivalent flux noise and showed that the slope a of the plateaus in the staircase pattern is related to the intrinsic energy sensitivity E (Jackel and Buhrman 1975):

where

E

is associated with the equivalent flux noise @,, in the SQUID loop

R. ILMONlEMl ET AL.

218

Fig. 4. The current-voltage characteristics of a hysteretic rf SQUID in the presence of thermal noise, for even and odd multiples of @,,0/2 of the externally applied flux 0".itf indicates the point of operation producing the triangular response illustrated in fig. 3. Q, is the quality factor of the tank circuit.

as

The intrinsic flux noise increases the noise temperature of the tank circuit. If noise from the preamplifier and the tank circuit is included as well, the experimentally determined value of a,a e x pcan , be used to estimate the equivalent input energy sensitivity (Jackel and Buhrman 1975): r

(2.7) where Ti @X/kgL is called the intrinsic tank circuit temperature; TT denotes the equivalent noise temperature of the tank circuit, and TAis the noise temperature of the preamplifier. Equation (2.7) is derived by assuming that the impedance of the loaded tank circuit equals the optimal impedance of the preamplifier (section 2.3) and that k2QT- 7r. Since the energy dissipation caused by the hysteretic loop is related to Ti simply by f k , T , a A E , the size of the loop should be as small as possible but sufficient to assure proper operation. TT is often determined by the preamplifier; thus T,, T,= TT and careful design of the preamplifier becomes imperative for a low-noise magnetometer. If wp = 27r x 20 MHz, L = 1 nH, and PL = 3, eqs. (2.5) and (2.7) predict that a 20.1 and T,= 300 K. Neglecting the tank circuit and the amplifier, an equivalent flux noise as low as (@:)"' = 1.8 x lo-' @,I& should be achievable, but without a cooled preamplifier this is impossible in practice. L-

L-

MULTI-SQUID DEVICES

279

A weak magnetic coupling results in a low flux-to-voltage conversion efficiency if it is not compensated for by a high Q-value of the tank circuit. On the other hand, tight coupling causes uncertainty in the flux transitions induced by tank circuit and preamplifier noise. Optimal choices of the mutual inductance and particularly of the product k2QT are discussed in detail by Jackel and Buhrman (1975). They argue that the best performance is obtained when k2QT exceeds unity. The same conclusion was drawn by Simmonds and Parker (1971) on the basis of computer simulations. Moreover, Ehnholm (1977) derived a small-signal model for an rf SQUID with complete input and output circuits and was able to show that the choice k2QT= 1 is a solid foundation for SQUID design.

Rf SQUIDS at high frequencies The sensitivity of rf SQUIDs can, in principle, be improved by increasing the pump frequency up,but the benefit is partly cancelled by the resulting higher preamplifier noise. When wp = w, = R / L,where w, is the characteristic frequency of the SQUID loop, the original absorption loop begins to deform, manifesting itself as a change in the tank circuit impedance. The flux sensitivity diminishes and the rf SQUID begins to resemble an LSQUID. Buhrman and Jackel (1977) concluded that proper adjustment of the SQUID parameters provides a low-noise rf SQUID even when w p > w , . High wp is tempting not only because it reduces the noise but also because it increases the signal bandwidth. It is, however, evident that a SQUID cannot reach the classical noise limit of the resistive loop, i.e., the thermal energy ( i k B T )divided by the noise bandwidth (aw,). SQUIDs operated at w,> w , have been studied both experimentally and theoretically by many authors (Kamper and Simmonds 1972, Kanter and Vernon 1977, Buhrman and Jackel 1977, Hollenhorst and Giffard 1979, Long et al. 1979, Seppa 1983, Vendik et al. 1983, Kuzmin et al. 1985, Likharev 1986). In principle, the high-frequency or microwave SQUID is suitable for multichannel applications since it can be made to a high-gain, low-noise magnetometer with a large signal bandwidth. The lack of reliable thin-film devices, the high cost of the electronics, and the existence of the dc SQUID, however, do not make it very tempting for applications where several channels are needed.

2. I .3. Discussion It is very important to keep in mind that a well-behaved rf SQUID can be constructed only by damping the junction properly. Incomplete damping may lead to multiple transitions and thus to excess noise. If the shunt

280

R. ILMONlEMl ET AL.

resistance R is adjusted so that the Stewart-McCumber parameter pc= 2 a R ’ C f c / @” remains less than 0.7 (Ketoja et al. 1984a), the I V characteristics of the junction is nonhysteretic and considered well damped. The choice Pc< 1 ensures stable operation of the rf SQUID as discussed by Jackel and Buhrman (1975). Unfortunately, the extra resonances or the parasitic capacitance introduced by a tightly coupled signal coil may substantially decrease the effective damping; this will be discussed in more detail in connection with the dc SQUlD (section 2.2.2). The rf SQUID is based fundamentally on the hysteresis loop traversed as a result of the rf excitation. Energy dissipations reduce the dynamic Q-value of the tank circuit, broadening the signal bandwidth and also making the SQUlD characteristics predictable. The latter consequence is especially important in multichannel applications. With low bias frequencies, the flux-to-voltage conversion efficiency remains moderate and careful design of the preamplifier becomes one of the most important issues in the development of low-noise rf SQUID systems. The intrinsic flux noise can be reached by cooling the first amplifier stage; this is a suitable method in some applications, but hardly in multichannel systems because of increased helium boil-off. Recent developments show that dc SQUIDs are replacing hysteretic rf SQUIDs at least in multichannel magnetometer applications. It seems, however, that the discovery of high- T, materials makes the hysteretic rf SQUlD interesting again. 2.2. DOUBLE-JUNCTION(DC) SQUlDs 2.2.1. Operation

Much lower noise levels than with rf SQUlDs have been obtained with dc SQUlDs (Clarke 1966, Clarke and Fulton 1969, Clarke et al. 1976, Tesche and Clarke 1977, Ketchen 1981). An ideal dc SQUlD is a superconducting loop that has two identical Josephson junctions with critical current I, (fig. 5 ) . In principle, the dc SQUlD can be operated by measuring either the average voltage as a function of the external flux #a with constant bias current f, or by monitoring the current f as a function of @a with constant bias voltage The dc-SQUID loop is in the superconducting state for bias currents below a flux-dependent critical value; at higher currents a voltage over the SQUID appears. The properties of an autonomous dc SQUlD are normally described by two dimensionless parameters P L = 27rLfJ O0 and p, = 2 a R ’ C I J Go,representing the normalized loop inductance and the damping of the junctions, respectively (see also sections 2.1.1 and 2.1.3). The voltage

MULTI-SQUID DEVICES

28 1

Fig. 5. Equivalent circuit of the dc SQUID. The ideal Josephson junctions are characterized by their critical current I , . Each is shunted by a capacitor C, by a resistor R, and by a thermal noise generator I , . I is the bias current and L is the loop inductance of the SQUID. 8 is the phase difference of the macroscopic wave function of the superconductor over the junction.

and the circulating current oscillate at high frequencies, typically on the order of l-lOGHz, depending on @a and Z, and only the time average of the voltage is monitored. The Iv and O a v characteristics are obtained by integration over the period of one oscillation (Tesche and Clarke 1977, Imry and Marcus 1977, Ben-Jacob and Imry 1981, Ketoja et al. 1984b). In fig. 6 we show the Z v characteristics for p L = n- and pc = 0.3. Figure 7 shows the periodic behavior of the voltage 7 as a function of @ a . A more detailed description of the dynamics is presented in section 4.1.3 of the companion paper (Ryhanen et al. 1989). Higher values of p L and pc create more complex behaviour (Ben-Jacob and Imry 1981, Ben-Jacob et al. 1983, 1985, Ketoja et al. 1984b, 1987, Kurkijarvi 1985). When pc> 0.7, the Z v curves are divided into different voltage branches connected by hysteresis loops, leading to multiple solutions at the same set of parameters (Kulik 1967, Imry and Marcus 1977). Qualitaand Zv characteristics by rounding tively, the thermal noise affects the the point where the voltage state emerges. Because of thermal noise, the hysteresis also disappears, or the hysteresis loops are rounded. On the other hand, fluctuations between different states increase the excess noise in the system. If the SQUID loop inductance and the junction capacitance are negligible, i.e., p L< 7r and BE= 0, the total flux in the ring @ = @a, and the dc SQUID behaves like a single Josephson junction with a resistance R / 2 and = 21, cos( n-Qa/ Oo).Integrating V over the an effective critical current Ic,eff period T of the Josephson oscillation, the average voltage for Z Zc,cn is obtained (Tinkham 1975): V d t = Y [ l - ( ~ c o s ~2 ) 1/2 .

v

@=v

v=Lj:

]

7

In comparison with figs. 6 and 7, an increased inductance reduces the

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282

3

2

1/24 1

0

Fig. 6. Current-voltage characteristics of the d c SQUID with PI. = T and f3, =0.3. The solid line is obtained when the external flux CDo = @"/2, the dashed line when @, = @,,/4, and the dash-dotted line when CDa = 0. The straight diagonal line is the resistive curve of the SQUID, V = R1/2.

-

I

0.8-

, /

,--.

00--.

,/--'\\

\

\\

'

L.'

I-

\ '' . 4 '

\

-

\ 'i

-U

Fig. 7. Voltage as a function of external flux of a dc SQUID with pL = n and p, = 0.3. The dash-dotted line is obtained when the bias current i = 1 / 2 4 = 0.8, the solid line when i = 1.0, and the dashed line when i = 1.2.

MULTI-SQUID DEVICES

283

modulation depth of the effective critical current AZc.c,; for @ a = 00/2, a voltageless state of supercurrent exists. Because thermal noise rounding was neglected in eq. (2.8), the approximate characteristics do not yet indicate the point of operation maximizing the transfer function 3 P/3@,. Differentiating eq. (2.8), one obtains an estimate at the practical point of operation in the flux-locked-loop mode: TRZ,

-@,=@,,0/4,1=2Ic

d @ O *

In a similar way, the dynamic resistance is (2.10)

To release the assumption pL=O,we note on the basis of RSJ-model simulation of fig. 6 that AZc,effi/3L = IT) = 0.5 x AZc,err(jlL= 0) = I,. Thus a V/a@,a AZ,,, is reduced approximately by a factor of 2 for BL= IT. Using PL = 2rLZ,/ Q0 = IT, we obtain R

z@.=@00/4.1=21,

zL'

(2.1 1 )

The equivalent spectral density of the voltage noise power is

(;:I2

Sv = 4kBT - L 2 / (2 R ) + 4k0 TRdyn+ 4kBTARdyn,

(2.12)

where the first term is the contribution of the fluctuations of the circulating current generated by thermal noise of shunt resistors, the second term represents thermal noise across the SQUID ring, and TA is the noise temperature of the amplifier. Applying approximations (2.10) and (2.1 l ) , the energy resolution (2.6) becomes (2.13)

In comparison to the rf SQUID, eq. (2.7), we note that the energy resolution of a dc SQUID depends on the characteristic frequency o,= R/ L, which is normally much higher than the pump frequencies in rf SQUID magnetometers. Since, in addition, TA tends to increase with frequency, the dc SQUID appears superior. Neglecting amplifier noise and setting PL/ r = PC= 1, we find E = 1 2 k 0 T m . For practical reasons, Bc must be set below 1 (Knuutila et al. 1988); in a dc SQUID with C = 1 p F and L =0.2 nH the shunt resistance is about 5 R implying o,= 27r x 4 GHz. The noise temperature of a state-of-the-art

R. ILMONIEMI ET AL.

284

0’04 0.02

t

1

I

I

HYSTERETIC JUNCTIONS-

0 0

0.5

1.5

1.0

2.0

2.5

3.0

Bc

-

Fig. 8. Dimensionless energy resolution E = J L / C O ; ~ Cas a function of / 3 c = 2 ~ R 2 C I c / @ , The solid line depicts the approximation (2.13), where the junction capacitance C and the loop inductance L are fixed, BL = n,and /3, is varied by changing R. The dashed vertical line refers to the critical value of p, for hysteresis. The squares are from hybrid computer simulations (de Waal et al. 1984). the circles from numerical simulations (Ryhanen et al. 1989).

amplifier can be as low as 2 K (section 2.3); its contribution is therefore negligible. According to eqs. (2.6) and (2.13), the flux noise in our example is ( CD;)~’*= 1.4 x O0/&. However, the presence of the coupling circuits deteriorates the performance (section 2.2.2). The sensitivity of the dc SQUID has been studied by computer simulations (Tesche and Clarke 1977, Bruines et al. 1982, de Waal et al. 1984). The optimized energy resolution was found to be nearly independent of p L and pc for.rr
2.2.2. Problems with practical devices In rf SQUIDS the coupling between the SQUID and the flux transformer circuit has normally been kept relatively loose. In SQUID magnetometers

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a good coupling between the SQUID loop and the signal coil is tempting because the energy resolution at the output of the device E , , ~ = e l k : , where E is given by eq. (2.13) and k, is the coupling constant (see eqs. (3.1) and (3.2)). In practical devices, this leads to the use of planar gradiometers discussed in section 3.5.3. Several studies (Ketchen et al. 1978, Jaycox and Ketchen 1981, Tesche 1982, Carelli and Foglietti 1982, Tesche et al. 1985, Muhlfelder et al. 1985, Enpuku et al. 1985a, de Waal and Klapwijk 1982, Knuutila et al. 1987a) indicate problems in coupling the dc SQUID to the flux-coupling circuit. Figure 9 illustrates a dc SQUID and its signal coil. The SQUID loop is represented by a thin square washer, with the junctions shown as projecting edges of the plate; the signal coil is shown as a two-turn stripline over the dc SQUID. The signal coil introduces stray capacitance across the junctions, and the signal-coil turns are capacitively connected mainly via the SQUID plate and at crossings of the signal coil stripline. High-Q resonators are created, as shown in fig. 10. The most prominent new feature introduced by the signal coil is the increased parasitic capacitance C,, appearing across the junctions (Tesche 1982, Enpuku et al. 1985a). Evidently, C, is roughly proportional to the number of turns in the signal coil; its effect becomes significant when it approaches C. If C,> C, the dynamics is completely determined by the stray capacitance, because of noise; the voltage-to-flux transfer ratio deteriorates and new sources of noise are introduced. Another important feature in the dynamics is caused by the parasitic capacitance C, in the signal coil. Resonances in the signal coil are far below the operation frequency and do not manifest themselves in the SQUID characteristics as CURRENT NODES

m

--I

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+ LOW FREQUENCIES HIGH FREQUENCIES

---

4

PARAS1 TIC CAPASITANCE

JOSEPHSON JUNCTIONS

Fig. 9. Simplified dc SQUID with its signal coil. The rectangular washer is the SQUID ring; the stripline over the washer is the signal coil. The tunnel junctions are illustrated as black layers interrupting the ring.

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Josephson junctions Fig. 10. Model of a dc SQUID with its signal coil. C, is the parasitic capacitance created by the signal coil, and C, is the stray capacitance in this coil. R, damps parasitic oscillations in the SQUID (Enpuku et al. 1985a), and R , and C, are for damping the signal coil oscillations (Seppa and Ryhanen 1987).

voltage plateaus. Thermal noise can, however, activate these resonances as shown by computer simulations (Seppa and Ryhanen 1987) and by experiments (Knuutila et al. 1987a). The dc-SQUID loop itself forms a A/2 transmission line. The Q-value of the oscillations of this resonator is, however, quite low, unless the signal coil prevents the SQUID from radiating energy into the surrounding space. Increased loop size moves the resonance frequencies near the Josephson oscillations. This is hazardous for proper operation of a dc SQUID, causing voltage plateaus in the Zv characteristics (Muhlfelder et al. 1985, Seppa and Ryhanen 1987, Knuutila et al. 1988). Another resonance is created by the spiral transmission line formed between the SQUID and the stripline. If the signal coil is long, i.e., there are many turns for tight coupling at low frequencies, the resonance is well below the Josephson frequency and the coupling at high frequencies becomes looser, which makes the device more independent of its surroundings. On the other hand, the demand for lower C , favors a short signal coil, and thus the h / 2 resonance of the signal coil has a tendency to be near the operating frequency. Thus, a compromise is necessary. 2.2.3. The state of the art

The desire for maximal coupling efficiency for the external flux favors a large loop area, i.e., a high SQUID inductance, especially at low frequencies, whereas maximum energy sensitivity calls for a small device with a small junction capacitance. Studies of autonomous dc SQUIDS have shown that it is possible to increase slightly the values of BL and / I c from their optima without severe deterioration of performance. In practice, however, increasing the dimensions of the d c SQUID has not been promising.

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Capacitive shunting of the Josephson junctions has been suggested (Paik et al. 1981, Tesche 1982, Tesche et al. 1985). In this structure, the loop inductance is divided into two parts: a small loop of inductance L on the junction side and a large loop of inductance L, on the signal coil side, enabling good coupling. High-frequency Josephson oscillations sense only the small loop, and the effective size of the dc SQUID is small. The parasitic capacitance over the junctions is in parallel with a larger shunting capacitance and thus its effect is negligible. The characteristics appear, however, double-valued, hysteretic and noisy, owing to the frequency-dependent inductance of the dc-SQUID loop. At higher bias currents, where the characteristics are smoother, the noise is mainly thermal; the flux-to-voltage transfer function is, however, much smaller than at low bias currents. The energy resolution turns out to be proportional to L,, and thus the doubleloop dc SQUID is noisier than an optimally adjusted conventional dc SQUID. Enpuku et al. (1985a,b, 1986, Enpuku and Yoshida 1986) have investigated thoroughly the effect of the damping resistance R, (see fig. 10). The main idea of these studies is to increase the SQUID loop inductance in order to secure better coupling to the flux transformer. The increased inductance enhances the current oscillations, and the effect of the internal and external resonances becomes stronger. If the resistive shunting is assumed to wipe out the beating resonance of the SQUID, the damping resistor must be set to R P = W . The damping resistor creates extra flux noise, limiting the energy resolution to E 3 2kBTL/R, = 2 k B T m , which is still determined by the loop inductance, and thus increased L deteriorates the performance. A structure analogous with the capacitively shunted dc SQUID is obtained by dividing the loop inductively into two parts (Ketchen et al. 1978, de Waal and Klapwijk 1982, Koch 1985). In this solution the large loop Le couples the flux while the smaller loop forms the dc SQUID. The is the flux threading flux in the smaller loop is Gi = (L/Lp)@,, where the large loop. The effectkflux noise of the smaller loop is @fn = 4kBTL2/ R, and the energy resolution (2.6) is (2.14) The advantage of this structure is that the parasitic capacitance across the junctions is avoided, which means that LC-resonances at low frequencies do not arise. However, as seen from eq. (2.14), the energy resolution remains modest and the large loop size may bring the transmission-line resonances down to Josephson frequencies.

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Following the idea of the fractional-turn SQUID, originally reported by Zimmerman (1971) for rf SQUIDs, Carelli and Foglietti (l982,1983a, 1985) have constructed multi-loop dc SQUIDs. The structure developed by them suffers from transmission-line resonances that are seen as voltage plateaus of the measured characteristics. This is not surprising considering the large size of the SQUID. The multi-loop structure is suitable for rf SQUIDs because the pump frequency is far below the transmission-line resonances, whereas in dc SQUIDs the Josephson frequencies are near these resonances. The result is good coupling, but there are problems in obtaining smooth characteristics, and hysteresis may cause excess noise. An efficient scheme to couple the low-inductance dc SQUID to a signal source was developed by Muhlfelder et al. (1983). This matching-transformer solution has proved to be excellent for several reasons (Seppa and Ryhanen 1987, Knuutila et al. 1988): (1) The number of turns over the dc SQUID is small; as a result, parasitic capacitances are small and transmission-line resonances are easier to control. (2) Good coupling between the high-inductance signal coil (1-2 pH) and the low-inductance SQUID loop (10-100pH) can be achieved. (3) The effect of parasitic resonances in the transformers can be controlled in the external circuit (see fig. 10). Recently, the design and fabrication of the first completely optimized dc SQUID was published (Knuutila et al. 1988). All resonances were damped or eliminated, and the device was designed to be independent of its surroundings. The problem of bringing the flux to the SQUID was solved by using an intermediate coupling transformer. The dimensions of the SQUID were optimized to obtain the lowest noise level allowed by the available fabrication technology. The energy resolution was limited by two values determined by the fabrication process: the junction capacitance C and'the stray inductance ofthe SQUID loop. The disadvantage is that the realization of the complete structure requires a total of 10 mask layers, complicating the fabrication. Comparison of different dc SQUIDs reveals that the devices are limited by the inductance of the flux-coupling loop. Whatever efforts are made in order to obtain better energy resolution, one is always limited by until the quantum noise limit is reached (see Likharev 1986). Best results have been obtained with low-p, junctions and by damping the signal-coil resonances, or by using relatively moderate values for all the SQUID dimensions in order to prevent the different resonances from interfering with the dynamics of the dc SQUID. The complicated structures needed in dcSQUID devices make the use of high-T, materials difficult, and it is surely going to take a long time before reliable, low-noise high- T, dc-SQUID systems can be produced.

a

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2.3. ELECTRONICS

General When a SQUID is used to monitor a low-frequency magnetic field, current, or voltage, a flux-modulation technique is almost invariably used (Forgacs and Warnick 1967). Usually a square-wave modulation, correspondi'ng to ad2 peak-to-peak, is applied on the SQUID and the output signal is detected by a demodulator circuit. This technique is applicable to any SQUID and it helps to eliminate some sources of low-frequency noise such as thermal EMFs, changes in the critical current of the junctions; drifts in SQUID-circuit parameters, and l/f noise from amplifiers. A further improvement is attained by feeding the detected signal through a resistor back to the SQUID ring. The high-gain feedback loop tends to maintain a constant flux in the SQUID loop, and thus the voltage drop across the feedback resistor is proportional to the external flux. In addition, the ambiguous SQUID response is removed and the output voltage becomes independent of variations in amplifier gain. In general, a well-designed lock-in electronics does not increase the flux noise in the system, but in SQUIDs with multi-valued V@ characteristics some loss in sensitivity may result. Because of the periodicity of the SQUID response, the feedback electronics can lock the intrinsic flux at any of many values. A strong external signal may kick the system from one stable point of operation to another, causing a sudden change in the output voltage. Evidently, the higher the feedback gain, the better the system is for preventing the external flux from entering the SQUID loop. The open loop gain is mainly limited by the filters necessary for stabilizing the feedback loop. In rf SQUIDs the modulation frequency must be less than the bandwidth of the tank circuit, and it is often limited by the bandwidth of the components in the lock-in circuit. In dc SQUIDs the utmost modulation frequency is set by the amplifier noise which tends to grow with increasing frequency. SQUID electronics operated in the lock-in mode is extensively discussed by Giffard et al. (1972) and Giffard (1980); only the most important features will be mentioned here. Usually the feedback loop contains an integrator in a form of a PI (proportional-plus-integrating) controller that provides a feedback gain G(ja)=jwmOd/u In. this construction the maximum rate of flux change but an overall loop stability (slew rate) is limited to the order of is easily attained. Electronics providing a higher slew rate can be constructed but these devices are not very stable against change of loop parameters (Giffard 1980). Anyway, under quiet conditions, such as inside a magnetically shielded room, a conventional PI controller is sufficient to ensure proper operation. In multichannel applications the electronics should

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contain a special unit to automatically reset the SQUIDs after a loss of lock-in. Although the low-frequency readout circuits are very similar, the rest of the electronics may differ essentially for rf and dc SQUIDs. We next discuss special features of both SQUIDs. Rf SQUID in the hysteretic mode Energy dissipation in the rf SQUID loop affects the tank circuit impedance and thus the signal bandwidth. On the other hand, the dynamic impedance is related to the dissipation of the unloaded tank circuit and thus also to the input impedance of the preamplifier. Therefore, the optimization of noise characteristics leads to the condition R,,, = a R T , where a is the slope of the staircase pattern (see fig. 2), Rop,is the optimal input impedance of the preamplifier, and RT is the effective tank-circuit resistance. If R T is limited by the input impedance of the amplifier, the above condition will be fulfilled only in a narrow frequency band. This problem is aggravated with cooled preamplifiers (Ahola et al. 1979). The low output impedance of the hysteretic rf SQUID can be utilized by increasing the flux-modulation frequency and thus by constructing a device with a high slew rate. The electronics of a typical rf SQUID is shown in fig. 11. The rf level is adjusted by the variable capacitor C,, and the tank circuit is tuned by the room temperature capacitor C , . The square-wave oscillator providing the flux modulation also controls the PSD circuit (phase-sensitive detector). The main design principles for obtaining a low-noise rf SQUID are a very careful design of the preamplifier, the elimination of all dissipative elements

P Y

* -

-

-

c,7&

TLZ K - i

i-


.

-1

I

I

AF GEN

-

c, I

__

PI CONTROLLER

-:.-_I

Fig. 11. Block diagram of the electronics for a hysteretic rf SQUID biased via the tank circuit by a radio-frequency (rf) drive and modulated by a square-wave audiofrequency ( a n signal; the amplified rf signal is monitored by the diode detector. The output of the phase-sensitivedetector ( E D )is fed back to the tank circuit and thus into the SQUID ring through the PI controller and the feedback resistor R,. The rfdrive level is adjusted by the tunable capacitance C, and the resonant frequency by the capacitor C,.

