superconductivity by kitterson & song

8/7/2019 superconductivity by kitterson & song

http://slidepdf.com/reader/full/superconductivity-by-kitterson-song 1/511



- J ,

This is a text on the subject of superconductivity, an area of intense research

activity worldwide. The book is in three parts: the first deals with

phenomenological aspects of superconductivity, the second with the microscopic

theory of uniform superconductors, and the third with the microscopic theory of

nonuniform superconductors.

The first part of the book covers the London, Pippard, and

Ginzburg-Landau theories, which are used to discuss a wide range of phenomena

involving surface energies, vorticity, the intermediate and mixed states,

boundaries and boundary conditions, the upper critical field in bulk, thin-film,

and anisotropic superconductors, and surface superconductivity. The second part

discusses the microscopic theory of Bardeen, Cooper, and Schrieffer. Finite

temperature effects are treated using the Bcgoliubov-Valatin transformation. The

theory is used to discuss quasiparticle tunneling and the Josephson effects from a

microscopic point of view. The final part of the book treats nonuniform

superconductors using the Bogoliubov-defiennes approach with which it is

possible to extract many important results without invoking Green's function

methods.

This text will be of great interest to graduate students taking courses in

superconductivity, superfluidity, many-body theory, and quantum liquids. Itwill

also be of value to research workers in the field of superconductivity.



Superconductivity



Superconductivity1. B. KETTERSON and S. N. SONGNorthwestern University

CAMBRIDGEUNIVERSITY PRESS

...-----.----..--.-~R U H R -U N lV E R S :T i 'T , - , . . : ' . '

Fak. f. Phys':':~; "', ,p.-:-,.;' i'" ~.

In",v:r~~.;;'" 6 0 ~ , · , 5 : s .



PUBLISHED BY TIlE PRESS SYNDIC."TE OFTHE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB21RP, United Kingdom

CA:VIBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, LK http://www.cup.cam.ac.uk

40 West 20th Street, New York. NY 10011-4211, USA httpr/www.cup.org

10 Stamford Road, Oakleigh, Melbourne 3166, Australia

< £ ) Cambridge University Press 1999

This hook is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 1999

Printed in the United Kingdom at the University Press, Cambridge

Typeset in 9.5/13pt Times New Roman [VN]

A catalogue recordfor this hook is available/rom the British Library

Library a/Congress Casaloquinq in Publication data

Ketterson, J. B. (John Boyd)

Superconductivity - J. B. Ketterson and S. N. Song.p. ern.

ISBN 0--521-56295-3 (hardcover). - ISBN 0-521-56562-6 (pbk.)

1. Superconductivity. 2. Superconductors. 1. Song, S. N.

(Shengnian N.) II. Title.

QC611.95.K48 1998

621.3'5-dc21 97-3060 CTP

ISBN 0 521 562953 hardback

ISBN 0 521 565626 paper hack

http://www.cup.cam.ac.uk/

http://httpr/www.cup.org

http://httpr/www.cup.org

http://www.cup.cam.ac.uk/



Contents

Preface Xlii

Part I Phenomenological theories of superconductivity

1 Introduction

2 The London-London equation 5

3 Pippard's equation 8

4 Thermodynamics of a Type I superconductor 1 2

5 The intermediate state 1 5

6 Surface energy between a normal and a supcrconducting metal 1 7

7 Quantized vorticity 1 9

8 Type II superconductivity 2 4

8 . 1 Magnetic fields slightly greater than H c1 2 4

8 . 2 The region Hc1 < < H < < Hc2 2 7

8 . 3 Microscopic magnetic probes of the mixed state 2 9

9 The Ginzburg-Landau theory 3 1

9 . 1 Basic equations 3 1

9 . 2 Gauge invariance 3 5

9 . 3 Boundaries and boundary conditions 3 6

10 The upper critical field of a Type II superconductor 4 3

11 The anisotropic superconductor 4 5

1 2 Thin superconducting slabs 4 7

13 Surface superconductivity 5 0

1 4 The Type II superconductor for H just below H c2 5 2

1 5 The Josephson effects 5 9

1 5 . 1 The Josephson equations 5 9

1 5 . 2 Magnetic field effects: the two-junction SQUID 6 1

1 5 . 3 The extended Josephsonjunction 6 4

1 5 . 4 Effect of an applied rf field 6 5

1 5 . 5 The resistively shunted junction (RSJ) model 6 6

1 5 . 6 The rf biased SQUID 73

1 6 The Josephson lattice in ID 77

1 7 Vortex structure in layered superconductors 8 2

1 7 . 1 3D anisotropic London model 8 2

vii



VIII ~onrenrs

17.2 Lawrence-Doniach model 85

17.3 Vortex struct ure in a 2D film 87

18 Granular superconductors: the Josephson lattice in 2D and 3D 91

19 Wave propagation in Josephson junctions, superlattices, and arrays 95

19.1 Wave propagation in a junction 95

19.2 Wave propagation in a superlattice 104

19.3 The Josephson transmission line 108

20 Flux pinning and flux motion 110

20.1 Nonideal Type II superconductors 110

20.2 Microscopic description 111

20.3 The Lorentz force 113

20.4 Pinning centers and pinning forces 115

20.4.1 The core interaction 11620.4.2 Surface magnetic interaction 117

20.4.3 Summation of the pinning forces 118

20.5 The equation of motion 118

20.6 The critical state 120

20.7 The elastic constants of a flux-line lattice 123

20.8 Collective flux pinning 127

20.9 Mechanisms of flux motion 131

20.10 Relaxation of the magnetization with time 134

20.11 Phase diagram of high To oxide superconductors 137

21 Time-dependent G~L theory 140

22 Fluctuation effects 144

22.1 The Ginzburg criterion 144

22.2 The diamagnetic susceptibility for T> t; 147

22.3 Paraconductivity for T> t; 149

23 G~L theory of an unconventional superfluid 152

23.1 The order parameter of an unconventional superfluid 154

23.1.1 Superfluid 3He: isotropic p-wave pairing 154

23.1.2 Isotropic d-wave pairing 157

23.2 Crystal-field and spin-orbit effects 158

23.3 TheG L theory of an unconventional superfluid 163

23.3.1 G L theory for 3He 163

23.3.2 G L theory for an isotropic d-paired superfluid 168

23.3.3 Unconventional G~L theory in metals 168

23.4 Inhomogeneities in the order parameter 171

23.5 Collective modes in an unconventional superfluid 172

23.5.1 Collective modes of 3He B 173

23.5.2 E state collective modes 177

24 Landau Fermi liquid theory 180

24.1 Basic equations 180

24.2 Collisionless collective modes 187

24.2.1 The kinetic equation 187

24.2.2 Collisionless longitudinal zero sound 189



24.2.3 Collisionless transverse zero sound

24.2.4 Collisionless spin waves

190

191

Part" The microscopic theory of a uniform superconductor

25 The Cooper problem: pairing of two electrons above a filled

Fermi sea 195

26 The BCS theory of the superconducting ground state 199

27 Elementary excitations: the Bogoliubov Valatin transformation 208

28 Calculation of the thermodynamic properties using the

Bogoliubov-Valatin method 212

29 Quasiparticle tunneling 216

30 Pair tunneling: the microscopic theory of the Josephson effects 222

31 Simplified discussion of pairing mechanisms31.1 The electron-phonon interaction

31.2 The spin fluctuation mechanism

32 The effect of Coulomb repulsion on To

33 The two-band superconductor

34 Time-dependent perturbations

34.1 Ultrasonic attenuation

34.2 Nuclear spin relaxation

230230

234

240

243

245

248

248

35 Nonequilibrium superconductivity 251

35.1 Elastic and inelastic scattering processes 251

35.2 Quasiparticle and phonon populations in a noncquilibrium

superconductor 254

Part '" Nonuniform superconductivity

36 Bogoliubov's self-consistent potential equations

37 Self-consistency conditions and the free energy

257

262

38 Linearized self-consistency condition and the correlation function 265

38.1 Treating the gap function as a perturbation 265

38.2 Relation to a correlation function 267

39 Behavior of the correlation function in the clean and dirty limits 272

39.1 A simple model for the clean limit 272

39.2 The dirty limit 273

39.3 The general case

40 The self-consistency condition

40.1 The dirty limit at zero magnetic field

40.2 The dirty limit at finite magnetic field

40.3 The clean limit at zero magnetic field

41 Effects involving electron spin

41.1 Spin generalized Bogoliubov equations

4l.2 The density matrix

4l.3 The linearized gap equation

41.4 Spin-dependent potentials

275

277

277

279

282

284

284

286

288

289



x Contents

41.5 Paramagnetic impurities, electron paramagnetism, and

spin orbit coupling 290

41.6 The Fulde-Ferrcll state 293

41.7 Gapless superconductivity 295

42 Boundary conditions 298

43 The proximity effect at zero field 304

43.1 Governing equations 304

43.2 The thin-film (Cooper-deGennes) limit 306

43.3 The general1D case 307

43.4 A microscopic theory of the IDJosephson superlattice 314

44 The proximity effect in a magnetic field 316

44.1 Governing equations in the presence of magnetic fields,

spin susceptibility, paramagnetic impurities, and spin -orbit

coupling 316

44.2 Representative numerical solutions 321

45 Derivation of the G-L theory 326

45.1 The first G-L equation 326

45.2 The gradient term in the clean limit 327

45.3 The gradient term in the dirty limit 328

45.4 The gradient term in the general case 329

45.5 The second G-L equation 33046 Gauge invariance: diamagnetism in the low field limit 332

46.1 Gauge invariance 332

46.2 The magnetic field as a perturbation 332

46.3 The diamagnetic current 335

46.4 Diamagnetism of the superconducting Fermi gas 338

46.5 Magnetic field behavior near a vacuum-superconductor

interface 341

46.6 Relation between normal-state conductivity and the

superconducting diamagnetic response 343

46.7 Calculations of the diamagnetic response using Chambers'

method 347

47 The quasiclassical case 350

47.1 Quasiclassicallimit of the Schrodinger equation 350

47.2 Quasiclassical limit of the Bogoliubov equations 353

47.3 Andreev scattering 355

48 The isolated vortex line 360

48.1 Bogoliubov's equations for the isolated vortex line 36048.2 The quasiclassical equations for a vortex line 363

48.3 A model calculation for the bound core states 365

49 Time-dependent Bogoliubov equations 369

49.1 Basic equations 369

49.2 The time-dependent, linearized, self-consistency condition 370

49.3 The linearized, time-dependent, G-L equation 372

50 The response of a superconductor to an electromagnetic field 374



Contents xi

50.1 The vector potential as a time-dependent perturbation 374

50.2 Relation between the current density and the vector potential 376

51 The Bogoliubov equations for an unconventional superfluid 380

52 Mean potentials, density matrices, and distribution functions in

a uniform superfluid

52.1 The mean potentials

52.2 Singlet-triplet separation

52.3 Density matrices and distribution functions

53 Superfluid 3He

53.1 Some experimental properties of super fluid 3He

53.2 Structure of the gap in an t=1, S =1 superfluid

53.3 The Balian-Werthamcr (BW) phase

53.4 The Anderson-Brinkman-Morel (ABM) or A phase

54 Collective modes in normal and supcrfluid Fermi systems

54.1 General formalism

54.2 Zero sound in a normal Fermi liquid

54.3 Collective modes in superfluid 3He B

55 Green's functions

55.1 Green's functions for the Bogoliubov equations: the

Nambu-Gorkov equations 421

55.2 Quasiclassical Green's functions in a normal system 42655.3 Quasiclassical Green's functions in a superfluid system 430

55.3.1 Keldysh formalism 437

55.3.2 Quasiclassical equations of motion

55.3.3 Eilenberger equations

55.4 Temperature Green's functions

55.5 Dirty superconductors

55.5.1 The dirty-normal-metal Green's function

55.5.2 The dirty superconductor Green's function

55.6 G-L theory revisited

55.7 The Usadcl equations

384

384

389

393

397

397

397

401

404

406

406

412

414

421

443

443

437

443

443

447

451

452

Appendix A Identical particles and spin: the occupation number

representation 458

A.l The symmetry of many-particle wavefunctions 458

A.2 Occupation number representation: Bose statistics 461

A.3 Occupation number representation: Fermi statistics 465

Appendix B Some calculations involving the BCS wavefunction 467Appendix C The gap as a perturbation through third order 470

Superconductinq transition temperature, thermodynamic criticaljield,

Debye temperature and specific heat coefficientfor the elements 475

References 476

Additional readinq 481

List of mathematical and physical symbols 483

Index 493



Preface

A preface is supposed to alert potential readers to the contents and style of what lies within so

they can decide whether to proceed further. Superconductivity is now a vast subject extending

from the esoteric to the very practical; the people who study or work on it have different

preparations, goals, and talents. No treatment can or should address all these dimensions.

Part I is devoted to the phenomenological aspects of superconductivity, e.g., London's and

Pippard's electrodynamics, the Ginzburg-Landau theory and the Landau Fermi liquid theory.

These theories allow a discussion of the effects of magnetic fields, interfaces and boundaries.

fluctuations, and collective response (which may all be thought of as different manifestations of

inhomogeneities).

Since there is currently much interest in unconventional (non-s-wave) superconductivity,

we have included a discussion of the associated Ginzburg-Landau theory (which then has a

multidimensional, complex order parameter). 3He is the only established example of an uncon-ventional superftuid (triplet p-wave) and therefore our discussions of this substance are somewhat

longer.

Part II is devoted to the microscopic theory of uniform superconductors: the theory of

Bardeen, Cooper, and Schrieffer (BCS) and the Bogoliubov-Valatin canonical transformation,

where the latter so greatly simplifies the discussion of excited states and finite temperature effects.

Although not strictly a uniform-superconductor phenomenon, the theory of tunneling and the

accompanying Josephson effects are also discussed in Part II .

Part III deals with the microscopic theory of nonuniform superconductors exclusively

through the apparatus of the self-consistent Bogoliubov equations as developed extensively by

deGennesand coworkers (and hence sometimes referred to as the Bogoliubov-decIennes theory).

Inhomogeneities associated with a magnetic field, impurities, and boundaries are discussed;

temporal 'inhomogeneities' (e.g., relaxation phenomena and collective modes) are also discussed

via the time-dependent Bogoliubov equations. Bogoliubov theory is extended to include uncon-

ventional BCS superl1uids, such as 3Hc and (possibly) high temperature superconductors and

heavy fermion superconductors (UPt3). Part II I ends with a simplified discussion of Green's

functions starting from the Bogoliubov theory. This formalism serves as an appropriate depar-

ture point for those wishing to go deeper into the theory of superconductivity. Real time Green's

functions, thermal or Ma tsu bara Green's functions, as well as q uasiclassical Green's functions are

all discussed.

Our goal in this book is to focus primarily on the physics of superconductivity. The

materials aspect is generally ignored; material properties enter the discussions only via idealized



xiv Preface

parameters. For this reason the book may not contain enough material on high temperature

superconductors to suit some readers' taste, although idealized layered systems are treated in

some detail, However, it is our view that much of the high T; literature does, in fact, involve the

systematics of the superconducting properties with regard to their chemical and physical make-

up. On the other hand, the underlying physical pairing mechanism and the order parameter

symmetry are still controversial at this time (1997).

The level of the treatment varies between sections. Much of Part I is accessible to fourth

year undergraduate students or first year graduate students in physics who have had exposure to

mechanics, electricity and magnetism, and elementary quantum mechanics. Parts II and III

involve some use of second quantization at a level usually arrived at by the end ofthe first year of

graduate school. With some omissions the book could form the basis of a one semester graduate

course on superconductivity; if all topics are discussed it would likely extend to a year.

The microscopic origin of the attractive electron-electron interaction involving the ex-

change of phonons is discussed (in Part II) only in terms of the jellium model which is sufficient to

bring out the basic physics of the electron-phonon interaction. A proper discussion, however,

requires solving the coupled Green's function equations of motion for the electrons and phonons,

which is beyond the scope of this book. A simplified discussion of the spin fluctuation pairing

mechanism is included since pairing in superfluid 3He likely arises from this eJTect.Other pairing

mechanisms, as have been proposed for high T; materials, are ignored entirely.

The character of many of our discussions was strongly influenced by deGennes' 1966

classic, Superconduct ivity of AIetals and Alloys; any similari ty of our discussions and his is likelyintentional. Both of the present authors are primarily experimentalists; their involvement in

writing a largely theoretical book about superconductivity was primarily a self-education excr-

cise. We have tried to use the simplest mathematical methods we can while minimizing the

(pedagogically useless) 'it-can-be-shown' approach. Our decision to use a uniform theoretical

approach to the topics in Part IIImeant that in many cases we had to devise our own methods; in

so doing we have strived to get things right but inevitably errors will creep in (herc and elsewhere),

for which we apologize in advance. Incidentally we would appreciate hearing about any errors

detected.

This book is not meant to be a treatise on superconductivity (for this kind of treatment we

recommend the well-known text edited by R. Parks); i.e., it is not our purpose to collect most of

what is known and useful, but rather to present the nuts and bolts used to obtain some of the

important results.

The literature on superconductivity is enormous; it is therefore not possible to be aware of

anything but a small fraction of it (this situation was 'worsened' by the discovery of high

temperature superconductivity). For this reason we decided to minimize the number of original

citations to those which are particularly important historically or pedagogically (a difficult and

dangerous decision) or to those where the reader may seek further details. We hope the manyworkers not cited will forgive us.

We have benefited from discussions with (or encouragement by) P. Auvil, K. Bennemann,

S. Doniach, A. Patashinskii, P. Wolfle, and S.K. Yip. The text was typed (again and again) by Ms

A. Jackson who deserves the most thanks.

Northwestern University J. B. Ketterson

S.N. Song



P A R T I PHENOMENOLOGICAL THEORIES

OF SUPERCONDUCTIVITY

Introduction

Many metals, alloys, and intermetallic compounds 1 undergo a phase transition at some (gen-

erally low) temperature, Tc' to a state having zero electrical resistance (see Fig. 1.1), a phenom-

enon which is call cd superconductivity. Superconductivity was discovered by Kamerlingh Onnes

in 1911; it has been sensitively probed by observing the magnetic field produced by a circulating

supercurrent using a sensitive technique such as nuclear magnetic resonance. Favorable ma-

terials exhibit no detectable decay of this current for periods limited by the patience of the

observer ( .~ years). However, superconductivity is better defined by the nature of the associated

phase transition and it then includes materials or measurement conditions where the resistance

may not vanish; here we postpone the discussion of these exceptions and initially restrict

ourselves to perfect superconductors.

0.001

I.>V j

1,

!,,

: ,'Hg

,,

I,,

< 10-6 ,I'

0.0015

~:::: 0.0010~

0.0005

0.0000-w o 4.10 4.20 4.30

T(K)

4.40 450

Figure 1.1 The original R vs T curve of Kamcrlingh Onnes showing the

superconducting transition in mercury. (After Kamerlingh-Onnes (1911).)

A second fundamental characteristic associated with a superconductor is the exclusion of

magnetic flux, discovered by W. Meissner and R. Ochsenfeld in 1933 (which we will often refer to

1. An alloy is a solid solution of two or more clements at least one of which is a metal. An intermetal1ic

compound involves a metal and one or more other elements which form a chemical compound

having nearly precise ratios of its constituents.



2 Part I Phenomenological theories

Perfect conductor Supercond uctor

I ) rf./r\\.

1 ,1

'\\, ~!. \''__j I

I . \ . ..1,I I : I

, illI ' .i / \ I

fr~

I\VJi\ \ , (! \ i I

Experiment 2:

Perfect conductor Supercondncto

: ' I

/ ; 1 t

l · r D . ii I I

i I !

i . :

II \,1

,( "I ! ~

\~Jr \ /

Figure 1.2 Hypothetical experiments showing the difference between a

superconductor and a perfect conductor. Experiment 1:sample cooled in zero magnetic

field after which a field is applied; experiment 2: sample cooled in applied magnetic field.

simply as the Meissner effect in what follows). The degree of flux exclusion can depend on the

material or measurement conditions; we will also assume this property is nearly perfectlydisplayed for our initial discussion.

