i/sc_20… · contents 1 introduction 1 1.1 a brief history of low-temperature physics . . . . . ....

Superconductivityand

Low Temperature PhysicsPart I: Superconductivity

Lecture Notes of the Academic Year 2013/14

Rudi Hackl and Dietrich Einzel

Walther-Meissner-InstitutBayerische Akademie der Wissenschaften

Walther-Meissner-Strasse 8D-85748 Garching

[email protected]

c© preliminary — Garching, November 21, 2013

Contents

1 Introduction 1

1.1 A brief history of low-temperature physics . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Present Status of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Areas of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.3 Solved and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Basic Experiments and Understanding 9

2.1 Key Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Zero Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Perfect diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Shubnikov phase (mixed state) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.4 Quantization of the flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.5 Josephson effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Condensation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3 Specific heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 The Drude model in the limit τ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Generalized London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.3 The London equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.4 Some conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

iii

iv R. HACKL AND D. EINZEL CONTENTS

3 Microscopic Theory 35

3.1 The Cooper Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Origin of the interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 The BCS wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Coherent states in a boson field . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.2 Properties of fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.3 A coherent state of fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Determination of the ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 The BCS Hamilonian in second quantization . . . . . . . . . . . . . . . . . . . 45

3.4.2 Some expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.3 Determination of the energy minimum at T = 0 . . . . . . . . . . . . . . . . . . 47

3.5 The general solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.1 Approximation of the four-operator term . . . . . . . . . . . . . . . . . . . . . 48

3.5.2 The Bogoliubov-Valatin transformation . . . . . . . . . . . . . . . . . . . . . . 49

3.5.3 Solution of the gap equation for T ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . 52

3.6 Connection to experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.1 Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6.2 Single particle response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Ginzburg-Landau Theory 59

4.1 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Application to superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 Density of the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.2 Functional of the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.3 The Ginzburg-Landau equations . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Two new length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.1 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.2 Ginzburg-Landau coherence length . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.3 Energy of the normal-superconductor interface . . . . . . . . . . . . . . . . . . 65

4.4 States with internal flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.1 The upper critical field Bc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.2 The nucleation field Bc3 on the surface . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.3 The thermodynamic critical field Bc . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.4 The lower critical field Bc1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.5 The Abrikosov lattice (1957) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

c© Walther-Meißner-Institut

CONTENTS SUPERCONDUCTIVITY v

5 The Josephson Effect 73

5.1 Weakly coupled superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 The Josephson equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 The RCSJ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Josephson contact in a microwave field . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Josephson effect in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5.1 Ring with a single weak link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5.2 Ring with two weak links: Quantum interference . . . . . . . . . . . . . . . . . 79

5.5.3 Quantum interference in a long junction . . . . . . . . . . . . . . . . . . . . . . 80

6 Unconventional Materials 83

6.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 The iron-age of superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.3 Copper-oxygen compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.3.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.3.3 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3.4 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3.5 Summary and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7 An overview of applications 113

7.1 Potential areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1.1 Economic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1.2 Areas of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2 Passive applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2.1 Physical and technical challenges . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3 Active devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

2013

Preface

This is a first version of lecture notes on superconductivity on an introductory level. It is intended to beused along with the lectures I deliver at the Technical University Munich. The manuscript is the result ofseveral courses on the subject during the last years and should not be considered final. In fact, it will needa few more iterations. In particular, the referencing is not yet complete. The notes are inspired by severaltextbooks and lecture notes on superconductivity or Condensed Matter Physics including the books by J.F. Annett, N. Ashcroft and D. Mermin, W. Buckel and R. Kleiner, P.-G. de Gennes, R. Groß and A. Marx,H. Kinder (notes of A. Heinrich), C. Kittel, A. Sudbø and C. Fossheim, V. V. Schmidt (edited by P. Mullerand A. Ustinov), and M. Tinkham. We profitted from numerous discussions with our colleagues workingin the field, in particular at the Technical University Munich and at the Walther-Meissner-Institut.

The notes are not for distribution.

