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Thermal Conductivity of Single Crystals of Yttrium-Based High Temperature

Superconductors

Vaughan William Wittorff

St John’s College

A dissertation submitted to the

University of Cambridge

for the degree of Doctor of Philosophy

To Zig and John, who never asked me whether I am a Physicist or an Engineer,

and saw the benefits of being both.

Acknowledgements

There are a number of people I would like to thank. I am indebted to Dr John Cooper,

my supervisor, for his high expectations, high standards and the criticism he offered,

as well as his later encouragement. It is rare and fortunate for a research student to

find his supervisor in and around the lab with him, most hours of the day and night,

seven days a week. His practical expertise and theoretical familiarity were always

nearby, and snippets given usually just when required. I was also fortunate to work

closely during my first year with Dr (now Professor) Nigel Hussey, when he was a

post-doc in the group, and he taught me the ropes in the lab, and accelerated the early

development of the experimental rig.

By the time I have come to write these acknowledgements, the IRC in

Superconductivity no longer exists in its former state, and all of my contemporaries

have graduated and gone their separate ways, but for their parallel pursuits of

knowledge, camaraderie, advice, ideas and friendship I would especially like to thank

Drs David Broun, Alex Moya, Ian Fisher, Christos Panagopolous, James McCrone,

and from the non-IRC crowd, Drs Christoph Bergemann and Prabhat Agarwal.

I am grateful for financial support from the University of Western Australia (through

a Hackett Studentship), the UK Government (through an Overseas Research Student

Award), the Interdisciplinary Research Centre in Superconductivity, the Cavendish

Laboratory, the Cambridge Philosophical Society and St John’s College.

I would like to thank my parents, brother and sister-in-law for their interest and

encouragement, even if for most of them this has for some years been reduced to:

“When are you submitting?” Thanks to Béatrice for her support and patience during

what were difficult times for both of us. Thanks to Aleida. Thanks to Eva.

I declare that this dissertation is the result of my own work and includes nothing

which is the outcome of work done in collaboration except where specifically

indicated in the text. I have not submitted any part of it to satisfy the degree

requirement at this or any other university.

Vaughan Wittorff

V. W. Wittorff iv

Foreword

The production of this dissertation and therefore conclusion of my PhD had to wait

until I came to certain realizations – not about superconductivity or thermal

conduction – but about the nature of academic research and the ability of a document

to impart understanding.

I began my time as a research student with the apparently naïve expectation that as a

result of my project the understanding of thermal conduction in the cuprates would be

complete and consequently a microscopic theory of high-temperature

superconductivity substantially closer. Studying at the Cavendish Laboratory with its

history of delivering such advancements in understanding did nothing to check my

optimism, although it quickly became clear to me from the calibre of others there just

how far I had to go to achieve anything close. This was a good thing.

Once I had realized that my contribution would necessarily be an incremental

contribution to a massive body of knowledge, and that it would be a piece in the

puzzle rather than a stand-alone epic, it became a matter of deciding how large that

increment would be. Or rather, it became a matter of my supervisor convincing me

that my increment was already large enough. I hope he does not mind me saying so.

The field of high-temperature superconductivity is still one in which the experiment

often leads the theory. In such a field it is easy when performing complicated, long

and difficult experiments, when the data are always controversial and open to various

interpretations, to concentrate on the making sure of the samples, experiment and

data. Sooner or later it becomes necessary to delve more deeply into the theory, to be

sure that one’s initial interpretations of (expectations for?) the results, cannot be easily

explained away by a critic, or have an alternative and simple explanation.

This lead me on a quest for complete understanding of superconductivity, so that I

would be able to honestly present myself as an expert at my oral examination. I also

thought that my dissertation had to reflect this complete understanding of all the

issues. The more I learnt, the more I discovered I didn’t know. I now believe that a

Foreword

v

PhD student is not ready to submit until they realize that they can only ever have

properly come to terms with a small fraction of the issues in their field of study, and

their examiners – who have been thinking about the issues for much longer – will

always be able to ask them about things they haven’t thought about. But the

examiners know this, and so have realistic and reasonable expectations.

Even so, I did a lot of experiments and substantial development of experimental

technique. For quite some time after I completed my experiments, I felt that it was

my duty to record everything that I had achieved – in terms of both experimental

technique and interpretation of my results. I thought it needed to be a complete

record, so that every one of my little breakthroughs were documented, without it

being tedious or verbose. I thought I needed to document the evolution of my

understanding that came from every stage of experimentation, so that even the most

malicious cynic would be convinced of everything I had to say without needing to

question me further. This lead me of course back to a quest for deeper understanding.

I also thought that I needed to draw out in a clear way all of the essential points of the

arguments, without omitting the details.

This made the writing of my dissertation seem an onerous task indeed. Eventually I

have come to the pragmatic decision that my dissertation can never accurately

document everything that I have achieved, and doesn’t need to. I think without

having taken this decision I would never have finished writing. I think now that,

while everything I have discovered seems worthy of publication, during my life I will

only ever publish a small part of what I come up with, and that this is the same for

everyone. (Hopefully I will choose wisely which parts!) Therefore it will be rare for

what I publish to be anything close to self-contained, and it will be a snapshot of the

research process. I will be relying on my audience to already understand much of

what I have discovered for myself but haven’t got time to explain (or at least believe

me in the absence of the explanation).

One reason that it took me so long to take this attitude is because I used to believe that

research writing was meant to enable an intelligent and infallible computer-controlled

robot, once having read the paper and background literature, to repeat the same

experiments, collect the same data and come to the same conclusions. I am now

Foreword

vi

happy to give up this view, relying on Gödel’s theorem of 1930 as justification. In the

words of Roger Penrose1:

“What [Gödel’s theorem] does tell us can be viewed in a … positive light,

namely that the insights that are available to human mathematicians – indeed,

to anyone who can think logically with understanding and imagination – lie

beyond anything that can be formalized as a set of rules. Rules can sometimes

be a partial substitute for understanding, but they can never replace it

entirely.”

V. W. Wittorff

May 2002

1 R Penrose, Shadows of the Mind, Vintage Random House, London, 1994, §2.4, p. 72.

Thermal Conductivity of Single Crystals of Yttrium-Based High Temperature Superconductors

Vaughan W. Wittorff

This dissertation largely concerns the measurement of the thermal conductivity of

single crystals of twinned and detwinned YBa2Cu3O7-! (Y-123) and

YBa2(ZnxCu1-x)3O7-! (Zn-doped Y-123), and untwinned YBa2Cu4O8 (Y-124), and the

interpretation of those results. These materials are high-temperature superconductors,

and their thermal conductivity in both the normal and superconducting states was

measured using a modified steady state method which accounts for radiation losses, in

the temperature range 4-300K. Various aspects of this method and the data analysis

are novel.

Thermal conductivity has a unique potential to be an informative probe of the normal

excitations in the large range of temperature below Tc. In addition its ability to probe

the phonon heat current might prove important in understanding superconductivity in

the cuprates. However it is precisely the fact that neither the electronic nor phononic

component dominates, which hampers analysis at all but the lowest temperatures.

Twinning in crystals imposes addition limits to analysis. All of these limitations are

identified and quantified.

Analysis of the measured data for twinned and detwinned Zn-doped Y-123 crystals

shows that most of the enhancement of the thermal conductivity that occurs below Tc

in clean materials is due to the electronic component, which is strongly suppressed by

Zn. This suppression uncovers a smaller secondary peak at 28K which is far more

resilient to Zn-doping, and which we conclude to be due to phonons. There is

superimposed an additional enhancement below 55K in the b-direction only, which is

a chain phenomenon, and as expected is relatively unaffected by the Zn content of the

crystals in comparison to the effect of Zn on the main (electronic) peak.

Measured data for untwinned Y-124 show a huge peak at 18K in

!

"a and at 22K in

!

"b.

Although there is large ab-plane anisotropy, the phenomenon responsible for these

large peaks is apparently not related to the chains, since peaks exist in both directions.

We have shown that the peaks are strongly suppressed by an applied magnetic field,

however this does not help to identify their origin (since there is reason to believe that

both electronic and phononic contributions are field-dependent). Analysis of the

measured data suggests that the peaks in Y-124 are due to a combination of both

electronic and phononic contributions, much like for Y-123, except that the relative

sizes of the two peaks are reversed, with the phonon contribution dominating for Y-

124. The electronic peak in Y-124 is far smaller than, and impossible to distinguish

from the main phonon peak, however it is possibly responsible for most or all of the

break in slope at Tc.

viii

Priority Dates

This dissertation was written over several years. So as to prevent questions about

priority or originality of ideas, given below are the dates on which certain Chapters

and Sections were completed in close to their final form, and disclosed to others.

Chapter 1: 5 June 2002

Chapter 2: 16 November 2001

Section 4.6: 11 June 2004

Chapter 5: 25 May 2003

Chapter 6 summary: 18 June 2002

Chapter 6: 29 September 2003

Chapter 7: 13 February 2003

Chapter 8: 29 October 2003

Chapter 9: 23 March 2004

Section 10.1: 23 March 2004

Section 10.2: 13 February 2003

V. W. Wittorff ix

Table of Contents

Chapter 1 : Introduction 11.1 Superconductivity...............................................................................................1

1.2 Superconductors in a magnetic field ..................................................................3

1.2.1 Type I superconductors ..............................................................................4

1.2.2 Type II superconductors.............................................................................7

1.3 BCS theory .......................................................................................................11

1.3.1 Pairing of electrons...................................................................................11

1.3.2 Excitations................................................................................................16

1.4 The cuprates .....................................................................................................18

1.4.1 High-temperature superconductors ..........................................................18

1.4.2 Unconventional pairing ............................................................................20

1.4.3 Modifications to the BCS theory and alternative theories .......................21

1.5 References ........................................................................................................25

Chapter 2 : Formal Transport Theory – the Wiedemann-Franz Law 272.1 Historical background ......................................................................................27

2.2 The Boltzmann Equation..................................................................................28

2.3 The macroscopic transport coefficients............................................................31

2.4 The Lorenz number ..........................................................................................33

2.5 The Lorenz number for semiconductors ..........................................................37

2.6 The Lorenz number for metals and cuprates ....................................................38

2.7 Deviations from the Wiedemann-Franz Law (small angle scattering).............39

2.8 References ........................................................................................................43

Chapter 3 : Sample preparation 443.1 Crystal structure and material properties ........................................................ 44

3.1.1 Superconductivity in the perovskite structure......................................... 44

3.1.2 YBa2Cu3O7-δ (Y-123).............................................................................. 44

3.1.3 YBa2Cu4O8 (Y-124) ................................................................................ 46

3.2 Thermo-mechanical detwinning...................................................................... 47

3.3 Solvents ........................................................................................................... 50

Table of Contents

x

3.4 Chemical etching............................................................................................. 52

3.5 Silver epoxy contacts ...................................................................................... 54

3.5.1 Dupont 6838 conductive epoxy............................................................... 55

3.5.2 Dupont 4929 conductive epoxy............................................................... 56

3.6 References ....................................................................................................... 58

Chapter 4 : The measurement of thermal conductivity 594.1 Thermal conductivity – an informative probe of normal excitations.............. 59