The combination of zero resistance and perfect diamagnetism results in a clear distinction

between a superconductor (which as we will see is in a thermodynamic state) and a hypothetical

'perfect conductor' (which has the unique transport property of zero resistance); this difference is

illustrated in Fig. 1.2 and involves the differing response each would have for different histories of

cooling below the transition temperature and applying a magnetic field. If we start by cooling

through T, and then apply a magnetic field, both the superconductor and the perfect conductor

would exclude the field, For the perfect conductor, induced currents arising from Faraday's law

would screen the flux, Flux exclusion in the superconductor could be assigned to the same

mechanism or (more fundamentally) to the Meissner effect itself. If we reverse the order by first

applying the field and then cooling through Te, the superconductor and the perfect conductor

beha ve differently: the superconductor excludes the flux (the Meissner effect); the perfect conduc-

tor would remain fully permeated by the field.

These experimental observations argue that the transition associated with superconductiv-

ity is indeed a phase transition since an equilibrium thermodynamic state is defined by its

independent thermodynamic variables (in this case T and H), and is independent of its history

(which as we see is not true for the perfect conductor).Superconductivity, and with it the Meissner effect, does not persist to arbitrarily high

magnetic fields, For each temperature there is a well-defined critical field, He(T) , at which

superconductivity disappears." Fig. 1.3shows a universal curve of the temperature dependence of

He vs T.

2. This statement is restricted to so-called Type I materials having a shape for which there is no

demagnetization effect, as will be discussed in later sections.



Introduction 3

H

Normal

Superconducting

Figure 1.3 The temperature dependence of the critical field H J T).

,8x 10-J I"

ca V g K I Ti n

6

<;Q)

si:

~U

'" '0en

2

I/

////

/

".~.:;': .,,'- : : ..'.'.'.'...

o3 4

Tempera ture

o

Figure 1.4 Specific heat of tin as a function of temperature: open circles without an

external magnetic field; filled circles with an external field H > He' Shown also are the

individual contributions from the electrons and the lattice for H > He: chain line

lattice contribution; dashed line - electron contribution. (After Keesorn and van Laer

(1938).)

A superconductor also exhibits a discontinuous increase in its specific heat at T; (there is no

latent heat at zero field) below which it drops rapidly (approaching an exponential dependence at

low temperatures), A discontinuity in the specific heat is a signature of a second order phasetransition providing added evidence that superconductivity is associated with a distinct ther-

modynamic phase. Fig. 1.4 shows the heat capacity of a typical superconductor.

There is a related low temperature phenomenon known as superfluidity which occurs in

liquid helium. There are actually two such liquids involving the isotopes 3He and 4He having

superfluid transitions near 2 x 10- 3 K and 2 K, respectively, depending on the pressure. For

some purposes a superconductor may be regarded as a superfluid having an electric charge. 4He

superfluidity involves a Bose condensation, a phenomenon which is related to superconductivity




in a somewhat subtle or indirect way and will not be discussed in this book. The superfluidity of

3He is intimately related to superconductivity and will be addressed in later sections. Another

related superfluid is the neutron liquid in the interior of neutron stars.



The London-London

equation

Wenow present two derivations of equations which are useful in describing many of the magnetic

properties of superconductors. The older approach, used by F. London and H. London (1935)(see also London (1950)), starts with the Drude-Lorentz equation of motion for electrons in a

metal, which is Newton's law for the velocity, v, of an electron with mass, m, and charge, e, in an

electric field, E, with a phenomenological viscous drag proportional to v/r:

(2.1)

For a perfect conductor r - - - + cc, Introducing the current density j = nev, where n is the conduc-

tion electron density, Eq. (2.1) can be written

dj ne1-=-Edt In '

(2.2)

which is referred to as the first London equation. The time derivative of Maxwell's fourth

equation (in cgs units) is

DH 4rr oj r. o2EV x -- =- - + - -.

a t C o t c D t2·(2.3)

where E is the dielectric constant. Taking the curl of (2.3) and using (2.2) we have

( C H ) (4rrne2

r. (2

)V x V x -- =_- + - - , V x E;a t m (' c D l-

(2.4)

using V x E =- ( l jc)DH/Dt we have

( a ) ( 1 I: a 1) ax V x -. H + - + - -.- - H = .a t ; . ~ ('2 D t2 a t .

(2.5)

where we introduced the London depth, i.L, defined by

--,-.me:

(2.6)

Eq. (2.5) has been obtained for a perfect conductor model. In order to conform with the

experimentally observed Meissner effect, we must exclude time-independent field solutions

5




arising from integrating (2.5) once with respect to time and we therefore write

(1 8 (

2)

V X (V x H ) + -:;-,----::;:;-. H =0;J'L C~ o r

(2.7)

this is referred to as the second London equation. In what follows we will refer to Eq, (2.7) simply

as the London equation.

An alternate derivation of (2.7) is motivated by the idea that some of the moving electrons

behave collectively as a superfiuid, a liquid possessing no viscosity. This concept is borrowed

from the physics of liquid 4He; below 2.l9 K this system behaves as if it were composed of a

mixture of two liquids: a superfluid, having no viscosity, and a normal liquid, having a finite

viscosity. We assume that the free energy of a superfiuid consists of three parts

(2.8)

where F N is the free energy associated with the normal liquid, Ekin is the kinetic energy of the

moving superfluid, and Emag is the magnetic field energy. We may write these latter two terms as

(2.9)

and

(2.10)

where p ( r ) is the mass density associated with the supcrfluid. Writing p = 1 m and v=l /ne)j and

using the fourth Maxwell equation V x H = (4rr/c)j, Eq, (2.l0) becomes

(2.l1)

n IS now interpreted as the density of supcrconducting electrons, We will assume that the

superconducting electrons adjust their motion so as to minimize the total free energy; thisrequires c 5 ( E m a g + E kin) =0 or

I fme? }

H(r)' bH(r) + --2 [V x H(r)] . [V x bH(r)] d3=0,4r rne

(2.12)

where bH(r) is a variation of the (initially unknown) function H(r). Integrating the second term by

parts (and placing the resulting surface outside the superconductor) we obtain

I[H(r) + A~Vx (V x H)]' c)H(r)d

3

r = . (2.13)

Since the variation c'iH(r) is arbitrary, the term in the square brackets must vanish; therefore

V x (V x H ) + ic~H =0, (2.14)

which is eq uivalent to (2.7) (including the displacement term in Maxwell's equation yields the last

term in (2.7), which is negligible for most applications).

As a simple application of Eq. (2.l4) we now discuss the behavior of a superconductor in a



The London-London equation 7

magnetic field near a plane boundary. Consider first the case of a field perpendicular to a

superconductor surface lying in the x y plane with no current flowing in the z direction. From the

second Maxwell equation, V' H = 0, we obtain DHz /u z = 0 or J J = const. From the fourth

Maxwell equation, V x H = (4n/c)j, the first term in (2.14) vanishes and hence H = 0 is the onlysolution. Thus a superconductor exhibiting the Meissner effect cannot have a field component

perpendicular to its surface.

As the second example consider a field lying parallel to the superconductor surface, e.g.,

H I I X,which we may write as H =H(z)x (which satisfies V' H =0). Using the vector identity

V x (V x H) = V (V 'H ) - V 2H , (2.15)

Eq. (2.14)becomes

(j2H 1__ x __ H =0(jZ2 A l x

(2.16)

or (for a superconductor occupying the region z > 0)

(2.17)

A field parallel to the surface is therefore allowed; however, it decays exponentially, with a

characteristic length, Au in the interior; ).L (T =0) ranges from 500 to 10000 A , depend ing on the

material. Accompanying this parallel field is a surface current density, which, from Maxwell's

fourth equation, is

c .j(z) =- -.-HAO)c-Z!'.LY.

4nl'L(2.18)

This current density shields or screens the magnetic field from the interior of the superconductor.



Gf----p-;-p-p-a-r-d-'-S-e-q-u-a-t-;-o-n---

At temperatures well below the superconducting transition temperature the heat capacity of a

superconductor displays an exponential behavior C ~ e - MkB T (see Fig. 1.4).This suggests that theconduction electron spectrum develops an energy gap, L '\ (not to be confused with the gap in a

semiconductor); recall that electrons in normal metals have a continuous (gapless) distribution o f

energy levels near the Fermi energy, fl. On dimensional grounds one can construct a quantity

having the units of length from L '\ and the Fermi velocity, 1!F; we define the so-called coherence

length by

(3.1)

This length bears no resemblance to the London depth, Au and hence represents a different length

scale affecting the behavior of a superconductor; it can be interpreted as a characteristic length

which measures the spatial response of the superconductor to some perturbation (e.g. the

distance over which the superconducting state develops at a normal metal-superconductor

boundary). Length scales of this kind were introduced independently by Ginzburg and Landau

(1950) and by Pippard (1953).1 We first discuss Pippard's phenomenological theory (which

scmiquantitatively captures the main features of the microscopic theory to be discussed later). We

begin by writing London's equation in an alternative form. Substituting V x H =4n/c)j (from

the fourth Maxwell equation) in the London equation yields

. cVXJ=---H

4n}~

ne2

=--Hme

(3.2)

We next write H = V x A . where A is the magnetic vector potential, and restrict the gauge to

satisfy

V·A = 0 (London gauge) (3.3)

and the boundary condition

I.These length scales are not identical, however; the Pippard length is temperature-independent while

the Ginzburg-Landau length depends on temperature. The Pippard coherence length is related to

the ReS coherence length.

8



Pippard's equation 9

(3.4)

where Anis the component of A perpendicular to the superconductor surface. London's equation

may then be written

ne2

i= --Ame

(3.5)

Note that the condition (3.4) builds in the reasonable boundary condition that the normal

component ofthc supercurrent.j., vanishes at a boundary (this is a good boundary condition at a

superconductor-insulator boundary but will require modification for metal-superconductor or

superconductor-superconductor boundaries).

To generalize (3.5), Pippard reasoned that the relation between j and A should be nonlocal,

meaning that the current j(r) at a point r involves contributions from A(r') at neighboring points r'located in a volume with a radius of order ~o surrounding r. The mathematical form he selected

was guided by the non local relation between the electric field, Etr'], and the current, j(r), which

had been developed earlier by Chambers (1952). The expression employed by Pippard was

f [ A(r')' R]R ..( ) C -R'<odJ r

J r =- c ,. rR4 '

(3.6)

where R= - r'. The constant C is fixed by requiring (3.6) reduce to (3.5) in the quasiuniform

limit where we may take A from under the integral sign; we then have

C fCO s2 (!dQ fe - R!~odR =ne2

,

me(3.7)

fromwhich we obtain C = 3ne2 /4n~omc. Pippard's generalization of London's equation is then

,. 3ne2.f[A(r')'R]R -R~ J,

J(r) =---- e ! ,Od r.4n;ome R4

(3.8)

Since Eq. (3.8) involves two functions. A(r) and j(r), a complete description requires a second

equation which is obtained by substituting H =V x A in the fourth Maxwell equation to obtain

4nVx(VxA)=~j

e(3.9)

(resulting in an overall integrodifferential equation for A).

Eq. (3.8)applies only to a bulk superconductor. An important question we would like to

examine is the behavior of a magnetic field near a surface, which will require a modification (or

reinterpretation) of (3.8). To model the effect of the surface the integration over points r' is

restricted to the interior of the superconductor. If the surface is highly contorted, then it can

happen that two points near the surface and separated by about a coherence length cannot be

connected by a straight electron trajectory, without passing through the vacuum; one then has to

account for this shadowing effect. We restrict ourselves here to plane boundaries which we take to

be normal to the z direction.

In the limit A L >> ~o Eq. (3.8) reduces to the London equation, as discussed above. (By

expandingA(r') in a power series in R, we may obtain corrections to the London equation due to

nonlocality.) In the opposite limit, A L < < ;0' A(r') varies rapidly. Let us assume that A(r) falls off




over a characteristic distance ;_ (which we will determine shortly through a self-consistency

argument). When )~< ~O, the value of the integral (3.8) will be reduced roughly by a factor ( A . / ( o ) ;

I.e.,

;. ne2j(r) =-~- A(r).

C ;o mc(3.10)

We may also write (3.10) in the 'London-like' form

(3.11)

This equation has solutions which decay in a characteristic length (I.~~0!;_)1/2; to achieve self-

consistency we set this length equal to A :

(3.12)

(A more rigorous derivation from the microscopic theory carried out in Sec. 46 yields

2=0.62i. t~0.) We conclude that in the Pippard limit the effective penetration depth A is larger

than the London depth, )'L: ;'!;'L =~oI)L)l/.l > 1. At the same time ), remains smaller than the

coherence length: }'/(o =(I'LgO)2/3 < l.

If our metal has impurities it is natural to assume the relation between the current and

vector potential will be altered. To account for the effects of electron scattering, Pippard modified

the coherence length factor in the exponent of(3.8) as 1!~0 --> (1/(0) + (lit) where tis the electronmean free path;' the coefficient in front of the integral was not altered. Eq. (3.8) then becomes

. 3ne2

f[A(r')'R]R -R(., '-+~) 1,J(r)=--- ..- e '" Id'r.

47 [~omc R4(3.l3)

In the limit ). >> (t, ~o) we may again take A(r) from under the integral sign; carrying out

the integration we obtain

ne2 1

j(r) =-- ..·--A(r).mcc;o 1 1-+-~o t

(3.14)

For the case of a very dirty metal, where l< < ~o,

ne? tj(r) =-- ~A(r).

me C ;o

(3.15)

The effective penetration depth is then obtained from the expression

(3.16)

If ;.< < (t, ~o) we continue to use Eq. (3.12).

The magnetic response of a superconductor depends on whether A ; ; : ; ~ or A . : S ~ , as will be

developed later." These regimes are designated in Table 3.l. A Type I superconductor displays a

2. Other expressions, such as ~=({;0)1/2, are sometimes used to estimate the effective coherence length.

Such ambiguities are removed by the microscopic theory to be discussed in Part III.



Pippard's equation 11

Table 3.1 Regimes defininq Type J and

Type II superconductors?

Type I (or Pippard)

Type II (or London)

complete Meissner effect (flux exclusion) up to some critical field H e' above which it becomes

normal. The magnetic response of a Type II superconductor is more complex and will be

developed later.

3. The precise criterion separating the regimes is A =ufi.



.I--~-h-e-r-m-O-d-y-n-a-m-ic-s-o-f-a--

Type I superconductor

In this section we consider the thermodynamics of a Type 1superconductor (Gorter and Casimir

1934a,b); our discussion follows that of London (1950). We restrict the geometry of the supercon-ductor to a form for which the external field, H, is not distorted by the presence of the

superconductor (examples being an infinitely long cylinder with H parallel to the axis or a plane

slab of infinite extent with H parallel to its surface). Far inside the superconductor (i.e., several

London depths from the surface) the magnetic field essentially vanishes in the superconducting

state and is equal to H in the normal state. In the thermodynamic identities that follow we identify

this interior field as B, the flux density. The H field will be taken as the applied external field. The

relation between Band H is shown in Fig. 4.l.

We recall the thermodynamic identity for the response of a system in a magnetic field

1dIS=TdY + -H·dB,

4n(4.1)

where C f · is the energy density and Y is the entropy density. When T and B arc the independent

variables we use the Helmholtz free energy density, Y =f f - TCf', and when T and H are the

independent variables we usc the Gibbs free energy density, '§ = . ' 1 7 - (1/4n)B· H; taking the

differential of these two quantities and using (4.1) yields

1dY = - 9'dT+ -H·dB

4n(4.2)

and

1d~1j= - .'f'dT - - B· dH.

4n(4.3)

A Type I superconductor displays the Mcissner-Ochsenfeld properties:

B =0,

B=H, H> H e.

(4.4)

(4.5)

H e is called the thermodynamic critical field. Since H and Twil l be our independent variables, we

integrate (4.3) at constant T to obtain

< fJ(T , H) =§(T ,O ) - L f B·dH;12



Thermodynamics of a Type I superconductor 13

B

/

/

/

o H

Figure 4.1 B vs H curve for a Type I superconductor.

therefore

1~fj (T,H) = 0'(T ,O ) - _(H 2 - H D

Sn( 4 . 6 )

or

~fj (T,H) =0' (T,O) ( 4 . 7 )

Note that qj is continuous at the transition. We define a function

1 0

'f Jo = = ' fJ(T,O) +-II~.Sn

(4 .8 )

Wemay then write

(H>HJ

(4 .9)

and

1' f J (T ,H) =fJ o - -H ~

8n

(4.10)

where qjN and qjs denote the normal and superconducting states, respectively. We may interpret

qjo as the Gibbs free energy of the normal metal at zero field (were it stable); hence the Gibbs

free energy density of the superconducting state is lower than that of the normal state by

( 1 /8n)H~(T) ; this quantity is referred to as the c on den satio n e ne rgy . Since ~ fj= =.'F - (1/4n)H' B,

§ is(T ,O) = jo - ( I /Sn)H~. The normal state Helmholtz free energy density is then

(4.ll)

From Eq. (4.3),

.r;'=_~ 'fJ ) ;6T H

(4.12)




thus

aqjoY' ---

N - D T(4.l3)

and

~,(f 1 ( ' H ):~ O,'lo () cJs= --+-Hc - .

a T 4n a T H

(4.14)

Note the entropy is discontinuous across the transition and hence we have a first order transition

(when H = F 0):

(4.l5

The heat of the transition is

(4.16

this equation corresponds to the Clausius-Clapeyron equation of a (P, V,T) system. The specifn

heat (at constant H) is defined as

(4.17

or

_ T [ 32Hc (OHc)2]---H--+-4n c cT2 a T .

(4.l8

At 11=0, where the transition is second order,

t;( 3 1 1 c)2

/I,.«jHIT=T'=-4n a T T~T';

this is sometimes called Rutgers' formula (Rutgers 1933).

(4.15



.I---~-'h-e-in-t-e-r-m-e-d-ia-t-e-s-t-a-t-e--

I f a superconducting body of arbitrary shape is placed in a magnetic field, the flux exclusion

associated with the Meissner effect will in general distort the magnetic field. Exceptions are aninfinite cylinder with the field parallel to the axis, or a sheet or half space with H 0 parallel to the

plane of symmetry. For situations involving lower symmetry the local magnetic field can vary

over the surface, being both higher and lower than the applied field, Ho. As a simple example

consider the case of a spherical superconductor shown in Fig. 5.1. From magnetostatics the field

willbe highest at the equator (on the cirele C in Fig. 5.1)where it is H =~Ho- Hence flux enters the

sample, not at the thermodynamic critical field, H c ' but at a value flo =~Hc . For magnetic fields

He> Ho > iH c the sample consists of alternating domains of normal metal and superconductor.

A superconductor in such a regime is said to be in the intermediate state (Landau 1937).

Figure 5.1 The magnetic field distribution about a superconducting sphere of radius a.

For an applied field H o < ~Hc ' there is a complete Meissner efTectand the field at the

equator (at any point on circle C) is ~H 0 ; the field at the poles (Q, Q') is zero. For

~Hc < Ho < He' the sphere is in the intermediate state.

15




Figure 5.2 Schematic of the magnetic field distribution in a slab in a perpendicular

field. For all f ields J J 0 < ll; the sample is in an intermediate state having a laminar

structure of normal and superconducting domains.

We will limit our discussion to the simple case of a plane superconducting sheet with H o

parallel to the surface normal. From our earlier discussion we know that a superconductor

cannot sustain a field component perpendicular to its surface. The field behavior is shown

qualitatively in Fig. 5.2. It has the following features.

