Progress in Low Temperature Physics, Volume 5

PROGRESS IN LOW T E M P E R A T U R E PHYSICS V

CONTENTS O F VOLUMES I - I V

VOLUME I

c. J.

The two fluid model for superconductors and helium 11 (16 pages)

GORTER,

R. P. FEYNMAN,

I. R. PELLAM, A.

Application of quantum mechanics to liquid helium (37 pages)

Rayleigh disks in liquid helium I1 (10 pages)

c. HOLLIS HALLET, Oscillating disks and rotating cylinders in liquid helium 11 (14 pages)

E. F. HAMMEL,

The low temperature properties of helium three (30 pages) and K. w. TACONIS, Liquid mixtures of helium three and four (30 pages)

J. I. M. BEENAKKER

B. SERIN,

c. F.

The magnetic threshold curve of superconductors (13 pages)

SQUIRE,

The effect of pressure and of stress on superconductivity (8 pages)

T. E. FABER and A. B. PIPPARD,

Heat conduction in superconductors (1 8 pages)

K. MENDELSSOHN, J. G : DAUNT, A.

H. COOKE,

The electronic specific heats in metals (22 pages) Paramagnetic crystals in use for low temperature research (21 pages)

N. I. POULIS and

D. DE KLERK

Kinetics of the phase transition in superconductors (25 pages)

c. I. GORTER, Antiferromagnetic crystals (28 pages)

and M.

J. STEENLAND, Adiabatic

demagnetization (63 pages)

L. N&L,

Theoretical remarks on ferromagnetism at low temperatures (8 pages)

L. WEIL,

Experimental research on ferromagnetism at very low temperatures (11 pages)

A. VAN ITTERBEEK,

J. DE BOER,

Velocity and attenuation of sound at low temperatures (26 pages)

Transport properties of gaseous helium at low temperatures (26 pages)

VOLUME I1 Quantum effects and exchange effects on the thermodynamic properties of liquid helium (58 pages)

J. DE BOER,

H.

c. KRAMERS, Liquid helium below 1 OK (24 pages) and D. H. N. WANSINK, Transport phenomena of liquid helium 11 in slits and capillaries (22 pages)

P. WINKEL

K. R. ATKINS,

Helium films (33 pages)

B. T. MATTHIAS,

Superconductivity in the periodic system (13 pages)

CONTENTS OF VOLUMES I-IV

VOLUME 11 (continued)

Electron transport phenomena in metals (36 pages)

E. H. SONDHEUIER.

v.

A. JOHNSON

and K.

Semiconductors at low temperatures (39 pages)

De Haas-van Alphen effect (40 pages)

D. SHOENBERG, The

c. J.

LARK-HOROVITZ,

Paramagnetic relaxation (26 pages)

GORTER,

and (46 pages)

M. J. STEENLAND

c. DOMB and J. s. F. H. SPEDDING,

H. A. TOLHOEK,

DUGDALE,

Orientation of atomic nuclei at low temperatures

Solid helium (30 pages)

s. LEGVOLD, A.

H. DAANE

and L.

D. IENNINGS,

Some physical properties of

the fare earth metals (27 pages) The representation of specific heat and thermal expansion data of simple solids (36 pages)

D. BIJL,

and M.

H. VAN DIIK

DURIEUX,

The temperature scale in the liquid helium region (34 pages)

VOLUME I11 w. F.

VMEN,

G. CARERI,

Vortex lines in liquid helium I1 (57 pages)

Helium ions in liquid helium I1 (22 pages) and w.

M. I. BUCKINGHAM

M. PAIRBANK,

The nature of the L-transition in liquid helium

(33 pages) E. R. GRILLY K.

and E. F.

HAMMEL,

Liquid and solid 3He (40pages)

w. TACONIS, aHe cryostats (17 pages)

I. BARDEEN

and J.

M. YA. AZBEL’

R. SCHRIEFFER,

and I.

w. J. HUISKAMP and (63 pages)

M. LIFSHITZ,

Recent developments in superconductivity (118 pages) Electron resonances in metals (45 pages)

H. A. TOLHOEK,

N. BLOEMBERGEN, Solid

Orientation of atomic nuclei at low temperatures I1

state masers (34 pages)

The equation of state and the transport properties of the hydrogenic molecules (24 pages)

J. J. M. BEENAKKER,

z.DOKOUPIL, Some solid-gas equilibria at low temperatures (27 pages)

CONTENTS O F V O L U M E S I - I V

VOLUME IV

v.

P. PESHKOV,

Critical velocities and vortices in superfluid helium (37 pages)

K.

w. TACONIS and R. DE BRUYN OUBOTER, Equilibrium properties of liquid and solid mixtures of helium three and four (59 pages)

D. H. DOUGLASS JR.

and

0. J. VAN DEN BERG,

Anomalies in dilute metallic solutions of transition elements (71 pages)

KEI YOSIDA,

The superconducting energy gap (97 pages)

Magnetic structures of heavy rare earth metals (31 pages)

c. DOMB and

A. R. MIEDEMA,

L. N ~ E L ,R. PALITHENET A. ABRAGAM

L. M. FALIKOV,

and M.

J. G. COLLINS

and G.

Magnetic transitions (48 pages)

and B.

DREYFUS, The

BORGHINI,

Dynamic polarization of nuclear targets (66 pages)

K. WHITE,

T. R. ROBERTS, R. H. SHERMAN,

rare earth garnets (40 pages)

Thermal expansion of solids (30 pages)

s. G. SYDORIAK and F.

temperatures (35 pages)

G. BRICKWEDDE,

the 1962 SHe scale of

PROGRESS I N LOW

TEMPERATURE PHYSICS EDITED BY

C. J. G O R T E R Professor of Experimental Physics Director of the Kamerlingh Onnes Laboratory, Leiden

VOLUME V

1967 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

0 1967 N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y

- AMSTERDAM

No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher

Library of Congress Catalog Card Number: 55-14533

PUBLISHERS:

NORTH-HOLLAND P U B L I S H I N G CO. - A M S T E R D A M SOLE DISTRIBUTORS FOR U.S.A. AND CANADA: INTERSCIENCE PUBLISHERS, A DIVISION OF

J O H N WILEY & S O N S , I N C . - N E W Y O R K

P R I N T E D I N THE N E T H E R L A N D S

PREFACE

Looking back on five volumes of Progress in Low Temperature Physics, it seems worth comparing this latest volume with the earlier ones, and thus draw attention to the trends which emerge. It is immediately evident that chapters in the early books were shorter; then they averaged 20-25 pages, whereas the present average approaches twice this figure. To maintain a reasonable size, it has therefore been necessary to decrease the number of chapters per Volume. Thus, in the rapidly expanding field of Low Temperature Research, succeeding Volumes in this series cover an ever-decreasing fraction of the body of knowledge. Turning to the general features of the presentation, the more recent books contain relatively less text, an equivalent number of figures and tables, but more formulae and more references. This increased use of formulae is not due to an increasing representation of theoretical physicists among the authors, but rather to a tendency among the experimenters to use formulae more often in the presentation of data and in its interpretation. It is instructive to compare papers in the present Volume with those covering similar or related topics in the earlier books. Feynman’s paper in Vol. I on “Application of Quantum Mechanics to Liquid Helium” may be contrasted with Anderson’s chapter in the present Volume. Entitled “Quantum Coherence”, this stresses the relationship between superconductors and liquid helium and concentrates on important, novel aspects of “quantum fluids”. The more experimental papers in this Volume on Liquid Helium 11, presented by De Bruyn Ouboter, Taconis and Van Alphen, Andronikashvilli and Mameladze, might be compared respectively with those of Winkel and Wansink p o l . 11) and Pellam and Hollis-Hallett (Vol. I). The wealth of experimental data has increased greatly, but it is the improvement in relating theory to experiment which is quite striking; a result of developments which have taken place in the last few years, mainly in the U.S.A. and the Soviet Union. The short chapter by Cribier, Jacrot, Madhav Rao and Farnoux on the neutron diffraction analysis of niobium crystals in the superconducting

WI

PREFACE

mixed state has no direct ‘ancestor’ in this Progress Series, though suggestions of a magnetic microstructure date back to 1935. Only the paper of Faber and Pippard in Vol. I can be said to contain ideas which anticipate the rapid expansion in investigations and applications of the mixed state since 1960. Ganthmaker’s chapter and that of Azbel and Lifshtz in Vol. I11 are in several respects branches of the same young tree - electron resonance in metals. Though one might say that they scarcely belong to low temperature physics in a narrow sense of the term, the use of very low temperatures is just as essential for them as is the use of magnetic fields and the purity of the metals investigated. This applies also to the chapter of Stark and Falicov which has several links with Shoenberg’s paper on the De Haas-Van Alphen effect (Vol. 11). All of these papers include new and valuable information on Brillouin zones and Fermi surfaces. Finally, one may regard the chapter of Beenakker and Knaap as in some respects supplementary to another Leiden paper on solid-gas equilibria - that written by Dokoupil in Vol. 111. It covers, however, a much wider field of Fluid Mixtures and discusses also the links with recent theory. Several important fields in Low Temperature Physics are missing from the present Volume. Magnetism and temperatures below 0.3 OK are the most striking omissions. These gaps are closely connected with the reduction, mentioned above, in the number of chapters per volume, but are also due to some delay in the arrival of promised papers. I hope to redress the balance later, as in the case of the missing papers on Liquid Helium from Vol. IV. I want to express my thanks to the Leiden physicists and the foreign guests who by their valuable assistance made it possible to edit this Volume; particularly to Drs. K. W. Mess, who, among other things, prepared the subject index of this book, as well as that of Volume IV. C. J. GORTER

CONTENTS

Chapter

Page

I P. W. ANDERSON, THE JOSEPHSON EFFECT

AND QUANTUM COHERENCE

MEASUREMENTS IN SUPERCONDUCTORS AND SUPERFLUIDS

. . . . . . . . .

.

1

1. Historical introduction, 1. - 2. Elementary perturbation theory of the Josephson effect, 4. - 3. Coherence properties of coupled superconductors and superfluids, 11. - 4. Statics of finite tunnel junctions: magnetic interference experiments, 20. - 5. Systems other than tunnel junctions showing interference phenomena, 33. - 6. A.c. quantum interference effects, 36.

I1 R. DE BRUYN OUBOTER, K. W. TACONIS and W. M. VAN ALPHEN, DISSIPATIVE AND NON-DISSIPATIVE FLOW PHENOMENA IN SUPERFLUID HELIUM. Introduction, 44. - 1. Superfluidity, the equation of motion for the superfluid, 45. - 2. The critical superfluid transport in very narrow pores between 0.5 OK and the lambda-temperature, and the impossibility to detect Venturi pressures in superfluid flow, 54. - 3. Superfluid transport in the unsaturated helium film, 62. - 4. Dissipative normal fluid production by gravitational flow in wide channels with clamped normal component, 64. - 5. The dependence of the critical velocity of the superfluid on channel diameter and film thickness, 72.

44

111 E. L. ANDRONIKASHVILI and YU. G. MAMALADZE, ROTATION OF HELIUM11

. .

.

.. .

.

.

.......

. .

.

. .

. .

.

. . . . . .

Introduction, 79. - 1. Solid body rotation of helium 11, 80. - 1.1. Angular momentum and meniscus of rotation helium 11, 80. - 1.2. The thermomechanical effect in rotating helium 11, 83. - 1.3. The theory of the phenomena, 85. - 2. Dragging of the superfluid component into rotation, 91. - 2.1. Peculiarities of dragging of a quantum liquid into rotation, 91. - 2.2. Development of quantum turbulence on dragging helium I1 into rotation, 93. - 2.3. Relaxation time for the formation of vortex lines at small angular velocities of rotation, 96. - 2.4. Relaxation time for vortex line formation in rotating helium I1 on transition through the I-point. The mechanism of vortex line formation, 97. - 3. Observation of vortex lines and their distribution in uniformly rotating helium 11, 100. - 3.1. Experiments on establishment of vortex lines, 100. - 3.2. Direct observations of vortex lines in rotating helium 11,

79

X

CONTENTS

103. - 3.3. Irrotational region, 104. - 3.4. Distribution of vortex lines under a free surface, 109. - 3.5. On the normal component motion in a rotating cylindrical vessel, 110. - 3.6. The structure of the vortex line array, 113. - 4. Elastic properties of vortex lines. Oscillations of bodies of axial-symmetric shape in rotating helium 11, 114. - 4.1. The modulus of shear in rotating helium 11, 114. - 4.2. Anisotropy of elastieviscous properties of rotating heliumlI,ll5. -4.3. Hydrodynamics of rotating helium 11, 122. - 4.4. Hydrodynamics of small oscillations of bodies of axial symmetry in rotating helium 11, 125. - 4.5.

Sliding of vortex lines and collectivization of vortex oscillations, 134. - 5. The phase transition in rotating liquid helium in the presence of vortex lines, 137. - 5.1. The central macroscopic vortex, 137. - 5.2. Relaxation of vortex l i e s for the transition helium 11-helium I in the state of rotation, 139. - 5.3. The order of the phase transition in rotating liquid helium, 141. - 6. Decay of vortex lines and their stability, 144. - 6.1. Decay of vortex lines on stopping of rotation, 144. 6.2. Relaxation of vortex lines on change of temperature of rotating helium II, 146. - 7. Persistent currents of the superfluid component, 149. - 7.1. Discovery of persistent currents and the first observations, 149. - 7.2. Dependence of a persistent current on temperature. Superfluid gyroscopes, 153.

IV D. GRIBIER, B. JACROT, L. MADHAV RAO and B. FARNOUX, STUDY OF m s ~ ~ p ~ ~ c o m MMED u cSTATE ~ ~ vBY~NEUTRON-DIFFRACTION . . . . 161 1. Introduction, 161, - 2. Theory of neutron scattering by vortex lines, 164. - 3. Experimental conditions, 166. - 4. Experimental results with niobium, 171. - 5. Analysis of the results obtained with niobium, 171. - 5.1. Line shape, 171. - 5.2. Position of the peak, 174. - 5.3. Intensity of the peak, 175. - 5.4. Observation of only one Bragg peak, 177. - 6. Conclusions, 178.

.

.

V V. F. GANTMAKHER, RADIOFREQUENCY SIZE EFFECTSIN METALS . . . . . 18 1 1. Introduction, 181. - 2. Principles of the theory, 183. - 2.1. Anomalous skin-effect in zero magnetic field, 183. - 2.2. Anomalous skin-effect in a magnetic field, 185. - 2.3. Application of the ineffectiveness concept to the study of size effects, 193. - 3. Various types of radiofrequency size effects, 197. - 3.1. Methods of detection of size effects, 197. 3.2. Closed trajectories, 201. - 3.3. Helical trajectories, 207. - 3.4. Open trajectories, 216. - 3.5. Trajectories with breaks, 216. 3.6. Trajectories of ineffective electrons, 218. - 3.7. Conclusion, 219. - 4. Shape of line and various experimental factors, 220. - 5. Applications of radiofrequency size effects, 225. - 5.1. The shape of the Fermi surface, 225. - 5.2. Length of the electron free path, 229.

-

-

CONTENTS

XI

VI R. W. STARK and L. M. FALICOV, MAGNETIC BREAKDOWN IN METALS . . . 235 1. Introduction, 235. - 1.1. Pseudopotentialsand the nearly-free-electron model, 236. - 1.2. Dynamics of the electronic motion in a metal, 238. - 1.3. A diffraction approach to magnetic breakdown, 241. - 2. The theory of coupled orbits, 244. - 2.1. Amplitudes and phases at a MB junction, 244. - 2.2. Semi-classical transport properties, 246. - 2.3. Quantization of coupled orbits, 251. - 2.4. Theory of the De HaasVan Alphen effects in a system of coupled orbits, 256. - 2.5. Oscillatory effects in the transport phenomena, 261. - 3. Analysis of experimental results, 265. - 3.1. Hexagonal lattice of coupled semi-classical trajectories in magnesium and zinc, 265. - 3.2. Semi-classical galvanomagnetic properties of magnesium, 267. 3.3. Quantum mechanical galvanomagnetic properties of magnesium and zinc, 272. - 3.4. De Haas-Van Alphen effect in magnesium and zinc, 279. VII J. J. M. BEENAKKER and H. F. P. KNAAP, THERMODYNAMIC PROPERTIES OFFLUIDMIXTURES

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

287

1. Introduction, 287. - 2. Quantum liquids: zero point effects, 290. - 2.1. General remarks, 290. - 2.2. Apparatus, 292. - 2.3. Theory, 295. - 2.4. Phase separation for the systems Ha-Ne, HD-Ne and Da-Ne, 297. - 2.5. Ortho-para mixtures, 299. - 3. Classical liquid mixtures, 299. - 4. Gaseous mixtures, 301. - 4.1. Experiment, 301. - 4.2. Theory, 307. - 4.3. Comparison between experiment and theory, 314. - 4.4. Gas-gas phase separation, 317.

. . . . . . . . . . . . . . . . . . . . . . . . . AUTHORINDEX..

323

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . .

330

This Page Intentionally Left Blank

CHAPTER I

TIHE JOSEPHSON EFFECT AND QUANTUM COHERENCE MEASUREMENTS IN SUPERCONDUCTORS AND SUPERFLULDS BY

P. W. ANDERSON BELLTELEPHONE LABORATORIES, MURRAYHILL,NEW JERSEY

CONTENTS: 1. Historical introduction, 1. - 2. Elementary perturbation theory of the Josephson effect, 4. - 3. Coherence properties of coupled superconductors and superfluids, 11. - 4. Statics of finite tunnel junctions: magnetic interference experiments, 20. - 5. Systems other than tunnel junctions showing interference phenomena, 33. 6. A. c. qua-ntum interference effects, 36.

1. Historical introduction The nomenclature, and to some extent the history, of the area of knowledge to be covered by this review has been confused by the very breadth of the achievement represented by the original letter by B. D. Josephson in which the first theoretical results were announced’. In this letter Josephson discovered not one but three (at least) distinguishable “effects”, to any one of which his contribution was sufficient that it could be - and has been named after him. As far as I can sort them out, these are: (1) The transmission of supercurrents through thin insulating or normal metallic barriers by means of quantum-mechanical tunneling. Without any intention to minimize the importance of Josephson’s other contributions, it seems to me that in order to reduce the amount of confusion in nomenclature, at least in this article, I should confine the term “Josephson effect” to this phenomenon, and let a “Josephson junction” be a device in which supercurrents flow by tunneling. In order to understand the implications of this discovery, Josephson had to investigate how such a junction would behave in the presence of applied electric and magnetic fields. In so doing, he realized two things : (2) That the supercurrents in different parts of the Josephson function References p . 42

I

2

P. W. ANDERSON

[CH.1, 8 1

could be forced to interfere destructively by application of an external magnetic field. This is an effect which I would like to call the “d.c. macroscopic quantum interference effect” or the “d.c. quantum interferenceeffect”, and it may be observed in many other situations than Josephson junctions. It is related to flux quantization and vorticity quantization in helium, and it may even have been observed unintentionally2 in some of the early experimental work on flux quantization, before Josephson’s letter. If it must have a name, it might appropriately be called the “Mercereau effect”, because of the many elegant applications of it which have been made3 by Mercereau and his colleagues. It is important to emphasize that, contrary to common opinion in the early days of the subject, this and the other interference effects have no necessary connection with the Josephson effect proper. (3) That the supercurrent in a Josephson junction under the right circumstances varies in time at a rate given by 2eV/h, and that this effect could be observed by irradiating the junction electromagnetically at this frequency or a subharmonic and measuring the d.c. current characteristic. Both of these have been called the a.c. Josephson effect, and the equation

hv

= 2eV

the Josephson frequency condition ;however, similar phenomena are of such wide occurrence that again it seems too confusing to call the whole complex of phenomena the a.c. Josephson effect. In this article we confine that name to the effect as observed in Josephson tunnel junctions, and use the wider term “a.c. macroscopic quantum interference” or “a.c. quantum interference” for the generalized phenomenon. A second circumstance which has complicated the literature and obscured the relationships among the various kinds of interference effect is that the second most important early paper in the field, again by Josephson, was never published; this is Josephson’s fellowship thesis4 in which he reexpressed his earlier results in more general and more physically satisfying terms, pointing out where the results were special to tunnel junctions and where they were a general property of coupled superconducting systems. It is in this paper that the generalized concept of quantum interference appears, as well as a number of special results and concepts of importance in the practical realization of these ideas. Many of these results were discovered independently and published by the present author 5, also unfortunately in a rather obscure place. Meanwhile the experimental demonstration of Josephson’s ideas proReferences p . 42

CH. 1, @ 11

JOSEPHSON EFFECT

3

ceeded fairly rapidly, after an initial lull of less than a year. The first published confirmation of the existence of tunneling supercurrents - the Josephson effect - and of the accompanying d.c. macroscopic interference effect was by Rowel1 and Andersons, though the effect had been observed but not recognized often previously. Rowel17 soon thereafter produced junctions which demonstrated the d.c. interference effect quantitatively. The a s . Josephson effect proper was demonstrated by S. Shapiro8 very soon thereafter (A. H. Dayem observed it independently but did not publish it), by the synchronization technique suggested by Josephson. Next occurred the elegant series of “two-junction” macroscopic interference experiments by Mercereau and colleagues 9, in analogy to the twoslit interference phenomenon in optics. Eventually, these experiments came to use simply superconducting contacts or bridges, verifying the theoretical indications that the tunneling phenomenon was inessential. In an attempt to demonstrate that experiments on thin film bridges by Parks10 were basically the d.c. interference effect, Anderson and Dayem11 demonstrated the a.c. effect on thin film bridges as well. No reasonable suggestion has yet been made for a physical realization of a true analogue to the ideal Josephson junction for superfluid helium. Nonetheless, the realization that Parks-Dayem bridges show the a.c. effect, and Mercereau contacts the d.c. interference effect, stimulated Richards and Anderson to attempt to demonstrate both using small orifices as the obvious helium analogue of the superconducting contact. So far, only the a.c. analogue experiment has given successful results12. The most recent chapter in the story has been the achievement of successful direct - i.e. incoherent, independent - detection of the radiated a.c. power from a Josephson junction. The Josephson-Anderson theories indicated that a stronger effect could be obtained with a synchronizing mechanism to monochromatize the frequency emitted, and this was realized in the “spontaneous step” phenomenon as observed by Fiskel3 in which internal electromagnetic resonances of the junctions become locked to the a.c. Josephson currents. These steps were used by Giaeverl4 in the first observation (by an indirect but perfectly genuine technique) and in two later more direct observations 15. If one may speculate on the “future history”, the direction of progress appears likely to be two-fold : first, more applications to specific devices, especially scientific instrumentation 16, and second, more variants of these phenomena are yet to be observed in liquid He.

References p . 42

4

P. W. ANDERSON

[CH.1,

52

2. Elementary perturbation theory of the Josephson effect proper It seems to lead the reader most gently into this novel conceptual structure to start with the rather straightforward theory of the tunnel junction as developed in Ref. 5. The basic calculation is extremely simple: one sets out merely to find the coupling energy between two superconductors which is caused by forming a tunnel junction between them. The tunnel junction is described by the “tunneling Hamiltonian” of Cohen, Falicov and Phillips 17

The states k and q are on the left and right sides of the tunneling barrier respectively; the phase relationship among the terms shown is required by time-reversal symmetry. We have not shown possible spin-flip terms which are, nonetheless, permitted by time-reversal invariance; they have no particularly striking effect, so far as is known. No completely satisfactory derivation of (2.1) from a rigorous point of view has yet appeared, to our knowledge. A number of correction terms are to be expected, representing tunneling with interactions (phonon, photon, or Coulomb) as well as multiparticle tunneling, but none of these appear at present to be very relevant to the Josephson effect, as opposed to ordinary incoherent tunneling, and their discussion does not belong in this review. Some derivations avoiding the tunneling Hamiltonian have been given 18119, especially the elegant and enlightening discussion by Josephson in terms of the temperature Green’s function theory of Gor’kov. The physical assumptions, though somewhat concealed by this technique, are essentially the same, as are the results: whether one wishes to introduce a tunneling Hamiltonian or a propagator (i.e. inverse Hamiltonian) which carries single particles across the barrier is simply a matter of taste. The former seems more suitable for the level of this article. and %, on left (1, k ) and right (2, q) We add to (2.1) the Hamiltonians sides respectively. For instance,

where Ek is the one-particle energy measured - for instance - from the bottom of the band, eV the mean electrostatic potential (including any long-range Coulomb effects of space charge) and Zin, the interparticle interactions reReferences p. 42

CH. 1,

8 21

JOSEPHSON EFFECT

5

sponsible, among other things, for superconductivity. Often, especially at finite temperatures, we may insert a fictitious chemical potential term p N to represent the effect of an attached reservoir of electrons, etc. Let us recall a few simple results from the B.C.S.theory20. Let

where in H:,ck is measured from the Fermi surface energy p. We assume for the purposes of simplicity - and powerful and sophisticated arguments are available to support that assumption in many cases - that 3‘:leads to a B.C.S. superconducting state of the conventional kind at or near absolute zero : (2.4) ‘y, ( A 1) = ( U k + ~ k c : c ~ kF)v a c (1) Y

n k

and similarly for (2)t. This ansatz and simple assumptions about Hint give us the usual results : k

E i = (ck - p)2 + 1A21. Here A is the energy gap, and Ek the quasi-particle energy. In one way (2.4)-(2.7) are simpler than the original B.C.S. - or for that matter the Gor’kov21 - theory, in that we choose the simple product eigenstate (2.4) rather than the state projected on a fixed number of pairs N,, which may be obtained by a transformation: 2n

e-iN1a!PI( A , e”) dq .