4.2 Basics of the measurement .............................................................................. 59

4.3 The three-thermocouple technique.................................................................. 60

4.3.1 Apparatus ................................................................................................ 60

4.3.2 Fabrication and preparation of differential thermocouples ..................... 64

4.3.3 Thermocouple physics............................................................................. 67

4.3.4 Contact resistance considerations............................................................ 68

4.3.5 Influence of the thermopower of the sample on the measurement ......... 70

4.3.5.1 Peltier and Thomson heating............................................................... 73

4.3.5.2 Experimental consequences ................................................................ 74

4.3.6 Details of analysis ................................................................................... 79

4.4 Radiation losses............................................................................................... 83

4.5 Alternative techniques..................................................................................... 85

4.5.1 Techniques with suspended heater(s)...................................................... 85

4.5.1.1 One thermocouple technique............................................................... 86

4.5.1.2 One heater, two thermometers ............................................................ 87

4.5.1.3 Two heaters, one thermometer ............................................................ 88

4.5.1.4 Comparative dynamic method using a reference ................................ 89

4.5.2 Differential method with null thermocouples ......................................... 90

4.5.3 Measurement of thermal diffusivity........................................................ 91

4.6 Montgomery’s method for measuring anisotropic electrical resistivity.......... 92

4.6.1 Consequences of the contacts being of finite size................................... 95

4.6.2 Details of use of Montgomery’s method on a detwinned

YBa2(Cu0.992Zn0.008)3O6.98 single crystal ............................................................. 96

4.7 References ..................................................................................................... 100

Table of Contents

xi

Chapter 5 : General expressions for ρρρρ and κκκκ of twinned crystals 1015.1 Introduction ....................................................................................................101

5.2 Twinned crystal ..............................................................................................103

5.3 Joule heating and its analogue........................................................................108

5.4 Use of the expressions with measured data....................................................109

5.5 The case of isotropic domains ........................................................................111

5.6 The case of average current density oblique to crystallographic axes ...........112

5.7 The case of an “unevenly” twinned crystal ....................................................114

5.8 The case of extreme intra-domain anisotropy ................................................115

5.9 Edge effects ....................................................................................................117

5.10 References ......................................................................................................118

Chapter 6 : Measurements of a series of twinned YBa2(ZnxCu1-x)3O6.98 singlecrystals 119

6.1 Thermal conductivity results ......................................................................... 119

6.2 Relationship between contributions to thermal conductivity of twinned and

untwinned crystals................................................................................................. 120

6.3 Of what use, if any, is the “separation” into components of the thermal

conductivity of a twinned crystal? ........................................................................ 123

6.4 Normal state .................................................................................................. 124

6.4.1 Scenario A: All crystals are of oxygenation 6.98; plρ obeys

Matthiessen’s Rule; chρ unaffected by Zn ....................................................... 126

6.4.2 “Matthiessen’s Rule” for abρ ................................................................ 133

6.4.3 Scenario B: All crystals are of oxygenation 6.98; both plρ and chρ obey

Matthiessen’s Rule ( chρ less sensitive to Zn)................................................... 135

6.4.4 Scenario C: The oxygenations of the samples are different, with more Zn

corresponding to lower oxygenation ................................................................. 139

6.5 Superconducting state ................................................................................... 144

6.6 References ..................................................................................................... 146

Chapter 7 : Zn-doped detwinned Y-123 1477.1 Normal state thermal conductivity ................................................................ 147

7.2 Anisotropic electrical resistivity and application of the Wiedemann-Franz Law

....................................................................................................................... 149

Table of Contents

xii

7.3 Superconducting state thermal conductivity ................................................. 156

7.3.1 Comparison of the detwinned data with twinned data for each crystal 156

7.3.2 Analysis of the difference in the detwinned data of the crystals........... 161

7.3.3 Cause of the sudden enhancement in chκ at Tonset................................. 167

7.4 References ..................................................................................................... 171

Chapter 8 : Separation of electron and phonon contributions to the thermalconductivity 172

8.1 Low temperature residual electronic thermal conduction in d-wave

superconductors..................................................................................................... 172

8.2 Use of the Wiedemann-Franz Law at elevated temperatures........................ 175

8.3 Use of magnetic fields to separate components at elevated temperatures .... 177

8.3.1 à la Zeini et al. ...................................................................................... 180

8.3.2 The theory of Franz and the results of Krishana et al. .......................... 186

8.4 References ..................................................................................................... 189

Chapter 9 : Untwinned YBa2Cu4O8 single crystals 1919.1 The advantages of Y-124 ...............................................................................191

9.2 Electrical resisitvity of Y-124 single crystals ................................................191

9.3 The anisotropic in-plane thermal conductivity of Y-124 ...............................194

9.3.1 Comparison of thermal conductivity and microwave conductivity data......

................................................................................................................196

9.4 The anisotropy of the change in slope of κ at Tc ............................................198

9.5 Normal-state W-F analysis and in-plane anisotropy of phκ ..........................201

9.5.1 Fits to ( )Tphκ and the value of the Lorenz number...............................204

9.5.2 Extrapolations of fits to phaκ into the superconducting state .................208

9.5.3 Anisotropy of the in-plane phκ ..............................................................210

9.5.4 The possibility of a temperature-dependent Lorenz number .................211

9.6 Magnetic field dependence of the thermal conductivity in the superconducting

state ........................................................................................................................211

9.7 References ......................................................................................................214

Table of Contents

xiii

Chapter 10 : Conclusions & Future Work 21510.1 On the temperature dependence and the magnetic field dependence of the

thermal conductivity...............................................................................................215

10.2 Y-124 as a candidate for the observation of quantum oscillations ............218

10.3 Thermal conductivity of Y-123 as a function of hole doping – influence of

the pseudogap.........................................................................................................220

10.4 Thermal conductivity in a magnetic field ..................................................221

10.5 References ..................................................................................................222

Appendix A : Calculation of the temperature profile in a thermal conductorsubject to radiation losses 224

Appendix B.1 :Code for running of experimental rig and calculating results 228

Appendix B.2 :Montgomery Numerical Analysis Mathematica Code 238

V. W. Wittorff 1

Chapter 1 : Introduction

1.1 Superconductivity

Superconductivity, along with superfluidity, is one of a few direct macroscopic

manifestations of quantum mechanics. A quantum mechanical wavefunction

!

" x( ) is

usually microscopic in extent, describing the behaviour of a single particle.

Macroscopic phenomena are usually the statistical superposition of the behaviour of a

huge ensemble of such interacting particles, and classical concepts such as dissipation,

resistance, decay and relaxation are often the result.

However the phenomenon of superconductivity defies such concepts, since a single

wavefunction extends across the entire superconducting region, and simultaneously

describes the behaviour of a large number of (paired) charge carriers. These carriers

can no longer be treated individually, and without altering the state of all such carriers

in this single ground state, nothing can be done to alter the quantum parameters of any

of them. The huge numbers of particles involved in macroscopic samples makes this

so improbable that under many circumstances we expect absolutely no change to

occur in times less than

!

101010

years.1 The phenomena of superconductivity and

superfluidity are the only known examples of the motion of systems of macroscopic

size that are not quickly destroyed by dissipative processes.2 In other words, such

systems are the only kinds to exhibit perpetual motion.

One of the properties of a superconductor then is that it exhibits perfect conductivity.

This property is the prerequisite for many potential applications, such as high-current

transmission lines, dissipationless energy storage systems, or high-field magnets.

Below a material-dependent temperature, known as the critical temperature Tc, a

superconductor loses all electrical resistance, and sustains currents without any

driving force. This was first observed in 1911 by H. Kamerlingh Onnes3 in Leiden,

where he observed an abrupt disappearance of the electrical resistance of pure

mercury below 4.15K. The effect was still observed when he deliberately increased

the scattering by adding impurities.4 These low temperatures were made possible

using the refrigerant liquid helium, which Onnes had first produced in 1908.5 Since

then about half of the elemental metals have been found to superconduct, with the


2

element with the highest critical temperature at standard pressure being niobium with

a Tc of only 9.25K.

It is perhaps counter-intuitive that the elemental metals with the highest values for Tc

are generally the poorest electrical conductors; whereas the noble metals silver,

copper and gold, which are the best electrical conductors, do not exhibit

superconductivity. However as explained in §1.3 this is a simple consequence of the

size of the electron-phonon coupling constant g, since the room temperature resistivity

is determined by electron-phonon scattering.6 The noble metals are good conductors

at room temperature, so that g is small, hence we would expect from BCS theory that

the Coulomb repulsion would dominate the phonon-exchange interaction, and that

superconductivity would not occur.

The second hallmark of superconductivity is perfect diamagnetism, discovered in

1933 by Meissner and Ochsenfeld.7 They found that surface currents occur

spontaneously to exclude an external magnetic field from entering the bulk of a

superconductor (see Figure 1.1). This might appear to be explained by perfect

conductivity, but they also found that surface currents occur spontaneously to expel

the field in an originally normal sample as it is cooled through Tc. This latter effect is

known as the Meissner effect and cannot be explained by perfect conductivity. A

simple perfect conductor would instead trap any flux that existed within its bulk at the

time the resistance dropped to zero, and if the external field changed thereafter would

produce surface currents to maintain that internal flux.

Figure 1.1: Schematic diagram of exclusion of magnetic flux from interior of massive superconductor.

! is the penetration depth, typically only 500Å. (From Ref. 8.)


3

This property of perfect diamagnetism means that a superconductor will repel either

pole of a magnet, and makes possible applications involving stable levitation in a

magnetic field.

1.2 Superconductors in a magnetic field

Both of these fundamental properties of perfect conductivity and perfect

diamagnetism have their limits. A superconductor can only support lossless electrical

conduction up to a certain maximum current density, known as the critical current

density Jc, and only subject to magnetic fields less than a certain maximum field

intensity, known as the critical field Hc. For currents and fields higher than these

limits, superconductivity is at least partly destroyed and the material ceases to be a

perfect diamagnet and perfect conductor. Both Jc and Hc vary with temperature,

decreasing with increasing temperature, and approach zero at the critical temperature

Tc (see Figure 1.2).

Figure 1.2: Temperature dependence of the critical field. (From Ref. 9.)

The penetration of the magnetic flux into the interior of the superconductor can be

either relatively catastrophic or gradual with increasing field. The nature of the flux

penetration is determined by whether a superconducting-normal boundary has a

positive or negative free energy. If the boundary free energy is positive, the system

tries to minimize the number of boundaries, and the flux penetrates relatively

catastrophically; and whenever there is a mixture of superconducting and normal

domains those domains are relatively large. If the boundary free energy is negative,

on the other hand, the system tries to maximize the number of boundaries by

dispersing the flux as finely as possible, and penetration is gradual with increasing

field.

Whether the boundary free energy is positive or negative is related directly to the

phenomenological parameter

!

" = # $ , where ! is the penetration depth, being the


4

characteristic range of penetration of magnetic fields into the superconductor, and

where ! is the Ginzberg-Landau coherence length*, being a characteristic distance

over which spatial changes in " occur.10 Figure 1.3 shows spatial variation of the

wavefunction " and magnetic flux density B near a boundary, for large #

(corresponding to a negative boundary free energy) and for small # (corresponding to

a positive boundary free energy).

Figure 1.3: Cross-sections through a superconducting-normal boundary for large and small #. (From

Ref. 11.)

1.2.1 Type I superconductors

In a so-called Type I superconducting material, a superconducting-normal boundary

has a positive free energy (which is the case for # < 1/!2), and so a stable system

configuration is one where the total area of boundaries is minimized.

For the case where every part of the surface of a Type I superconductor is subject to

the same value of external magnetic field, the field is excluded until it reaches the

critical field value Hc, above which superconductivity can no longer be supported.

Increasing the field just above Hc causes a catastrophic penetration of the magnetic

flux throughout the bulk, destroying superconductivity completely. The sample is

either superconducting everywhere (the Meissner state) or normal everywhere, so

there is never a superconducting-normal boundary.

* In a clean superconductor, the G-L coherence length is related to the BCS coherence length !0 by

!