(i) For a magnetic field 0 < H o < He the sample consists of adjacent domains which are

wholly superconducting (with no internal flux) or normal (with Hloeal =HcJ,

(ii) In the interior of the superconductor and far from the surface, the domain walls are parallel

to the applied field direction; the fraction of the cross section that is supercond ucting is fixed

by Ho and H, such that the total flux through the sample is conserved

normal cross section H 0

total cross section He(5.1)

Hence the superconducting fraction is 1 - HolHe (which vanishes, as it should, for

H o =H c l .

(iii) Near the surfaces the flux sheets 'flare out' (which reduces the field curvature, which would

otherwise raise the local fields at the interface). Were the local field to remain fixed at He' to

sustain the average flux, needle-like superconducting domains would have to encroach on

the normal domains in the vicinity of the surface. Such needles do not occur since the field

profile is controlled by a minimization of the surface energy (between supcrconducting and

free surface or the normal regions, see next section), the superfluid kinetic energy, and the

total magnetic field energy.

A detailed treatment of the domain structure is mathematically complex and will not be dealt

with here (see Landau, Lifshitz, and Pitaevskii (1984), deGcnncs (1966), London (1950)).



8 f - - - - s-u - r - , . - a - c - e - e - n - e - r - g - Y - b - e - t w - e - e - n -anormal and a

superconducting metal

Consider a slab parallel to the x-y plane in a perpendicular magnetic field parallel to z. Assume

we have a phase boundary perpendicular to the x axis with the superconductor occupying theregion x> O . The total free energy, F, in the London model is:

IYJ l H2(r),i2 ]

F =sf dx .'1's + -- + - - - - ' = : [ V x H(r)]2 ,o Sn Sn

(6.1)

where s1 is the interface area,:Fs is the condensation energy density, H~(T)!8n, and the second

and third terms are the magnetic field energy density and the superfluid electron-kinetic energy

density, respectively. At our phase boundary in the intermediate state, where H =He for x < 0,

we have (see Eqs. (4.6) and (4.7))

(6.2)

By definition

e W I ,< ,

G =F - - H e' B(x)dx.4n 0

(6.3)

According to the discussion at the beginning of Sec. 4 w e set B(x) =H(x); from Eq. (2.17) we have

H(x) =Hcz e -X

/AL

• Inserting this form in (6.3) and using (6.1) we obtain

IYc

= "I' d3" + " _4- .7" S I ,yY,

o

(6.4)

where y is the surface energy per unit area, i.e., the 'surface tension', given by

(6.5)

Note this surface energy is negative. This suggests the system can lower its energy by maximizing

17




the interfacial area (i.e., the system is unstable to the formation of multiple domains with

associated interfaces),

Type I superconductors (for geometries not possessing an intermediate state) display a

single domain for H :::;;H e; hence they must have a positive domain wall energy. Physically this

positive surface energy arises because superconductivity is destroyed over a region of order ~

perpendicular to the interface; i.e., we lose the condensation energy over a volume of order .#~,

where ( is a coherence length. This is equivalent to a positive contribution to the surface energy of

order

H2},=~_c.

Sn(6.6)

In a Type I material ( > )'L and hence the positive contribution (6.6) outweighs the negative

contribution (6.5) and the interface is stable. In Sec, 9.3 we will continue this discussion and derive

an expression for the surface tension.

For a Type II material the system does, in some sense, try to maximize the amount of

internal interfacial area above some field (referred to as the lower critical field); however, it is

subject to a constraint imposed by quantum mechanics, as we discuss in the next two sections.



8I---o-u-a-n-t-;z-e-d-v-o-r-t-ic-;-t-y---

The previous discussion of the surface energy of a normal metal/superconductor interface

suggests that Type II materials, where ¢ : : S A are unstable to the formation of domain structureswhichin some way maximize the amount of interface area. Two possible domain geometries are:

(i) an array of nested sheets (closed or open, depending on the geometry 1) and (ii) a two-

dimensional lattice of flux filaments. Calculations show the latter domain structure to be more

stable.

Since the filaments (are presumed to) admit flux into the interior of the superconductor, we

envisionthem as having a normal core with a diameter of order ¢, outside of which supercurrents

flowin a diameter of order }, which produce the internal field via Ampere's law.

Asa primitive model of a single flux filament we consider the extreme limit ¢ -- > 0 for which

the London approach should provide a good description. We recall Eq. (2.5) associated with our

first derivation of the London equation

a l 4nne2

]- v x (V x H) + -2- H = 0;D t m e (7.1)

we next integrate (7.1) over an area s~ intersecting the filament (for convenience we choose a

plane perpendicular to its axis) and use Ampere's law (Maxwell's fourth equation) in the form

a f l m e ];;- -2 V X + H . d,xi = O.o t ne

(7 .2)

In integrating (7.2) with respect to time we now allow the possibility of a nonzero constant of

integration (since the flux filament phenomen violates the Meissner behavior); thus

(7.3)

Applying Stokes' law to the first term in (7.3) yields

m e 1 fne2jj'dt + H·d.xi = 4 > . (7 .4)

Ifwe choose the contour to enclose a large area, we may expect the first term to be exponentially

1. For a superconducting slab we envision an array of interfaces parallel to the surface and for a

cylindrical sample an array of coaxial cylinders. Other shapes would have more complex structures.

19




small (since the currents fall off exponentially with a characteristic length ),);we can then identif

the constant of integration, ¢, as the total flux contained within the filament (most of which alsr

falls inside a radius of order A ) .

To gain further insight we substitute H = V x A into (7.4) and again apply Stokes' law tc

obtain

f[:~jA 1 M = ¢; (7.Sa

writing j=nev =ne(p/m), (7.Sa) becomes

l [ p + : A ] . dt = : ¢.J C C

(7.Sb

Wc identify the integrand as the canonical momentum associated with the motion of a charged

particle in the Hamiltonian formulation of mechanics. F. London correctly concluded that

superconductivity was a macroscopic quantum phenomenon, and guided by this insight he

suggested that Eqs. (7.S) must conform with the Bohr-Sommerfeld quantization rule for the

(quasiclassical) motion of an electron, i.e. (I e I/c)¢ = nh or ¢ = n(he/I e I),where n is an integer.

However, this assumes the orbiting entities are single electrons; Ginzburg and Landau allowed

for a marc general case where e -> e* ; we then have

(7.6a)

where

he

¢o=le*l·(7.6b)

From the Bardeen-Cooper-Schrieffer (ECS) theory it is now known that e* = 2e; i.e.,

he

¢o = 21 e 1

= 2.07 X 10-7 G crrr' (7.6c)

which is referred to as the flux quantum. Hence flux enters a Type II superconductor as an array

of quantized flux filaments; the lowest energy situation corresponds to singly quantizied (1 1 = )

filaments eaeh carrying a flux quantum ¢o.

In what immediately follows we adopt cylindrical coordinates (r, 0, z) and write the in-plane

radius vector as r. Let us next examine Eqs. (7.S) for a contour of radius A >> r >> ¢; the

amount of flux contained is then vanishingly small and the first term in the integrand of (7.Sb)

dominates yielding the condition 2npr = h or p =h/r; BCS theory also dictates that the mass ofthe orbiting entity is m* = 2m; hence

h ~v(r)=-{}

2mr(7.7)

where (j is an azimuthal unit vector. This velocity profile corresponds to the large r behavior of a

vortex in a fluid, although with the vorticity quantized. One then refers to the filaments as

qu antiz ed vortex lines or vortex lines for short.




Figure 7.1 Structure of an isolated Abrikosov vortex line in a type IIsuperconductor

or

(7.14

where we choose a radius r =(for the (inner) line integral; this corresponds to the physicallj

reasonable assumption that the field divergence is removed at the coherence length scale (as,

more complete theory confirms). We expect that the l/r divergence of the superfluid velocitj

ultimately destroys superconductivity in the vortex core; we may model this effect by assumim

the density of supcrconducting electrons, n(r) , approaches zero (sufficiently rapidly) as r -- > O.Fig

7.1 shows the qualitative behavior of H(r) and Il/I(r) I near the center (core) of a vortex filamen

where I l/I(r) I = = [n(r)]t .

In the next section, where we discuss the mixed state of a Type II superconductor, we wil

require an expression for the interaction energy of two vortices. Returning to our London-lik:

model, we generalize Eq. (7.8) to the case of two (parallel) vortices

(7.15

Since this equation is linear the resulting magnetic field will be the sum of two terms having th

same form as (7.10):

(7.H

The total energy (which is quadratic in H) will involve three terms: two of these correspond to th

'self-energies' of the individual vortices (as given by (7.13)) and the third results from thei

interaction, The interaction energy in the extreme London limit can be evaluated from the secor»term in (7.12) along with (7.10) (we assuoe the path of the first integral involves a single circle c

very large radius encircling both vortices); the interaction energy is then given by



Ouantized vorticity 23

= 2 ( ~ ) 2 K o ( l r l ~ r 2 1 ) ,4n)·L AL

where,due to our assumptions, we require 1 r 1 - r 2 1 ;< ; ~ . The sign of (7.17) is positive and hence

the force (per unit length) between two vortices is repuls ive . We may rewrite Eq. (7.17) in the form

(7.17)

E(l2) '"

__ =~H(l2)

L 4n '(7 .18 )

whereH(12) =0 ll~l)(r = r2) = H~2)(r = r 1) with the latter given by (7.10); i.e., H(12) is the contribu-

tionto the field at one vortex resulting from the presence of another. Regarding ( - ¢oL/4n)z as a

magneticmoment, p, we could write E(12) = - p"H.where H = H(12)Z.



v Type II superconductivity

8.1 Magnetic fields slightly greater than He1

Superconductors with ~ < )"termed Type II materials, behave differently in a magnetic field fron

those with ~ > I " (Type I materials). Fig. S.1 shows the field dependence of the magnetization of a:

ideal Type II superconductor (the sample geometry is assumed to be a plane slab or a cylinder tl

avoid geometrically induced field inhomogeneities as noted earlier in our discussion of th

intermediate state). For low fields the magnetization is - H/4n; i.e., the sample displays,

Meissner-like behavior. However, at a field H =He1, called the lower criticalfield, flux abrupt!

enters the sample (the susceptibility X =dM/dH) --+ +xfor an ideal sample at this field). Th

magnetization increases (becomes less negative) continuously above H el and reaches zero at ;

field H = H c2 called the upper critical field. The regime H el < H < H c2 represents a nev

thermodynamic superconducting state called the mixed state or Shubnikov phase (Shubnikov eal. 1937).

We now show that the Meissner state of a Type II superconductor becomes unstable to tb

entry of vortex filaments at a field which we identify with H c1• For magnetic fields which are onl

slightly above H e! we may write the Gibbs free energy per unit volume as

1 .. B·HI# (H) = I#(H = 0) + nLEIL + - IE(I,JI - --.

Li<i 4n(S.1

The second term represents the self-energy of the individual lines where nL is the number of lineper em? with energy E/ L per em, the third term is the (repulsive) interaction energy between th:

lines (and is summed over a unit of area), and the last term is the usual field term relating tl u

Gibbs and Helmholtz free energies. In the presence of a uniform array of flux lines the magnetn

induction is

(S.2

Near He! (above which flux first enters), we will initially neglect the interaction term and, usiru

(S.2),write

(S.3

For H < 4nE /¢oL , (S.3) is minimized by setting nL=O. However, for H > 4nE/4>oL Eq. (S.3

24



I ype " euperconaucttvtty

-41TM

(a) 0 H

-41TM

MeissnerPhase

AbrikoSOlJMixed Phase

Normal

Phase

Hb) 0

Figure 8.1 The reversible magnetization curves for a Type I (a) and a Type II (b)cylindrical superconductor. The magnetic field is applied parallel to the cylinder axis.

would suggest that q}ex - nL; i.e., the system can lower its free energy indefinitely simply by

creating more flux lines. It is clear that we can identify He Ias

(8Aa)

or

(8Ab)

where we used (7.14) for the vortex self-energy in (SAb).1 The negative divergence of C I J for

H> Hel is eliminated if we include vortex-vortex interaction (repulsion) effects; i.c., we must seek

the minimum with respect to B of the quantity

B l 1 4 > 0 ( d ) ]J.q}=- H -H+-z--K -4n c I 2 2n}~ O}L .

( 8 .5 )

The first two terms are the same as in (S.3). The last term is the result of the vortex-vortex

1. We can now make contact with the comment made at the end of Sec. 6. Taking the classical limit as

h - - -> 0, we see that the lower critical field would approach zero. At any finite field we would then

have a divergent number of flux lines; i.e., the system would have a divergent internal interface area.

Thus quantum mechanics imposes the constraint on the maximal amount of internal interfacial area.




0.10

0.05

0.00

!J.'§ -0.05

-0.10

-0.15

-0.20

o

-50

-100

!J.,§-150

-200

-250

-3000.0

H/H =cl

(a)

1.1

1.2

1.3,_

2 3 4 5 6 7 8

dI", l

(b)

I

\ H/H z:

\cl

\ 8\ ->

\//-

,,/-

9 /-:

/

\ .> //

\ -~

///\\

\

\

\////10

\_-

0.2 0.4 0.6 0.8

diAL

Figure 8.2 Normalized Gibbs free energy density calculated from Eq. (8.5)as a function

of the reduced vortex lattice spacing in the low applied field regime (a) and high applied

field regime (b).

repulsion; the factor 1/2 assigns half of the interaction energy to each vortex (in Eq. (7.18)), z is the

number of nearest neighbors in the vortex lattice (the exponential fall-off of Ko justifies including

only nearest neighbor interactions at low fields where nL is small), and d (appearing in the

argument of Ko) is the vortex-vortex spacing, which depends on the symmetry of the vortex

lattice. Calculations show that a triangular (centered hexagonal) arrangement of lines has the

lowest energy. For this lattice nL = 2/J3d2 =BN o or d2 = 24>o/J3B, and z = 6. On substitu-

ting these values of d and z in Eq. (8.5) we obtain !J.~tJ=J.'§(B); this function is shown qualitatively

in Fig. 8.2 and the minimum value of M tJ yields the magnetic induction, B , for a given external

field, H. Carrying out the minimization process for each field H we can develop the function

B = B(H ) (or M = M(H ) through B = H + 4n:M) . (Further analysis of this model shows that

X = (3M/aH)T does diverge as H --> He! from above.)

The above analysis breaks down as nL increases (i.e., as B increases), and more powerful

techniques are required. The simplest method involves the Ginzburg-Landau (G-L) theory, a

discussion of which we begin in the next section.

In discussing the thermodynamics of a Type II superconductor the concept of the ther-



Type" superconductivity 27

modynamic critical field, H e' is retained in terms of the condensation energy, H~ /Sn ; however,

He! < He < Hc2' From the discussion in Sec. 4 (where we showed (l/4n)Sg,B'dH =0) we

identified H;/Sn as the condensation energy. The fact that M has a different field dependence in a

TypeIIsuperconductor does not alter this identity.

Therefore

fH 1

c M(H )dH = __ H ~.o Sn

(S .6 )

Hc2 greatly exceeds He in certain alloys and intermetallic compounds. This fact, coupled with the

experimental observation that a zero resistance state often persists up to H e2, makes Type II

materials of great importance in the manufacture of magnets and related technologies.?

8 . 2 The region H C 1 < < H < < H C 2

In the region He! < < H < < He2 there is a densely packed array of vortices; however, the

spacing between vortices ( ~ n C 1/2), d , satisfies the inequality ~ < < d < < A (here we assume that

( < < A so as to guarantee the existence of such a regime). When d >>~,H(r) is accurately given

by the solution of the inhomogeneous London equation

( S . 7 )

where iruns over all vortices in the lattice. Assuming the vortices lie on a periodic lattice:' H(r)

may be expanded in a two-dimensional Fourier series as

H(r) =HGe-iG,

c

(S.Sa)

where

(S.Sb)

here G denotes all vectors of the two-dimensional reciprocal lattice associated with the real space

lattice of the vortex array and nL is the reciprocal of the area of the unit cell. Inserting (S.Sa) into

(8.7),using the fact that V x [V x H(r)] =-V2H(r) (since V' H(r) =0), multiplying by eiG"', and

integrating over d2r. and noting G'·ri is a multiple of 2n we obtain

, s ¥ 'I1+ A 2G

2

)H(A;G' = ¢oNLzC

2. In the presence of a transport current (as one has in a magnet) the nux lines arc subjected to a

Lorentz force which would tend to make the flux lines move, which is a dissipative process.

However, flux lines are usually pinned (immobilized) by various inhomogeneities present in the

material (c.g., defects, grain boundaries, ctc.),

3 . In an inhomogeneous superconductor, vortices may be attached to so-called pinning sites, resulting

in deviations from periodicity. On the other hand, thermal agitation may cause the lattice to melt in

a high temperature superconductor.




or

(S.9

where sf is the total cross sectional area of the vortex lattice, IVL is the total numbers of lines ant

n, = = NLied; henceforth we assume that H(r) and Hc are directed along the z axis and denote onl-

their amplitudes, H( r ) and H c '

The total energy of the vortex lattice follows from Eq. (7.11) as

Defining the energy density I f f ' =E/sf L we obtain

(S.10

Now the minimum nonzero value of I G I ~ 2 n /d( ~ 2 nn~ !2 ) and with our assumption d < < ;.w r

have ; . 2 G ; ; : ' i n >> 1; therefore we may approximate 1/(1 + }.2G 2) by 1/).2G 2 . The sum in (S.10

depends on the particular lattice adopted by the vortices. However, following deGennes (1966)

we will limit ourselves to a semiquantitative estimate and replace the sum by an integral;

(S.11

From the above argument Gmill ~ 2n /d ; on the other hand we expect Gmax to be cut off at the scale

of an inverse coherence length and write Gmax =2n /¢ . We then obtain the total energy density 0

the lattice as

(S.12

here we used Eq. (S.4b) for He! and included a numerical factor f 3 to offset partially the various

approximations used to obtain (S.12). Matricon (see deGennes (1996)) finds f 3 =0.3S1 for a

triangular lattice where d 2=2¢o /J3B . We note in passing that by considering the changes in the

free energy arising from small distortions of the lattice from its equilibrium form, we rna)

calculate the magnetic contribution to the elastic constants of the lattice. We will return to this

topic in Sec. 20.7. Ignoring the entropy contribution we may replace If f by IF ; using the definition

1'! J =ff - -BH

4n(8.13;

we may then calculate B =B(H) from the condition aG/3B = 0 which is appropriate for a

constant-H environment. Carrying out the calculation (including the contribution from the

implicit dependence of d on H ) we obtain



Type" superconductivity 29

(S.14)

where{ J ' = = {Je-I!2. From the definition B =H + 4nM we have

tn ( { J ' ~ )He! c ,

M=-4;- ...( l ) .tn -

(

(i\.15)

Sinced X. B-I!2 we see that M increases logarithmically (becomes less negative) with increasing

magneticfield. This behavior agrees well with experimental data for materials with )_>>~.

8 . 3 Microscopic magnetic probes of

the mixed state

Wewill now briefly discuss two experimental probes of the inhomogeneous magnetization in the

mixedstate: neutron diffraction and nuclear magnetic resonance. Other probes which have been

applied or proposed include: decorating the surface with magnetic atoms (the Bitter technique)

and magnetic force microscopy. (For more coarse grained magnetic structures magneto-optic

and scanning Hall probe microscopy are useful.)