Y(N,) = 0

This is easily verified by substituting in (2.4) and using (2.6). The questions involved in why we use (2.4) rather than (2.8) are rather deep and basic to the whole subject, so aside from remarking that since we have tunneling it is not necessarily suitable to use (2.8) we will postpone them briefly. We follow the convention that - k ++ - k&,k tt k?. References p . 42

6

P. W. ANDERSON

[CH.1, 8

2

On the other hand, it is essential that we treat A (as we have) as complex *, if only because of (2.8): we must have a set of states complete enough to describe the fixed-iV, as well as fixed-A situation. The phase of A gives, in fact, the phase relationship between the component of (2.4) with N particles and those with N + 2, as we see from (2.5). The wave function (2.4) is essentially different from that of any normal substance in that it does contain a coherence between components with different total numbers of particles, and as will be described later it is this coherence which is vital to the whole concept of superfluidity and superconductivity. It is only the part of the Hamiltonian .Xy which is consistent with a timeindependent A , however. It was first noticed by Gor’kov21 that A has an intrinsic time-dependence which is different for samples at different electrostatic potentials. Josephson showed its relevance to tunneling, and since then several more complete discussions4~23,24 have been given. We follow Anderson, Werthamer and Luttinger23. They simply introduced the last term of (2.3) and noted that its only effect on a wave function of type (2.4) is to make A time-dependent according to the equation of motion

so that A

- lAl

e-

( 2 i / l ) (w

+ eY)t

(2.9)

Again, this merely signifies that while the last term of (2.3) has no effect on the components of Y of fixed N , , clearly the energies of states of different N differ by (2.10)

by definition, and therefore that a state with a definite phase relationship between components with different numbers of pairs must allow that phase

* It is necessary to distinguish carefully between this use of a “complex” energy gap and that of the Nambu-Eliashberg formalism used extensively in single-particle tunneling work (see, e.g., Schriefferaz).The “real” and “imaginary” components of A used here are, in that formalism, the “ri” and “r8” spinor components, dl and Aa, each of which may itself be complex when lifetime effects are included, the complex part representing decay in time. The true quasi-particle energy may be complex but is given by Ek2 = &k2 A l 2 As2. In most Josephson phenomena lifetime effects are not very important because of the low frequencies, and a proper calculation including them has not yet been attempted to my knowledge.

+

References p . 42

+

relationship to vary in time according to (2.9). Note that (2.6) ensures that the coefficient of &:k varies in time like A , which is of course what is necessary to given the proper time-dependence of A. Now let us do perturbation theory using the tunneling Hamiltonian (2.1) as the perturbation. Any reasonable barrier is tens of Angstr6ms thick, and or me-'' at best, so that the tunneling matrix elements are of order the only phenomenon of any interest must appear in the lowest non-vanishhaving no diagonal matrix elements. ing order, which is the second - ST Applying a typical term of (2.1) to the wavefunction (2.4), we get T~(C:C~

+

C ? ~ C - ~ )Y , ( A , ) yZ(dz)= T k q ( u k u q

+ ~,p~)(c:c?~) x (rest of Y ) . (2.11)

When there is no relative voltage between sides (1) and (2), the energy of this intermediate state is just the quasi-particle energy &+ Eq; it is obviously the state with an extra quasi-particle on each side. Thus the total coupling energy is just:

(2.12) The first two terms in the bracket are of no particular interest because they do not depend on the phase of A. By (2.6), however, the last one does, in fact its value is

k, 4

The relevant integral in (2.13) has been done elsewheres; the result is

(2.14)

K is a complete elliptic integral. The most trivial and yet most important consequence of (2.14) is just the idea that the energy does depend on the relative phase (rp1 - q Z )on the two Referencesp . 42

8

P. W. ANDERSON

[CH.

1, 5 2

sides, and in fact is a minimum when this relative phase is zero. This implies that if the coupling energy is strong enough these two approximately independent pieces of metal, connected only by very weak tunneling matrix elements, will find it energetically favorable to occupy coherent states like (2.4) with (ql-cp,)=O, rather than fixed-number states like (2.8). In the states (2.8), the phase-dependent coupling energy simply vanishes identically. Characteristically,if the tunneling current is appreciable this coupling energy is large relative to helium-temperature as well as zero-point fluctuations 5, and so, as we will review in the next section, the calculation using coherent states given here is correct. Before going on to that let us complete the perturbation theoretical discussion. The calculation of the energy perturbation in case there is a voltage difference across the junction is somewhat more complicated. The most complete calculation in the literature is that of Werthamer 25, following Riedel26. We will give here a simpler one related to Josephson’s original work. Of course, there is one major difference which is not discussed in any detail in any of these papers including Werthamer’s: that where the existence of the coupling energy provides a motivation for using the coherent states (2.4) in the zero voltage case, here the coupling energy is necessarily periodic in time and the assumption that the starting states should be the coherent ones (2.4) is not obviously valid. What should by rights be calculated is the current-current (or energy-energy) correlation function, which would, however, then depend on details of external circuitry, applied a.c. voltages, etc. The calculation we do should perhaps be thought of as follows: we calculate the energy - and, shortly, the current - using the coherent states (2.4) as a basis. Any state reasonably close to the equilibrium ground state can be made up as a wave packet of such coherent states (this situation is discussed in Refs. 24 and 5 and will be further elucidated in Section 3). In fact, it often turns out that even in the a.c. case the coherent states are approximately the correct ones. Even assuming the coherent states the calculation is not quite straightforward because the quasi-particle energy in the intermediate state depends on whether the quasi-particles were created by moving an electron from left to right or vice versa - i.e. whether a given quasi-particle, which is a coherent mixture of electron and hole, is in its “electron” or “hole” aspect. The first term of (2.11) creates a state of energy Ek+Eq+e(Vl- V,), the second Ek+Eq- e( V, - V,), relative to the starting state. The Hermitian conjugate terms have energy denominators which are the reverse, giving just the same References p . 42

CH. 1, 8

21

JOSEPHSON EFFECT

9

form of result but with a sum of energy denominators:

(2.12')

This expression increases as the voltage difference approaches the double energy gap, becoming singular there. Above that point, both the energy and the current contain complex parts, the interpretation of which has not been fully discussed in the literature. (But see Scalapino27.) Expressions for the current which probably suffice in all regions have been given by Werthamer26. The increase in current in the intermediate region has probably been observed by Grimes, Richards and Shapiro 28. So far we have merely done a second-order energy calculation. Of course, for finite voltage the quantity we calculate is time-dependent - it is simply the mean value of the tunneling Hamiltonian X T in the assumed states. In any real physical situation other contributions to the energy - external sources necessary to maintain the currents, etc. - will also be present. It will be shown in the next section by very general arguments that a phasedependent energy implies the existence of a supercurrent (2.15)

This relationship may be derived directly from second-order perturbation theory if we realize that the second-order energy is just the mean value of the tunneling Hamiltonian XTin the perturbed state:

(~E)~=((~Y,+~~),~,(~Y,+~'Y))~((~~Y,~,Y/~)+(~Y, As in (2.1 l), a typical term of XTis c:cq

+ c?qc-k.

(2.16)

A typical term of the tunneling current is given by using (as suggested by Cohen, Falicov and Phillips 17)

and the corresponding term of this to (2.16) is

References p . 42

10

P. W. ANDERSON

[CH.1,

2

Thus when calculating the current rather than the energy we must replace the coherence factor in (2.1 1) by

On the other hand, 6Y is unchanged, so that only one of the two coherence factors in (2.12) should have its sign changed. Thus we get for the current

and this is exactly the same as (2.12) and (2.13) except that the phasedependent factor Re (Ad,) = ldld2l C 4 P l - (P2) (2.17) This is precisely what we would get by applying (2.15). It is easily verified that the same expressions work in the frequency-dependent case of (2.12‘). The tunneling current, like the energy, is phase-dependent; this is in fact the point from which Josephson startedl. The interpretation can only be that the actual current which flows is determined by the external circuit, and that experimentally what we expect to see is a “tunneling supercurrent” phenomenon: we can draw any current consistent with (2.17) through a tunnel junction at zero voltage, or where there is a finite voltage there will be an a.c. tunneling supercurrent. Both of these predictions were verified shortly after Josephson’s paper came out. In order to understand these observations, it is first necessary to fully understand the meaning of the use of coherent states in our calculations, which is the purpose of the next section. The most interesting and most convincing demonstrations of these coherence effects all involve modulation of the supercurrents by electromagnetic or other forces, so that in the final sections, where the actual experimental work is explained, the emphasis will be on that aspect. The purpose of this section has been merely to make the References p . 42

CH.

1,o 31

JOSEPHSON EFFECT

11

most straightforward calculations on this simplest microscopic model which would give the reader a feeling for the source of these phase-dependent energies and currents with which the rest of the article will deal. 3. Coherence properties of coupled superconductors and superfluids

The first reasonably careful discussions of the basic theory of coherence in coupled superconductors were given independently by Josephson and Anderson5. Since then these have been expanded, the most complete, with a discussion of the similar case of superfluidity in He, being the recent review of Anderson 24. In the preceding section we showed that there exists a phase-dependent coupling energy between two superconductingsamples connected by a tunnel junction (assumed infinitely small spatially), when the states of the two superconductors are assumed to be coherent states, i.e. coherent superpositions of states of different numbers of pairs of particles: Y = C a NeiNqY,. N

(3.1)

We also showed that this phase-dependent energy implies the possibility of a supercurrent flowing between the two superconductors:

in this case. As was demonstrated in Refs. 5 and 25, this connection between phase and current is a consequence of the commutation relationships of the field operators : (Josephsonls gives another, thermodynamic, derivation)

“,$I

= -$9

(3.3)

which is equivalent for coherent states like (3.1) to the canonical commutation relationship between number and phase,

“,cpl=

-i,

(3.4)

which in turn implies the operator equivalences

As we said, the proviso that this holds only for states for which N is large and a fortiori for states which are superpositions of coherent states like (3.1)References p . 42

12

P. W. ANDERSON

[CH.

1, 5 3

must be made. Certain well-known difficulties occur for small N . The derivations of (3.4), (3.5), and (3.2) as given in Ref. 24 are almost trivial. First, we verify that (3.1) obviously has a definite fixed phase of the field operator ( IY, $ lU) = ei‘p u N -N‘l (IYN- 17 $ y N ) N

and we assume (as we are free to do) that the phases of a, and Y Nare chosen to make the former real as well as the matrix element. Second, it is clear that acting on (3.1) the operator -ia/acp has the same effect as N . The converse statement in (3.5) follows only if N may be taken as a quasicontinuous variable, which is the source of the small-N difficulties, of course. Finally, (3.2) is just the Hamiltonian equation of motion which follows from the fact that N and cp are canonical conjugate variables:

(3.6) or

( U = ( S ) ) . Of course, all these statements are equally valid for liquid He, where N is the number of helium atoms, $ their quantum field, as for superconductors where the reference is to number of electron pairs and to the pair field $$. The more explicit discussion given in Section 2 may remove any residual doubts in the reader that the pair field may be so treated. From this point of view, the Gor’kov-Josephson frequency equation (2.9) or dcp (3.7) dt aN is simply the other Hamilton’s equation of the pair implied by the canonical commutation relationship between number and phase. These various relationships hold just as well between small but macroscopic cells within a bulk superfluid as between fully macroscopic samples connected through a weak “Josephson” link. The corresponding coupling energy is the energy which maintains the internal coherence of the superfluid, the currents which flow according to (3.6) are the usual supercurrents nev, =ne(tt/m)Vcp, and (3.7) becomes the acceleration equation which controls the dynamics of the superfluid. Within the bulk of the superconductor, however, the various interference effects which Josephson suggested are not References p . 42

CH.

1, 8 31

JOSEPHSON EFFECT

13

easily observed because the coupling is too strong. The Meissner effect represents the successful attempt of the superconductor to exclude from its interior all phase differences which lead to supercurrents: Correspondingly, the zero-resistance phenomenon is the successful attempt to prevent temporal phase differences according to (3.7): dp/dt=O implies d U / a N is constant, i.e. the chemical potential p is constant. When, in type I1 superconductors, the magnetic field is increased to the point that phase differences and supercurrents do exist in the interior, the energies which control the resulting vortex structure are still so great that the structure becomes nearly microscopic in scale and has various types of rigidity which makes the direct observation of interference effects difficult. Similar considerations hold in liquid helium. Thus, while the coherence-interference phenomena in superfluids are perfectly general, the Josephson “weak link” - originally a tunnel junction as discussed in Section 2 - is essential to their direct observation, primarily because in such a weak link the response of the superfluid to phase perturbations is not controlling and we can read off the current corresponding to an “externally” applied phase difference. This discussion then brings us to the most fundamental question of principle with regard to these coherence effects. This is the following: we understand very well that in a strong-coupled situation such as the interior of a bulk superconductor the phasedependent coupling energy can maintain the phase coherence between different parts of the material. Only thermal agitation at a sufficiently high temperature, T,, can break this coherence. Whether we describe the coupled system mathematically, as a whole, by a totally coherent state, in which we assume (+) exists, or whether we use a fixed N or an incoherently mixed grand canonical ensemble is irrelevant, since the phase of the system as a whole is not observable. What processes, on the other hand, affect the phase coherence between two samples connected by a weak link in which we are modifying the phase relationship intentionally in order to observe interference? What maintains the coherence in opposition to the perturbations we may introduce? I believe the answer is threefold, depending on the type of interference experiment which is attempted. These are basically of three kinds: d.c., a.c. with synchronization, and a.c. without synchronization. The first two have been well understood in the past; the third is not complicated but no previous discussion has been given. In the first case, coherence in d.c. interference experiments is maintained by the phase-dependent coupling energy itself. As has been shown elsewhere24, a d.c. current source may be represented by a term in the MamilReferences p . 42

14

P. W. ANDERSON

[CH. 1,

83

tonian of the system depending linearly on the phase: &source

=

- 5 1 2 (cpl - cpd

and the Josephson current which flows is then obtained by minimizing the Since the phase sum of the phase-dependent internal energy and Nsource. relationships between different parts of a Josephson junction or between two junctions connected in parallel can be influenced by a magnetic field (by gauge invariance physical quantities can depend on the phase of the electron field only through q ( r ) - ( e / c ) r A .dl, where A is the vector potential) the phase-dependent energy can depend on the magnetic field, as first demonstrated by Rowel17 and exploited by Mercereau and his colleagues in many beautiful experiments. The question of what mechanisms may break up the coherence in the d.c. case was discussed in Ref. 5 as well as by Josephson29and Zimmermann and Silver30. The ultimate limitation by the zero-point fluctuations implicit in the fact that ( A N A q ) - l according to the commutation relation (3.4) is actually always completely negligible. The relevant frequency was first estimated in Ref. 5 but is most simply understood by pointing out, as Josephson 2o did, that the Josephson current is essentially inductive; i.e.

for small signals, where J = J1 sin cp cs: J,cp

This inductance per inverse unit area* matches the capacitance per unit area to give an LC resonance at a frequency

the “Josephson plasma frequency”, which is typically a few megacycles. The zero-point energy is then -Am, for a single small junction, which is much less than usable kT’s. That is, the electrostatic energy (involving C ) which tends to reduce the charge fluctuation A N is not strong enough to outweigh even a small coherent Josephson coupling which tends to make Acp small instead.

*

This inductance was actually measured directly by Silver et aL31.

References p . 42

CH. 1,

8 31

15

JOSEPHSON EFFECT

-

Much more relevant is the question of thermal fluctuations. Josephson currents of less than 0.1 PA correspond to coupling energies kT at helium temperatures and so are intrinsically unobservable at those temperatures. One or two orders of magnitude higher than that should still be quite hard to observe because of the rapid rate of phase slippage by thermal activation. That is (see Fig. 1) the combination of the current source energy and the CONSTANT CURRENT SOURCE J e

Fig. 1. Energy vs. phase for a Josephson junction with constant applied current. Thermal or zero-point fluctuation and/or magnetic fields can cause instability and “phase slippage”.

periodically phase-dependent energy can be represented as a “washboardlike” energy surface, with local minima but no absolute ones. Only if the local minima are very deep compared to kT will the supercurrent-carrying state be stable relative to one in which the phase slips and - by (2.9) - a voltage appears. References p. 42

16

[CH.1,s 3

P. W. ANDERSON

IC

SUPERCONDUCTING OR FILM

,/WIRE

JOSEPHSON JUNCTION

0

0

0

IC

0

AV (1) - AV (2) = f (H, I A p p

(a)

GENERAL CASE

SUPERCONDUCTOR

,

/I

'\'.

., - __-

(b)

/

/

SCHEMATIZED

Fig. 2. Schematic diagrams of the geometry of the Mercereau interferometer. (a) shows a superconducting loop containing two junctions to which may be applied fields or drift currents in order to change the relative phases &(l) and &(2) of the two junctions; (b) shows a possible path for the integration in the text, Section 4.

In the early experiments noise from the external circuit at room temperature was of importance. Josephson29 has given a thorough discussion of how to avoid that if desired. A final consideration which has been emphasized by Zimmerman and Silver30 is the question of the relative coherence between two Josephson or other junctions which are part of an interference experiment. CharacteristicReferences p. 42

CH.

31

17

JOSEPHSON EFFECT

ally, the Mercereau geometry (shown in Fig. 2) involves a pair of Josephson junctions connected in parallel in a superconducting loop around which one wishes to measure the phase difference caused by a current or vector potential. When the phase difference is 2nn, the junctions are in phase and twice the critical current of one can flow; when it is (2n+ 1) n , the currents are necessarily opposite, the energy coupling the phases of the two halves of the loop is reduced drastically, and a much smaller critical current is observed. The thermal fluctuation limit here, it is suggested, is the thermal fluctuation of the loop integral of phase. This may be simply computed from the point of vieuw of one of the junctions, which sees a superconducting loop of inductance L. The fluctuating voltage V obeys : dl V=Ldt ’ but also V controls the phase across the junction dq 2e dt

I/=--*

Thus we find

or 8e2L

( q 2 ) = fi2 ( 3 L I 2 ) = kT x

8e2L -* h2 ’

(3.9)

the fluctuation is proportional to the inductance of the loop. When ( q 2 ) 1, phase coherence will not easily be observed; this corresponds to an inductance of h2 x 2.5 x cgs = 2 cm N 2 x L--millihenry. 8e2kT N

Thus loops for use in Mercereau interferometers must either be wound noninductively (he has observed interference for non-inductive loop lengths of 1 m) or must be rather small. It is, however, not clear (Mercereau, private communication ; also Ref. 30) that this limitation cannot be evaded in practice, by sufficiently careful averaging procedures, by using the Josephson current in moderately stronglycoupled junctions to “tie down” the phase during most of the measurement process, or by appropriate filtering. N

References p . 42

18

P. W. ANDERSON

[CH.1,s 3

The second type of interference experiment which may be used to demonstrate coherence in superfluids is the “synchronized a.c.” experiment. This was suggested in the original letter of Josephson1 and first carried out by Shapiro * on Josephson junctions. Later the experiment was extended to superconducting contacts by Anderson and Dayemll, and it is so far the only successful interference experiment in superfluid He as carried out by Richards and Anderson 12. The principle of the experiment is to apply an alternating force and to observe simultaneously the junction’s d.c. I-V characteristic. At a voltage such that the Josephson frequency v, = 2eV/h

is in synchronism with the applied a.c. signal, power can be transmitted between the a.c. source and the Josephson system and therefore singularities in the I-V characteristic are to be expected. We will discuss the rather primitive state of the theory of this phenomenon elsewhere; our concern here is with the fact that in such an experiment, apparently, a definite phase relationship is maintained across the junction in spite of the presence of a finite d.c. voltage. The mechanism is, of course, one of synchronization; the coupling energy to the a.c. signal can be sufficiently great to maintain coherence against both thermal and zero-point fluctuations. Detailed study of even a simple case of the synchronization effect is an exercise in nonlinear, largesignal theory of noise and irreversible processes which has not as yet been carried out and is probably not at all simple; yet one can make some qualitative statements. One is that even a rather small synchronizing signal is adequate to outweigh thermal and zero-point fluctuations under the proper circumstances. For instance, if we think in terms of an externally applied a.c. supercurrent, which can be schematized as a term in the Hamiltonian like

the gain in energy when cp - rp changes by 2n in favorable phase witho, relative to unfavorable is of order hJ,le which is >> kT if J, is as much as a microamp6re. J, is of course limited to be less than the critical Josephson current. One can see no physical reason why the synchronization should not be essentially perfect, and more perfect the stronger the applied signal. All experimental evidence points in this direction. References p . 42

CH. 1, Ej 31

JOSEPHSON EFFECT

19

A special case is the “self-synchronized” Josephson system. In many cases, the Josephson junction itself may be able to sustain high Q electromagnetic resonant modes. In such a junction, the a x . Josephson current itself may be thought of as generating its own synchronizing signal, the relative phase being controlled by the resonant circuit. The result is the “step” phenomenon first reported by Fiskel3. It is notable that where with external synchronizing signals V is absolutely fixed by the applied frequency, in the “step” phenomenon the frequency and thus voltage may be “pulled” by varying the current. A fascinating variant of the self-synchronized technique has been given by Mercereau and Vant-Hull4I. Two junctions are connected in a loop with a long resonant strip line, and the harmonics ok the strip line frequency observed as voltage singularities across the pair of junctions. This technique seems to demonstrate coherence between junctions separated by over one meter of superconductors, and may be the most sensitive interference effect so far suggested. The final type of experiment is the “free-running” a.c. Josephson effect. It is hard to distinguish very strongly in principle between this and the “self-synchronized” case, since it certainly is true that there always will be a greater or lesser degree of feedback. In the case of point contacts or Dayem bridges, where the electromagnetic coupling to free space is relatively strong and the Q of any resonance reasonably low, we do not expect a very strong synchronizing effect and we can idealize the junction as simply running free at a mean voltage determined by the external circuitry. The measurements we can make all amount basically to measuring the correlation function of the Josephson current. If we measure the current a t any instant, we will get a finite value with some arbitrary phase, and in making such a measurement we automatically put the system in a wave packet of definite phase difference ‘pl - q 2 : since J = J , sin (ql -cpz), a measurement of J is equivalent to a measurement of ‘p1-’p2. Again, the dissipation of this wave packet is controlled by two kinds of perturbation: quantum-mechanical, and classical thermal fluctuations. The quantummechanical fluctuation is controlled by the electrostatic “kinetic energy”like term

which causes a wave-packet which starts out as a &function in phase to dissipate, just as the kinetic energy (A2/2m)/(i32/i3~2) causes a particle wave References p . 42

20

P. W. ANDERSON

[CH.

1, 5 4

packet to dissipate in x. The equation is

(3.10) where C is the capacitance. For a point-contact device such as is used by Dayem and Grimes32 C - lo2 cm and the time involved in complete dissipationis only -lo-' sec; they can never observe a spectrum narrower than 1 megacycle or so. For Josephson devices, on the other hand, C-104 and the spectra may in principle be very narrow, especiallyif the cavity resonances of the junction are employed to still further reduce the effective C (which is an equivalent result from a different point of view to the idea of selfsynchronization). If now a second measurement of current is made, the full amplitude of Josephson current will again be measured but its phase relationship to the previous measurement will depend on whether the measurement is made before or after the wave packet was dissipated. Thus the breadth of the spectrum depends on the mean time of dissipation of the wave packet. The thermal noise problem for this case is a fairly complicated non-linear one, but in practice the frequency spectrum will be controlled by the sum of the various noise voltages due to the resistance in the external circuit and the effective resistance of the junction device, and these noise voltages will have a role - probably the controlling one - in breaking up the coherence of the wave packet. Another way of doing the problem which would give similar answers would be simply to calculate the current correlation function ( J ( 0 ) J ( t ) ) , using if desired ordinary incoherent superpositions of states. This calculation, however, has to be carried to higher order than the type of calculation so far done. The major conclusion of this section is that it is correct in almost all cases to discuss the quantum coherence effects by means of coherent states with definite values of order parameters A or (I)) and definite phases. Almost all deviations from perfect coherence can be treated as thermal noise rather than as an intrinsic breakdown of the coherence assumption. Only in the last case of the free-running point contact device is there any possibility that this breakdown may be observed. 4. Statics of finite tunnel junctions: magnetic interference experiments

So far we have been discussing only an extremely small Josephson junction in which the phases q l and cppZon the two sides could be assumed to be References p . 42

CH. 1,f 41

21

JOSEPHSON EFFECT

single constants. Most of the interesting effects, starting from Rowell’s first observations 61 7, result when electromagnetic fields are allowed to enter the junctions and modulate the phase. The Josephson current at a particular point in a tunnel junction cannot depend only upon the phase difference of the order parameter on the two sides because that is not a gauge-invariant quantity. The current must not change if we add a gradient Vx to the vector potential A and at the same time add (2elhc)x to the phase. The most elegant way to write down the gauge-invariant Josephson current is in terms of the so-called “gaugeinvariant phase” w (called @ in Ref. 25), which is defined as the quantity which acts like a velocity potential for the supercurrent: (4.1) w is not single-valued, nor even the phase of a single-valued function; only its gradient is defined and unique. Nonetheless it is often a formally useful quantity. The Josephson current at a given spot x , y in a junction is given by

J ( x , Y ) = 51 sin [ w , - w2 (x, Y ) ]

(4.2)

9

where w1- w2 can only be defined in terms of a specific path, the shortest one from one superconductor to the other - and thus presumably the one followed by tunneling electrons WI

- ~2

=

s

s 2

2

Vw(x,y,z)dz = ~ p 1- ~2

1

- 2e PlC

A, dz .