Garching, October 16, 2013 Rudi Hackl

vii

Chapter 1

Introduction

Imagine a macroscopic object which obeys the rules of quantum mechanics and can be described by asingle wave function. What would be the consequences? This question is at the heart of the phenomenato be discussed here. An intuitive answer can be found by considering the usual microscopic quantumworld. The wave function describing the electron in a hydrogen atom, for instance, has an amplitudeand a phase. The amplitude corresponds to the probability of finding the electron in a volume elementaround the pointr. The phase determines the type of the stationary state dictated by the uniqueness andthe number nodes of the wave function. In the stationary state there is no dissipation. The superpositionof stationary states leads to interference phenomena and transitions with typically very sharp energies.

Superconductivity and superfluidity along with Bose condensation or density wave order (periodic mod-ulations of the magnetization or charge density) are macroscopic quantum phenomena. In all cases theestablishment to a coherent quantum state facilitates charge, mass or spin transport without energy dis-sipation. For appropriately prepared experimental settings interference phenomena can be observed. Inthe case of superconductivity the characteristic energy is solely determined by constants of nature. Elec-tromagnetic radiation can be created and extremely sensitive detectors can be constructed which exploitthe rigid phase of the wave function.

The emergence of quantum phenomena on a macroscopic scale, i.e. in a large ensemble which can bedescribed by statistical methods, can be expected when temperature is not the dominant energy scale anyfurther. Therefore, the discovery of macroscopic quantum phenomena in the early 20th century is closelyrelated to the development of low temperature techniques although meanwhile a variety of manifestationshave been found to exist also at room temperature and above.

In this chapter we first sketch briefly the hallmarks of physics at low temperature (with the focusplaced on superconductivity), then qualitatively describe examples of condensates and finally providean overview over the current status of superconductivity.

1.1 A brief history of low-temperature physics

The liquefaction of air in 1895 [?] can be considered the starting point of low-temperature physics. It tookonly 13 years until Heike Kamerlingh 1 Onnes in Leiden succeeded to liquefy Helium, the gas with thelowest boiling point [?]. Not only does He have the lowest boiling point of all elements but it is also theonly material which does not solidify upon further cooling. Only at an applied pressure of approximately30 bar the solidification can be induced. This phenomenon is among the first manifestations of quantum

1Privy Councillor (“Hofrat”)

1

2 R. HACKL AND D. EINZEL Introduction

(a) (b)

Figure 1.1: Resistances of metals at low temperature. Note the different temperature scales in (a) and(b). (a) Proposals for the low-temperature resistivity of metals as of 1900. Temperatures below 15 Kwere unaccessible. Curves 1 through 3 were suggested by Dewar, Matthiessen, and Kelvin, respectively.From W. Buckel, Supraleitung VCH Weinheim 5th Edition (1994), Fig. 1. (b) Resistance of Hg close tothe superconducting transition. From Ref. [1].

effects directly determining the macroscopic properties. If one estimates the amplitude of the zero-pointoscillations on the basis of the uncertainty principle values on the order of 10 to 20% of the atomicdiameter are found which are big enough to prevent the establishment of a crystal lattice.

Only three years later Onnes observed vanishing resistance upon cooling mercury (Hg) below 4.2 K. [1]At that time the low temperature properties of metals were discussed intensively since the experimentaldata ended at some 15 K being close to the solidification point of liquid hydrogen. There were threemain concepts Fig. 1.1 (a): (1) Probably inspired by the Drude model2 Dewar argued in 1904 that ρ

should asymptotically approach 0 in the limit T → 0 since all scattering mechanisms might freeze out.(2) Matthiessen3 observed that the increase of the resistivity of a metal as a consequence of a smallconcentration of another metal in a solid solution was temperature independent. This implies that theresistivity of a metal can stem from more than one source, for example a constant one from impurities,ρ0, and a temperature dependent one of different origin, ρ(T ), which add up.4. As a consequencethe resistivity saturates at low temperature at a finite ρ0, the residual resistivity. (3) Alternatively, theresistivity was proposed to diverge due to electron localization (Lord Kelvin 1902).5

In the experiment on very clean Hg, Onnes and his collaborators unexpectedly found the resistance tobecome abruptly unmeasurably small at a finite critical temperature Tc as shown in Fig. 1.1 (b) ratherthan asymptotically in the limit T → 0.