We begin with neutron diffraction. The neutron has a magnetic moment fln = (gn /2)(eh/

2M pc) where gn is the anomalous 9 factor of the neutron (gn/2 = - 1.91354) and eh/2M pc is the

nuclearmagnet on with Mp the proton mass. The neutron interacts with a magnetic field via the

usual spin Hamiltonian

f1 = - f i n · H(r), (S.16)

where f i n = fln!! where g is the vector Pauli matrix. The coherent scattering cross section, (J c -followsfrom the standard Born approximation expression

wheref(q) is the scattering amplitude given by

A 1 n f ~ · ·3f(q) = -. H(r)e'Q"rdr ;

2nh1(S.17)

here Mn is the neutron mass and q = k - k' with k and k' the wave vectors of the incident and

scattered neutrons ( I k I = I k' I for a diffraction experiment). Inserting Eq. (S.Sa), with H G given by

(8.9),into (S.17)we have

(S.lS)




where V is the sample volume and the Kronecker delta function reflects the usual Ewald

diffraction condition, q=G. Writing B =1L< :P o , < :P o =2nnc/2e and Jin ~ (gn/2)(elJ/2Mnc) we have

(g .19)

Note that as discussed above ; c 2G2 >> 1; hence 6(q) = If(q) 12 o: (AG)-4. For this reason it is

very difficult to observe more than the lowest order peak. Experiments were first performed on

Nb by Cribier et al. (1964). Since high T; and heavy fermion materials have anomalously large

London depths neutron diffraction has not been a useful probe of the vortex lattice in these

materials.

Wc now discuss the expected behavior of the line width, ~w, measured in an nmr

experiment. For many purposes the line width may be inferred from the root mean square

deviation of the magnetic field from its average value"; i.e.,

(S.20)

and we take ~w =y~H , where y is the gyromagnetic ratio of the nuclei which are in resonance.

Here the bar implies an average over the sample volume. For simplicity we assume the external

(applied) magnetic field is strictly uniform and that the sample has the shape of a long cylinder

parallel to the applied field so that demagnetization effects may be ignored. Now H ( r ) in (S.20) is

simply B =1r< : P o ; H 2 ( r ) is calculated as follows using (S.Sa) and (S.9):

where "I is the area of the sample perpendicular to the magnetic field. Separating ofTthe G =0--2

term (which cancels the [ H ( r ) ] term in (S.20)), restricting to the limit A 2G 2 >> 1, and again

replacing the sum by an integral we obtain

The integral is convergent at the upper limit so we extend Gmax to cc and we again take

Gillin ~ 2n/d (valid for a square lattice) yielding

(S.23)

Note that our result is independent of magnetic field and within a factor of order unity is equal to

H c L' Therefore nmr is readily observable in extreme Type II materials containing nuclei with

sufficiently large moments. We emphasize that this line width is not to be interpreted as thereciprocal of a magnetic relaxation time; it is referred to as an inhomogeneous broadening. The

relaxation of the magnetization will be discussed briefly in Subscc. 34.2.

4, For a discussion ofmoments in nmr seeAbragam (1961),p. 106.



. f ---~- 'h-e-G-in-Z-b-U-r-g---L-a-n-d-a-u--

theory

9.1 Basic equations

In 1937Landau developed a model to describe second order phase transitions (those involving

nolatent heat). Central to this theory was the introduction of the concept of an order parameter.

The order parameter is an appropriate quantity which vanishes in the high temperature phase

(T> Te) but which acquires a nonzero value below the transition (T < Te) ' The identification of

theorder parameter is often obvious from the nature of the second order transition. Thus for the

ferromagnetic transition it is natural to identify the spontaneous magnetization, M, as the order

parameter.

The ferromagnet brings out a fundamental property of second order phase transitions: the

development of a spontaneous magnetization is accompanied by a reduction in the symmetry of

the system. Thus if the material is iron, where the high temperature (nonmagnetic) phase has a

body centered cubic (bee) structure, on passing through the transition (Curie) temperature

(T c =1043 K) the material chooses one of several symmetry-equivalent crystal axes along which

themagnetization develops; the choosing of one among several directions (for iron the directions

ofeasy magnetization are the cube axes and there are six such directions, ± x , ± y , ± z ) lowers

thesymmetry of the crystal. The system is then said to have 'spontaneously broken symmetry'. It

isa general property of second order phase transitions that the low temperature phase always has

a lower symmetry.

Since the order parameter evolves continuously from zero below Te, it is natural to expand

thefreeenergy as a power series in the order parameter. The free energy is a scalar but the order

parameter may be a higher-dimensional object (e.g., a vector, tensor, complex numher). For our

exampleof the ferromagnet, the order parameter is the magnetization, M, a vector. Thus in

making the expansion of F (M ) we can admit only symmetry-invariant combinations of the

c om pon ents, M i; since the magnetization may develop along any ofthe (easy) crystal axes the free

energy expansion must preserve the full symmetry of the high temperature phase. For a cubic

ferromagnet the expansion of the free energy satisfying the above requirements has the form

F (M , T) =F (O , T) + r x (M ; + M ; + M ;) + ~f:il(M ; + M ; + M;)2

1 ?

+ 2f:i2(M xM y + M yM z + MzMx l - · (9.1)

Expression (9.1) is invariant under all the symmetry operations of a cube. However, to simplify

31




F-Fo

F-Fo

0: < 0 j

(a)M

~~o\~ LLM

···~~a2/2~

(b)

Figure 9.1 G L freeenergyfunctions for T> T , (ex> 0)and [or T < T , (e t.< 0).

the discussion consider a (hypothetical) isotropic ferromagnetic liquid. The only rotationally

invariant scalar that can be formed is then M· M and (9.1) simplifies to

(9.2)

Limiting the expansion to fourth order will lead to an adequate description of ferromagnetism.

Thermodynamic stability requires / J >0 (otherwise the system would seek a divergingly large

magnetization). However, (f.may have either sign. If o . > 0 the minimum of(9.2) occurs at M =0;

if : x < 0 the minimum is for M #. O . These two situations are shown in Fig. 9.1. Hence we can

model a second order phase transition simply by arranging for the sign of (f. to change at T;which

is easily done by writing

(9 .3 )

it is sufficient to regard f J as a constant. Setting i J F / a M = 0 yields

(9.4)

This equation has the two solutions discussed above:

e x > 0, M =O (T> Tel;

(T < Te).

(9.Sa)

(9.Sb)x < 0,

In terms of Eq. (9.3) we have

1 \ . , f 2 =° (T> TJ; (9.6a)

(T< TJ. (9.6b)

It is important to recognize that the ferromagnetic state is degenerate. For the case of the

crystal it is degenerate with respect to the number of independent crystal axes (or directions) the

magnetization might orient along (six for our case of bee iron with easy (100) axes). For our

ferromagnetic liquid the ordered state is continuously degenerate: M can point in any direction.

There is an important aspect of the second order phase transition which is neglected in the

above model: fluctuations. All thermodynamic systems undergo fluctuations. As an example the

density of a liquid may fluctuate by an amount o p. However, this is at the expense of an increase

in the local pressure, ()P, governed by the bulk modules. This is a general feature: the 'restoring

force' (pressure in our example) is linear in the 'displacement' (here the density). For a system



The Ginzburg-Landau theory 33

whichundergoes a second order phase transition the order parameter becomes a thermodynamic

variable below the transition. For our magnetic system the generalized restoring force accom-

panying a change in the magnetization 3M follows from expanding the free energy about its

equilibrium value, M , However, as the transition is approached the linear restoring force goes to

zero:just at the transition the restoring force is proportional to (()M)3. We see that fluctuations

can have a profound effect on the system near its transition temperature. The Landau model

therefore assumes that the temperature is far enough from the transition temperature that

fluctuation effects may be ignored. This is referred to as the mean field model.

We now take up the case of the supercond uctor. Our earlier discussion has brough t out the

idea that superconductivity is some kind of macroscopic quantum state. Ginzburg and Landau

built this idea into the Landau second order phase transition theory by assuming the existence of

a macroscopic 'wave function', 1 / 1 , which they took as the order parameter associated with

superconductivity. Since wavcfunctions can be complex, only the form 1 / 1 1 / 1 * may enter the free

energyexpansion; we therefore write

(9.7)

Theminimization proceeds exactly as above, i.e.,

1 1 / 1 1 =0,

[a( T c - T)]1!2I I / f 1 = { 1 ,

(9.8a)

(9.8b)

To describe situations where the superconducting state is inhomogenous we must generalize (9.7).

F is then interpreted as a free energy density, .?(r), and we write

F = F o + l?(r)d3r

=Fo+ fd3r[xll/I(rW+~fill/l(r)14J(9.9)

F is now the total free energy. Eq. (9.9) in its present form does not model the increase in energy

associated with a spatial distortion of the order parameter, i.e., effects associated with a coherence

Iength.Z. To account for such effects Ginzburg and Landau added a 'gradient energy' term to (9.9)

ofthe form

(9.10)

with m* as a parameter; the choice of the coefficient f j2 /2m* makes (9.9) mimic the quantum

mechanical kinetic energy (introduced earlier in Eq. (2.10)). Ginzburg and Landau assumed that

if(9.10) was to be regarded as the kinetic energy contribution to the Hamiltonian density of the

superconducting electrons, then (as in Hamiltonian mechanics) the interaction of the electrons

with an electromagnetic field would be accomplished by the Hamiltonian prescription

e*p-->p--A

c




or, since p --->(~) V in quantum mechanics,

ie*

V--->V--k he ' (9.11J

the use of e* aIlows the superconducting entities to carry a different charge (e * = 2e in BCS

theory). Combining the above we have

f 1 i 2 / [ ie* ] / 2G = _ V - - A(r) ljI(r ) d3r.2m * he

(9.12)

Finally we must add the contribution of the magnetic field to the energy density

(2.9)

Combining the above we have

F = F 0 + f d 3r { x IIjI(r) 12+!i 1~J(r) 14 + 112 / [V _ ie* A(r)] ~J(r) / 22 2rn* he

+ ~H 2 (r)l. (9 .13)

gn 5The minimization of (9.13) must be carried out using the methods of the calculus of variations

since F is a functional involving the free energy density ff(l jI(r), I/J*(r), VIjI(r), V~J*(r), H(r)) which in

turn involves the unknown functions l jI( r) , 1 jI* (r ), and H(r)( = V x A(r)).

Minimizing (9.13) with respect to 1jI*(r) yields

f { 1 z 2 [ ie * ] 2 })F = d 31' - - V - - A(r) l jI( r) +x lj l( r) + fJll/J(r) 121j1(r) bl j l*(r)

2m * lie

f lil [ ie * ]+ d2r'- V - --:-A(r) l jI(r)bl j l*(r)2rn* he

(9.14)

(variation with respect to 1jI, which is an independent variable, yields the complex conjugate of

(9.14)). To minimize iF we set the integrand of the first part of (9.14) to zero; this yields the first

G-L equation

1 1 2 [ ie*]2- -* V - --:- A(r) l jI(r) + cxl j l (r) + Ii IIjI(r) 121j1(r)= O .

2m he

(9.15)

The surface term (which was generated by an integration by parts) can be used (with caution) to

establish certain boundary conditions and will be discussed later.

Variation of(9.13) with respect to A (with H = V x A(r)) yields Ampere's law

v006b;0114n

V x H(r) = -j(r)c

(9.16)

prov ided we identify j as



TheGinzburg-Landau theory 35

e* {[ e*] [ e* J }(r)=* 1jJ*(r) - ihV - - A(r) l jJ(r) + l jJ(r) + ihV - - A(r) 1jJ*(r) ,~n c c

(9.17a)

or equivalently,

- ie*h e*2j(r)=-- [~J*(r)V IjJ(r) - ljJ(r)VIjJ*(r)]- --IIjJ(r) 1 2 A(r).

2m* m*c(9.17b)

Eq.(9.17)is the second G-L equation; we note that (9.17) is the same as the expression for the

current density in quantum mechanics. Note the current density satisfies the equationj(r) = e(I)F /

bA(r));i.e., it is the variable conjugate to A(r).

9 . 2 Gauge invariance

Thesimplest solution of (9.15) is for the case of a uniform superconductor, IjJ #. l jJ(r), with A = 0,

as givenearlier in Eqs. (9.8). However, (9.15) possesses a continuum of other solutions having the

same free energy, which we now show. As we can with any complex function, we write

~(r)= a(r)ei<l>(r)hereatr) and cJJ(r)arc the position-dependent amplitude and phase. respectively.

Letus examine a class of solutions which satisfy the complex equation

(ie* )

V - -A(r) l jJ(r) = 0,he

(9.18)

whichis equivalent to the two real equations

Va(r) = 0 (9.19a)

and

[VcJJ- e* A] = O.

he(9.19b)

From (9.19a)we see that the only allowed solutions of(9.18) involve a constant amplitude, ,,; Eq.

(9.19b),on the other hand, has infinitely many solutions involving a vector potential and a

position-dependent phase (which does not affect the free energy) related by

heA = -VcJJ.

e*(9.20)

Anyvector potential satisfying (9.20) results in a uniform free energy and (on substituting (9.18)

into (9.17))a vanishing current density (note that H = V x A = 0 for all A of the form (9.20)).

The above exercise shows that the symmetry broken in superconductivity is qauqe symme-

try, or equivalently, phase symmetry. Superconductors having different phase functions, (D(r),are

in a real sense physically distinct/ this arbitrariness of the phase is the analogue for a supercon-

1. Strictly speaking we cannot determine the absolute phase of a superconductor, but in our later

discussion of the Josephson effects we will show that phase differences can be measured.




ductor of the property that the magnetization may point in any direction in an isotropic (liquid

ferromagnet.

9.3 Boundaries and boundary conditions

We first examine a simple case involving an inhomogeneous order parameter generated by the

presence of a boundary, in the absence of a magnetic field. Assume we have a superconducting

half space occupying the region x > O . We further assume that the order parameter is driven tc

zero at this interface. Experimentally this can be accomplished by coating the surface of th e

superconductor with a film of ferromagnetic material.' We then seek a solution to the one-

dimensional G-L equation

(9.21;

Noting o : is negative in the superconducting state (r x = - I xl), defining the G-L coherence length

as

l i 2[2= __ -

- - 2 rn* I o : I'

and writing ( f J / I o : l ) t j J 2 =? we may rewrite (9.21) as

- ~2r -f + f3 = O .

(9.22)

(9.23)

Multiplying by f' we may rewrite (9.23) as

(9.24)

hence the quantity in square brackets must be a constant. Far from the boundaryJ' =0 and

[2 = 1 (equivalent to t j J 2 = I x l I f J ) ; then (9.24) becomes

(9.25)

which has the solution( = tanh(x/J2~) or

(9.26)

From (9.26) we see that ~ is a measure of distance over which the order parameter responds to a

perturbation. Since x = a(T - T e l we have

" ( l i 2 ) 1 , 2 ( T ) - 1 / 2dT)= -- 1--

2 rn*aT c t;

2. Paramagnetic impurities (those bearing a spin in a host material) or interfaceswith a ferromagnetic

metal strongly depress superconductivity. A normal metal interface has a much smaller effectand an

insulator or vacuum has a negligibleeffect formost purposes.

(9.27)



Ine l2mzburg-LanClau theory 37

We see that the G L coherence length diverges as 1/(1 - T/TjiZ; this divergence is a general

property of the coherence length at all second order phase transitions (although the exponent

differs in general from this 'mean field' value of 1/2 close to TJ

We now examine Eq. (9.17b). In a limit where the first two terms are negligible, we

recognize the remainder as the London equation, as written in the form (3.5), provided we identify

n = I IjI IZ ; this supports the identification ofl j l as a (condcnsate) wave function associated with th e

superconducting electrons. We can immediately caleulate the London penetration depth as

(9.2Sa)

or

(9.28b)

Comparing Eqs. (9.27) and (9.2S) we see that in the G-L theory ),1. and ~ both diverge as

(1 - T/TJ-1i2. Their ratio, called the G L parameter, is therefore a constant which we write as

(9.29)

Let us now return to the discussion of the surface tension of a normal-superconductor

phase boundary begun in Sec. 6. From Eq. (6.3) the total Gibbs free energy, which is a constant, is

given by

G =W f + dx [§(X) - _ _ 1 _ _ H e .B(X)J;-"0 4n

(9.30)

here .9 1 is the interface area, : 1 F is the G-L free energy density and the integral must now be

extended into the region x > 0 since, due to the gradient term in the G-L model, the supercon-

ducting properties do not turn on abruptly at the interface (which we still locate, nominally, at

x =0) . Inserting (9.13) into (9.30), employing a gauge where A = A(x)z, and using the definition of

igiven by (6.4), we obtain the one-dimensional equation

f+ [ r x 2 1 7

z1 dljl(x) 1 2 e*2

}'= dx -+- ~- +--.zA(x)lljJ(xW_ < C D 2 f3 2m* dx 2m*c

1 BZ(x) H B(X)J+ r x I I j I ( xW + - f 3 1 1 j 1 ( x ) 1 4 + ~. __ ~e _._ •

2 Sn 4n(9.31)

The vanishing of the cross term iA' V in both (9.31) and the first G-L equation (9.15) allows us tochoose I j I real; it then follows from (9.17) that

(9.32a)

and

i,=,= . (9.32b)




To compute (we must simultaneously solve the first and second G-L equations, (9.15) and (9.16),

for ~J(x) and B(x) ( = - (dA(x) /dx)) , subject to boundary conditions which will be specified

shortly, and insert the results in (9.31). To eliminate various parameters in the subsequent

calculations we rewrite all equations in terms of the scaled variables:

(9.33)

In terms of these variables the first and second G-L equations become (where we now drop the

bars over the variables)

(9.34)

and

(9.35)

The solutions of (9.34) and (9.35) for arbitrary I(must be obtained numerically. The appropriate

boundary conditions are

~J = 0 , B =A' = 1 at x =- e x ) (9.36a)

and

t / J =1, A' =0 at x = + ex; (9.36b

(where x > 0 is nominally the superconducting side). 3 The behavior of B(x) and t jJ(x) in the small k

regime where the field varies more rapidly than the order parameter is shown in Fig. 9.2. Tc

obtain a first integral ofthcse equations we multiply (9.34) by tjJ',which leads immediately to

1 d ' 2 d [(1 2 ) 2 1 4 ] 2 d (A2)-(~J )=- -A -1 tjJ +-tjJ -t jJ - -,1(2 dx dx 2 2 dx 2

(9.37a

and multiply (9.34) by A', which yields

~(A '2 ) _ tjJ2~ (A2)dx ,2 - dx, 2 .

Combining (9.37a) and (9.37b) we have for our first integral

(9.37b

(9.38

the value of the constant was fixed by the boundary conditions at either + 00 or - o : » . Usin:

(X2 !2~ =H:/8n and the scaled variables of Eq. (9.33) we may rewrite Eq. (9.31) as

) H Z f +c(=--

Sn _

3. From the structure of Eqs, (9.34) and (9.35) it follows from the boundary conditions (9.36a) and

(9,36b) that t i t ' =0 at x =± 00. Our restriction to real t / I requires that the constant A (which is

equivalent to a phase < 1 » vanishes at x=+ Xl.




r-----s--~

K ll> I

Figure 9.2 Schematic diagram of the variation of Band 1 / / in a domain wall. The case

/{< < 1refers to a Type I superconductor with a positive surface energy; the case

/{>> Iefers to a Type II superconductor with negative surface energy.

Combining this expression with Eq, (9.38) we have

) H 2 f " % [ 2 ]=~ _1jJ'2 + A'(A' -1) dx;

4n K2(9.39)

givenA'(x)( = - B(x) ) and tj/(x) we can compute (.

Returning to Eq. (9.38) we verify that in the limit A ' =0 we recover the dimensionless form

ofEq. (9.25);

(9.40)

Thesolution of this equation is

K(X - x o)tj/(x) = tanh-~,------'--

)2(9,41)

(whichis the analogue of Eq. (9.26)); here Xo is the nominal position of the boundary. The

characteristic length scale of the order parameter variation associated with this normal/super-

conductor phase boundary is ~ (in un scaled units), as discussed earlier in connection with Eq.