" T( ) #"0Tc

2 Tc$T( )

, where !0 is the characteristic size of the pair wavefunction.


5

As noted, the behaviour just described requires that every part of the surface be

subject to the same value of external magnetic field. For a uniform applied magnetic

field this requires that the superconductor be in the shape of a long thin rod, aligned

parallel to the field (see Figure 1.4a). For other geometries, exclusion of flux from the

interior of the superconductor requires that the field lines be deformed in the region of

the superconductor in order to bend around it. This concentrates the external

magnetic field adjacent to some parts of the surface, so that it exceeds the ambient

field value, and this effect is most pronounced at the extremal points of the surface in

directions normal to the ambient applied field.

For example, the magnetic field intensity at the equator of a superconducting sphere

in the Meissner state is a factor of 1.5 higher than the ambient field (see Figure 1.4b).

Flux will begin to penetrate the sphere when the value of the field at the equator

reaches Hc, which only requires the applied magnetic field to be two-thirds of Hc. At

this point normal regions invade the sphere, not, as one might have expected, as a thin

equatorial belt of normal material (which can be shown to be unstable) but as a

uniform intermediate state which fills the whole sphere.12

Figure 1.4: Contrast of exterior-field pattern (a) when demagnetizing coefficient is nearly zero and (b)

when it is

!

1

3 for a sphere. (From Ref. 13.)

For other ellipsoidal shapes (for which alone a demagnetizing coefficient is well-

defined14), this geometric effect is formalized by the concept of the demagnetizing

coefficient !, where in gaussian cgs units:

!

H = B "#4$M (1.1)

For geometries having non-zero demagnetizing coefficients, there will be a range of

field below Hc where normal and superconducting laminae coexist throughout the

bulk of the superconductor (with a relatively large spatial period of typically a few

tenths of a millimetre15): this is the so-called intermediate state. The range of applied

fields for which we expect an intermediate state is


6

!

1"# <H

Hc

<1 (1.2)

Hence the two extremes are: (i) a long rod aligned parallel to the field (having ! = 0)

where there is no intermediate state except at the field Hc at which superconductivity

disappears; and (ii) an infinite flat slab in a perpendicular field (having ! = 1) in

which an intermediate state will always exist when superconductivity is present.

Figure 1.5 shows magnetization curves, as a function of applied field H, for a long rod

and a sphere of a Type I superconductor. The curve for the sphere is obtained by

shifting the curve for a long rod horizontally by

!

"4#M . The shaded area is the

difference in the Helmholtz free energy densities of the superconducting and normal

states (which is the condensation energy

!

Hc

28" ), and is not affected by this shift, so

is independent of ! and therefore independent of the shape of the sample.

Figure 1.5: Ideal reversible magnetization curves as a function of applied field H for a long rod (! = 0)

and a sphere (! =

!

1

3) of a Type I superconductor. (Modified from Ref. 16.)

In a long Type I superconducting wire of circular cross-section with radius

!

a >> " , Jc

is the transport current density at which the self-field at the surface

!

H = 2I ca reaches

a value of Hc, and above which the wire develops a resistance. This is the so-called

Silsbee criterion. Application of the London and Maxwell equations in this geometry

shows that the current flows within a depth " of the surface.17 The cross-sectional

area of this surface layer is

!

2"a# , and so the critical current density Jc in this layer

where the current flows is

!

Jc

=Ic

2"a#=c

4"

Hc

# (1.3)


7

It can be shown using a different argument17 that this value of Jc also holds for wires

that are much thinner than !, where the current density is nearly uniform.

1.2.2 Type II superconductors

In a so-called Type II superconducting material, a superconducting-normal boundary

has a negative free energy (which is the case for " > 1/!2), and so a stable system

configuration is one where the total area of boundaries is maximized, meaning that

any flux that penetrates is dispersed as finely as possible.

In Type II materials, the penetration of the magnetic flux into the interior of the

superconductor is gradual with increasing field. A lattice of non-superconducting

regions begins to develop at a lower critical field Hc1 and grows in density with

increasing field, but the matrix of still-superconducting material can support lossless

dc conduction until all superconductivity is destroyed at a much higher upper critical

field Hc2. This coexistence of finely dispersed normal regions and a superconducting

matrix that continues to support dissipationless transport is known as the mixed state,

or the Schubnikov phase.

Flux penetration in Type I and in Type II superconductors with the same

thermodynamic critical field Hc is compared in Figure 1.6. The thermodynamic

critical field Hc is not itself identified with any critical behaviour in a Type II

superconductor, and is simply defined with respect to the condensation energy

!

Hc

28" .

Figure 1.6: Comparison of flux penetration behaviour of Type I and Type II superconductors with the

same thermodynamic critical field Hc. (From Ref. 18.)

It was shown by Abrikosov19 that in the mixed state between Hc1 and Hc2, flux

penetrates in a regular array of flux tubes, each carrying a quantum of flux


8

!

"0

=hc

2e= 2.07 #10

$7Gcm

2= 2 2%&'H

c (1.4)

where the final expression comes from the derivation of the Ginzburg-Landau

coherence length ! (Ref. 20). Within each unit cell of the array, there is a vortex of

supercurrent concentrating the flux towards the vortex centre, wherein the core is

normal (non-superconducting). The lower critical field Hc1 is therefore the field at

which it becomes energetically favourable for the first vortex to enter the sample. By

equating at a field of Hc1 the Gibbs free energy for the Meissner state, with that of a

single vortex in the sample, we find21,22

!

Hc1

=4"#

1

$0

(1.5)

where23

!

"1

=#0

8$h %( ) (1.6)

is the extra free energy per unit length of the vortex line, and

!

h r( ) "#

0

2$%2ln%

rr << %, and & >>1 (1.7)

is the local value of the magnetic flux density a radial distance r from the centre of the

vortex core. Substituting Equations 1.6, 1.7 and 1.4 into 1.5 gives

!

Hc1"Hc

2#ln# # >>1 (1.8)

The upper critical field Hc2 is the highest field at which superconductivity can

nucleate in the interior of a large sample in a decreasing external field. An expression

for Hc2 is determined by solving the linearized Ginzburg-Landau equation.24,25 The

solutions are the Landau orbitals, which correspond to charged particles spiralling

around the magnetic field, with levels separated by the cyclotron energy

!

h"c

= 2ehH mc . Equating these energy levels with the energy eigenvalues of the

linearized GL equation, the highest field for which a solution exists is defined as Hc2

and is given by

!

Hc2

="0

2#$ 2 T( )=4#%2H

c

2

"0

= 2&Hc (1.9)


9

where the second and third equivalent expressions for Hc2 are obtained with the use of

Equation 1.4. We see that for a Type II superconductor Hc1 < Hc < Hc2, and in a

decreasing field such materials become superconducting in a second-order phase

transition at Hc2.

A larger value of ! is advantageous for applications, since it means a persistent

transport current can be supported in the mixed state to higher fields. Since ! is

inversely proportional to the mean-free path for dirty materials, we have the unusual

requirement that to remain superconducting in high magnetic fields, the

superconducting material in magnets needs to have many imperfections.

If we use the crude approximation to Hc1 where the ln ! is omitted,26 we see that

!

HcHc1

= Hc2Hc

= 2" , so that Hc is approximately the geometric mean of Hc1 and

Hc2:

!

Hc" H

c1Hc2

(1.10)

For a Type I superconductor, Hc1 has no meaning, but the field Hc2 remains the

highest field at which superconductivity can nucleate in the interior of a large sample

in a decreasing external field. However since Hc2 < Hc for a Type I superconductor,

this implies that in a decreasing field these materials “supercool”, remaining normal

even below Hc, ideally until Hc2 is reached (in the absence of surface effects).

The qualitative change in shape of the magnetization curve with the value of ! is

sketched in Figure 1.7. Despite these changes in shape, however, the area under the

curve is in all cases given by the condensation energy

!

Hc

28" , equal to the difference

in the Helmholtz free energy densities of the superconducting and normal states.

The above analysis is further complicated when the sample has a non-zero

demagnetizing coefficient " (see §1.2.1). The range of applied fields for which we

expect a mixed state is then

!

1"#( )Hc1< H < H

c2 (1.11)


10

Figure 1.7: Comparison of magnetization curves for three superconductors with the same value of

thermodynamic critical field Hc, but with different values of !. For ! < 1/!2, the superconductor is of

Type I and exhibits a first-order transition at Hc. For ! > 1/!2, the superconductor is Type II and shows

second-order transitions at Hc1 and Hc2 (for clarity, marked only for the highest ! case). In all cases, the area under the curve is the condensation energy

!

Hc

28" . (From Ref. 27.)

Hence the onset of the mixed state occurs at an applied field lower than Hc1 for a

sample with a non-zero demagnetizing coefficient, and in the extreme case of an

infinite flat slab in a perpendicular field (having " = 1), a mixed state will always

exist when superconductivity is present. Figure 1.8 shows magnetization curves, as a

function of applied field H, for a long rod and a sphere of a Type II superconductor.

The curve for the sphere is obtained by shifting the curve for a long rod horizontally

by

!

"4#M . The shaded area is the difference in the Helmholtz free energy densities of

the superconducting and normal states (which is the condensation energy

!

Hc

28" ),

and is not affected by this shift, so is independent of " and therefore independent of

the shape of the sample.

Figure 1.8: Ideal reversible magnetization curves as a function of applied field H for a long rod (" = 0)

and a sphere (" =

!

1

3) of a Type II superconductor. (Modified from Ref. 16.)


11

In §1.2.1 we found that in a Type I superconductor, Jc is the transport current density

at which the self-field at the surface reaches a value of Hc.* For currents higher than

this the magnetic field can no longer be excluded from the interior of the

superconductor and superconductivity breaks down. However a Type II

superconductor can support lossless electrical conduction in the mixed state above a

field of Hc1, and so for this type, Jc can be defined differently. The presence of

magnetic flux and a lattice of non-superconducting regions within a superconductor in

the mixed state do not prevent lossless electrical conduction, provided that the flux

remains at rest. However if the flux moves, dissipation occurs. So for a Type II

superconductor, Jc is the transport current density at which pinning can no longer hold

the flux at rest in the face of the driving force.28 The pinning of the flux is caused by

impurities and structural imperfections in the crystalline structure of the

superconducting material, and so once again we have the unusual requirement, that to

support high current densities, superconducting conductors need to have many

imperfections.

1.3 BCS theory

1.3.1 Pairing of electrons

The macroscopic wavefunction that describes the common ground state comprising

the superconducting electrons is not consistent with the Fermi-Dirac statistics of

single particles. Somehow the Coulomb repulsion between individual electrons must

be overcome so that bound states each comprising two electrons may form, thus

enabling many of these paired Fermions to populate a superconducting ground state

according to Bose-Einstein particle statistics.

Figure 1.9: Phonon exchange between electrons. (From Ref. 29.)

* This argument only holds for a wire of radius

!

a >> " , but it turns out that when the wire is much

thinner than ! the expression for Jc is the same.


12

A possible mechanism for an effective attractive interaction between electrons is one

where one electron polarizes the medium by attracting positive ions (emitting a

phonon); and these excess positive ions in turn attract a second electron (which

thereby absorbs the phonon). This is illustrated in Figure 1.9, and the equivalent

direct interaction for the exchange of one phonon turns out to be6

!

V k, " k ,q( ) =g2h#q

$k +q %$k( )2

% h#q( )2

(1.12)

where, following Figure 1.9, the momenta of the incoming electrons are

!

hk and

!

h " k ,

and the momentum of the exchanged phonon is

!

hq. In Equation 1.12

!