He called this qualitatively new phenomenon superconduction. It was clear from the beginning that noneof the three scenarios discussed above was capable of describing the observation. It took until 1957 untilBardeen, Cooper, and Schrieffer succeeded in presenting a microscopic theory of the superconducting

2P. Drude 19003A. Matthiessen, Ann. Phys. Chem. (Pogg. Folge) 110, 190 (1860) and A. Matthiessen and G. Vogt, Ann. Phys. Chem.

(Pogg. Folge) 122, 19 (1864)4Matthiessen’s rule is purely empirical and valid only if the different scattering mechanisms are independent and isotropic.

For a discussion see Ashcroft and Mermin Solid State Physics (Holt-Saunders International Edition 1981)5Anderson showed in 1958 (Phys. Rev. 109, 1492) that even a small amount of impurities can localize electrons due to

interference of Bloch waves. Even in a completely pure system (with low carrier density) the resistivity can diverge due to thecrystallization of the electrons (Wigner).


A brief history of low-temperature physics SUPERCONDUCTIVITY 3

state [2]. A discovery of enormous importance and influence was the demonstration of thermodynamicnature of the superconducting phase by Ochsenfeld and Meissner in 1932. Soon thereafter in 1935,Fritz and Heinz London suggested the first phenomenology [?] and proposed a first quantum mechanicalapproach in 1950. In the same year Ginsburg and Landau applied Landau’s theory for phase transitionsto superconductors using a complex order parameter rather than a real one such as for magnetism orlattice distortions. In the year of publication of the microscopic theory Abrikosov used the GinsgurgLandau phenomenology to derived the existence of a mixed state accommodating superconductivity anda magnetic field on a mesoscopic scale [?] as observed by Shubnikov and coworkers already in 1936 [?].This discovery paved the way towards a huge variety of applications since electric currents and very largemagnetic fields can coexist here with superconductivity facilitating the quasi-dissipationless maintenanceof magnetic fields in excess of 20 T. Finally in 1960, Eliashberg made connection to real materials in hiscelebrated strong-coupling approach [3] which is used to date for analyzing interaction potentials insuperconductors.

In 1938 a phenomenon analogous to superconductivity was found in an uncharged system. Here, massflows without viscosity. This property is called superfluidity and was independently discovered byKapitsa6 and by Allen and Misener7 in Helium below 2.17 K. Rotation corresponds to the magnetic fieldand vortices of electric current around lines of the field (see section ??) to vortices of the liquid.

If two lumps of a superconductor or two reservoirs of a superfluid are weakly connected in a way thatonly the supracomponent can pass, the eventually occurring voltage and pressure drops, respectively,correspond. The spectacular fountain effect (Fig. 1.2) is a result of the pressure difference across a so-called weak link (realized by fine powder in the lower part of the nozzle which is transparent only forthe superfluid) induced by a temperature difference between the reservoirs (by heating the upper partclose to the narrow nozzle). The corresponding phenomenon in a superconductor is the Josephson effectpredicted 1962 [?] and observed one year later [?]. It is one of the most spectacular manifestation of thequantum nature of the superconducting condensate and became the basis of a highly sensitive detectorof magnetic fields with a broad range of applications [?].

In 1972, Lee, Osheroff, and Richardson discovered superfluidity also in the lighter isotope 3He. Thetransition temperature Tc = 2.2 mK is three orders of magnitude smaller than that of the of the heavierisotope.8 At first glance, superfluidity in the two isotopes appears to be only quantitatively different.However, quantum effects dominate the behavior at these temperatures and the nuclear spin of 1/2 pre-vents 3He from Bose condensation occurring in S = 0 4He. As opposed to a condensate of stronglyinteracting 4He atoms 3He forms a pair condensate similar to that of a superconductor. In either casethe condensate is neutral. Only one year later A. Leggett proposed a full theoretical description of thevarious phases of 3He [4].