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from the cooled tank circuit, which is coupled to the room temperature via a coaxial cable, and choosing the feedback resistor to prevent the increase of the effective temperature of the resonant circuit. Dc SQUID A matching circuit is required to monitor the voltage over a dc SQUID with a low-noise JFET amplifier, which has an optimal input impedance of a few kilo-ohms: the SQUID impedance is typically only a few ohms. The modulation frequency must be high enough to avoid l/f noise from the preamplifier but low enough to exclude input current noise which increases drastically with increasing frequency. Many switching transistors with large gate areas provide excellent noise characteristics for reasonable source impedances and are thus suitable. The low output impedance of dc SQUIDS can be increased by feeding the signal through a cooled inductor into a capacitor, set in parallel with the preamplifier or by using an ordinary tuned transformer (with or without ferrite core) in a helium bath (Clarke et al. 1975a,b, 1976, Danilov et al. 1977). The tuned transformer is recommended since the increase of the output capacitance reduces the bandwidth of the reactive transforming circuit. Wellstood et al. (1984) improved the slew rate by placing another transformer at room temperature and increasing the modulation frequency up to 500 kHz. One practical construction for the dc SQUID readout circuit is illustrated in fig. 12 (Knuutila et al. 1988). The signal from the SQUID is fed through the cooled resonant transformer into the preamplifier consisting of two FETs in parallel (Toshiba SK146). The demodulator circuit contains, in

I

INSIDE SHIELDED ROOM

I

OUTSIDE SHIELDED ROOM

Fig. 12. Block diagram of the dc SQUID electronics based on phase-sensitive detection (Knuutila et al. 1988).

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addition to the switching circuit, also an integrator and a sample-and-hold circuit, which can be used to attenuate effectively the modulation frequency and its harmonics, without introducing a phase shift at signal frequencies. The output of the demodulator is fed via the PI controller and the transconductance amplifier back to the modulation coil. The electronics is operated at 100 kHz, and the cooled input circuits are designed to transform a 10 R SQUID impedance to 4 kR, which is the optimal input impedance of the amplifier stage including two Toshiba SK146 FETs. The noise temperature of the amplifier was found to be 2 K at 100 kHz. A digital-to-analog converter is used for automatic zeroing of the output voltage after flux jumps. It has been found that l/f noise in dc SQUIDs can be reduced by reversing the bias current synchronously with flux modulation. This technique requires complicated electronics and is thus not tempting in devices where the simultaneous operation of many SQUIDs is required. One elegant solution, with effective reduction of low-frequency noise, is discussed by Foglietti et al. (1986).

3. Applications: biomagnetism In biomagnetism, the magnetic field produced by biological events is recorded. The field can be measured non-invasively without any physical contact to the subject; therefore, the technique is well-suited for the study of activity in the human body. Biomagnetic fields are produced by three main mechanisms. First, active electrical currents in the brain, the heart, muscles, or in other parts of the body give rise to fields that convey information about the functions of these organs. Second, magnetized contaminants or foreign objects in the body produce steady fields, which may reveal the amount and distribution of the contaminants or the locations of magnetic objects. Third, diamagnetic or paramagnetic tissues modify externally applied fields; a technique has been developed to determine the content of iron in the liver using this effect. Typical amplitudes and spectral densities of biomagnetic fields and noise sources of various origins are shown in fig. 13. The external magnetic disturbances are several orders of magnitude larger than biomagnetic fields. To illustrate this, consider a flux transformer coil parallel to the earth’s 5 0 p T magnetic field. Tilting the coil by only 0.0002 seconds of arc will change the field sensed by it by 50fT. If the diameter of the coil were 20 mm, this rotation would correspond to a movement of the wire by only 0.01 nm, i.e., by a fraction of an atomic radius! Since many biomagnetic fields of interest are of the order of tens of femtotesla, it is evident that successful measurements require a careful elimination of vibrations of the magnetometer and shielding against external disturbances.

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10’0

-

108

E

1

-t

106

x

! i

E

n

10‘

; d

10’

U

a u)

1

10-2

Fig. 13. Peak amplitudes (the arrows on the left) and spectral densities of various biornagnetic and other fields. The laboratory noise level is adapted from Kelhi et al. (1982). the geomagnetic noise from Fraser-Smith and Buxton (1975). the brain noise from Knuutila and Hamalainen (1988).

3.1. MEASUREMENT TECHNIQUES

3.1.1. Magnetically shielded rooms

The most straightforward and reliable way of reducing the effect of external magnetic disturbances is t o perform the measurements in a magnetically quiet space. Several shielded rooms have been constructed for this purpose. An effective shielded enclosure was first built at the Massachusetts Institute of Technology by Cohen (1970). This room has three ferromagnetic layers and two layers of aluminium in the form of a rhombicuboctahedron, a polyhedron of 26 faces. A shielding factor of 66 dB at very low frequencies was obtained when active shielding and so-called shaking were used to enhance the shielding factor. Shaking is performed by applying a strong continuous 60 Hz field on the ferromagnetic layers so as to increase the effective permeability at other frequencies. At the Low Temperature Laboratory of the Helsinki University of Technology a cubic shield with inner dimension of 2.4 m was built in 1980 (Kelha et al. 1982). This room (see fig. 18) has three layers of mu-metal, sandwiched between aluminium plates; it attenuates external fields by 90-1 10 dB above the frequency of 1 Hz. 10-20dB of additional shielding below 10 Hz is obtained with active shielding, which is based on measurement of the

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external magnetic field and its compensation in a volume that contains the shielded room. A very heavy shield, consisting of 6 layers of mu-metal and one layer of copper, was built at the Physikalisch-Technische Bundesanstalt in Berlin (Mager 1981). Although this room has much thicker walls, its shielding factor is comparable to that of the Helsinki room in the frequency range 1-100 Hz (Emi et al. 1981). The reason is, evidently, that eddy-current shielding is not effective in the six layers of the Berlin room, whereas in Helsinki the aluminium plates in each layer were welded together at their edges to form closed shells for effective eddy-current flow. Recently, several shielded rooms for biomagnetic use have been constructed commercially by Vacuumschmelze'; these enclosures have two ferromagnetic layers and inner dimensions of 3 x 4 x 2.4 m'. The shielding factor varies from about 50 dB at 1 Hz to about 80 dB at 100 Hz. The Vacuumschmelze room is a compromise between performance and price; with suitable second-order gradiometers (see section 3.1.2) it offers a sufficiently quiet space for practically all types of biomagnetic measurements. Less expensive shielded rooms can be built from thick aluminium plates; Zimmerman (1977) constructed such an eddy-current cubic shield with inner dimensions of 2 m. Several similar rooms have been built later (Malmivuo et al. 1981, Stroink et al. 1981, Nicolas et al. 1983, Vvedensky et al. 1985b). The wall thickness in these rooms is about 5 cm; the shielding factor is proportional to frequency, being about 50 dB at 50 Hz. Magnetically shielded rooms are easily made tight against rf fields. One might think that this would facilitate the design and operation of SQUID magnetometers because of reduced demands on the radio-frequency insensitivity of the instruments. However, in many practical situations, at least in a research laboratory, it is difficult to avoid radio-frequency fields inside the shielded room. Some experiments and, in particular, preparations for experiments are easiest to perform when the door of the room is open. Since the room acts as a resonance cavity, strong rf fields may occupy its interior. Even with the doors closed, cables to the stimulus equipment, EEG leads, and other wires act as antennae, bringing external rf fields inside. It is, therefore, essential that the SQUID system operates even in the presence of rf fields. The ultimate noise level in magnetically shielded rooms is determined by Johnson noise in the conducting walls. The effect of this noise source is most significant in eddy-current shields (Varpula and Poutanen 1984); it can be reduced by mu-metal sheets (Maniewski et al. 1985).

' Vacuumschmelze GmbH, Griiner Weg 37,6450 Hanau, West Germany.

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3.1.2. Gradiometers

In addition to or instead of shielded rooms, external magnetic disturbances may be partly cancelled with gradiometric coil configurations. A simple tirst-order gradiometer, where two oppositely wound coaxial coils are connected in series with the signal coil, is shown in fig. 14c. This arrangement is insensitive to a homogeneous magnetic field, because it imposes the same flux on the lower (pickup) and the upper (compensation) coils. On the other hand, the first-order gradiometer is effective in measuring magnetic fields produced by nearby sources. If the pickup coil is close to.the head and if the distance between the two coils (the baseline) is at least 4-5 cm, the magnetic field produced by the brain is sensed essentially by the lower coil only. Other possibilities of arranging the first-order gradiometer are shown in figs. 14b, d and e. In fig. 14d, an asymmetric first-order gradiometer is illustrated; its main advantage is that the compensation coil inductance is reduced; therefore, a better sensitivity is obtained than with a symmetrical gradiometer. In fig. 14e, the pickup coil and the compensation coil are connected in parallel instead of in series so as to reduce the inductance of the detection coil by a factor of four from the configuration in fig. 14c. This parallel connection is easier to match to a SQUID signal coil with a small inductance; the disadvantage is that homogeneous magnetic fields give rise to shielding currents in the detection coil that couple to neighbouring

t

Fig. 14. Different types of flux transformers. (a) Simple magnetometer that measures the flux threading the loop. (b) Planar 2-D gradiometer that is sensitive to the difference of flux in the two loops. (c) Symmetrical gradiometer; the difference of the field between the two coils, connected in series, is measured. (d) Asymmetric gradiometer; the homogeneous magnetic field sensed by the pickup coil is compensated for by a single large turn in the upper coil. (e) Symmetrical gradiometer, in which the pickup and compensation coils are connected in parallel. (f) Symmetrical second-order gradiometer.

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gradiometer channels. In fig. 14b, a planar off -diagonal, so-called double-D gradiometer is shown. To obtain insensitivity to homogeneous fields, one has to make the turns-area products of the pickup and compensation coils equal. In wirewound gradiometers, an initial balance of a few percent is typical; it can be improved by adjusting the effective areas of the coils with movable superconducting tabs. Three balancing tabs are needed to balance all three field components; a final balance of about 1-10 ppm can be achieved. One must remember that even after careful balancing at low frequencies, the balance at rf frequencies is not guaranteed. Therefore, it is important to have good rf shielding surrounding the pickup coils on an rf shunt across the signal coil; an R C shunt is appropriate (Ilmoniemi et al. 1984, Seppa and Ryhanen 1987). Figure 14f shows a configuration where two gradiometers are connected together in opposition so that the detection coil is insensitive to both homogeneous fields and to uniform field gradients. This arrangement further improves the cancellation of magnetic fields produced by distant noise sources; successful experiments with second-order gradiometers have been performed in unshielded environments. A disadvantage of gradiometers of high order is that the signal energy coupled to the SQUID is reduced. Although superconducting tabs are suitable for balancing single-channel instruments, their use is too complicated with multichannel devices. In New York University a system was adopted where balancing was done electronically, as described in more detail in section 3.5.2. 3.2. NEUROMAGNETISM At present, neuromagnetism, the study of magnetic fields that originate in the human brain, is the branch attracting most attention in biomagnetism. Magnetoencephalography (MEG) is closely related to electroencephalography (EEG): both are produced by the same cerebral events; a great advantage of MEG is that the local irregularities of the skull and the scalp do not essentially affect the magnetic field whereas they have a major effect on the electric field. The current that gives rise to the magnetic field is confined by the poorly conducting skull to flow within the cranial cavity. As a result, the determination of locations of sources from the measured electric potential or magnetic field distribution is more accurate in MEG than in EEG. Another advantage of MEG is that no electrode leads need be attached to the head. This facilitates the preparation of experiments especially in multichannel measurements; putting on, say, 32 EEG electrodes can take about one hour. A fundamentaljustification for the magnetic technique is that MEG and EEG are sensitive to different configurations

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of active currents in the brain; MEG conveys information not contained in EEG signals. A more detailed view on neuromagnetism can be obtained from several recent reviews (Hari and Ilmoniemi 1986, Romani and Narici 1986, Kaufman and Williamson 1987, Williamson and Kaufman 1987). 3.2.1. Origin of neuromagnetic fields

Interpretation of MEG signals requires an understanding of the origin of the neuromagnetic field. It is believed that a major portion of the measured field is produced by currents of neurons whose alignment is approximately perpendicular to the cortical surface. This primary current gives rise to volume currents, and the primary and volume currents together produce the measured magnetic field. In a spherically symmetric conductor, the magnetic field due to the radial primary current is cancelled by the accompanying volume currents. Thus, the externally observable magnetic field is produced by tangential primary currents alone, i.e., mainly by activity in the fissures (Grynszpan and Geselowitz 1973). If a small area of cortex is active, one may model the associated primary current as a dipole. Vector formulae for the magnetic field produced by a dipole in a spherically symmetric volume conductor have been developed (Ilmoniemi et al. 1985, Sarvas 1987). The sphere model for the head is, of course, a severe simplification. However, in studies where the spherical head model was compared with a realistically shaped multilayered head model (Hamalainen and Sarvas 1987), it was found that in regions where the deviation from sphericity is small, e.g. at occipital areas, the sphere model gives correct field values to an accuracy of a few percent, provided that the radius of the sphere is fitted to the local radius of curvature of the skull’s inner surface. In frontotemporal locations, however, the sphere model failed to reproduce the correct field pattern. The realistically shaped head model is, of course, computationally much more demanding than the sphere model, since explicit formulae for the magnetic field are not available. At present, realistically shaped models are too tedious for use in least-squares fitting. A possible solution might be in applying the homogeneous head approximation (Hamalainen and Sarvas 1987), where the skull and the scalp are neglected and the head is replaced by a uniform conductor of the shape of the inner surface of the skull. This is justifiable since only a small amount of the volume current flows in and through the poorly conducting skull, thus giving negligible contribution to the magnetic field measured outside the head. The homogeneous head approximation performs substantially better than the sphere model in nonspherical areas and yet is not computationally as demanding as the three-

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layer model. However, at present only the sphere model is in routine use. Once the forward problem, i.e., the calculation of the magnetic field from a given primary current distribution, is mastered, one must tackle the inverse problem by determining such primary current distributions that can explain the observed field. The simplest source model is the current dipole, which in a spherical conductor geometry is described by five parameters, i.e., three location coordinates and two tangential components of the dipole moment vector. Figure 15 shows the pattern of the radial component of the magnetic field due to a current dipole in a spherically symmetrical conductor. To determine the location of the dipole on the basis of noisy measurements, it is necessary to measure the pattern at points that cover both polarities of the field. Otherwise, small amounts of noise may put the estimated location of the dipole in a totally wrong region. The inverse problem can be solved only if the number of independent measurements is equal or exceeds the number of parameters that are to be calculated. In practice at least twice this number is needed, because the magnetometers are usually not optimally located for the purpose of determining the source parameters from noisy measurements. Further, to evaluate whether the model itself is reasonable, more measurements are required than there are adjustable parameters. 3.2.2. Spontaneous activity Neuromagnetic fields were first measured by Cohen ( 1968) who detected the magnetic alpha rhythm with an induction-coil magnetometer. The

Fig. 15. A calculated isofield contour map of the radial component of the magnetic field due to a current dipole in the brain. The dipole points in the direction of (he arrow; its depth is roughly equal to the distance between the field extrema divided by J2. Solid lines indicate field emerging from the head, dashed lines field entering it. A measured pattern like this can be used for estimating the tangential part of the current dipole vector and its three-dimensional location.

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measurements were carried out in a shielded enclosure and the signals were averaged using a trigger from the EEG. The polarity of the signal was opposite over the left and right hemispheres; this suggested that the primary currents associated with the magnetic alpha rhythm are oriented along the direction of the sagittal midline and located in the occipital area. Subsequent experiments with SQUID magnetometers have confirmed these results (Carelli et al. 1983, Chapman et al. 1984), the major new results being the determination of locations of equivalent sources (Vvedensky et al. 1985a) and the finding that different spindles of alpha waves are generated by different configurations of cortical activity (Ilmoniemi et al. 1988). The latter two studies would not have been possible without multichannel SQUID magnetometers (see section 3.5.2): the signals are not repeatable and, therefore, measurements performed at different times over different locations are not directly comparable.

3.2.3. Evoked fields

In the most common type of neuromagnetic measurements, suitable sensory stimuli are given to the subject and the magnetic field due to the subsequent cerebral activity is measured. Because of noise, including that produced by the background activity of the brain itself, the same stimulus is repeated dozens of times, and the measured signals are averaged. An example of a sequence of magnetic field patterns evoked by an auditory stimulus is shown in fig. 16 ( S a m et al. 1985). The stimulus in this experiment was a 30-ms tone burst of 1000-Hz frequency and of 70-dB amplitude delivered to the left ear. The magnetic field maps are shown at intervals of 20 ms beginning at the time of stimulus onset. There is some evidence of a response 40 ms after the onset of the stimulus; at 100ms the field amplitude is maximal. Dipoles that best explain the measured patterns were determined and are shown as dark arrows in the figure. The sites of the dipoles agree with the known location of the auditory cortex. A number of evoked-field studies have been performed on the auditory as well as other modalities (see, for example, Romani and Williamson 1983, Hari and Kaukoranta 1985, Weinberg et al. 1985, Hari 1989, Atsumi et al. 1988). These studies have amply demonstrated the locating power of M EG and provided new information about the origin of several evoked-field components. Recently, interest has grown among psychologists in using MEG as an indicator of the location of activity changes that are related to mental operations. One of the results in this area is the demonstration that conscious attention has a marked effect on the amplitude of the 100ms deflection of the auditory evoked response at or near the primary auditory cortex (Curtis et al. 1988). Of special relevance appear to be neuromagnetic

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Fig. 16. Magnetic field isocontour maps obtained in response to a 30 ms tone burst of lo00 Hz frequency, delivered to the left ear. The field was measured with three channels simultaneously and the measurement was repeated at a number of locations as shown in the upper left panel. The frames are labeled by the time, in ms, from the onset of the stimulus. In maps obtained after 80 ms, an equivalent dipole is drawn as an arrow; the amplitude Q of the dipole is shown as well as the proportion E of field energy explained by the dipole model. Solid lines indicate field emerging from the head and dotted lines field entering the head. The dots show the locations of magnetometer channels. The difference in field amplitudes between adjacent isocontour lines is 40 IT. Modified from Sams et al. (1985).

studies related to processing of speech sounds (Kaukoranta et al. 1987); the role of speech in communication is unique to the human species. While many other cerebral phenomena are amenable to detailed investigations in animals, language can be studied in humans only. 3.2.4. Clinical aspects of MEG

Clinical applications of MEG are just beginning. One of the reasons for the slow start is that with single-channel instruments, measurement sessions have simply taken too long for being practical for studies of patients. Multichannel instruments will remove this problem and it is expected that several clinically oriented MEG laboratories will start operation in the near future. MEG has obvious merits as a potential diagnostic tool. There are several other methods to probe brain anatomy and functions, but the only noninvasive techniques available for the real-time monitoring of brain activity are

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MEG and EEG. The latter is already in wide clinical routine use, but its capability for locating brain activity is poor, so MEG seems to have unique possibilities. While the EEG is mostly used to detect abnormal wave forms of electrical cerebral activity, the locating power of MEG offers a possibility to characterize the spatiotemporal operation of the brain in a more specific manner. Proper analysis of EEG signals would require detailed knowledge about the conductivity as a function of location in the head, particularly in the scalp and the skull. Although the numerical description of the detailed shape of the skull and the scalp is cumbersome and computational analysis of EEG with the acquired shape information requires a great deal of computer time, it is probable that realistic models for the conductivity distribution of the head will be used increasingly. The spatial accuracy of EEG will thereby improve, but because MEG contains information that is complementary to EEG and because its use involves no tedious attachment of electrodes, MEG will become clinically increasingly attractive, particularly when large magnetometer arrays become available. Epilepsy The first clinically relevant application of MEG was with focal epilepsy, where progress has been made in locating spots in the brain where seizures originate. In cases where drugs are not effective in controlling seizures, the location of the foci is needed for the planning of their surgical removal. Modena et al. (1982) reported simultaneous electric and magnetic recording of epileptic activity from 12 patients with generalized epilepsy and 15 with focal epilepsy; in 9 patients out of the latter group sharply localized magnetic field patterns were observed. Locations of epileptic foci in three dimensions were determined by Barth et al. (1982), who used electrically measured interictal spikes as triggers for averaging, and by Chapman et al. (1983), who applied the relative covariance method (Chapman et al. 1984) in the formation of magnetic field maps for subsequent determination of equivalent dipole parameters. In addition to finding epileptic foci, MEG seems appropriate for establishing patterns of the subsequent spread of the activity as well as for studying the dynamics of non-focal epilepsy. Other clinical applications There is much interest in other clinical applications, especially in the most common disorders of the brain and in disorders that are difficult to diagnose. So far practically nothing has been done with MEG in studies of this type, and it is premature to judge what the chances for success are. However, because of the immense significance of any progress in understanding or diagnostics of brain dysfunction and because of the lack of alternative noninvasive real-time monitors, it seems worthwhile to put effort into this

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area. If the dynamics of the disorders in question will be understood, one will probably also be able to develop diagnostics using MEG, possibly together with EEG. However, because work in this area is just beginning and multichannel instruments will come into widespread use only in the next decade, one should not expect many significant results earlier than in the mid-90s. Another line of development that might lead to clinical applications of MEG is the monitoring of the integrity of sensory pathways. Most of current MEG research is of the evoked-field type, in which stimuli are given to the subject and the neural response is measured. Abnormalities in the pathways can be objectively evaluated using this paradigm. A study along these lines was performed by Peliuone et a]. (1987) who found, on the basis of neuromagnetic measurements, how auditory pathways had been modified in a patient with a cochlear prosthesis. The ability of MEG to determine the location of brain activity has recently attracted interest in its application to the study of effects of drugs on cerebral function. Sinton et al. ( 1986) administered the psychoactive substances diazepam and triazolam to an experimental subject in order to investigate the feasibility of using MEG to monitor central effects of drugs. They found that these substances modify the amplitudes of magnetic deflections at 100 ms and 150 ms after an auditory tone stimulus; the time course of the effect is an indicator of clearance of the drug from the body. In another study, Ribary et al. (1987) investigated the effect of antidepressive drug treatments on MEG response to 40 Hz auditory stimulation. The advantage over EEG in these studies is that one knows that the observed change in activity is at the auditory cortex. This spatial specificity is of importance in testing the location-specific effects of drugs. 3.3. CARDIAC STUDIES A potentially very large area of clinical applications of SQUID magnetometers is in cardiac monitoring and in the diagnosis of various pathological conditions. Magnetocardiography (MCG) is the oldest branch of biomagnetism, but progress towards clinical practice has been slow. A major obstacle is the difficulty of taking into account the complicated conductivity of the heart and its surroundings. The changing blood volume in the heart has a conductivity that differs from that of the heart muscle, and the conductivity of the lungs is much smaller than that of the surrounding tissue. These effects can be taken into account by using multi-compartment conductivity models (Horacek 1973, Cuffin and Geselowitz 1977a,b, Miller and Geselowitz 1978); even these attempts suffer from insufficient knowledge of the conductivity distribution. The conductivities of the different tissues

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are generally well-known, but there is uncertainty about the locations of boundaries between the different compartments. A further complication is the marked anisotropy of the conductivity of the heart muscle. In addition, it is not yet known how much information is contained in MCG that is not conveyed by the ECG (MacAulay et al. 1985). Progress in MCG has been made, for example; in the diagnostic evaluation of the conduction system: extra conduction pathways, which are excited in the Wolff -Parkinson-White syndrome, have been found to give rise to magnetic field patterns that make it possible to find their location (Fenici et al. 1985, Katila et al. 1988). Estimation of locations of active tissue has been attempted also for arrhythmias and so-called late fields, which are created by the circular activation of the heart muscle in pathological tissue. A recent area of interest is the study of so-called micropotentials that have been found to correlate with the risk of sudden heart failure (Montonen et al. 1988). At present, the uncertainty in locating pathological tissue or function in the heart on the basis of MCG measurements is at best of the order of 2 cm (Ern6 1985, Katila et al. 1987); improved accuracy can be expected with better conductivity models, which, for example, take into account the anisotropy of the heart muscle. With advances in the analysis of MCG signals and with the projected availability of multichannel systems suitable for hospitals, one may expect increased clinical use of this methodology. In MCG, no electrodes need be attached to the body like in ECG; the convenience of the magnetic measurement is valuable in this context. One must bear in mind that the patients in question are often in an unstable condition. It is therefore important to complete the study quickly so that the state of the patient does not change during the field mapping. Many channels are also necessary for capturing possible one-time events. Standard MCG is done without an attempt to determine source locations. Its widespread acceptance into hospitals to essentially just replace ECG would require an inexpensive multichannel system for field measurement over the chest. A large-area array is necessary and it seems that one must wait for the high- T, superconductors to help bring down the cost of refrigeration. On the other hand, even with present magnetometer technology, further progress in conductivity models would make MCG attractive for preoperative determination of pathological sources. Also here, a multichannel system is needed. 3.4. OTHERBIOMAGNETIC APPLICATIONS

Dust contamination in the lungs can be assessed by measuring its remanent field after magnetization. Metal dust usually contains iron, oxidized to

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magnetite or hematite. Magnetopneumography offers a non-invasive determination of the inhaled dust load among steel workers, welders etc. (Cohen 1973, P.-L. Kalliomaki et al. 1983) but also a means for studying lung functioning as small amounts of these particles serve as a tracer (Brain et al. 1985). To avoid complicated models, the sensitivity of the measuring system should be roughly constant over the chest. A straightforward method would be to use both a uniform magnetizing field and a detector with a uniform sensitivity pattern. However, it is impractical to make detector coils large enough to fulfill this criterion; instead, the field is measured at several points with smaller coils. Usually, this is accomplished by moving the subject with respect to the sensor (K.Kalliomaki et al. 1983). For lung measurements, SQUID sensors are not absolutely necessary; good results have been obtained with flux-gate sensors. Detecting excessive accumulation of iron in the liver, which is an intermediate storage of iron in the body, has significance in the diagnosis of the diseases hemochromatosis and thalassemia. Assessment of the iron overload of the liver has been traditionally carried out using biopsy, which is not totally free of risk and causes discomfort to the patient. The magnetic susceptibility of the liver provides a quantitative, noninvasive measure of hepatic iron concentration. An external field is applied and the distortion in the field caused by the liver is then sensed with a SQUID magnetometer. The liver gives a paramagnetic response to the applied field, whereas the surrounding body tissue is diamagnetic with a susceptibility close to that of water. The diamagnetic background normally outweighs the paramagnetic signal of the liver; hence a reference measurement is necessary. Usually this is accomplished by moving the sensor with respect to the body. Typical resolution of the method is of the order of 100 ppm of iron in the tissue; therefore, iron deficiency cannot be measured accurately and the method is best suited for detecting iron overload (Farrell 1983, Farrell et al. 1983). The biomagnetic technique has also been applied to studies of steady magnetic fields caused by dc currents in the human body. These currents can be due to an injury or they may be produced by normal electrophysiology of an organ; noninvasive in vivo information may thus be obtained (Cohen 1983, Grimes et al. 1983). In addition, biogenic fernmagnetism naturally occurring in some organisms has been studied (Kirschvink 1983). NEUROMAGNETOMETERS 3.5. MULTICHANNEL

Until fairly recently, all biomagnetic measurements were carried out using single-channel magnetometers, with their pickup and compensation coils

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wound of superconducting wire on three-dimensional coil formers. The optimization of the coil parameters, i.e., the coil diameter and length, the number of turns, and the baseline of the gradiometer, for good sensitivity and discrimination against external disturbances has been discussed by several authors (Romani et al. 1982, Vrba et al. 1982, Duret and Karp 1983, 1984, Farrell and Zanzucchi 1983, Ilmoniemi et al. 1984). Besides maximizing the field sensitivity, the spatial distribution of the biomagnetic signal and of the disturbances must be taken into account in the design. Since locating current sources is one of the main objectives of biomagnetic studies, multichannel devices designed to map the spatial pattern of the magnetic field are very desirable. Such instruments not only make the measurements faster but also give more reliable data. So far, devices have been designed for brain measurements; multichannel systems for cardiac measurements may emerge in the near future. In this section we shall mainly consider magnetometers intended for neuromagnetic research. After discussing the design of these devices, with remarks about the applicability of the methods to other biomagnetic studies, we present existing biomagnetic multichannel systems. Problems encountered in their construction using conventional solutions and the introduction of integrated SQUID sensors indicate that in the near future multichannel devices may be made using planar technology. We end the section by discussing some specific questions about multichannel devices in biomagnetic research.