(4.3)

1

The equations satisfied by w were written down in an elegant form by Werthamer25, making use of the fact that rp is the phase of a single-valued function so that V x V p is zero except possibly at vortex lines where the phase changes by 2n or 2nn on traversing the line: (VxV)w= - V x

2e +27cC6(r-rvorter)=- - H ( + 2 n Z 6 ) hC

(4.4)

and

a

2e

- ( V W )= - E at h

.

(4.5)

The second is just London’s acceleration equation for vs, leaving out as small thermal and Bernoulli terms. (As pointed out by London33, the ( u / c ) x H and References p . 42

22

[CH.1,

P. W. ANDERSON

84

SUPERCONDUCTOR 1

SUPERCONDUCTOR 2

motional terms cancel.) The presentation of Werthamer seems to be accurate for his purposes but does not make clear that w (his @) is not really the phase of any order parameter or Green’s functions (nor does the time equation have quite the same form as we have given). In component form, referring to a typical junction experimental geometry (Fig. 3) these two equations are

aZw a2w 2e ---- - - H , (+ &functions),

(4.4‘)

aZw --EE,. 2e --

(4.5‘)

azax

,ic

axaz

A

ataZ

Let us consider first a simple junction with field H, through the barrier region in the y direction. We want to compute the rate at which a field changes w1 - w 2 : 2

-ax (aw l

- w2)

a

= ax -

(J3) 1 2

2

dz

By London’s equation, (t&

a ax

-(wl References p . 42

= -(I&

=(2eH/rnc)A, so that 2e

- w 2 ) = -(2A Ac

+ d)ff,.

a. 1,8 41

23

JOSEPHSON EFFECT

Thus in the presence of a constant magnetic field in the plane of the junction, the maximum current it can sustain follows a single-slit diffraction pattern: +I

J,,, = J1

cos[32A -9

+ d)H,x

1

dx

for a junction of length I and width (in the y direction) W. This pattern was first observed in detail by Rowel17; it had been predicted by Josephson’. It is of course no coincidence that the dependence is on the number of flux quanta contained in the junction region, as we shall show. Since a flux quantum 2e/hC is -2 x lo-’ gauss-cm2, and 21even the earth’s field is adequate to perturb reasonable size tunnel junctions, which accounts for most of the early failures to notice the effect (as suggested by Anderson1.6). In large or high-current junctions the tunnel current is big enough to influence H,,: there is a kind of one-dimensional Meissner effect noted independently by Josephson4, Anderson6 and Ferrell and Prange34. If we insert the tunnel current J, = sin(w, - w 2 ) into Maxwell’s equation dHy 4nJ2 -=-ax c ’ we obtain

Linearizing the sin for small signals, this is d2A, --

ax2

1 --Az,

A;

(4.9)

in close analogy to London’s equation for the Meissner effect, with

(4.10) References p . 42

24

P. W. ANDERSON

[CH.1,s 4

Then (4.8) can be written wz)

aZ(W1-

1

sin(w, - w2). (4.8‘) 2; A junction larger than 2, excludes the field for small fields, and may break down entirely or exhibit more complicated effects for larger ones, rather than showing the full interference pattern of (4.8), a phenomenon also observed by Rowel17. It seems not to have been noted previously (except see De Gennes35 and Josephsonls, Appendix) that (4.8) has solutions corresponding to a onedimensional version of a single quantized vortex, as well as to a one-dimensional “Abrikosov array” of quantized vortices. That is, it exhibits in a particularly simple way the complete range of type I and type I1 superconducting behavior. It may be amusing and rather instructive in understanding the relationship between the interference phenomena and macroscopic superconductivity to describe these solutions briefly. Let w 1 - w2 =q. The first integral of (4.8‘) is

axz

AJ

=

dqo - = [2(1 dx

~~

+ 2 c - cos q)]*

(4.11)

which is a differential equation which may be solved in terms of an elliptic integral:

-- xo - [d(&p) (C + sin2 &p)-*.

(4.12)

J

We may distinguish two cases, which turn out to represent type I and type I1 behavior. TypeI: ascp+Oor2nn, dq/dxcCH-0. This is the only case in which, in a largejunction, the field can actually decay to a negligible value within the junction, i.e. J(a sin cp) and H+O together. By (4.1 1) this means C=O. The solution of (4.12) is then a degenerate limit of the elliptic function: x - xo (4.13) = log tan i c p . ~

AJ

(4.13) represents an isolated “linear vortex” at the point xo. The field ccdqldx and current ccdZq/dx2for such a solution are shown in Fig. 4. Note that the total flux J(dq/d.x)dx=2n; this corresponds to exactly one quantum of flux Q0 =hc/2e in the junction. References p . 42

CH. 1,s 41

JOSEPHSON EFFECT

25

Fig. 4. “Vortex” solution of Josephson equation for junction with applied fields. “H” and “J” are proportional to the field and current respectively: H is in units of Hci, J in units of UI.

To satisfy the boundary condition at the edges of the junction imposed by an external field, we may also use the solution (4.13) if H,,, is less than the maximum shown in Fig. 4, which is

(4.14)

This maximum field was noted by Ferrell and Prange34, who gave an equivalent solution to (4.13), as well as by Josephson 18. It represents one quantum of flux in a length nA,. Type 11: once H>H,, we cannot fit the boundary conditions with C=O. Set (4.15)

References p . 42

26

P. W. ANDERSON

[CH.

1, 8 4

and (4.12) is transformed into a standard Jacobian elliptic function : (4.16) or (4.17a)

T) kn, -

dq‘

2 x-xo dx =% dn(

=

2

(1

k2 sin2 $q’)*.

(4.17 b)

(4.17b) tells us that H,,

HC 1 =Happ= - = (1 k

+ C)*Hcl, (4.18)

Hmin= Hap,(1 - k2)* = Happ

The period of the vortex structure is given by the complete elliptic integral K, which is a logarithmically decreasing function as H increases above Hcl, approaching a constant as H+ co.

(x-xdh, Fig. 5. A “type 11” solution of Josephson equation. The parameters are Hex$ = Hms, 1.118 Hci, period = 2.257 15 = 1.44 H / @ o t . References p . 42

=

CH. 1,

41

JOSEPHSON EFFECT

27

As an illustration we have plotted out the structure in Fig. 5 for the case

k Z = l / ( l+C)=O.8; C=0.25. This corresponds to H,,,=1.118 !Icl.As Fig. 5 shows, H fluctuates by slightly more than SO%, but y ( x ) is surprisingly linear. An interference experiment at this field would lead to qualitatively similar results to k2+0, or +a, while for H/H,, greater than 1.5 or 2,H would be virtually constant and the screening effect unimportant. The biggest effect is on the period of the structure: K(0.8) is 1.44 times bigger than K(O), so the period is modified by 50%. The one dissimilarity with type I1 superconductivity we observe is that there is no natural Hcz. The supercurrents cannot be altogether stopped as long as the superconducting samples on the two sides retain their coherent fields, so the flux structure will last up to the critical field of the bulk samples. The net result is to point up the fact again that the Josephson effect is not a unique phenomenon unrelated to superfluidity as a whole but rather a kind of microcosmic manifestation of superconductivity in a particularly simple and understandable form. In this discussion it becomes clear that the Josephson interference pattern can be viewed as a question of fitting a vortex pattern into the appropriate boundary conditions, much as one might wish to discuss the thin film experiments of Tinkham36 and Parkslo. As in the optical case, while the simple “onsslit” interference pattern establishes the nature of the phenomenon quite adequately, much more beautiful and useful experiments can be done with more complicated systems, and in particular with the “two-junction” interferometer of Mercereaus. The principle here is again to study the critical current as a means of examining the relative phasing of the different parts of the Josephson current, but now two or more junctions are placed in parallel and their relative phasing controlled by electromagnetic fields in the loops connecting them. A rather general and simple way to understand such effects is to use Stokes’ theorem on the basic equation (4.4) for the gauge-invariant phase difference:

-

2e $Vw*dl= - - JH.dS( + 2nn). Ac

(4.19)

If we wish to compare the phase-differences w 1- w 2 at two junctions or two parts of a junction, AB and CD, in any interferometer circuit (see Fig. 2b on p. 16), we apply (4.19) to any circuit we may choose which includes the Referencesp . 42

28

P. W. ANDERSON

[CH.1, 8 4

(4.20) A

B

ABCD

For instance, in the single junction, 0, is everywhere parallel to the junction so that if we extend our circuits AC and BD perpendicular to the junction into the field and current-free interior of the superconductors, the line integrals on the right in (4.20) vanish and we recover the result we have already noted, that the phase difference depends only on the total flux in the junction between the points AB and CD, measured in flux quanta. Again,

Fig. 6. Mercereau interferometers (after Jaklevic et a1.9.s). (a) Cross-section of a Josephson junction pair vacuum-deposited on a quartz substrate (d). A thin oxide layer (c) separates thin (- loo0 A) tin films (a) and (b). The junctions (1) and (2) are connected in parallel by superconducting thin film links forming an enclosed area (A) between junctions. Current flow is measured between films (a) and (b). (b) The junctions (I) and (2) are connected in parallel by superconducting thin film links enclosing the solenoid (A) embedded in Formvar (e). References p . 42

CH.

1, B 41

29

JOSEPHSON EFFECT

in an interferometer circuit such as Mercereau's with a thick-film loop connecting two junctions, zlsl is very small and the total flux enclosed by the loop controls the phase difference. Fig. 6 shows the apparatus used by Jaklevic, Lambe, Mercereau and Silvers, 99 37 to demonstrate two-junction interference due to f H . d S in such a loop either from an external H or a small solenoid (the latter to demonstrate the reality of the AharanovBohm38 vector potential effect). The corresponding interference patterns are shown in Fig. 7. 4

I

>

d

H

-&

l

I

-500

l

l

-400

I

l

-300

l

l

-200

: l -100

I

I

0

I l 100

I

l

200

I

300

~ I

I I 400

I 500

I

I 800

MAGNETIC FIELD (MILLIGAUSS)

Fig. 7. Maximum supercurrent versus magnetic field for configuration similar to that of Fig. 6a with junctions of Sn-SnOa-Sn. For (a) the field periodicity is 39.5 mG, for (b) 16 mG. Approximate maximum currents are 1 rnA (a) and 0.5 mA (b). Configuration of Fig. 6b gives similar results but without the modulation envelope. (After Jaklevic et a1.s)

More interesting were two experiments to demonstrate the reality of the Su;dl terms in (4.20). In ones, one of the links AC was a very thin film of length I,, which was part of a loop exterior to the interferometer. The field of the exterior loop was well contained in it and contributed virtually no IH-dS to the interferometer loop. Then the phase difference is controlled by the surface current on the interferometer side :

s C

u;dl

=(use)-' lACj,= (n,e)- lAc($)(sinh

:)-',

(4.21)

A

where A is the penetration depth, t the thickness, and w the width of the thin film. Since n s N A - 2 , this gives a very strong dependence of the periodicity on A and thus on T which was observed. Fig. 8 shows the apparatus and results. References p . 42

-5

-10

-

0 5 10 DRIFT CURRENT

15

20

25

(ma)

7

Fig. 8 . (After Jaklevic et al.s) (a) Schematic of a junction pair (1) and (2) similar to Fig. 6, where the base film strip b carries a drift current which is returned beneath itself by a second base film b' designed to keep the field due to the drift current from the area enclosed by the junction loop. The insulating layers d are of Formvar. (b) Experimental trace of Zma, versus the drift current showing interference and diffraction effects. The zero offset is due to a static applied field. Maximum current is 1.5 mA. (c) Variation of observed driftcurrent period d h with temperature for two junction pairs of identical dimensions ( w = 0.5 mm and N.' = 8 mm). The curves are theoretical. The cross-section dimensions of the base film are 3 mm by 1100 *SO A.

CH. 1,O

41

31

JOSEPHSON EFFECT

Finally, a demonstration of the effect simply of mechanical motion was given, where us was caused by rotation of a circuit containing a junction pairso. It is interesting to go into the theory of this effect in a little more detail. If we rotate a bulk cylinder of superconductor, it is essential that over the sample as a whole us=ulaIIicc,because otherwise very large currents would flow. Since $Vq=O (or 2 n ~this ) means that

or

e

This is equivalent to the statement that the g-factor of the superconducting diamagnetism is exactly 2. The same B would appear in a rotating superconducting ring. Of course, under some circumstances B could be indeterminate to one or more additional flux quanta. In a ring containing Josephson junctions, finite currents can flow,and if in fact leA, we can make the opposite assumption that the magnetic field due to these currents is negligible, and that all the phase differencewill appear across the junctions. Then

s

Vq*dl- ( q l - q 2 )= 2nx

n

= 0,

k 1, ...

loop

where 'pi and q2 are the phase differences across the two Josephson junctions. This gives (m/A)2nR * ulaIIice 2nn = A q

+

so that the critical current of the rotating interferometer is a periodic function of Aq/2n = R 2 w m / k . (4.23) As Zimmerman and Mercereau30 point out, this experiment measures A/m and thus can be considered as a measurement of hjmc, the Compton wavelength; the value they quote is h/mc = (2.4 References p. 42

_+

0.1) x lo-'* cm.

32

[a. 1,g 4

P. W. ANDERSON

Josephson (private communication) has observed that there is an interesting “relativistic” effect which might be measured in this type of experiment. In both (4.22) and (4.23) the mass of the electron enters; the question is whether this mass is the rest mass or some kind of average inertial mass including kinetic energy corrections. Josephson argues that the correct mass is the rest mass corrected by the work function W,the energy difference between free space and the Fermi surface, i.e. that it is the total energy

2m0c2 - 2 w = 2m”c2 necessary to create a pair in the metal which determines the relevant mass m* which is to be used in (4.22) or (4.23). The argument is that the only gauge in which a Lorentz transformation gives simple results is that in which the potentials are zero outside the metal; then the frequency of J/ at rest (given by Aw =aE/aNpairs =2m*c2) is related to the wavelength in motion by a Lorentz transformation A la de Broglie. One nice way to see that not only the kinetic energy correction EF but also the potential energy V of the electron in the metal ( W =V-E,) must enter was also observed by Josephson. V may be thought of as the consequence of a surface electric field E in a layer d thick. The corresponding magnetic flux caused when this layer is rotated at a velocity u is @ = 2~

Rdv

-E C

RU

= 2n - V . c

The relative phase change caused by this is A-_q - _@J_ - R’w(2eV) 2n @o AC2 ’

the ratio of which to (4.23) is 2eV/mc2. As Josephson also points out, a rather larger effect can be induced by charging up the interferometer to high voltages, but this effect is rather trivially caused by the magnetic field of the moving charges. If it were wished to measure either of these effects, a basic limitation pointed out by Josephson is the difficulty of defining R of the interferometer in the rotating-interferometer experiment. Thus clearly the simple gyromagnetic experiment using an interferometer only as a fluxmeter and measuring the flux due to a rotating cylindrical sample will be much more accurate. The basic limitation there is thermal fluctuation of the flux, as discussed in the last section. The amount of flux is about 1 quantum/cps References p . 42

CH. 1,O 51

JOSEPHSON EFFECT

33

for a 1 cm3 sample, and the fluctuation in the same sample about 0.1 quanaccuracy we need very rapid and accurate rotation. tum; to get Perhaps more fundamentally interesting as an instrumental use of the interferometer is the measurement of gyromagnetic ratios in general magnetic materials. With conventional flux meters this is a very difficult experiment. Thermal noise is also a very severe limitation here, but it ought to be possible to get two or three-figure accuracy on ordinary paramagnetic materials. Of course, the interferometer has many other interesting device possibilities, especially as an extremely sensitive fluxmeter. One use which is not quite so obvious is as a sensitive ammeter. Because the input impedance is so low and its sensitivity to current so small, the energy sensitivity is many orders better than any other device, as demonstrated by CIarke 40.

5. Systems other than tunnel junctions showing interference phenomena Before going into the field of a.c. interference effects, a majority of which seem to have been observed in systems other than uniform tunnel junctions such as we have so far discussed, it may be worthwhile to have a brief section on these other types of “weak superconducting” or “weak superfluid” systems. In Josephson’s second paper 4, ZQ, the generalization from the pure Josephson effect to the general idea of “coupled superconductors” had already been made; and from the first it has been clear that it is not very easy to tell experimentally whether one is dealing with a tunneling supercurrent or a supercurrent flowing through a number of tiny metallic shorts. In fact many different systems have shown the extreme magnetic field sensitivity of critical current characteristic of the interference devices: tubular thin filmsz, flat long thin film bridges10, short thin film bridges in various geometries10.11, various kinds of point contacts32 and pressure contacts30, and even rather thick, wide films driven near critical current (Mercereau, private communication). Only one interference phenomenon has been observed in superfluid helium, the reason being that there are no good analogues to the Josephson effect except for the orifice geometry, and even that is either too strong a coupler or carries critical currents too small to measure conveniently12724. In the past there has been controversy in some instances about whether a given experiment on thin films in magnetic fields was better explained in terms of the static equilibrium of a system of vortices in the presence of the appropriate boundaries, or of a Josephson-like critical supercurrent resulting from interference between different parts of the supercurrent-carrying path 10. It must be clear from the theory we worked out in the last section, of the Referencesp . 42

34

P. W. ANDERSON

[CH.1,

55

Josephson junction in a magnetic field, that there is no distinction in principle between these two points of view. A Josephson junction exhibiting interference is a system of vortices in equilibrium under the appropriate boundary conditions. Conversely, the general theorem24 that the current is & = ( d U / d ( q , -qp,)>, the derivative of the energy with phase, indicates that a stable system of vortices - and therefore one which has a lower energy relative to other possible configurations with different phase difference ( q l - q Z )- can carry a larger supercurrent and thus will exhibit a lower effective resistance. As we deal with these systems we have also to lean more and more on the concept of “phase slippage”24*12. This concept proceeds from the Gor’kov-Josephson frequency condition (2.9) or, more generally, (3.7) :

That is, in any coupled superconducting system the appearance of a voltage (electron chemical potential ,u) difference can only mean that there is a corresponding rate of “phase slippage”. An applied current source represents a driving force causing the phase to slip, while the coupling energy tends to lock the phases together. In general, because of thermal and other fluctuations the phase will slip at a finite rate, which may be infinitesimal, as in a reasonably strong superconducting contact, or quite rapid, as in a thin film bridge very near T,. The more stable the structure of the vortex system at the contact and the less current is being forced through the slower will be the slippage, and thus the smaller the resistance shown - often, unmeasurably small. From this point of view all d.c. interference experiments are measurements of the rate of phase slippage vs. current and applied field. In the Rowell or Mercereau experiment, one has a sharp distinction between subcritical and supercritical currents, because the free energy barriers are rather high and the system is either stable or breaks down into very rapid flow above J,. In the Parks-Mochel10 experiments on thin film bridges, the barriers are relatively low (because near T,)and the dissipation high when phase slippage starts, so that one appears to be studying resistance in the transition region. In these experiments the resistance of a very narrow thin film bridge near T, is measured as a function of magnetic field, and found to exhibit quasiperiodic structure reasonably closely related to the fields at which an Abrikosov structure containing n whole vortices might fit in the film’s width (in a perpendicular field all thin films are type 11). The analogy with the References p . 42

CH. 1,

8 51

JOSEPHSON EFFECT

35

Rowel1 type of interference is clear: when a structure containing n vortices fits exactly into the film, the free energy is minimized, the structure is stable and phase slippage is hard. One might even suspect that the original Little-Parks experiment on thin film cylinders2 might well have a phase-slippage explanation. The usual explanation, that T,is modulated by the magnetic field, does not discuss the heart of the matter, which is by what mechanism the sample shows simultaneously resistive and superconducting properties. Phase slippage is occurring - then why not discuss directly how it is modulated by a magnetic field? Intermediate in character between point contacts and the Parks bridges is the Anderson-Dayernll thin film constriction. D.c. interference phenomena of great complication have been observed with these, but the main advantage, as with the point contacts, is the strong coupling to external electromagnetic fields which this geometry allows. The difficulty with this geometry for d.c. magnetic fields is that the entire film becomes type I1 in very small fields, and that therefore some resistance is developed almost everywhere. Only in nearly zero magnetic fields is the weak link clearly localized at the constriction. Virtually no theoretical work has been done on these thin film constriction devices. The critical currents have the right order of magnitude and temperature dependence to be explained by simple depairing, but a theory in terms of vortex motion across the constriction has not been ruled out. Particularly interesting is the strange and unexplained increase in critical current with a.c. power -the “Dayem effect”42. Now, we come to the various point contact devices. Many of the interferometer experiments of the Ford group31.39.44 were carried out actually not with tunnel junctions but with small Nb screws in pressure contact with Nb wires. Presumably, such a contact contains a thin oxide layer squashed between the metals (though a tiny true contact would probably serve as well) through which very local tunneling takes place. Point contact junctions have also been used by Grimes et al. in studies of the a.c. Josephson radiati0n29.9~. Clarke’s sensitive ammeter40 simply used blobs of solder on superconducting wires, making poor contacts which exhibited Josephson behavior. The final system possible is that in which the weak link is artificially created out of a strong one by driving high current, through it part of the time. This type of device shades imperceptibly into the measurement of flux quantization in truly macroscopic systems. While it can be quite useful as an interferometric device it does not really belong in the present article. References p . 42

36

[CH. 1,s

P. W.ANDERSON

6

6. Ax. quantum interference effects

In his original letter1 Josephson laid the groundwork for the theory and observation of the two closely-related alternating current interference effects: the appearance of a.c. supercurrents through weak links when a finite potential difference is applied across them, and the appearance of singularities in the d.c. I vs. Y characteristic at voltages such that: nhw = n’(pl - p 2 )

(6.1)

when external a.c. currents are applied at angular frequency w . Both of these phenomena are manifestations of the basic Josephson-Gor’kov (and one might as well add Einstein) relationship (3.7): dq dt

A-=p=-

aE dN

which simply states that the frequency of oscillation of the coherent matter field is given by the chemical potential (for electron pairs in the case of superconductivity). Since the supercurrent is a periodic function of q1- q2 (by the general principle of gauge invariance, all physical properties must be periodic functions of the phase with period 2n) this means that a.c. supercurrents must be associated with any voltage difference. In Section 3 we discussed rather fully why we assume that the state is a coherent one in these a.c. experiments; and the deeper background for the fundamental equations has been reviewed elsewhere24. Here we should like to first give an elementary discussion of the driven Josephson effect; write down the basic dynamical equations of the Josephson junction; and then to discuss briefly various experiments which have so far been done. By far the easiest, most general, and at the same time most accurate of the various interference experiments is the “driven” or “synchronized” a.c. effect. In the elementary derivation as given by Josephson1 one imagines a simple tunnel junction across which is maintained both a d.c. voltage Vo and an alternating voltage V, cos (w,t qo).The current (if w, and Yoare reasonably small, so that the Josephson-Riedel variation of 5, with o is not important) is given by

+

2e

= J~ sin -

(h

References p . 42

vat + 2e V, sin(o,t ~

hwa

+ cpo) .

)

CH. 1,

5 61

JOSEPHSON EFFECT

37

This is a simple frequency-modulated a.c. current which may be Fourier analyzed in terms of Bessel functions. In particular, whenever 2eV0 =nhw,, the current is perfectly periodic with frequency 0,/27c, so that all of the energy is in harmonics of 0,.One of these is the zeroth harmonic, or d.c., the amplitude of which is given by

when 2eV0 = nhw, . Jn is the nth order Bessel function. The important thing about (6.3) is that it is indeterminate: the mean current J, and thus the total power which is being fed from the as. source which maintains V , to the d.c. battery which maintains Vo

depends on the arbitrarily assigned phase angle cpo between the applied a.c. field V, and the spontaneous Josephson current at frequency 2eV0/h=coo. This means in turn that as we modify the characteristics of the external circuit which is supplying Vo so that the power drawn from it varies, the system can accommodate itself to such modifications by changing the phase cpo; thus for some large class of possible external circuits we expect to be able to observe different values of the current J f o r the same value of voltage V,: there will be a segment of the I-V characteristic where V is fixed at a finite value (1) and I varies, at many of the harmonics V0=nhwa. This phenomenon was indeed observed by Shapirog, who even verified the Bessel function dependence on V, in a rough way. For an example of the theory of this observation, let us imagine that the d.c. source is a constant voltage source V, applied through a resistance R which determines the load line:

As we vary V,, Vo will stay fixed and J will satisfy J=( V, - Vo)/Runtil J is greater than the appropriate critical current, which under ideal conditions will be given by

References p . 42

38

P. W. ANDERSON

[m. 1, Ei 6

To this simple discussion we need to make two addenda. First, suppose that the device involved is not a simple Josephson junction but, just for an example, a junction wide compared to Ap Then, the phase on the two sides can be varied only by passing “one-dimensional vortices” such as shown in Fig. 4 through the junction. The total current from such a vortex deep inside the junction is zero, but when it is near the surface a surface current will flow in the appropriate direction as discussed in our theory of the large junction. Thus the Josephson cursent as a function of phase or of time will be highly anharmonic. The equations of motion of such a system will be discussed briefly later; they are of such a complexity that no time-dependent solutions have been given, to my knowledge. In any case (6.2) will be replaced by

where f is some anharmonic but periodic function. Iff is anharmonic it is easy to see that the nth harmonic of 2eV0/h may beat with the mth harmonic of w, and give a d.c. current at “subharmonics” as well as “harmonics” of the fundamental voltage hwa/2e. This phenomenon was noted by Shapiro in some tunnel junctions, but is particularly striking in the measurements of Dayern11s42 on thin film bridges, as might well have been expected, since in the bridges the motions are almost certainly more vortex-like than sinusoidal. We show a typical I-V characteristic for an irradiated bridge in Fig. 9. Dayem makes the point (private communication) that it is probably because of the large harmonic content of his a.c. Josephson currents that radiation from his bridges was not directly observed. The second important point is that the basic nature of this type of effect is analogous to two rather similar nonlinear devices with whose operation we are fairly familiar, though in fact in each case the mathematical theory, beyond the simplest semi-intuitive considerations, is extremely complicated : the a.c. to d.c. power converter, and the synchronization circuit. We have already pointed out the a.c. to d.c. power conversion idea, and this analogy was extensively discussed in another review 24 : the “locomotive” analogy. The second analogy is to a synchronization circuit such as is contained in all cathode-ray oscilloscopes; we can think of the Josephson current as a freerunning oscillator, the frequency of which is read as the voltage Vo, and the a.c. signal applied as a “synch.” signal which triggers each cycle of the freerunning oscillator in phase synchronism with the external frequency, by means of the nonlinear coupling inherent in the Josephson effect. Both References p . 42

M. 1,9 61 2.

39

JOSEPHSON EFFECT

iiir I I

11

I

I

I

I

I

I

I

!