"k is the energy

of the electron state with wave number k,

!

h"q is the phonon energy, and g is the

coupling constant for the interaction between the electrons and the phonons. It is

apparent that superconductivity is less likely for a material with a small electron-

phonon coupling constant g, since the exchange interaction will be smaller and less

likely to overcome the Coulomb repulsion between the two electrons.

It was first suggested by Fröhlich30 in 1950 that the electron-lattice interaction might

explain (conventional) superconductivity in this way. This suggestion was confirmed

experimentally by the discovery31 of the isotope effect, that is, the proportionality of

Tc and Hc to

!

M"0.5 for isotopes of the same element. This dependency on the mass of

the ions in the lattice was seen as direct confirmation that the effective electron-

electron interaction was being mediated by phonons.

The effective interaction between electrons given by Equation 1.12 is attractive for

!

"k+q #"k < h$q . However for a pair of electrons considered in isolation it is unlikely

that this attractive interaction would exceed the regular screened Coulomb repulsion

between the electrons by enough of a margin to form bound pairs, even for a large

coupling constant g. This is because in three dimensions two particles must interact

with a certain minimum strength to form a bound state,32 that is, there is a barrier to

binding which must be overcome by a net attractive interaction.

It was Cooper33 who first considered the effect of the other electrons, which through

Fermi-Dirac statistics and the Pauli exclusion principle, and conservation of


13

momentum, restrict the states from which and into which the two interacting electrons

may scatter to those in the neighbourhood of the Fermi surface. Cooper considered

two electrons added to a Fermi sea (having a spherical surface) at T = 0, with the

stipulation that the extra electrons interact with each other but not with those in the

sea except via the exclusion principle, so that these two extra electrons were forced to

occupy levels outside the Fermi sphere. Cooper showed that for such a system there

was no minimum interaction strength for binding, so that a bound state, known as a

Cooper pair, would form whenever the attractive interaction exceeded the Coulomb

repulsion. With no barrier to pair formation, Cooper showed that pairing as an

explanation of superconductivity is plausible.

To derive the main features of the transition to superconductivity, the subsequent

theory of Bardeen, Cooper and Schrieffer (known as BCS theory)34 used the following

simplified interaction, where the energies are measured with respect to the Fermi

surface:

!

Vk " k

=#V if $

k< h%

D and $ " k

< h%D

0 otherwise

& ' (

(1.13)

and where

!

h"D

is the Debye energy for phonons in the lattice, and

!

Vk " k

is the

interaction between electrons which before scattering have momenta

!

hk and

!

h " k (see

Figure 1.9). Conservation of momentum requires that the centre-of-mass momentum

(call it

!

hK ) is not altered by the phonon exchange. Equation 1.12 gives an attractive

interaction provided

!

"k+q #"k < h$q , so the simplified interaction of Equation 1.13 is

taken as attractive if both

!

"k and

!

" # k are less than an average phonon energy, for

which the Debye energy serves.35 States within this energy of the Fermi surface

comprise a thin shell, and pairs of points in momentum space which lie within the

interaction shell and have centre-of-mass momentum

!

hK are represented by the

shaded region of intersection in Figure 1.10.

Figure 1.10: Construction to find levels within energy shell and with centre-of-mass momentum

!

hK .

(From Ref. 36.)


14

It can be seen that the number of electron levels involved in the interaction increases

as K decreases, and has a sharp maximum for K = 0 when the whole shell is

involved.35 The effects of the interaction are therefore by far strongest between

electrons of opposite momentum, k and –k, and we can restrict our attention to pairs

having this relationship.

Consider then the following two-electron singlet wavefunction, where the electrons

have equal and opposite momenta:

!

"0r1# r

2( ) = gkcos k $ r

1# r

2( )[ ]k>k

F

%&

' ( (

)

* + + ,1-2#-

1,2( ) (1.14)

where the gk are the weighting coefficients, where

!

"1 refers to the “up” spin state of

particle 1, and

!

"1 refers to its “down” state, and where in the summation we have used

Cooper’s original simplified state-space restriction. We are interested in the binding

energy Eb = 2EF – E, and solving the Schrödinger equation with 1.14 yields:

!

" 2#k

+ Eb( )gk

= Vk $ k

g $ k

$ k >kF

%

i.e. gk

k>kF

% = V

g $ k

$ k >kF

%

2#k

+ Eb

&

'

( ( (

)

*

+ + + k>k

F

%

i.e.1

V= 2#

k+ E

b( )"1

k>kF

%

(1.15)

When we replace the summation by an integration, with N(0) denoting the density of

states at the Fermi level for electrons with one spin orientation, this becomes

!

1

V= N 0( )

d"

2" + Eb0

h#D

$ =1

2N 0( ) ln

2h#D + Eb

Eb

i.e. exp2

N 0( )V

%

& '

(

) * =2h#D

Eb

+1

i.e. Eb =2h#D

exp2

N 0( )V

%

& '

(

) * +1

, 2h#D exp +2

N 0( )V

%

& '

(

) *

(1.16)


15

where the final expression makes use of the weak-coupling approximation, valid for

!

N 0( )V <<1. This shows that indeed there is a bound state with respect to the Fermi

surface, for two electrons both outside the Fermi sphere, regardless of how small V is.

Cooper’s argument applies to a single pair of electrons in the presence of a normal

Fermi distribution of additional electrons. The BCS theory took an essential further

step, constructing a ground state in which all electrons form bound pairs. Cooper

pairs in this theory are highly overlapping and therefore highly interacting. They

cannot be considered independently; indeed the binding turns out to be cooperative –

the binding energy of any one pair depends on how many other pairs have condensed,

and, in addition, the external centre of mass motions of all the pairs are coupled

together so that each pair is in exactly the same state.37

The fractional occupation number of paired states at T = 0, denoted

!

vk

2 , is plotted in

Figure 1.11. There is a startling resemblance between

!

vk

2 for the BCS ground state at

T = 0 and the normal-metal Fermi function at T = Tc, also plotted in Figure 1.11 for

comparison. From this comparison we see that, contrary to the early ideas of

Fröhlich, Bardeen, and others, the change in the metal on cooling from Tc to T = 0

cannot be usefully described in terms of changes in the occupation numbers of one-

electron momentum eigenstates. In particular, no gap opens up in k space. Rather,

the disorder associated with partial occupation of these states with random phases is

being replaced by a single quantum state of the system, in which more or less the

same set of many-body states with various one-electron occupancies are now

superposed with a fixed phase relation.38

Figure 1.11: Plot of BCS occupation fraction

!

vk

2 versus electron energy measured from the chemical

potential (Fermi energy). To make the cutoffs at

!

±h"D

visible, the plot has been made for a strong-

coupling superconductor with N(0)V = 0.43. For comparison, the Fermi function for the normal state at Tc is also shown on the same scale using the BCS relation !(0) = 1.76kBTc. (From Ref. 39.)

In the choice of the form of the two-electron wavefunction we made in Equation 1.14,

we anticipated singlet coupling to have lower energy because the cosinusoidal


16

dependence of its orbital wavefunction on

!

r1" r

2( ) gives a larger probability

amplitude for the electrons to be near each other.40 In triplet pairing, on the other

hand, the orbital wavefunction has a sinusoidal dependence on

!

r1" r

2( ) , preventing

the two electrons from existing in close proximity. Whilst possible, triplet pairing

would imply characteristic magnetic properties that are not observed (e.g. in the

Knight shift) in most superconductors.41 However triplet pairing has been observed

unambiguously in very clean Sr2RuO4, and in liquid helium-3, albeit at very low

temperatures, as might be expected. The critical temperatures at atmospheric pressure

for these two systems are about 1.5K for superconductivity in Sr2RuO4,42 and about

1mK for superfluidity in helium-3.43

1.3.2 Excitations

At non-zero temperature, there exists in a superconducting system elementary quasi-

particle excitations having momentum

!

hk , and the energies of these excitations are

given by44

!

Ek

= "k

2+ #

k

2

(1.17)

where

!

"k plays the role of an energy gap or minimum excitation energy since even at

the Fermi surface , where

!

"k

= 0,

!

Ek

= "k

> 0.

The total Cooper pair wavefunction must be antisymmetric under particle exchange;

therefore since singlet pairing provides an antisymmetric spin wavefunction, the

orbital wavefunction must be symmetric (l = 0, 2, …).* Phonon interactions usually

favour s-wave states45 (l = 0) in which the energy gap

!

"k may be taken to be real and

without nodes. The energy gap then has the full crystal symmetry and only relatively

weak anisotropy.

In the BCS theory the zero-temperature energy gap is given by46

!

" 0( ) =h#D

sinh1

N 0( )V

$

% &

'

( )

* 2h#D exp +1

N 0( )V

$

% &

'

( ) (1.18)

* For triplet pairing, which is symmetric under particle exchange, the orbital wavefunction is

antisymmetric and in Sr2RuO4 and liquid helium-3 has been observed to be p-wave (l = 1).


17

where the final expression makes use of the weak-coupling approximation, valid for

!

N 0( )V <<1. A similar formula is predicted for the (zero field) critical temperature:47

!

kBTc =2e

"

#h$D exp %

1

N 0( )V

&

' (

)

* + (1.19)

where ! = 0.577… is Euler’s constant and so the coefficient

!

2e" # $1.13. Taking the

ratio of Equations 1.18 and 1.19 gives a fundamental formula independent of the

phenomenological parameters "D, N(0) and V:

!

" 0( )kBTc

=2

1.13=1.764 (1.20)

which has been found experimentally to be accurate to within 10% for

superconductors for which the weak-coupling approximation holds.

The energy gap is temperature dependent, and decreases monotonically with

increasing temperature, as shown in Figure 1.12. It is insensitive to T at low

temperatures, where the number of thermally excited quasi-particles is insignificant.

However the elementary theory predicts that near the critical temperature (in zero

field) the energy gap vanishes with a vertical tangent according to the universal (mean

field theory) law:48

!

" T( )" 0( )

=1.74 1#T

Tc

$

% &

'

( )

1

2

T * Tc (1.21)

Figure 1.12: Temperature dependence of the energy gap in the BCS theory. Strictly speaking, this

universal curve holds only in a weak-coupling limit, but it is a good approximation in most cases.

(From Ref. 49.)

The physical reason50 for the temperature dependence of ! is that the energy gap

arises from a cooperative smearing of the Fermi surface which allows the electrons to


18

take advantage of the attractive phonon interaction. However, this cooperative

behaviour is undermined by the thermal rounding of the Fermi surface edge,

eventually disappearing at Tc with

!

kBTc~ " 0( ). Secondly, for T > 0 there is some

thermal excitation of quasi-particles, so not all particles are in the ground state.

1.4 The cuprates

1.4.1 High-temperature superconductors

Until 1986 it had been widely believed that superconductivity of the usual type could

not exist at temperatures above about 30K. There was therefore great excitement

when in that year Bednorz and Müller51, working for IBM Zürich in Switzerland

discovered superconductivity in a lanthanum-doped barium cuprate at 36K, and the

following year Wu et al.52 at the Universities of Alabama and Houston found it in a

related oxygen-doped yttrium-barium cuprate at 93K. Since then superconductivity

has been found in a large number of similar cuprate materials at temperatures up to

138K at ambient pressure, and up to 155K under pressure.

The cuprates are so-called because they all contain flat layers of copper and oxygen

atoms, and it is these copper oxide layers that carry the supercurrent.

Superconductors in this class of materials are also known as high-temperature

superconductors (HTS).

Since many of these materials are superconducting above 77K, the boiling point of

liquid nitrogen, these discoveries were widely expected to lead to a great flowering of

applications, and much effort was poured into superconductivity research. This effort

is now bearing fruit in the area of high-power electric cables, transformers, motors

and high-field magnets for energy storage.