3.5.I . Optimization of multichannel neuromagnetometers Because of the large variety of possible source current distributions in the human brain, general criteria for optimality in magnetometer design do not exist. Figures of merit for different configurations can be obtained only by making assumptions about the signal sources and their characteristic field distributions. The ability to separate field patterns caused by sources at different locations and of different strengths requires a good signal-to-noise ratio; maximum sensitivity in every channel is thus desirable. At the same time, one has to find the sensor configuration that gives the best locating accuracy. Unfortunately, these two tasks are not independent. For example, an increase of pickup-coil diameter improves field sensitivity, but sacrifices spatial resolution. In practice, the freedom of the designer is restricted by several constraints: the size of the dewar, the properties of the SQUID sensors, the feasible upper number of channels, the distribution and strength of external noise sources, the characteristics of the cerebral signals.

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The design of flux transformers In optimizing a gradiometer coil, the problem is to maximize the flux (3.1)

coupled to the SQUID loop by the flux transformer circuit. Here L,, L, and L, are the inductances of the pickup, compensation, and signal coils, respectively. L is the SQUID inductance, L, denotes the parasitic inductance of the connecting leads, k, is the coupling coefficient between the signal coil and the SQUID. the difference of fluxes threading the pickup and compensation coils, depends on the field distribution. The field sensitivity of a magnetometer is defined as the locally homogeneous magnetic field at the pickup coil that would produce a signal at the output of the system of equal magnitude with the system noise. If @" is the flux noise in the SQUID, the corresponding field noise level is

where np is the number of turns in the pickup coil and A, is their average area. As far as the SQUID and its signal coil are concerned, the key for high field sensitivity is the energy resolution at the output, E , , ~ = l ~ ; ~ ( @ ; ) / 2 L . However, as the SQUID and its signal coil are usually fixed, only the pickup and compensation coil parameters can be adjusted. Then, the inductancematching condition, L , = L,+ L,+ L,,

(3.3)

does not generally maximize the sensitivity. Furthermore, as will be discussed later, the diameter and length of the pickup coil are often determined by other considerations and the compensation coil dimensions are fixed by the gradiometric balance condition and by space limitations. The optimum number of turns np in the pickup coil is found by differentiating eq. (3.1), where Qne, is assumed to be proportional to n,, (Ilmoniemi et al. 1984):

a

L,+L,+L,+L,-n,-(L,+L,+L~)=O. (3.4) an, Therefore, the inductance-matching condition, eq. (3.3), maximizes the sensitivity for tightly wound coils only. In general, no simple analytical expression exists for the dependence of the inductance on the number of turns in the pickup coil; thus the optimum number of turns and other coil parameters must be determined numerically. Then, feasible maximum dimensions must be specified as constraints, quite often dictated by the space available in the dewar.

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Enlarging the pickup coil increases field sensitivity but at the same time the loop integrates the field from a larger area; thus the coils can no longer be approximated by point magnetometers. In practice, this effect becomes significant when the coil diameter exceeds the distance to the source (Romani et al. 1982, Duret and Karp 1984). Similarly, an increased length of the coil, i.e., less tight winding, allows more turns without excessive increase of inductance; the sensitivity improves, but because of increased distance from the source the signal becomes weaker. For cortical current dipoles these two effects tend to cancel, leading to a broad maximum in the signal-to-noise vs. coil length curve. For successful numerical analysis of neuromagnetic data it may be necessary to take into account the finite volume of the coils. The first significant correction to point magnetometers, namely the effect of the diagonal second-order derivatives of the field, may be calculated by evaluating the average field at the vertices of a tetrahedron, ( * r / 2 , *r/2, h / 2 h ) , ( * r / 2 , T r / 2 , - h / 2 h ) , located symmetrically around the center of a cylindrical coil of radius r and height h. The base length of the gradiometer should be chosen 1-2 times the typical distance to the sources. This provides adequate rejection of distant noise sources, without significantly attenuating the signal (Vrba et al. 1982, Duret and Karp 1983). One should note, however, that noise from nearby sources such as the dewar, cannot be avoided with gradiometric techniques. With modern SQUIDS, noise from the dewar has turned out to limit the field sensitivity. Thus, further improvement requires the reduction of dewar noise, which is probably dominated by thermal currents in metallic insulation layers. Locating error calculations and simulations A useful figure of merit of multichannel magnetometers is the uncertainty in locating cortical current sources. To estimate this uncertainty, a suitable source current model and a volume conductor model must be chosen. A current dipole in a spherically symmetric volume conductor is appropriate; because of computational limitations, as discussed in section 3.2.1, it is the only model presently used in the least-squares search of cortical sources. The locating ability of the magnetometer may be simulated numerically by adding noise to the calculated field values and by then fitting an equivalent dipole source to the data by means of a least-squares search. Repeated simulations then reveal the average locating error. A more effective way to estimate the error is to directly determine the confidence regions of the least-squares fit (Kaukoranta et al. 1987, Hari et al. 1988). The confidence region in the parameter space may be defined as being the smallest set of parameter values with a given total a posteriori probability.

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Since the field depends in a nonlinear way on the source parameters, the confidence set in the parameter space is tedious to compute exactly (Hamalainen et al. 1987). However, linearization in the vicinity of the least-squares solution makes the problem tractable (Kaukoranta et al. 1986, Sarvas 1987). The resulting confidence region is an m-dimensional hyperellipsoid, where m is the number of model parameters, centered at the estimated value. The half-axes of the ellipsoid are given by the eigenvectors and eigenvalues of the matrix ( J T 2 - ' J ) - ' ,where J is the Jacobian of the function giving the dependence of the field on the model parameters and Z is the covariance matrix of the measurement errors. The confidence intervals for the model parameters are then given by the edges of a rectangular box containing the confidence ellipsoid and having its faces parallel to the coordinate planes. The confidence limits for a current dipole are generally largest in the depth direction and smallest in the direction perpendicular to the dipole in the tangential plane. Their values depend on the signal-to-noise ratio, but also on the depth of the dipole and on the measurement grid. Therefore, the calculated confidence limits can be applied for comparing various magnetometer arrays (Knuutila et al. 1985, 1987b). Other considerations If the spatial frequency content of the signal to be measured is known, the spatial analog of sampling theory can be applied to determine the best spacing for the measurement grid (Romani and Leoni 1985). Field distributions caused by current dipoles in the brain have maximum spatial frequencies between 10 and 30 m-', depending on the depth of the sources. Thus, a suitable grid spacing is 20-30 mm. To apply this criterion, a large area of coverage must be assumed. If .the number of channels is fixed in advance, the optimal channel separation may turn out to be larger than predicted by the spatial-frequency analysis. With a fixed number of channels, it is also instructive to calculate the angles between the lead fields of the channels, as discussed by Ilmoniemi and Knuutila (1984) and by Knuutila et al. (1987b). This angle is a measure of the amount of diferent information conveyed by the channels. In a multichannel magnetometer, compact modular construction is a virtue. When many channels are simultaneously used, the possibility for computer control of the electronics is important. Furthermore, to avoid cross talk between channels, cables must be well shielded or symmetrized; this is also important for the noise immunity of the system. For highsensitivity magnetometers, a robust grounding system is essential. In addition, materials must be carefully chosen; for example, careless cabling to

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room temperature may increase the liquid helium boil-otf rate to an intolerable level; in a 30-channel dc-SQUID system, 90 wires are needed if no special techniques are used. Making dewars for wide-area multichannel magnetometers is problematic, too. Because of considerable variations in head shapes, no dewar bottom can be made optimal for them all; one may use a spherical bottom to approximate the average curvature, but then the maximum area is obviously limited. The fact that some of the channels are rather far from the scalp imposes high demands on the sensitivity of the SQUIDS. In addition, a large sensing area requires thick dewar walls for strength and thus a wide gap between the room temperature surface and the 4.2K interior. High-T, materials may make it possible to get the sensors closer to the head, resulting in increased signal strength. A good combination of high sensitivity and close-to-the-head sensors might be obtained with a hybrid structure, with a high-T, pickup coil and a SQUID at 4.2 K. Design of multichannel systems for other biomagnetic studies The previous discussion was primarily about magnetometers for neuromagnetic research. However, the same general principles can be applied to cardiac studies as well. The main new factors affecting the design are the different conductor geometry and the different source configuration. In the brain, the tissue can be modeled successfully as a homogeneous conductor, whereas in the chest the lungs introduce a major inhomogeneity which must be taken into account. As a result, simple models do not give sufficiently accurate results and computationally tedious finite-element methods must be used. Estimating the locating accuracy of various magnetometer configurations is thus difficult. The body surface of interest is larger in magnetocardiographic than in neuromagnetic studies; typically, the field is measured over the whole chest (for a standardized MCG grid, see Karp 1981). Therefore, a device for clinical use which would cover the entire area at once, requires a very large helium container. The dewar does not need a curved tail, but the gap in the tail should be as small as possible. Also, the optimal interchannel separation is larger in a cardiac multichannel device than in a neuromagnetometer. Although the cardiac signals are two orders of magnitude stronger than the cortical magnetic fields, a very high sensitivity is required for highresolution measurements, which detect possible disorders in the heart’s conduction system. A small signal of interest, superposed on the strong cardiac field, naturally requires electronics with a high slew rate.

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3.5.2. Existing multichannel systems The double-D sensor at M I T The double-D gradiometer of Cohen (1979) was the first multichannel neuromagnetometer. It consisted of two mutually orthogonal off -diagonal gradiometers, shown schematically in fig. 14b, measuring the components a E : / a x and aE,/ay. The diameter of the coils was 2.8 cm and the gradient In addition to the two 2-D sensitivity of the 2-D detectors 16 fT/(cm&). channels, the system has a 2.8 cm diameter coil measuring B,;the sensitivity of this third channel was lOfT/&. In each channel, a commercial rf SQUID was used. Since a 2-D detector is sensitive to currents directly under it, and since the orthogonal channels (see also the discussion in section 3.5.3) measure orthogonal current components, the system was used to display the distribution of currents as a function of surface location. In studies of the magnetic alpha rhythm over the occipital lobe, the currents were found to flow preferably along the direction of the longitudinal fissure (Cohen 1979). The MIT system has recently been augmented with a duplicate assembly of a two-channel 2-D detector and a E, channel in a separate dewar (Kennedy et al. 1988). The four-channel gradiometer in Helsinki A multichannel neuromagnetometer measuring the field at several locations

simultaneously was first used by Ilmoniemi et al. (1984). Their 4-channel first-order gradiometer (4M), developed for use in a magnetically shielded room (see section 3.1.1), consisted of three elliptical pickup coils inside a 30 mm diameter dewar tail and a fourth circular 2 I mm diameter coil 12 mm above the lower coils. The field sensitivities of the channels in this rf-SQUID were 22 and instrument, obtained with flux noise @ , = 7 x 15 fT/JHz in the lower and upper coils, respectively. The compensation coils for all four channels are coaxial; thus the signs of mutual inductances of adjacent and coaxial coils are opposite, reducing the coupling between the lower channels. By suitably adjusting the distance between the individual turns in the compensation coils, the mutual inductances can, in principle, be made arbitrarily small. The fourth pickup coil of 4M was placed above the lower three in order to get additional information from the variation of the field strength as a function of distance from the brain. The magnetic field outside a surface enclosing all the sources is uniquely determined by the values of the normal field component on this surface; however, when the detection coils are constrained to a narrow dewar tail, useful additional information is obtained from a more distant channel. The location of the fourth channel was

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determined by maximizing its sensitivity to source current patterns that are not seen by the lower three channels (Ilmoniemi and Knuutila 1984). An experiment with 4M that would not have been possible with fewer than the four channels was performed by Vvedensky et al. (1985a). They measured the magnetic field produced by the spontaneous human alpha rhythm, concluding that the sources of alpha bursts on the left and right sides of the occipital lobe could be active separately. The analysis was based on normalized gradients that were obtained by dividing the gradient calculated from signals in the lower three channels by the amplitude in the fourth channel. Near a field extremum, to a first approximation, these gradients point toward the extremum and their amplitudes are proportional to the distance. The four-channel magnetometer was in use from mid-1983 to the end of 1986; most experiments were of the evoked-field type, and usually, only the three lower channels were used (see, for example, Hari et al. 1987, Huttunen et al. 1987, Kaukoranta et al. 1987, Makela and Hari 1987).

The five-channel installation in New York A 5-channel dc-SQUID system was installed at New York University (NYU)

in 1984 by BTi' (.Williamson et al. 1985). The Neuromagnetism Laboratory of NYU is situated in Manhattan; it may be the noisiest location where MEG is recorded. Each detection coil is a symmetrical second-order gradiometer, with 15 mm diameter and a 40 mm baseline. The pickup coils lie on a spherical surface of 10 cm radius, four of them at the comers and one slightly above the center of a 28 by 28 mm square; the axis of each coil passes through the center of the sphere. The dewar is moved in a special scanner; with the subject's head in its center, each channel is approximately radial and the probe can be quickly repositioned, the dewar moving on a spherical surface around the head. A novel attempt was made in New York to tackle the external noise problem: in addition to the five signal channels, the system has four SQUID channels far away from the head that measures the three magnetic field components and the axial derivative of the radial component. These compensation channels are insensitive to brain sources but sensitive to distant noise sources. When the compensation signals are suitably weighted and added to each signal channel, rejection of external noise is improved by about 20 dB. Still, the noise level could not be reduced sufficiently for some experiments, although many successful measurements were performed with this system (see, for example Pelizzone et al. 1985, Kaufman and Williamson 1987). A magnetically shielded room was installed at NYU by Vacuum-

'

Biomagnetic Technologies, Inc., 4174 Sorrento Valley Blvd., San Diego, California 92121, USA.

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schmelze in 1987 (see section 3.1.1); the electronic balancing system may not be necessary any more. An exciting addition to the five-channel system is a pair of so-called CryoSQUIDs (Buchanan et al. 1988). Each consists of a standard BTi second-order SQUID gradiometer in a dewar kept cold by a commercial Gifford- McMahon cooler and a helium Joule-Thomson refrigerator. A noise level of 25 f l / a was obtained while operating the cryocooler. The obvious advantage with this closed-cycle system is that helium refills are avoided; in addition, the CryoSQUlD has been designed to operate in virtually any orientation, even upside down. If reliable cryocoolers suitable for high-sensitivity magnetic measurements will become available, SQUID systems might find more acceptance in hospitals and in applications or locations where the supply of liquid helium is limited.

The four-channel sysfem in Rome The rf-SQUID system in Rome (Romani et al. 1985) has four parallel and symmetrical, second-order gradiometers with a baseline of 53 mm; the coils are 15 mm in diameter and they are located at the corners of a 21 by 21 mm square. The system is designed to operate in an unshielded environment. The permanent balancing of the gradiometers is done separately for each channel before the final assembly of the system. Thermal cycling deteriorated the original balance only by a factor of 2-3. The noise level obtained in the wooden hut at the lstituto di Elettronica dello Stato Solido in Kome is 40-50ff/& (Romani et al. 1985). A number of successful studies with this system have been reported (see, for example, Romani and Narici 1986). Particular attention has been paid to positioning the probe accurately: two optic fibers emitting narrow light beams have been attached to the bottom of the dewar; this helps locating the probe with respect to a predetermined grid on the subject's head; an accuracy of about two millimeters is achieved. Commercial seven-channel gradiometers A 7-channel second-order gradiometer system with dc SQUIDS is commercially available from BTi. The arrangement is similar to that of the 5-channel system in New York;the main difference is just that the four detection coils surrounding the center coil are replaced by six. Several of these units are now in use; a pair, named Gemini, has been installed at the NYU Medical Center in Manhattan (see fig. 17) and is operated in a Vacuumschmelze shielded room. There the emphasis is in clinical studies, but the system was first used to investigate the alpha rhythm (Ilmoniemi et al. 1988). This is the first multichannel system allowing measurements over both hemispheres simultaneously; it is suitable for studies in which interhemispheric correlation or the transfer of signals is investigated. A novel feature of the system

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Fig. 17. The Gemini system of two seven-channel gradiometers (Biomagnetic Technologies, Inc.) in the magnetically shielded room of the New York University Medical Center. One transmitter coil of the probe position indicator system (see section 3.5.4) is seen on the subject’s head; the receiver coils are mounted on the dewars.

is an automated probe-positioning indicator (see section 3.5.4). As of December 1987, BTi had installed their 7-channel MEG system at 9 sites. A special version of this instrument with 2.7cm channel separation and flat-bottomed dewar was installed at the University of California in Los Angeles (Sutherling and Barth 1987). Another commercially available 7-channel second-order gradiometer is manufactured by Shimadzu Corporation3. This instrument has 25 mm diameter pickup coils at a mutual separation of 37.5 mm in a flat-bottomed dewar, connected to rf SQUIDS. The quoted noise level is less than 100 fT/&.

The seven-channel device in Helsinki A sensitive 7-channel first-order dc-SQUID gradiometer, covering a spherical cap area of 93 mm in diameter, was put into use at the Helsinki University of Technology in 1986, in collaboration with the IBM Corporation (Knuutila et al. 1987b). The experimental setup is shown in fig. 18.

’ Shimadzu Corp., Nakagyo-ku, Nishinokyo-Kuwabara-cho 1, Kyoto 604, Japan.

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Fig. 18. The magnetically shielded room of the Low Temperature Laboratory of the Helsinki University of Technology; the 7-channel first-order gradiometer (Knuutila et al. 1987b) is seen through the open door. The electronics of the magnetometer is on the left part of the top row on the instrument rack in the tf-shielded cabinet; most of the rack is occupied by a commercial 32-channel E E G amplifier.

The coils of the device (see fig. 19) are located at the vertices and slightly above the center of a regular hexagon, and the coil system is placed inside a dewar having a curved, tilted bottom4. The inner diameter of the tail section is 140 mm, and the radius of curvature of the spherical cap is 125 mm, with its symmetry axis slanted 30" from vertical; the isolation gap at the tip of the dewar, when cold, is less than 15 mm. The pickup-coil diameter is 20 mm, length 7 mm, and the number of turns is six; the compensation coils have 4 turns, a diameter of 24.5 mm, and a length of 15 mm. The baseline of the gradiometers is 60 mm. The design of this device aims at very high sensitivity and coverage of a large area. The instrument enables both quick, coarse mappings without The dewar was made by C T F Systems. Inc.. Port Coquitlam, British Columbia, Canada V3C I M9.

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Preamplifiers Top flange Thermal radiation shields Support tube Impedance matching transformers SQUIDS Dewar Gradiometer coils

Fig. 19. The Helsinki 7-channel gradiometer.

moving the dewar and more detailed field distribution measurements by recording successively at different dewar positions. The system was designed for use inside a magnetically shielded room (Kelha et al. 1982) so that first-order gradiometers without precise balancing can be used. The coil configuration in this instrument was selected on the basis of calculations discussed in section 3.5.1. The locating uncertainty decreases with increasing interchannel distance within the fixed inner space of the dewar; the pickup-coil dimensions were selected to give maximum sensitivity without significant deterioration of the spatial resolution due to the integrating effect. The d c SQUIDS in the 7-channel device were developed by the IBM (Tesche et al. 1985). The sensitivity of each channel is 5 ff/&, mainly limited by thermal noise from radiation shields in the dewar, and the l/f noise is very low with the comer point at a few tenths of hertz. The intrinsic noise of the complete magnetometer, measured inside a superconducting shield, is only 1-2ff/&, corresponding to a flux noise of 3-6x Having the SQUID noise clearly below dewar noise is advantageous: small excursions from the optimal operation point of the SQUID do not appreciably increase the noise level of the system. In the construction, particular care was exercised to eliminate excess noise from resonances in the flux transformers and to ensure insensitivity to rf fields (see section 2.2.2). Each flux transformer is shunted with a capacitor and a resistor to lower the Q-value of signal-coil resonances. All cables are carefully shielded and properly grounded to minimize interchan-

@,/a.

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nel interference. Since the channel separation in this instrument is relatively large, coupling between the channels via the pickup coils is only a negligible 0.7%. The electronics was made compact and modular to simplify its use in a multichannel instrument (see section 2.3). The preamplifiers, with optimum source impedances of a few kilo-ohms, are carefully matched to the dcSQUID impedances of only a few ohms. Besides impedance transformation, a reasonably high bandwidth is needed at the same time, suggesting the use of a cooled resonant transformer. The high sensitivity of the 7-channel instrument was necessary in a study of the middle-latency (about 30-60 ms after the stimulus onset) deflections in the auditory evoked response following short noise bursts (Pelizzone et al. 1987).These deflections are very weak, only 40-50 fT. The corresponding deflections have been studied electrically, but without certainty about the source location; both subcortical and cortical sources have been suggested. The successful field mapping suggested a source at supratemporal auditory cortex, slightly anterior to the source of the main auditory response at l00ms after the stimulus onset. A state-of-the-art display of auditory responses recorded with the 7-channel system is shown in fig. 20. Another example is given in fig. 21, where traces of alpha rhythm, recorded in real time with the Helsinki 7-channel instrument are shown. The traces were recorded over the occipital lobe with the subject having his eyes closed. Comparison of multichannel neuromagnetometers The key parameters of the multichannel magnetometers discussed above have been summarized in table 1. The noise levels reported in the table depend on the particular installation and on testing conditions and are not directly comparable. Besides these existing instruments, several other projects of constructing multichannel instruments based on wire-wound gradiometers with more than 10 channels have been started (Becker et al. 1987, Knuutila et al. 1985, 1987c; CTF, Inc.; BTi, Inc.). Vector magnetometers In addition to SQUID instruments for mapping one component of the field pattern, vector magnetometers have been introduced, the first of them by Shirae et al. (1981). In their device, three rf SQUIDS are connected to first-order gradiometer coils with a baseline of 11 cm. The sensors share a common tank circuit, an rf preamplifier and a detector, but use different audio modulation frequencies. The signals of the individual channels are then separated by phase-sensitive detection. This approach resulted, Later, frequency multiplexing however, in a high noise level, 1 pT/(cm&). was applied in a six-SQUID system (Furukawa et al. 1986) having a

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317

stimuli

- left eor _ _ _riaht ear

1st

2;

NlOOm

step 20 f T

NlOO m'

step 20 f T

Fig. 20. Responses evoked by left-ear (solid lines) and right-ear (dashed) sound stimuli as recorded with the Helsinki 7-channel SQUID magnetometer. The stimulus begins with bandlimited white noise (marked black on the time axis) and ends with a 250 Hz square wave (vertical-line shading). The passband is 0.05-45 Hz; 120 responses were averaged to get these traces. The locations of the seven first-order gradiometers are shown as circles on the head profile. On the right, magnetic field isocontour maps are shown for the deflections that occur approximately 100 ms after the beginning of the noise burst (N100m) and after the transition from noise to square wave (N100m'). The separation between the isocontour lines is 20 ff. These maps, although from the same subject, do not include data from traces shown on the left (Joutsiniemi 1988).

*L ~

-_

ptuVMflbpw~ -

Fig. 21. Alpha rhythm recorded with the Helsinki 7-channel magnetometer over the occipital lobe of a subject with his eyes closed; the bandwidth was 0.05-40 Hz.Note the polarity reversal between, for example, Channels 1 and 4 at the time instant marked with a vertical bar.

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318

Table 1 Comparison of multichannel magnetometers. llmoniemi Williamson et al. et al. (1984) (1985) Channels Pickup diam. [mm] Gradiometer order Baseline [mm] Channel sep. [mm] Noise [ I T l G ]

5

4 20/12",2Ih

15

1 60 16' 22d. 15'

2 40 20 20

Romani et al.

BTi Corp.

Shimadzu Cop.