I !

!

I

1

SAMPLE NO.

9Ini

1

1.

I.

4

E

0

0

0

0.

2

’

’

2 2 3

4

1

1 5 4 3

.2L

r 3

2

3

Fig. 9. Experimental I-V curves for a Dayem bridge irradiated with microwaves at a frequency of 4.62 gigacycle. Relative microwave power in dB is the parameter. Voltage fluctuations in the steps are an experimentalartifact. (After Dayern and Wiegar~d~~.)

analogies indicate that the stronger the synchronizing signal, the more tightly controlled will be the Josephson frequency and therefore the voltage, and that for strong signals thermal noise will not be a factor. For this reason we can expect the relationship between V, and w, to be as exact as can be measured: this is, as originally emphasized by Josephson, by many orders the most accurate possible way to measure e/A. The synchronizationconcept (originally introduced in Ref. 5 ) indicateswhy the a.c. effect is so easy to see and so general. The relationship (6.1) between References p. 42

40

[CH.1,

P. W. ANDERSON

56

chemical potential and frequency is perfectly general, whether the phase changes with time by means of vortex motion, direct phase slippage as in a Josephson junction, flow of an Abrikosov structure, or whatever. Equally, we can expect that with strong enough applied synch. signals, almost any such motion can be forced to be regular and periodic. This is the reason we felt the best bet for an interference experiment in liquid He would be the driven ax. effect, and indeed the experiment did give positive results 12.This is the only interference experiment as yet successfully carried out in a precisely analogous way on both superconductors and helium 11. In view of the numerous experiments on the “synchronized” a.c. Josephson effect, there was no doubt that the a.c. Josephson currents existed, but the possibility of direct detection of radiation from them seems still to have stimulated very considerable effort and interest. The first successful attempt was by Giaeverl4 utilizing a second, different thin film device as his detector; thereafter two other groups reported completely external detection of the radiated powerl5. Intrinsic limitations on this power (coming essentially from the self-screening effect) are such that it can never be of much practical importance except possibly as a spectroscopic source in the infrared 29. All of these observations utilized the phenomenon of “Fiske steps”l3 which is undoubtedly a self-synchronization of the Josephson current to a.c. electromagnetic signals fed back from cavity modes of the tunnel junction structure. In discussing these it may be well to write down the equations of the dynamics of tunnel junctions. The dynamical term is easily inserted if we return to the derivation of equation (4.8) from the Maxwell’s equation for curl H . We must include the displacement current term in that equation if we are to allow time-dependence, so we start from

aH, 4x -=--J,+ ax c

1 ao,

--.

c

at

But now by (4.5’)

a% 2e -- -&,

ataZ

A

so that we get for the time-dependence of the phase shift a (wl

at References p. 42

- w 2 ) = -2ed E A

*

CH.

1,

5 61

JOSEPHSON EFFECT

41

Now we introduce the relationship (4.6)between phase shift and H,, into (6.5):

a2

kc

=

47T -

C

J1sin(w, - w2) +

he d2 -- (wl - w 2 ) 2edc at2

(e is the dielectric constant of the layer) which is the equation first written down by Josephson18 for cp =w1 - w 2:

where I , has been defined already (4.lo), and v, the effective electromagnetic wave velocity in the planar structure, is v=(

)e .

d e(2I d)

+

(6.7)

In general, this is 10-100 times slower than c, so that for usual junction sizes resonant modes of the purely electromagnetic 1.h.s. of (6.6)are to be expected in the microwave region. Josephson has given a very succinct discussion of two types of resonances which can be expected to follow from (6.6).Both may be understood best if we consider I , to be rather large. First is the Fiske step, which occurs at resonances of the 1.h.s. plus boundary conditions. We can think of the sin rp term as simply a weak driving term which generates a fairly large electromagnetic response, which in turn reacts back via the nonlinearity of sin cp to give a kind of self-synchronization at 2eV0 =hwres.Since the effective d.c. impedance is very low at a Fiske step, the breadth of the spectrum due to voltage fluctuations is relatively small and the radiation is easier to detect. A second type of resonance was discovered by Eck et al.43. In a magnetic field, as noted in Section 4, the junction takes on a periodic structure of wavelength about I QO/H,,(2A+ d ) . When this wavelength is equal to the electromagnetic wavelength A = 2 n v l o = hv/2eV

-

we can again expect a resonance, which turns out to be rather broad, and to show up only as a broad extra dissipation dependent on magnetic field. Dayem and GrimesS2 have observed radiation from point-contact tunnel junctions; here the resonant, low-impedance structure is avoided and relatively good coupling may be achieved to the radiation field. (That was one References p . 42

42

P. W. ANDERSON

[CH.

1

of the original purposes of the Anderson-Dayem structure l1, but the point contacts do not have the disadvantage of large anharmonicity.) Most interesting, however, is the inverse performance of these junctions as infrared detectors when biased near their critical currents so that the appropriate dynamical equations are near a point of instabilityzs. The appropriate equation for such a system (Werthamer and Simon, private communication) is azq drp + wo2 sin rp = J,, sin (oat+ q 0 )+ Jdc- R , (6.7) atz dt ~

where J,, is the applied signal of frequency o,,Jd, the bias current, and R drp/dt is the radiation and other resistance, When Jdo=o$, (6.7) may be expected to be extremely sensitive to the J,, term. It remains, in this as in most cases, my opinion that the great future of the a.c. Josephson effect lies not in the straightforward direction of using the radiated power but in their unique possibilities as nonlinear driven devices, as detectors, as phase-locked discriminator systems which convert an a.c. signal directly into a voltage, or in yet more interesting and complicated devices. Notes added in proof. The following are two developments of special interest occurring since the article was sent to press: 1 ) Kharana and Chandrasekhar45have repeated the measurements of Ref. 12 with improved equipment and results. 2) Parker, Taylor and Langenberg46 have determined 2e/h to a precision greater than previous measurements (2e/h=483.5913 f0.0030 megacycles/ microvolt), and great enough to be relevant to quantum electrodynamical effects. REFERENCES

5

8

B. D. Josephson, Phys. Letters 1,251 (1962). W. A. Little and R. Parks, Phys. Rev. Letters 9 , 9 (1962). R. C. Jaklevic, J. Lambe, J. E. Mercereau and A. H. Silver, Phys. Rev. 140, A1628 (1965). B. D. Josephson, Trinity College Fellowship Thesis (unpublished). P. W. Anderson, in: Lectures on the Many-Body Problem, Ravello 1963, Vol. 2, Ed. E. R. Caianello (Academic Press, 1964) p. 115. P. W. Anderson and J. M. Rowell, Phys. Rev. Letters 10, 230 (1963). J. M. Rowell, Phys. Rev. Letters 11,200 (1963). S. Shapiro, Phys. Rev. Letters 11, 80 (1963); S. Shapiro, A. R. Janus and S. Holly, Rev. Mod. Phys. 36, 223 (1964). R. C. Jaklevic, J. Lambe, 3. E. Mercereau and A. H. Silver, Phys. Rev. Letters 12, 159 (1964).

CH.

11

JOSEPHSON EFFECT

43

R. D. Parks, J. M. Mochel and L. V. Surgent, Phys. Rev. Letters 13, 331a (1964); R. D. Parks and J. M. Mochel, Rev. Mod. Phys. 36,284 (1964). P. W. Anderson and A. H. Dayem, Phys. Rev. Letters 13, 195 (1964). l2 P. L. Richards and P. W. Anderson, Phys. Rev. Letters 14, 540 (1965). l3 D. D. Coon and M. D. Fiske, Phys. Rev. 138, A744 (1965). l4 I. Giaever, Phys. Rev. Letters 14,904 (1965). l5 I. R. Yanson, V. M. Svistunov and I. M. Dmitrenko, Zh. Eksperim. i Teor. Fiz. 47, 2091 (1964) [English transl.: Soviet Phys.-JETP 20, 1404 (1965)l; D. M. Langenberg, D. J. Scalapino, B. N. Taylor and R. E. Eck, Phys. Rev. Letters 15, 294, 842 (1965); Proc. IEEE 54,560 (1966). IR J, E. Zimmerman and A. H. Silver, Phys. Rev. 141, 367 (1966); Solid State Commun. 4,133 (1966); and Refs. 3 and 30. l 7 M. H. Cohen, L. M. Falicov and J. C. Phillips, Phys. Rev. Letters 8, 316 (1962). B. D. Josephson, Advan. Phys. 14,419 (1965). l8 P. G. DeGennes, Phys. Letters 5,22 (1963). 20 J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108,1175 (1957). 21 L. P. Gor’kov, Zh. Eksperim. i Teor. Fiz. 36, 1918 (1959) [English transl.: Soviet Phys.-JETP 9, 1964 (1959)l. 22 J. R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1964). P. W. Anderson, N. R. Werthamer and J. M. Luttinger, Phys. Rev. 138, A1157 (1965). a4 P. W. Anderson, Rev. Mod. Phys. 38,298 (1966). 26 N. R. Werthamer, Phys. Rev. 147,225 (1966). 26 E. Riedel, Z. Naturforsch. 19a, 1634 (1964). 27 D.J. Scalapino, Phys. Rev, Letters, to be published. 28 C. C. Grimes, P. L. Richards and S. Shapiro, Phys. Rev. Letters 17, 431 (1966). B. D. Josephson, Rev. Mod. Phys. 36, 216 (1964). 30 J. E. Zimmerman and A. H. Silver, Phys. Rev. 141, 367 (1966). 31 A. H. Silver, R. C. Jaklevic and J. Lambe, Phys. Rev. 141, 362 (1966). A. H. Dayem and C . C. Grimes, Appl. Phys. Letters 9, 47 (1966). 33 F. London, Superfluids, Vol. 1 (Wiley and Sons, New York, 1950) Section 8. 34 R. A. Ferrell and R. E, Prange, Phys. Rev. Letters 10,479 (1963). 35 P. G. DeGennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966) p. 240. 3~3M. Tinkham, Rev. Mod. Phys. 36,268 (1964). 37 R. C. Jaklevic, J. Lambe, A. H. Silver and J. E. Mercereau, Phys. Rev. Letters 12, 274 (1964). 8* Y. Aharanov and D. B o b , Phys. Rev. 115,485 (1959). 99 J. E. Zimmerman and J. E. Mercereau, Phys. Rev. Letters 14, 887 (1965). 4o J. Clarke, Phil. Mag. 13, 115 (1966). 41 J. E. Mercereau and L. L. Vant-Hull, Phys. Rev. Letters, to be published. A. H. Dayem and J. J. Wiegand, Phys. Rev., to be published. These results were repeated by A. F. G. Wyatt, V. M. Dmitriev, W. S. Moore and F. W. Shard [Phys. Rev. Letters 20, 1166 (1965)l. R. E. Eck, D. J. Scalapino and B. N. Taylor, Phys. Rev. Letters 13, 15 (1964). 44 J. E. Zirnmerman and A. H. Silver, Solid State Commun. 4, 133 (1966). QS B. M. Kharana and B. S. Chandrasekhar, Phys. Rev., to be published. 46 W. H. Parker, B. N. Taylor and D. N. Langenberg, Proc. Xth Conf. Low Temp. Phys., Moscow, 1966, to be published; Phys. Rev. Letters, to be published. la

C H A P T E R I1

DISSIPATIVE AND NON-DISSIPATIVE FLOW PHENOMENA IN SUPERFLUID HELIUM BY

R. DE BRUYN OUBOTER, K. W. TACONIS AND W. M. VAN ALPHEN KAMERLINGH ONNES LABORATORIUM, LEIDEN

CONTENTS: Introduction, 44. - 1 . Superfluidity,the equation of motion for the superfluid, 45. - 2. The critical superfluid transport in very narrow pores between 0.5 “K and the lambda-temperature, and the impossibility to detect Venturi pressures in superfluid flow, 54. - 3. Superfluid transport in the unsaturated helium film, 62. - 4. Dissipative normal fluid production by gravitational flow in wide channels with clamped normal component, 64. 5. The dependenceof the critical velocity of the superfluid on channel diameter and film thickness, 72.

-

Introduction Complementarily to earlier work reviewed in Progress in Low Temperature Physics on the properties of superfluidity (see also Andronikashvili and Mamaladze, and Anderson, in this volume) a research program of some interest has been accomplished in the Kamerlingh Onnes Laboratory. A number of flow experiments performed with He11 will be reported here. It concerns sub- and super-critical flow of the superfluid. The experimental work will be treated in four Sections (2-5) and is preceded by a short summary of the properties of a superfluid, including the equations of motion (Section I). The picture may facilitate the discussion of the presented material. The first series of experiments (Section 2) concern a new attack on the characteristics of the flow through pores. The gravitational flow is determined under the stringent condition that the temperatures at both sides of the “superleak” are known as precisely as necessary for an eventual fountain pressure correction. It was found that indeed real critical flow exists which is pressure and temperature independent, as it should be ; in contradiction References p . 76

44

CH.

2,p 11

FLOW PHENOMENA IN SUPERFLUID HELIUM

45

with many earlier experiments in which apparently small unobserved temperature differences gave rise to misleading results. The second study (Section 3) is directed to still smaller channel dimensions formed by the thickness of the helium film. It substantially increases the data on flow of the saturated and unsaturated films, especially at lower temperatures, down to 0.7 O K . In Section 4 a new technique is described which is introduced to detect succesfully the critical velocity in wide channels by means of measurements of the onset of energy dissipation in the superfluid flow using a calorimeter, whereas the dissipation above the critical velocity as a function of u, is found to obey a very simple formula for the observed friction. Finally in Section 5 a proposal is made for a general empirical relation between the critical velocity and channel dimension. It originates to a certain extent from the observation that earlier trials in this respect have been mislead by the fact that in wide channels often the turbulence of the normal fluid is taken as the appearance of a critical velocity of the superfluid. 1. Superfiuidity, the equation of motion for the superfluid

The equation of motion for the superfluid can, according to Landaul, be derived in the following way: in a simply connected amount of superfluid, the change in energy dU, caused by transport of a small quantity of superfluid dM, in or out of a considered part of the system, keeping the momentum p of the normal fluid constant with respect to the superfluid, is equal to (aU/aM), dM. Only superfluid is transported and the distribution of the normal fluid remains unchanged. Since the superfluid carries no entropy, one has (dUjdM),,, d M = & dM, in which p& is the chemical potential of He11 per unit of mass. The change of energy by a supeAuid transport of a unit of mass is equal to the chemical potential p. This quantity can be considered as the potential of the force acting on the superfluid; hence one gets the following equation of motion for the superfluid per unit of mass dvs -=dt

0 = - -Vp 1 VpM P

or

P S

= - -Vp

P

References p. 76

+ p,SVT,

+ SVT

R. DE BRUYN OUBOTER et

46

al.

[CH.

2, 5 1

whereas for duJdt =O (or V& =O), eq. (2) delivers the fountain relation of H. London: dp/dT=pS. However, superfluidity is restricted to certain possible types of flow by the restriction that the flow is irrotational (Landau', London2 and Feynman 3)

v A Us = 0.

(3)

This implies that the superfluid velocity us is the gradient of the classical velocity potential p; us=Vp and aus/at=(a/af)Vp=V dp/at. This gives us the following equation of motion (per unit of mass) for an ideal irrotational liquid

pM is the generalized chemical potential defined by pM = p k The equation of motion per 4He atom becomes

m -a U=S at

V(p0 + imo,2) = - VP,

+

41):.

(5)

in which rn is the mass of the 4He atom, p o =m& its chemical potential and p = m p M its generalized chemical potential (eqs. (4) and ( 5 ) are valid for a steady flow, when the normal fluid velocity u, =O), if du,/at =O the equation of motion gives us the theorem of Bernoulli VpM = v ( p i

+ 30:)

(6)

=0

(and if V A us was initially zero it will always remain zero). Instead of eq. (4) Landau considers also the more general case in which u,#O and pn#O ( p , ps= p ) and gets the equation

+

The general chemical potential p contains in this case also a kinetic energy term of the relative motion of the superfluid and normal excitations. For a steady flow

1

=o.

A unique feature of the superfluid is its single macroscopic quantum state for a11 condensed helium atoms. As a consequence of the Bose-Einstein condensation an enormous number of helium atoms are in exactly the same quantum state and have exactly the same wave function $,. According to References p . 76

CH.2,

5 11

FLOW PHENOMENA IN SUPERFLUID HELIUM

47

Feynman4 and Anderson5 t,bzt,bs can be interpreted as the density of the All condensed helium atoms move as a condensed helium atoms p s / p =l1,9~1~. unit rather than individually. Another way of reasoning is to interpret yS=,/(p,/p) e*@ as the internal order parameter of the condensed superfluid, which is a completely determined, single valued, complex quantity (Anderson5, Ginzburg and Landaus, Ginzburg and Pitaevskii?) with a phase factor ei@(Beliaevs) - irt/r *s = J(Ps/P) ei4= J(Ps/P) e (7) The frequency of the internal order parameter is given by v = - - =1 - 84 -

c1

h'

2.rcat

A new feature appears when a superfluid occupies a multiply-connected region. For instance, suppose that the superfluid moves in circles around a cylinder (in principle this quantum state can be reached by cooling down liquid helium in a rotating bucket, containing a cylinder in the middle, through its lambda-temperature). Such a flow has a circulation K =

fus*dsZ 0 S

around the cylinder, although V A us-0 everywhere in the superfluid. The phase 4 of the wave function t,bs = J ( p , / p ) eibchanges by an integral multiple of 2n in going around the cylinder

S

S

This implies that the circulation K around the cylinder is quantized, as first was predicted by Onsagero and Feynman3

s

S

(9) h cm2 sec-' . I C = ~ - = 0.997 x m In this case of cylindrical symmetry the superfluid velocity us is given by u , = ~ / 2 n r(r > radius of the cylinder). Vinen10 and later Whitmore and Zimmermanl1 have experimentally verified this relation for the quantized circulation around a small wire by measuring the Magnus force (or lift force) on the wire. Reppy and Depatielz References p . 76

48

R. DE BRWN OUBOTER

et al.

[CH.2, $ 1

and Mehl and Zimmerman 13 have observed long-living persistent currents of superfluid in powder-filled vessels and have observed that the total amount of angular momentum present in such a sphere can be changed reversibly solely by warming up or cooling down. The amount of angular momentum of the persistent current is proportional to the superfluid density ps ( T ) , which means that the circulation around each flow loop within the sphere remains constant at varying temperatures. Only the amplitude J [ p , ( T ) / p ] of the wave function $s changes by warming up or cooling down, and the condensation (or evaporation) of the helium atoms occurs in a state of definite phase, that of the superfluid condensate wave function. Up to now we have mainly considered regions that are multiply-connected in a straightforward macroscopic manner, for example superfluid flow around a solid cylinder, but as we will discuss now it is possible to have a region that is indeed multiply-connected by having a line through the fluid on which the amplitude of the wave function $s vanishes. From experiments one knows that it is possible to have, well enough above absolute zero and below lambda, normal excitations (phonons and rotons) as well as macroscopic excitations, of highly localized regions of vorticity, in the form of singularities in the velocity field. The “normal” core of such a vortex or vortexring (radius approximately a few A) is excludedfrom what we call the superfluid, in this way forming a multiply-connected region. Except for these “normal” cores the irrotational condition must hold throughout the superfluid. The pressure in the vortex near its centre can maintain a very small cylindrical hole in the superfluid against the action of the surface tension (Feynman3). Rayfield and Reif14 have shown that ions in superfluid helium at low temperatures can be accelerated to create freely moving charge carrying vortexrings, the ions being probably trapped in the core of the vortexring. They measured their energy E and velocity v and found a welldefined relation of v versus E (v E - ’) from which the circulation u and the core radius a can be determined by means of the relations

-

’

[

E = i p s ~R In

raR)-(:)] -

and v = L4nR [1.(s>-(a)],

(10)

in which R is the radius of the vortexring, u is found to be one quantum of circulation, u,, =hlm and the core radius a=1.3 A. These freely moving charge carrying vortexrings are only observed at sufficiently low temperatures (0.3 OK).At higher temperatures a pronounced energy loss is observed due to collisions with the normal excitations, especially the rotons, and the vortexrings are slowed down. References p . 76

CH. 2,

0 11

FLOW PHENOMENA IN SUPERFLUID HELIUM

49

Since V A v,=O in the superfluid, its velocity u, can be defined as the gradient of a scalar velocity potential in such a way that the potential may be identified with the phase 4 of the internal order parameter (or wave function) I),(Onsager and Penroseg, Feynman3) by means of the relation A us=

m

v4,

where (h/m) 4 can be identified as the classical velocity potential q =(h/rn) 4 per unit of mass. This can be verified since by substituting eq. (10) into eq. (9) one gets eq. (8). Eq. (11) is satisfied provided the superfluid velocity us does not change significantly over a distance less than some characteristic distance (“healing length” or “coherence length”) of the order of the atomic distance in the liquid. Sometimes one also uses the integral relation 2

(12)

The well-known De Broglie wavelength of the moving superfluid follows from eq. (1 2) as

A=-.

il

mu, Using eq. (1 1) one can transform eq. ( 5 ) in the following way

v

[ ::

h-+po++rnu, 2]

=v

[ :: ] = o . A-+p

Eq. (14) has the same form as the general Bernoulli equation (4) of classical hydrodynamics for an ideal irrotational liquidl5. Integrating eq. (14) gives that the term between the brackets is at any instant equal to the same constant throughout the fluid, which may be still a function of the time. However, this constant will appear to be time independent and equal to zero. Instead of starting with the classical equation of motion (l), one can better derive directly from the time dependent Schrodinger equation

(describing the single macroscopic quantum state, with the same wave function $s =,/(p,/p) eie for all condensed Bose particles and a potential References p . 76

R. DE BRLIYN OUBOTBR et al.

50

[CH.

2,8 1

energy p) the equation of motion5

a4

+ po + at

h-

34

=h at

+ p = 0.