In addition to the advantages and savings gained by only requiring liquid nitrogen for

cooling, the higher values of Tc mean higher values of Hc. Furthermore, these

materials are Type II superconductors, with extraordinarily large values of the

parameter ! and therefore Hc2, and so superconductivity is supported at 77K up to

very much higher values of magnetic field than in other classes of superconductors.

Correspondingly high critical current densities are also achievable.


19

There are a variety of cuprate superconductors, and some of the more well-known are

shown in Table 1.1. For HTS electric cables, the material of choice until recently was

Bi-2223. However a major obstacle that prevented the rapid deployment of HTS

electric cables after the discovery of HTS in 1986 was that the cuprate materials are

not ductile like a metal, but instead are brittle ceramics. They cannot be drawn into a

wire. The most popular technique that was developed to produce Bi-based HTS

cables is to pack the Bi-2223 into a silver tube that is then heat-treated, drawn, and

rolled to make a thin ribbon several millimetres wide. These tapes, almost 70%

silver, are then woven around a core to form the cable.

Table 1.1: Formulae, names and critical temperatures of some of the best-known cuprate

superconductors.53,54

Cuprate family Chemical formula Shortened name(s) Critical temp. Tc

La/Sr cuprate La2-xSrxCuO4 LSCO 38K

YBCO YBa2Cu3O7-! YBCO or Y-123 93K or yttrium

YBa2Cu4O8 Y-124 81K

BSCCO Bi2Sr2CaCu2O8+x Bi-2212 94K or bismuth

Bi2Sr2Ca2Cu3O10+x Bi-2223 110K

TBCCO TlBa2Ca2Cu3O9+x Tl-1223 123K or thallium

Tl2Ba2Ca2Cu3O10+x Tl-2223 127K

mercury HgBa2CaCu2O6+x Hg-1212 127K

HgBa2Ca2Cu3O8+x Hg-1223 133K

Hg0.8Tl0.2Ba2Ca2Cu3O8.33 Hg(Tl)-1223 138K

The main drawback of using Bi-2223/Ag tapes is their cost. Consequently, since

1999 various companies and agencies, especially in the US, have been undertaking

projects to develop the second generation of HTS tapes, using YBCO-coated

substrates. These YBCO-coated tapes promise to be much cheaper, with a quoted

target of US$10 per kA per metre, and it is expected that they will be ready for

commercial trials by 2006.

1.4.2 Unconventional pairing

In conventional BCS theory, an isotropic system has !k independent of k, and because

of the spherical symmetry, this is often referred to as s-wave pairing. However, if the

material is anisotropic (weakly, like Sn, or strongly, like YBCO), then one expects


20

that !k will no longer be isotropic, but that its dependence on k will have the same

symmetry as the underlying crystal symmetry. This situation is sometimes referred to

as anisotropic s-wave pairing: It lacks spherical symmetry but has the full symmetry

of the underlying crystal. The term “unconventional pairing” refers to a situation in

which the symmetry of the energy gap !k is lower than that of the underlying

crystal.55

The fact that the Knight shift in NMR measurements in the cuprates is observed56 to

go smoothly to zero at T = 0 implies that the pairing is of the spin singlet form. Since

singlet pairing provides an antisymmetric spin wavefunction, the orbital wavefunction

must be symmetric (l = 0, 2, …). Phonon interactions usually favour s-wave states45

(l = 0), however since 1997 there has been consensus that the pairing in the cuprates is

d-wave (l = 2).

The consensus for d-wave pairing was reached as more and stronger experimental

evidence was accumulated. Initial experimental evidence was indirect, and showed

gaps that were not isotropically devoid of states for quasi-particle excitations. One

interpretation was a d-wave gap, but the difficulty of sample preparation meant that

another likely interpretation was impurity states in the gap. An example of early

evidence was in low temperature specific heat data from Loram et al.57 for YBCO of

various hole dopings, that showed that the entropy does not fall as fast as predicted for

an s-wave BCS superconductor with a clean gap.

Perhaps the first highly convincing evidence comprised measurements showing for

the penetration depth ! of especially high-quality crystals of YBCO that

!

1 "2 (which

is proportional to the superfluid density ns) has a linear temperature dependence at

low temperatures. These measurements implied an anisotropic gap with nodes (d-

wave). Surface and bulk measurements produced highly consistent results: Hardy

and Bonn et al.58 measured microwave surface impedance at "1GHz, and later Sonier

et al.59 measured the penetration depth at 0.5T using the muon spin-rotation

technique, which probes the bulk via penetrating flux lines.

Soon after, more direct evidence came from phase-sensitive measurements, where the

phenomena involved directly reflected the symmetry of the paired state. Evidence


21

came from measurements of the quantization of flux in SQUID rings, first performed

by Wollman and Harlingen et al.60 The possibility of errors from undetected stray

trapped flux was addressed in subsequent experiments by Tsuei et al.61 and by Mathai

et al.62 Later evidence came from measurements by Aubin and Behnia et al.

63 of the

variation of the thermal conductivity of a detwinned YBCO crystal as a function of

the relative orientation of the crystal axes and a magnetic field rotating in the CuO2

planes.

1.4.3 Modifications to the BCS theory and alternative theories

It is evident64 that the cuprates seem to have the usual sort of order parameter

associated with electron pairs, and in particular that the superconducting ground state

is comprised of zero-momentum singlet Cooper pairs (see §1.4.2). This raises the

question of whether a pairing theory of BCS type can fit the data. If this is to be

attempted, there are various modifications of the theory required,65 outlined below:

a) The planar conducting CuO2 structures mean almost two-dimensional conduction

electron states. The form of BCS theory used must reflect this anisotropy.

b) For YBCO the one-dimensional CuO chain conduction electrons may have little

coupling to those on the planes, and the participation of the chain conducting

subsystem to the superconducting condensation might have to be considered

separately.

c) The c-direction localization also leads to a very short coherence length in the c-

direction, of the same order as the unit cell dimension. Therefore it may be

appropriate to treat the layers as two-dimensional superconductors coupled by

weak links described by a Josephson model.

d) The coherence length within the layers is also short, of order 2.5nm, very much

shorter than in conventional superconductors. Therefore thermal fluctuations

become much more important, and local defects and local atomic discreteness

have strong local effects on the order parameter. The superconductor may no

longer be treated as uniform. The effects of grain boundaries, spread over a few

unit cells, are no longer averaged out, but behave like weak links.

e) The chief scattering mechanism for excitations may be electron-electron

scattering, rather than impurity scattering as in conventional superconductors.

Impurity scattering is elastic and so for this scattering mechanism the mean free

path is the same as in the normal state and is independent of temperature. Many


22

calculations based on BCS theory assume that this is the case, and the assumption

might not be valid in the cuprates.

f) Unlike conventional BCS superconductors, in the cuprates there are low-lying

excitations at low temperatures. Consensus is that the cause is an anisotropic

energy gap in the cuprates, having a d-wave symmetry contrary to the isotropic s-

wave symmetry of conventional superconductors.

g) Magnetic order is present. The parent compound is an antiferromagnetic

insulator. As hole doping is increased and conduction occurs, long-range

magnetic order disappears, but short-range magnetic fluctuations persist. The

excitation spectrum will therefore differ from the BCS spectrum. These effects

will be more important in underdoped cuprates where the magnetic ordering is

stronger.

h) The attractive mechanism might not be exchange of phonons as assumed by BCS,

and the BCS weak-coupling regime, for which many of the standard calculations

have been done, might not be applicable.

With reference to the final point (h) above, what is the motivation for considering

other attractive mechanisms? Might not a phonon-mediated interaction in a BCS

strong-coupling regime be able to account for superconductivity in the cuprates, once

modifications to BCS theory are made to account for some or all of points (a)-(g)?

There are at least two arguments against this scenario. Firstly, if the electron-phonon

interaction is strong enough to produce a Tc of the order of 100K, then it should be

strong enough to make the supposed superconducting phase unstable against a

transition to some other crystal structure.66 Secondly is the nature of the isotope effect

in the cuprates.

Initial reports in 1987 were that YBCO lacks an isotope effect, in the sense that

substitution of oxygen-18 for oxygen-16 does not alter Tc (Refs. 67, 68). This was

interpreted by many observers as showing that “BCS theory cannot apply”, despite the

more careful statements of the authors.69 In fact, more precise measurements by the

same groups and others in the months that followed showed a small but measurable

effect of about 0.5K for optimally-doped materials (where optimally-doped means

having a hole doping to maximize Tc).


23

Since then extensive measurements have been taken70 of the isotope effect in the

cuprates, so that well defined results are available across the doping phase diagram (T

versus hole doping p). These show that the superconducting region in the phase

diagram is slightly narrowed while remaining almost exactly the same height. In

other words, in terms of the empirical expression

!

Tc~ M

"# for the isotope effect,

raising the isotopic mass has little effect on Tc for optimally doped material (! is

about 0.1), but in both overdoped and underdoped material Tc falls (! approaches and

in some cases exceeds 0.5). By comparison, ! = 0.5 in a conventional weakly-

coupled BCS superconductor.

This strong dependence on doping cannot be explained by the simple BCS

expectation of Tc scaling with the phonon frequencies (which would scale Tc down

uniformly across the phase diagram); nor by a systematic change in the effective

doping of the CuO2 planes because of the different isotope (which would shift the Tc

curve sideways on the phase diagram).71 While the dynamics of the O atoms are

apparently somehow involved in the superconducting condensation, it is not in the

manner expected in any simple BCS picture, and the effect may be more indirect than

lattice polarization providing the dominant attractive mechanism.

One alternative microscopic mechanism to a phonon-mediated interaction, as

proposed by Pines72, is based on the exchange of antiferromagnetic spin

fluctuations73. This mechanism would lead to

!

dx2"y

2 pairing, compatible with

experimental evidence.

Other theories are framed in the very non-BCS situation where the interaction has

been made strong enough that Fermi-liquid theory has collapsed (non-Fermi-liquid

theories), or is on the point of collapsing (marginal Fermi-liquid theories, e.g. as

proposed by Varma74).

Two other non-BCS theories are the polaron-bipolaron model of Mott and

Alexandrov75, and the resonating valence bond (RVB) model of Anderson.76 In the

polaron-bipolaron model the pairs are small and non-overlapping, so pairing is in real-

space and each pair has its own binding energy. Unlike BCS pairing, in this model,

because of the lack of overlap between pairs, there is no cooperation between pairs in


24

the pairing process. Superconductivity occurs when Bose-Einstein condensation of

the pairs occurs, which may happen at a lower temperature to that at which pairing

commences. The RVB model describes a normal state in which the electrons are

bound together in pairs on neighbouring sites in singlet states like the electrons in a

Heitler-London covalent bond,77 with a lowered energy due to resonance. One

consequence of this model (amongst others) is so-called spin-charge separation,

where spin and charge are carried separately by different excitations. Spin-charge

separation has not (yet) been observed in any system.