One of the most important technical innovations, proposed in 1962 and first demonstrated in 1965, wasthe realization of the dilution refrigerator using mixtures of 3He and 4He. Here, the cooling is achievedby dissolving 3He in 4He. The minimal temperature of this continuously working cryo-system is below10 mK. The dilution refrigerator is the basis of many ultra-low-temperature experiments and replaced thecooling by paramagnetic salts almost completely. If a nuclear demagnetization stage is attached a recordbase temperature of 8-10 µK was realized. Lower temperatures can be reached only in cold gases (seebelow). In the range between 10 µK and 1 K quite a few new superconductors and unexpected phases ofmatter were found as described in the textbooks and to be discussed partially in the second part of thelecture series.

In these days dilution refrigerators do not need pre-cooling with liquid 4He any further as in the earlydays. With the continuous improvement of closed-cycle refrigerators µK can now be generated by

6P. Kapitsa, Nature 141, 74 (1938)7J.F. Allen and A.D. Misener, Nature 141, 75 (1938)8Lee, Osheroff, and Richardson, Phys. Rev. Lett. vv, ppp (1972)

2013


Figure 1.2: Bose-Einstein condensation. (left) Demonstration of the fountain effect in superfluid He.The Photograph was taken by Jack Allen in 1970 who co-discovered superfluidity. From R.J. Don-nelly, Physics Today, July 1995, p. 30. (right) Bose-Einstein condensation of 87Rb at 500 nK. Theseries of pictures shows the momentum distribution of Rubidium atoms in a trap at different tempera-tures. If the majority of the atoms is in the lowest state the total momentum approaches zero. Fromhttp://www.mpq.mpg.de/cms/mpq/en/departments/quanten/homepage cms/projects/bec/.

combining a pulse-tube unit [?, ?] with an optimized dilution system. The prototype of this innovationwas developed at the Walther-Meissner-Institut [?] and is now commercially available.

Another important discovery at low temperature is the integer and fractional quantum Hall effect which,in connection with the Josephson effect, completes, for instance, the metrological triangle. Spin andcharge density waves SDW/CDW are ordering phenomena very similar to the superconducting state.Here, the ordering produces a periodic modulation of the carrier or spin density [5]. The modulationamplitude can vary between a fraction of a percent and several ten percent of the average charge density.While in a superconductor two electrons form a Cooper pair having vanishing net momentum densitywaves are characterized by a finite ordering vector connecting particle and hole states.

The most spectacular discoveries of the late 20th and the early 21st century are the Bose-Einsteincondensation of trapped gases [?], the observation of superconductivity above 100 K and 40 K in thecuprates [?, 6] and, respectively, in iron pnictides [7, 8]. These most recent developments demonstratethat low-temperature physics in general and superconductivity in specific remain vibrant fields of con-densed matter physics.

A timetable displaying the history in consecutive order is planned to be provided soon in the appendix.

1.2 Condensates

Schrodinger was the first to elaborate on a coherent state of Bosons [?] and constructed a coherentwave function from a superposition of an infinite number of harmonic oscillator wave functions |ψn〉 =1/√

n!(a†)n|0〉 with a† the creation operator and |0〉 denoting the ground state,

|α〉= e−|α|2

2

∞

∑n=0

αn√

n!|ψn〉= e−

|α|22

∞

∑n=0

(α a†)n

n!|0〉. (1.2.1)


Present Status of Superconductivity SUPERCONDUCTIVITY 5

Here, α = |α|eiϕ is a complex number in polar representation having the amplitude |α| and the phaseϕ . As shown in problem 1 |α〉 obeys a Poisson distribution which is sharply peaked at the average oc-cupation number 〈n〉= N. In addition, one finds an uncertainty relation between the average occupationnumber N and the phase ϕ to hold implying that for a large number of particle ∆ϕ/ϕ = 1/

√N.

Coherent states have this structure in general. Here the wave function is constructed for Bosons and candirectly applied for the description of the laser field or for Bose-condensed cold atoms. In principle, itis also applicable for 4He but strong coupling effects must be considered in this case and make affairsmuch more complicated. Is the wave function also useful for Fermionic systems?