(1985)

4 I5 2 53 21 50

Knuutila et al. (1987b)

7

I

I

15 2 40 20 20

25 2 35 37.5

20

(100

1 60 36.5 5

Axes of the elliptical lower coils. Diameter of the upper circular coil. ' Distance between the centers of the coils, located at the vertices of a regular tetrahedron. Lower coils. Upper coil.

second-order vector gradiometer plus a vector magnetometer for monitoring noise fields; the signals from these reference channels were used for electronic balancing. The vector gradiometer of Seppanen et al. (1983) had square coils with 27 mm side length and two turns separated by 15.6 mm in each; the baseline was 12 cm. The detection coils were connected to rf SQUIDs operated at 20-30 MHz; the resulting sensitivity was 28 fT/&. Time-division multiplexing of the SQUID readout is used in the vector gradiometer of Lekkala and Malmivuo (1984), which consists of three first-order asymmetric gradiometers with pickup-coil diameters of 28 mm and compensation coils of 48.5 mm diameter. The outputs of the rf SQUIDs are fed to separate preamplifiers, then multiplexed for the main amplifier using phase-sensitive detection. The output of the lock-in amplifier is then routed to sample-and-hold circuits and integrators, separate for each channel. There thus exist three separate parallel feedback loops. The noise level is 2.5 times that obtained in nonmultiplexed of the device, 85-105 fT/&, mode. Multiplexing of the SQUIDs is not recommended, because aliasing associated with the sampling of the different channels increases the noise level. Although the input signal to the magnetometer may be filtered to avoid aliasing, intrinsic SQUID noise and the wide-band noise in the preamplifier and in the tank circuit remain; their contributions to the noise at the output will increase at least by the square root of the number of multiplexed channels. In principle, multiplexing the detection coils would not increase the noise level, provided that the switching frequency would

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be high enough and the switching would not involve additional noise mechanisms. Multiplexing in multichannel dc-SQUID magnetometers has been proposed also by Shirae et al. (1988). In their scheme the SQUIDs are connected in series, and the sum voltage is read out using an extra dc-SQUID. Each channel is modulated with a different frequency, and the signals are separated by lock-in detection.

Multichannel susceptometers Measurements of the magnetic susceptibility of the human liver and of the remanent field of contaminated lungs have been carried out using singlechannel devices; the patient and the magnetometer are moved with respect to each other. At most 2-3 SQUIDs are used simultaneously, as in the susceptometer of Farrell et al. (1983), which had two second-order gradiometers, or in the forthcoming 3-SQUID liver measurement system made by Dornier Systems GmbH (Ludwig and Sawatzki 1986). 3.5.3. Planar gradiometer arrays At present, seven channels in a hexagonal array seems to be the de facto standard for MEG instrumentation; most of the devices are second-order gradiometers. The 7-channel instruments are only one step towards magnetometers with enough channels to cover the whole area of interest. Only then routine clinical studies become feasible. Several plans for devices having 20-30 or even 100 SQUIDs have been put forward. However, the construction of multiple-SQUID systems is not just multiplication of previous designs; many new problems are introduced (Knuutila et al. 1985). The use of 20-30 conventional, wire-wound axial gradiometers leads to a bulky, elaborate, and expensive construction. In addition, the conical space required by axial gradiometers aggravates space problems in dewar design. Axial gradiometers, measuring the radial component of the field, have been popular because .of the easy intuitive interpretation of the results. Since the baseline is usually longer than the distance to the source, the signal measured by the gradiometer is well approximated by the field at the pickup coil; the effect of the compensation coil is merely to cancel uniform fields. However, other components and derivatives of the magnetic field are, in principle, not more difficultto analyze and should thus not be rejected a priori. With magnetometers the construction would be simpler and there would be no problems in the interpretation of data; in practice, however, simple magnetometers are not feasible, even inside shielded rooms: mechanical vibrations of the dewar in the remanent magnetic field and nearby noise sources such as the heart may disturb the measurement.

3 20

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To avoid problems of wire-wound gradiometers, there has been growing interest in the use of planar integrated sensors (Knuutila et al. 1985, ErnC and Romani 1985, Carelli and Leoni 1986). The compact structure and excellent dimensional precision, allowing a good intrinsic balance, are the main advantages of thin-film gradiometers. The first integrated gradiometer was reported by Ketchen et al. (1978). In their design, the SQUID loop shares a common conductor with a larger field-collecting loop, thus forming an inductively shunted dc-SQUID structure as discussed in section 2.2.3. The flux coupling to the SQUID is poor, because of the mismatch between the pickup loop inductance and the inductance of the SQUID. For a pair of 24x 16 m m coils, connected in parallel, the gradient sensitivity was 20 fT/(cm&). Later, several other integrated first- and second-order gradiometers, with the pickup loop an integral part of the SQUID loop itself, were introduced (de Waal and Klapwijk 1982, van Nieuwenhuyzen and de Waal 1985, Carelli and Foglietti 1983b); their performance was similar to that reported by Ketchen et al. ( 1978). Integrated magnetometers have been discussed in several review articles (Donaldson et al. 1985, Ketchen 1985, 1987). The best results with integrated gradiometers have been achieved by connecting the pickup loop to the SQUID via a signal coil (see Jaycox and Ketchen 1981), which allows a good signal coupling and the necessary inductance matching (see also section 2.2.2). The first such fully-integrated magnetometer, built for geomagnetic applications, was reported by Wellstood et al. (1984); their device had a pickup-coil area of 47 mm’, a n a t h e sensitivity of this instrument was 5 fT/& above 10 Hz and 20 f T /JHz at 1 Hz. Quite recently, new devices based on the same signal coupling principle have been reported by groups working at Karlsruhe University (Drung et al. 1987), at the Electrotechnical Laboratory in Japan (Nakanishi et al. 1987), and at the Helsinki University of Technology (Knuutila et al. 1987~). The Karlsruhe device (see fig. 22) is a first-order gradiometer with coil areas of only 4 mm’ and a baseline of 3.7 mm; its gradient sensitivity is 3 8 f T / ( c m m ) . A 1-bit D/A converter is on the same chip for a digital feedback readout scheme (Drung 1986). The integrated magnetometer of ETL, s h E n in fig. 23, uses a square-loop area of 64 mm’; its sensitivity is 11 1T/JHz. A twelve-channel device, measuring all three field components at four different locations is currently under construction (M. Nakanishi, personal communication). The structure of the integrated sensor of the Helsinki group is shown schematically in fig. 24. It consists of two orthogonal off-diagonal gradiometers in a rectangular figure-8 configuration, measuring a B , / a x and aE,/ay. The pickup-coil size is 28x28mm2; the baseline is 14mm. The signals are coupled via intermediate transformers to two dc SQUIDS,located

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Fig. 22. The Karlsruhe integrated gradiometer (Drung et al. 1987).

Bonding

8 x 8 mm2

JJ

DC SOUID

Input coil

LOOP

Fig. 23. The integrated magnetometer of the Electrotechnical Laboratory in Japan (Nakanishi et al. 1987).

/ /

/ , /

\ \

\ \

Pickup coil

Prim. Intermediate transformer

Input coil

Fig. 24. Schematic structure of the integrated SQUID gradiometer of the Helsinki group (Knuutila et al. 1988). See text for details.

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on separate 6 x 9 mm2 chips. The three chips are mounted together on a fiber-glass holder to form a single two-channel module. The intermediate transformer allows a small SQUID with only a few signal coil turns on top of it; this results in a low intrinsic SQUID noise and reduced parasitic capacitances. Yet it is possible to match the small-inductance SQUID to a practical-sized coil; in addition, the control of resonances is easier (Knuutila et al. 1988). This separate-chip construction, with inductive coupling, allows flexibility for changing the pickup coil design, because it is not necessary to redesign the whole SQUID when the pickup coil is changed. The intermediate transformer and the SQUID have two oppositely wound coils to avoid spurious signals caused by homogeneous fields. The gradient sensitivity of this sensor, measured in a fiber-glass dewar, is about 4.5 fT/(cmJHz) at frequencies above 3 Hz. The sensor has been designed for use in a 32-channel device measuring the gradient of the radial field component at 16 locations (see fig. 25). Uncertainties in locating cortical current dipoles with off-diagonal gradiometers have been calculated for planar arrays (Ern6 and Romani 1985, Knuutila et al. 1985, Carelli and Leoni 1986). It was found that the locating accuracy of first-order planar gradiometers is essentially the same as that of axial first-order gradiometers; the higher-order gradiometers investigated in the first and last references were in some cases better than second-order axial gradiometers. The short baseline of off -diagonal sensors may make them better than conventional axial gradiometers for measurements in noisy environments. In addition, the spatial sensitivity pattern of off-diagonal gradiometers is narrower. These sensors thus collect the brain’s background activity from a restricted area near the sources of interest. The “field patterns” measured by off-diagonal gradiometers are morphologically different from those of axial gradiometers. For example, the 2-D gradiometer (Cohen 1979) gives the maximum response directly over the source, the direction of the current being perpendicular to the direction of the gradient. This is shown in fig. 26 for a current dipole.

Fig. 25. W e planned configuration of the 32-channel planar gradiometer at the Helsinki University of Technology. Each square module consists of two orthogonal off-diagonal gradiometers connected to planar SQUIDS (see fig. 24).

MULTI-SQUID DEVICES

I

I -100

I

- 50

I 0

I

50

323

I 100

d (mm)

Fig. 26. The off-diagonal gradient a B z / a x of a 10 nAm tangential cerebral dipole as a function of position on a spherical surface of 125 mm radius. The distance from the gradiometer pickup coil to the scalp is 15 mm, and the dipole is 40 (solid line), 30 (dashed line), and 20 m m (dash-dotted line) below the scalp, respectively.

It is seen that the response is well localized, the 50% width of the peak being 35-55 mm for dipoles 20-40 mm below the scalp. The spatial frequencies at which the amplitudes of the spectral components have dropped under lo%, are 20 and 30 m-', respectively. Therefore the size of the sensors should be less than 30 mm and the grid spacing ideally no more than 30 mm. 3.5.4. Use of multichannel magnetometers

Calibration and Dewar position indication Shielding currents in superconducting flux transformers distort the magnetic field; an individual channel does not sense the original external field. If the inductances of flux transformers and their mutual inductances are known, a correction can be calculated easily.' Assume that there are n channels and that the mutual inductances are described by an n x n matrix M. Let = (@I . . . @k)T, where T means the transpose, be the net fluxes further 9' measured by the flux transformers and let @ be the undistorted applied flux. Then,

@=(z+ML-')9',

(3.5)

' Note that the effective inductance of a flux transformer coil is affected by feedback.

R. ILMONIEMI ET AL.

324

-

where L = diag( L , . . L , ) is a diagonal matrix containing the total inductances of the flux transformers and I is the identity matrix. In practice, one measures the output voltages V = (V, . . * Vn)T, which are related to @' via calibration coefficients K, . . K,. Since in MEG measurements the field distribution generally differs from that used in the calibration, the effects of coupling also differ. Consequently, the calibrations K, cannot be simply calculated from @,/ V,, where now the applied flux CP is calculated from the geometry of the experimental setup. Instead, the true calibration @:/ V, is obtained from eq. (3.5). Of the n( n - 1)/2 independent mutual inductances between n channels, often only the nearest neighbours need to be taken into account. Nevertheless, the situation becomes complicated if the mutual inductances cannot be measured directly. The best way is then to calibrate the device with a phantom whose field distribution resembles that generated by the brain, thus taking the interchannel interference corrections into account during the calibration of coefficients. An elegant alternative solution to these problems has been presented by ter Brake et al. (1986). In their method, flux feedback is applied to the flux transformer rather than to the SQUID. Now, the feedback keeps the transformer effectively currentless, thus eliminating cross talk between the detection coils. In locating cortical sources on the basis of externally measured magnetic fields, the accuracy with which the position of the magnetometer can be determined affects the results in an essential way. In single-channel measurements, the uncertainty in the location of the field point may be considered as a source of extra random noise. A Gaussian distributed positioning error, with a standard deviation of 5 mm, may cause an uncertainty of up to 20% in the magnitude of the measured field and, correspondingly, an error in the position of the current dipole as determined on the basis of the experimental field values (Romani et al. 1985). In multichannel devices this difficulty is, in principle, simpler to solve because the relative positions of the different channels are accurately known. However, the problem is fully resolved only when the device has enough channels to record the whole field pattern of interest at the same time; even then the position and orientation of the dewar with respect to some fixed points on the head must be known. Traditionally, the magnetometer has been positioned with some alignment marks on the dewar and on the subject's head. The shape of the skull may also be digitized to provide 3-D information. When the multichannel magnetometer covers an extended area, however, the dewar bottom is quite large and its accurate positioning becomes difficult. The accurate location and orientation of the dewar with respect to the head can be determined by measuring the magnetic field produced by

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current in small coils attached to the head (Knuutila et al. 1985, 1987b, Ern6 et al. 1987). This approach was taken with the seven-channel magnetometer in Helsinki with three small coils, mounted on a thin fiber-glass plate. The plate is attached to the subject's head at some point in the measurement area, the dewar is positioned, the coils are energized separately, and the resulting field is measured. The location and orientation of the magnetometer are found by a least-squares fit of magnetic dipoles to the test data; the dewar location can be determined with rms error of about 1 mm. The method provides direct information on the locations of the measuring sensors themselves and does not rely on knowing the relative position of the magnetometer with respect to the dewar. BTi uses three orthogonal coils in the receiver and in the transmitter, attached to the dewar and to a stretch band on the head, respectively. From the measured mutual inductances the position and the orientation of the magnetometer can be found. The locations of the head coils are first found by a reference measurement: the three-dimensional coordinates of landmarks on the head are determined by touching them with a pointer having a similar orthogonal-coil transmitter. The probe position. indicator system is shown in a measurement situation in fig. 17.

Correlated noise The background activity of the brain produces magnetic fields that are not relevant to the experiment; this so-called subject noise limits the signal-tonoise ratio in experiments made with the best SQUID magnetometers. In contrast to instrumental noise, the subject noise is correlated between the different channels; this correlation can be taken into account in the analysis to reduce the effects of subject noise. If the noise source is different from the signal source, the two can be separated in multichannel recordings. The spectral density of the subject noise is typically 20-40 fT/& below about 20 Hz,decreasing towards higher frequencies (Maniewski et al. 1985, Knuutila and Hamalainen 1988). An example of the field spectrum measured over the head for one subject is shown in fig. 27. The contribution of instrumental noise is negligible at low frequencies. Since the subject noise' is often not time-locked to the stimulus, its effect can be reduced by signal averaging. The normalized coherence functions show substantial correlation even in channels located 73 mm apart; the correlation extends up to 5060 Hz. The observed long-range correlation has important practical consequences for the analysis of evoked responses. Multiple, simultaneously measured data can be used to estimate the covariance matrix for the field errors at different sites. This matrix can then be used for advantage in the least-squares search of the equivalent current dipole (Sarvas 1987). Its effect is to reduce the variance of the maximum-likelihood estimate, which can be considered

R. ILMONIEMI ET AL.

3 26 80

@. t5

60

-

>

t

$ W

40

-

n

FREQUENCY (Hz)

FREQUENCY ( H Z )

Fig. 27. (a) Subject noise measured with the Helsinki 7-channel gradiometer. The upper trace was recorded over the left temporal lobe with the subject having his eyes open; the bottom trace shows the corresponding noise spectrum without a subject. (b) Coherence for channels 36.5 mm (solid line) and 73 mm (dashed line) apart, recorded at the left postero-temporal lobe, with the subject having his eyes open. The dash-dotted bottom curve shows the coherence without a subject (Knuutila and Hamalainen 1988).

as the change in the volume of the five-dimensional hyperellipsoid describing the confidence limits of the maximum-likelihood estimate. In the study of Knuutila and Hamalainen (1988), an example is calculated; it was shown that this reduction is significant, demonstrating the advantages of simultaneously measured multiple recordings in obtaining more accurate estimates for the source parameters.

4. Other multi-SQUID applications 4.1.

G EO MA G N ETI S M

In addition to biomagnetism, multiple-SQUID systems are used in geomagnetism. The main advantages of SQUID magnetometers are low noise even with compact detection coils, a large dynamic range, a wide frequency band that starts from dc, and selectivity to specific field or field gradient components, allowing vector measurements. However, the need of liquid helium may cause problems at remote observation sites that are often necessary in geomagnetic studies; large dewars are needed to allow long maintenancefree operation. Small, closed-cycle cryocoolers would be most welcome; they must be battery operated and must not cause magnetic or mechanical disturbances in the measuring system, which has to operate in the open without shielding. The sensitivity of SQUIDS to rf interference and mechanical vibrations complicates their field use. Besides, the high cost of

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adequate multi-SQUID systems is perhaps the main factor limiting their use in geological research and surveying. The SQUID sensors for geophysical applications must have a very large dynamic range. The signals of interest are often only small variations superposed on large field fluctuations not relevant to the particular event under study. The slew rate must be high, even in the frequency range of several tens of kHz, to ensure operation without losing the flux lock. In addition, since the signals of interest extend to very low frequencies, low l/f noise and stable operation, free from flux jumps and drifts, are required. A compact integrated magnetometer with a high slew rate, .intended specifically to geomagnetic studies, has been introduced by Wellstood et al. (1984). In geomagnetic studies, usually all three components of the magnetic field are measured, and thus three SQUIDs are required. Measurements of the magnetic field gradient tensor may also be performed to detect possible anomalies; a gradiometer gives a strong signal when the sensor is moved over the boundary of the anomalous region. In magnetotelluric studies, the magnetic field and the electric field on the ground, caused by incident electromagnetic waves, are measured simultaneously. These waves, typically from 0.1 mHz to 100 Hz, are generated by ionospheric and thunderstorm activity. From the measured electric and magnetic field components as functions of frequency, the impedance tensor Z ( w ) can be determined. The analysis must usually be restricted to simple models such as plane waves and one- or two-dimensional structure of the ground. Because of noise, the estimates of the impendance tensor are, however, unreliable. To overcome this problem, a second measurement is carried out simultaneously at a site several kilometers away and transmitted to the first site by radiotelemetry. Since the noise sources at the two sites are uncorrelated, an unbiased estimate is obtained (Gamble et al. 1979). Correlation techniques can also be applied to improve the estimates, when an artificial variable-frequency electromagnetic source is used in magnetotellurics (Wilt et al. 1983). In addition to the distant reference signal, a phase reference signal is transmitted to the measurement station for lock-in detection. In this case, the reference measurement site senses mainly naturally-occurring geomagnetic fluctuations, which in this controlled-source paradigm constitute noise; this background noise scales roughly as l/f below about 1 Hz, easily outweighing the signal. Measuring the remanent magnetization or magnetic susceptibility of rock samples was perhaps the first application of SQUIDs to geological studies. In paleomagnetism one investigates the history of the earth’s magnetic field; another important area of interest is the identification of magnetic phases in samples. In both cases, a dewar with room temperature access to the

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measurement volume is needed; the devices have a coil to generate a magnetizing field and pickup coils to measure the remanent field components in the axial and transverse directions. Use of SQUID magnetometers in geophysics has been reviewed in detail by Clarke (1983). Several applications have been proposed and some feasibility studies made, although no full-scale tests have been carried out. Yet, for example, stress changes in the ground prior to earthquakes give rise to observable changes in the local magnetic field. The changes of the field are due to the piezomagnetic effect when the domains are reoriented under pressure. I n an experimental feasibility study, the gradients due to seismic waves were found to be typically on the order of 1 pT/m and thus easily detectable with a SQUID (Czipott and Podney 1987). Such a system requires an array of vector magnetometers with telemetry and data acquisition and processing capabilities; the high cost, however, limits the popularity of the method. The measurement of the magnetic field vector could also be applied to locate artificially induced fractures in the ground. These are made by injecting high-pressure propellant into a borehole; such fractures enhance oil or gas flow to a collecting borehole in weakly permeable rock regions. Similarly, in geothermal power plants water is pumped into the ground and high-pressure steam is collected via another borehole. If a magnetic tracer fluid is injected in the fractures, the resulting magnetic anomaly can be detected on the ground with SQUID magnetometers. In addition to measurement of the earth's magnetic field, SQUIDs are used in gravimeters to detect gravitational anomalies (see section 4.2.1). Geomagnetic magnetometers have been made commercially by BTi, CTF, and Cryogenic Consultants, Ltd.6

4.2. PHYSICAL EXPERIMENTS

SQUIDs have many applications in metrology and in various physical experiments where high sensitivity is required. Many of these rather nonstandard applications have been reviewed in detail by Cabrera (1978) and Fairbank (1982), and more recently by Odehnal (1985). Usually such measurement setups contain only a single SQUID; since we are discussing systems with multiple SQUIDs, we consider only a few specific examples where simultaneous recordings are important. In these experimental systems multiple SQUIDs are used mainly to measure different components of the magnetic field. Normally, no mapping

'Cryogenic Consultants, Ltd., Metrostore Building, 23 1 T h e Vale, London W3 7QS.

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of the field pattern is necessary; in some cases a simultaneous reference reading outside the experimental cell is needed. In many of the physics applications, SQUIDs are used to detect very small changes, usually requiring ultra-low-noise sensors. 4.2.I . Accelerometers and displacement sensors

SQUIDs can also detect very small displacements and accelerations. Devices of this type are based on a superconducting proof mass, suspended by springs or by magnetic levitation to allow free movement with respect to a superconducting coil. The inductance of the coil becomes very sensitive to its distance from the proof mass; the measurement coil is connected to a second, auxiliary superconducting coil, and a circulating current is persisted in the circuit. Changes in the inductance are then sensed by measuring the flux changes in the auxiliary coil (see fig. 28). In practical devices, the test mass is usually a superconducting diaphragm, and the coils are flat, as in the original work of Paik (1976). The design may then include several proof masses and coil sets connected to different SQUIDs. This enables the simultaneous measurement of sum and difference phases, different gravitational gradient tensor components, angular accelerations etc. The devices can be applied for detecting anomalies and for measuring the gradient tensor to improve inertial navigation (see, for example, Paik et al. 1978, Colquhoun et al. 1985); sensitivities better s-* HZ-’’~can be achieved. These instruments have been used for than sensing the movements of gravitational wave antennae (see, for example, Kos et al. 1977; McAshan et al. 1981) and for measuring transmitter movements in Mossbauer spectroscopy with subatomic precision of the m/& (Ikonen et al. 1983). Besides this inductance-moduorder of lated technique, piezoelectric and capacitive sensing with SQUIDs as pre-

-Proof mass

Detector coil

Auxiliary coil

Pickup

Input

coil

coil

I

Fig. 28. Schematic illustration of a superconducting accelerometer. A current I is persisted in the detector coil - auxiliary coil loop; changes of the detector coil inductance due to movement of the superconducting proof mass cause in the auxiliary coil a flux change, which is sensed by the SQUID.

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amplifiers have been used in gravitational wave antennae as well (Kos 1978, Cosmelli et al. 1987). 4.2.2. Monopole detectors

Superconducting detectors connected to SQUIDS have been applied in the search for magnetic monopoles predicted by grand unification theories. These particles are predicted to be very massive, too heavy to be produced at accelerators, and extremely penetrating. A magnetic monopole passing through a superconducting loop changes the flux threading the ring by 2Q0, providing a means of detection that is independent of the particle mass or velocity. These detectors should have as large an area as possible to enhance the probability for the passage of a monopole; then, however, the ambient field variations have to be carefully eliminated since an event in a detector, having an effective area of 1 m2 for example, would correspond to a 4fT field change. SQUID sensors with stable operation and freedom from flux trapping are thus required. So far, only one candidate event has been reported (Cabrera 1982). The apparatus consisted of a four-turn 5 cm diameter coil connected to an rf SQUID; it was surrounded by a 20cm diameter superconducting shield and a mu-metal cylinder, providing an ambient field as low as 5 pT and an 180 dB shielding factor for external magnetic fields. Ideally, a monopole traversing the four turns would give rise to a current of 8@,/L. in the loop for coupling to the SQUID, where L is the total inductance of the coil. However, a monopole traversing the superconducting shield will leave doubly quantized vortices at the intersections of its trajectory with the walls. This change in trapped flux reduces the observed signal; the reduction becomes a significant fraction of 2Q0 when the detector dimensions are close to those of the shield. In addition, the degree of degradation is dependent on the trajectory of the monopole, which creates a spread of magnitudes for possible signals instead of a single well-defined value. On the other hand, a small transient is found even from near-miss events. Later, the Stanford group reported a three-channel monopole detector having orthogonal coils with an area of 71 cm2 and a corresponding equivalent near-miss area of 476cm2 (Cabrera et al. 1983). Reliability in detection is based on a coincidence technique: a monopole passing through the detector should be seen in all loops. In a period of 150 days, however, no candidates for monopole events were found. The requirement of eliminating the effect of ambient magnetic field variations severely restricts the cross-sectional area of the sensor. However, the variations in the external field and the effects of the surrounding superconducting shield can be suppressed by twisting the coils to a

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1x1

33 1

2x2

h g . 29. Schematic illustration of the structure of two-dimensional planar gradiometers. The order of the gradiometer is given below the coils, and the clockwise and counterclockwise windings are indicated by pluses and minuses. The dashed lines show adjacent wires that can be left out.

gradiometric configuration. A monopole passing through only one of the loops still gives a full signal. Design of planar high-order gradiometers has been discussed by Tesche et al. (1983). Generally, an ( N + 1) x ( N+ 1) rectangular two-dimensional array is obtained by taking four N x N units with adjacent blocks oriented oppositely; a zeroth-order sensor is a simple loop (see fig. 29). An N x N array is insensitive to gradients of order P < 2N. In superconducting structures, adjacent loops having the same polarity may be joined. The construction may also be extended to polar and cylindrical geometries. Planar gradiometric sensors provide adequate rejection of external disturbance in ambient fields of the order of 1 FT (Tesche et al. 1983, Incandela et al. 1984). The latter device had two identical 60 cm diameter sensors. To be able to differentiate between real events and spurious signals even better, multiple planar detectors that cover the surface of a certain volume have been built. The six-channel device of IBM (Chi et al. 1984, Bermon et al. 1985) has independent fifth-order sensors on the sides of a 15 x 15 x 60 cm3 rectangular parallelepiped, providing a total area of 0.4 m2. A monopole passing through the volume causes signal in two and only two of the six coils; this kind of coincidence detection provides rejection also to events that might excite all of the detection coils in other geometries, as in the closely spaced set of parallel plates. Quite recently, a new dc SQUID detector of the same parallelepiped geometry of dimensions 26 x 26 x 380 cm3 and an effective area of 1 m2 (averaged over the solid angle 4 n ) has been reported (Bermon 1987); the Stanford group has constructed an octagonal-shaped detector with eight independent 17 x 521 cm2 gradiometer coils, having an effective area of 1.25m2, averaged over 4 n solid angle (Huber et al. 1987).