(9.26).This form docs not satisfy the boundary conditions (9.36), as expected, since a stable phase

boundary in an unbounded superconductor can exist only in the presence of a field, but rather

satisfiesthe boundary conditions I j J =+ 1 or - 1 as x - - - > + cc , However, assuming the presence

ofan order parameter quenching mechanism at Xo =0 (e.g., a thin ferromagnetic plane embed-

ded in an otherwise homogeneous superconductor) we may evaluate the associated surface

tensionby substituting (9,41) into (9.39); carrying out the integration (with A = 0) we obtain

(9,42)

From the definition of K given in (9.29) we see that :' ~ ~(H; /8n) as anticipated earlier in Eq. (6.6).

WhenA is nonzero the characteristic length scale of a field variation may be estimated from Eq.

(9.35).In a Type I material at a vacuum/superconductor interface I j J may be regarded as constant

overdistances where the field varies. Since the length scale of Eq, (9.33) is unity this corresponds

to the field varying over a distance A L (the London depth) in unsealed units. However, near a




normal-metal-superconductor phase boundary the field variation occurs in a region where I / ! is

small. Ifwe seek an approximate solution to (9.35) as an exponentially decaying form, A .~ e~X!,j

where () is a characteristic length, then l/(F ;:::: I / ! 12. From an expansion of Eq, (9,41) about

x =xo, I / ! ~ K (X - x

o). The average value of 1 / ! 2 in a region of width b would be of order (K3f

Combining these forms we see that the field decay would be governed by (K6)2 ;:::: l/ b 2 or

() .~ K~ 1/2. This would result in a (negative) surface tension contribution of order - Kl/2y which,

although less than the expression (9,42) in a Type Imaterial, decreases slowly with K and hence

limits the accuracy of this expression.

With increasing K the surface tension continues to decrease and further analysis (see

Lifshitz and Pitaevskii (1980), Sec. 46) shows that it passes through zero for K =1//2.

We end this section with a discussion of the boundary conditions at an interface between

two dissimilar materials at least one of which is superconducting. A boundary condition which is

appropriate for the case when no current flows parallel to the surface normal, i i , is (see Fig.

9.3(a))4

[e* J 1- ihV - ~A . i i I / ! = - -I/f,C b

(9,43)

as is easily verified by substituting in Eq. (9.17a); here b is a parameter having the units of length.

The superconductor insulator case corresponds to the limit b - co,

Lastly we consider contact between two superconductors in the presence of a current

normal to the boundary. In the most general case the boundary, which we will locate at x =0,

may have properties different from both superconductors, an example being a thin layer of a third

(normal) metal, which is referred to as an SNS (supcrconductor-normal-superconductor)junc-

tion. The most general (Cauchy) boundary conditions connecting the two sides of the junction,

denoted 1 (x < 0) and 2 (x> 0), have the form (deGennes 1964)5

(9,44a)

and

(9,44b)

Ifwe choose a single gauge for regions 1 and 2 the vector potential will also be continuous, The

coefficients Mij are not independent but arc constrained by the requirement of current conserva-

tion through the boundary:

(9.45)

as a result we may choose the M ij to be real and they must satisfy the condition

M llM 22 - M 12M 21 = 1.

The resulting expression for the current through the boundary.j., is

4. We emphasize that Eq. (9.43) is a macroscopic boundary condition. The behavior of the microscopic

order parameter may differ substantially near the interface.

S . A boundary condition of this type wil l be discussed from the microscopic point of view in Part III,

Sec. 42.

(9,46)




t \ 1 I f ( x ) \

I

hll}jx o

(b)

x

(a)

Figure 9.3 Schematic of the order parameter behavior: (a) near an NS boundary; (b) in

an SNS junction.

e*h= -* - 1 1 / 1 1(0) 1 1 1 / 12(0) I sin [(f)2(0) - (f)1 (0)].m J W l2

(9.47)

The maximum value of this current is

e*/iJ m " x =~ 1 1 / 1 1(0) 1 1 1 / 12(0) I ·

m 1Vl12

(9.48)

If jmax is much less than the bulk critical currents in the two superconductors then we may

approximate the I / I j ( x ) by the A = 0 forms analogous to (9.26)

(9.49a)

and

(9.49b)

here Xl and X2 are free parameters and ~1.2 are the G-L coherence lengths in the two media.

From Eqs. (9.49) we have immediately

(9.50a)

and

(9.50b)



Of--~-'h-e-u-p-p-e-r-c-rl-·t-ic-a-'-~-ie-'-d-o-f-

a Type II superconductor

Asthe magnetic field is raised above He1, and the density of flux lines nL increases, a point is

eventually reached where the distance between flux lines becomes of the order of the vortex corediameter;i.e. n e 1/2 ~ ~. One would then expect a transition to the normal state, and the field at

whichthis occurs, called the upper critical field, is designated He2• In the region just below H c2 the

superconducting order parameter must be small and this is the regime where the G-L approach

shouldprovide a good description.

We may calculate Hc2 by linearizing the first G-L equation, since the order parameter is

vanishingly small just at Hc2' Eq. (9.15) then becomes

h2

( ie*)2_ - V _ - A ,I, + (X'/' = ·2m* he 'I' 'I' ,

(10.1)

thisequation is the same as the Schrodinger equation for a particle with energy, _ a, mass, m",

andcharge, e* , in a magnetic field, H,with an associated vector potential, A. We will assume a

uniformfield Ho I I i.The solution of(10.1) is easiest in the so-called Landau gauge where we write

thevector poten tial as

A =HoY x ; (10.2)

theresulting G-L equation is then

(10.3)

We may separate the variables in (10.3) by writing I j J n . k z . k J X , y , Z ) =; k z z + ; k y Y u n ( x ) (we do not

normalize I j J at this point as this property follows only from solving the nonlinear G-L equations,

whichwe address in Sec. 14). Inserting this wavefunction in (10.3) we obtain:

[1 1 2 d

21 ]

_ -_ --2 + - m*w~(x - XO)2 u n i x ) = n U n ( X ) ,

2m* dx 2

( lOA)

whereXo = ( l1e/e*H o)k y = (¢0 /2nHo)ky , and f,n = - rx- (h 2k; /2m*) . Eq. ( lOA) is the Schrodinger

equation of a harmonic oscillator with frequency we = I e* I H o/m *e (the cyclotron frequency of

theparticle) with energies

(10.5)

43




having its origin at the point xo' The full (unnormalized) wavefunctions are

'(k +k) ( )'2 2 ( x - x o )/, (x V z) = e' ) " y . ,z e " X-Xo~! " H H - - -V ' n.k;:,kv· ,~, n'

. all

(10.6)

where af r = = h /m*wc = hc /e*H o =¢o /2nHo·

Only the smallest eigenvalue, n =0, k; =0 solution (corresponding to the highest field at

which superconductivity can nucleate in the interior of a large sample) is meaningful since our

linearized theory is valid only as a description of the onset of superconductivity. Hence

(10.7a)

or

2m*caH ----c2 - he"

2m*c=--a(T -T)

ne* c

(10.7b)

or, as it is more commonly written,

H -~.c Z - 2n~2'

(10.7c)

in terms of the length aH, Eq. (1O.7c)may also be rewritten as aHel=~From (1O.7b)we see that

lld goes to zero at T ; and increases linearly below that temperature. Recalling nL= / 4 > o , we see

that (10.7c) is in accord with the order-of-magnitude estimate of H cz made at the beginning of this

section. However, we are then struck with the fact that in our particle analogy, we have

considered only a single quantum state (n =0, the 'ground state'), and yet the material contains

the largest possible number of flux lines. The resolution to this apparent contradiction is that our

n = 0 quantum state is highly degenerate (as evidenced by the fact that the quantum number k;

does not affect the energy).

The relation between this high degeneracy and the high density of flux lines will be dealt

with in Sec. 14, where we discuss the region of fields just below H cz .



.I--~-'h-e-a-n-i-s-o-t-ro-p-ic-----

superconductor

Anisotropy may be incorporated in the phenomenological G-L theory simply by introducing an

'effective-mass tensor' into the 'kinetic energy' term in Eq. (9.15) in the form

[ - in ( v - i:* A ) J ~[~* J [ - in ( v - i;: A ) ] ~ + a~ + P I ~ 12~ = 0,

(11.1)

wherel/m* is an effective-mass tensor.

For a system which may be regarded as uniaxial, the reciprocal effective-mass tensor may

bewritten as

0 0

[ ~ * l jmx

0 0 (11.2)mx

0 0mz

wherethe z axis is the symmetry direction. The inverse of Eq. (11.2) yields the mass tensor m*.

We now obtain an expression for the upper critical field Hc2(T). Near the field-dependent

transition temperature the order parameter is vanishingly small and we may again neglect the

nonlinear term. Our equation is then formally identical with the Schrodinger equation of a

particlewith charge e* and an anisotropic mass tensor m* in a uniform magnetic field Ho and the'energylevels' have the harmonic oscillator form

- a = (n + 1/2)nwc(O), (11.3)

wherewJ8) is the angle-dependent cyclotron frequency which is encountered in the effective-mass

theory of cyclotron resonance in semiconductors. The latter can be worked out classically simply

by solving Newton's equations of motion for our 'particle' moving under the influence of the

Lorentz force:

e*m* 'v = -v x Ho;c

(11.4)

here v is the velocity. The solution of this equation involves elliptical orbits which are traversed at

a frequency

(11.5)

45




here 0 measures the angle of the magnetic field from the z axis. The solution with the lowest free

energy corresponds to n = 0 in Eq. (11.3) and the upper critical field is thus given by

2ea(Tc - T )

Hd8) = . 2 2 2 12'ne*[slll 8jrnxrnz + cos 8/mJ ! (11.6)

Ifwe define two coherence lengths,

and

and recall the 'flux quantum' 1 > 0 = he/ I e* I, then Eq. (11.6) may be written as

(l1.7a)

in particular, for fields parallel and perpendicular to the symmetry plane of the material we have,

respectively,

(11.8a)

and

1 > 0HCH =~.

2n<,x(11.8b)

The angular dependence of the upper critical field is thus characterized by the two parameters Cand ((Lawrence and Doniach 1971). Note that (11.8a) and (11.8b) both lead to a linear behavior

of the upper critical field in T; - T. In terms of the definitions (11.8a) and (11.8b) we may rewrite

Eq. (l1.7a) as

H~2 (O) 2 LJ H~2(8) . 2 LJ _ 12 cos ()+ 2 Sill () - ,

HcH He211

(l1.7b)

which is the equation of an ellipse.

Eqs. (9.17) can be generalized to the case of an anisotropic superconductor by replacing

rn* -1by Ijm* from Eq. (11.2); an anisotropic London penetration depth then results.




The behavior ofEq. (12.4) is radically different from that of Eq. (11.8aJ.The reduced dimensional-

ity has the effect of replacing ~z by d/(12)l!2. Since only one factor of'; enters Eq. (12.4), the

temperature dependence of He21i is altered from H c2 'I «: T; - T to He21i C£ iT; - T)112 . We can

refer to these two behaviors as 3D and 20, respectively.

Let us now consider the case where the field is at an arbitrary angle with respect to the thin

slab. We take the vector potential in the form

A = y(xcos8 - zsinO)Ho . (12.5)

Writing ljJ(r) =u(x,z)ei\y, the linearized G-L equation becomes

.itu + au=0, (12.6)

where

(12.7)

with

(12.8a)

and

- 112 e 2 1

.it' =-- ~ + - m*(j)~(z - Zo)2 s in ? 0 + m*(!)~x(z - zo)sin Ocos 8; (12.8b)2m* oz 2

here we have defined

Continuing our analogy to charged particle motion in a magnetic field, we note that there is no

constraint on the in-plane motion and thus the term in x2 would take on arbitrarily large values,

were we to treat it as a perturbation on an x-independent trial wavefunction; we have therefore

regarded this term as part of a zero order Hamiltonian, .ie' o - The eigenvalues of yt'o are simply

harmonic oscillator levels with a frequency We cos 8. The requirement that eu/ez vanishes at

z = ± d/2 again requires that (in lowest order) the wavefunction has no z dependence; i.e.,

u(x, z) =u(x). Since particle motion in the z direction is restricted, we can treat .ft' as a

perturbation. The eigenvalue is again minimized by setting k; =0 (the perturbation produced by

the last term in Eq. (12.8b) then vanishes by symmetry). The resulting perturbation produced by

x' is of the same form as Eq. (12.3). The total eigenvalue is then

(12.9)

Defining H e211 = (12)1/2cPO/2n(d and HcH =cP o!2n(2 we may rewrite Eq. (12.9) as

1 - H e2( O ) I III + H ~2 (8) " 2 O '- cos (J 2 Sill ,

».; He211

(12.10)

note that the angular dependence of this expression, first suggested by Tinkham (1963), is

radically different from that of Eq. (11.7b); in particular, it has a cusp at 8 = n/2.

We next treat the problem where the boundary condition is altered such that u(z) I z s= +dl,



I tun superconaucnng slaos

=O.T n practice, this condition is approximated by a superconducting film which is sandwiched

between two films containing a large concentration of paramagnetic impurities (such as fer-

romagnetic films) where 'pair breaking' effectively drives the order parameter to zero at the

boundary. We return to Eq. (12.1). For Ho = 0 (w e = 0) the eigenfunction satisfying our bound-

ary condition is

u = uocos (k z ) (12.11 )

with kdl2 =± n l2 for the ground state. This leads to a suppression of the transition temperature

governed by

h 1n2-rx=--

2m*d 2

or

(12.12)

here TeO is the bulk transition temperature and T, is the film transition temperature in the

presence of the new boundary condition. Application of a parallel magnetic field induces an

additional reduction in T; which we calculate from perturbation theory (Wong and Ketterson

1986):

<u( z ) 1 1 m *w~ z2 1 u (z )- brx=-'----__:_:-=----_.::_---'----'--

<u( z ) 1 u ( z )

1 2,(1 1)=2 m*w c d' 12 - 2n2

= 0.0l63m*w;d 2.

Combining Eqs. (12.12) and (12.13) gives

(12.13)

(12.14)

Eq. (12.14) may be written in terms of T', as

2 (m*C )2 2a(Tc - T)

Helll = 1 e* 1 0.0327m*d2·(12.15)

Ifwe define a coherence length involving Tc ' rather than Tco, by

(

h2 ) 1 / 2~= 2ma(Tc - T) ,

(12.16)

then we may write

5 . 53CPoH ---

e 2 1 1 - 2n,;d '(12.17)

whichis to be compared with Eq. (12.4). The angular dependence follows from inserting (12.17)

into (12.10).



~~s-u-r-t-a-c-e----------------

superconductivity

A superconducting half space in a parallel field may exhibit the phenomena of surface supercon-

ductivity (Saint-James and deGennes 1963). We take the interface to be the plane z =O.Thecritical field is still determined by the eigenvalue of Eq. (12.1) (k x =0 with the requirement that

a u / u z =0 at z=0). For values of Zo (i.e., k y ) such that the 'orbit center' is deep in the bulk of the

superconductor (z o -> - 00) the corresponding eigenvalue yields the bulk critical field; the

boundary condition is adequately satisfied since the ground-state wavefunction of our harmonic

oscillator is a Gaussian (centered on zo ) having negligible amplitude at the boundary. For the

opposite extreme, where we choose Zo = 0, the same Gaussian wavefunction also satisfies the

boundary condition, since the derivative at the peak of the Gaussian (now centered on Zo =0)

vanishes. The question arises whether there is a smaller eigenvalue (than the bulk eigenvalue) for

some value 0 > Zo > - 00. To examine this situation, we reflect the harmonic oscillator potential

V= tm*w;(z - zo)2 through the plane z = 0 such that

(13.1)

Note that this is not a harmonic oscillator potential but rather has two minima (located at

z = ± I Zo I ) and the slope a V/cz is discontinuous at z = O.Ifwe seek a nodeless solution for the

motion in this potential (which is symmetric under a sign change of z), the boundary condition at

z=0 will automatically be satisfied, and it should correspond to a possible ground state.

We can see immediately from Eq. (13.1) that an eigenvalue smaller than that of the bulk

must exist, since the potential described by Eq. (l3.1) is always less than or equal to that obtained

by extrapolating the potential from either half space into the other. The above-defined eigenvalue

problem admits an exact solution in terms of Weber functions (Saint-James, Thomas, and Sarma

1969). However, an adequate solution is obtained using a trial wavefunction u ~ e-rz2 (deGennes

1966). Minimization of o : with respect to rand Zo yields

(

2 ) 1 ! 2 I i W e I e* I I i H c 3- C i =1-- -=060---IT 2 . 2m*c' (13.2)

where we have defined the upper critical field associated with this surface superconducting state

as H e3. The exact solution yields 0.59 rather than 0.60 for the numerical factor. The upper critical

fields for bulk and surface superconductivity are then related by

(13.3)

50



Surface superconductivity 51

Surface superconductivity persists to higher magnetic fields (in a Type II material) and thus

may (in a conductivity measurement) mask the bulk transition. The state may be suppressed by

the following:

(1) rendering the surface rough (by sand blasting);

(2) using sufficiently high measuring currents to destroy surface superconductivity;

(3) depositing a normal-metal overlayer.

We now briefly discuss the case in which we have a thick film, of thickness d (with the origin

in the film center), rather than a superconducting half space. For films with d ; : ; ; ( , the amplitude

oftheG-L wavefunction at nucleation would be highest in the immediate vicinity of the surfaces

at z=± d/2 for He211 < H < H e3• As d is reduced, the eigenvalues (satisfying the boundary

conditions at z =± d/2 associated with bulk and surface superconductivity) approach each

other and, below some distance d =dc, only a state which has its wavefunction centered on z =0

isstable (Saint-James et al. 1969).

As we saw above, when the G-L equation for the magnetic field at an arbitrary angle is

written in terms of coordinates for which thc boundary conditions are conveniently applied, it is

nonseparable. Thus, for thick films where u ~ u(x, z) (rather than thin films where u ~ u(x)), we

must resort to a numerical, variational or perturbational technique. This complicated, and to

some extent not fully explored, problem has been discussed by Minenko (1983), Saint-James

(1965),Yamafuji, Kawashima, and Irie (1966) and Saint-James et al. (1969). However, we note

that, at some angle 8 = 8c' we expect the eigenvalues associated with the surface and bulksuperconducting states to merge.



4 I > ~ ~ - h - e - T - ~ - p - e - ' - ' - - - - - - - - - - -superconductor for Hjust

belowHc2

The discussion of the upper critical field given in Sec. 10 involved the linearized G L theory and is

strictly valid only for the transition line in the H-Tplane where 11= HdT). For fields below thisvalue we must include the nonlinear terms in Eq. (9.15). T n this section we obtain an approximate,

but analytic, solution for the G-L equations which is useful for a small range of fields just below

He2. This problem was first examined by Abrikosov (1957).

We recall the solution of the linearized G-L equation obtained in Sec. 10 for a 'nucleation

wavefunction' at 11<2(for H I I z) which follows from Eq. (10.6) with n = 0 and kz = 0:

(14.1)

where we dropped the subscript from k; and Xo = = (hc/e*Hc2)k = 4Jok/2rrHc2 = (2k; here C is a'normalization constant', which vanishes for H =He2. As noted in Sec. 10, this wavefunction is

highly degenerate in that I < can have any value (so long as Xo lies within the superconductor).