It is the achievement of Schrieffer to construct a coherent state similar to that in Eq. (1.2.1) from theFermionic wave function of electrons. The nature of Schrieffer’s state will be derived in detail in chap-ter 3. The origins of a superconducting and a Bose condensate are quite different. Superconductivityoriginates from pair correlations between the electrons 9 In the case of Helium or, similarly, for vaporsof alkali metals10 the individual atoms undergo a Bose-Einstein condensation at sufficiently low tem-peratures, which means that a macroscopic number of particles assumes the lowest possible state. Thedifference between the alkalis and Helium is just the interaction strength Ve f f . In the weak-couplinglimit the first excited state has essentially the same energy as the ground state. With increasing couplinga gap develops between the ground state and the first excited state. Then, following the argumentationof Landau, condensed particles can be accelerated to a finite critical velocity vc before they leave theground state and dissipate energy. In spite of the fundamental differences between superconductors andBose condensed systems all form a coherent ground state which can be described by a single quantummechanical wave function.

1.3 Present Status of Superconductivity

1.3.1 Materials

The development was continuous but quite slow in the beginning. While many elements become su-perconducting as shown in Fig. 1.3 the transition tempertures are moderate, and Nb having Tc = 9.26 Kholds the record. If pressure is applied even semiconductors may become superconducting. At pres-sure values in the range of 20-150 GPa (0.2-1.5 Mbar) even insulators or alkali metals occasionally reachTc values between 15 and 25 K (see Table ??). In Fig. 1.3, the elements with pressure-induced super-conducting transitions are marked with red points. On the other hand, compounds soon exceeded thetransition temperatures of the elements as shown in Fig. 1.4 and Table ??. This was the time periodwhen materials sciences flourished and many compounds and new properties and effects were discov-ered which demonstrated the potential of applications while the transition temperatures Tc increased at avery low rate of 0.4 K/year (Fig. 1.4). The discovery of superconductivity in the copper oxygen systemLa2−xBaxCuO4 was a qualitative change in many respects.11 The slow increase of Tc which in a sensewas the linear analogue of Moore’s law was interrupted in the non-Murphyian direction. Most impor-tantly, the superconducting state turned out to be unconventional meaning that in addition to the gaugesymmetry also the rotational symmetry is broken at the transition. In other words, the order parame-ter and the energy gap have a lower symmetry than the crystal. Unconventional superconductivity wasnot completely new12 but was considered an exception occurring only in a few exotic systems such as

9We note here, that Bogoliubov had results equivalent to those of BCS at approximately the same time but the publicationwas delayed.

10Cornell, Ketterle, and Wieman11J.G. Bednorz and K.A. Muller, Z. Phys. B (1986)12F. Groß et al., Z. Phys. B (1986). It is a coincidence with some charm that the article of Bednorz and Muller on

La2−xBaxCuO4 directly follows the first clear evidence of an unconventional gap with non-zero orbital momentum in UPt3and UBe13.

2013


http://www.spring8.or.jp/en/news_publications/press_release/2011/110705_fig/fig1.png[18.10.2013 13:56:26]

Figure 1.3: Transition temperature Tc of the elements. The latest record is Tc≈ 25 K in Ca at 210 MPa [?].Nb having Tc = 9.26 K is the elements with the highest transition under normal conditions. Higher valueswere obtained only with applied pressure. From http://www.spring8.or.jp.

150T c (K

)

HgBa2Ca2Cu3O8+δ150

ture

T

g 2 2 3 8+δ

Bi Sr Ca Cu O

Tl2Ba2Ca2Cu3O10+δ

100

pera

t

YBa2Cu3O6+x

Bi2Sr2Ca2Cu3O10+δ

3 NdFeAsO1-xFx

50

n te

m

La2-xBaxCuO4 MgB23

2 Ba1-xKxFe2As2

1 LaFePO

1 x x

nsiti

on

HgNbTi V3Si Nb3Ge

Ba K BiOCs2RbC60

2

1900 1925 1950 1975 2000 20250tra

n

f di

Ba0.6K0.4BiO3 1

year of discovery

Figure 1.4: Maximal transition temperature Tc vs. year of discovery. The discovery of superconduc-tivity in the cuprates is a historical hallmark. Superconductivity in Fe-based compounds was similarlyunexpected.

the heavy fermion compounds (e.g., UPt3) or, more recently, in the ruthenates (e.g., Sr2RuO4)13 or insuperfluid 3He.