In addition, several anticoincidence detectors such as accelerometers, field sensors, cosmic ray detectors, power line analyzers, and rf monitors have been used. All the devices have been operated several hundreds of

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days; the null results imply an observed upper limit of 1.5 x lo-' m-* sr-2s-' (90% confidence level) for the magnetic monopole flux (Berrnon 1987). This is still three orders of magnitude higher than the theoretically predicted Parker bound (Turner et al. 1982). With the new superconductors operating at liquid nitrogen temperature it might be feasible to fabricate sensor arrays with a total area a few orders of magnitude larger than in the present devices; then the theoretical bound could be achieved during an operation of a couple of years.

5. Conclusions

Recent progress has made it posssible to build arrays of reliable, low-noise SQUIDs. First such systems, although with rather few channels, are already in use in neuromagnetism. It is expected that arrays with more than 20 channels will become available in the near future. In the state-of-the-art biomagnetic systems, SQUID noise is already negligible in comparison with noise from the dewar materials and from the subject itself. Therefore, there is no immediate need for further reduction of SQUID noise. However, reliability and insensitivity to rf interference will probably be improved as a result of further development. Also, the price of SQUIDs and the readout electronics is expected to drop significantly. The new high- T, superconductors offer some hope in eventually reducing the cost of operating SQUIDs, because liquid helium as a coolant could be replaced by liquid nitrogen. There are many problems on the way to practical high-T, SQUIDs, but one should keep in mind that up to 100 times inferior energy resolution than that in present state-of-the-art SQUIDs would be sufficient for neuromagnetometers. One reason for this is that the noise level in present SQUIDS is already significantly below that from other sources. In addition, the pickup coils in liquid nitrogen could be brought closer to the head, where the signals are stronger and the field profile carries more information about cerebral activity. Even if no high-7, SQUIDs will prove practical, detection coils made of these materials could be used to advantage. Simultaneously with improvements in instrumentation and as a result of it, the feasibility of using SQUIDs in fundamental brain research and in clinical applications has been demonstrated. Although the measurement of electric and magnetic fields are the only methods for non-invasive real-time monitoring of brain function, the eventual significance of neuromagnetism can only be speculated upon. MEG seems superior to EEG in that its interpretation is simpler and because the need of attaching electrodes to

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the scalp is avoided. However, because these two methods convey complementary information, they probably will often be used together. We live in a period of rapid change in the use of SQUIDS: the present 7-channel magnetometers will be replaced in a few years by systems with several dozens of SQUIDS. Faster data collection, provided by the possibility to record the whole field pattern at once, enables more complicated exoeriments and studies of patients, and improves the reliability of results. In addition to technical development of instruments, new ways of thinking are needed in the signal analysis and in the inverse problem studies to optimally utilize the measured signals. For the wide acceptance of the biomagnetic technique, it is crucial that clinically relevant results will be obtained.

Acknowledgements

We thank Olli V. Lounasmaa for suggestions and criticism; Toivo Katila, for sharing his views on cardiomagnetism; Riitta Hari, Matti Hamalainen, Matti Kajola, and Jyrki Makela for reviewing the manuscript. Financial support of the Academy of Finland and of Telecommunications Laboratories of Posts and Telecommunications of Finland is acknowledged.

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AUTHOR INDEX Abel, W.R. 152, 189 Abel, W.R., see Wheatley, J.C. 124, 194 Abell, J.S., see Gough, C.E. 14, 43 Abragam, A. 83, 90, 94, 111 Achiezer, A.I. 26, 41 Adamenko, I.N. 179, 189 Adamenko, I.N., see Khalatnikov, I.M. 179, 191 Aeppli, G., see Harshman, D.R. 34, 43 Agranovich, V.M. 27, 28, 33, 41 Ahlers, G. 106, 111 Ahlfors, S., see Knuutila, J. 308, 313, 314, 318, 337 Ahlfors, S.,see Pelizzone, M. 302, 316, 338 Ahola, H. 290, 333 Ahonen,,A., see Knuutila. J. 285, 286, 308, 313, 314, 318, 337 Ahonen, A.I. 124, 125, 189 Ahonen, A.I., see Knuutila, J. 308, 316, 319, 320, 322, 325, 337 Aittoniemi, K., see Kalliomaki, K. 304, 336 Aittoniemi, K.,see Kalliomaki, P.-L. 304,336 Al-Shibani, K. 85, 111 Alexander, S . 144, 159, 189 Alexandrov, A S . 23, 41 Allen, P.B. 18, 41 Allender, D. 28, 37, 41 Alstrom, P. 248, 266 Ambegaokar, V. 232, 266 Ambegaokar, V., see Arani, M.R. 232, 266 Anayama. T.,see Ogushi, T. 8, 44 Ancill, R.J., see Ribary, U. 302, 338 Anderson, A.C. 126, 127, 166, 167, 179, 189 Anderson, A.C.. see Abel. W.R. 152, 189 Anderson, A.C., see Folinsbee, J.T. 126, 127, 166, 167, 190 Anderson, A.C.. see Peterson, R.E. 167, 179, 193 Anderson, A.C.. see Reynolds, C.L. 126,127, 167, 193 Anderson, C.H., see Sabisky, E.S. 169, 193

Anderson, P.W. 25,26,41, 181, 189,230, 266 Anderson, P.W., see Cohen, M.L. 17. 37. 42 Anderson, P.W., see Inkson, J.C. 37, 38, 43 Anderson, P.W., see Lee, P.A. 205, 206, 267 Andreev, A.F. 167, 181, 189, 190 Andres, K . 124, 134, 140, 147, 190 Ansaldo, E.J., see Harshman, D.R. 34, 43 Antervo, A., see Sams, M. 299, 300, 338 Aoyagi, M., see Nakanishi, M. 320, 321, 338 Arani, M.R. 232, 266 Archie, C.N., see Halperin, W.P. 54, I12 Ashburg, J.R., see Wu, M.K. 10, 44 Atsumi, K. 299, 333 Avenel, 0. 124, 125, 127, 186, 190 Azbel, M.Ya. 248, 266 Aziz, R.A. 49, 51, 52, 69, 75, 111 Bain, R.J.P., see Donaldson. G.B. 320, 334 Bak, Per 248, 249, 266 Bak, Per, see Azbel, M.Ya. 248, 266 Barbanera, S.. see Chapman, R.M. 299, 301, 334 Barbanera, S., see Modena, I. 301, 337 Barbi, R. 88, 111 Bardee, T.W.. see Faltens, T.A. 20. 42 Bardeen, J. 12, IS, 37, 38, 41, 42, 199, 219221, 237, 238, 253, 266 Bardeen. J., see Allender, D. 28, 37, 41 Bardeen, J., see Thorne, R.E. 219, 222. 226. 234, 237, 244. 250-253, 268 Bardotti, G. 96. 111 Barone, A. 274, 333 Barranco, M.,see Stringari, S. 110, 113 Barth, D.S.301. 333 Barth, D.S., see Sutherling, W.W. 313, 339 Barton, R., see Kos, J.F. 329, 337 Bashkin, E.P. 71, 111 Basso, H.C. 169, 183, 190 Batey, G.T.187, I90 Batlogg, B. 20, 21, 42 Batlogg, B., see Cava, R.J. 10. 19, 42

341

342

AUTHOR INDEX

Batlogg. B., see Harshman, D.R. 34, 43 Baym, G., see Hoddoson, L. 12, 43 Btal-Monod, M.T.. see Mills, D.L. 152, 155, 192 Beall, J.A., see Muhlfelder. B. 285, 286, 338 Beasley, M.R. 232, 266 Beatty, J., see Barth, D.S. 301, 333 Beauchene, P. 219, 266 Becker. W. 316, 333 Bednort, J.G. 9, 42 Beenakker, J.J.M. 165, 190 Beenakker, J.J.M., see Gorter, C.T. 117, 191 Behringer, R.P.. see Ahlers, G. 106, 111 Bekarevich, I.L. 123, 128. 135, 190 Ben-Jacob, E. 281, 333 Ben-Jacob. E., see Bardeen, J. 219-221, 266 Beno, M.A., see David, W.I.F. 32, 42 Bera, J., see Veuoli, G.C. 8, 44 Berg, M., see Kingery, W.P. 142. 191 Berglund, M.P., see Avenel, 0. 124, 125, 127, 186, 190 Berglund, P.M.. see Ahonen, A.I. 124, 189 Bergman, D.J., see Ben-Jacob, E. 281, 333 Berlincourt, T.G. 7, 42 Bermon, S. 331, 332, 333 Bermon, S.. see Chi. C.C. 331, 334 Bernstein, R.B. 55, 61, 111 Berthier. C., see Segransan, P. 221, 268 Bertotti, B., see Bardotti, G. 96. 111 Betts, D.S. 48, 99, 103, 111, 140, 190 Betts, D.S.,see Leduc, M. 98, 100, 104, 105, 107, 112 Beyermann, W. 215. 266 Bhattacharya, S. 225,226. 239, 260, 261, 263, 264. 266 Bierstedt, P.E., see Sleight, A.W. 8, 44 Bigelow. N., see Johnson, B.R. 98, 112 Bird, R.B.. see Hirschfelder, J.O. 55, 74, 77, 112 Black, W.C., see Abel, W.R. 152, 189 Blackford, B., see Stroink, G. 294. 339 Blatt, J., see Schafroth, M.R. 14, 44 Bleher, M. 235, 266 Bloom, S.B., see Brain, J.D. 304, 333 Bloomfield, P.E., see Lefkowitz, 1. 8, 9. 44 Bogdan, T.J., see Turner, M.S. 332. 339 Bohr, T., see Bak, Per 248, 249, 266 Bonn, D.A., see Kamaris, K. 32-34, 43 Borisov, N., see Neganov, B.S. 140, 193 Borodin, D.V. 211, 215, 263. 266

Bouchaud, J.P. 108, 111 Bouffard, V., see Jacquinot, J.F. 187, 191 Bourg, J., see Cabrera, B. 299, 330. 334 Bourne, L.C. 21, 42, 265, 266 Bourne, L.C., see Faltens, T.A. 20, 42 Boutaud, P., see Segransan, P. 221, 268 Bowley, R.M., see Graf, M.J. 117, 191 Bowley, R.M., see Sheard, F.W. 179, 193 Boyd, M.E. 74. 75, 111 Bozler, H.M., see Ytterboe, S.N. 187, 194 Brain, J.D. 304, 333 Brandt, N.B. 8, 42 Bray, J., see Allender, D. 28, 37, 41 Brewer, D.F. 180, 190 Brewer, D.F., see Ritchie, D.A. 132, 158,159, 162-165, 193 Brewer, J.H., see Harshman, D.R.34, 43 Breymayer, H.-J. 232, 266 Breymayer, H.-J., see Wonneberger, W. 232, 269 Brickett, P., see Ribary, U. 302, 338 Brickett. P.A., see Vrba, J. 305, 307, 339 Brittenham,G.M., see Farrell, D.E. 304,319, 335 Bron, W.E. 127, 168, 190 Brossel, J., see Barbe, R. 88, 111 Brossel, J., see Lefhre-Seguin, V. 85, 89.93, I12 Brown, B.. see Stroink, G. 294, 339 Brown, E. 8, 42 Brown, K.H.,see Tesche, C.D. 285,287,315, 339 Brown, S.E. 216,221,222.224.225,239,240, 242. 246, 249, 250, 256, 257, 262, 266 Brown, S.E., see Thorne, R.E. 222, 226, 250, 251, 268 Bruines, J.J.P. 284, 333 Brush, S.C. 55, 111 Buchal, C., see Owers-Bradley, J.R 71, 113, 141, 193 Buchal, Ch.. see Chocholacs, H. 58, 159, 162-164, 190 Buchanan, D.S. 312, 334 Buckingham, R.A. 61, I l l Buckingham, R.A., see Massey. H.S.W. 61, 113 Buhrman, R. 279. 334 Buhrman, R.A., see Jackel, L.D. 276-280.336 Bulaevskii, L.M. 40, 42 Bulaevskii, L.N. 24, 25, 30, 35, 42

AUTHOR INDEX Bulaevskii, L.N., see Buzdin, A.I. 19, 42 Burbank, M.B., see Vrba, J. 305, 307, 339 Burger, S. 175, 179, 190 Burger, S., see Mok. E. 169, 192 Burton, D. 157.. 194 Busch, P.A., see Greywall, D.S. 158, 191 Busse, F.H. 106, 111 Butler, S., see Schafroth, M.R. 14, 44 Buxton, J.L., see Fraser-Smith, A.C. 293, 335 Buzdin, A.I. 19, 42 Byrne, J. 63, 111 Cabrera, B. 299, 328, 330. 334 Cabrera, B., see Huber, M.E. 331, 336 Cais, M., see Gamble, F.R. 30, 36, 42 Callegari, A.C., see Tesche, C.D. 285, 287, 315, 339 Cameron, J.A. 75, 111 Campbell, M., see Incandela, J. 331, 336 Campitelli, F., see Chapman, R.M. 301. 334 Candela, D., see Masuhara, N. 71, 113 Capone, D.W., see David, W.I.F. 32, 42 Capone, D.W., see Kwok, W.K. 19, 43 Carelli, P. 285, 288, 320, 322, 334 Carelli, P., see Cosmelli, C. 330, 334 Carelli, P., see Modena, I. 301, 337 Carley, J.S., see Aziz, R.A. 51, 52, 69. 111 Castaing, B. 109, 111, 117, 190 Castaing. B., see Peach, L. 117, 193 Castellano, M.G., see Cosmelli, C. 330, 334 Castellano, M.G., see Paik, H.J. 287, 338 Castiel, D., see Shen, T.J. 179, 193 Cava, R.J. 10, 19, 40,42 Cava, R.J., see Batlogg. B. 20, 21, 42 Cava, R.J.. see Fleming, R.M. 219, 246, 267 Cava, R.J., see Harshman, D.R. 34, 43 Chaikin, P.M., see Griiner, G. 198, 227, 267 Challis, L.J. 123, 127, 152, 165, 166, 190 Challis, L.J., see Toombs, G.A. 179, 194 Chang, K.1.. see Bourne, L.C. 21, 42 Chapman, R. 90, 1I1 Chapman, R.M. 299, 301, 334 Chapman, S. 5 5 , 58. 111 Chaudhari, P., see Bermon, S. 331, 333 Chaudhari, P., see Chi, C.C. 331, 334 Chaudhari, P., see Tesche, C.D. 331, 339 Cheeke, J.D.N. 179, 190 Cheeke, J.D.N., see Challis, L.J. 123, 190 Chen, H.S.. see Batlogg, B. 21, 42

343

Chen, M.M., see Tesche, C.D. 285, 287, 315, 339 Chi, C.C. 331, 334 Chi, C.C., see Bermon, S. 331, 333 Chi, C.C., see Tesche, C.D. 331, 339 Chocholacs, H. 158, 159. 162-164, 190 Chocholacs, H., see Owen-Bradley, J.R. 71, 113, 141, 193 Choen, M.L., see Bourne, L.C. 21, 42 Christianson, R.V., see Bak, Per 248,249,266 Chu, C.W. 8, 42 Chu, C.W., see Huang, C.Y. 11, 43 Chu, C.W., see Wu, M.K. 10, 44 Clark, T.D. 242, 266 Clark, T.D., see Long, A. 279, 337 Clark, W.G., see Weger, M. 217, 222, 230, 231. 269 Clarke, G.R., see London, H. 140, 192 Clarke, J. 276, 280, 291, 328, 334 Clarke, J., see Gamble, T.D. 327, 335 Clarke, J., see Ketchen, M.B. 285, 287, 320, 336 Clarke, J., see Tesche, C.D. 280,281,284,339 Clarke, J., see Wellstood, F. 291, 327, 339 Clement, J.R. 101, 111 Cohen, D. 293, 298,304, 310, 322, 334 Cohen, D., see Kennedy, J.G. 310, 336 Cohen, E.G.D., see De Boer, J. 61, 111 Cohen, M.L. 17, 37, 42 Cohen, M.L., see Faltens, T.A. 20, 42 Cohen-Tannoudji, C. 80, 111 Colclogh, M.S., see Cough, C.E. 14, 43 Colegrove, F.D. 78, I I I Collins, R.T. 34, 42 Collins, T.C. 9, 42 Colquhoun, A.B. 329, 334 Combescot, R., see Masuhara, N. 71, 113 Compaan. K., see De Boer, J. 61, 111 Connolly, J.I., see Anderson, A.C. 126, 127, 167, 189 Cooper, L. 15, 42 Cooper, L.N.. see Bardeen, J. 12, 15, 42 Coppersmith, S.N. 234, 238. 253-255, 259, 265, 266, 267 Corruccini, L.R. 71, 111 Corruccini, L.R., see Osheroff, D.D. 158, 162-164, 193 Cosmelli, C. 330. 334 Cowling, T.G.. see Chapman, S. 55, 58, 111 Crabtree, G.W., see Kwok, W.K. 19, 43

344

AUTHOR INDEX

Crampton, S.B., see Leduc, M. 8 3 , 8 5 4 7 . 8 9 , 112 Crampton, S.B., see Nacher, P.-J. 72.86.9395,97, 113 Crocoll, E., see Drung, D. 320. 321, 335 Cromar, M.W., see Muhlfelder, B. 285, 286. 288. 338 Crommie, M.F., see Bourne, L.C. 21, 42 Cross, M., see Sneddon, L. 233,236,253.264, 268 Crum, D.B., see Williamson, S.J. 311, 318, 339 Cuffin. B.N. 302, 334 Cuffin, B.N., see Kennedy. J.G. 310, 336 Cunsolo, S.. see Kinder, H. 183, 191 Curtis, S. 299, 334 Curtiss, C.F., see Hirschfelder, J.O. 55, 74, 77. 112 Czipott, P.V. 328, 334 Daniels, J.M. 78, 81, 84, 85, 99. 111 Daniels, J.M., see Leduc. M. 98, 100, 104, 105, 107, 112 Daniels, J.M., see Timsit, R.S. 89,91,93, 114 Danilov, V.V. 291, 334 Danish. E.H..see Farrell, D.E. 304, 319, 335 David, W.I.F. 32, 42 Day, P.. see David, W.I.F. 32, 42 I>e Boer, J. 5 1 , 61. 1 1 1 De Ninno, A.. see Kinder, H. 183, 191 de Waal, V.J. 284, 285, 287, 320, 334 de Waal, V.J., see Bruines, J.J.P. 284, 333 d e Waal, V.J., see van Nieuwenhuyzen, G.J. 320, 339 DeConde, K., see Perry,T. 124,151-153, 193 Denker, J.S., see Johnson, B.R. 98, 112 Deptuck, D. 144. 157, 159. 190 Derka, I., see Maniewski, R. 294, 325. 337 Derrida, B. 159, 190 Diekmann, V., see Becker, W. 316, 333 Dietsche, W., see Basso, H.C. 169, 183. 190 Dietsche, W., see Kinder, H. 168, 191 Dietsche, W., see Weber, 1. 168-170. 194 Dinger, T.R. 17. 32, 42 Dinger, T.R., see Collins, R.T. 34. 42 Dinger. T.R., see Worthington, T.K. 32, 44 Disalvo, F.J., see Gamble, F.R. 30, 36, 42 Doettinger, S., see Mok, E. 169, 192 Dokoupil, Z., see Beenakker, J.J.M. 165, 190 Dolgov. O.V. 17-19, 29, 42

Donaldson, G.B. 320, 334 Donaldson, G.B., see Colquhoun, A.B. 329, 334 Donaldson, G.B., see Ketchen, M.B. 285, 287, 320, 336 Doniach, S., see Inoue, M. 265, 267 Doss, M.G., see Kamaras, K. 32-34, 43 Dransfeld, K. 165, 190 Dransfeld, K., see Challis, L.J. 165, 166, 190 Drazin, P.G. 106, 1 1 1 Dries, L.J., see Huang, C.Y. 11, 43 Drung, D. 320, 321, 335 Duggan, D.D., see Ong, N.P. 224, 268 Dumas, J., see Beauchene, P. 219, 266 Dupont-Roc, J. 81, 1 1 1 Dupont-Roc, J., see Himbert, M. 85, 86, 112 Duret, D. 305, 307, 335 Duret. D., see Nicolas, P. 294. 338 Duret, D., see Seppanen, M. 318, 338 Durieux, M. 101, 1 1 1 Durieux, M., see Matacotta, F.C. 75, 77, 113 Dynes, R.C., see Allen, P.B. 18, 41 Early, S., see Chu, C.W. 8, 42 Ebisawa, H., see Saito, S. 149, 150, 152, 153, 193 Eckert, J.C., see Ong, N.P. 217, 223, 268 Eckert, M., see Hoddoson, L. 12, 43 Edwards, D.O. 8 5 . I12 Edwards, D.O., see Masuhara, N. 71, 113 Efetov, K.B. 209, 267 Ehnholm, G.J. 279, 335 Ehnholm, G.J., see Ahola, H. 290, 333 Einstein, A. 12, 13, 42 Eisenmenger, W. 169, 190 Eisenmenger, W., see Burger, S. 175, 179,190 Eisenmenger, W., see Heim, U. 169, 191 Eisenmenger. W., see Koblinger, 0. 169,171, I92 Eisenmenger, W.. see Marx, D. 169, 175, 192 Eisenmenger, W., see Mok, E. 169, 192 Eisenmenger, W., see Trumpp, H.J. 168, 194 Eliashberg, G.M. 25, 40, 42 Emery, V.J. 26, 42, 70, 112 Emoto, H., see Furukawa, H. 316, 335 Endo, T., see Nakanishi, M. 320, 321, 338 Engel Jr, J., see Barth, D.S. 301, 333 Enpuku, K. 285-287, 335 ErnC, S.N. 294, 303, 320, 322, 325, 335 Estes. W.E., see Lefkowitz, 1. 8. 9, 44

AUTHOR INDEX Etienne, P.E., see Sinton, C.M. 302, 338 Ettinger, H., see Cheeke, J.D.N. 179, 190 Ezawa, H. 171, 172, 190 Fack, H. 232, 242, 267 Fairbank, H.A. 165, 190 Fairbank, H.A.. see Lee, D.M. 117, 192 Fairbank, W.M. 328, 335 Fairbank, W.M., see McAshan, M.S. 329,337 Faltens, T.A. 20, 42 Faltens, T.A., see Leary, K.J. 32. 44 Farago, P.S., see Byme, J. 63, 1 1 1 Farrell, D.E. 304, 305, 319, 335 Fenici, R.R. 303, 335 Fermon, C., see Jacquinot, J.F. 187. 191 Fife, A.A., see Vrba, J. 305, 307, 339 Fiory, A.T., see Gurvitch, M. 34, 43 Fisher, D. 234, 236, 264, 267 Fisher, D., see Sneddon. L. 233, 236, 253, 264, 268 Fjeldly, T.A., see Ishiguro. T. 168, 191 Fleming, R.M. 197, 198, 215-219, 237, 240, 246,257, 258, 267 Fleuren, F.H., see ter Brake, H.J.M. 324, 339 Flokstra, J., see ter Brake, H.J.M. 324, 339 Fogan, E.M., see Cough, C.E. 14, 43 Foglietti, V. 292, 335 Foglietti, V., see Carelli. P. 285, 288, 320, 334 Foglietti, V., see Cosmelli, C. ,330, 334 Folinsbee, J.T. 126, 127, 166-169, 190 Fomin, LA. 123, 190 Ford, P.J.. see Hall, H.E.140, 191 Forgacs, R.L. 289, 335 Frankel, R.B., see Huang, C.Y. 11, 43 Fraser-Smith, A.C. 293, 335 Freed, J.H., see Johnson, B.R. 98, I12 Freitas, P., see Collins, R.T. 34, 42 Friedman, L.J., see Ytterboe, S.N.187, 194 Friend, R.H. 30, 42 Frisch, H., see Incandela, J. 331, 336 Frisken, B. 142, 144, 190 Frohlich, H. 197, 267 Frolov, V.V., see Latyshev, Yu.1. 221, 267 Frossati, G. 124, 141, 158, 162-164, 190 Fuchs, I.M.,see Adamenko, I.N. 179, 189 Fujii. Y. 187, 191 Fukuda, T., see Ishimoto, H. 71,98, 112, 158, 191 Fukuyama, H. 209,232,253, 267

345

Fukuyama, H., see Ishimoto, H. 71, 98, 112, 158, 191 Fukuyama, H., see Mamiya. T. 158, 192 Fulton, T.A., see Clarke, J. 280, 334 Furayama, T., see Maeda, A. 216, 267 Furukawa, H. 316, 335 Furukawa, H.,see Shirae, K. 316, 318, 338 Gabovich, A.W. 8, 20, 42 Gallagher, W.J., see Collins, R.T. 34, 42 Gallagher, W.J., see Dinger, T.R. 17, 32, 42 Gallagher, W.J., see Foglietti, V. 292, 335 Gallagher, W.J.. see Worthington, T.K. 32.44 Gamble, F.R. 30, 36, 42 Gamble, T.D. 327, 335 Gammic, G . , see Lyding, J.W. 224, 225, 267 Gao, L., see Wu, M.K. 10, 44 Gardner, R., see Cabrera, B. 299, 330, 334 Gardner, R.D., see Huber, M.E. 331, 336 Gavoret, J. 123, 128, 135, 191 Geballe, T.H., see Chu, C.W. 8, 42 Geballe, T.H., see Gamble, F.R. 30, 36, 42 Geballe, T.H., see White, R.M. 16, 35, 44 Gehr, P., see Brain, J.D. 304, 333 Geilikman, B.J. 30, 42 Gershenson, E.M. 34, 42 Geselowitz, D.B., see Cuffin, B.N. 302, 334 Geselowitz, D.B.. see Grynszpan, F. 297.335 Geselowitz, D.B., see Miller 111, W.T. 302, 33 7 Gianolio, L., see Bardotti, G. 96, 111 Giffard, R.P. 276, 277, 288, 335 Giffard, R.P., see Hollenhorst, J.N. 279, 335 Gill, J.C. 211. 215, 263. 267 Gillson, J.L., see Sleight, A.W. 8, 44 Ginsberg, D.M., see Thomasson, J.W. 96, I14 Ginzburg, V.L. 5-7,9, 12-16, 18-20, 22, 2731, 35, 36, 38-40, 42, 43 Ginzburg, V.L., see Agranovich, V.M.27.28, 33, 41 Ginzburg, V.L., see Bulaevskii, L.M. 40, 42 Giri, M.R., see Morii, Y. 158, 192 Glukhov, N.A., see Kagan, Yu. 117, 191 Godfrin, H., see Frossati, G. 124, 141, 190 Godfrin, H., see Lauter, H.J. 180, 192 Goldman, V.V. 117, 191 Goldstein, N.E., see Wilt, M. 327, 339 Golovashkin, A.I. 9-11, 43 Gonzales, O.D., see White, D. 165, 194 Goodstein, D., see Kinder, H. 183, 191