Eq. (15) is valid if the superfluid velocity varies slowly with position and if the amplitude of the wave function I)s is practically constant. If one also considers a gradient in the internal order parameter, one has to add a term (Ginzburg and Pitaevskii ')

to the left-hand side of eq. (15), a more general situation. Such a term is of importance if we have a boundary between the superfluid and the wall or normal core of a vortex. Moreover, sometimes is added an additional term involving the total density, for example in the helium film near the wall. Following Andersonl6.5, one may take the line integral of this equation along an arbitrary path which is stationary through the superfluid from a point 1 to a point 2. Suppose that the end-points 1 and 2 lie in a region where the fluid is nearly at rest and undisturbed, then one may neglect the term +mu: and one gets the integrated form of this equation

which describes, even when dissipation is present, the connection between the change in the chemical potential and the average rate of production of vorticity. When two baths of superfluid helium are connected with each other by means of a superleak, in every path in the superleak which connects both baths and which remains superfluid, the phase of the order parameter, will slip relative to the other bath as long as there is a difference in the chemical potential Ap. On the average in 1 sec the phase will slip Av =A p / h times 2n. The phase slippage takes place by means of motion of vortexrings (or by vortices). Since I), is single-valued the phase 4 of a vortex will change by n . 2 on ~ going around its normal core. Experiments indicate that free vortexrings have n = 1 as quantum number of circulation. If Av = d p / h vortexrings (excited states) are produced during 1 sec the phase slippage can take place in a proper way (Andemonla). The vortexrings produced in this way stream into one of these two baths and are slowed down by collisions with rotons (most likely radiating second sound). The vanishing vortexrings become References p . 76

CH. 2,g 11

FLOW PHENOMENA IN SUPERFLUlD HELIUM

51

probably eventually rotons by themselves’@‘. In this way the difference in the chemical potential energy is converted into heat by producing normal fluid. According to Andersonle*5 dissipative effects in superfluids follow from this concept of phase slippage and the gradient in the generalized chemical potential p=po++mu: gives the average rate of vortex production. Normal fluid counter flow and the dissipation phenomena have probably no influence on the phase slippage concept. Richards and Anderson16 have observed this rate of vortex production (phase slippage) at a small orifice by synchronizing the vortex motion to an ultrasonic frequency (the so-called a.c. Josephson effect) in which an a.c. flow is superimposed on the d.c. flow. Furthermore one can formulate a law of induction of circulation (vorticity) by integrating the equation of motion (1) along a closed integration line in the superfluid

S

S

when superfluid flows through a We now determine the critical velocity small orifice, with radius r, from one helium bath to another when there is a difference in its chemical potential A p between both baths, by supposing that A v = A p / h vortexrings (eq. (16)) are produced in 1 sec. The amount of energy which flows per unit of time from one bath to the other is equal to ijss

2

n, A y = E Av or &, =

E n,nr2h ’

n, is the number of condensed helium atoms per unit volume (ps=nsm). Suppose E is the energy necessary to form one vortexring with radius R =r, which is given by eq. (10)’ substituting for E eq. (lo), for Av eq. (16), and for K = K ~=h/m in eq. (17), gives

This expression gives an explanation for the critical transfer rate, and is nearly equal to the familiar Feynman’s expression17

We would like to remark that eq. (18) is independent of LIP (see Section 3). The values predicted by these equations are, however, not in agreement with References p . 76

52

R. DE BRWN OUBOTER et

[CH.2, 5 1

al.

the experimentally observed critical velocities. We come back to this point later in Section 5, in which is discussed the dependence of the critical velocity of the superfluid on the channel diameter and film thickness. The boundary condition for the superfluid gives a special problem (Ginzburg and Pitaevskii 7). When a superfluid is moving along a solid wall its tangential velocity us cannot gradually decrease to zero since V A v, =O. Just the opposite case we have in a viscous normal fluid flow, where v, =O at the wall. In the bulk superfluid the internal order parameter p , / p = l$J2 in general is constant and unequal to zero. However, generally on a solid wall and at the free liquid surface and on the axis of a vortex line the amplitude of the order parameter p , / p =1t,!~,1~ =O is zero. Summarizing the boundary condition one gets: in the bulk

at the wall PAP = l$,I2 = 0

A m

us= - V 4 # 0

superfluid t

=

I$,(T)I’

z0

fi

v A V, = 0 +

normal fluid

PnlP = 1

P,(T)ip

us = - v4 # 0 m pn/p

+ p,/p = 1 ,

p,/p # 0

(normal component) u, = 0

+

Poiseuille’s law

--f

v, # 0

For a normal fluid (and also for the normal component of HeII) the equations are much more complicated, because the effects of viscosity ( q ) have to be included and the liquid is no longer rotation-free V A u = 0. The flow of an ordinary incompressible viscous fluid is governed by the Navier-Stokes equation of motion

When all acceleration terms are zero (du/dt=O) (or nearly equal to zero) one gets Poiseuille’s equation for laminar flow vp = qv2v,

(21)

for which the exact solution is known. At high enough velocities in the completely turbulent region (du/dt#O) no exact solutions are known. In retrospect of a later discussion of effects of turbulence (Section 5) which are References p. 76

CH. 2,

5 11

FLOW PHENOMENA IN SUPERFLUID HELIUM

53

observed in He11 by Staas, Taconis and Van Alphenl*, one can define two dimensionless Reynolds numbers in the way introduced by these authors for a type of flow with a circular cross-section 2rpB Pr3 Re,=and Re,= t ~ p , 1 41

in which ij is the mean velocity over a cross-section and r is the radius of the tube. The relation between Re, and Re, for a turbulent flow can be obtained from the empirical equation by Blasius and is then represented by Re, = 4.94 x loL3

(23)

In laminar flow through a channel with circular cross-section the law of Poiseuille becomes rz (or: ij = - -Vp).

Re, = Re,

8rl

(24)

The transition from laminar to turbulent flow has been studied by Reynolds. The value of the critical velocity for onset of turbulence corresponds usually to a Reynolds number Re, =2rpij/q between 1200 and 2300. For the normal and superfluid component together one has an equation of motion of the form of eq. (20) du, dun + Pn- = - v p dt dt

Ps-

+ qnv2un,

in which q,, is the viscosity of the normal component. Combining eq. (25) with the equation of motion for the superfluid (eq. (2)) one obtains 'the Navier-Stokes equation of motion for the normal component dun dt

P -=--

Pn P

Vp

- p,SVT + ~ , , V 2 ~ , ,

(eq. (14) remains unchanged). Furthermore, sometimes one introduces the idea of mutual friction. The first description of the interaction between the normal and superfluid component was made by Gorter and Mellinkl9Y2Oin order to describe some aspects of thermal conduction in moderately narrow channels. They proposed a mutual friction force of the following form F sn = ~ ~ n ~ s I unIz u s (8s

References p . 76

-uJ*

R. DE BRWN OUBOTER et al.

54

[a. 2,§ 2

Adding this extra friction force to the hydrodynamical equations, they obtain

du, p,dt

+ pndun = - Vp + qnV2un dt ~

(eq. (25)).

However, in classical hydrodynamics the phenomenon of turbulence, for instance; does not require a revision of the fundamental hydrodynamical equations, but rather a consideration of special types of solution of the ordinary Navier-Stokes equation. The phenomenon of mutual friction is present in HeII, but this does not imply the necessity of adding those mutual friction terms in the equation of motion (eq. (25)). One may expect to find mutual friction between the normal and the superfluid when vortices are present in the superfluid, since the excitations constituting the normal fluid will be scattered by a vortex. With respect to this picture of mutual friction Hall and Vinen20 have investigated the attenuation of second sound in rotating helium. These experiments are fully discussed by Vinen20 in Progress of Low Temperature Physics, Volume 111, Chapter 1. 2. The critical superfluid transport in very narrow pores between 0.5 OK and the lambda-temperature, and the impossibility to detect Venturi pressures in superfluid flow An advantage in studying the characteristics of superfluid flow in narrow pores (“superleaks”) is the possibility to obtain a pure superfluid flow as the normal component is quite immobile due to its shear viscosity qn. Moreover very high superfluid velocities (us 1/42, the surface energy is negative and the material is of the second kind. Fig. 3 shows a schematic diagram of

Fig. 3. Structure of an isolated vortex line.

a vortex line. Typical values of 1, and ( are 100-1000 A. The ratio K is 0.026 for aluminium, and as high as 40 for V,Ga for instance. In 1950 London11 proved that the flux trapped in a superconductingring is quantized; this was experimentally codrmed by Doll and Nabauer12 and by Deaver and Fairbank13.As a special case one can show that a vortex line as defined above should carry a quantum of flux cDo = ch/2e =2 x 10- gauss cm2,or a multiple of this quantum. Saint-James and Sarmal4 have proved that for a large range of conditions a line should carry only one quantum. Thus, for an induction B in the sample the number of lines equals N = BIGo. Abrikosov also showed that these vortex lines form a regular array. In his model the structure in the mixed state can be seen as a regular array of fibrae parallel to the field. The distance between these fibrae decreases when the field increases from H,,up to Ifo2.This model gives a good explanation of

’

References p . I79

164

[CH.4,s

D. CRIBIER ef a[.

2

the behaviour of superconductors of the second kind, but up to 1964 there was no proof of the existence of these vortices. A splitting in laminae alternatively normal and superconducting has been proposed by Goodman and Gorter 15. This model explains the general macroscopic behaviour almost as well as the vortex model. De Gennes and Matricon16 suggested to check the existence of a regular array of these vortex lines by neutron diffraction techniques. The neutron, being sensitive to the magnetic field, will be diffracted by the regular structure of the field which is present in the mixed state.

2. Theory of neutron scattering by vortex lines The theory of this scattering16917is merely a special case of the usual theory of magnetic scattering of neutrons 18. The fundamental interaction takes place between the magnetic moment p a of the neutron and the magnetic field H . The scattering amplitude is given by m a (4)= 2ntiz j p n. H ( r ) eiqrd 3 r ,

(1)

where m is the mass of the neutron and q the scattering vector or momentum transfer in the scattering process. H ( r ) is the spatial field distribution. In the Abrikosov model the structure of the field has to be considered only in a plane perpendicular to the applied field, and it is always possible to describe it by a two-dimensional Fourier series

~ ( r=)A , C cos M\ r i

+ A , C cos M: + ..., o r

i

where MI,M,, ... are the vectors of the reciprocal lattice and the summation over the index i is extended to vectors of the same length. The diffraction pattern of such a field structure is similar to that of a crystal: the scattering amplitude is zero except for q =M , ...,where this amplitude is proportional to A,. In general it is not possible to solve the field distribution from the GinzburgLandau equations, so that explicit expressions for A,, A,, ,.. in terms of the fundamental parameters cannot be obtained except for two cases which will be considered now. a) Vortex lines of infinitely small core (t=0). Then the field distribution is given16 by

H

+ A:

rot r o t H = djoXd2(r - q ) , i

References p . I79

(2)

CH.4,1,2]

SUPERCONDUCTIVE MIXED STATE

165

where d2 is the Dirac function for two dimensions and ri the coordinate of the ith line. This case has been considered in detail by Kemoklidzel7. If we forget about the numerical constant we find for the scattering amplitude

In this case 1

For a polycrystalline structure* one finds17 for the integrated intensity of the Debye-Scherrer line

R2

1

‘“.“M,(l + R ,2M ,2)

2’

If we assume that A,lM,J 9 1, which is the case in all practical circumstances, the ratio of the intensity of the first Bragg peak to that of the next one is about 16 for a triangular lattice and about 6 for a square lattice. This calculation is correct for vortices of very small core. For the case of niobium which was used in the experiments, 5.-1,-400 A, so that the ratio of the first Bragg peak intensity to the second one should be even larger than these values. b) Quasi-sinusoidal field distribution. Kleiner et al.19 have calculated the ratio A 2 / A , , in the region close to Hcz, for any value of K . For the triangular lattice, which is found to have a lower free energy than the square one, this ratio is 2.5 x so that close to HCz the sinusoidal field distribution is a good approximation for any value of K. With such a ratio the intensity diffracted in the second Bragg peak is less than 1% of that in the first one, even for a perfect lattice. So far we have considered only perfect lattices. In the case of a liquid-type structure, the intensity of the first Bragg peak is much lower and is likely to be unobservable. The intermediate case of a distorted lattice gives an intensity lower than for a perfect lattice and a broadening of the peak. The lowering and the broadening are dependent of the degree of distortion. It is not to be expected that the vortex lines are arranged as a monocrystal throughout the whole sample. In fact, we observed diffraction phenomena for any orientation of the sample with respect to the incident neutron beam; this implies that the vortex lines are, at best, distributed on numerous small two-dimensional monocrystals which have a random orientation. References p . 179

166

D. C R I B I E R ~al. ~

[CH.4,

3

3. Experimental conditions The neutron cross-section considered above is, for the available wavelengths, very small (lo4 times smaller than for diffraction by an ordered magnetic material like iron). It is therefore advisable to select a material which will give a relatively high intensity. As we do not have a correct expression for the cross-section in the general case, we can only use the expression derived in the case of an infinitely small core as an indication. Then AP and M must be small. This means that we will obtain the highest intensity for materials of small IC.For these reasons we have chosen niobium and Pb0.98Bi0.02 which both have a K of the order of 1. The distance between the vortices in Nb is more than 1000 A, in PbBi more than 2000 A. The highest intensity of our long wavelength neutron source lies about at 5 A; this means that the first Bragg peak is to be expected at about 10' in Nb, and at about 5' in PbBi. It is therefore essential for the experiment to work with a well-collimated beam of long wavelength neutrons. a) The neutron source is the reactor EL3, with a cold moderator20 which increases the ffux of long wavelength neutrons. The neutron beam passes a 20 cm beryllium filter, which filters out neutrons with a wavelength smaller than 4 A. The spectrum which is obtained with this mter is represented in Fig. 4. In some of the experiments, this spectrum has been used as such. The large wavelength spread is of little consequence in determining accurately the Bragg peak, since the peak is observed near forward direction. It has taking into account the 1 ' dependence of the cross-section, a mean wavelength of 4.5 A. For some other experiments, the beryllium filter is followed

Fig. 4. Spectrum of the incident neutron beam filtered by beryllium. This spectrum is obtained with a liquid hydrogen moderator shifting the Maxwell spectrum towards long wavelengths. References p . 179

CH. 4,

I 31

SUPJlRCUNDUCTIVE MIXED STATE

167

by a lead flter, which acts like the beryllium filter, but with a cut-off at 5.7 A and a mean wavelength of 6.18 A. To improve the wavelength resolution a mechanical monochromator, using a helical obturator, has been built21 which can give a variable wavelength with a resolution A l / l of about 10% (Fig. 5). With this monochromator the more precise results are obtained. The collimator for the neutrons consists of a row of 9 plates of cadmium, placed perpendicularly to the beam direction and well spaced, which have horizontal slits, 0.5 mm high and 20 mm long. Each plate contains 12 such slits separated by 0.5 mm. The distance between the first and the last plate is 1 m. With this configration a beam consisting of 12 identical and parallel parts is obtained, each of them with a total divergence of 1.5 minute of arc.

Fig. 5. Schematic view of the mechanical monochromator.

We preferred this system to the usual Soler-slits, because it gives no extra broadening due to reflection on the walls of the slits. Behind the sample another collimator is placed identicalto the first one, which can rotate around the axis of the sample. Using the monochromator the overall resolution due to angular and wavelength dispersion is about 3.5’. b) The sample is mounted inside a liquid helium cryostat. By reducing the vapour pressure of the liquid helium, the temperature of the sample can be References p. 179

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D. CRIBIER et al.

[CH.

4,

03

lowered to 1.9 "K. The cryostat is plr -ed inside the gap of an electromagnet, which provides a well-homogeneous horizontal field, parallel to the slits of the collimator. Fig. 6 shows a diagram of the set-up, which is also shown on the photograph of Fig. 7. c) The samples. Niobium sample. The material is the commercially available niobium of serni-

j.-

Cryostot Detector

Monochromator

\

\

Reactor core

Beryllium

I

I l l I l l

i

Collimators

I

I

Sample

Fig. 6. Schematic diagram of the experimental set-up.

Fig. 7. Photograph of the experimental set-up. The monochromator is in the shielding at the right of the picture. The detector is on the left, outside the frame. References p . 179

CH. 4,g 31

SUPERCONDUCI'IVE MlxED STATE

169

element. It has a resistivity ratio P ~ O O - K / P ~ ~of ~ K80. The sample consists of 9 cylinders placed horizontally and parallel to the applied magnetic field. The cylinders have a 3 mm diameter and a 24 mm length; they are arranged in 3 rows. There is no direct contact between the cylinders, they are separated by about 0.2 mm*. -1 K M

c

Fig. 8. Magnetization curve of the niobium sample, consisting of an assembly of 9 cylinders.

The magnetization curve at 4.2 "K (T/7',=0.53) of this sample is represented in Fig. 8. It shows that at this temperature Hc2= 3060 Oe. Maki22 has shown that for T f T,, one should introduce two parameters K~ and x2, with the two relations I C ~= Hc2/d2Hc

and, near Hc2 -4nM

~-

H,, - H

--1

1 1.16 ( 2 ~ ;- 1)"

This last relation corresponds to a magnetization linearly approaching zero near Hc2. We found for our sample ~1

= 1.38,

I C = ~

1.58.

These values are not too different from those of McConnille and Serin23. Other samples of niobium have also been used (with less success). Lead-bismuth alloy. This is an alloy with 2%(atomic) bismuth. Fig. 9 shows its magnetization curve at 4.2 OK.The analysis of the curve yields K 1 = x* = 0.91.

* The demagnetizing factor of such a sample is unknown but small, of the order of 1 "/,. It has therefore been neglected throughout the analysis. References p. 179

170

D. CRIBIER et al.

[a. 490 3

Fig. 9. Magnetization curve of the lead-bismuth alloy.

Neutron diffraction

by niobium a t L.2OK

H= 1L75

Oe

h neutron= ~ . A3

2l

05

5

ld

15' Scattering angle

Fig. 10. Bragg peak obtained with niobium. The background which has been substracted is represented as a dashed line. References p . 179

CH.4,g 51

SUPERCONDUCTIVEMIXED STATE

171

4. Experimental results with niobium

Fig. 10 shows a typical angular distribution obtained with monochromatic neutrons of 4.3 A, AA/A= 10% and a niobium sample in an applied field between H,, and Hc2. The background, measured before the sample is magnetized for the first time or by application of a field larger than Hcz, is also represented in the graph. This graph shows the two characteristicsobserved with rather pure niobium samples: one observes a very well defined peak with a full width at half maximum of 3.5‘ and essentially no scattering outside of this peak. In particular, no other peak at a larger scattering angle, is observed. Efect of the magneticfield. When the magnetic field on the sample is increased from zero nothing happens in the lowest field region. Then, at some value Ha, a peak begins to appear, which increases in height and shifts to larger angles ; the intensity passes through a maximum and then decreases, while the peak still shifts to larger angels. This is illustrated in Fig. 1I which shows the angular distribution for increasingvalues of the applied field. These angular distributions have been obtained with an incident spectrum which is filtered only by the beryllium. The shift of the peak is just what is to be expected for scattering by a pattern of vortex lines, in which the line distance becomes smaller as the field increases from H,, to HCz. 5. Analysis of the results obtained with niobium 5.1. LINESHAPE

In fig. 10 a typical result obtained with niobium using the best resolution available in our experimental set-up has already been shown. The line is perfectly symmetrical and has a width of 3.5‘. The instrumental resolution is determined by the two collimators and the wavelength spread of 10% of the monochromator; a crude estimate of the total instrumental width, for a Bragg peak at small angles, gives with these conditions 3.5’. The observed width of 3.5’just equals the instrumental resolution under the best conditions. The natural width of the line must therefore be smaller than 1‘. As the peak lies between 10’ and 15’, the ratio of the widths to the position of the peak is rather small and one can really speak of a Bragg peak associated with some long range order, and not of a scattering of a liquid type. Clearly experiments with still better resolution would be useful in order to increase the information about the value of the natural width, and so about the nature of the order. References p , 179

172

D. CRIBIER

eta/.

[CH.

I

I

I

I

5,

10'

15'

2d

4, 0 5

I

Scstering angle

(a)

Fig. Ila. Diffraction pattern obtained with niobium as a function of the magnetic field. The neutron spectrum is not monochromaticbut beryllium-filtered. Each point is obtained counting during about 30 minutes. (1) 1476 Oe, (2) 1435 Oe, (3) 1397 Oe, (4) 1353 Oe, (5) 1312 Oe, (6) 1271 Oe, (7) 1230 Oe, (8) H = 0. References p. I79

CH. 4,

5 51

173

SUPERCONDUCTIVE MIXED STATE

Intensity

liI

1oooo

I

moo0

NIOBIUM a t L.~'K moo0

loo00

1

I

5'

lo'

I

6'

I

M'

I

25' S c s t e r i n g angle

(b) Fig. 11 b. Diffraction pattern obtained with niobium as a function of the magnetic field. (1) 1476 Oe, (2) 1517 Oe, (3) 1588 Oe, (4) 1599 Oe, (5) 1650 Oe, (6)1722 Oe, (7) 1804 Oe, (8) 1886 Oe.

It should also be noticed that the scattered intensity by this niobium sample really goes to zero at angles smaller than the Bragg peak. There is no observable diffuse scattering at very low angles, associated with the applied magnetic field. References p . 179

174

D. CRIBIER et

d.

[CH.4,

5

This experimental fact pleads strongly against the possibility of distorted vortex lines around the field direction. As distorted vortices would give a more isotropic scattering than straight lines, the scattering pattern would be closer to a Debye-Scherrer circle, than to a straight line. In our experimental set-up (long size of the slits in one direction compared to the diameter of the Debye-Scherrer circle) scattering would occur in all directions between 0 and flB= RD/L, where RD=radius of the Debye-Scherrer circle and L = 1 m, the length of the second collimator. In samples with a strongly irreversible magnetic behaviour we could only observe a central scattering, as occurs in a disordered material. 5.2.

POSITION OF THE PEAK

This position is independent of the resolution, and is perfectly reproducible for the same sample from one experiment to the other. From the analysis of the line shape, we have concluded that this peak can be considered as a real Bragg peak, so we can relate its position OB to some interplanar distance d

N

0';

-9 -1%

@ ~

Results obtained w i t h a wide incident spectrum Ax = 18 Results obtained with a narrow incident spectrum ah ~ 0 5 8 .

@ F o r trapped flux in zero field

Fig. 12. The vortex lattice parameter versus the induction for niobium. The two lines are the expected variations for a square lattice (Bda = 90)and a triangular lattice (Bd2 = fl/J@O).

References p . 179

CH. 4,g 51

SUPERCONDUCTIVE MIXED STATI?

175

by the relation 1= 2d sin OB, where 1is the wavelength of the neutrons. In Fig. 12 we represent the variation of l/dz so obtained as a function of the induction in the sample. This induction has been measured as a function of the applied magnetic field by Vivet in an independent experiment. All data reported in the graph have been obtained during the first magnetization of the sample. The graph shows that, except for small induction, a linear variation is found 1 43 B -=-d2 2 Q0’ as is expected from the theory of vortex lines. From the proportionality

constant it follows that each of these vortex lines carries one quantum of flux. More precisely it follows that the line density is the density expected

for a triangular lattice. The departure from the dependence one would expect for a square lattice is definitely outside the experimental accuracy. In preliminary reports on this subject24325 we drew and published the opposite conclusion. This was due to a trivial mistake in the evaluation of the Scattering angle. After recalibration, the former results agree well with these much more accurate new results. Experiments with neutrons of longer wavelength (lead filtered) have been used to check the position of the peak. The departure from linearity for small values of the induction, is undoubtedly due to a partial penetration of the sample by the magnetic field, which effect is well known2e for rather impure and consequently not reversible samples. So the induction as derived from the measurements, assuming that the whole sample is homogeneously penetrated by vortices, is underestimated as compared to the local induction effective for the scattering process. In fact, one observes that in this range of applied fields the intensity of the peak is too small compared to that for higher values of the field. This intensity reaches a maximum for a field of about 1500 Oe. Above this value one also gets the linear variation. So, it is very reasonableto assume that homogeneous penetration of the field is achieved only above this value. One can expect a linear variation of l/dz versus B, starting from very small values of B, only in experiments on samples with a perfectly reversible behaviour and a good field penetration.

5.3. INTENSITY OF THE PEAK As we have remarked in Section 2, the intensity of the peak must decrease when the scattering angle increases. So, one expects a decrease of the intensity when the field increases. In fact, the observed general behaviour of the inReferences p . I79

176

D. CRIBIER et

at.

[CH.4,

55

tensity as a function of the applied field is shown in Fig. 13. As already mentioned, the increase just above H,, is due to the fact that only a fraction of the sample is permeated by vortex lines. When one increases the field beyond the point of homogeneous penetration, one observes in fact a rapid decrease of the intensity ; however, one also observes two unexpected effects: 1) The intensity of the peak becomes practically unobservable for an applied Jield (about 2600 Oe) much lower than HC,x3O6O Oe, even if we use an experimental device with a rather poor angular definition in order to increase the luminosity of the spectrometer. Intensity

(arbitrary units)

t

Fig. 13. Intensity of the Bragg peak as a function of the applied fiield. Only the shape of this curve is reproducible from one experiment to the other.

A possible explanation of this fact is that at higher fields (small intervortex distance, dxlOOOA in Nb), the vortex lines are more and more free of pinning by the impurities or dislocations, and better and better arranged in a few large monocrystallites which have no reason to be well oriented in front of our neutron beam; so we shall not observe any diffraction at high fields without changing the orientation of the sample itself. 2) The intensity of the peak corresponding to a given jield is reproducible References p . I79

CH. 4,

B 51

SDERCONDUCTIVE W

D STATE

177

only fi we use an angular collimation of 3' or more. For a collimation of 1' the intensity of the Bragg peak can vary by a factor of 3 when one repeats the experiment in the following way: switch off the field, heat the sample above T,,cool it down again and switch on the same field as before. This irreproducibility of the diffracted intensity observed with the best angular collimation only, seems to us to be an argument against the hypothesis of a liquid-type order of the vortex structure, as this type of order must always give a reproducible scattering intensity. But if we think of a polycrystalline vortex structure we can explain this irreprodi1::bility: if the crystallites are big enough their number will be small and, with a good angular collimation, only a few of these crystallites will be well-oriented in order to diffract the neutrons; let (n) be the mean value of the number of crystallites, suitable oriented. If (n) is too small, it will largely fluctuate from one experiment to another and this explains the irreproducible intensity. Let us calculate (n). The number of crystallites N in the useful cross-section S of the sample for a given d, as a function of the number of vortex lines x along each side of the crystallites, is N = S (s); s is the effective area of the crosssection of the sample in the scattering plane; (s) is the mean surface of one crystallite. Sm.nl0" A'; ( s ) x J 3 x 2 / 2 x lo6 A2 for a triangular lattice when d m 1000 A. In the case of an incident beam collimated to 1' of arc the mean value (n) of the number of suitably orientated crystallites in the diffraction should be < n ) = 6 x N x 3 x 10-4/2n

(the multiplicity factor of 6 arises from the symmetry of the triangular lattice) and we find ( n ) m 106/x'. In order to explain the observed variation in the peak intensity by a fluctuation of n, one is led to a choice of x of the order of 200 to 300 giving ( n ) between 10 and 25. This gives a length of the edge of the crystallites of the order of 20 microns. Further experiments would be advisable to support these conclusions. 5.4. OBSERVATION OF ONLY ONE BRAGGPEAK

We have undertaken several attempts to observe the second Bragg peak at the angle d2 =J30el,where 0, is the angular position of the first Bragg peak and J3 a coefficient valid for the triangular lattice. These experiments were sensitive enough to detect a signal of 2% of the intensity of the first Bragg peak but gave a negative result; so, in the range of observation, the magnetic Referencesp , 179

178

D. CRIBIER et

al.

[CH.4, !j 6

field distribution seems to be well described by

H(r)-

c

COSM;.r, 1=1.2.3

where M fare coplanar vectors of the same modulus Mi =4nJdJ3 such that C,Mf =O. This describes a perfect sinusoidal distribution on a two-dimensional triangular lattice. This field distribution is not surprising in the case of niobium with K w 1 and 5 w 1, not small compared to the intervortex distance d. We remark h t from the value of Hc2one can deduce the coherence length 5 by the relation 5


p. The magnitude $ is evaluated in the following way: for the deepest splashes which may be reached by electrons directly from the surface ( Z N I sin p- lp), the dispersion of the displacement Au should be less than 6.