The wavefunction which is the solution to the full G-L equations for H just below 1102 is

expected to resemble some particular solution ofthe linearized Eq. (10.3). At the same time, based

on our earlier discussion, we also expect some sort of vortex lattice to be present. We therefore

seek an approximate solution for t/Jas a superposition of terms of the form (14.1) chosen so as to

be appropriately periodic in both the x and y directions (thus forming the lattice). Eq. (14.1) is

periodic in the y direction as it stands with a period b = 2rr/l<. We may retain this periodicity in

the y direction while simultaneously forming a function which is quasiperiodic in the x direction

by substituting nk for k in (14.1) and summing over all n to yield

n= + o:

t/J(x,y) =C I einkY e-(X-XY/2I",

n~

(14.2)

where Xn =n~2k = 2r rn~2 / b and C is a (common) normalization constant. We define the period in

the x direction as a = = x, + 1 - Xn =2rr~2 lb. and the area of the unit cell is constrained by

ab = 2rr~2. Eq. (14.2) is not, however, strictly periodic in x: if we substitute x + a for x this is

equivalent to replacing n by It + 1 in the summation, which chanqes the phase of the waiefunction b y

eiky

; i.c.,

t/J(x+ a,y) =e-ikyt/J(x,y). (14.3)

This behavior is inherent in our choice of the gauge Ay =11 x ; under a coordinate displacement

x -> x + a, A y -> Ay + H a leading to a change in the canonical momentum nhk --+ nhk - (e* / c ) H a

52




We must also ineludc the effect of a change in the internal field, the magnetic induction Hir]. W

write this as a sum of two contributions arising from: (i) an explicit shift in the external fiel.

relative to He2 (since we have an exact solution for the latter), and (ii) a (screening) shift H(S)(I

arising from circulating supercurrents associated with the vortices (this latter effect is vanishingl

small for H = Hc2 and was therefore neglected in Sec. 10). We denote the sum of these contribu

tions by

(14.11

with

Associated with the internal field H(l) is a vector potential A( 1). Expanding (14.9) to first order i

A(l) yields (in a London gauge)

he" [( i e * ) ( i e * ) lI I j J 1 4+-. __ A(1)(f)· 1 j J * V--, Ao 1 j J - 1 j J V+-Ao 1 jJ * = 0 , ( 1 4 . 1 :21m*c he he

where the remaining terms cancel (since they satisfy the linear Schrodinger-like equation a

H =He2) '

The last two terms may be written as (l/c)j(S)(f)' A(ll(f) where we identify j(S)(f) as th

screening current density associated with the vortex lattice. Introducing V x H(s) s: (4rr!c)jls)(r

integrating by parts, and writing H<l)(f) =V x A(l)(f), Eq. (14.13) becomes

(14.14

where H(s) is parallel to H(l) by symmetry. To the accuracy we are carrying the calculation, H(s) i

the diamagnetic response field (H(s) = rrM(s)).

To obtain the field H(s) we first calculate the associated magnetization current, j(sl(f). Th

calculation of jIg) is facilitated by noting that the contours of constant 1 ~/(f) 1 and the (two

dimensional) streamlines of j(s) coincide, which we now demonstrate. We introduce the canonica

momentum operator n(=m*V where V is the velocity operator)

~ h e*II =-;-V--A.

Ie'(14.15

which has the commutation relations [IIx, I I y ] =(e*h/c)H. In analogy with the standard har

monic oscillator problem we introduce the raising (creation), n + =Ix + illy, and lowerin

(destruction), II_ = IIx - illy, operators." The ground state has the property I I _ 1 j J = 0 frorr

which we obtain

[(h a e* ) ( I i a e*

) l;----Ax -i -;----A, 1jJ=0.1 O X C l O Y C }

(14.16

Ifwe substitute th e wa vefunction in the form I j J =I j J 1 e;<!>nto (14.16), and identify the current ( in

lowest order) as

( 1 4 . 1 7 !

2. The linearized Schrodinger-like G-L Eq. (10.4) may be written Il + Il _ if ; =O.



Type" superconductor tor Hjust below HC2 55

weobtain (on equating the real and imaginary parts separately to zero)

e* /1 Gj~ ) = I 1 / 1 I 2

2m* o y (14.1Sa)

and

(14.1Sb)

Fromthe structure of (14.1Sa) and (14.1Sb) we see that V I!/II.lj(s), The magnetization field, H(s),

followsfrom Maxwell's fourth equation, V x H(s)(r)= (4rr/c-jj<S\r);omparing this equation with

Eqs.(14,IS)yields

(14.19)

where the constant of integration was fixed such that H(s) vanishes when II/II= . We may

interpret H(s) as a magnetization through the usual definition B=H + 4rrM(s) where

M(S)(r)=- ( e*h /2m*c) I!/I1 2 z ; note it is negative as expected (since superconductors are diamag-

netic).

We now combine (14,12), (14.14), and (14.19) to obtain

---- e* /1 [ e*h

lJ I!/I1 4 + --I !/I1 2 Ho - H c2 - 4rr--I!/I 1 2 = .2m*c 2m*c (14.20)

Writing!/I(r)=~I of(r), where !/I6=Ix ] /fJ, and using the definitions (9.29) and (10.7) for I(and H c2

wemay write (14.20) as

4 ( 1 ) 2 ( Ho)1 - - - f 1 - - = .21(2 Hc2

(14.21)

The quantities p and r may be calculated provided we choose a symmetry (!/In or !/ID

from(14.6)or (14.S)) and a lattice spacing. It turns out that near H c2 the ratio

(14.22)

(introduced by Abrikosov (1957)) is independent of H 0' Introducing this quantity into (14.21) we

have

{2

. ( 1 )'r i A 1 - 2/,;2

(14.23a)

(14.23b)




-- - -- ,--,2

From the above I V I I 2 =I e x l / f J ) f 2 and I if ; 1 4 =J A ( I if ; I 2 ) .

We now proceed to a consideration of the thermodynamics of a Type II superconductc

We begin with the magnetic induction which is defined as the position average of the intern

field, H(r): B = = Ho + I I ( s ) ; using (14.19) and the definitions of K and H c2 we have

(14.2

where we used the relation He2 = 2 K H c in the second step. Substituting (14.23a) for f2 v

obtain the thermodynamic magnetic induction B=B(H 0) as

(14.25

We may invert this relation to obtain Ho =Ha(B) as

(14.25

From (14.25a) we obtain the magnetization, (B - Ho)/4n, as

(14.25

note this is linear near H e2 as discussed earlier.

We now calculate the free energy. Comparing the variational expression (14.9) with E

(9.13) for the full free energy we have

Rewriting the first integral and substituting the unaveraged form of (14.24) this equation bccom

(14.2

Substituting Eqs. (14.23) we obtain F = F(Ho) as

( 1 4 . 2

Alternatively, using Eqs. (14.25), we may calculate F = F(B) for which we find

V[ 2 (B-Hc2)2 JF =Fo + gn B - - 1 - + - r - ' J

A

- ( 2 -K - - ; : . 2 - - - 1 ) . ( 1 4 . 2

For the case of a long cylinder with an external field, Hex!' parallel to the axis or a thin film w i



Type 1 / superconductor for Hjust below Hc2 57

000oo

(a)

o o(b)

Figure 14.1 Schematic diagram of square and triangular vortex lattices. The dashed

lines show the basic unit cells.

Figure 14.2 The spatial configuration of I if ; I 2 near H 02 for a triangular vortex lattice.

The numbers labeling the contours specify the square of the reduced order parameter.

(After Kleiner, Roth, and Autler (1964).)




Hex ' lying in the plane (where Hex ' = Ho) wc would use (14,27): for the case where the external fiek

is perpendicular to the film (where Hex ' =B) we would use (14,28),

The only remaining problem is to calculate the Abrikosov parameter f 3 A' This calculation i

somewhat tedious and we refer the reader to Saint-James et al. (1969) for the details. The result

are

f 3 ~ = 1.18 (square lattice) (14.29a

and

r J ~ =1.16 (triangular lattice). (14.29b

The triangular lattice is thus slightly more stable. A schematic diagram of the square anc

triangular vortex lattices is shown in Fig. 14.1. The spatial configuration of 1 if ; 1 2 near Hc2 for,

triangular vortex lattice is shown in Fig. 14.2. The above treatment has neglected the elTectsof an]

in-plane anisotropy which, given that the structure is sensitive to f 3 A' can (and in some cases does

alter the observed symmetry.



.,.1-------~ TheJosephson effects

15.1 The Josephson equations

Letus recall our expression for the G-L current density

ie*h e*2j = - -(lp*Vif; - if;Vif;*) - -I if;12A.

2m* m*e(9.17b)

Wewrite if; in the form if; = I if; I ei<I>s before, obtaining

e* fJ [ e* J=-1if;12 V<l>--A ;m* he

(ISJ)

i.e.,the current in a superconductor involves the gradient of the (gauge-invariant) phase (recall

that real G- L wavefunctions carry no current).

The G-L boundary condition at an insulator-superconductor interface was given earlier as

(1 1 e* )

fi· iV - ~ A if; = 0, (12.2)

whichis equivalent to no current flow through the boundary. Ifwe have a 'junction' which weakly

couplestwo superconductors (formed, for example, by a thin insulating layer between the two

superconductors through which electrons can tunnel), we must modify this condition. Taking fi

II, we replace (12.2) by more general (yet linear) boundary conditions 1

oif;! ie* I P 2-..,- - -h A,if;1= ---;-ox e A

(lS.2a)

an d

(1S.2b)

theparameter i is the same in both equations since it is a property of the boundary and not the

superconductor designations, 1 and 2. We insert (IS.2a) into Eq. (9.17b) yielding

. ie*h [ * oif;1 Oif;iJ e*2 2

t.= - -2 * if;! --;-. - if;! --;-. - -*-1 if;! I A,m uX ox m c

1 . This form is identical to Eq. (9.44) with Mll =M22 = 0, - MI2 = M';-/ = A .

59




ie*h [ (if; ie* ) (if; * ie* ) J e*2= _._ VI* __2. + -A if; - if; ~ - - A 1/1* - --11/1 12A .2m* 1 i " he xli J.* he x 1 m*e 1 x

(1 5

In the absence of magnetic a toms, the supercond ucting properties are in variant under til]

reversal, which results in if ; --> 1 / 1 * , j --> - j, and A --> - A. Under these operations both sides

Eqs. (15.2) turn into their complex conjugates and hence J . must be reaL We then obtain fro

(15.3)

(15.

Writing if;; = if;; I ei<I\ and assuming both sides arc prepared from the same kind of superconduc

ing material, I if ;1

I = I VI 2 I , we have

j=irnsin< l > 2 1 (15.:

where

(15.6,

and

(15.6t

Note j.; is the maximum current density that may be carried by the junction.

In deriving Eq. (15.5) we have assumed that no electric field and magnetic flux density an

present in the junction. When no electric field is present in the junction, the phase is time

independent. We generalize to the case in which a field is present by using a gauge invariano

argument. Under a gauge transformation (recall H =V x A and E = - V V - (l/c)(c'A/at))

A --> A + Vx (15.7

and

1 a XV-->V-----'-

c D t'(15.8

where x ( r , t) is an arbitrary single-valued function and Vis the potential. Since the G-L equation

contains the form V - (ie* / l7e)A, we must change the phase of the G-L wavefunction by

e*< l > --> < l> + ~ x( t).

he(15.9)

Comparing Eqs. (15.8) and (15.9) we see that the form

a < l > e*---V=Oat n

(15.10)

is qauqe-incariant. If V is (initially) assumed to be independent of time and denoted as V21> then

integration of (15.1 0) yields

(15.11)



The Josephson effects 61

or

[e* J.' (0) .

J = J rn Slll 11>21 - Ii V21t . (15.12)

Introducing the frequency w] =ol1>21!ot we see that (15.11) leads to

(15.13)

Theoscillating currentj(t) given by (15.12) will be associated with an oscillating voltage which will

be superimposed on the static voltage, V2 I'

We next examine the Josephson effects in the presence of a magnetic field. We will restrict

ourselvesto the case of a relatively weak field where a quasielassical description is adequate; i.e.,

the dominant effect of a field, which is described by a vector potential A, is to make the phase

position-dependent. From the discussion surrounding Eqs. (9.19) we know that for a 'pure' gauge

field (one not involving a field H(r)) the only effect of the vector potential is to produce a

position-dependent phase; this suggests that in the presence of a vector potential associated with

aweakfield, H(r), the effect may be approximately incorporated in the phase of the wa vefunction.

Comparing Eqs. (15.7) and (15.9) we have the gauge-invariant form analogous to (15.10) as

e*VI1>--A=O.

he(15.14)

Therefore, the gauge-invariant phase difference is given by

_ (0) 2n f 2 .

11>21 -11>21 +- A dt.1 > 0 1

(15.15)

where 1 > 0 = = hel2e is the flux quantum.The fundamental equations governing the behavior of Josephson junctions are the cur-

rent-phase relation (15.5) , the voltage-phase relation (15.11) , and the gauge-invariant phase

relation (15.15) . They are believed to be exact. In the subsequent sections these equations are

applied to some simple junction structures and circuits. For example, electromagnetic radiation

isemitted from a Josephson junction in the presence of a potential (sec Sec. 17). Eq. (15.13) now

forms the basis for defining the standard volt in terms of a measured frequency and the

fundamental constants, e and h (Taylor, Parker, and Langenberg 1969).

15.2 Magnetic field effects:

the two-junction SQUID

Consider the superconducting circuit shown in Fig. 15.1 involving two Josephson junctions

connected by superconducting leads. The loop formed by this circuit is assumed to contain a




' l l = 'musin~<l>ll C

-- --a b

@H

d c-- ---

't = 'mtsin~<t>t

Figure 15.1 Schematic of a de SQUID consisting of two Josephson junctions

connected in parallel by supcrconducting links. The path of integration C is shown by

the dashed line.

magnetic flux, <D,arising from a position-dependent field, H(r), with some associated vector

potential A(r).

On entering the loop on the left in Fig. 15.1 (point 1), the current I splits into two

components III and It, where the subscripts refer to the upper and lower paths, respectively. From

Eq. (15.5) the total current arriving at point 2 is

I=Iu+1t

=ImusinL1l1>u+ Im t sin L1<D t, (15.16)

where Irnu and Imt correspond to the maximum currents associated with the upper and lower

junctions, and L1l1>llnd L111>{re the corresponding gauge-invariant phase shifts. Assuming a

matched pair of junctions for simplicity (lmu= Imz" = 1m), we may rewrite (15.16) as

(15.17)

The gauge invariant phase shifts may be obtained by integrating VII>around the elosed path

C shown in Fig. 15.1. Noting that <Dis a multivalued function that can change by 2nn upon

completing the path we have

where n is an integer. The phase differences across the upper and lower Josephson junctions are

given by Eq. (15.15) as

? n f bII>b- (D=L1(D+ . _ _ A .dt

a u c P o a

and

The second and fourth terms in Eq. (15.18) are phase differences in the superconducting leads

themselves and are found by using the supercurrent equation (15.1) and the expression for the

London penetration depth Eq. (2.6):

i c 2n i c ( 4n).2

)<Dc- <Db= VII>·dt =- A + _ _!j 'dtb ¢o h C




and

f

a 2n f a ( 4n ..[ 2 )<Pa-<Pd= V<D·dl=- A+ __ l.j ·dt.

d c P o d C

Substituting the above four equations in Eq. (15.18) gives

2n~ . 2n 4nill rl<Du-L1<Dt=2nn+- A'dt+--- .j·dt.

c P o c c P o C • C'

(IS.19a)

Theintegration of A is around a complete closed path C and is equal to the total flux c P inside the

area enclosed by the contour. The integration ofj follows a path C' which excludes the integration

over the insulators. If the superconducting leads are thicker than the London penetration depth.

the integration path can be taken deep inside the superconductors where the integral involvingthe supercurrent density is negligible. The phase difference is then simply related to the total flux

by

(IS.19b)

Using this equation to eliminate L1cI\,rom Eq. (15.17), the total current is

(15.20)

When the inductance L of the loop is taken into account, the total flux in Eq. (IS.19b)

consists of the externally applied flux c P C X L and the flux generated by the screening circulating

current Idr; i.e..

(15.21)

For the identical junction case,

(15.22)

In general, Eqs. (1 5.2 0), (1 5.2 1), and (15.22) must be solved self-consistently to describe the

behavior of the two-josephson-junction loop. For simplicity, we assume the loop inductance is

negligible and consider only the effect of the externally applied flux on the characteristics of the

loop.The maximum supercurrent density which can be carried by the loop is found by maximiz-

ing Eq. (15.20) with respect to L1<D{;.e.,

L1<Dt+ n ~ : X L =n + 1/2)n.

Hencethe maximum supercurrent density, I m a x , is given by

I(n c P C X L ) II m a x = 1 m cos To ' (15.23)

whichis periodic in the external flux. Since c P o ~ 2.07 X 10-7 Gcm ', it is clear that the device

pictured in Fig. 15.1 can be used to measure very small changes in magnetic field. It is sometimes




z

H

@ax

Figure 15.2 Schematic diagram of an SIS Josephson tunnel junction.

referred to as a two-junction SQUID where the latter is an acronym constructed from the words

superconducting quantum interference device.

15.3 The extended Josephson junction

We next discuss the behavior of a single planar Josephsonjunction in a magnetic field. We refer tothe junction cross section depicted in Fig. 1S.2. The magnetic field is directed into the page along

y . The middle of the junction is taken as the origin and the vector potential in the three regions is

_ { _ Hxe-(o-a,2)'AL

Az - - Hx

_ Hxe(o+1I'2)/'L

(z > a/2)

(a/2 > z > - a/2)

(z < - a/2)

(lS.24a)

(1S.24b)

(lS.24c)

where the exponential dependencies arise from the Meissner effect in the two superconductors.

The phase difference encountered in crossing the junction from z =-::I:) to z =+ C fJ at a given

horizontal coordinate x is

(1S.25)

Assuming a rectangular junction of dimensions L, and Lv, the total current, I , through the

junction is

f1.,12

1= jmLv . sin L1<D(x)dx-Lx/2

or




1.2 ,---,---,--r--,-,--.,.--,--,-----,,---,

1.0

0.80'

: = - E 0.6: : : E

0.4

0.2

Figure 15.3 J m vs r P characteristics for a short Josephson junction when the self-field

effect is negligible.

sin(rr¢l¢o). (0)

1 =1m sin < 1 >2 1 ,rr¢l¢o

where¢ = = H(a + 2),dLx , and 1m =jmLJ~y . The maximum value of the junction current, 1m""

occursfor sin < I > ~ O != ± I(depending on the sign of sin (rr¢l¢o)); thus

I'llax =1 I sin(rr¢l¢o) Im rr¢l¢o '

(15.26)

W e note we obtain a 'diffraction-like' (sin x / x ) pattern (see Fig. 15.3) involving the variable ¢ I ¢a

where¢ is the flux contained within the junction.

The effective junction area entering the definition of ¢ involves an effective thickness (made

u p of the insulator thickness plus a contribution of one London depth in each superconductor)

timesthe junction width, Lx '

15.4 Effect of an applied rf field

An interesting behavior results if an external rf voltage is applied to a de voltage-biased

Josephson junction, as first observed by Shapiro (1963). We take the applied voltage to have the

form

v =V o + V I cos tot. (15.27)

FromEq. (15.12) the resulting current in the junction will be

1=Imsin [;~ I ( V a + V I cos wt)dtj

= 1m sin [ < 1 > ( 0 ) + OlJt + ()sinwt], (15.28)

where ( ! J ( O ) is an arbitrary phase, WJ is given by Eq. (15.13) and b = = e* Vdhm is called the

modulation (or deviation) index. Using expressions which are well known in the theory of

frequencymodulatiorr' we may rewrite (15.28) as

2. SeeAbramowitz and Stegun (1970), Eqs, (9.1.42) and (9.1.43), p. 361.




I(t) =i:{sin[<D(O) + wjtJ [ J 0(3) + 2 J I J 211(6)cos(2nwt) J

+ cos[(D(O) + wjtJ2 n~o .!2n+1

((»)sin[(2n + l)wtJ },(15.29)

where the Jn are Bessel functions of first kind of integer order. Using sin a cos b =

t[sin(a + b ) + sin(a - b)J we may rewrite (15.29) as

I(t) =t;(.! o((»)sin[w/ + <D(O)J

+ J l .!11(()){sin[(wj + nCU) l + <D(O)]+ ( - 1)"sin[(wj - nw)t + <D(O)J}).