346

AUTHOR INDEX

Goodstein. D., see Taborek. P. 168, 169, 175, 176, 193, 194 Gorbatsevich. A.A., see Ginzburg, V.L. 9, 43 Gorin, Yu.N.. see Vendik, O.G. 279, 339 Gor’kov, L.P. 35, 36, 43 Goner, C.T. 117, I91 Goubau, W.M.. see Clarke, J. 280, 291, 334 Goubau. W.M., see Gamble, T.D. 327. 335 Goubau, W.M., see Ketchen, M.B. 285, 287, 320, 336 Gough. C.E. 14, 43 Gould, C.N., see Ytterboe. S.N. 187. 194 Grace, J.D., see David, W.I.F. 32, 42 Grad, H. 5 5 . 112 Graf, M.J. 117. 191 Grant, P. 4, 10, 43 Greedan, J.E., see Kamaras, K. 32-34. 43 Gregg, J.F., see Jacquinot, J.F. 187, 191 Greiner, J.H., .see Tesche, C.D. 285,287.3 15, 339

Grest. G.S. 159, 191 Greywall, D.S. 158, 191 Grimes, C.C., see Fleming, R.M. 198, 215, 217, 218, 237. 267 Grimes, D. 304, 335 Grimsrud, D.T. 75, 112 Griiner, G . 198, 205, 215, 218, 227, 267 Griiner. G.. see Bardeen, J. 219-221, 266 Griiner, G., see Beyemann, W. 215, 266 Griiner, G., see Brown, S.E. 221, 222, 224, 225, 239, 240, 242, 246, 249. 250, 256, 257, 262, 266 Griiner, G., see Kim, Tae Wan 213, 267 Griiner, G . , see Mihaly, L. 216, 267 Griiner, G . , see Mozurkewich, G. 218, 222, 223. 227, 234, 260, 263, 264, 268 Griiner, G., see Reagor, D. 212, 213, 268 Griiner, C . , see Sridhar, S. 212. 213,235, 268 Gruner, G., see Thorne, R.E. 222, 226, 250, 25 1, 268 Gruner,G., see Weger, M. 217,222,230,231, ,769

Griiner, G., see Wu,Wei-Yu 215. 238, 269 Griiner, G., see Zettl, A. 215, 216, 224, 226, 239. 240, 243-246, 260, 263-265, 269 Grynszpan. F. 297, 335 Guenault, A.M. 141, 191 Guillon, F., see Frisken. B. 142, 144, 190 Guillon, F., see Robertson, R.J. 160. 161. 193 Gully, W.J. 71. 98, 1J2

Gully, W.J., see Osheroff, D.D. 128, 193 Gumnit, R.J., see Kennedy, J.G. 310, 336 Gunn, J.M.F., see David, W.I.F. 32, 42 Guo. C.J. 168, 191 Gurvitch, M. 34, 43 Gustafson, H.R., see Incandela, J. 331, 336 Guyer, R.A. 152, 158, 191 Gylling, R.G., see Avenel, 0. 124, 125, 127, 186, 190 Haario, H.. see Hamalainen, M.S. 308, 335 Hahlbohm, H.-D., see Erne, S.N. 294, 335 Haikala, M.T., see Ahonen. A.I. 124, 189 Hakuraku, Y., see Ogushi. T. 11, 44 Hall, H.E. 140, 191 Hall, R.P. 247, 264, 267 Hallstrom. J.. see Hari, R. 311, 335 Hallstrom, J., see Knuutila, J. 308, 313, 314, 318. 337 Halperin, B.I., see Ambegaokar, V. 232, 266 Halperin. B.I., see Anderson, P.W. 181, 189 Halperin, W.P. 54. 112 Ham, W.K., see Faltens, T.A. 20, 42 Ham, W.K., see Leary, K.J. 32, 44 Hamalainen, M., see Kaukoranta, E. 308,336 Hamalainen, M., see Pelizzone, M. 302,316, 338

Hamalainen, M., see Sams, M. 299, 300, 338 Hamalainen, M.S. 297, 308, 335 Hamalainen, M.S., see Ilmoniemi, R.J. 297, 336

Hamalainen, M.S., see Knuutila, J. 293, 308, 316, 319, 320, 322, 325, 326, 337 Hamilton, J.. see Buckingham, R.A. 61, 1 1 1 Hammel, E.F., see Kilpatrick, J.E. 61.74, 112 Happer, W. 80, 112 Harding. T., see Zimmerman, J.E. 276, 339 Hardy, W., see Harshman, D.R. 34, 43 Hardy, W.N.. see Lefevre-Seguin, V. X5, 89, 93, 112 Hari, R. 297, 299, 307, 311, 335 Hari, R., see Huttunen, J. 3 I I , 336 Hari, R., see Ilmoniemi, R.J. 296, 305. 306, 310, 318, 336 Hari, R., see Kaukoranta, E. 300, 307, 308, 311, 336 Hari, R., see Makela, J.P. 31 I , 337 Hari, R.. see Pelizzone, M. 302, 316, 338 Hari, R., see S a m , M. 299. 300, 338 Harris, J.W., see Farrell, D.E. 304, 319, 335

AUTHOR INDEX Harrison, J.P. 118, 124. 142, 187, 191 Harrison, J.P., see Deptuck, D. 144, 157,159, I90 Harrison, J.P., see Folinsbee, J.T. 168, 169, 190 Harrison, J.P., see Frisken, B. 142, 144. 190 Harrison, J.P., see Maliepaard, M.C. 144, 157, 159, 160, 192 Harrison, J.P., see Robertson, R.J. 160, 161, I93 Harrison, J.P., see Rutherford, A.R. 147,148, 159, 193 Harrison, W.T.A., see David. W.I.F. 32, 42 Harshman, D.R. 34, 43 Hatfield, W.E., see Leflowitz, I. 8, 9, 44 Haug, H. 167, 179, 191 Haug, R. 180, 191 Haught, J.R., see Wilt, M. 327, 339 Hayakawa. K., see Iwama, S. 142, 191 Hayashi, C. 141, 191 Hayashi, M. 142, 144, 191 He, Da-Run, see Yeh, W.J. 249, 250, 269 Hebral, B., see Frossati, G. 124, 141, 190 Hebral, B., see Peach, L. 117, 193 Heiden, C., see Wellstood, F. 291, 327, 339 Heim, U. 169, 191 Heim, U., see Koblinger, 0. 169, 171, 192 Heino, J.J., see Kelha, V.O. 293, 315, 336 Heinonen, P., see Malmivuo, J. 294, 337 Herr, S.L., see Kamaris, K. 32-34, 43 Henvig, R., see Drung, D. 320, 321, 335 Higgins, M.J., see Bhattacharya, S. 225, 239, 260, 261, 263, 266 Higo, S., see Ogushi, T. 11, 44 Himbert, M. 85, 86. 112 Hinks, D.G., see David, W.I.F. 32, 42 Hinks, D.G., see Kwok, W.K. 19, 43 Hirschfelder. J.O. 55, 74, 77, 112 Hoddoson, L. 12, 43 Hoen, S., see Faltens, T.A. 20, 42 Holiday, S., see Ribary, U. 302, 338 Hollenhorst, J.N. 279, 335 Holste, J.C., see Radebaugh, R. 124,141,162, I93 Holzman. G.N., see Gershenson, E.M. 34,42 Homan, C.G. 9, 43 Homan, C.G., see Brown, E. 8, 42 Honjo, Y., see Ogushi, T. 11, 44 Hood, K. 152, 191 Hor, P.H., see Huang, C.Y. 11, 43

347

Hor, P.H., see Wu, M.K. 10, 44 Horacek, B.M. 302, 335 Horacek, B.M., see MacAulay, C.E. 303,337 Horacek, M., see Stroink, G. 294, 339 Horstman, R.E. 168, 191 Hoshino, R., see Ido, M. 149, 191 Hostetler, W., see Ilmoniemi. R.J. 299, 312, 336 Hoyt, R.F., see Masuhara, N. 71, 113 Huang, C.Y. 11, 43 Huang, C.Y.. see Chu, C.W. 8, 42 Huang, S., see Chu. C.W. 8, 42 Huang, Z.J., see Wu, M.K. 10, 44 Hubacek, J.S., see Lyding, J.W. 224,225,267 Hubacek, J.S., see Thorne, R.E. 248, 268 Huber, M.E. 331, 336 Huber, T.E. 117, 191 Huber, T.E., see Mans, H.J. 117, 192 Huberrnan, B.A., see Beasley, M.R. 232, 266 Hundley, M.F., see Hall, R.P. 264, 267 Huttunen, J. 311, 336 Huttunen, J., see Peliuone, M. 302.3 16, 338

Ido, M. 149, 191 Ikonen, E. 329, 336 Ilmoniemi, R. 308, 311, 336 Ilmoniemi, R., see Ryhanen, T. 273,276,281, 284, 338 Ilmoniemi, R.J. 296, 297, 299, 305, 306, 310, 312, 318, 336 Ilmoniemi, R.J., see Chapman, R.M. 299, 301, 334 Ilmoniemi, R.J., see Hari, R. 297, 335 Ilmoniemi, R.J., see Kelhl, V.O. 293, 315, 336 Ilmoniemi, R.J., see Knuutila, J. 308, 316, 319, 320, 322, 325, 337 Ilmoniemi, R.J., see Sinton, C.M. 302, 338 Ilmoniemi, R.J., see Vvedensky, V.L. 299, 311,339 Imry,Y. 281, 336 Imry, Y., see Ben-Jacob, E. 281, 333 Incandela, J. 331, 336 Inkson, J.C. 37, 38, 43 Inoue, M. 265, 267 Irie, F., see Enpuku, K. 285-287,335 Ishiguro, T. 168, 191 Ishimoto, H. 71, 98, 112, 158, 191 Iwama, S. 142, 191

348

AUTHOR INDEX

Jackel. L., see Buhrman, R. 279, 334 Jackel, L.D. 276-280, 336 Jacquinot, J.F. 187, 191 Jaklevic, R.C. 274, 336 Janossy, A., see Beauchene, P. 219, 266 Jinossy, A.. see Brown, S.E. 224, 225, 266 Janossy. A.. .see Segransan, P. 221, 268 Janossy, A., see Wu, Wei-Yu 215, 238, 269 Jayaraman, A., see Batlogg, B. 20, 21, 42 Jaycox, J.M. 285, 320. 336 Jensen, M.H., see Alstrom, P. 248, 266 Jensen, M.H., see Bak, Per 248, 249, 266 Jerome, D. 36, 43 Jerome, D.. see Gor'kov, L.P. 35, 43 Jing, T.W. 224, 267 Johnson, B., see Ribary, U. 302, 338 Johnson, B.R. 98, 112 Johnson, R.C. 166, 191 Johnson, R.T., see Wheatley, J.C. 141, 194 Johnson, W., see Muhlfelder, B. 288, 338 Johnson, W.L., see Anderson, A.C. 126, 166, 167, 179, 189 Johnson, W.W., see Muhlfelder, B. 285, 286, 338 Johnston, H.L., see White, D. 165, 194 Jones, H.C., see Tesche, C.D. 285, 287, 315, 339 Jordan, R.G., see Gough, C.E. 14, 43 Jorgensen, J.D., see David, W.I.F. 32, 42 Jorgensen. J.D., see Kwok, W.K. 19, 43 Joutsiniemi, S.-L. 317, 336 Joutsiniemi, S.-L., see Hari, R. 307. 335 Jutzi, W., see Drung, D. 320. 321, 335 Jutzler, M. 187, 191 Kado, H., see Nakanishi, M. 320, 321. 338 Kagan, Yu. 117, 191 Kahan, B.G., see Shapligin, 1.S. 10, 44 Kaiser, M.B., see Zettl, A. 224, 269 Kajanto, M.J. 232, 267 Kajola, M., see Knuutila, J. 283, 286, 288, 291, 308, 313, 314, 318,320, 321, 337 Kajola, M.J., see Knuutila, J. 308. 316, 319, 320, 322, 325, 337 Kajola, M.J., see Vvedensky, V.L. 299, 311, 339 Kalem, C.B., see Ong, N.P. 2 17,223,224,268 Kalliomaki, K. 304, 336 Kalliomaki, K., see Kalliomaki, P.-L. 304, 336

Kalliomaki, P.-L. 304, 336 Kalliomaki. P.-L., see Kalliomaki, K. 304, 336 Kamaras, K. 32-34, 43 Kamper, R.A. 279, 336 Kanter, H. 279, 336 Kao, Y.H., see Yeh, W.J. 249. 250, 269 Kapitza, P.L. 117, 191 Karasik, B.S., see Gershenson, E.M. 34, 42 Karp, P. 309, 336 Karp, P., see Duret, D. 305, 307, 335 Karp, P., see Ikonen, E. 329. 336 Karp, P., see Seppanen, M. 318, 338 Kasai, N., see Nakanishi, M. 320, 321, 338 Kastler, A., see Cohen-Tannoudji, C. 80, 111 Katayama, M., see Furukawa, H. 316, 335 Katayama, M., see Shirae, K. 318, 338 Katayama, T., see Shirae, K. 318, 338 Katila, T. 303, 336 Katila, T., see Atsumi, K. 299, 333 Katila, T., see lkonen, E. 329, 336 Katila, T., see Maniewski, R. 294, 325, 337 Katila, T., see Montonen, J. 303, 338 Katila, T., see Seppanen, M. 318, 338 Katila, T., see Weinberg, H. 299, 339 Kaufman, L. 297.31 1, 336 Kaufman, L., see Curtis, S. 299, 334 Kaufman, L., see Pelizzone, M. 311, 338 Kaufman, L., see Romani, G.L. 305,307,324, 338 Kaufman, L., see Williamson, S.J. 297, 311, 318, 339 Kaufmann, R.,see Becker, W. 316, 333 Kaukoranta, E. 300, 307, 308, 311, 336 Kaukoranta, E., see Hari. R. 299, 335 Kaukoranta, E., see Sams, M. 299, 300, 338 Kawano, I . I . , see Ogushi, T. 1 1 , 44 Keene, M., see Gough, C.E. 14, 43 Keith, V., see Guenault, A.M. 141, 191 Kelha, V.O. 293, 315, 336 Keller, S.W., see Bourne, L.C. 21, 42 Keller, S.W., see Faltens, T.A. 20, 42 Keller, S.W., see Leary, K.J. 32, 44 Keller, W.E. 48, 61. 74, 75, 103, 112 Keller, W.E., see Kilpatrick, J.E. 61, 74, 112 Kennedy, C.J., see Guenault, A.M. 141, 191 Kennedy, J.G. 310, 336 Kennedy, J.S., see Ribary, U. 302, 338 Kemsk. J.F., see Keller, W.E. 103, 112 Ketchen, M.B. 280, 285, 287. 320, 336

AUTHOR INDEX

349

Ketchen, M.B., see Clarke, J. 280, 291, 334 Koch, R.H., see Foglietti, V. 292, 335 Kohjiro, S., see Enpuku, K. 287, 335 Ketchen, M.B., see Foglietti, V. 292, 335 Kojima, H.. see Morii, Y. 158, 192 Ketchen, M.B., see Jaycox. J.M. 285,320,336 Kopaev, Yu.V. 22, 43 Ketchen, M.B., see Tesche, C.D. 285, 287, Kopaev, Yu.V., see Ginzburg, V.L. 9, 43 315, 339 Kornhuber, H.H., see Becker, W. 316, 333 Ketoja, J.A. 280, 281, 336, 337 Khalatnikov,J.M. 117,120,123, 165,179,191 Kos, J.F. 329, 330, 337 Khalatnikov, I.M., see Bekarevich, I.L. 123, Kosaka, S., see Nakanishi, M. 320, 321, 338 Kose, V., see Fack, H. 232, 242, 267 128, 135, 190 Kosevich, Yu.A., see Andreev, A.F. 181, 190 Khomskii, D.I., see Bulaevskii, L.N. 24, 25, Kotani, M.. see Atsumi, K. 299, 333 35, 42 Khomskii, D.J., see Zvezdin, A.K. 24, 25, 44 Kourouklis. G., see Batlogg, B. 20, 42 Kourouklis, G.A., see Batlogg, B. 21, 42 Kilpatrick, J.E. 61, 74, 112 Kilpatrick, J.E., see Boyd, M.E. 74, 75. 111 Koyanagi, M., see Nakanishi, M. 320, 321, 338 Kim, K.K., see Tesche, C.D. 285, 287, 315, Kreitzman, S.R., see Hanhman, D.R. 34, 43 339 Krusius, M.,see Ahonen, A.J. 124, 189 Kim, Tae Wan 213, 267 Kuang, W.-Y. 165, 166, 192 Kinder, H. 168, 179. 183, 191 Kubo, R., see Teranishi, N. 234, 268 Kinder, H., see Basso, H.C. 169, 183, 190 Kubota, M., see Chocholacs, H. 158, 159, Kinder, H., see Schubert, H. 183, 193 162-164, 190 Kinder, H., see Weber, J. 168-170, 194 Kubota, M., seeOwen-Bradley, J.R. 71, 113, Kingery, W.P. 142, 191 141, 193 Kirschvink, J.L. 304, 337 Kubota, M., see Rogacki, K. 124. 141-143, Kirzhnits, D.A. 17, 37, 43 193 Kirzhnits, D.A., see Ginzburg, V.L. 12, 16, Kuchnir, M., see Incandela, J. 331, 336 18-20, 22, 27-30, 36, 38, 43 Kukharenko, Yu.A., see Bulacvskii, L.N. 30, Kishida, K., see Shirae, K. 316, 338 42 Klapwijk, T.M., see d e Waal, V.J. 285, 287. Kulik, 1.0. 281, 337 320, 334 Kleinsasser, A.W., see Foglietti, V. 292, 335 Kummer, J.T. 149, 192 Kleinsasser, A.W., see Tesche, C.D. 285,287, Kurkijarvi, J. 277, 281, 337 Kurkijarvi, J., see Ketoja, J.A. 280, 281, 336, 315, 339 337 Klemm, R.A. 233, 267 Klemm, R.A., see Bhattacharya, S. 225, 226, Kuvshinnikov, B.V., see Brandt, N.B. 8, 42 Kuzmin, L.S. 279, 337 239, 260, 261, 263, 264, 266 Kwok, W.K. 19, 43 Klemm, R.M., see Robbins, M. 233,235,268 Klitsner, T. 169, 183, 191 Knight, W.D. 150, 192 Kniittel, A., see Drung, D. 320, 321, 335 Laibowitz, R.B., see Collins, R.T. 34, 42 Knuutila, J. 283,285,286,288,291,293,308, Lake, G.M., see Hanhman. D.R. 34, 43 313, 314, 316, 318-322, 325. 326, 337 Laloe, F. 47, 48, 83, 86, 112 Knuutila, J., see Ilmoniemi, R. 308, 311, 336 Laloe, F., see B a h t , R. 88, I I I Knuutila, J., see Ilmoniemi, R;J. 297, 336 Laloe, F., see Dupont-Roc, J. 81, 111 Knuutila, J., see Ryhanen, T. 273, 276, 281, Laloe, F., see Himbert, M. 85, 86, 112 284. 338 Laloe, F., see Leduc, M. 81, 83, 85-87, 89, Kobayashi, S. 149, 192 98, 100, 104, 105, 107. I12 Koblinger, 0. 169, 171, 192 Laloe, F., see Lefkvre-Seguin. V. 85, 89, 91, Koblinger, 0.. see Heim, U. 169, 191 93.94, I12 Koch, H. 287, 337 Laloe, F., see Lhuillier, C. 48, 59. 60,62,63, Koch, R.H., see Collins, R.T. 34, 42 66, 67, 70-72, 74, 77, 108, 110, 112, 113

350

AUTHOR I N D E X

Laloe. F., see Nacher. P.-J. 72, 84-88,93-95, 97, 113 Laloe, F., see Pavloviac, M. 83, 104. 113 Laloe, F., see Pinard, M. 60, 113 Laloe. F., see Stringari, S. 110, 113 Laloe, F.,see Tastevin, G . 93.97-99, 109, 113 Lambe, J., see Jaklevic, R.C. 274, 336 Lambert. C.J. 157, 159, 192 Lambert, C.J., see Burton, D. 157.. 194 Landau, L.D. 13, 43, 120, 128, 192 Landau, L.D., .see Ginzburg, V.L. 14, 43 Larkin. A.I., see Efetov, K.B. 209, 267 Larsen, S.Y., see Boyd, M.E. 74. 75, 1 1 1 Lasarev, V.B., see Shapligin. I.S. 10, 44 Lassmann, K., see Burger, S. 175, 179, 190 Lassmann, K., ree Mok. E. 169, 192 Lassmann, K., see Trumpp. H.J. 168, 194 Latyshev, Yu.1. 221, 244, 267 Lauter, H.J. 180. 192 Lazarus, R.B. 102, 112 Leary, K.J. 32. 44 Leary, K.J., see Faltens, T.A. 20, 42 Leduc. M. 81,83,85-87.89,98,100,104,105, 107, 112 Leduc, M., see Betts, D.S. 48, 1 1 1 Leduc, M.. see Daniels, J.M. 78, 84. 85, 99, 111

Leduc, Leduc, Leduc, Leduc,

M., see Dupont-Roc, J. 81, 1 1 1 M.. see Himbert, M. 85, 86, 112 M., ree Laloe. F. 48. 83, 86, 112 M., see Lhuillier, C. 48.50.63-65.98,

113

Leduc. M., see Nacher, P.-J. 72.84-88.93-95, 97. 1 1 3 Leduc, M., see Schearer, L.D. 78, 98. 113 Leduc, M., see Tastevin, G . 93, 97-99, 109, I 1.3 Leduc, M., see Trenec, G . 8 1, 82. 114 Lee, D.M. 117, 192 Lee, D.M.. see C o m c c i n i , L.R. 71, 1 1 1 Lee, D.M.. see Johnson, B.R. 98, 112 Lee, D.M., see Osheroff, D.D. 128, 193 Lee, P.A. 205, 206. 223, 232, 253, 267 Lee, P.A.. see Fukuyama, H. 209, 253, 267 Lee, P.A., see Ong, N.P. 224, 268 Lefivre-Seguin. V. 8 5 , 89, 91, 93, 94, 112 LeTevre-Seguin. V., see Himbert. M. 85, 86, 112

Lefkowitz, 1. 8. 9, 44 Leggett, A.J. 71, 112, 151, 152, 192

Lehtinen, M.S., see Hamalainen, M.S. 308, 335 Leiderer, P.. see Basso, H.C. 169, 183, 190 Leiderer, P., see Schubert, H. 183, 193 Leinio, M., see Montonen, J. 303, 338 Leinonen, L., see Hari, R. 311, 335 Leinonen, L., see Huttunen, J. 311, 336 Lejus, A.-M., see Schearer, L.D. 78, 98, 113 Lekkala, J., see Malmivuo, J. 294, 337 Lekkala, J.O. 318, 337 Lennard, R., see Grimes, D. 304, 335 Lennard-Jones, T.E. 180, 192 Leoni, R., see Carelli. P. 320, 322, 334 Leoni, R., see Chapman, R.M. 301, 3.74 Leoni, R., see Modena, 1. 301, 337 Leoni, R., see Romani. G.L. 308, 312, 318, 338 Leory, K.L., see Bourne, L.C. 21, 42 Levinsen, M.T., see Alstrom, P. 248, 266 Levy, L.P., see Johnson, B.R. 98, 112 Lhuillier, C. 48, 50. 59-67, 69-72, 74, 77. 98, 105, 108, 110. 112, 113 Lhuillier, C., see Bouchaud, J.P. 108, 1 1 1 Libchaber, A. 106, 113 Libchaber, A., see Maurer, J. 106, 113 Liburg, M., see Neganov, B.S. 140, 193 Lifshitz, E.M., see Landau, L.D. 120, 192 Likharev, K.K. 274, 279, 288, 337 Likharev, K.K.. see Danilov, V.V. 291, 334 Likharev, K.K., see Kuzmin, L.S. 279. 337 Lindelof, P.E. 229, 242-244, 255, 267 Lindelof, P.E.. see Clark, T.D. 242, 266 Link, G.L. 225, 226, 267 Little, W.A. 28, 34, 36, 44, 165, 179, 192 Little, W.A., see Johnson, R.C. 166, 191 Littlewood, P.B. 235, 252, 267 Littlewood, P.B., see Coppersmith, S.N. 234, 238, 253-255, 259, 265, 266, 267 Liu, P.S. 40, 44 Llurba, R., see d e Waal, V.J. 284, 334 Lockerbie, N.A., .see Colquhoun, A.B. 329, 334 Lockerbie, N.A., see Sherlock, R.A. 168, 193 Lockerbie, N.A., see Wyatt, A.F.G. 169, 194 Lombroso, C.T.. see Kennedy, J.G. 310, 336 London, F. 13, 15, 44 London, H. 140, 192 Long, A. 279, 337 Lopez, L., see Fenici, R.R. 303, 335 Lounasmaa, O.V. 140, 192, 274, 337