25

1.5 References

1 M. Tinkham, Introduction to Superconductivity (2nd ed.), McGraw-Hill, 1996, §1.1, p. 2. 2 D. R. Tilley and J. Tilley, Superfluidity and Superconductivity (3rd ed.), Institute of Physics Publishing, Bristol, 1990, p. ix. 3 H. Kamerlingh Onnes, Leiden Comm. 120b, 122b, 124c (1911). 4 J. R. Waldram, Superconductivity of Metals and Cuprates, Institute of Physics Publishing, Bristol, 1996, §1.2, p. 2. 5 Tinkham, op. cit., §1, p. 1. 6 Tilley and Tilley, op. cit., §4.1, p. 120. 7 W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933). 8 Tinkham, op. cit., §1.1, Fig. 1.2, p. 3. 9 ibid., Fig. 1.3, p. 4. 10 Waldram, op. cit., §§4.4-4.5, pp. 47-49. 11 ibid., §4.8, Fig. 4.4, p. 54. 12 ibid., §5.2, p. 65. 13 Tinkham, op. cit., §2.3, Fig. 2.1, p. 22. 14 ibid., §2.3.1, p. 25. 15 Waldram, op. cit., §5, p. 61. 16 ibid., §5.1, modified from Fig. 5.1, p. 63. 17 Tinkham, op. cit., §2.2.2, p.21. 18 ibid., §1.6, Fig. 1.5, p. 12. 19 A. A. Abrikosov, Zh. Eksperim. I Teor. Fiz. 32, 1442 (1957) [Sov. Phys.–JETP 5, 1174 (1957)]. 20 Tinkham, op. cit., §4.2.1, p. 119. 21 ibid., §5.1, p. 149. 22 Waldram, op. cit., §5.6, pp. 73-76. 23 Tinkham, op. cit., §§5.1.1-5.1.2, pp. 151-154. 24 ibid., §§4.7-4.8, pp. 132-135. 25 Waldram, op. cit., §4.11, pp. 58-60. 26 Tinkham, op. cit., §5.1.2, p. 154. 27 ibid., §5.3.3, Fig. 5.2, p. 161. 28 Waldram, op. cit., §5.10, p. 84. 29 Tilley and Tilley, op. cit., §4.1, Fig. 4.1, p. 120. 30 H. Fröhlich, Phys. Rev. 79, 845 (1950). 31 E. Maxwell, Phys. Rev. 78, 477 (1950); C. A. Reynolds, B. Serin, W. H. Wright and L. B. Nesbitt, Phys. Rev. 78, 487 (1950). 32 N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College Publishing, 1976, §34, p. 740. 33 L. N. Cooper, Phys. Rev. 104, 1189 (1956). 34 J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). 35 Tilley and Tilley, op. cit., §4.1, p. 121. 36 ibid., §4.1, Fig. 4.2, p. 122. 37 Waldram, op. cit., §2.1, p. 11. 38 Tinkham, op. cit., §3.4.1, pp. 56-57. 39 ibid., §3.4.1, Fig. 3.1, p. 56. 40 ibid., §3.1, p. 44. 41 Ashcroft and Mermin, op. cit., §34, footnote 44, p. 741. 42 Y. Maeno et al., Nature 372, 532 (1994). 43 D. D. Osheroff, R. C. Richardson and D. M. Lee, Phys. Rev. Lett. 28, 885 (1972). 44 Tinkham, op. cit., §§3.4-3.5, pp. 53-62. 45 Waldram, op. cit., §16.10, p. 314. 46 Tinkham, op. cit., §3.4.1, p. 56. 47 ibid., §3.6.1, p. 63. 48 ibid., §3.6.2, p. 63. 49 ibid., §3.6.2, Fig. 3.2, p. 64. 50 Tilley and Tilley, op. cit., §2.7, p. 68.


26

51 J. G. Bednorz and K. A. Müller, Z. Phys. B64, 189 (1986). 52 M. K. Wu et al., Phys. Rev. Lett. 58, 908 (1987). 53 Waldram, op. cit., §12.1, p. 223. 54 http://superconductors.org/type2.htm 55 Tinkham, op. cit., §9.9.1, p. 375. 56 S. E. Barrett et al., Phys. Rev. Lett. 66, 108 (1991) and Phys. Rev. B41, 6283 (1990); M. Takigawa, P. C. Hammel, R. H. Heffner and Z. Fisk, Phys. Rev. B39, 7371 (1989). 57 J. W. Loram et al. in Research Review 1994, Cambridge: University of Cambridge, IRC in Superconductivity (1994). 58 W. N. Hardy et al., Phys. Rev. Lett. 70, 3999 (1993); D. A. Bonn et al., Phys. Rev. B47, 11314 (1993). 59 J. E. Sonier et al., Phys. Rev. Lett. 72, 744 (1994). 60 D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg and A. J. Leggett, Phys. Rev. Lett. 71, 2134 (1993); also see later work by D. A. Wollman, D. J. Van Harlingen, J. Giapintzakis and D. M. Ginsberg, Phys. Rev. Lett. 74, 797 (1995); D. J. Van Harlingen, Revs. Mod. Phys. 67, 515 (1995). 61 C. C. Tsuei, J. R. Kirtley, C. C. Chi, L. S. Yu-Jahnes, A. Gupta, T. Shaw, J. Z. Sun and M. B. Ketchen, Phys. Rev. Lett. 73, 593 (1994). 62 A. Mathai, Y. Gim, R. C. Black, A. Amar and F. C. Wellstood, Phys. Rev. Lett. 74, 4523 (1995). 63 H. Aubin, K. Behnia, M. Ribault, R. Gagnon and L. Taillefer, Phys. Rev. Lett. 78, 2624 (1997). 64 Waldram, op. cit., §15, pp. 267-292. 65 ibid., §16.1, pp. 293-295. 66 Tilley and Tilley, op. cit., §11.1, p. 425. 67 B. Batlogg, R. J. Cava, A. Jayaraman, R. B. van Dover, G. A. Kourouklis, S. Sunshine, D. W. Murphy, L. W. Rupp, H. S. Chen, A. White, K. T. Short, A. M. Mujsce and E. A. Rietman, Phys. Rev.

Lett. 58, 2333 (1987). 68 L. C. Bourne, M. F. Crommie, A. Zettl, H.-C. zur Loye, S. W. Keller, K. L. Leary, A. M. Stacy, K. J. Chang, M. L. Cohen and D. E. Morris, Phys. Rev. Lett. 58, 2337 (1987). 69 Tilley and Tilley, op. cit., §11.7, p. 455. 70 J. P. Franck, Physical Properties of High Temperature Superconductors (vol. 4), ed. D. M. Ginsberg, World Scientific, Singapore, 1994, p. 189. 71 Waldram, op. cit., §16.11, p. 318. 72 See, e.g., N. E. Bickers, D. J. Scalapino and S. R. White, Phys. Rev. Lett. 62, 961 (1989); P. Monthoux, A. V. Balatsky and D. Pines, Phys. Rev. Lett. 67, 3448 (1991) and Phys. Rev. B46, 14803 (1992); P. Monthoux and D. Pines, Phys. Rev. B49, 4261 (1994); P. Monthoux and D. J. Scalapino, Phys. Rev. Lett. 72, 1874 (1994). 73 J. R. Schrieffer, Physica C185-189, 17 (1991). 74 C. M. Varma et al., Phys. Rev. Lett. 63, 1996 (1989). 75 N. F. Mott, Physica C205, 191 (1993); A. S. Alexandrov and N. F. Mott, High Temperature

Superconductors and other Superfluids, Taylor and Francis, London, 1994; A. S. Alexandrov, V. V. Kabanov and N. F. Mott, Phys. Rev. Lett. 53, 2863 (1996). 76 P. W. Anderson, Science 235, 1196 (1987). 77 Waldram, op. cit., §17.9, p. 338.

V. W. Wittorff 27

Chapter 2 : Formal Transport Theory – the Wiedemann-Franz Law

2.1 Historical background

The empirical law of Wiedemann and Franz, proposed in 1853, states that for a great

number of metals1

LT=

σκ (2.1)

where L is known as the Lorenz number and is to a fair accuracy the same for all

metals. Drude proposed his model of electrical and thermal conduction2 in 1900, by

applying the kinetic theory of gases to a metal, considered as a gas of electrons. At

the time the Drude model appeared to successfully explain the Wiedemann-Franz

Law, but this was fortuitous since both the thermal and electrical conductivities are

really about 100 times smaller than the classical prediction made by Drude. These

errors cancel in the ratio, yielding a value L = 1.11 × 10-8 WΩ/K2, however a further

factor-of-two mistake made by Drude in his original calculation meant that he found a

value L = 2.22 × 10-8 WΩ/K2, in extraordinary agreement with experiment.3

The modern version of the Wiedemann-Franz Law relates the electrical conductivity

and the electronic component of the thermal conductivity elκ under conditions where

carriers are scattered elastically. Under such conditions it gives

2B

2

0 3

==ekL

T

el πσκ (2.2)

where L0 = 2.45 × 10-8 WΩ/K2 is the free electron value (Sommerfeld value) of the

Lorenz number.

Since much of the analysis in this dissertation makes use of the detail of this result, it

seems worthwhile to derive it formally, and without restricting ourselves to the free-

electron simplifications used in the Sommerfeld Theory of conduction. The

derivation given below draws together various chapters and sections from the book by


28

Ziman4, as indicated, with certain parts paraphrased and other parts expanded upon, as

considered necessary.

2.2 The Boltzmann Equation

The Boltzmann equation allows us to find the distribution function kf in the steady

state, where ( )rkf measures the number of particles in the kth state in the

neighbourhood of the position r.* This is achieved by relating the total rate of change

of the distribution function to the rate of change arising from diffusion, external fieldsand scattering (respectively), according to5

0scatt,field,diff, =++= kkkk ffff (2.3)

Substituting the standard expressions for diff,kf and field,kf yields

scatt,1

kk

kk

k kHvE

rv ff

cef

−=∂∂

•

×+−∂∂

•− (2.4)

where kv is the electron group velocity, and E and H are the electric and magnetic

fields, respectively.

When this has been solved for kf we may calculate the electric current density†

∫= dkvJ kk fe (2.5)

and the flux of energy, ∫ dkv kkk fε (2.6)

where kε is the energy of an electron in the kth state, and we have ignored

mechanisms of energy transfer other than electron transport.

The general form of the Boltzmann equation given by Equation 2.4 is difficult to

solve because of the complexity of the scattering term.5 It depends on the sum of the

transition rates from all other states k′ into the kth state, which depend, in turn, on the

occupation k′f of those states. An integro-differential equation can be the result.

* For carriers, being Fermions, this means that ( )rkf is simply the probability of occupancy of the state

k.† Assume without loss of generality that we have unit volume, so nf =∫ dkk , the number of carriers

per unit volume.


29

However, if the deviation from equilibrium is small in the steady state, we may

linearize the Boltzmann equation to aid analysis. Ignoring the magnetic term, we may

write5

( ) ( )[ ]∫ ′−−−=∂

∂•−∇

∂

∂•− ′

′′ kdEvv kkkkkk

k

kk

kk QfffffeT

Tf 00

00

ε(2.7)

where 0kf is the equilibrium distribution, and where k

k′Q is the intrinsic transition

probability from k known to be full to k′ known to be empty, and according to the

principle of microscopic reversibility

kk

kk ′′ = QQ (2.8)

The left-hand side of Equation 2.7 is a function of k and depends linearly on the

electric and thermal gradients in the specimen. The solution for ( )0kk ff − must then

be, itself, a linear function of E and T∇ . Therefore, since there can be no currents in

the equilibrium state 0kf , Equations 2.5 and 2.6 require that the electric and thermal

fluxes must also be linear functions of E and T∇ .

The so-called elementary solution to the linearized Boltzmann equation 2.7 is valid

when there exists a relaxation time ( )kτ , depending only on the speed of the carrier

and not its direction of motion. This elementary solutions is6

( )

∂∂

+∇∂∂

•−= Evk

kkkkk ε

τ00

0 feTTfkff (2.9)

An underlying assumption in the analysis to obtain this solution is that energy is

conserved in the scattering process, that is, that collisions are elastic. Without this

assumption, we cannot look at one energy surface at a time, and the simple analysis of

the elementary solution is lost.