(15.30)

I f we vary Wj (by sweeping Va) such that the condition OJ j = ± nco is met, then the time

dependence associated with this term in (15.30), having the amplitude I(()) and phase <D(O),would

be 'transformed to zero frequency' and would appear as a 'spike' in the current-voltage character-

istics.

The impedance of a Josephson junction is usually much smaller than the resistance of the

leads extending into the cryostat and hence a constant current source is a more accurate

representation than the constant voltage source assumed above. Further analysis shows (and

experiment confirms) that this results in a step-like (rather than a spike-like) current-voltage

behavior.

15.5 The resistively shunted junction

(RSJ) model

In analyzing the behavior of circuits involving small Josephson junctions, one may model the

effects of various dissipative processes and the distributed junction capacity with so-called

'lumped' circuit parameters. Fig. 15.4 shows such a model; here C and R represent the capacity

and effective resistance of the junction, 3 where the latter is represented by a cross. V j is the voltage

across the junction while I is the total current flowing through all three circuit elements which is

given by"

(15.31a)

hC .. I z .=Imsin<D + -<D + -<D,

2e 2eR(15.31b)

where we have used the Josephson equations

3. The capacitance is largely determined by the geometry of the junction and may he regarded as

constant. The resistance on the other hand may depend strongly on the junction voltage; for this

reason an additional shunt resistance, which is much smaller than the junction resistance, is

sometimes incorporated to bypass this effect.

4. We use MKS units in this subsection.




I

+--- •

•

Figure 15.4 Equivalent circuit of a resistively shunted Josephson junction.

I~<P I

I

m ..

Figure 15.5 The simple pendulum with an applied torque.

(15.32)

and

I J =1m sin c D (15.33)

in obtaining (15.31b).

Let us examine the behavior of an RSJ to which a constant current is applied. If I is slowly

increased the voltage across the junction will remain zero (VJ = 0), implying <!>= sin -1 (1 1 1m ),

until I=m' Above this point a voltage develops and the junction becomes resistive with a

time-dependent phase. To examine the beha vior in this regime it is useful to note that the phase,

C l l ( t ) , ofajunction described by Eq. (15.31b) is in one-to-one correspondence with the angle of

rotation (which we will also call c I » ) of a damped pendulum, driven by a constant torque, in a

constant gravitational field (see Fig. 15.5): the applied torque, .r=mwyr , gravitationally induced

torque, mgl sin <!>,moment of inertia, .f=mr", and damping coefficient, k, are identified with I,

lmsin<!>,zeI2e, and hl2eR, respectively. The regime I < 1m corresponds to a situation in which

theapplied torque .r is less than a critical torque :Y m necessary to raise the pendulum to an angle

Cll=n/2 (where the opposing gravitational torque is maximal). for .'Y > :Ym the pendulum

rotates in a manner such that the average energy dissipated per rotation is equal to the average

work per rotation (were there no damping the average angular velocity c b would increase without

limit in this regime). We now go to a rotating reference frame by writing

< ! > ( t ) =cot + < ! > ' ( t ) , (15.34)

where

(15.35)

isthe average angular velocity and c D ' is periodic (with an average value of zero); Eq. (15.31b)then

becomes




he .. f z .-$+-$ +msin[wt +D'(t)J = .2e 2eR

(15.36)

In the limit where the departure of c I ) from a steady state rotation, oit, is small, i.e. <D'( t) < < 1, we

may neglect the $contribution to the last term in (15.36). We then have the equation for a driven

harmonic oscillator.

Writing $'(t) = < D ~ sin(wt + fi) l e a d s to the pair of equations

hC hI - - w2<D~ cos fi - - w$~ sin e =0m 2e 2eR

(15.37)

and

hC Ii

- - U )2$~ sin fi + - w$~ cos (J = .2e 2eR (15.38)

From the second condition we obtain

1tan f=-_

wRC(15.39a)

or

1

sin (J = 7 2 12(1+orr) /

where, = = RC. Substituting (15.39) into (15.37) yields

(15.39b)

, 2e ImR 1

< D o = Ii -:;;1 + (J)2,2)1/Z'(15.40)

Now

-s- 2e _ 2e< l > = - V =- IR.

h h

(15.41)

In the limit, (liT < < 1, $~=ImlI). In the opposite limit, OJ, >> 1, (15.40) becomes < D ~ =1/

(liT)/m/l. The requirement $;) < < I is therefore best satisfied in the limit of large I and large WI.

The limit OJ, ~ 0 may be obtained by eliminating the shunting capacitor in Fig. 15.4; the

differential equation (15.31b) then reduces to

. h.I =1 m Sill < D + - < D

2eR(15.42)

or

I z f < l > « ) d$

t= i R l m < 1 > 1 0 ) III", - sin < D '

which upon integration yields

[(

!X 2 - 1 ) + ( n t ) ](t) = 2tan-

1~ tan T -'X , (15.43)




a

~~

'/

4.0 B "'/

"/3.0

"/

/

/

/

/

2.0 "/

"/

"/

A "/

"/

1.0 /

/

/

/

1.0 2.0 3.0 4.0 5.0 T)

Figure 15.6 Time-averaged nonhysteretic I-V characteristic computed in the RS.J

model in the zero capacitance limit; a = 111m, '1 = VIRlm. Points A and B give the lime-

averaged voltages corresponding to o : = 1.2 and rJ . =4, respectively.

where we have defined

(15.44)

whereIX

is a parameter and Tis the period (which is the minimum time required for < D ( t ) to returnto its initial value <D(O)).The voltage across the junction from Eq. (15.5) is (h<D/2e). Ifwe assume the

voltage is sensed with a device which records the time average of V(t) (over many periods T) then

- Iz.. h 1 f T d<D h 2n 2 1/2V=-c))=--- -dt=--=Rlm(1X -1)·.

2e 2e T 0 dt 2e T(15.45)

Note that for I >> 1m ( I X >> 1) (15.45) approaches (15.41). Fig. 15.6 shows the J- V characteris-

tics f rom Eq. (15.45); note that the J- V characteristic is non hysteretic.

Wenow give a qualitative discussion of the general case when C * ' O.We begin by rewriting

Eq. (15.31b) in dimensionless form. We measure time in units of Wo and introduce a parameter f Jwhere

(

7 I ) 1 / 2__ C n . ' .

Wo =he ' (15.46)

• woisthe natural frequency for small oscillations (referred to as the Josephson plasma frequency).

Then Eq. (15.31 b) may be rewritten as

< i > + { 3 < D + sin C [) ='1, (15.4 7)

where the derivatives are now with respect to the dimensionless time variable. We define an

angular velocity O(t) = = < 1 > ; the angular acceleration is then < i> = n = (dO/dcD)ci>= (dO/d<D)O = (dj

dCll)(Q2/2)and Eq. (15.47) becomes

d ( 0 2

)- - + fin + sin <D=X .d<D 2

( 15.48)

The 'orbits' associated with the solutions of (15.48) for various initial conditions and parameter




t"J7

------~----~--~~--~$

Figure 15.7 The Q-1tJ plane trajectories (lower panel) associated with the solution of

Eq. (15.49). There are two kinds of orbits: an open orbit for (1/2)Q2(<ll =0) > 2 and a

closed orbit for (1/2)Q2(<ll =0) < 2. The upper panel shows the potential as a function of

<ll.

values (z and m are trajectories in the (HI> plane; we may think of this plane as the t w o -

dimensional 'phase space' associated with the motion. We now discuss the nature of these orbits.

We begin by discussing the orbits when : x = { J = O . Our equation is then

d ( 0 2

)d < 1 > 2+ sin c D = o . {15.49)

This equation may be integrated analytically in terms of Euler elliptic functions, as discussed in

mechanics textbooks. Wc will restrict ourselves to some qualitative statements. Since (15.49)is

invariant under the transformation c D ~ - < 1 > , we expect extrema at < 1 > = 0 and (when they exist]

± tt. For the rotating case, < t > advances continuously in time with 0 having a maximum at < 1 > =0

(where the kinetic energy of the pendulum is largest) and a minimum at c D =± tt (where thepotential energy is maximal); the phase space trajectory is shown in Fig. 15.7. Such 'open' orbits

req uire that the kinetic energy always be larger than the potential energy; i.e.,

(15.50)

When this condition is not satisfied the motion is oscillatory and the orbit in phase space is closed

as shown in Fig. 15.7.

We now turn on a small damping (Ii> 0), but still keeping the drive current (torque) zero

(« = 0); we will then have 0 ( < 1 » > 0 ( < 1 > + 2n) > 0 ( < 1 > + 4n), etc., i.e., the rate of rotation slows

down. Eventually a point is reached where 0passes through zero before the system (pendulum)

reaches the 'top' after which it reverses its motion. This point marks the transition between rotary



- I T it -it it -it it

A 8 c

Figure 15.8 The qualitative phase-plane trajectories, corresponding to the three

different situations (A, 8, C) in the fi e x plane in Fig. 15.10. (After Belykh, Pedersen, and

Soerensen (1977).)

andoscillatory motion. After the transition the phase point proceeds along a spiral, asymptoti-

callyapproaching the origin.

The above described sequence of events is shown as a phase space plot in Fig. 15.8. (Note

that sincedissipation is present the arrow of time is relevant.)

We now consider the general case;« > 0 , 1 3 >O.When a >> 1 (l>> 1m), Q is practically

constant and given by Q =a/f3. As a is lowered through unity from above the behavior of the

systemdepends on the magnitude of 1 3 . Since '- x is starting at a value greater than 1, the pendulum

motionmust evolve from a rotary state. I f we now decrease a to a value less than 1 (J < 1m) the

pendulumcan continue to rotate (i.e., we can maintain a steady state motion) provided the work

performedper cycle by the external torque, 2na, can compensate for the energy dissipated per

cycle;the latter is given by

f

T f 2 1 r dr{ 3 Q2(t)dt =3 Q2(<D)-d<D

o 0 d<D

f2 1 r

=3 0 Q(clJ)dclJ. (15.51 )

Thebehavior ofQ(<1»is not known without actually integrating the equation of the orbit (15.48).

However,we can estimate the critical torque, ac, by assuming Q(n) =0 and Q(O)is the angular

velocityresulting from 'free-fall' rotation under the influence of gra vity, which from Eq. (15.50) is

Q ( C l l=) = ; we linearly interpolate between 0 and tt by writing Q(<D)=Q(O)<D/n . We then have

f1 r <D

2n a c ~ 2jiQ (0) - d<Do n

or

Amore accurate expression can be obtained from the exact treatment of the free pendulumwhichyields

(15.52)

whichis valid for ji : : s 0.2 (Stewart 1968, 1974). Wc see that as fl ~ O,«, ~ 0; i.e., the junction is

hysteretic, switching from a constant-phase, zero-voltage state to a phase-precessing, finite-




/

r.t d~o_ _/(

[3 = 10

/

/

/

2.0

/

//

~ / 0.1

1.0 2.0 T)

Figure 15.9 Analog computer results for the time average of:x( = I / I m ) vs 1 1 ( =V/Rlm)

characteristics for different values of the parameter f J = l/woRC. (After Johnson (1968).)

1.0

.\

I

IIII

II

Bel---':"-1

II

IIIIII

I

0.5

0.5 1.0

Figure 15.10 Analog computer results for the hysteresis parameter !X c as a function of

{i. Tnthe region above the c x J f J ) curve only one stable solution exists. Tnthe region below

the curve two stable solutions exist. (After Johnson (1968).)

voltage state as a is increased above 1, but returning to the original (constant-phase) state ata

value a < 1. The J- V behavior of a hysteretic junction is shown in Fig. 15 .9 . The junctioninstability line in the a-Ii plane obtained by computer is shown in Fig. 15.10 (Johnson 1 9 6 8 ) .

For a discussion of the case in which the junction resistance is voltage-dependent s e e

Barone and Paterno (1982) .




Figure 15.11 A single junction SQUID coupled to a tank circuit via a mutual

inductance M.

15.6 The rf biased SQUID

Rather than measuring a magnetic flux with a two-junction interferometer (a de SQUID), as

discussed in Sec. 15.2, an alternative method involves the use of only a single junction (Silver and

Zimmerman 1967).

The device involves three circuits (see Fig. 15.11): (i) the primary SQUID, which is a

superconducting loop connected to a resistively shunted Josephson junction; (ii) a second loop(not shown in the figure), which is inductively coupled to the first loop and introduces an external

flux<Pex!=11;[ex!Iex! as a result of a mutual inductance, M ex!' coupling the two loops and a current,

lex!' applied to the second loop; and (iii) a third loop which is driven by an external oscillator

(typically operating at a frequency of order 107 Hz) which applies a periodically changing flux,

¢rf' through a mutual inductance, M.

To explain the operation of the device, which is somewhat subtle, we must examine the

action of CPex! and C P r f on the response of the primary SQUID loop. We begin by discussing the

equation of motion when CPex! = P r f = . The equivalent circuit consists of an RSJ connected to

an ind uctor, L(represented by the loop in Fig. 15.11). The equa tions of motion are"

(15.53a)

where

(15.53b)

Combining these equations with the forms in Eq. (15.31b) we have (where we multiply through by

L)

h<D hLC .. «i.:- + LIm sin < D + -- < D + - < D =O.2e 2e 2eR

(15.54)

The phase variable, (D , in Eq. (15.54) may be replaced by the internally generated flux, c P ,

according to the relation c P =cI)/2n)cpo' Incorporating the contribution of the external flux, CPex!,

Eq. (15.54) then becomes

5 . We use MKS units in this subsection.




3(a)

2

3

3 4

2

(c) q ,

Figure 15.12 The static behavior of a single-junction superconducting loop in an

applied external flux ¢ex, : (a) c j ) m N > o =0.25; (b) ¢m/¢O = 1.25; (c) a 'linearized' version of

Fig. lS.12(b) showing the path traced out by ( p and ¢ext for the case ¢ =n c j ) o involving

three branches.

(IS.S5)

where ¢m = = LJ me w~=1/LC, and T =L/R. We examine the static solutions of this equation for

the responses ¢ =¢(¢ex,), which are shown in Fig. IS.12(a) and (b). Fig. IS.12(a) shows the

behavior when ¢m/¢O=0.2S, where a continuous, single-valued evolution of ¢ with ¢ext is

obtained; a line with a slope of 1 is shown for reference. Fig. IS.12(b) on the other hand shows the

behavior when ¢rn!c/JO= 1.2S. We now have a multi-valued behavior. Between successive points

4 ) c at which dcp/d¢ext diverges (indicated by the dashed vertical lines in Fig. IS.12(b)) there are

alternating regions with positive and negative slopes; the former are stable while the latter are

unstable and the system spontaneously oscillates. However, the frequency involved is typically in

the microwave regime and if T is sufficiently short the oscillations damp quickly (relative to the

period associated with ¢rr)' The system then, effectively, switches from one stable (positive slope)

region to an adjacent one. (By Taylor expanding in the region c/J/¢o ~ n (an integer) one sees that

the slope of the plateau at this point is (2n¢m/¢O + 1)-1.) The upward and downward directed i



, ne Josepnson eTTeCrS /'0

arrows in Fig. IS.12(c) indicate how the system switches for increasing and decreasing ¢ex ! '

respectively.

To use these characteristics to make measurements we need a means of locating a stable

branchand the position alongit. Todo this we introduce the third loop, which is driven sinusoidally

byan rfgenerator, and induces an additionalflux ¢rf( t ) = ¢ rf sin wrft into the primary loop. We will

assumethat a capacitor is associated with this additional loop and that the pair resonate at the rf

frequency.The quality factor, Q, of this resonant circuit along with the resonant frequency fixes its

bandwidth, ~OJ =wrrlQ, which in turn fixes the response time Trf ~ (~w) -1. Suppose that at t =0

therfgenerator is connected to the resonant circuit. The amplitude, ¢rf, of the oscillatory flux will

then increase with a characteristic time T rf . When the total externally induced flux, ¢ e x, + ¢ rf( t),

exceedsone of the thresholds, ¢e' (associated with the primary SQUID flux, ¢, depicted in Fig.

15.l2(b)),the SQUID loop becomes resistive, since it is then in a finite voltage state (oscillating at

somecharacteristic frequency much higher than Wrf' which we ignore). This results in a rapiddissipationleading to a partial quenching of the amplitude, ¢ r6 however, the process then repeats

itself.The peak amplitude of ¢rf, denoted rPrf (which can be measured with the appropriate

electroniccircuitry), is then a measure of ¢ex! as we will now discuss.

Referring to Fig. IS.l2(b), we identify two 'special' points: (i) the midpoint of a stable

branch,where ¢ e x , =n¢o (with n an integer), and (ii] the midpoint of an unstable branch, where

fex!=n + ! )¢o ' We first examine case (i).Assuming sufficient rf drive, as the oscillations build up

intime, the thresholds, ¢u, connecting the 11, n + 1and 11,11 - 1 branches are encountered and the

SQUIDswitches symmetrically between the three branches.Ifthe rf drive is further increased the

thresholds connecting the 11 + 1,11 + 2 and the 11 - 1 ,11 - 2 branches also are encountered and

theSQUID then oscillates symmetrically between five branches and so on. An example of a path

traversed by the system for a circuit involving three stable regions is shown in Fig. IS.12(c).

For case (ii), where ¢ex, =n + ~)¢o, the path traversed by the SQUID is nominally

centeredat the midpoint of an unstable region. As the rf drive is increased from zero, a level is

reachedwhere a complete circuit involves traversing the portion of the two stable branches lying

directlyabove, n + 1, and below, 11 , the unstable 11 + ~section. The threshold amplitude corre-

sponds to the extent of the unstable branch. Since this extent is smaller than that of the stable

region,the threshold for first completing circuits in which the SQUID switches irreversibly is

lowerfor case (ii) than for case (i).

Fig. IS.13 shows the peak amplitude of the rf tank circuit, VT, as a function of the rf drive

current, Irf• The slope is initially high but drops abruptly (nominally to zero for a peak reading

detector) when the thresholds for SQUID transitions are encountered, involving 3, S , ... stable

branches for case (i) and 2,4,6, ... for case (ii). From the symmetry it follows that cases (i) and (ii)

representthe maximum and minimum values of VT (or rPrf) for a given drive level, Irf'

Itis clear from the above discussion that if ¢ex, is continuously varied (at fixed Irf), the peak

amplitude (fjrf will move back and forth (in a triangular fashion) between maxima, at «;=l1¢o,andminima, at ¢ e x t =n +})¢o ' By observing the change in the number of flux quanta, one can

accurately measure a change in a current, Iext, associated with the loop generating ¢ o x t .

Fig. IS.14(a) shows the characteristic 'staircase' dependence on Trf for an rf SQUID

operated at 27 MHz that was shown schematically in Fig. IS.13. The two curves correspond to

thelimiting cases for the de external flux. Fig. IS.14(b) shows the triangular dependence of the

detectedrf voltage Vr vs Iex tcharacteristics. The different curves refer to different values of the rf

drivecurrent, T rf'




< il =(n+ 1/2)$0

t;--/

/

Figure 15.13 Relation between the tank circuit voltage Vr and the rf current amplitude

I"f for the cases of integral and half integral numbers of nux quanta. For intermediatevalues of flux, the voltage steps occur at intermediate values of VT•

f - <>. , -Cl

s'0>

- g. . . .o

2l.,o

(a) rf drive, Irf (b) de field, c/Jdc

Figure 15.14 Experimental responses of an rf SQUID in the dissipative mode: (a)

detected tank circuit voltage Vr vs rf current amplitude J rf (the two curves are the two

limiting cases for external dc flux ( I Jde = 11(/>0 and (/>dc = (n + })¢o); (b) VT vs appliedmagnetic flux curves at different values of I,.f. (After Zimmerman (1972).)