AUTHOR INDEX Lounasmaa, O.V., see Ahonen, A.I. 124, 125, 189 Lounasmaa, O.V., see Andres, K. 140, I90 Lounasmaa, O.V., see Hari, R. 311, 335 Lounasmaa, O.V., see Kaukoranta, E. 300, 307, 311, 336 Lounasmaa, O.V., see Knuutila, J. 308, 313, 314, 318, 337 Loye, H.C., see Bourne, L.C. 21, 42 Loye, H.C., see Faltens, T.A. 20, 42 Ludwig, W. 319, 337 Ludwig, W.,see Becker, W. 316, 333 Lusznynski, K. 70, 71, 113 Lyding, J.W. 224, 225, 267 Lyding, J.W., see Thorne, R.E. 222,226,234, 237, 238, 244, 248, 250-253, 268 Lyding, J.W., see Tucker, J.R. 235, 269 Lynton, E.A., see Beenakker, J.J.M. 165, 190 Lyons, W.G., see Thorne, R.E. 222,226,234, 237, 238, 244, 248, 250-253, 268 Lyons, W.G., see Tucker, J.R. 235, 269 MacAulay, C.E. 303, 337 MacCrone, R.K., see Brown, E. 8, 42 MacCrone, R.K., see Homan, C.G. 9, 43 Madekivi, S., see Montonen, J. 303, 338 Maeda, A. 216, 226,267 Mager, A. 294, 337 Main, P.C., see Batey, G.T. 187, 190 Makela, J.P. 311, 337 Makela, J.P., see Han, R. 311, 335 Makela, J.P., see Pelizzone, M. 302, 316,338 Maki, K., see Ong. N.P. 217, 223, 268 Maki, M., see Mozurkewich,G. 218,223,227, 234, 268 Maki, M., see Reagor, D. 213, 268 Makijarvi, M., see Katila, T. 303, 336 Makijarvi, M., see Montonen, J. 303, 338 Maksimov, E.G., see Dolgov, O.V. 17-19.29, 42 Maksimov, E.G., see Mazin, 1.1. 20, 44 Maliepaard. M.C. 144, 157, 159, 160, 192 Malmivuo, J. 294, 337 Malmivuo, J.A.V., see Lekkala, J.O. 318,337 Mamiya, T. 158, 192 Mandl, F. 61, 113 Maniewski, R. 294, 325, 337 Maniewski, R., see Katila, T. 303, 336 Manning, J.S., see Lefkowitz, I. 8, 44 Mapoles, E.R., see Paik, H.J. 329, 338

351

Maradudin, A.A. 178, 192 Maradudin, A.A., see Shen, T.J. 179, 193 Marcus, J., see Beauchene, P. 219, 266 Marcus, J., see Segransan, P. 221, 268 Marcus, P.M., see Irnry, Y. 281, 336 Marek,D. 187, 192 Mans, H.J. 117, 179-182, 192 Maris, H.J.. see Graf, M.J. 117, 191 Maris, H.J., see Guo, C.J. 168, 191 Mans, H.J., see Huber, T.E. 117, 191 Marsden, J.R., see Williamson. S.J. 311, 318, 339 Marshall, R., see Betts, D.S. 99, 103, 111 Marx, D. 169, 175, 192 Mason, E.A., see Monchick, L. 61, 113 Masselli, M., see Fenici, R.R. 303, 335 Massey, H.S.W. 61, 113 Massey, H.S.W., see Buckingharn. R.A. 61, 111

Masuda, Y., see Mamiya, T. 158, 192 Masuhara, N. 71, 113 Matacotta, F.C. 75, 77, 113 Mathews, R.H., see Paik, H.J. 287, 338 Matkowsky, B.J., see Ben-Jacob, E. 281, 333 Matsukawa, H. 234, 236, 252, 253, 260, 267 Mattheiss. L.F. 20, 44 Matthias, B.T. 11, 38, 44 Maurer, J. 106, 113 Maurer, J., see Libchaber, A. 106, 113 May, A.D., see Timsit, R.S. 89, 91.93, 114 Mazin, 1.1. 20, 44 McAdams, H.H. 86, 113 McAshan, M.S. 329, 337 McColl, D.B., see Harrison, J.P. 142, 191 McConville, G.T., see Aziz, R.A. 51, 52, 69, 111

McConville, G.T., see Matacotta, F.C. 75,77, I13 McCourt, F.R.W., see Aziz, R.A. 49, 51, 52, 75, 111 McCullen, T., see Hood, K. 152, 191 McCulloch, R.D., see Page, J.H. 144, 157, 159, 160, 193 McCullough, J.R., see Sinton, C.M. 302, 338 McTaggart, J.H. 101, 113 Meerschaut, A., see Monceau, P. 204, 268 Mendoza, E., see London, H. 140, 192 Meng, R.L., see Huang, C.Y. 11, 43 Meng, R.L., see Wu, M.K. 10, 44 Mercereau, J.E. 276, 337

352

AUTHOR I N D E X

Mercereau. J.E., see Jaklevic, R.C. 274, 336 Metropolis, N., see Kilpatrick. J.E. 61, 74, I12 Meyerovich. A.E. 47, 53.60, 71, 113 Meyerovich, A.E., see Bashkin, E.P. 71. 111 Michaels, J.N., see Faltens, T.A. 20, 42 Michaels, J.N., see Leary, K.J. 32, 44 Michels, A., see De Boer, J. 51, 61, 111 Michelson, P.F.,see McAshan, M.S. 329,337 Migulin, V.V., see Kuzrnin, L.S. 279, 337 Mihaly. G. 215, 267 Mihaly, L. 216, 267, 268 Mihaly, L., see Beyermann, W. 215, 266 Mihaly, L., see Brown, S.E. 216, 224. 239. 249, 256. 257, 266 Mihaly, L.. see Mihaly, G. 215, 267 Miller, J.E., see Guenault, A.M. 141, 191 Miller, J.M., see Thorne, R.E. 222, 226,238, 250, 251, 268 Miller, J.M., see Tucker. J.R. 235, 269 Miller 111, W.T. 302, 337 Mills, D.L. 152, 155, 192 Mills, D.L., see Maradudin, A.A. 178, 192 Mills, N.G. 169, 192 Mills, N.G., see Sherlock, R.A. 168, 169, 193 Minakova.V.E.,see Latyshev.Yu.1. 221,244, 26 7 Miura, Y., see Ishirnoto. H. 71. 98, 112, 158, 191 Mizutani. N. 187, 192 Modena, I. 301, 337 Modena, I., see Chapman, R.M. 301, 334 Moilanen, M., see Kalliomaki. K. 304, 336 Moisseyev, D.P.. see Gabovich, A.W. 8, 20, 42 Mok, E. 169, 192 Molchanov, V.N.,see Sirnonov, B.I. 32, 44 Mollenauer. L.F. 81, 113 Monceau, P. 198,204,215,220,221,227,234, 239, 240, 263, 268 Monceau, P., see Richard, J. 226, 264, 268 Monchick, L. 61, 113 Montonen, J. 303, 338 Montonen, J., see Katila, T. 303. 336 Mooij, J.E., see Bruines, J.J.P. 284, 333 Morii. Y. 158, 192 M o m s , D.E., see Bourne, L.C. 21, 42 M o m s , D.E., see Faltens, T.A. 20, 42 Morrison, H.F., see Wilt, M. 327, 339 Mota, A.C., see Jutzler, M. 187, 191

Mota, A.C., see Marek, D. 187, 192 Moze, 0.. see David, W.I.F. 32, 42 Mozurkewich, G. 218,222,223,227,234,260, 263, 264, 268 Mozurkewich, G., see Brown, S.E. 225, 239. 246, 250, 266 Mozurkewich, G., see Link, G.L. 225. 226, 26 7 Mueller, R.M., see Chocholacs, H. 158, 159. 162-164, 190 Mueller, R.M., see Owers-Bradley, J.R. 71, lJ3, 141, 193 Mueller, R.M., see Rogacki, K. 124,141-143, 193 Muhlfelder, B. 285, 286, 288. 338 Muirhead, C.M.. see Gough, C.E. 14, 43 Mujsce, A.M., see Batlogg, B. 21, 42 Miiller, K.A.. see Bednorz, J.G. 9, 42 Mullin, W.J., see Gully, W.J. 71, 98, 112 Munn, R.J., see Monchick, L. 61, 113 Murphy, D.W., see Batlogg, 8. 21, 42 Muta, T., see Enpuku, K. 287, 335 Mutikainen, R., see Knuutila, J. 283, 286, 288, 291, 320, 321, 337 Nacher, P.-J. 72, 84-88, 93-95, 97, 113 Nacher, P.-J., see Daniels, J.M. 78, 84, 85, 99, I 1 1 Nacher. P.-J., see Himbert, M. 85, 86. 112 Nacher, P.-J., see Laloe, F. 48, 83, 86, 112 Nacher, P.-J., see Leduc, M. 83. 85-87, 89, 98, 100, 104, 105, 107, 112 Nacher, P.-J., see Lefkvre-Seguin, V. 85, 89, 91, 93, 94, I12 Nacher, P.J., see Stringari, S. 110, I13 Nacher, P.-J., see Tastevin, G. 93,97-99, 109, 113 Nacher, P.-J., see T r h e c , G. 81, 82, 114 Nad', F.Ya., see Borodin, D.V. 211,215,263, 266 Nain, V.P.S., see Aziz, R.A. 51, 52, 69, 1 1 1 Naito, M., see Maeda, A. 226, 267 Naito. N., see Maeda, A. 226, 267 Nakagawa, H.,see Nakanishi, M. 320. 321, 338 Nakajirna, S. 152. 192 Nakanishi, M. 320, 321, 338 Nakayama, T. 118, 120, 121, 149, 151-156, 159, 161, 162, 164, 169, 173, 174, 178-181, 183, 184. 189, 192

AUTHOR INDEX Nakayama, T., see Nishiguchi, N. 129, 133, 134, 144, 147, 149, 152, 189, 193 Nakayama, T., see Saito, S. 149, 150, 152, 153, 193 Nakayama, T., see Yakubo, K. 159, 194 Narici, L., see Emt, S.N. 325, 335 Narici, L., see Romani, G.L. 297, 312, 338 Naurzakov, S.P., see Vvedensky, V.L. 294, 339 Neganov, B.S. 140, 193 Nemoto, I., see Brain, J.D. 304, 333 Nenonen, J., see Katila, T. 303, 336 Nenonen, J., see Montonen, J. 303, 338 Neuhaus, M., see Drung, D. 320, 321, 335 Nicolas, P. 294, 338 Nisenoff, M. 276, 338 Nishida, N., see Ishimoto, H. 71, 98, 112, 158, 191 Nishiguchi, N. 129, 133, 134, 144, 147, 149, 152, 189, 193 Nishiguchi, S., see Nakayama, T. 120, 121, 189, 192 Noakes, D.R., see Harshman, D.R. 34, 43 Norberg, R.E., see Lusznynski, K. 70, 71, I13 Northrop, G.A. 176, 193 Notarys, H.A., see Tesche, C.D. 285, 287, 3 15, 339 Novikov, L.N., see Laloc, F. 48, 83, 86, 112 Nozibres, P., see Castaing, B. 109, 1 1 1 Numata, T., see Ogushi, T. 11, 44 Obara, K., see Ogushi, T. 8, 44 Obennayer, D.E., see Lauter. H.J. 180, 192 Ogawa, S., see Ishimoto, H. 71.98, 112, 158, I91 Ogg, R.A. 13, 44 Ogushi, T. 8, 11, 44 Ohtsuka, T., see Mizutani, N. 187, 192 Okada. Y., see Williamson, S.J. 311,318,339 Ong, N.P. 217, 223, 224, 268 Ong, N.P., see Jing, T.W. 224, 267 Ong, N.P., see Monceau, P. 204, 268 Ong, N.P., see Venna, G. 224, 269 Ono, R.H., see Muhlfelder. B. 285, 286, 338 Opfer, J.E., see Luszczynski, K. 70, 71, 113 Opsal, J.L. 166, 193 Orbach, R. 144, 193 Orbach, R., see Alexander, S. 144, 159, 189 Orbach, R., see Demda, B. 159, 190

353

O’Reilly, A.H., see Kamaris, K. 32-34, 43 Osheroff, D.D. 125, 126, 128, 152-154, 157, 158, 162-164, 193 Osheroff, D.D., see Corruccini, L.R. 71, 111 Osiecki, J.H., see Gamble, F.R. 30, 36, 42 Ostman, P., see Ahola, H. 290, 333 Ott. H.R. 26, 44 Owers-Bradley, J., see Chocholacs, H. 158, 159, 162-164, 190

Owers-Bradley, J.R. 71, 113, 141, 193 Ozhogin, V.I., see Vvedensky, V.L. 294, 339 Ozono, Y.,see Ogushi, T. 11, 44’ Paalanen, M., see Ahonen, A.I. 124, 189 Page, G.J., see Wyatt, A.F.G. 169, 170, 194 Page, J.H. 144, 157, 159, 160, 193 Page, J.H., see Frisken, B. 142, 144, 190 Page, J.H., see Maliepaard, M.C. 144, 157, 159, 160, 192 Paik, H.J. 287, 329, 338 Papoular, M., see Castaing, B. 117. 190 Papoulas, H., see Richard, J. 226, 264, 268 Panlla, P. 219, 268 Parker, E.N., see Turner, M.S. 332, 339 Parker, W.H., see Simmonds, M.B. 279, 338 Paterno, G., see Barone. A. 274, 333 Paterno, G., see Kinder, H. 183, 191 Paulson, D., see Buchanan, D.S. 312, 334 PavloviE, M. 83, 104, 113 Peach, L. 117, 193 Peierls, R.E. 197, 201, 268 Pelizzone, M. 302, 311. 316, 338 Pelizzone, M., see Hari, R. 311, 335 Pelizzone, M., see Williamson, S.J. 311, 318, 339 Peltonen, R.S., see Kelha. V.O. 293, 315,336 Penttinen.A.A., see Kelha,V.O. 293,315,336 Pergrum. C.M., see Donaldson, G.B. 320,334 Perry, T. 124, 151-153, 193 Peshkov, V.P. 149, 193 Peterson, R.E. 167, 179, 193 Pethick, C.J. 26, 44 Phillips, N.E., see Avenel, 0. 124, 125, 127, 186, 190 Phillips, W.A. 181, 193 Pickett, G.R., see Guenault, A.M. 141, 191 Pietronero, L. 234, 236, 268 Pinard, M. 60, 83, 113 Pines, D., see Pethick. C.J. 26, 44 Pisharody, R., see Gamble, F.R. 30, 36, 42

3 54

AUTHOR INDEX

Pizzella, V., see Erne, S.N. 325, 335 Plaskett. T.S., see Collins, R.T. 34, 42 Pobell. F., see Chocholacs, H. 158, 159, 162164, 190 Pobell, F.. see Owers-Bradley, J.R. 71, 113, 141. 193 Pobell. F., see Rogacki, K. 124. 141-143, 193 Podney, W.N., see Czipott, P.V. 328, 334 Pohl, R.O., .see Klitsner, T. 169, 183. 191 Pohl, R.O., see Zeller, R.C. 103, 114 Pollack, G.L. 127, 193 Pollack, G.L., see Opsal, J.L. 166, 193 Polls, A., ree Stringan, S. 110, 113 Polunin, E.A., see Kuzmin, L.S. 279, 337 Pomeranchuk, f.Ya., see Achiezer. A.I. 26,41 Porter, C.D., see Kamaris. K. 32-34, 43 Portis, A.M., see Monceau. P. 204, 268 Potter, W.H. 149, 193 Poulder, D., see Renne, M.J. 248, 268 Poutanen, T., see Varpula, T. 294, 339 Prance, R.J., see Long, A. 279, 337 Proto. G., see Tesche, C.D. 285,287.3 15.339 Pukki, J.M., see Kelha, V.O. 293, 315. 336 Quader 40. 44 Quinnell, E.H., see Clement, J.R. 101, 1 1 1 Radebaugh, R. 124. 141, 162, 193 Rae, A.I.M., see Gough. C.E. 14. 43 Raider. S.I., see Foglietti, V. 292. 335 Raivio, M., see Katila, 7. 303, 3-16 Raksheev, S.N., see Mazin, 1.1. 20, 44 Ramadan, B., see Kos, J.F. 329. 337 Ranninger, J., see Alexandrov, A.S. 23. 41 Rantala, B.. see Ahola, H. 290, 233 Rapp, R.E.. see Wheatley, J.C. 141, 194 Rasmussen, F.B., see Halperin, W.P. 54, 112 Rawling, K.C. 166. 193 Reagor, D. 212, 213. -768 Reagor, D., see Sridhar, S. 212,213, 235, 268 Reid, W.H., .see Drazin, P.G. 106, I 1 1 Reinikainen, K., .see Ilmoniemi, R.J. 296, 305, 306. 310, 318, 3.76 Reinikainen, K., see Sams, M. 299, 300, 338 Renard, M.. see Monceau, P. 198, 215, 220, 221, 227. 239, 240, 263, 268 Renard. M., see Richard, J. 226, 264. 268 Renne. M.J. 248, 268 Reynolds, C.L. 126, 127. 167, 193 Ribary, U. 302. 338

Ricci, G.B., see Chapman, R.M. 301, 334 Ricci, G.B., see Modena. 1. 301, 337 Rice, M.J. 179, 19.1 Rice, M.J., see Leggett. A.J. 71, 112 Rice, M.J., see Toombs, G.A. 128. 135, 138, 140, 148, 194 Rice, T.M., see Lee, P.A. 205, 206, 223, 232, 253, 267 Richard, J. 226, 264, 268 Richard, J., see Monceau, P. 198, 215, 220, 221, 221, 239, 240, 263, 268 Richards, M.G., see Long, A. 279, 337 Richardson, R.C. 140. 141, 193 Richardson, R.C., see Corruccini, L.R. 71, 111

Richardson, R.C., see Halperin, W.P. 54, 112 Richardson, R.C., see Osheroff, D.D. 125, 126, 128, 152-154. 157, 158, 193 Rietman, E.A., see Batlogg, B. 20, 21, 42 Rietman, E.A., see Cava, R.J. 10, 19, 42 Risken, H., see Breymayer, H.-J. 232, 266 Riski, K., see Ikonen, E. 329, 336 Ritala, R.K., see Ketoja, J.A. 280, 281, 336 Ritchie, D.A. 132, 158, 159, 162-165, 193 Robaszkiewicz, S., see Alexandrov, A.S. 23, 41 Robbins, M. 233, 235, 268 Robbins, M.O., see Bhattacharya. S. 226,264, 266 Roberts, T.R. 75, 113 Robertson, R.J. 160, 161, 193 Rogacki, K. 124, 141-143. 193 Roinel, Y., see Jacquinot, J.F. 187, 191 Romani. G.L. 297, 299, 305, 307. 308, 312, 318. 324. 338 Romani, G.L., see Chapman, R.M. 299. 301, 334 Romani, G.L., see E d , S.N. 320, 322, 325, 335 Romani, G.L.. see Modena, I . 301, 337 Rouxel, J . , see Monceau. P. 204, 268 Rupp, L.W., see Batlogg, B. 20, 21, 42 Rusakov, A.P., see Brandt, N.B. 8, 42 Rusakov, A.P., see Chu, C.W. 8, 42 Rusby, R.L., see Durieux, M. 101, I l l Rusinov, A.I.. see Kopaev, Yu.V. 22. 43 Rutherford, A.R. 147, 148, 159, 193 Ruvalds, J., see Tua, P.F. 234, 252, 268 Ryhanen, T. 273, 276, 281, 284, 338 Ryhanen, T., see Ketoja, J.A. 281, 327

AUTHOR INDEX Ryhanen, T., see Maniewski, R. 294,325,337 Ryhanen, T., see Seppa, H. 286,288,296,338 Saam, W.F., see Edwards, D.O. 85, 212 Sabetta, F., see Fenici, R.R. 303, 335 Sabisky, E.S. 169, 193 Saint-Lager, M.C., see Monceau, P. 215,221, 263, 268 Saito, S. 149, 150, 152, 153, 187, 293 Sakai, H., see Hayashi, M. 142, 144, 191 Salamon, see Quader 40, 44 Salmi, J., see Knuutila, J. 283, 286, 288, 291, 320, 321, 337 Salomaa, M.M., see Kajanto, M.J. 232, 267 Saltzman, B. 106, 113 Salustri, C., see Romani. G.L. 312, 318, 338 Salva, H., see Monceau, P. 221, 268 Sams, M. 299, 300, 338 Sandmann, W., see Weber, J. 168-170, 194 Sandstrom, R.L., see Collins, R.T. 34, 42 Sandstrom, R.L., see Dinger, T.R. 17,32, 42 Sandstrom, R.L., see Foglietti, V. 292, 335 Santikaya, Ya.S., see Latyshev, Yu.1. 221,267 Sarvas, J. 297, 308, 325, 338 Sarvas, J., see Hamalainen, M.S. 297, 335 Sarvas, J., see Hari, R. 307, 335 Sarvas, J., see Kaukoranta, E. 308, 336 Sasaki, W . , see Kobayashi, S. 149, 192 Sauls, J.A., see Perry, T. 124, 151-153, 193 Saunders, J., see Ritchie, D.A. 132, 158, 159, 162-165, 193 Saundry, P.D., see Ytterboe, S.N. 187, 194 Savitskaya, Ya.S., see Borodin, D.V. 21 1,215, 263, 266 Savrasov, S.Yu., see Mazin, 1.1. 20, 44 Sawada, Y., see Mamiya, T. 158, 192 Sawatzki, G., see Ludwig, W. 319, 337 Scaramuzzi, F., see Kinder, H. 183, 191 Schafroth, M.R. 14, 44 Schearer, L.D. 78, 90, 91, 98, 113 Schearer, L.D., see Colegrove, F.D. 78, 2 2 1 Scheer, H., see Erne, S.N. 294, 335 Schiff, L.I. 61, 223 Schlenker, C., see Beauchene, P. 219, 266 Schlesinger, Z., see Collins, R.T. 34, 42 Schneemeyer, L.F., see Fleming, R.M. 219, 240, 246, 257, 258, 267 Scholz, H.N., see Masuhara, N. 71, 113 Schomer, D.L., see Kennedy, J.G. 310, 336 Schrieffer, J.R. 25, 26, 44

355

Schrieffer, J.R., see Bardeen, J. 12, 15, 42 Schrieffer. J.R., see Klemm, R.A. 233, 267 Schrijner, P., see de Waal, V.J. 284, 334 Schubert, H. 183, 193 Schuller, I.K., see David, W.I.F. 32, 42 Schulz, H.J., see Jerome, D. 36, 43 Schumacher, G., see Frossati, G. 124, 141, 190 Schuss, 2.. see Ben-Jacob, E. 281, 333 Schweizer, R.J., see Heim, U. 169, 191 Scott, M.J., see Rutherford, A.R. 147, 148, 159, 193 Segransan, P. 221, 268 Segre, C.U., see David, W.I.F. 32, 42 Seidel, G.M., see Cameron, J.A. 75, 1 1 1 Semenov, A.D., see Gershenson, E.M. 34.42 Semenov, V.M., see Brandt, N.B. 8, 42 Senba, M., see Harshman, D.R. 34, 43 Seppa, H. 279, 286, 288, 296, 338 Seppa, H., see Ketoja, J.A. 281, 337 Seppa, H., see Knuutila, J. 283,286,288,291, 321, 337 Seppa, H., see Ryhanen, T. 273, 276, 281, 284, 338 Seppanen, M. 318, 338 Seppanen, M., see Maniewski, R. 294, 325, 33 7 Sera, M. 40, 44 Shabanov, S.Yu., see Vvedensky, V.L. 294, 339 Sham, L.J. 29, 37, 44 Shapiro, S. 239, 250, 268 Shapligin, I.S. 10, 44 Sheard, F.W. 179, 193 Sheard, F.W., see Toombs, G.A. 128, 135, 138, 140, 148, 194 Shearer, L.D., see Daniels, J.M. 78, 84, 85, 99, 1 1 1 Shen, T.J. 179, 193 Sherlock, R.A. 168, 169, 193 Sherlock, R.A., see Mills, N.G. 169, 192 Sherlock, R.A., see Wyatt, A.F.G. 169, 194 Sherman, R.H., see Roberts, T.R. 75, 113 Sherrill, D.S., see Masuhara, N. 71, 113 Sherwin, M. 239, 260, 268 Sherwin, M.S., see Bourne, L.C. 265, 266 Shigi, T., see Fujii, Y. 187, 191 Shinoki, F., see Nakanishi, M. 320, 321, 338 Shirae, K. 316, 318, 338 Shirae, K . , see Furukawa, H. 316, 335

3 56

AUTHOR INDEX

Shiren, N.S. 179, 193 Shlyapnikov, G.V., see Kagan, Yu. 117, 191 Shoji, A., see Nakanishi, M. 320, 321, 338 Short, K.T., see Batlogg. B. 20, 21, 42 Siegwanh, J.D., see Radebaugh, R. 124, 141, 162. 193 Sigmund, E.. see Haug, R. 180, 191 Silin, V.P. 60. 71, 113 Siltanen, P., see Katila, T. 303, 336 Siltanen, P., see Montonen. J . 303, 338 Silver, A.H. 274, 338 Silver, A.H., see Jaklevic, R.C. 274, 336 Silvera, I.F. 47, 113 Simmonds, M.B. 279, 338 Simmonds. M.B., see Kamper, R.A. 279, 336 Simonov. B.I. 32, 44 Simonov, N.A., see Kuzmin, L.S. 279, 337 Sinton, C.M. 302, 338 Sitnikova, V.I., see Zinov'eva, K.N. 167, 194 Slack, G.A., see McTaggan, J.H. 101, 113 Sleight, A.W. 8. 44 Slichter, C.P. 83. 94, 113 Smith, E N . , see Richardson, R.C. 140, 193 Smith, F.J., see Monchick, L. 61, 113 Sneddon, L. 233, 236, 253, 264, 265, 268 Sniguiriev, D.V.. see Danilov, V.V. 291, 334 Sobyanin, A.A., see Bulaevskii, L.M. 40, 42 Sobyanin, A.A., see Bulaevskii, L.N. 24, 25, 35, 42 Sobyanin, A.A., see Ginzburg, V.L. 35, 43 Soderholm, L., see David, W.I.F. 32, 42 Sokoloff, J.B. 234, 268 Soldatov, E.S., see Danilov, V.V. 291, 334 Somalwar, S., see Incandela, J. 331, 336 Soper, A.K., see David, W.I.F. 32, 42 Sprenger, W.O., see Andres, K. 124, 134,147. 190