\

/

\ 1-

/

Fig. 10. Vicinity of the limiting point 0 at a Fermi sphere in an inclined magnetic field.

Since the electron velocity along the field is u,-u,(l-+$’), lq$’-6, hence $-(6/1p)* and q 4 (S/1)*.

we shall obtain (39)

The conditions (38) and (39) determine the interval of angles p a t which the group of effective electrons will be successively focussed at depths u,=nulim (n= 1, 2, ...). It should be emphasized that the constant a which determines the significance of this group from the viewpoint of contribution to the skin current enters only once into the expression for the amplitude, irrespective of the splash number. The decrease of the amplitude during the transition from n-splash to (n+ 1)-splash is determined only be the length of path A,-aexp (-u,/fp). That is why the focussing at a depth of distant splashes was required in (39). Violation of (39) means that the limiting point is far from the effective region. It has been shown in Ref. 54 that splashes do exist in this case too, but the electrons which form such splashes do not belong to the vicinity of the limiting point. (About forming the splashes when q ~ % (6/1)*, see Section 3.3.3.) References p . 232

210

[CH.5,

V. F. G M M A K H E R

63

An example of the size effect from the limiting point is shown in Fig. 11. This effect may be easily distinguished as the period AH greatly depends on the inclination angle 2nc AH = -K - 3 9. eH The existence of a line at a certain magnetic field means that the trajectories in the vicinity of the limiting point contain integer revolutions between the surfaces of the plate at this field. It should be noted, that transition from one size effect line to the other is related to the change of the field while the length of the trajectory from one surface of the plate to the other remains unchanged. When passing from one number to the other the amplitude decreases due to the dependence a(k), as well as due to the change of the absolute value of the line width H. The latter is essential for the measurements of a derivative.

I

I

I

I

I

2

3

4

I H(kOe)

Fig. 11. Records of the limiting point size effect lines in indium. The point is near [11 11. 1 1 10111, Ell[111], d = 0.3mm, p = 7"15', f = 1.6 Mc/s.

As it is shown in Fig. 11, the temperature increase, i.e. the decrease of the length of path I, influences only the amplitude of the lines but not the line width. This phenomenon may be easily understood if we try to retrace how the same result follows from the theory [eqs. (11)-(19)]. Remember that the sharp inhomogeneity of the field in the skin layer can References p.

232

CH.

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RADIOFREQUENCY SIZE EFFECTS IN METALS

211

be described with the aid of a superposition of monochromatic plane waves whose wave numbers k have a continuous spectrum of width A k - h - ' . The electrons drifting inside the metal interact most effectively with those field-spectrum harmonics whose wavelength il is contained an integer number of times in the distance ulim: ulim= Nil. This condition separates from the continuous spectrum a discrete series of wavelengths which can penetrate to an anomalously large depth into the metal. The interference of these waves gives rise to quasi-periodic field maxima at depths z,=nu,im. The widths of these maxima are determined by the width of the initial wavelength spectrum and therefore they are of the order of 6 . The height of the maxima and their decrease with distance are determined by non monochromaticity of the waves with anomalous deep penetration, i.e. in final analysis by the length of the electron free path 1. 3.3.2. Extremal non-central orbits As was stated above, in the case of a complicated dispersion law the extremal non-central orbits may exist. Let us consider the behaviour of the size effect lines caused by such orbits in an inclined field51. It is necessary, of course, to account again for the drift of electrons along the field. The lines are observed when d=u, where u=sZ-' j::uz dt and z1 and t2 are two effective points of the orbit. When the pitch of the helical line is h g D (h=uHTO= 2n&/Q), we have the following approximate formula for u ua-+ = D cos rp

+ (n - + ) h sin rp,

( n = 0,1,2, ...).

(41)

The factor ( n -4) is equal to a number of revolutions in electron paths from one surface of the plate to the other. The value n = 0 corresponds to the case where the component of the drift velocity uH sin p is opposite to the main motion along the turn and its projection on the z axis is negative. The values n = 1, 2, ... correspond to the positive projections of the velocity (see Fig. 12, where a helical trajectory is once more pictured in the lateral projection). Using (41) at small values of q, we obtain for the line position H,-+=Ho

D

(42)

It is clear from the meaning of the factor ( n - 4 ) that the amplitudes of the first two lines should be of the same order, while the others should decrease rapidly with number due to the increase of the path A from one surface to the other. Thus, the size effect lines caused by the non-central orbits should split as References p . 232

212

V. F. GANTMAKHER

[CH.

5, 5 3

the field inclines. Such a splitting was experimentally observed in indium51 (see Fig. 8 ; under favorable circumstances the line corresponding to n + 2 is seen also; on the curve of Fig. 8 it is obscured by the line “a+g”). The splitting would exceed the line width only if h a, S 6, i.e. uH should be sufficiently large. This condition is satisfied in indium due to the form of its Fermi surface.

Fig. 12. A helical trajectory in an inclined magnetic field with two effectivepoints through a period.

As always, the radiofrequency size effect may be brought in correspondence with a picture of splashes inside a metal. The helical trajectory, shown in Fig. 9, contributed to both the skin current and to the splash at a depth D. When the condition ha,% 6 becomes satisfied, the splashes u, =An sin a, (n= 1, 2,. ..) split from the skin layer and a set of splashes u, -+ determined by the formula (41) occurs instead of the splash at a depth u= D.The splash amplitude will certainly decrease with the increase of the index along the exponential curve A k a exp ( - A k b ) , where Ak-RnD, ( k = + , l , $ ,...) N

and the factor Q, as previously, determines the contribution to the skin current made by the focussing electrons. Generally speaking, each of these splashes may be initial for a chain and the damping along the chain will be determined by the factor an. When the exponential factor in Ak extinguishes all splashes but aft, then the size effect line in the double field will split into 3, the triple field into 4, etc. (see Fig. 8). References p . 232

CH.

5, 31

RADIOFREQUENCY SIZE EFFECTS IN METALS

213

3.3.3. The spherical Fermi surface in an inclined magnetic field In an inclined field, two functions D and h depending on p H are contained in the expression for u. In the case of indium, the extremum u(pH)is determined by the extremum &,). However, this is not necessary. The function u(pH) may have an extremum when D(p,) and h(p,) are not extremal if taken separately. In particular this is true for the spherical Fermi surface. Let us determine the orbit position on the sphere with the aid of an angle 8 (see Fig. 13). The value 8=0 corresponds to the central orbit and 8= corresponds to the limiting points. Effective points exist in orbits at 18118,,,., where Om,, =+n: - q ;at f3 = Om, the trajectory has only one effective point through a period (see Fig. 3).

++

Fig. 13. Different orbits on the Fermi sphere in an inclined magnetic field.

In the previous description of the slowly damping splashes caused by the closed trajectories it was assumed that I B D . It gave us the possibility to formulate conditions (34) and (35) under which the contribution to the skin current made by the helical trajectories could be neglected. Let us now consider another case: conditions (34) and (35) are not satisfied. Then all the orbits within the range (-Om,,, Om,,) are equivalent. Their contribution to the skin current is proportional to a number of electrons contained in them. In Fig. 12 a trajectory with 8#8,,, is shown. Sphericity of the Fermi surface permits us to write instead of (41) precise formulae for u, (0) and References p . 232

214

V. P. GANTMAKHER

[m. 5, § 3

The effective points u, cannot be observed in the size effects of the first order since un0, may have extrema. For example, the function u8 has two extrema with positions determined by the equation x

- CqJl-

q2 =arccos q ,

c = tg-’cp.

(45)

One of them is a minimum which is very shallow for all values of .I Another is a maximum located in the region q < 1/42. It may exceed u+(O) by more than 20% (see Fig. 14). Existence of an extremum of u(0) means that focussing of a group of effective electrons takes place at the depth uext.Hence a splash must arise at that depth, and when d=uexta size effect must also occur. We consider first u=umax.It appears to be a quite peculiar situation. When the conditions (33)-(35) are satisfied, the conductivity is determined mainly by the vicinity of the central orbit. However, when H decreases [i.e. when Do increases and the condition (35) ceases to be fulfilled], another splash should appear and this splash should be followed by the appearance of the second size effect line satisfying the condition d = urn,,. Possibly, just such a structure of the H.F. field within a metal explains the splitting of the lines of the cyclotron resonance in sodium in the presence of an inclined magnetic field55, though the splitting in this case occurs at much smaller tip angles than those which may be expected on the basis of the simple calculations given above. The magnitude umi, practically does not differ from

where the effective point of the boundary effective trajectory with 0 = emaris References p . 232

CH. 5,031

21 5

RADIOFREQUENCY SIZE EPPECTS IN METALS

1.'

&

I.(

,\--"\

7bo

\

O.!

0

0.5

9

1.0

Fig. 14. Graphs which illustrate eq. (43) for U , ( n = 1) ;(Y = tg QI.

located. It follows from the comparison of (46) and (37) that if the field is inclined the limiting point splashes convert into the boundary effective trajectory splashes at the depth nu, (n = 1,2,. .,). Subsequentinclination of the field leads to the decrease of the difference umax-umin(see Fig. 14). At q- 20-25 the splashes(and corresponding size effect lines) should amalgate. Their intensity under these circumstances should increase because a focussing of a comparatively wide layer of orbits takes place. A further increase References p. 232

216

V. F. GANTMAKHER

[CH.5,

3

of the angle p results in deterioration of the focussing and the size effects lines should disappear.

3.4. OPEN

TRAJECTORIES

In the case of open trajectories, the most essential is the direction of the field relative to the crystallographic axes, but not relative to the metal surface, since it may occur that the open trajectories exist only in a small range of angles. When the average electron velocity 5 is directed along the surface, the effect does not differ from that of the closed trajectories. The line position determines the extremal “amplitude of crimping” of the open trajectories. If the angle y between ii and n differs from go”, we have an analogy with the helical trajectories. The pictures shown in Figs. 3, 9 and 12 may be used for a schematic illustration of the behaviour of the open trajectories ifH is assumed to be perpendicular to the picture plane (and y =+z- 9).At small values of y, when the “crimp” of the trajectory is insignificant, or, to be more precise, when we have only one effectivepoint during the period, as it is shown in Fig. 3, the value u of the average drift of electrons inside the metal is the same for all open trajectories and is given by u = (cb/eH)cos y ,

(47)

where b is the period of the open orbit (reciprocal lattice). Thus, the “focussing” in this case is effected automatically due to the dispersion law. The requirement for the presence of effective electrons among the focussing ones means that trajectories having at least one effectivepoint should be contained among the open trajectories (concerning the opposite case, see p. 219). The size effect from the open trajectories was observed in tin 0nly38. To save space we shall not demonstrate here the curves which are given in the original paper. We only note that at y=O (which is the condition of the experiment) the velocity u in the effective points should be perpendicular to the mean velocity 5.

3.5. TRAJECTORIES WITH BREAKS It seems that we have already discussed all possible types of electron orbits. However, when the size effect in indium was studied53, a number of “superfluous” lines was observed (the indium Fermi surface had already been known approximately from both experimental and theoretical investigations). None of the mechanisms discussed above could explain the existence of these ‘‘supeTfluous” lines. The detailed study of the anisotropy of these lines, their behaviour at an inclination as well as the form of the Fermi surface as a whole References p . 232

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RADIOFREQUENCY SIZE EFFECTS IN METALS

217

made it possible to show that the occurrence of these lines was caused by the presence of breaks in the electron trajectories. The example of an orbit with breaks existing in indium is shown in Fig. 15. The occurrence of size effect lines caused by breaks in the electron trajectories indicates the presence of an electromagnetic field not only near the splashes, but also in the intervals between them. To estimate qualitatively

Fig. 15. Fermi surface of indium in the second zone in accordance with the nearly-freeelectron model. Circled part of the orbit a denotes a break; b is a non-central orbit with an extremal caliper in [loo] direction.

this field let us consider the distribution of the electromagnetic field in the depth of a metal with a Fermi surface in the form of a circular cylinder. An electron, which obtained the addition to the velocity Au in the skin layer, passes into a depth along the circular trajectory. If at a depth z this electron moves at an angle K to the surface, the horizontal component of the current produced by this electron is equal to e Au cos K . (The vertical component is compensated by the electrons moving towards the surface along the analogous trajectory at an angle I C = ~ 272 - K . ) Taking into account the fact that the number of electrons belonging to the given trajectory at a depth z is References p . 232

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proportional to l/sin K , we obtain the current produced by these electrons at z B 6 and D - z B d , i.e. far from splashes, equal to j(2) -j(O)

JG cotg K .

(48)

Let us assume now that the trajectory has a break, i.e. the condition D% S is still valid but at the same time within some part of the trajectory at a distance AzS6, the change of the angle K is A K - K . It is clear that in this case the jump of the functionj(z) Will also occur at a distance AzS6. 3.6. TRAJECTORIES OF INEFFECTIVEELECTRONS Let us consider first the magnetic field directed along the normal to the surface56. The closed, helical and open effective trajectories with the effective points located at various depths are certainly possible in the case of a nonspherical Fermi surface in the field perpendicular to the metal surface. These possibilities occur due to the drift of electrons along the field [the component uH(z)]which is absent in eq. (9). Naturally, the radiofrequency size effects similar to those which were discussed above should occur on these trajectories. The size effects of such a type were not yet observed experimentally, though there were works devoted to the study of the cyclotron resonance for such orbits 57*5*. However, there may be no such effective trajectories in a metal. For a spherical Fermi surface, for example, the region of effective orbits, shown in Fig. 13, constricts into a strip whose width has an angular size fl-S/l, as in a zero field. The whole free path of these effective electrons is within the skin layer. The electrons which penetrate into the metal are moving along the helical trajectories at an angle to the field. The ineffective electrons in a zero magnetic field determine the small but deeply penetrating component of the field which damps as e-{/t2, where c=z/Z. In the presence of a field, an ineffective electron also carries information concerning the instantaneous value of the skin layer electric field into the interior. However, as its velocity component uI =(u: uy”)* perpendicular to H rotates in the plane (x,y), the field component e-{/c2 converts into a helical one [we would remind the reader that the conditions (l), (8) and (20)are assumed to be satisfied]. The translation period of this helix may depend only on the extremal values cTo(pH),such, for example, as values u,To in the vicinity of the limiting point. Thus, refusing the requirement for the electron effectiveness,we obtain a harmonic distribution of the field inside a metal instead of splashes and, respectively, a sinusoidal component in the plate impedance instead of narrow size effect lines (see Fig. 16).

+

References p . 232

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RADIOFREQUENCY SIZE EFFECTS IN METALS

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Fig. 16. Record of a size effect related to ineffectiveelectrons in tinss. n [I [loo], Ell [OlO], d = 0.96 m,f = 5.2 Mc/s, 4(n, H ) = 17'.

Such a radiofrequency size effect at the normal field has been observed in tin only. As it was mentioned in Ref. 56, there are a number of obscurities in the experimental results. We wil not dwell upon them here. For the magnetic field directed along the surface a similar effect exists owing to electrons in open orbitsbg. If the corresponding trajectories have q#O, but at the same time have no effective points, again a sinusoidal component in the impedance appears instead of sharp size effect lines. Such an effect was detected in cadmium59. 3.7. CONCLUSION Here at the end of this section we present Table 1 summarizing all known size effects in the anomalous skin condition. The following remarks should be made about Table 1. 1) We have discussed the experiments with a plane-parallel plate only, In principle the radiofrequency size effects may occur in samples of other form too. No work dealing with such samples was done up till now. 2) Azbel60 noted that in principle observation of the cutoff of the quantum oscillations in the high-frequency surface impedance is possible. A peculiar mixing of quantum and quasi-classical effects takes place in this case: the quantum oscillationsmay be observed only when the corresponding trajectory fits inside the plate. Since the quantum oscillations appear usually only in fields of several kilooersteds, the single-crystal planeparallel films with a thickness of about 10-4cm are required for observation of such cutoff effects. That is hardly feasible now. References p . 232

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[CH.

5,$ 4

TABLE 1 Cutoff of cyclotron resonances35 Cutoff of non-resonant orbits at high frequencies (w Q) 34* 47 low frequencies (w < Q)32,48*48,58 Chain of trajectories caused by the extremal orbit^^^.^^ at cyclotron resonance28 at cylindrical Fermi surface36 at inclined magnetic field37 Extremal helical trajectories 51 Vicinity of limiting points38-62 Open trajectories with effective points38 Breaks of electron trajectories j3 Ineffective trajectories at HI1 n 5 6 Open ineffective trajectories59 N

4. Shape of line and various experimental factors In the previous section we discussed various cases of occurrence of size effects and the position of the line in the scale of magnetic fields. These problems may be considered to some extent as a first approximation in the solution of the size effect problem. The next approach from this viewpoint consists of the solution to the problem of the line shape. In most of the theoretical works use is made of the first approximation only. The solution of (18) is not simple but feasible, since one of the four integrations concerned is trivial as dfo/d& may be considered as a &function and integration over z and zo may be done with the use of the stationary phase method. However, in the course of transformation (19) from the Fourier representation to the distribution of the field inside a metal, the calculations become so complicated that it is necessary to introduce some simplificationsin which the shape of the line is practically dropped out. Thus, the problem of the line shape is considerably less studied than those problems discussed above. For this reason, the material presented in this section may be considered possibly as a statement of the problem only. We will limit ourselves to a brief description of all the known experimental factors influencing the line shape. 1) Mode of excitation. Undoubtedly the line shape of the size effects should References p . 232

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greatly depend on whether the excitation of the electromagnetic field in the plate is of one- or double-side type (see pp. 197-200 above). At present, however, the experiments in which various modes of excitation are directly compared with one another are not yet known. All experimental results used below belong to the case of double-side excitation. 2) The relation between the resistance Rand the reactance X. Since the position of the line in the size effect is independent of the frequency o,the functions R ( H ) and X ( H ) are not interrelated through differential expressions arising from the Kramers-Kronig relations as in the case of resonance effects. It seems, therefore, that there is no reason to expect the line shape to depend on what function is being studied in the case (the function aR/aH or the function aX/aH). The experiment carried out by Krylov61 showed, however, that such a dependence really existed. In all the experiments the extrema of the function BR/aH within the line width always correspond to the places of the greatest changes of the function aX/aH and vice versa. Thus, the qualitative relations between AR and A X are the same as those, for example, observed within the lines of the nuclear resonance. A good quantitative agreement is observed in some cases toosl. There is, therefore, a basis to assume that the observed relation between AR and A X in the size effect may be caused by some common properties of the equations for the high-frequency current distribution in the metal.

3) Type of size effect and the dispersion law. Under similar conditions the line shape varies of course for the size effects of various types (cf., for example Figs. 8 and 11). The line shape of the limiting point size effect is well reproduced for various limiting points even in different metals. In the case of the cutoff of the closed trajectories it is quite different for different extremal orbits. This is obviously related to the behaviour of the dispersion law in the vicinity of the extremal cross-section. It is possible to put forward the following two obvious influencing mechanisms of the dispersion law. First, the line shape may depend on the character of the extremum of the function D(pH):the measurable caliper may reach a minimum or a maximum on the extremal orbit (see Fig. 17). Second, the shape of the line should depend on the shape of the extremal trajectory in the skin layer, i.e. it should depend on the length of the electron drift in the skin layer J ( R 6 ) related to the value 6 ( R is the radius of the trajectory curvature at the effective point). Presumably the influence of the trajectory form is more important. Let us illustrate this by an example. From the topological viewpoint the References p . 232

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Fermi surface of tin in the fourth zone consists of two planes connected by “tubes”. According to the experimental datass.47~49, their deviation from a cylinder does not exceed several per cent and D reaches the maximum at the central cross-section;the central cross-section of the tube practically does not differ from a square. When the magnetic field is applied along the cylinder

Fig. 17. Different types of the extremal calipers at Fermi surfaces.

axis (crystallographic axis C4),the central cross-section of the cylinder causes a square closed trajectory. Depending on the orientation of the normal n, either side or diagonal of this square may be parallel to the sample surface. In the latter case the trajectory enters the skin layer at an angle. The shape of the line is different in these two cases (see Ref. 49, curves 1, and 1J . 4) Frequency. When choosing the frequency w it is necessary to satisfy the

quasi-static condition (20) to exclude the influence of the relation between D and w upon the shape of the line. The frequency in this case may influence the shape of the line only through the depth of the skin layer, through the ratio 6/d. The dependence of the line width on the frequency in bismuth is shown in Fig. 18 (the same line as that on curve 1 in Fig. 7). It follows from Fig. 18 that the relative width of the line d H / H , is fully determined by the value S/d, at least atf- 3 Mc/s (generallyspeaking, at the maximum frequency studied one can expect an influence of the non-uniformity in the sample thickness). The line presumably stretches to the right of H , up to those values of the field at which the electron trajectory still can brush against both skin layers. Using the distance between two minima as A H , we shall see that at f-3 Mc/s, dH/Ho-26/d=0.25 and hence6- lo-’ cm. At the same time, if we recalculate the result of the measurements of the real part of the impedance at high frequencies with the use of formulae (4), we shall obtain 6cm. Discrepances of the same sign were observed in indium-62 and tin also. These discrepances are obviously related to the non-exponential distribution of the field within the skin layers. 18920: the field considerably References p. 232

CH.

5,g 41

RADIOFREQUENCY SIZE EFFECTS IN METALS

12 M C / S

I

223

\ I

I

Fig. 18. The dependence of the width of the size effect line in bismuth on the frequency. The experimental conditions are the same as for the record in Fig. 7, curve 1.

differs from zero at distances exceeding the value derf, evaluated from the impedance with the aid of eq. (28). 5 ) Sample thickness. The possible range of thickness values is determined by the inequalities (1). Practically, however, this range is not fully used due to experimental diffculties which increase rapidly as the sample thickness decreases. Thus, for example, at d = lo-’ cm it is very difficult to obtain a plane-parallel plate, i.e. to obtain the same d along the whole sample with an accuracy of at least several per cent. It: is interesting that for a wedge shaped sample the size effect line does not smear out but splits: two lines corresponding to dmin and d,,, appear instead of one line (see Fig. 19). The position of the limiting point size effect line greatly depends on the angle p [see eq. (40)]. For this reason these lines may split also due to bending of the References p . 232

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sample; such a split was observed experimentally in indium. For samples that are free of such defects, the shape of the line does not depend on d if the latter satisfies (1).

I 400

450 H(Oe)

Fig. 19. Splitting of the size effect lines in indium due to a wedge shape of the sample. (1) a size effect line in a plane-parallel shaped sample (d = 0.4 mm, f = 3 Mc/s, n 11 [Oll], Hll[Oil]); (2) the same line in a 5'-wedge shaped sample, (dd/d = 7%); (3) line in the same wedge shaped sample at another direction of the field { a ( H , [Osl]) = So>. In all three cases, His parallel to the sample surface.

6 ) Smoothness of the surface. The influence of the surface smoothness upon the size effectlines may be imagined if we assume that the specular scattering of electrons at the surface occurs under some conditions. The specular scattering may be expected, for example, near the left part of the size effect lines in the region 0 I ( H , - H ) / H oI b/d, where effective electrons reach the surface at very small angles. The shape of the lines, however, remains quite Referencesp. 232

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the same when samples with etched surfaces are used instead of samples with cast mirror surfaces. This may be explained either by the complete absence of specular scattering or by the fact that disturbances of the surface smoothness (etch pits) are considerably smaller than the depth of the skin layer. In bismuth the second reason is quite obvious.