If the loop gencratingcjJext is coupled to another loop which, in turn, is coupled to a

magnetic sample in a fixed field (via an all superconducting circuit), changes in the magnetization

with temperature or some other parameter may be observed. (Note that current changes induced

through magnetization changes arc persistent.) The self-inductance of the two loops, together

with the mutual inductances coupling them to the SQUID and the sample, make up a 'flux

transformer' the characteristics of which can be optimized for maximal sensitivity.

For further discussion of the rf SQUID see Zimmerman (1972) or Barone and Paterno

(1982), where many additional references may be found.



. f-----~-'h-e-J-o-S-e-p-h-s-o-n-,a-t-t-ic-e--

in 1D

Certain intermetallic compounds have a layered structure in which the in-plane conductivity

greatlyexceeds the conductivity normal to the layers. This applies for most of the high tempera-ture superconductors. Such structures may also be made artificially by depositing alternating

layersof a superconductor and a low conductivity material,

The simplest model is to assume infinitesimally thin superconducting layers which arc

coupledvia order parameter tunnelinq (Josephson coupling) through insulating layers of thickness

s.Following Lawrence and Doniach (1971), we introduce a modified free-energy functional of the

form

(16.1)

where

(16.2)

herewe have defined ,4 = = (lIs) J~~1),Azdz. The structure of this equation is similar to Eq. (9.13)

withrespect to the in-plane components while the intcrplane coupling is seen (on expanding theexponent) to be a generalization of the operator

a ie*---A_D z he ~

to the case of finite differences. Our order parameter J j J "(r)has a discrete dependence on the index

n and a continuous dependence on r =x x + y y : the total vector potential A I + Azz is, however,

definedat all points. Variation with respect to J j J * yields

1 1 2 ( ie* ) 2-- V--A ! If2m* he - "

(16.3)

77




Variation with respect to A _Land Az yields Eq , (9 .16 ) with

(16 .4)

and

. i 1 1 . , , + 1 . n =

Eq. (16.5) for the tunneling current flowing between planes nand n + 1 is equivalent to Eq. (15.4)

(written in a gauge-invariant form).

Solutions of E q. (16 .3) were first examined in detail by Klemm, Luther, and Beasley (1975) .

As before, we confine our interest to the evaluation of H c2, in which case we may neglect the last

(nonlinear) term. The external magnetic field will be assumed to lie in the x z plane. In order to

take advantage of the layer periodicity, we choose a gauge for which the vector potential has no z

dependence:

(16.6)

We seek a solution of the form

noting that I f z = - Hx sin 0, we find that

{

1 i2 d2 1- -- __ + -m*w2(x - x )2 cos? e2m* dx? 2 c 0

1 i

2

[ ( e* H s )]}-;;2cos kz s + --' x sin 8 - 1 + e x u,,(x) =0,mz s lie

(16.7)

where Xo = l1 eky /e * H cos e is an orbit center.For the field perpendicular to the layers, we again have a harmonic oscillator problem and

we recover our earlier result - c : = ~ 1 1 ( J ) e ( O ) . For H parallel to the planes Eq. (16.7) becomes

{1 1 2 d 2 / 1

2

[( e* H S ) ] }- -- __ - -- cos kz s + -, ,-x-I + c : un = O.2m* dx ' m:s2 he

(16 .8 )

Changing k z has the effect of shifting the origin on the x axis. We shall initially fix the origin such

that the argument of the cosine vanishes when x =O.Ncar the zero-field transition temperature,

where e x is small, we expect the critical magnetic field, H e 2 1 1 ' also to be small and further that thelowest eigenfunction u,which by Floquet's theorem is periodic (since the potential is periodic), is

concentrated ncar the minimum of the 'potential '

li2

[ (e*H S )](x ) =--2 1 - cos -, _. x .m:s he

(16.9)

Expanding Eq. (16 .9 ) to second order, we obtain a harmonic oscillator potential which, o n

substitution into Eq. (16.8), yields



The Josephson lattice in 1D 79

_! i__ u l! + [~m*w2 (~)X 2 + r x J u = 02m* 2 c 2 '

(16.10)

where

Thelowest eigenvalue is - o : = ±hm e(n/2), yielding an upper critical field H e 2 1 1 = cjJo!2n((z where

~ = = (h 2 /2m* I o : I )112 and (z = = (h 2!2m;I , :x I )112. For temperatures further from T e, where the eigen-

value- c : and the associated parallel critical field are both large, u becomes less concentrated at

thepotential minimum and our expansion of Vex) breaks down. To examine this limit, we write

Eq . (16.8)in its 'canonical' form (Abramowitz and Stegun 1970) by introducing new variables

e*H s 2cjJ~ m*2v = = - . - x + tt., 2q = = --0 - -

hc n2H-~m:

and

whichyields

d2u-2 + (a ' - 2qcos2v)u = .dv

(16.11a)

For large H, both at and q are small and arc related by the expansion (Abramowitz and Stegun

1970,p. 722)

(16.11b)

Inserting our expressions for a' and q into Eq. (16.11b) we obtain

(16.12)

Sincethis is a high field expansion, Eq, (16.l2) is valid only for temperatures such that (z ~ s/)2;

however, H c2 '! is real only for (z > s/j2. No (real) solution exists for (z < s/j2 and hence thecritical field becomes formally infinite (in this model) at a temperature T* such that (z(T*) = s /

)2. In a real system, this divergence would be removed by the effect of paramagnetic limiting (see

Part III, Eq. (41.45)).

Deutcher and Entin-Wohlman (1978) generalized the above model to the case of a

superlattice of thin slabs of thickness d separated by Josephson-coupled insulating layers of

thickness s. We shall continue to take the z axis normal to the layers, and we place z = 0 at the

interface with the lower surface of a metallic layer. The free energy is given by




f { f IlD t d [ h 2 1 ( ie*) 1 2F = I dxdy dz ~ V----:-A !11,,(X,y,Z)11 11f) _In I I (

, ( J

+ C t . 1 0 / l x , y, z ) 1 - + 2 1 o /,,(X , y , Z ) 1 4

+ ___ 2 _ , 1 o/l l+l(x,y,(n + I)D)exp( - ie*. All_ I S )2m;s" he

(16.13)

here , 4 , , + I = = (I/s) J ~ n D \ I . l D AJx, y, z )dz and D = d + s. We seek a solution to E q, (1 6.1 3) in the limit

d < < I C L and d < < ¢. The first of these assures us that we may neglect the diamagnetic screeningcurrents and hence we shall take the magnetic field as uniform and given by the external field, H.

We consider only the case where H is parallel to y and work in the gauge A = - Hxz. The limit

d < < ¢ implies that the magnitude of the order parameter is constant along z [or a given x ;

however, we do allow the phase to vary. We thus choose a solution of the form

The z component of the gradient energy in Ell. (16.13) is

1(

C < D e* ) 1 2--+-Hx 0 /jjz he

To avoid an arbitrarily large growth in this term for large x, we choose the phase in the form

e*( D = ---:-l1x(z - Zll);

he

21 1 is chosen to minimize the overall free energy. For an isolated slab it has the value Z/1 =D +

d/2 , i.c., the origin must be taken to be in the center of the nth slab. For the ground state, we assert

that, by symmetry, Zll will have the same value in the coupled superlattice. We include no

variation in the phase along y since this results in an increase in the free energy. Varying Eq .

(16.13) with respect to ! I I * yields the linearized equation

172

[ (e*llDx) l }--- cos ~-~ - 1 u + C t . 1 I dz = O.m;s2 he

(16.14)

Integration over z (noting that the second term in the first parentheses vanishes b y

symmetry) yields

(16.15)

where we h.ave introduced the notation we =e*H/m*c. Using Eqs. (12.3) and (12.4) we m a y I'

rewrite Eq. (16.15) as



The Josephson lattice in 1D 81

1 72

d2

11 112 [ (e*HD ) J ( H2)-- --0 - -- cos ----- x-I u + a 1- - u = 0,2m* d.x" m;s2 he H~

(16.16)

where H II is the parallel critical field of the isolated slab. The equation is identical with Eq. (16.X)

except for the additional term multiplying a.

To obtain the low-field behavior, we again expand the cosine term; this time we shall

include the H 4 term, the effect of which we calculate by perturbation theory. We require the

matrix element of X4 with respect to the unperturbed Gaussian ground-state wavefunction [/0

which is given by (Landau and Lifshitz 1977, p. 136)

(16.17)

Our eigenvalue equation is then

1 . D (m"'.') 112 m*(!)~ d2 ( 3 D 2)- ':i =-hwe- - + 1 - - - - - - - 0 + ...

2 s m~ 24 4 d:(16.18)

or

( 16.19)

For H -> 0 (T---7

Tel , we have the limiting behavior

H _ ¢o .e211 - 2nr;~z (D!s) '

(16.20)

thiswe refer to as 3D behavior with He211 o: T; - T.

To obtain the high-field behavior, we must write Eq. (16.15) in the canonical form of Eq.

(16.11).The constants now become

and

4¢2 ( H2 1'2 m*)

a' = (2n)2H 2oD 2r;2 1 - 1 1 1 i - 2 : 2 m; .

Using Eq. (16.12), we have

(16.21)

(16.22)

From Eq. (16.22), as H ---7 cc; we obtain He211 = HII; i.e., we obtain the upper critical field of an

isolated slab, where IIelll o: tT; - T)112, which we refer to as 2D behavior. Thus as the tempera-

ture is lowered a 3D to 2D 'crossover' occurs. Note that the divergence encountered with the

Lawrence-Doniach superlattice model involving infinitesimally thin superconducting layers is

avoided in the Deutcher-Entin-Wohlman model where d # 0.



Vortex structure in

layered superconductors

Dichalcogenides ' of transition metals such as NbSe2' superconductor/insulator supcrlattices,

and high T; oxide superconductors all have a layered structure. Most of these systems arc Type II

superconductors with relatively large I( values. One of the fascinating characteristics of the

layered superconductors is their strongly anisotropic magnetic properties. Usually, the coherence

length perpendicular to the layer plane ((J is much smaller than that parallel to the layer (SI) '

The anisotropy can be characterized by an anisotropy ratio, y, defined as )' = ~II/¢ L'Other length

scales of interest are the penetration length, A, and the scale of intrinsic inhomogeneities.

Depending on the relative size of ~j_ and the layer spacing, s, we may identify three different

regimes for the vortex structure in the layered, high-x superconductors: (1) If ¢j_(T) >> s, then

the layered structure is largely irrelevant and the superconductor may be regarded as three-

dimensional: anisotropic, but uniform. Since the coherence length diverges as T, is approached,

this regime will always occur sufficiently close to T" The vortex structure can be described using

the London or G-L theories by introducing an anisotropic mass tensor as discussed in Sec. 11.( 2 )

With decreasing temperature, we may have a regime where ¢J(T) < < s (especially in high T,

oxides). If both regimes can be entered by sweeping the temperature a 3D-2D crossover will

occur at some temperature. Below this temperature we have a quasi-2D regime; the layered

structure is then relevant and the Lawrence-Doniach-Iike model discussed in Sec. 16 is adequate

to describe the vortex structure. (3) For the case of extreme anisotropy, as in Bi- and Tl-based

oxide superconductors, with }' < : 10, the interlayer Josephson coupling is very weak. The flux

'lines' are then better viewed as stacks of 2D vortices residing in the superconducting layers

(so-called pancake vortices).

In this section we will give a brief account of the vortex structure in each of these three

regimes. We will focus on the weak field region and derive expressions for Hel, which contain

information on the anisotropy in addition to that contained in H c2'

17.1 3D anisotropic London model

As discussed in Sec. 11, the upper critical field, Hd in layered Type IIsuperconductors can be

described by the G-L equations with a phenomenological anisotropic mass tensor. Due to thelinearity of the equations near Helo a relatively simple solution can be found in this region (see

1. The chalcogenides are the group VI elements S, Te, and Se.

82



Vortex structure in layered superconductors 83

Sec.11). However, close to the lower critical field, H c1, it is difficult to solve the nonlinear G-L

equations to obtain H c l in the anisotropic case. Furthermore, the G-L expansion itself is

problematic in this regime since we are far from T'; For this reason, we will use the less accurate

London model, which provides a reasonable description, at least for a large G-L parameter K

(see,for example, Kogan (1981)). For K ( = = AR ) > > 1, the amplitude of the order parameter, 1 V I I,

variesconsiderably only for distances smaller than ~; one may then assume 1 1 / 1 1 to be constant

everywhereexcept in a narrow core of radius ~.

As in Sec. 11, we introduce a phenomenological effective mass tensor. In the reference frame

alignedwith the principal axes of a crystal with orthorhombic symmetry or higher, this mass

tensor is diagonal with elements which we now write as M j • For convenience we define a

geometricmean mass M=M 1M2M 3)1/3 and normalize the mass tensor such that mj =M J M.

Interms ofMwe define a mean penetration depth, A , and mean coherence Iength.z, by using Eqs.

( 2 . 6 ) and (9.22), respectively. The penetration depths, )'j =Aml/2 , describe the decay of the

components of the screening supercurrent along the principal directions, i. The coherence

lengths,~j =~/mj1/2, characterize the spatial variation of the order parameter along these direc-

tions.For a uniaxial crystal, m3 = 111j_,1 = m2 = m il (in high T, oxides, the anisotropy within the

Cu-O layers is very weak). The anisotropy ratio is defined as :! ~ ~ II/~j_= ; ' 1 / A ll = (M ljMII)1!2.

The free energy, given in Eq. (2.8) for the isotropic case, may be generalized to the

anisotropic case as

(17.1)

whereB is the local field. Straightforward minimization of the energy (17.1) yields the anisotropic

London equation

B + A 2V x [m- (V x B)] =O . (17.2)

Takinginto account the boundary condition, the local magnetic field around a single flux line

directedalong z and carrying a flux quantum C P o is determined by

B+ A 2V x (m V x B)= C P o z b ( x ) b ( y ) . (17.3)

In the isotropic case where mij = bij, Eq. (17.3) coincides with the usual London equation (2.14).

Todealwith an arbitrarily directed flux line, we define two reference frames as shown in Fig. 17.1.

Thevortex frame (x , y , z) is obtained from the crystal frame (x o, Y o, 1 .0) by rotating by an angle e

fromZoabout Y o axis. Restricting ourselves to the case of uniaxial symmetry, the components mij'

transformedto the vortex frame, are

(

ml COS2 e + m3 sin? () 0

m(8) = 0 1111

(m1 - 1113)sincos e 0

(17.4)

The algebra is greatly simplified by working in the vortex frame, since both Band j are

independentof z. Substituting (17.4) into Eq. (17.3) results in a set of equations for B x, B y , and B ;

Theseequations lead to a more complicated vortex structure in the anisotropic superconductor

thanthe isotropic Abrikosov vortex. For example, transverse fields, B x•y , are present even when




Figure 17.1 In the crystal lrame (xo , Y o , 7 .0) the plane xoYo coincides with the basal

crystal plane. The vortex frame (x, y , z) is obtained lrorn the crystal frame by rotating by

an angle 0 from 1 . 0 about the Y o axis. The axis 1 . is parallel to the vortex axis.

the applied field is along the z direction. For simplicity, we consider two special cases in which the

magnetic field is applied perpendicular to or parallel to the layer plane.

F or the perpendicular field case, ()=O. The mass tensor is diagonal;

mxx = myy = m1,H1zz = H13• Eq. (17.3) then becomes

2 ( a 2

B z a 2

B z ) .B , -}. ml- ..~2 + ml- ..~2 =<Po()(x)6(y).

A X o y(17.5)

This equation is equivalent to Eq. (7.S) and the lower critical field is given by (S.4b) as

< P o . ( A )clj_ ::::::-, tn -;:-.

4nJ~ C ;

When the applied magnetic field is parallel to the layer plane, e =n12 . The mass tensor is

(l7.6)

also diagonal: mxx = 1113, myy = mz z = mi' Eq. (17.3) now reads

(17.7a)

LC.;

(17.7b)

By interchanging Zo <-> x , this equation can be transformed back to the crystal frame with B along

the X o axis:

( 1 7 . 8 )

Eq. (17.S) can be solved by a Fourier transform method. Alternatively, we may map Eq, (17.8) to

the isotropic case by defining

(17.9)

From Sec. 7 we know that the solution of Eq. (17.S) at large distances is given by

(17.10)

where Kn(f j ) is a modified Bessel function of the second kind of order n . The free energy per unit



Vortex structure inlayered superconductors 85

length of the flux line, Ep can be calculated from Eq. (17.1). To logarithmic accuracy it is given by

(17.11)

Note for this case the normal core radius ~ j _ serves as the cut-off scale. From the usual

thermodynamic relation E1 =CPoHcl/4n, the parallel lower critical field is

(17.12)

Comparing Eqs. (17.6) and (17.12), we see that important information about the anisotropy

can be obtained by measuring H c 1 in both parallel and perpendicular fields.

17.2 Lawrence-Doniach model

InSec. 16 we discussed the upper critical fields for layered superconductors which can be modeled

asJosephson lattices in 10.We saw that whenthe temperature is lowered to a value T*such that

(,{'P) = s/J2, a 30-20 crossover occurs. Above T*, the layered superconductor behaves as an

anisotropic 30 superconductor. At temperatures lower than 1'*, the barriers of the layered

structure dominate, resulting in a 20 behavior of the temperature dependence of He211 (T). In this

subsection we will show that below 1'* the vortex structure in layered superconductors is also

drastically modified.

Wc will start with Eqs. (16.4) and (16.5) which were derived from the Lawrence-Doniach

freeenergy functional (16.2). For high tc supercond uctors in a weak magnetic field, the influence of

thefield on the value of 1 1 / 1 n I can be neglected, and we can regard 1 1 / 1 " I as a cons tan t; i.e., we use the

London approximation. We confine ourselves to the parallel field case. We take the field along

t h e x direction and choose a gauge such that A z = O . Writing 1 / 1 " = I V I " I ei<D",rom Eqs. (16.4) and

(16.5)we obtain

4nA~. CP oA j _ = - --h + - V < P n

c 2n(17.13)

and

(17.14)

where)'L is the bulk London penetration depth of the superconducting layers, < P n is the phase in

thenth superconducting layer and i;~e*h 1 1 / 1 1 2 /m;s is the maximum Josephson supercurrent

density.Note that in this model it is assumed, for simplicity, that each superconducting layer is

isotropic with an intrinsic bulk penetration depth AL• We use the rectangular contour C shown in

Fig.1 7 . 2 to compute the phase difference across one unit cell: < P = = < D " + 1 - ( D n . From E q . (17.13)

wehave




z

Figure 17.2 Schematic of a Josephson-coupled superconductor insulator superlattice.

The insulating layers of thickness d , alternate with superconducting layers

(crosshatched) of thickness d.. The modulation wavelength s=d, + d.. The rectangular

contour C is used to compute the magnetic flux in Eq. (17.15).

(17.I~

fy + "y f (n + 1), ~

dy' dz'B(y',z ')= "Aj_·dly ns (

C P o A

+ 2n [<D (y ) - ( D ( y + tly)].

The left-hand side in the above equation can be approximated as sLiyB(y,z) , and we obtain

41TA£ C P osLiyB (y , z ) ~ -- (j y n+ 1- i.n)Liy + - [<D (y ) - ( D ( y + Liy)] .

c' jO 2n

Hence

where we have used the following approximations: o jjoz ~ Uy ,n+ 1- .iy,n)!s a n d

c<D jD y ~ [<D (y + L 1 y ) - cD(y)] /L iy, For small < D the Josephson current relation (17.14) may b e

approximated as j, ~j m c D and substituting this form into Eq. (17.15) we obtain

4nA£ ojv 4n).; ojz- ~ - - ~ = B(y,z) ,

C o z C oy(17.1~

superconductivity by kitterson & song

Documents