Sridhar. S. 212. 213, 235, 268 Sridhar, S., see Reagor, D. 212, 213, 268 Stacy. A.. see Bourne, L.C. 21. 42 Stacy, A.M., see Faltens, T.A. 20, 42 Stacy. A.M., see Leary, K.J. 32, 44 Stager, C.V., see Kamaras, K. 32-34, 43 Stark, M., see Wilt, M. 327, 339 Stein, D.L., see Perry, T. 124. 151-153, 193 Stepanova, M.G.. see Vendik, O.G. 279, 339 Steur, P.P.M., see Matacotta, F.C. 75.77, I13 Stokes, J.P., see Bhattacharya, S. 225, 226, 239, 260. 261, 263. 264, 266

Slrachen, C., see Lennard-Jones, T.E. 180, I 92 Strassler, S., see Pietronero, L. 234, 236, 268 Stringari, S. 110, I13 Stroink, G . 294, 339 Stroink, G., see MacAulay, C.E. 303, 337 Stroink, G., see Weinberg, H . 299, 339 Stubbs, R.J., see Maliepaard, M.C. 144, 157, 159, 160, 192 Sueoka, K., see Enpuku, K. 285-287, 335 Sunshine, S., see Batlogg, B. 21, 42 Suresha, G.N., see Ogushi, T. 1 1 , 44 Sutherling. W., see Barth, D.S. 301, 333 Sutherling, W.W. 313, 339 Sutton, S., see Cough, C.E. 14, 43 Suzuki, H., see Mizutani, N. 187, 192 Swanenburg, T.J.B. 168, 193 Swithenby. S., see Grimes, D. 304, 33.5 Sydoriak, S.G., see Roberts. T.R. 75, 113 Symonds, A.J., see Brewer, D.F. 180, 190 Synder, N.S. 166, 193 Syskais, E.G., see Rogacki, K. 124, 141-143, I 93 Taber, M., see Cabrera, B. 299, 330, 334 Taber, M.A., see Huber, M.E. 331, 336 Taber, R.C.. see McAshan, M.S. 329, 337 Taborek, P. 168, 169, 175, 176, 193, 194 Taconis. K.W., see Beenakker, J.J.M. 165, I90 Taconis, K.W.. see Gorter, C.T. 117. 191 Takada, S.. see Nakanishi, M. 320, 321, 338 Takahashi, T. 40, 44 Takahashi, T., see Kobayashi, S. 149, 192 Takano, Y., see Ishimoto, H. 71,98, 112, 158, I91

Takayama, H., see Matsukawa, H. 234, 236, 26 7 Tamura, I., see Hayashi, M. 142, 144, 191 Tanaka, S.. see Maeda, A. 216, 226, 267 Tanner, D.B.. see Kamaras, K. 32-34, 43 Tastevin, G. 93, 97-99, 109, 113 Tastevin, G., see Laloe, F. 48, 83, 86, I12 Tastevin, G., see Leduc, M. 98, 100, 104, 105, 107, 112 Tastevin, G., see Nacher, P.-J. 72, 86, 93-95, 97, 113 Taylor, W.L., see Aziz, R.A. 51, 52, 69, 111 Tazaki, T., see Ishimoto, H. 71, 98, 112, 158, 191

AUTHOR INDEX ter Brake, H.J.M. 324, 339 Teranishi, N. 234, 268 Tesche, C., see Knuutila, J. 285, 286, 308, 313, 314, 318, 337 Tesche,C.D. 280,281,284,285,287,315,331, 339 Tesche, C.D., see Bermon, S. 331, 333 Tesche, C.D., see Chi, C.C. 331, 334 Tessema, G.X., see MihLly, L. 216, 268 Teszner, D., see Nicolas, P. 294, 338 Thkry, J., see Schearer, L.D. 78, 98, 113 Thiene, P., see Zimmerman, J.E. 276, 339 Thomas, N., see Cough, C.E. 14, 43 Thomasson, J.W. 96, 114 Thompson, K.,see Hall, H.E. 140, 191 Thomson, A.L., see Brewer, D.F. 180, 190 Thorne, R.E. 219,222,226,234,237,238,244, 248, 250-253, 268 Thorne, R.E., see Lyding, J.W. 224,225. 267 Thorne, R.E., see Tucker, J.R. 235, 269 Thoulouze, D., see Frossati, G. 124, 141, 190 Thoulouze, D., see Peach, L. 117, 193 Threlfall, D.C. 106, 107, 114 Tiky, C., see Lauter, H.J. 180, 192 Timsit, R.S. 81, 89, 91, 93. 114 Timsit, R.S., see Daniels, J.M. 81, 111 Timusk, T., see Kamaris, K. 32-34, 43 Ting, T.W., see Ong, N.P. 224, 268 Tinkham, M. 274, 281, 339 Toombs, G.A. 128, 135, 138, 140, 148, 179, 194 Toombs, G.A., see Rice, M.J. 179, 193 Toombs, G.A., see Sheard, F.W. 179, 193 Torng, C.J., see Wu, M.K. 10, 44 Tracht, A.E., see Farrell, D.E. 304, 319, 335 TrCnec, G. 81, 82, 114 TrCnec, G.. see Leduc, M. 81, 112 Trtnec, G., see Nacher, P.-J. 84, 85, 87, 88, 113

Tripp, J.H., see Farrell, D.E. 304, 319, 335 Trontelj, Z., see ErnC, S.N. 294, 335 Trumpp, H.J. 168, 194 Tsuei, C.C., see Bermon, S . 331, 333 Tsuei, C.C., see Chi, C.C. 331, 334 Tsuei, C.C., see Tesche, C.D. 331, 339 Tua, P.F. 234, 252, 268, 269 Tucker, J.R. 235, 238, 269 Tucker, J.R., see Thorne, R.E. 219, 222,226, 234, 237, 238, 244, 248, 250-253, 268 Tuomisto, T., see Nicolas, P. 294, 338

357

Tuomisto, T., see Seppanen, M. 318, 338 Tuomola, M., see Malmivuo, J. 294, 337 Turner, C.W., see van Duzer, T. 274, 339 Turner, M.S. 332, 339

Uchinokura, K., see Maeda, A. 216, 267 Ulfman, J.A., see ter Brake, H.J.M. 324, 339 Uspenskii, Yu.A., see Mazin, 1.1. 20, 44

Valberg, P.A., see Brain, J.D. 304, 333 van der Linde, J., see Pinard, M. 83, 113 van der Sluijs, J.C.A., see Rawling, K.C. 166, I93 van Dover, R.B., see Batlogg, B. 21, 42 van Dover, R.B., see Cava, R.J. 10, 19, 42 van Duzer, T. 274, 339 van Kranendonk, J., see De Boer, J. 61, 111 van Nieuwenhuyzen, G.J. 320. 339 van Soest, G., see Beenakker, J.J.M. 165, 190 Varma, C.M., see Anderson, P.W. 181, 189 Varpula, T. 294. 339 Varpula, T., see Seppanen, M. 318, 338 Veinstein, B.K., see Simonov, B.I. 32, 44 Vendik, O.G. 279,339 Veno, S., see Atsumi, K. 299, 333 Verma, G. 224, 269 Verma, G., see Ong, N.P. 217, 223, 268 Vernon Jr, F.L., see Kanter, H. 279, 336 Vetsleseter, A., see Avenel, 0.124, 125, 127, 186, 190 Veuro, M.C., see Ahonen, A.I. 124, 125, 189 Vezzoli, G.C. 8, 44 Vilches, A.E.. see Anderson, A.C. 126, 167, I89 Vilches, O.E., see Wheatley, J.C. 124, 194 Vilkman, V., see Knuutila, J. 308, 313, 314, 318, 337 Villain, J. 156, 194 Vivien, D., see Schearer, L.D. 78, 98, 113 Volkov, B.A., see Ginzburg, V.L. 9,43 Vollmer, H.D., see Breymayer, H.-J. 232,266 Vrba, J. 305, 307, 339 VujiwciC, G.M. 37, 44 Vuono, M. 185, 194 Vuono, M., see Avenel, 0. 124, 125,127,186, I90 Vuorio, M., see Leggett, A.J. 151, 152, 192 Vvedensky, V.L. 294, 299, 31 1, 339

358

AUTHOR INDEX

Waldmann, L. 60. 114 Waldram, J.R. 248, 269 Walraven, J.T.M., see Silvera, I.F. 47, 113 Walters, G.K., see Colegrove, F.D. 78, 1 1 1 Walters, G.K.. see McAdams, H.H. 86, 113 Walters, G.K.. see Schearer. L.D. 90.91. 113 Wang, K.Y., see Paik, H.J. 329, 338 Wang, R.H.. see Tesche, C.D. 285, 287. 315, 339 Wang, Y.Q., see Wu, M.K. 10, 44 Wang, Z.Z., see Monceau, P. 221. 268 Warnick, A., see Forgacs, R.L. 289, 335 Webb, R.A.. see Giffard, R.P. 276, 277, 288, 335 Webb, W.W., see Kurkijirvi, J. 277, 337 Weber, J. 168-170, 194 Weber, J.C., see Marek, D. 187, 192 Weber, W. 19, 44 Weber. W., see Batlogg, B. 20, 42 Webman, I . , see Grest, G.S. 159, 191 Weger. M. 217, 222. 230, 231, 269 Weinberg, H. 299, '339 Weinberg, H.. see Ribary, U. 302, 338 Weinberg, H., see Vrba, J. 305, 307, 339 Weiss. K., see Haug, H. 167, 179, 191 Weiss, K., see Haug, R. 180, 191 Weiss, K., see Kinder, H. 179, 191 Wellstood, F. 291, 327, 339 Welte, M., see Heim, U. 169, 1Y1 Welte, M., see Koblinger. 0. 169, 171, 192 Werntz Jr. J.H., see Grimsrud. D.T. 75, / I ? Westfall, R. 5 , 44 Wheatley. J.C. 124, 133, 141, 149, 194 Wheatley, J.C., see Abel, W.R. 152, 189 Wheatley. J.C., see Anderson, A.C. 126, 127, 167, 189 Wheatley, J.C., see Giffard, R.P. 276, 277, 288, 335 White, A,, see Batlogg, B. 21, 42 White, A.E.. see Batlogg, B. 20, 42 White, D. 165, 194 White, R.M. 16. 35. 44 Wiechert, H., see Lauter, H.J. 180, 192 Wiesenfeld, L., see Tastevin, G. 109, 113 Wilks, J., see Challis, L.J. 165, 166, 190 Wilks, J., see Dransfeld, K. 165, 190 Wilks. J., see Fairbank, H.A. 165, 190 Williamson, S.J. 297. 311, 318, 339 Williamson, S.J., see Atsumi, K. 299, 333 Williamson, S.J., see Buchanan, D.S. 312,334

Williamson, S.J., see Curtis, S. 299, 334 Williamson, S.J., see Ilmoniemi, R.J. 299, 312, 336 Williamson, S.J., see Kaufman, L. 297, 311, 336 Williamson, S.J., see Pelizzone, M. 311, 338 Williamson, S.J., see Romani, G.L. 299, 305, 307, 324. 338 Wilt, M. 327, 339 Wolfe, J.P., see Nonhrop, G.A. 176, 193 Wolter, J., see Horstman, R.E. 168, 191 Wolter. J.. see Swanenburg, T.J.B. 168, 193 Wong. C.C.K.. see Aziz, R.A. 49, 51, 52, 75, 111

Wong, Zheng-yu 17, 18, 44 Wonneberger. W. 232, 269 Wonneberger, W., see Breymayer, H.-J. 232, 266 Worthington, T.K. 32, 44 Wonhington, T.K., see Dinger, T.R. 17, 32, 42 Wu, Hang-sheng, see Wong, Zheng-yu 17, 18, 44 Wu, M.K. 10, 44 Wu, R.H., see Waldram, J.R. 248, 269 Wu, Wei-Yu 215, 238, 269 Wyatt. A.F.G. 127, 169, 170, 194 Wyatt, A.F.G., see Mills, N.G. 169, 192 Wyatt, A.F.G., see Sherlock, R.A. 168, 169, 1 93 Yakubo, K. 159, 194 Yakubo, K., see Nakayama, T. 161, 192 Yeh, W.J. 249, 250, 269 Yoffe, A.D., see Friend, R.H. 30, 42 Yogi, T., see Tesche, C.D. 285. 287, 315, 3.39 Yoshida, K.,see Enpuku, K. 285-287, 335 Ytterboe, S.N. 187, 194 Yu, Kin-Wah, see D e m d a , B. 159, 190 Zaitsev-Zotov, S.V., see Borodin, D.V. 21 1, 215, 263, 266 Zanzucchi, P., see Farrell, D.E. 305, 335 Zaremba, E., see Hood, K. 152, 191 Zawadowski, A., see Griiner, G . 198,227,267 Zawadowski, Z., see Tua, P.F. 234, 252, 269 Zawadzki, P., see Deptuck, D. 144, 157, 159, I90 Zeller, R.C. 103, 114

AUTHOR INDEX Zettl, A. 215,216,224,226,239,240,243-246, 260, 263-265, 269 Zettl, A., see Bardeen, J. 219-221. 266 Zettl, A., see Bourne, L.C. 21, 42, 265, 266 Zettl, A., see Faltens, T.A. 20, 42 Zettl, A., see Hall, R.P. 247, 264, 267 Zettl, A,, see Parilla, P. 219, 268 Zettl, A,, see Sherwin, M . 239, 260, 268 Zhang, K., see David, W.I.F. 32, 42

359

Zhang, K., see Kwok, W.K. 19, 43 Zhanov, Ya.A., see Latyshev, Yu.1. 244, 267 Zimmerman, J.E. 276, 288, 294, 339 Zimmerman, J.E., see Silver, A.H. 274, 338 Zinov’eva, K.N. 127, 167, 194 Zubarev, D.N. 136, 194 zur Loye, H.C., see Leary, K.J. 32, 44 Zvezdin, A.K. 24, 25, 44

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absorption processes 91-93 ac Josephson effect 239 accelerometers 329 acoustic mismatch theory 117, 120, 167 acoustic mode 205 active shielding 293 adsorption processes 91-93 alpha rhythm 298, 310, 311, 312, 317 amplifier noise 289 arrhythmias 303 asymmetric gradiometer 295 auditory evoked fields 299 auditory evoked response 316 auditory stimulus 299 autonomous dc SQUID 280, 286 Aziz 69, 75 Aziz interaction 51 Aziz potential 105

charge density wave (CDW) 197 charged Bose gas 14 circle map 248, 249 circular polarkation 83, 84 classical particle model 228 classical transport theory 46 clinical applications of MEG 300 coherence 200 coherence enhancement 200, 259 collision cross section 49, 55, 64 collision integrals 64,68, 73 colour-centre laser 82, 83, 85, 95 compensation coil 295, 306 computer simulations of dc SQUlDs 284 conductivity 70 confidence regions 307 convection 46, 106, 107 cooled preamplifier 280, 290 cordial resistance R 226 correlated noise 325 coupling 285, 288, 306 critical current 281 critical temperature in the BCS theory I5 cross section 72. 81 cryocooler 312 Cryogenic Consultants, Ltd 328 cryoSQUlDs 312 CTF 328 current carrying state 200 current-current correlations 200 current dipole 297-300, 307 current oscillations 198, 200, 281, 287 current-voltage characteristics 277,278,281 282 currents of neurons 297

balance 296, 306 balancing 296, 312 baseline 295, 305 beating resonance 287 P c 286 P L 286 P L 275 bias currents 280 binary collisions 54, 55, 63, 72, 81, 90 biomagnetic technologies 31 1 biomagnetism 292 bipolaron model of superconductivity 23 blue bronze b.,MoO, 204 Boltzmann equation 54, 55, 59. 60 broad band noise 200 BTi 328 bulk relaxation 85

damping 280, 288 damping constant 199 damping resistance 287 dc currents 304 dc SQUID 280, 291, 309

calibration 322 capacitive shunting 287 characteristic frequency 279, 283 characteristic lengths 46 361

362 de Broglie 52 de Rroglie wavelength 50 definition 48 demodulator circuit 291 density of states 144, 148 dewar noise 315 diazepam 302 dielectric 215 differential conductivity d l / d V 231 diffraction effect 53 diffuse signal 168, 176, 179 dilution refrigerator 124. 140 dipole 297, 299 dipole-dipole coupling 54 displacement sensors 329 Dornier Systems GmbH 319 double-D gradiometer 295. 310 double-loop dc SQUID 287 dye laser 81-83 dynamic impedance 290 dynamic resistance 283 dynamics of the CDW 205 eddy-current shield 294 EEG 296. 297, 299, 301 effective mass 199 electroencephalography (EEG) 296 electronic balancing 296, 3 1 I , 3 18 electronic circuit analogs 230 electronic polarisation 78 energy resolution 283-285. 287, 288, 306 epilepsy 301 equation of state 46 equilibrium polarisation 53 equivalent flux noise 277 evoked fields 299 excess noise 274, 279. 281, 284 exciton mechanism 21 exciton mechanism or HTS 26 external magnetic disturbances 292 l/f noise 289, 291, 292 Fabry-Perot etalons 82 Faraday cell 82 faVOUrdbk pressures 88 feedback loop 289 Fermi liquid theory 118, 128 ferromagnetic shield 293 field noise 306 first-order gradiometer 295, 310. 313

SUBJECT INDEX fluctuation suppression 262 flux feedback 324

flux-locked-loop 283 flux modulation 290. 292 flux-modulation 289 flux noise 274, 284 flux quantum 274 flux-to-voltage conversion efficiency 279,280 flux transformer 284, 306 flux transition 276 flux-voltage characteristics 277, 281 focal epilepsy 301 forward problem 298 fractional-turn SQUID 288 fracton 144, 148, 159 frequency dependent conductivity 198 gas of 'He 63 Gaussian noise 225 Gemini 312 geomagnetism 326 Gifford-McMahon cooler 312 gradiometer 295. 305. 306 gradiometer balance 296 gravitational wave antennae 330 grid spacing 308

harmonic mixing 199 harmonic mode locking 239 'He gas 59, 89.90, 94, 99, 100. 108, 109 'He-4He mixture 158. 162 hemochromatosis 304 high-frequency phonon 127, 168, 186 high-frequency rf SQUID 279 high-T, materials 288 high-temperature superconductivity (HTS)3 homogeneous head approximation 297 hydrogen isotopes 47 hyperfine structure 79-81, 84 hysteresis 215 hysteretic rf SQUlDs 280 hysteretic SQUID 275 IBM SQUlDs 315 identical spin rotation 72 identical spin rotation effect 63. 64, 65, 71 "identical spin rotation" effects 62 impedance matching 291 impurity pinning 21 1 incommensurate 198

SUBJECT INDEX inductance matching 306 inductive shunting 287 inductive SQUID 275 input current noise 291 integrated gradiometer 320 integrated magnetometer 321, 327 integrated sensors 319 integrity of sensory pathways 302 interaction potential 46, 49, 51, 56, 75 interatomic potential 52 interference experiments 200 interference “features” 247 interference filter 83 interhemispheric correlation 312 intermediate coupling transformer 288 intermediate transformer 322 intrinsic energy sensitivity 277 inverse problem 298 iron in the liver 304 isofield contour map 298 Josephson junction 232. 274 Joule-Thomson refrigerator 312 Kapitza conductance 118, 133, 137, 146 Kapitza resistance - above 1K 126, 165 -between ’He gas and a solid 137, 162 -between ‘ H e 4 H e mixture and sintered powder 158, 162, 164 -between liquid ’He and Ag particles 134, 152 - between sintered powder and liquid ’He 140 - magnetic field dependence 125, 153 kinetic equation 128 Knight shift 150, 152 krypton ion laser 81-83

A12 resonance 286 A/2 transmission line 286 Landau parameter 123, 129, 133, 139 late fields 303 lattice distortion 201 layered compounds 28 layered materials 39 Lazarus theorem 102 LC-resonances 287 Lennard-Jones interaction 51, 66 Lennard-Jones potential 105

363

linear chain compounds 197 liquid-gas equilibrium 46, 109 LNA laser 78, 98 local pairs 23 locating accuracy 322 lock-in detection 318 lock-in electronics 289 low-frequency noise 289 Lyot filter 82 magnetic coupling 149, 152, 155, 163 magnetic dipole coupling 90 magnetic disturbances 293 magnetic field isocontour maps 300 magnetic flux quantum 274 magnetic monopoles 330 magnetic shielding factor 293 magnetic susceptibility 327 magnetically shielded room 293, 310, 314 magnetocardiography 302 magnetoencephalography (MEG) 296 magnetometer 295 magnetopneumography 304 magnetotelluric studies 327 magnetotellurics 327 matching-transformer 288 Maxwell-Boltzmann distribution 55 MCG 302,309 MEG 299, 301, 313 metal-oxide ceramics 8 metastable atoms 78, 80 micropotentials 303 microwave SQUID 279 mode locking 200, 221,253 modulation depth 283 modulation frequency 289, 291, 292 monopole detectors 329 Mossbauer spectroscopy 329 multi-compartment model 302 multi-loop dc SQUIDS 288 multichannel neuromagnetometers 304 multiple solutions 281 multiplexing 318 mutual inductances 323 narrow band noise 217 Navier-Stokes equations 55. 56 NbSe, 204 negative differential resistances 215 “neon” superconductors 19

364

SUBJECT INDEX

neuromagnetism 296 NMR 78, 83, 95-98, 100, 103-105 noise temperature 278. 283, 292 nonlinear 198 nonlinear feedback circuits 232 “nonreproducible” superconductivity-type anomalies 9 normal liquid ’He 123, 132 normalized inductance 275 normalized loop inductance 280 nuclear magnetic moment of ‘He 54 nuclear polarisation 71,78,81,83,84,87,96, 98 Nusselt number 106, 107 off-diagonal gradiorneter 295, 309. 321 optical mode 205 optical polarisation 78, 87 optical polarisation of ’He nuclei 46 optical pumping 78, 79, 87, 88, 94, 98 optimization of magnetometers 305 onho-helium 79, 80 para-helium 78, 79 parasitic capacitance 280, 285, 287, 288, 321 parasitic resonances 288 Pauli principle 48, 50, 54, 69 Peierls transition temperature 197 peleomagnetism 327 percolation 160 phase-phase correlation length 21 I, 223 phase-sensitive detector 290 phase shift 49, 61, 74 phase-shift 69 phasons 206 phonon mechanism 34 PI controller 289, 290, 292 pickup coil 295, 306 pinning 215 pinning frequency 198 pinning potential 21 1 planar gradiometer 295 planar gradiometer arrays 320 planar gradiometers 331 polarisation 48 polarisation methods 46 position indication 322 preamplifier 278 preamplifier noise 278, 279, 283 primary current 297

probe-positioning indicator 312, 325 product k2QT 279 PSD 290 pulse drives 255 pulse duration memory effect 259 pump frequency 276, 279 quantum mechanical transport theory 46 quantum noise limit 288 quarter-wave plate 83 quasiparticle 128, 135, 147 Rayleigh-Binard convection 106 Rayleigh mode 172, 177. 184 Rayleigh number 106, 107 realistically shaped head model 297 rectification 199 relaxation oscillator 231 relaxation processes 46, 83, 89 relaxation time 8 5 , 88-90. 92-94, 96, 109 remanent 303 remanent magnetization 327 reproducible “neon” superconductivity 9 resistively-shunted-junction (RSJ) 274 resonance 280, 285, 286 resonant transformer 29 1 rffields 294 rf interference 294 rf shielding 294, 296 rf shunt 296 rf SQUID 274, 290 RSJ model 274 Schafroth model 24 Schafroth model (with localized pairs) 15 second-order gradiometer 296, 311, 312, 313 second virial coefficient 74-76, 77 shaking 293 Shapiro steps 241 shielding factor 293 shunt resistance 283 signal bandwidth 279 signal coil 285, 295, 306 signal-coil resonances 3 15 signal coupling 285. 288 single-particle excitation 128, 135 sintered powder 142, 160 slew rate 289-291, 309, 327 small particle 120, 124 spectral response 200

SUBJECT INDEX spherical head model 297 spheroidal mode 133, 188 spin diffusion 46. 62, 70, 71 spin magnetism 72 spin polarisation 94 spin-polarised 'He gas 47 spin-polarised 'He in liquid form I 0 9 spin rotation effects 46,93, 95 spin waves 46, 64,71, 93,95, 97, 98 spiral transmission line 286 staircase pattern 277, 290 staircase structure 248 Stewart-McCumber parameter 280 stray capacitance 285 subharmonic mode locking 246 subject noise 325 Superconducting Quantum Interference Devices (SQUIDS) 273 superdiamagnetism 9 surface 215 susceptometer 319 switching 215 symmetrical gradiometer 295 symmetry effect 53 synchronization 200, 247 tank circuit 274, 276, 290 temporal fluctuations 261 thalassemia 304 the tunneling model 237 thermal conductivity 46, 54, 56, 58, 62, 66, 69, 70, 98, 102, 103, 105

365

thermal noise 277, 281, 283, 286 thermal wavelength 52 threshold field ET 198 toroidal mode 133, 188 transfer function 283 transient oscillations 220 transmission coefficient 119, 165 transmission-line resonances 287, 288 transport properties 46 transport properties of dilute gases 54 transport properties of the gas 54 transport theory 56. 59 trazolam 302 triangular pattern 276 "true" or "genuine" superconductivity 10 tuned transformer 291 two-fluid description 215 two-level tunneling states 181. 183 Vacuumschmelze 294, 311, 312 vector magnetometers 316, 3 18 viscosity 46, 54, 56, 58. 62, 66, 67, 70 voltage plateaus 286, 288 voltage-to-flux transfer ratio 285 vplume currents 297 VT/,, characteristics 277 wall relaxation 85 Wol ff - Parkinson- White syndrome 303 zero sound 123, 128, 131, 132

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