7 ) Polarization of the electromagneticfield. Each size effect line in a given sample has its own most advantageous polarization when the electric field is applied along the velocity vector of the electrons at the effective point of the trajectory. Formally the influence of the polarization is related to the fact that the Boltzmann equation (5) as well as eq. (7) contain the electric field in the form of the scalar product E - u. For the limiting point size effect, for example, the most advantageous direction of E is along the projection of H on the sample plane (see Fig. 3). As far as the shape of the line is concerned, it remains unchanged as the electric field turns at least within +60° from the most advantageous polarization. 5. Applications of radiofrequency size effects. One of the possible applications of the size effects actually was already mentioned in the previous section. The size effects presumably may give a valuable information concerning the structure of a high-frequency field in a metal in general, and in the skin layer in particular, in the presence of a constant magnetic field. In the present section we will expound on the other two applications of the size effects: the study of the shape of the Fermi surfaces and the study of the length of the electron free path. SURFACE 5.1. SHAPE OF THE FERMI

The use of size effects makes it possible to obtain various information concerning the Fermi surface. For the sake of convenience we shall divide the following material into several items approximately corresponding to the succession of the discussion of the experimental data. 1) The cutoff effect of the extremal orbit is usually observed when a field is applied parallel to the metal surface. The position of the size effect lines on the scale of the magnetic fields determines, in this case, the calipers of various sections of the Fermi surface. The caliper of the Fermi surface in the [nH] direction, which is of interest to us, is calculated in accordance with eq. (30). The thickness of the sample plate and the intensity of the magnetic field may References p . 232

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be measured with an accuracy up to one per cent. For this reason the determination of the caliper being measured is limited by the arbitrary determination of the Ho value within the width of the line, i.e. in the final analysis it is limited by insufficient knowledge of the line shape. Formally the H,,value is known with an accuracy up to the line width A, and, therefore, in such metals as indium and tin the error in measurements of the momentum absolute value does not exceed five per cent. Generally speaking this result is not so bad, if we bear in mind that the absolute measurements are usually not so important as the relative ones (the shift of lines while the field is rotated within the sample plane). In the case of bismuth, however, where just the absolute momentum values are of special interest, A,/Ho reaches the value 0.25 due to the considerable thickness of the skin layer. Nevertheless, even in this case the measurements are possible if some additional considerations are taken into account during the determination of the Ho value. As it was shown in Fig. 18, the line asymmetricallywidens as the frequency decreases. Taking into account the mechanism of the line widening discussed in the previous section, we may affirm that the Ho value to be found is determined by the left edge of the line which does not shift when the frequency changes. The Ho values determined with the aid of the left minimum of the lines in bismuth samples with thickness 0.97 mm and 1.20 mm gives the magnitude for the momentum along the binary axis p 1 =0.54 x lo-’’ g -cm/sec. This agrees well with the results of measurements camed out with the aid of other methods41963. Using the position of the line central maximum for Ho, we would obtain the value pi =0.58 x lo-’’ g*cm/sec,which disagrees with Refs. 41 and 63, and is beyond the possible errors of these works. Presumably it is thus possible to affirm that for all types of size effects in the Ho measurements the left (low-field) edge of the size effect line should be used instead of its centre, which has been used in Refs. 49, 52 and 53. It is interesting to note that such a correction removes a three per cent discrepance noted in Ref. 38 between the measurable period AH on the plots of the size effect from the open trajectories and the period calculated by the aid of eq. (47) with the reciprocal vector & known from crystallography. 2) Rotating a constant magnetic field within the sample plane, we may obtain the angular dependence of the value Ho and hence that of the value p with the aid of the relationship (30). This gives us the information on the shape of the corresponding section of the Fermi surface. In some specially symmetrical cases the dependence of p on an angle of turning x, constructed in References p. 232

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polar coordinates, is the central section of the Fermi surface of a plane normal to n. The extremal orbits, however, on which the size effect is observed, may show a variety of shapes and the surfaces themselves may be differently turned relative to n. For this reason, in the general case p determines the caliper (in the direction x++n) of the shaded projection of the Fermi surface on the plane perpendicular to n (see Fig. 20). Therefore, strictly speaking the interpretation of the experimental data is usually not unique ;some additional considerationssuch as the nearly-free-electronmodel would be required for the reconstruction of the Fermi surface. 3) The behaviour of the lines in an inclined magnetic field may give additional information, which is very useful for the interpretation of the experimental data. The linear shift of the line with a tip angle growth shows, for example [according to eq. (31)], that the Fermi surface is inclined relative to the sample surface. This phenomenon was used to study the Fermi surface

Fig. 20. A measurement of a caliper of a Fermi surface which has a complicated form.

References p. 232

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in indium. According to the nearly-free-electron model, the Fermi surface in indium in the third zone should consist of tubes a ong the edges of the Brillouin zone. The study of the behaviour of the lines at the inclinat on made it possible to separate the lines on the tubes from those connected with other sections of the Fermi surface, and to determine in each particular case the position of the tube in the reciprocal lattice. The split of the size effect lines at the field inclination may be interpreted in two different ways. There may be, for example, two sections of the Fermi surface (e.g., two tubes) which are symmetrically inclined relative to n ; such a situation was observed in indium (see Ref. 53, line g2). On the other hand, the split may indicate that the orbit being observed is not the central one [see eqs. (41) and (42)]. In the case of indium such orbits are present in the second zone (see Fig. 15). The behaviour of the appropriate size effect lines was previously illustrated in Fig. 8.

4) The size effect from breaks of the electron trajectories (of course, ifitis possible to show that the lines are really connected with the breaks) indicates the presence of sharp edges on the Fermi surface or, to be more precise, the presence of edges with a bend radius p5(G/d)po. In the case of indium this upper limit for p/poN 3%. It is interesting how often such sharp edges occur in the Fermi surfaces. It is clear now that the same effect caused by breaks of the electron trajectory had been observed previously in tin also on the square trajectory mentioned above when the diagonal of the square was parallel to the sample surface (Ref. 49, HIIC4, nll[l lo], curve 4J. 5 ) The presence of the open trajectories may also be shown with the aid of size effects. From the outward appearance of the plots Z ( H )some conclusions may be drawn concerning the form of the open trajectories, in particular concerning the existence of the effective points in them.

6 ) The size effect from the limiting points makes it possible to measure the curvature at certain points of the Fermi surface. In comparison with the measurements of calipers of the extremal orbits, we have here an additional source of errors which is an absolute value of the angle q of the field inclination contained in (40). Since only the changes of the angle of inclination A q may be measured experimentally, an absolute value of the angle q should be obtained by linear extrapolation to dH(cp)=O. It is assumed that the radius of curvature K - * does not depend on cp. If the angular region of the existence of the limiting point is sufficiently large, the curves with opposite References p. 232

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cp signs may be used to increase the accuracy of the determination of the absolute value of the angle cp.

7) The impedance oscillationsin the perpendicular magnetic field may be used for detecting helical trajectories with extremal displacement along the field56. OF THE ELECTRON FREE PATH 5.2. LENGTH

From the viewpoint of the possibilities of studying the length of the electron free path in a metal, various types of size effects are probably not of one and the same value. The effect from the extremal orbits in a magnetic field applied parallel to the surface (the cutoff size effect) involves several changes of the number ,u of the electron passage piercing the skin layer: to the right of the line at H>H,, p,-Z/nd, to the left of the line ,u= 1 and within the width of the line in the case of double-side excitation p-2Z/nd. This makes the dependence of the line amplitude on Z/d more complicated, especially as the condition 1% d practically cannot be satisfied in experiments. The size effects related to the drift of a group of electrons from one skin layer to the other, i.e. effects caused by the vicinity of the limiting point, extremal non-central trajectories and open trajectories, are more convenient for the measurements of 1. As it was stated above, the line amplitude of these effects contains the factor e-’’’, where A is the length of the path from one skin layer to the other. For a limiting point trajectory A =d/sin p, for an extremal non-central helical trajectory A , ~ ( n - + J n d ( n = O 1, , 2...) [see eq. (41)] and for an open trajectory A=kd/cos y, where the factor k is used for taking into account the depth of “crimping” of the open trajectory. Such a simple dependence on 1 is a consequence of the geometry of the experiment: the electron may pass through the second “receiving” skin layer only once even at I= 00. The line amplitude is proportional to the probability that the electron is not scattered in the path A. (Strictly speaking the effect caused by breaks of the trajectories also belongs to this group of effects, since in this case the electron may only once pass through the second skin layer.) The mean free path Z may be presented in the form

1/1 = 1/20

+ 1/lph,

(49)

where ,Z is the free path at T=O and I,,, is the free path connected with the electron-phonon collisions. The most convenient method of measuring the value lo gives us the limiting point size effect: by changing the tip angle cp we change A. A simple mathematical treatment of the dependence of the amplitude A(cp) measured at approximately the same values of H makes it References p . 232

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possible to obtain at once lo. Such measurements were carried out in tin for two different limiting points with the use of the same sample and in the same region of directions of the magnetic field H (Ref. 38). The ratio of the lo values was approximately four and the obtained values of lo were compared with effective masses and Fermi velocities of electrons at the limiting points. The measurements of the average value of lo along the extremal noncentral orbits require the comparison of line amplitudes corresponding to different A, at a certain direction of the magnetic field. Such measurements were performed in Ref. 51. Measurements of I,, on open trajectories require changes of samples, which is less convenient and gives lower accuracy. Turning to the problem of the temperature dependence of the free path, it is necessary to emphasize first the principal difference between collisions of electrons with static defects and electron-phonon collisions at low temperatures. This difference becomes important under anomalous skin effect conditions 27. Electron scattering on static imperfections of the lattice causes deflections through large angles as often as deflections through small anglesS - the scattering is an isotropic one. For this reason a single act of scattering exerts the same influence upon both the static conductivity and the size effect. On the other hand the interaction of an electron with vibrations of the lattice is influenced by the fact that the electron may emit or absorb a phonon. When the temperature T < 8 D (8, is the Debye temperature) the electron will scatter only through the small angle

b

Pph/PO

T/8D

9

where pphis the average absolute value of the phonon momentum at the temperature T and p o is the curvature radius of the Fermi surface. The effectiveness of such a scattering varies from case to case. In the static conductivity, (T/OD)' collisions are required for an essential scattering of the electron by the phonons, since only after such a number of collisions the electron scatters through the angle of unit order, As a result the supplementary factor of T2is introduced into the temperature dependence of the static resistance p (as it is known, p T 5and the factor of T 3 is determined, roughly speaking, by the dependence of the number of phonons on 2"). In the case of size effects, the scattering through a small angle may be sufficient for the conversion of the electron into an ineffective one due to the small skin layer thickness S. The value of this angle q and hence the value Iph depend on the type of effect, the relation S/d, and even on the value of the magnetic field. Let us estimate, for example, q for the case of the limiting point64. The path L which an effective electron may travel in the skin layer amounts

-

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RADIOFREQUENCY SIZE EFFECTS IN METALS

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approximately to L= S+d*v-'n-* [see Fig. 3 andp. 911; n ( H ) is the number of the line showing the number of revolutions made by the electrons. A considerable reduction in this path occurs in the case of scattering through the angle 6/L. Thus, 'I = S/L = (nS/d)*. (50) cm (and 6 - lo-' cm In actual experiments carried out int in with derrcm, I- lo-' and n = 2 , 'Ievaluated from the line width), d=4 x was obtained. At the same time, at helium temperatures the interaction of an electron with a phonon in tin deflects the electron through an angle of the order of The difference between 'I and this angle of deflection amounting to one order of magnitude is certainly small, but, as it may be considered from these estimates, even a single collision is sufficient for making an electron ineffective. For this reason the dependence of jph on the temperature should be proportional in this case to T 3, but not to T ', and this phenomenon was observed experimentally64. As it is shown in (50), a dependence fph(H)is also possible. Since at large values of n (i.e. in large fields) 'I increases, then at n & 1 several collisions may be required to make an electron ineffective. This problem, however, is not yet studied sufficiently. Some attempts were made also to apply the effect of the cutoff type to the study of the temperature dependence I ( r ) (see Refs. 32 and 64). These attempts proved not to be very successful, probably due to the difficulties mentioned above in the interpretation of the results. In both experiments, however, in tin and gallium, various temperature dependences of the amplitude were observed for various lines and this indicates the dependence of Iph on the position of the extremal orbit on the Fermi surface. Acknowledgements The author is very much indebted to Prof. Yu.V. Sharvin, Dr. 8. A. Kaner and Mr. I. P. Krylov for valuable discussions while writing this article.

4

Note added in proof. Since the article was written a number of new works65-69 have been done in the field of radiofrequency size effects. We shall restrict ourselves here to enumeration of new results : Ref. 65-a RF size effect from a chain of trajectories was observed in rubidium. Inclination of the magnetic field amplified the lines for which the conditions (33)-(35) were satisfied. So the mechanism of a weak damping chain predicted by Kaner 37 was detected experimentally. References p . 232

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Ref. 66-a RF size effect was observed in potassium. Inclination of the magnetic field led to a line shift toward higher fields just in accordance with eqs. (43)-(45) and Fig. 14. At tip angles ( ~ 2 2 5 the ” size effect lines disappeared. Ref. 67 - the shape of the limiting point size effect line was computed. The results are in good agreement with experimental curves62. Ref. 68 - a RF size effect was observed in cadmium. It was used for an investigation of the Fermi surface. Ref. 69 - a RF size effect was observed in ahminiurn. Thus R F size effects of different types have been observed so far in Alas, Bi50, Cd59.68, Ga32, In53,62, K66, Rb85, Sn38,49 and W44.

REFERENCES REPORTS AND MONOGRAPHS 1

2

5

7

8

I. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 69, 419 (1959) [English transl.: Soviet Phys.-Usp. 2, 831 (1960)l. I. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 78, 411 (1962) [English transl.: Soviet Phys.-Usp. 5, 878 (1963)l. I. M. Lifshitz and M. I. Kaganov, Usp. Fiz. Nauk 87, 389 (1965) [English transl.: Soviet Phys.-Usp. 8, 805 (1966)l. E. H. Sondheimer, Advan. Phys. 1, 1 (1952). A. B. Pippard, Advan. Electron. Electron Phys. 6, 1 (1954). A. B. Pippard, Rept. Progr. Phys. 23, 176 (1960). M.Ya. Azbel’ and I. M. Lifshitz, Progress in Low Temperature Physics, Vol. 3, Ed. C.J. Gorter (North-Holland Publishing Co., Amsterdam, 1961) p. 288. R. E. Peierls, Quantum Theory of Solids (Oxford at the Clarendon Press, 1955). ORIGINAL PAPERS

K. Fuchs, Proc. Cambridge Phil. SOC.34, 100 (1938). E. R. Andrew, Proc. Phys. SOC. (London) A 62,77 (1949). D. K.C.MacDonald, Proc. Phys. SOC.(London) A 63,290 (1950). l2 E. H. Sondheimer, Phys. Rev. 80,401 (1950). 13 J. Babiskin and P. G. Siebenmann, Phys. Rev. 107, 1249 (1957). l4 N. H. Zebouni, R. E. Hamburg and H. J. Mackey, Phys. Rev. Letters 11, 260 (1963). 15 Yu. V. Sharvin and L. M. Fisher, Zh. Eksperim. i Teor. Fiz. Pis’ma 1 (5), 54 (1965) [English transl.: JETP Letters 1, 152 (1965)). A. B. Pippard, Proc. Roy. SOC.(London) A 191,385 (1947). l7 A. B. Pippard, Proc. Roy. Soc. (London) A 224,273 (1954). 1 8 G.E. H. Reuter and E. H. Sondheimer, Proc. Roy. SOC.(London) A 195, 336 (1948). 19 R. G.Chambers, Proc. Phys. SOC.(London) A 65,458 (1952). 2 0 A. B. Pippard, G. E. H. Reuter and E. H. Sondheimer, Phys. Rev. 73,920 (1948). lo

CH.

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RADIOFREQUENCY SIZE EFFECTS IN METALS

233

A. B. Pippard, Proc. Roy. SOC.(London) A 203,98 (1950). R. G. Chambers, Proc. Roy. SOC.(London) A 215,481 (1952). 23 G. E. Smith, Phys. Rev. 115, 1561 (1959). 24 P. N. Dheer, Proc. Roy. SOC.(London) A 260, 333 (1961). 25 E. W. Johnson and H. H. Johnson, J. Appl. Phys. 36, 1286 (1965). 26 M. Ya. Azbel’ and 8. A. Kaner, Zh. Eksperim. i Teor. Fiz. 32, 896 (1957) [English trans].: Soviet Phys.-JETP 5,730 (1957)l. 27 M. Ya. Azbel’ and E. A. Kaner, Phys. Chern. Solids 6, 113 (1957). 2 8 M. Ya. Azbel’, Zh. Eksperim. i Teor. Fiz. 39,400 (1960) [Englishtransl.: Soviet Phys.JETP 12,283 (1961)l. I. M. Lifshitz, M. Ya. Azbel’ and M. 1. Kaganov, Zh. Eksperirn. i Teor. Fiz. 31, 63 (1956) [English trans].: Soviet Phys.-JETP 4, 41 (1957)l. 3 0 M. S. Khaikin, Zh. Eksperirn. i Teor. Fiz. 39,212 (1960) [English transl.: Soviet Phys.JETP 12, 152 (1961)l. 3 1 V. F. Gantmakher and Yu. V. Sharvin, Zh. Eksperim. i Teor. Fiz. 39,512 (1960) [English trans].: Soviet Phys.-JETP 12, 358 (1961)l. 32 J. F. Cochran and C. A. Shiffrnan, Phys. Rev. 140, A 1678 (1965). 33 V. Heine, Phys. Rev. 107,431 (1957). 34 8. A. Kaner, Dokl. Akad. Nauk SSSR 119, 471 (1958) [English trans].: Soviet Phys.Doklady 3, 314 (1958)l. 35 M. S. Khaikin, Zh. Eksperirn. i Teor. Fiz. 41, 1773 (1961) [English trans].: Soviet Phys.JETP 14, 1260 (1962)l. 38 V. F. GantrnaKher, Zh. Eksperirn. i Teor. Fiz. 43, 345 (1962) [English trans].: Soviet Phys.-JETP 16, 247 (1963)l. 37 8. A. Kaner, Zh. Eksperirn. i Teor. Fiz. 44, 1036 (1963) [English trans].: Soviet Phys.JETP 17, 700 (1963)l. 3 8 V. F. Gantrnakher and 8. A. Kaner, Zh. Eksperim. i Teor. Fiz. 45,1430 (1963) [English transl.: Soviet Phys.-JETP 18, 988 (1964)l. 39 C. C. Grimes, A. F. Kip, F. Spong, R. A. Stradling and P. Pincus, Phys. Rev. Letters 11,455 (1963). 40 M. S. Khaikin, Pribory i Tekhn. Eksperim. 3, 95 (1961) [English trans].: Instr. Exptl. Tech. (USSR) (1962)l. 4 1 M. S. Khaikin and V. S. Edel’man, Zh. Eksperirn. i Teor. Fiz. 47, 878 (1964) [English trans].: Soviet Phys.-JETP 20, 587 (1965)l. a2 R. T. Mina and M. S. Khaikin, Zh. Eksperirn. i Teor. Fiz. 48, 111 (1965) [English trans].: Soviet Phys.-JETP 21, 72 (1965)l. 43 W. M. Walsh, Jr. and C. C. Grimes, Phys. Rev. Letters 13, 523 (1964). 44 W. M. Walsh, Jr., C. C. Grimes, G. Adams and L. W. Rupp, Jr., Proc. IXth Intern Conf. LowTemp.Phys.,Columbus,Ohio, 1964(PlenumPress,NewYork, 1965)part B, p. 765. 45 R.B. Lewis and T. R. Garver, Phys. Rev. Letters 12, 693 (1964). 46 S. Schultz and C. Latham, Phys. Rev. Letters 15, 148 (1965). 47 M. S. Khatkin, Zh. Eksperim. i Teor. Fiz. 43, 59 (1962) [English transl.: Soviet Phys.JETP 16, 42 (1963)l. 4 8 V. F. Gantrnakher, Zh. Eksperirn. i Teor. Fiz. 42, 1416 (1962) [English transl.: Soviet Phys.-JETP 15, 982 (1962)]. 4 Q V. F. Gantmakher, Zh. Eksperirn. i Teor. Fiz. 44, 811 (1963) English transl.: Soviet Phys.-JETP 17, 549 (1963). 21

22

234 50

V. F. GANTMAKHER

[CH.5

V. F. Gantmakher, Zh. Eksperim. i Teor. Fiz. Pis'ma 2, 557 (1965) [English transl.: JETP Letters 2, 346 (1965)].

V. F. Gantmakher and I. P. Krylov, Zh. Eksperim. i Teor. Fiz. 47,2111 (1964) [English transl.: Soviet Phys.-JETP 20, 1418 (1965)l. 5z V. F. Gantmakher, Zh. Eksperim. i Teor. Fiz. 46, 2028 (1964)[English transl.: Soviet PhyS.-JETP 19, 1366 (1964)l. 53 V. F. Gantmakher and I. P. Krylov, Zh. Eksperim. i Teor. Fiz. 49,1054 (1965)[English transl.: Soviet Phys.-JETP 22, 734 (1966)l. 54 8.A. Kaner and V. L. Fal'ko, Zh.Eksperim. i Teor. Fiz. 49, 1895 (1965) [English transl.: Soviet Phys.-JETP 22, 1294 (196Q.l 55 C.C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963). 5 8 V. F. Gantmakher and 8. A. Kaner, Zh. Eksperim. i Teor. Fiz. 48,1572 (1965) [English transl.: Soviet Phys.-JETP 21, 1053 (1965)l. 5 7 J. F. Koch and A. F. Kip, Phys. Rev. Letters 8,473 (1962). W.M. Walsh, Jr., Phys. Rev. Letters 12, 161 (1964). 5 9 A. A. Maryakhin and V. P. Nabereshnykh, Zh. Eksperim. i Teor. Fiz. Pis'ma 3, 205 (1966) [English transl.: JETP Letters 3, 130 (1966)l. 'O M. Ya. Azbel', Phys. Chem. Solids 7, 105 (1958). I. P. KryIov, Zh.Eksperim. i Teor. Fiz. Pis'ma 1,4, 24 (1965)[English transl.: JETP Letters 1, 116 (1965)l. *2 I. P. Krylov and V. F. Gantmakher, Zh. Eksperim. i Teor.Fit. 51, 740 (1966)[English transl.: Soviet Phys.JETP 24 (1967)l. 83 A. P. Korolyuk, Zh. Eksperim. i Teor. Fiz. 49, 1009 (1965) [English transl.: Soviet PhyS.-JETP 22, 701 (1966)l. 84 V. F. Gantmakher and Yu.V. Sharvin, Zh.Eksperim. i Teor. Fiz. 48,1077 (1965)[English transl.: Soviet Phys.-JETP 21,720 (1965)l. 85 P. S. Percy and W. M. Walsh, Jr., Phys. Rev. Letters 17, 741 (1966). 66 J. F. Koch and T. K. Wagner, Report at the Xth Intern. Conf. Low Temp. Phys., Moscow, 1966. 67 E. A. Kaner and V. L. Fal'ko, Report at the Xth Intern. Conf. Low Temp. Phys., Moscow, 1966. V. P. Nabereshnykh and A. A. Maryakhin, Report at the Xth Intern. Conf. Low Temp. Phys., Moscow, 1966. Og J. F. Koch and T.K. Wagner, Bull. Am. Phys. Soc. 11, 170 (1966). 51

CHAPTER VI

MAGNETIC BREAKDOWN IN METALS' BY

R. W. STARK* AND L. M. FALICOV* DEPARTMENT OF PHYSICS AND INSTITUTE FOR THE STUDY OF METALS, UNIVERSITY OF CHICAGO,CHICAGO, ILLINOIS AND

CAVENDISH LABORATORY, UNIVERSITY OF CAMBRIDGE, CAMBRIDGE, ENGLAND CONTENTS: 1. Introduction, 235. - 2. The theory of coupled orbits, 244. - 3. Analysis of experimental results, 265.

1. Introduction

The concept of magnetic breakdown (MB) was first proposed by Cohen and Falicov to explain the very high frequency ascillation (the so-called giant orbit) observed by Priestley2.3 in his pulse-field De Haas-Van Alphen (DHVA) investigation of magnesium. Since then several authors have discussed extensions of the theory dealing with both the fundamental physical aspects of MB4-12 and its effects on experimentally measurable quantities such as the galvanomagnetic properties 13-16, the DHVA effect17-18, ultrasonic attenuation l9, etc. Experimentally, MB effects have at present been observed in several metals including magnesium 20-23, zinc22-33, cadmium 34-38, beryllium 37, white tin 38, aluminum 39-41, gallium42243, thallium44-47, and [email protected] one must anticipate that the effects of MB are very likely to appear in experiments on the electronic band structure of many metals. The purpose of the present contribution is to provide a detailed and unified review of the current status of the experimental and theoretical work on the subject. 2 9 3 9

t Work supported in part by the National Science Foundation, the U.S.Office of Naval Research, the U.S. Army Research Office (Durham) and the Advanced Research Projects Agency. * Alfred P. Sloan Research Fellows.

References p . 285

235

236

1.1.

[CH.6,s 1

R. W. STARK AND L. M. FALICOV

PSEUDOPOTENTIALS AND THE NEARLY-FREE-ELECTRON MODEL

We begin by considering some fundamental ideas of the one-electron theory of metals. A free atom of a metal-forming element can be considered, for our purposes, to be made up of a compact, tightly bound ion core surrounded by a few loosely bound valence electrons. The electronic states of the ion core remain essentially unchanged when an ensemble of free atoms condenses to form a periodic crystal lattice. The electronic states of the valence electrons, on the other hand, overlap with the valence electrons of neighboring sites to such an extent that the new wave functions for the crystal show little resemblance to those of the free atom. A new Schrodinger equation for the crystal

should be set up and solved. The wave functions $B are now Bloch states and V ( v ) is the periodic potential due to the ion cores and the self-consistent field of all the electrons. If an arbitrary potential probe is passed through the crystal, V ( r ) will appear as a rapidly varying function with a strong attractive part close to the ion cores; V ( r )becomes infinite at each lattice site. A schematic representation is shown in Fig. la. The valence states t+hB, however, are not arbitrary functions; they should be orthogonal to the electronic states t,hc of the ion cores. Thus, because of this additional constraint, the valence electrons occupying $B form very special probes which, in fact, sample the potential V ( r ) in a completely different way. The net result can be represented by an effective or “pseudo” potential V ’ ( r ) which is much weaker than V ( r ) and whose variations are also much smoother. Such a pseudopotential is schematically shown in Fig. Ib. In practice V(Y)is such that its Fourier expansion (1 3

V’(r)=xV,expiG-v G

converges rapidly, and a very adequate representation can be obtained by a truncation of the series after only a few terms. ln fact, a reasonable first approximation to the electronic structure of some simple metals can be obtained by keeping only the leading constant term V ’ ( r ) = V,. This, of course, leads to a spherically symmetric distribution in k-space with plane wave solutions of (1.1): $Br

= S2-*exp (iker),

where i2 is the volume of the crystal. References p. 285

A2k2 E(k) = - V , , 2m

+

(1.3)

CH. 6,I

11

Fig. 1.

A schematic representation of: (a) the lattice potential, (b) the pseudopotential.