On the other hand, when the elastic processes do dominate, we can make an important

generalization. If we relax the other assumptions that are made in obtaining the

elementary solution, there will still exist a solution of the form given by Equation 2.9,

but with a relaxation time of the form ( )kτ , which may depend very much on the

direction of motion of the carrier as well as its speed.6 However, since this expression

for ( )kτ will appear in the formulae for both electrical conductivity ( )kσ and thermal


30

conductivity ( )kκ , the simple existence of a relaxation time (even an anisotropic one)

means that the Wiedemann-Franz law should still hold.

Hereafter in our analysis we will restrict ourselves to the elementary solution.

The elementary solution of the Boltzmann equation shown in Equation 2.9 does not

account in any explicit way for the temperature dependence (and hence spatial

variation) of the chemical potential ζ (Ref. 7.) Consider that 0kf is a function of

( ) TkBζεη −= k . Then we may write

( )ζ

εεζε

ζζεε

ηηε

ε

∇∂∂

−∇∂∂−

−=

∇

∂∂

−−

−⋅∂∂

=

∇

∂∂

∂∂

∂∂

=∇∂∂

k

k

k

kk

k

k

k

k

k

kk

00

B2

BB

0

00

1

fTfT

TTTkTk

Tkf

TT

fTTf

(2.10)

Substituting this into the elementary solution Equation 2.9 gives

( ) ( )

∇−∂∂

+∇∂∂−

−•−= ζεε

ζετ

efeTf

Tkff 100

0 Evk

k

k

kkkkk (2.11)

The thermodynamic force ζ∇e1 has exactly the same effect as an external electric

field, and so we may simply make the replacement

ζ∇−=′e1EE (2.12)

which will be the observed electric field. Hence a consequence of the variation of ζ is

the possibility of a current of particles even when there is no applied electric field E.

We wish to calculate the electric current J and the heat current U, using Equations 2.5

and 2.6. However, a second consequence of the variation of ζ is that the flux of

energy 2.6 may not be equal to the flow of heat. We must subtract the free energy of

a current Je1 of electrons carrying their chemical potential ζ, i.e.8

( )∫∫∫ −=−= dkvdkvdkvU kkkkkkkk ffee

f ζεζε1 (2.13)


31

Now, combining Equations 2.5, and 2.11 to 2.13, we have the following linear

relations connecting electric and thermal forces and currents:

( ) ( )( )

( )( ) ( )( )

∇•∂

∂−+′•

∂

∂−−=

∇•∂∂

−+′•∂∂

−=

∫∫

∫∫

TfkT

fke

TfkTefke

dkvvEdkvvU

dkvvEdkvvJ

k

kkkk

k

kkkk

k

kkkk

k

kkk

εζετ

εζετ

εζετ

ετ

02

0

002

1

(2.14)

2.3 The macroscopic transport coefficients9

From the linearized Boltzmann equation 2.7 it follows that the electrical and thermal

currents are linear functions of electric field and temperature, i.e.

∇+′=

∇+′=

TLLTLL

TTTE

ETEE

EUEJ

(2.15)

where we have used E′ , the observed electric field.

We are interested here in the measurable properties of electrical conductivity σ (data

for which are usually presented as electrical resistivity σρ 1= ), thermal conductivity

κ, and the thermopower S. We will neglect the Peltier coefficient Π.

To measure the electrical conductivity σ, the specimen is kept at constant temperature

whilst an electric field E is applied. With 0=∇T , ζ∇ is also zero, and EE =′ . The

electric current J is measured, and the coefficient σ determined by Ohm’s law

EJ σ= (2.16)

Putting 0=∇T in Equation 2.15 gives simply

EEL=σ (2.17)

To measure the thermal conductivity κ, the specimen is electrically insulated to

prevent any electric current flowing through it. A thermal gradient is maintained, and

a flux of heat measured. The required coefficient is κ in the relation

T∇−= κU (2.18)


32

The negative sign is necessary because κ is defined to be positive, and the heat current

flows down the thermal gradient.

We see that the technique of measurement ensures J = 0, and substituting this into

Equation 2.15 gives

TLL

EE

ET ∇−=′E (2.19)

This can be written in terms of the absolute thermoelectric power, or thermopower Sof the specimen thus

TS∇=′E (2.20)

whereEE

ET

LLS −= (2.21)

The transport coefficient S, also called the Seebeck coefficient, is a measure of the

electric field developed in an electrically insulated specimen subject to a thermal

gradient. Direct measurement of this electric field is difficult because it would entail

the same temperature gradient in the measuring apparatus, accompanied by an

additional thermoelectric voltage. The usual technique then is to measure the

difference of the thermopower of two specimens, by measuring the voltage at any

break in a circuit comprising the two specimens with the two junctions at different

temperatures, as shown in Figure 2.1.

VSpecimen

ASpecimen

B

T0T0

T1

Figure 2.1: Circuit for measuring the difference in thermoelectric voltages developed in two differentspecimens.

Returning to the thermal conductivity, using Equation 2.19 in Equation 2.15:

TL

LLTLEE

ETTETT ∇−∇=U (2.22)


33

i.e.

−−=

EE

ETTETT L

LLLκ (2.23)

We see that the thermal conductivity is not simply –LTT in Equation 2.15. The

physical explanation is that we measure κ under the condition J = 0, not 0E =′ . The

electrical insulation of the specimen necessitates a buildup of charge due to the net

flux of carriers driven by the thermal gradient. The charge buildup results in an

electric field E′ in opposition to the net flux, and in the steady state must balance it.

This field (the magnitude of which is proportional to the thermopower S) slightly

reduces the thermal current U.

2.4 The Lorenz number10

Let us directly compare the linear relations given in Equation 2.14 connecting

electrical and thermal forces and currents, with the corresponding relations given in

Equation 2.15, in which we introduced the macroscopic transport coefficients.

If we define a general integral expression

( )( )∫ ∂∂

−−= dkvvk

kkkk ε

ζετ0fkK n

n (2.24)

then we see that

−=

−=−=

=

2

1

02

1

1

KT

L

LT

KTeL

KeL

TT

TEET

EE

(2.25)

The second of these equations is the Kelvin relation (1854), later included as one of a

whole class of Onsanger relations between the coefficients describing irreversible

processes.11 It is equivalent to the statement that the thermopower of a metal is equal

to its Peltier coefficient divided by the absolute temperature, that is TS Π= . If we

combine the Kelvin relation with Equation 2.21 for the thermopower, as well as

Equation 2.17 for the electrical conductivity and Equation 2.23 for the thermal

conductivity, we now see that


34

−=−−=

+−=

0

21

22

2 1KKK

TSTL

LTLL TTEE

ETTT σκ (2.26)

where the final expression makes use of all of the relations given in Equation 2.25.

To proceed we must evaluate the integrals Kn, which in general will be tensors.

However if we assume (without loss of generality) cubic symmetry, so that the tensor

kk vv is isotropic, the expression for Kn reduces to the scalar form

( )( )∫ ∂

∂−−= dk

k

kkk ε

ζετ0

2

31 fkvK n

n (2.27)

where vk is the magnitude of vk, and the appearance of the factor of 31 is no surprise

if we consider that the integral in Equation 2.24 gives a weighted average of the

isotropic tensor vkvk over all directions.

The expression for Kn represents a summation over all states in k-space. This can be

transformed to a summation over the energies of the states as follows (with the

integrand represented by the function F):

( ) ( ) ( )∫∫ = kkkdkk εεε dNFF (2.28)

where ( )kεN is the energy-density of states, so that ( ) kk εε dN is the number of states

(per unit real volume V) between the energy surfaces kε and kk εε d+ , which are a

distance dkn apart in k-space.

The density of states per unit volume of k-space is simply 382 πV , giving the

expression12

( )( )∫ ∫∫ =

∂∂== k

kk

kkk ε

πε

εππεε d

vdSd

kdSdSdkV

VdN

nn

333 41

41

821 (2.29)

where here S is the area of these constant energy surfaces.

Combining Equations 2.27 to 2.29 gives

( )( )∫∫ ∂

∂−−= k

k

k

kkk ε

εζετ

πdf

vdSkvK n

n

02

341

31

(2.30)


35

To evaluate the integrals Kn, we can make use of the effect of the derivative of the

equilibrium Fermi function in the integrand, which acts like an approximate negative

delta function in that it confines our attention to the neighbourhood of the Fermi

surface. For n > 0, a Taylor Series expansion about the chemical potential ζ gives13

( ) ( ) ( )

ζε

τπζε

επ

τπζε

=

+

−

∂∂

+−

= ∫∫

dSvTkdSvKnn

n 32

22

B

2

3 12612(2.31)

But since the electrical conductivity tensor can be written in the case of cubic

symmetry14

∫∫ == dSvee 3

2

3

2

124 πτ

πτ

σ dSv (2.32)

we have (for n > 0)

( ) ( ) ( ) ( )ζε

εσζε

επεσ

ζε=

+

−

∂∂

+−= 22

222

B

2

2 6 eTk

eK nnn (2.33)

where ( )εσ means ‘the electrical conductivity which one would calculate if the Fermi

energy of the metal were ε’.

Using Equation 2.33, we may evaluate K2 and K1:

( ) ( ) ( ) ( )

( ) ( )

( )ζσπ

ζεεσπ

εεσ

ζεεσ

ζεε

π

ζε

ζε

2

22B

2

222

B

2

22

222

B

2

2

3

26

126

0

eTk

Oe

Tk

eeTkK

=

−+=

∂∂

−+−∂∂

+=

=

=

(2.34)

and( ) ( ) ( )

( ) ( ) ( ) ( )

( )ζε

ζε

ζε

εεσπ

εεσζε

εεσ

εεσπ

εεσζεεσ

επ

=

=

=

∂∂

=

∂∂−

+∂

∂+

∂∂

=

∂∂−

+∂∂

+=

2

22B

2

2

2

22222

B

2

2222

B

2

1

3

116

60

eTk

eeeTk

eeTkK

(2.35)


36

We may evaluate K0 directly using Equations 2.30 and 2.32, remembering that the

derivative of the equilibrium Fermi function in the integrand acts like an approximate

negative delta function at the Fermi surface:

( )

( )2

3

02

30

12

41

31

e

dSv

dfvdSkvK

ζσπτ

εε

τπ

=

=

∂

∂−=

∫

∫∫

k

k

k

kk

(2.36)

Alternatively this expression for K0 can be obtained by simply combining Equations

2.17 and 2.25.

Now it is only necessary to combine Equations 2.26, and 2.34 to 2.36 to obtain an

expression for the thermal conductivity:*

( ) ( )( )

( )( )

( )

−=

−=

−=

=

=

222

B2

2

2B

2

22

2

22B

2

2

22B

2

0

21

2

33

331

1

ζε

ζε

εεσ

ζσπ

ζσπ

ζσεεσπ

ζσπ

κ

ddTk

eTk

edd

eTk

eTk

T

KKK

T

(2.37)

Comparing this with Equation 2.26 we see that we can rewrite it as

( ) ( )20 SLT −= ζσκ (2.38)

where as before2

B2

0 3

=ekL π

and where the thermopower ( )ζεε

εσπ

=

=d

deTkS ln

3

2B

2

(2.39)

* Note that the comment following (9.9.10) in Ziman about the relative magnitudes of K1, K0 and K2 is

incorrect. The corrected phrase should read: “We note that K1 is of order ( )ζTkB smaller than

20KK .”