The cuprates, on the other hand, cover a very broad class of materials which have maximal transition tem-peratures T max

c in a range from about 20 K in Bi2Sr2CuO6 to 160 K in HgBa2Ca2Cu3O8. Surprisinglyenough, apart from T max

c the temperature versus doping phase diagrams are similar: all are antiferro-

13G.M. Luke et al., Nature 394, 558 (1998)


Present Status of Superconductivity SUPERCONDUCTIVITY 7

magnetic insulators at half filling and superconductivity exists for 0.05 < p < 0.27 with p the numberof holes per CuO2 formula unit (away from the Cu3d9 configuration).14 While the first applications arealready on the market, the orgin of superconductivity is still an open question at least for the majority ofthe people working in the field. High-Tc research is indeed among the most vibrant fields in solid statephysics in these days, and we devote a complete chapter to it.

Since the discovery of the cuprates several other quite interesting materials were found to superconduct.The biggest surprise was certainly MgB2 with Tc = 39.5 K.15 Very recently, also elements were foundto become superconducting at temperatures close to 20 K under sufficiently high pressures.16 In January2008 superconductivity was reported in Nd(O1−xFex)FeAs compounds.17 By exchanging the rare eartha maximal Tc of 55 K was obtained.18

1.3.2 Areas of Application

It is immediately clear that the state of zero resistance offers enormous opportunities for applications.The first realization was a solenoid for fields in the range of 5 T in a volume of a few cubic-centimeters.In these days, the maximal permanent field maintained by a superconductor is slightly above 21 T, and thelargest volume is several m3 in coils for fusion experiments. Applications include now power transmis-sion, generators, motors, energy storage, fault-current limiters, and filters. The most popular applicationwhich many people have already encountered are superconducting coils for magnetic resonance imaging.

Beyond the state of R = 0 there are various other properties of superconductors which can be exploited.For instance, the behavior of weakly coupled superconductors opens a wide field of active devices whichrest on the Josephson effects. For sufficiently low currents there is no voltage drop across the weaklink (dc Josephson effect), and the phase difference between the two superconductors remains constant.For higher currents there is also a voltage drop making the phase difference time dependent. It followsfrom the Josephson equations that the current has now oscillating components (ac Josephson effect).Exploiting the dc effect extremely sensitive detectors for magnetic fields can be made (SuperconductingQUantum Interference Device) which allow one to observe and to locate the currents in the heart and inthe brain. The ac effect can be used to produce and detect electromagnetic waves in the hundred GHzrange. Other very important applications are the voltage standard and, along with the quantized Halleffect, the determination of Planck’s constant h and the elementary charge e.

1.3.3 Solved and Open Problems

The London brothers could clarify the phenomenology and electrodynamics of superconductors rela-tively soon after the discovery. In 1935 they proposed a model which was capable of describing theexclusion of magnetic fields from bulk superconductors and discussed the exponential decay of the fieldin a surface sheeth of thickness λL which is now called the London penetration depth. They understoodthat the effect was quantum mechanical in origin and describable in terms of a single wave function be-ing closely related to coherent states as proposed by Schrodinger in 1926.19 However, the microscopicfoundation of this wave function was completely unclear in particular since Schrodinger’s proposal wasfor Bosons and not for Fermions. It took another 22 years until Schrieffer could derive a coherent statewave function also for Fermions which finally completed the BCS theory of superconductivity [2].

14We note that the AF ordering (Neel) temperature TN and T maxc do not scale.

15Akimitsu et al., Nature vvv, ppp (2001).16J.J. Schilling, Schrieffer’s book, ppp (2006).17abc et al., J. Am. Chem. Soc. vvv, ppp (2008).18abc et al., Nature vvv, ppp (2008).19E. Schrodinger, Naturwissenschaften 28, (1926).