231

MAGNETIC BREAKDOWN IN METALS

A better approximation to the electronic structure of a real metal (the socalled nearly-free-electron approximation) arises from the fact that the typical De Broglie wave length of a metallic electron near the top of the distribution 2n 2nh 2nh 1, = - = - = (1.4) kF PF J ( 2 m G ~

(where E , is the Fermi energy) is of the same size as the typical spacings of the crystal lattice. Hence, even though the pseudopotential is very smooth and its Fourier components small, diffraction in particular is of paramount importance in determining the electronic structure. An electron of wave vector k impinging upon a crystal lattice will be coherently scattered from a set of crystallographic planes defined by a reciprocal lattice vector G, provided the Bragg diffraction condition

2k.G

f

G - G= 0

(1.5)

is satisfied. This condition is shown schematically in Fig. 2. The primary effect of the Bragg diffraction is the mixing of the states k and k G, which in

+

References p . 285

238

R. W. STARK AND L. M. FALICOV

f

[CH.6 , 8 1

tky

Fig. 2. Bragg diffraction of an electronic wave of vector k by a set of lattice planes of reciprocal lattice vector G.

turn results in an energy discontinuity or gap in the energy spectrum &(k) for values of k satisfying (1.5). The spherically symmetric distribution in k-space which resulted from (1.3) is now broken into a series of zones whose basic symmetry is that of the crystal, with ~ ( kexhibiting ) the well-known band structure. This has a profound effect on the dynamical behavior of the electrons, even for those cases in which the magnitude of the energy discontinuity is very small.

1.2. DYNAMICS OF THE ELECTRONIC MOTION IN

A METAL

Throughout the remainder of this article we will be primarily concerned with the dynamical response of the conduction (valence) electrons to applied external electric and magnetic fields. It has been repeatedly shown319s2 that in the presence of a periodic lattice potential, the dynamics of an electron are governed by the equation

where E and H are the external electric and magnetic fields respectively, and u p is the group velocity of the electron state defined by the wave vector k 1 grad k & ( k ) . A

l+ =-

References p . 285

(1.7)

CH. 6 , 8 11

MAGNETIC BREAKDOWN

IN METALS

239

By convention the magnetic field is chosen to define the z-direction, i.e.,

H

= (O,O,H).

(1.8)

Let us consider first the case E=O. Eq. (1.6) simplifies to

from which two constants of the motion are immediately obtained: dk, = k, dt = 0 , (2) dEk =gradk &,.dk = h k * k dt = 0

(1)

'

(1.10)

i.e., k, and &k are constant. Thus the electron trajectories in k-space are defined by the intersections of the planes perpendicular to H with the surfaces of constant energy. Due to the nature of the Fermi-Dirac statistics, which the electrons obey, we are interested in the constant energy surfaces only in the neighborhood of the Fermi energy, &k=EF. If the velocity vk in (1.9) is rewritten as i , the resulting equation can be integrated to obtain:

k, = - a ( y - y o ) ,

k, = a(x - xo),

(1.11)

where (1.12)

Eq. (1.1 1) shows that the trajectories of the electrons in r-space normal to the direction of the magnetic field are equal to the trajectories in k-space after a rotation by in and a multiplication by a scale factor p. One point is worth clarifying at this stage. The periodicity of the lattice potential induces a periodicity in k-space ; consequently a given vector k is only defined within a unit cell of reciprocal space, the first Brillouin zone, for instance. This implies that two vectors k and k + G separated by a reciprocal lattice vector G should be considered identical. Therefore, the representation of the surfaces ~ ( k=)constant is not unique and three possible schemes can be chosen51: the extended zone scheme, the reduced zone scheme, and the repeated zone scheme. Each of these has its advantages and serves to illustrate a given point more clearly. In order to avoid confusion in going from one representation to another, we shall only use the extended zone scheme in this paper. This choice makes some trajectories in k-space appear discontinuous ; it should be remembered that this is not so and that the discontinuities are only due to the representation and would disappear in one or both of the other two schemes. References p . 285

240

R. W.STARK

AND

L. M. FALICOV

[CH.

6, 8 1

As an example that we shall use to illustrate the properties of MB throughout this article, let us consider a hypothetical metal with low symmetry such that its unperturbed Fermi surface is given by the spherical slab (1.13)

Ikzl Ikzm < k,

9

and such that it intersects only one pair of zone boundaries, (1.14) where Bragg diffraction can take place. Such a model is depicted in Fig. 3.

t

ky

IA Fig. 3. A simple model for a metal whose Fermi surface intersects only one zone boundary. The diagram shows the section at k , = 0.

Restricting ourselves for the time being to electrons with k,=O and magnetic fields parallel to the z-axis, it is easy to see that for non-vanishing potentials (V,, # 0) two types of electron trajectories are possible : those which under the influence of the magnetic field and Bragg diffraction have a net drift velocity along the x-axis and describe an open trajectory (Fig. 4a) and those with zero net velocity which close back upon themselves (Fig. 4b). If, however, V,, =0, no Bragg diffraction will take place at the points indicated 1 and 2 in Fig. 3, since in this case no interaction between electron and periodic lattice is operative. Consequently only one trajectory is possible, namely the free-electron orbit shown in Fig. 4c. It is evident that the dynamics of the electron motion on the trajectories (a) and (b) of Fig. 4 are quite different from the dynamics on trajectory (c). References p. 285

CH. 6, I 11

MAGNETIC BREAKDOWN IN METALS

241

However, the former result from a “segmenting” of the latter via the Bragg diffraction which takes place as the periodic part of the lattice potential is ideally “switched on”. If the electron-lattice interaction is very weak, i.e., if V,, is very small compared with some other energies in the system, we should question the validity of the dynamics as expressed by eqs. (1.7) and (1.9) and

Fig. 4. Possible trajectories in the x-y plane: (a) an open orbit, (b) closed orbit, (c) the free-electron orbit when the lattice potential is made to vanish.

Figs. 4a and 4b. It is intuitively obvious that an intermediate regime should exist such that an electron has a probability greater than zero but less than one of being Bragg diffracted at points 1 and 2 of Fig. 3. This effect, which should depend on the magnetic field strength H, is called magnetic breakdown. 1.3. A DIFFRACTION APPROACH

TO MAGNETIC BREAKDOWN

The following “gedanken” experiments illustrate the physical processes involved in MB. Suppose that a metallic single crystal slab of thickness d of our hypothetical metal is cut such that the crystallographic planes corresponding

,G

Fig. 5. (a) A single crystal slab of thickness d is bombarded by a beam of electrons of wave vector k which satisfies the Bragg condition, (b) the current reaching the detector D, normalized to the incident current l o , as a function of thickness d. References p . 285

242

R. W. STARK AND L. M. FALICOV

[CH.6 , § 1

to the reciprocal lattice vector GIlie parallel to the surface of the slab, i.e., GIis normal to the slab as shown in Fig. 5a. A beam of electrons of welldefined wave vector k impinges upon the crystal. The wave vector k is chosen to satisfy the Bragg reflection condition (1.5) ;in particular it is chosen to be the vector k,. shown in Fig. 3. If we ignore surface effects which would partially reflect and scatter the electron beam, and look only at the effects of Bragg diffraction, we know that the intensity of the transmitted beam should decrease exponentially with the thickness of the slab as shown in Fig. 5b. Consequently the probability P which a given electron in the beam has of reaching the detector D of Fig. 5a is given by P = exp [ - d/lsin O,] ,

(1.15)

where 1 is the penetration depth characteristic of the crystal and dlsin O , is the true distance the electron travels through the crystal. P can be defined as the “tunneling probability” for the given Bragg diffraction.

Fig. 6. (a) Experiment for investigating the line width of the Bragg diffraction minimum in transmission. The energy of the incident beam of electrons is kept constant and the angle 0 varied. The Bragg diffraction condition is satisfied for 0 = 00 f +A0. The actual line shape (b) can be approximated by a rectangular model (c). References p . 285

CH. 6, I 11

MAGNETIC BREAKDOWN IN METALS

243

A second “gedanken” experiment .with the same system requires the thickness d as well as the energy ~ ( kof ) the electron beam to be kept constant. This second requirement is equivalent to keeping the magnitude of k constant: lkl = k , . (1.16) The angle 6 between k and the surface of the slab is then varied as shown in Fig. 6a and the intensity of the transmitted beam measured at the detector D. The resulting curve should be similar to that shown in Fig. 6b, i.e., there should be a minimum at the angle O0 corresponding to the Brag diffraction condition and the line shape should show a characteristic width do. For our purposes the actual line shape is somewhat irrelevant and can be replaced by a rectangular one, which assumes that Bragg diffraction takes place as long as 8 = Oo & 366.

Fig. 7. A diffraction interpretation of magnetic breakdown. dB is the width of the diffraction line, d is the effective “slab thickness” for Bragg diffraction and R is the cyclotron radius.

Although the two experiments described above cannot be performed in practice, a similar experiment is performed by the conduction electrons of metals in the presence of a magnetic field. An electron with a given k-vector at the top of the Fermi distribution precesses in the presence of a magnetic field and changes its direction of motion continuously while keeping its energy E~ constant. Such an electron trajectory which is valid for our hypothetical metal is shown in Fig. 7; it satisfies the Bragg diffraction condition at the point. 2’ on its trajectory. From the considerations of our “gedanken” experiment, such an electron sees an effective slab of Bragg-diffracting metal References p . 285

244

R. W. STARK A N D L. M. FALICOV

[CH.

6,s 2

of thickness d. By simple geometrical considerations apparent in Fig. 7

d = R A 9 sin 8, ,

(1.17)

where R is the cyclotron radius of the orbit

(1.18) w, the cyclotron frequency and vF the (Fermi) velocity of the electron. From (1.15) it is evident that the electron will follow the trajectory A2‘B of Fig. 7 with a probability ,=ex,[--1 A9 mcvF (1.19) 1 I4 H

while the Bragg-diffracted trajectory A2’C will be followed with a probability Q=l-P.

(1.20)

In (1.19) all quantities involved except H are characteristic parameters of the metal, and consequently the probability P can be rewritten in a simpler form P = exp As we shall see in the next section4>

7 9

[“R].

(1.21)

9918

(1.22)

where E~ is the energy gap at the relevant zone boundary.

2. The theory of coupled orbits In this section we discuss some theoretical aspects of MB. This discussion will include: (a) the theory of a single MB process; (b) semi-classical transport theory in the presence of MB; (c) the quantization of a system of coupled orbits; (d) the theory of the DHVA effect in the presence of MB; (e) MB oscillatory effects in the transport phenomena. Whenever possible, we will illustrate the theory with the simple hypothetical metal descibed in the last section. 2.1. AMPLITUDES AND PHASES AT A MB JUNCTION We have seen in Section 1.3 that a point on the electron semi-classical trajectories, where the Bragg diffraction condition (1.5) is satisfied, correReferences p . 285

CH. 6 , # 21

245

MAGNETIC BREAKDOWN IN METALS

sponds to a junction where the electron will choose between two different paths: the free-electron path, with probability P, and the Bragg-diffracted path, with probability Q. The junction, however, can be better considered as the meeting point of four channels: two incoming and two outgoingchannels. This is shown in Fig. 8. It is also important to consider these channels as “quantum mechanical”, in the sense of the Bohr-Sommerfeld quantization scheme, and determine not probabilities, but amplitudes and phases instead. These are also indicated in Fig, 8 for two waves reaching the junction from

Fig. 8. A magnetic breakdown junction showing amplitudes and phases.

the two incoming channels with amplitude 1 and phase 0. The symmetry of the junction (in first approximation) with respect to the two channels manifests itself in the symmetry of the amplitudes and phases in Figs. 8a and 8b. Conservation of the number of particles (conservation of theprobabilities) imposes p z + 4’ = P Q = 1 , (2.1)

+

while orthogonality between the two waves yields for the phases exp N ’ p p ‘pp

- 40411 + exp Ci(Cp4 - Cpp)l = 0

- p‘pq = (. +

*

7

(2.2)

The information would be complete if two quantities, say P = p 2 and ( p p , were known. The tunneling probability P has been calculated using perturbation theory by several authors. It has been done in the low field limit4, where it essentially reduces to the theory of Zener (electric) breakdown53; in the high field limit5.7, where the lattice potential can be treated as a perturbation; and at intermediate fieldsn. In all cases the same formula is obtained

(2.3) References p. 285

246

[CH.6,g 2

R. W. STARK AND L. M. FALICOV

where E~ is the relevant energy gap, given in the nearly free-electron model by Eg

= 2VG,

(2.4)

Y

and u, and o,, are the two components of the electron (Fermi) velocity perpendicular to H,u, being parallel to the Bragg planes and uy perpendicular to them. Comparison of (2.3) with (1.21) shows that

where K is a numerical constant, of order 1, which depends on the geometry. Regarding qp,its actual value depends sensitively on the details of the electron orbits and the lattice potential. For most physical phenomena, however, it is of little importance and its influence can be ignored. For the sake of uniformity we make the arbitrary choice

vp=tn,

(2.6)

qq=o,

which agrees with the convention assumed by several authors516$17. 2.2. SEMI-CLASSICAL TRANSPORT PROPERTIES

As a first approximation when studying electronic transport properties, all quantum mechanical phase coherence of the wave function can be neglectedJ1.62. In this case, the electrons can be considered to be classical particles with well-defined trajectories which (a) satisfy the equation of motion given by (1.6)-( 1.7) and (b) satisfy Fermi-Dirac statistics. With these approximations all stationary, transport phenomena can be interpreted in terms of the electron distribution functionf(r, k) which satisfies Boltzmann’s equation 51 : uk-grad, f

-le‘ tz

+ f uk x H]*grad,f

=-

Since we are primarily interested in discussing the influence of MB, we shall assume that: (1) there is no spatial variation of any physical quantity, i.e., we neglect surface and boundary effects and we assume that the temperature is uniform throughout the sample ; hence, grad,. f = 0 ; References p . 285

(2.8)

CH. 6, $21

MAGNETIC BREAKDOWN IN METALS

247

(2) the scattering term on the right hand side of (2.7) can be treated in the uniform relaxation time approximation

where fo (k) is the equilibrium Fermi-Dirac distribution ; (3) only terms linear in the electric field E are considered (Ohm's law regime). With these assumptions, which in general are satisfactory for discussing experiments at high magnetic fields, the Boltzmann equation can be reduced to an integration6116~ (2.10)

(2.11) -m

In eq. (2.1 1) vk(t)is the time-dependent velocity of that electron which at time t o ( k ) is at the state k, obtained by integrating eqs. (1.7) and (1.9). The vector n k can be called the effective path of the electron state k, and the integral in (2.1 1) is usually referred to as the path integral. It is worth remarking that the factor [ - afo/ae] in (2.10) is essentially non-zero only in a range of energies kT about the Fermi energy eF. When studying only semi-classical effects at low temperatures ( 5 4 OK) it is a very good approximation to replace that factor by a delta function 6(e(k)- E ~ ) consequently ; (2.1 1) needs to be calculated only for k-vectors on the Fermi surface. The presence of MB makes the calculation of the path integral (2.11) much more involved; in that case every time the electron arrives at a MB junction, two possible paths, both with non-zero probability, appear in the time evolution of Vk(t). The integral then has to be followed consistently through an infinite multiplicity of paths and becomes in fact a non-trivial system of coupled equations. Several mathematical techniques can be applied to the solution of this problem13-169559 56; the details of these methods, however, would take us too far outside the scope of the present review. We therefore refer the interested reader to the original contributions. It is, however, interesting to describe in detail a very simple example which exhibits the interestingfeatures caused by MB. The model corresponds to the hypothetical metal described in Section 1 and shown in Fig. 3; we further References p . 285

248

R. W. STARK AND L. M. FALICQV

[CH.

6, 0 2

make the initial assumption that the relaxation time z is infinite; all contributions to n k come from MB. In Fig. 9 we have depicted schematically a realspace network corresponding to the k-space diagram of Fig. 3. Consider the four segments 1‘A2‘, 2’B2, 2C1, 1Dl’; there is a one-to-one correspondence in both diagrams. We are interested in determing A , for each point on the

*X

I

I

R=pG, Fig. 9. The network of coupled orbits in real space corresponding to the Fermi surface of Fig. 3.

circle in Fig. 3, for example, point D ; this is equivalent to finding the “effective path” that the electron, presently at D in Fig. 9, has traversed since being scattered into the network, or equivalently, since being “created” at another point XD. Consequently AD

=D

-XD,

(2.12)

and since there are no other scattering mechanisms (z+oo), XD should be the same for all electrons on the arc 1’Dl. In addition, the symmetry of the net-

(2.13)

where the points ABCDCD’ are indicated in Fig. 9 and R is the vector between two neighboring orbit centers related to the reciprocal lattice vector Gi by R = PC, (2.14) and fl is defined in (1.12). References p . 28s

CH.6 , s 21

249

MAGNETIC BREAKDOWN IN METALS

Consider X,. Since the electron emerging from junction 1 may come from C or D' with probabilities P and Q respectively, it is evident that

+ XA = PXD+ Q x ~ v . X D = PX,

and similarly

QXD,,

Solution of eqs. (2.15) and (2.13) yields

Q

and X - - R. ,-P

XA = O

(2.16)

This result, when replaced in (2.12) gives the effective path Ak and consequently the Boltzmann function (2.10). The conductivity can now be calculated (2.17) If we now integrate over our spherical slab, the conductivity tensor is given by15

l o

o c o

i

where the angle O0 is defined in Figs. 3 and 9, and n is the number of eIectrons in the metal. It is interesting to plot the p 2 2 component of the resistivity tensor m 8 (2.19) sin2 (eo) [exp - 1 1 o, pZ2 = ne where l e l H and coo= lelffo w=---. me mc

(z)

~

It is easy to see that as w+m, i.e., at very high fields, p Z 2 can be rewritten as p22(H

--f

co) =

m l -, ne2 T~

(2.20)

where 1 - 8- sin'

-.

zM

References p . 285

(eo)o,

(2.21)

250

R. W.STARK AND L. M. PAL-ICOV

[CH.6, fi 2

is an effective inverse scattering time, due to the presence of MB. This scattering mechanism is easy to understand. When H+ m the free-electronlike orbits will be dominant; however, at each MB junction there will be a probability of scattering off that orbit given by

and since the frequency with which the electron returns to each junction is proportional to a,the probability of scattering per unit time is

as found in (2.21). When finite relaxation times are considered, the calculation of the galvanomagnetic tensors becomes rather cumbersome and, in general, has to be done numerically. In Fig. 10 we show p z z ( H ) for our hypothetical metal14 for various values of z and for 8, =+n. For finite z the magnetoresistance pzz starts increasing quadratically, reaches a maximum at a value H < H , and

I' 01

1

2

Fig. 10. The transverse magnetoresistance for the model of Fig. 9 for three relaxation

times t. References p . 285

CH. 6,E 21

25 1

MAGNETIC BREAKDOWN IN METALS

then decreases to a saturation value post = p ( H = 0)

[ +: I 1

(2.22)

- oor

This case which we have described in detail is only one of the possible new regimes in the galvanomagnetic properties originated by the presence of MB, namely a transition from open trajectories at low fields (P=O) to electronlike closed orbits at high fields (P=1). It is well known that, in the absence of MB, the galvanomagnetic properties of metals in high magnetic fields can be classified according to four regimes 577 58. These, with their corresponding magnetoresistance and Hall behavior, are listed and described in Table 1. Since MB can change the

Types of orbits and state of compensation

Transverse magnetoresistance

Transverse Hall resistance

Saturates

u -

cc H 2

X H

(A) All closed orbits; uncompensated

> nh (A2)ne < nh (A1)ne

(B) All closed orbits; compensated (C) Open orbits in direction perpendicular to H making angle a in real space with the current J

a

H asin2&

H ne - n h

uH

connectivity of the orbits, transition between any regime in Table 1 to any other regime is thus possible, with a consequent change in the galvanomagnetic tensors. The H dependence of p for all these possibilities has been studied and the relevant curves published in Ref. 14. In particular, the case (C)+(A,) corresponds to the simple model discussed above, and the case (B)-+(A,) with hexagonal symmetry will be discussed in detail in the next section in connection with the experimental measurements of the magnetoresistance and Hall effect in magnesium and zinc. 2.3. QUANTIZATION OF COUPLED ORBITS The quantization of electronic motion in the presence of MB poses a difficult and still not completely solved problem. This is related to the choice of gauge for the vector potential A, which is necessary for calculating the quantum mechanical phases. References p . 285

252

R. W. STARK A N D L. M. FALICOV

[CH.

6, 8 2

It has been proved6718359 that for the relevant electronic states in a metal, the wave functions in a magnetic field can be thought of as being confined in the equivalent semi-classical network formed by segments of more or less circular tracks; the effective width is in general less than 4% of the radius of the circle. In the case of free electrons, if we consider the vector potential as being given by A=+Hxr (2.23) and restrict ourselves to those orbits centered at the origin of the gauge, the wave function can be represented by 59 $

= exp

[ilql,

(2.24)

which corresponds to a classical radius

In these equations r and cp are polar coordinates normal to H . A change in angle d q = 2 n , i.e., a complete circular orbit, yields a change in the phase Q, of the wave function (2.26) i.e., as many times 2n as the number of flux units hc/lei that the electron has enclosed in its orbit. This change of phase in a closed orbit can be generalized to arbitrary orbits, and constitutes the basis for Onsager’s scheme60 of quantization. The presence of MB introduces difficulties in calculating the variation of phase along the various arms of the network. The problem arises from the fact that the same arm can be part of several different closed orbits, and, if Onsager’s rules are to be satisfied for each of these, the phase change along the arm (or equivalently at the junctions) should depend on the orbit considered. Pippard6 has proposed a consistent scheme of computing phases around the network which can be summarized as follows: (i) Choose the center 0, of one of the circular arms as the origin of the gauge and start measuring phases from an arbitrary point on that arm. (ii) The phase change for a path along any arm is equal to the (positive) area swept by the radius vector of the corresponding circle multiplied by the factor a defined in (1.12). References p . 285

CH. 6,

8 21

MAGNETIC BREAKDOWN IN METALS

253

(iii) The phase change in going across one MB junction continuing along the free-electron circular path is a constant $71 corresponding to our convention (2.6). (iv) The phase change at an arbitrary MB junction J, when the Braggdiffracted path is followed from a circular arm of center Oi to another arm of center OF, is equal to ct times the area OiJOF, considered positive (negative) if the letters as written have the same (opposite) sense of rotation as the particles in their circular orbits. (v) When returning from another circle to any arm of the circle centered at the origin of the gauge 01,in addition to (iv), add a phase equal to the product of CI and the area of the polygon OiO 2...0,01 determined by all the centers concerned, and considered positive (negative) if the sense of rotation 010, 0,01is the same (opposite) as that of the particles in their circular orbits. In the absence of MB these rules, except for the constant factors (iii), exactly reproduce Onsager's scheme. The quantization of an arbitrary system of coupled orbits can now, in principle, be achieved by requiring the wave function to be single-valued. This is equivalent to requiring that, starting from any point in the metal, the addition of all possible wavelets which follow all possible paths through the network add up to the original amplitude and that the phase change be an integral multiple of 27r. These conditions restrict the radii of the circles to a discrete set of values, and consequently restrict the energies to a discrete set of bands of allowed values. As an example, we now once more consider our hypothetical metal. The network shown in Fig. 9 is a one-dimensional periodic structure, and the condition of single-valuedness can be achieved by requiring that any t w o

...

c

Fig. 11. A unit cell in the periodic network and the areas relevant in the quantization of the coupled orbits. References p. 285

254

[CH.6 , § 2

R. W. STARK A N D L. M. FALICOV

equivalent points in neighboring cells (e.g. D and D', C" and C in Fig. 9) have the same amplitude and an arbitrary constant phase difference w. The allowed energies will thus be a function of the Bragg-diffraction amplitude q and of w, and, as expected for any periodic system, will exhibit a characteristic band structure. In order to obtain the quantized levels we have drawn one unit cell of the network in Fig. 11, and have labeled a few relevant points and areas. The phases used in what follows correspond to these areas multiplied by the factor a. The five areas shown are not independent, since 5=21-e,

a=22+e.

(2.27)

The amplitudes and phases of the wave functions at a given point in the network are denoted by $. In this way $(K) represents the wave function at K, while $(I,) stands for the wave function at the junction 1 along the arm of the network which contains the point K. If the following definitions are introduced: (2.28) application of the method discussed above yields a = qeiB - 2 2il 2 Zit - 1 P e Cr + 4eitl c1 - 4 e 1 j? = qye" - p2e2"[1 + qye"] [I - q 2eZ i t1- 1 Y

.

(2.29)

The periodicity of the system imposes two additional conditions $(K)=e'"@(K')+a=e i(m-211) 9

$(L) = e'"$(L')+y

= fiei(OtZX).

(2.30)

Eqs. (2.29) and (2.30) constitute a set of four equations with three unknowns (a, fi, y), which can only be solved if cos 0 =

sin(5 + B ) + q2 sin(5 - c) 2q sin g

(2.31)

Since the magnitude of the left-hand side of (2.31) is less than or, equal to one, it is apparent that the energy can only have values such that t and a make the right-hand side also less than or equal to one. In particular, for q = O it is evident that solutions can be found only if

g + B = nn, References p . 285

(2.32)

CH. 6, fj 21

255

MAGNETIC BREAKDOWN IN MFXALS

which corresponds to Onsager's rules for the free-electron circle (Fig. 4c). In the other limit, q= 1, solutions can be found far arbitrary values of i , corresponding to the non-quantized open orbits (Fig. 4a). There are, however, some singular solutions which correspond to

i

(2.33)

= nn,

which are the discrete levels due to the lens-like orbit of Fig. 4b. Since for fixed ratio &T (i.e. for fixed angle O,, in Fig. 3), 5 is proportional to the energy and inversely proportional to the magnetic field strength, a diagram of allowed