37

The expressions given above in Equations 2.37 to 2.39 are quite general. There is no

restriction to cubic symmetry, nor even to the existence of a single simple relaxation

time.15 What is required is that the Boltzmann equation should have a so-called

elementary solution of the form of Equation 2.9, that is, with only this explicit

temperature dependence. As explained in §2.2, this only relies on the electron

scattering being elastic. This is almost certainly the case when scattering is

predominantly due to impurities or imperfections in the lattice, for example.15

In addition, the validity of the derivation requires degenerate Fermi statistics. This is

equivalent to requiring that the Taylor Series expansion of the integrals Kn given in

Equation 2.31 converge. This requires that the density of states is not appreciably

gapped at the Fermi surface; that is, that if there is a gap or suppression of the density

of states around ζε = , it is restricted to an energy range that is small compared with

kBT. Otherwise, we must go back to the general expression for Kn given by Equation

2.30, and the general expression for the thermal conductivity given by Equation 2.26.

2.5 The Lorenz number for semiconductors

Ziman16 gives an analysis for non-degenerate semiconductors, in which the chemical

potential is buried deep within the energy gap between the valence and conduction

bands, and so for which the expansions represented by Equation 2.31 will not

converge. Ziman’s analysis provides expressions for the thermopower S as well as

Lorenz number ( )TL el σκ= .

S is typically two or three orders of magnitude larger than that of a metal, the leading

term is independent of temperature and larger than S for a metal (see Equation 2.43)

by approximately the factor ( )TkBζ .

L for an extrinsic semiconductor will be temperature-independent, and by coincidence

very similar in magnitude to L0. However since the electrical conductivity of a

semiconductor is small compared with that of a typical metal, we would not normally

expect to see the electronic component of the thermal current against the ordinary

lattice conduction.

For an intrinsic semiconductor (where the currents carried by the electrons and holes

are about equal), L is dominated by a much larger temperature-dependent term.


38

Hence the electronic component of the thermal conductivity for an intrinsic

semiconductor should be not insignificant against lattice conduction.

2.6 The Lorenz number for metals and cuprates

A test for degenerate Fermi statistics is probably that the thermopower can be ignored

in Equation 2.38. So in the case that electron scattering is elastic, we see that we

recover the Wiedemann-Franz Law from Equation 2.38 when the thermopower is

negligible compared with 0L = 157µV/K. This is the case for the metals, as well as

cuprates across most of the phase diagram. However it is not the case for

semiconductors or heavily-underdoped cuprates, where there is an appreciable gap

(for the latter called a pseudogap) in the density of states at the Fermi surface, and a

correspondingly large thermopower. In this case Equations 2.37 to 2.39 are not

applicable, as discussed above.

However for metals (a term which we will use to also describe cuprates well-

described by degenerate Fermi statistics), our derivation above culminating in

Equations 2.38 and 2.39 is valid, as long as scattering is elastic. In fact we can go on

to make a further simplification, which we have already predicted in the above

discussion in saying that a small thermopower in Equation 2.38 is a good indication of

degenerate Fermi statistics, and therefore necessary and probably sufficient for the

expressions to be valid. We will now show that the thermopower of a metal can

indeed be neglected in Equation 2.38.

For metals any change in the Fermi energy would produce a change in the electrical

conductivity only via a change in the electron density n. The electron density in turn

is to a good approximation the integral of the electron energy-density of states up to

this Fermi energy, that is

( ) ( ) ( )∫∫ ≈=∞ ζ

εεεεε00

dNdfNn (2.40)

If we recognize that N(ε) is a monotonically increasing and slowly-varying function

of energy, then we may write (with the Fermi energy of the metal allowed to vary

slightly around ζ and written as ε)

( ) ( ) µεµεεσ eNen ≈= (2.41)


39

where N is the average electron energy-density of states up to ε = ζ, and µ is the

electron mobility. Then17

( ) ( )ζζσ

µεεσ

ζε

=≈∂

∂

=

eN (2.42)

Using Equation 2.39, this gives a thermopower of

ζπ

eTkS

2B

2

3≈ (2.43)

For a metal, ζ is typically18 1.5 to 15eV, so in the solid state at even the highest

temperatures, 02 LS << , and Equation 2.38 for the thermal conductivity reduces to

( ) ( )ζσζσπ

κ TLeTk

02

2B

2

3== (2.44)

This is just the modern version of the Wiedemann-Franz Law as stated in Equation

2.2, where L0 = 2.45 × 10-8 WΩ/K2 is the free electron value (Sommerfeld value) of

the Lorenz number, and where we have assumed that only electron transport is

responsible for the thermal current.* The exact agreement of the Lorenz number

calculated in the derivation given here with the value calculated naïvely using

Sommerfeld Theory is remarkable, since this derivation is far more general. The only

two requirements of the derivation given here are that (i) electrons in the solid are

described by degenerate Fermi statistics, and (ii) the electron scattering is elastic.

2.7 Deviations from the Wiedemann-Franz Law (small angle scattering)

We have emphasized that the Wiedemann-Franz Law holds for a solid described by

degenerate Fermi statistics (e.g. a metal) when the electron scattering is elastic, that is,

when the energy of each electron is conserved in each collision. This condition of

elastic scattering is of course guaranteed in the relaxation-time approximation, but

may not be the case in general.

* If not, our derivation remains valid, but throughout we should replace the thermal conductivity κ with

its electronic component κel, which we need to be able to extract from the total κ in analysing

experimental data. Note that in our previous discussion of the Lorenz number for semiconductors, for

which the lattice conduction might dominate, we have already made this explicit.


40

The physical reason for this requirement of elastic scattering is straightforward.19

Equation 2.5 shows that the only way collisions can degrade an electrical current is by

changing the velocity of each electron vk. However, Equation 2.13 shows that there

are two mechanisms by which an associated thermal current can be degraded –

collisions can alter the electron’s energy εk as well as its velocity vk. Therefore

inelastic collisions will have a substantially different effect on thermal and electrical

currents, in which case there is no reason to expect a simple relation to hold between

electrical and thermal conductivities. On the other hand, if energy is conserved in

collisions, then thermal and electrical currents will be degraded in precisely the same

manner and to the same extent, and the Wiedemann-Franz Law will hold.

Let us consider metals, and leave aside the cuprates for a moment: If we assume it is

safe to ignore electron-electron scattering,20 then collisions can only arise from

deviations from perfect periodicity of the crystal lattice.21 Such deviations are the

result of either: (i) impurities and imperfections in the lattice; or (ii) phonons (thermal

vibrations of ions in the lattice). At low temperatures phonons are frozen out and the

dominant source of collisions is scattering from impurities and lattice imperfections,

which is elastic. At high temperatures, phonon scattering of electrons changes the

energy of each electron in a collision by a small amount compared with kBT, and so

energy is conserved to a good approximation. Therefore the Wiedemann-Franz Law

is generally well obeyed in a metal at both low and high temperatures.

However in the intermediate temperature range (roughly ten to a few hundred degrees

Kelvin)22, where inelastic collisions are both prevalent in a metal and capable of

producing electronic energy losses of order kBT, one expects and observes failures of

the Wiedemann-Franz Law. Such failures are always exhibited by a Lorenz number

that is smaller than the Sommerfeld value (L < L0), due to the additional degradation

of the thermal current. The Lorenz number will also be temperature dependent in the

temperature range where the Wiedemann-Franz Law fails, at least at both extremes of

this range.

The scattering of electrons by phonons at high and intermediate temperatures deserves

further explanation. Consider the situation in Figure 2.2. For each of (a) electrical


41

and (b) thermal conduction is shown the two-dimensional Fermi surface, and the

corresponding distribution function plotted against both positive and negative energies

(equivalent to momenta parallel and anti-parallel respectively to the transport

currents), at non-zero temperature. The equilibrium distributions are shown, as well

as the steady state transport distributions. Superimposed on the Fermi surfaces is

shown where additional electron states are occupied under steady state transport

compared with thermal equilibrium (filled circles), as well as where fewer electron

states are occupied under steady state transport compared with thermal equilibrium

(vacant circles).

Figure 2.2: The Fermi surface and distribution functions: (a) in electrical conduction; (b) in thermalconduction. (From Ref. 23.)

In the case of electrical conduction, the distribution function does not change shape

but is simply shifted in the direction of J. This creates an excess of electrons both

inside and outside the Fermi surface on the right-hand side of the figure and fewer

inside and outside on the left. In the case of thermal conduction, the distribution

function is sharpened at the cold end (up the thermal gradient), and spread at the hot

end (down the thermal gradient). It can be seen that this creates an excess of electrons

outside the Fermi surface at the hot end, and inside at the cold end; and an electron

deficiency inside the Fermi surface at the hot end, and outside at the cold end.

Therefore, when scattering is dominated by “vertical” processes, indicated in Figure

2.2, the thermal current is degraded significantly24 since the distribution undergoes a

relaxation towards equilibrium (the electron is scattered from an region of excess to a

region of deficit). However vertical processes do not degrade the electrical current to


42

nearly the same extent, since no such relaxation of the distribution takes place (the

electron is scattered from one region of excess to another). Hence we expect failure

of the Wiedemann-Franz Law when vertical processes dominate, which will occur for

temperatures below the Debye temperature (a regime which we have described as

intermediate and low temperatures). However at low temperatures where elastic

scattering by impurities and lattice imperfections dominates, this effect of electron-

phonon scattering is irrelevant and the Wiedemann-Franz Law is recovered.

The “horizontal” processes in Figure 2.2, on the other hand, cause relaxation of the

distribution for both electrical and thermal transport, and degrade both currents in the

same way. Hence when such processes dominate, the Wiedemann-Franz Law is well-

obeyed. This is the case at high temperatures, above the Debye temperature, where

there are many short-wave phonons available for such large angle scattering.

In the cuprates, in addition to standard scattering of electrons by lattice vibrations,

various exotic scattering mechanisms have been proposed for the normal state. If

there is no reason to expect such scattering to be restricted to small angles, then at

high temperatures the same arguments as above for metals can be used to predict

observation of the Wiedemann-Franz Law. Impurity scattering will dominate at low

temperatures, and so again for the same reasons as for a metal the Wiedemann-Franz

Law is expected to dominate at low temperature, which can be checked

experimentally only provided the electrical conductivity of the normal excitations can

be observed in spite of superconductivity (e.g. by measuring ( ) ( ){ }ωσωσ Re1 = at

microwave frequencies).


43

2.8 References

1 N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College Publishing, 1976, §1, p. 20.2 Annalen der Physik 1, 566 and 3, 369 (1900).3 Ashcroft and Mermin, op. cit., §1, p. 23.4 J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, OxfordUniversity Press, London, 1960.5 ibid., §7.3, pp. 264-267.6 ibid., §7.4, pp. 267-270; note that the second term in the right-hand side of (7.4.3) has the wrong signin Ziman.7 ibid., §9.9, p. 383.8 ibid., §9.9, p. 384.9 ibid., §7.5, pp. 270-271.10 ibid., §9.9, pp. 384-385.11 ibid., §7.6, p. 273.12 ibid., §2.11, p. 106.13 ibid., §2.10, pp. 103-104.14 ibid., §7.2, pp. 261-262.15 ibid., §9.9, p. 385.16 ibid., §10.2, pp. 423-428.17 This result, without explanation or reference to the density of states, is also given in Ashcroft andMermin, op. cit., §13, p. 255.18 Ashcroft and Mermin, op. cit., §2, pp. 37-38.19 ibid., §16, pp. 322-323.20 ibid., §17, pp. 345-351.21 ibid., §16, p. 315.22 ibid., §16, p. 323.23 Ziman, op. cit., §9.10, Fig. 122, p. 386.24 ibid., §9.10, p. 387.

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