2013


The BCS theory considered only the limit of weak coupling. Eliashberg succeed to establish a con-nection between the interaction potential Vk,k′ and spectra density of phonons. This became possibleafter Migdal’s observation that the electron-phonon vertex is of order

√m/M with the electron and ion

masses m and M. Physically speaking this means that the electron-phonon interaction is retarded or thatthe respective energy scales differ by orders of magnitude and can be considered independent. Eliash-berg’s approach can be generalized to other type of interaction as long as the interaction between twoelectrons is bosonic. Therefore, the scheme is used today to analyze the superconducting properties evenof cuprates and pnictides. However, while qualitative conclusions are enlightening a quantitative de-scription including the theoretical derivation of the transition temperature of an existing superconductoror the prediction of a new superconducting material are still at their infancy. Some progress was madefor conventional systems recently [9]. In spite of enormous progress the understanding of the cuprates isstill limited. One of the major problems is the similarity of all relevant energy scales such as the Fermienergy EF , the exchange coupling J, the energy gap ∆k, or the phonon energy hωq. The solution of thesetype of problems is among the most tantalizing questions in condensed matter physics.


Chapter 2

Basic Experiments and Understanding

Many of the basic observations in superconductors to be described in the first section of this chaptercould be understood phenomenologically long before the microscopic theory was finally presented in1957. In particular the London theory highlights the different aspects of the zero-resistance state and theperfect diamagnetism. If derived from the Schrodinger equation time-dependent phenomena such as theJosephson effects follow directly highlighting the quantum mechanical nature of the phenomenon in asimple though instructive way.

2.1 Key Experiments

2.1.1 Zero Resistance

After Onnes succeeded to liquefy Helium (1908) he started immediately to study the resistivities ofmetals. Hg was the purest material available at that time. Instead of finding support for one of theproposals being discussed at his time (see chapter I) he discovered a state of vanishingly small resistance(R < 10−5 Ω) below T = 4.2 K, the boiling point of liquid He (Fig. ??).1

The determination of the actual resistivity of a superconductor requires new techniques since the pre-cision of a direct four-probe measurement is orders of magnitude too low. Already Onnes designed anexperiment which uses the magnetic moment of a persistent current I in a ring to determine the decay rateof I rather than the resistance itself. In the experiment there are two concentric superconducting rings.The outer one is fixed, the inner one is suspended on a quartz filament. As long as there is no current inthe rings the inner ring turns with the filament. Now we get supercurrents Ii and Io to flow in the rings.To this end a (for simplicity) homogeneous field B0 Bc(T) is turned on with the axis parallel to theaxes of the concentric rings. If the rings are above their transition temperature Tc the field penetrates thecross sections Si and So as well as the material homogeneously. Now the rings are being cooled below Tc

and the field is expelled from the material due to the Ochsenfeld-Meissner effect (see next paragraph),i.e. is slightly distorted around the solid superconductors but more or less unchanged in most of the freespace.

What happens if the field is switched off? This can be derived easily from Faraday’s law for either of therings,

∇×E =−∂B∂ t

. (2.1.1)

1H. Kammerlingh Onnes, Leiden Comm. 120b, 122b, 124c (1911)

9

10 R. HACKL AND D. EINZEL Basic Experiments and Understanding

We integrate over the cross section S of the respective ring and apply Stokes’ theorem,

−∫

S

∂B∂ t·dA =

∫S

∇×E ·dS (2.1.2)

=∮

Γ

E ·dl.

Here Γ is the edge of S and runs approximately along the center of the ring. Since the resistance of thesuperconductor vanishes for sufficiently small fields (see below) there is no voltage drop along the ringand, consequently,

∂

∂ tΦ = 0 (2.1.3)

In the last equation Φ ≈ BS is the flux penetrating the ring, and the voltage drop for one circulationaround the ring is zero since R = 0. Hence the flux through a closed superconducting loop is constantand cannot escape after the external field is switched off. The related field is maintained by a current I inthe ring. In turn, we would be unable to get the current running when applying the field at T < Tc. Forthis reason it is misleading to think of inducing a current by changing an external field. The current israther a result of maintaining the flux fixed.

If the filament is now twisted there will be a restoring force proportional to µi× µo with the momentsgiven by µ = SI. Once the equilibrium position is reached any turn of the inner ring as a consequence ofa variation of Ii,o can be measured with great precision. In order to determine a limit for the resistivity ρ

from ∆I we consider the ring as a loop with a resistance R and an inductance L for which

RI +LdIdt

= 0. (2.1.4)

This differential equation can e integrated by separating the variables, and we find

R =−Lt

lnI(t)I(0)

, (2.1.5)

where I(t) is directly proportional to the torque. Onnes and his collaborators kept the experiment runningfor a year - quite heroic at that time. With modern NMR techniques the field produced by a current canbe measured with much higher precision than with a torque meter, and a lower limit of 105 years wasfound for ∆I to become significant.

2.1.2 Perfect diamagnetism

Given the perfect conductivity of a superconductor it is obvious that a magnetic field is completelyshielded when it is applied below the transition temperature. The induced screening currents do notdecay as in conventional metals with finite resistivity. However, what happens if the superconductingstate is entered by lowering the temperature in a finite external field?

The question was answered by Meissner and Ochsenfeld in 1933 when they studied the force betweentwo parallel tin wires. If the currents were applied above and below Tc the forces were different. In thelatter case the mutual exclusion of the fields due to screening explained the result. However, the sameeffect was observed when the wires were cooled through Tc with the currents on implying that the fieldwas also expelled without induction to start the screening currents 2.

2W. Meißner and R. Ochsenfeld, Naturwissenschaften 81, 787 (1933)


Key Experiments SUPERCONDUCTIVITY 11

Bi=0

Bi= Ba

( )

Type I SCT 0

B0

Bi

Bc

Bi = B0

Bi = 0

Meissner state

normalstate

(b)

TTc

B

Bc(0) Bc(T)

zero field cooled (z.f.c.)

field cooled (f.c.)

(a)

Figure 2.1: Field-temperature phase diagram of a superconductor. A given point in the superconduct-ing can be reached independent of the path. In this way the superconducting state is established as athermodynamic phase.

Therefore, the second actually defining property of a superconductor is its perfect diamagnetism: a mag-netic field which is sufficiently small to not destroy the new state is expelled from a singly connectedspecimen irrespective of the route the superconducting state is entered (Fig. 2.1). Hence, superconduc-tivity is a thermodynamic phase which, in turn, implies the existence of a critical field Bc(T ) defining thephase boundary between the normal and superconducting states in the presence of a field and determin-ing the condensation energy (see section 2.2). The field free state is usually called Meissner state. Themagnitude of critical field can be approximated by a parabola,

Bc(T ) = Bc(0)

[1−(

TTc

)2]. (2.1.6)

The microscopic theory arrives at a similar phenomenology but there are deviation which depend on theelectron-phonon coupling strength.

The path-independent complete exclusion of magnetic flux from the interior of a bulk superconductorin the Meissner state raises several questions: (i) How does the field (induction B = µ0H) change at thesurface? (ii) How are the screening currents set off if not according to Faraday’s law? (iii) Is it possiblethat the flux penetrates partially to reduce the magnetic energy? (iv) What happens in a material whichis not simply connected, i.e. has voids?

The level of complication in finding answers to these questions differs remarkably. (ii)-(iv) can be an-swered only qualitatively here and will be discussed in detail in the following chapters. (i) follows fromthe phenomenological London theory (see section ??). Electrodynamics shows directly that the appliedfield decays exponentially away from the surface into the bulk. The same holds true for the screen-ing currents. The characteristic length scale is called the London penetration depth λL and is of order20 . . .200 nm. The range can be traced back to the carrier concentration. Pippard observed a dependenceof the penetration depth on the purity of a material and concluded that non-local electrodynamics mustbe used for properly explaining this effect. This means that the response at a point r depends on theperturbation in a material dependent volume of order |r− ξ0|3. The new characteristic length scale ξ0was explained microscopically 20 years later by Bardeen, Cooper, and Schrieffer (BCS theory) but alsoanticipated in the framework of the Ginzburg-Landau phenomenology.

(ii) is also a consequence of the London equation but the microscopic foundation was found only in 1957by BCS. The reason for the Meissner effect is the existence of a uniform phase θ of the superconductingelectrons being established at the transition. The current is proportional to this phase modified by thevector potential A of the applied field B0 = ∇×A, j ∝ (∇θ − eA). From a classical point of view theforce on the electrons is transverse as for the Lorentz force but exits only in quantum-mechanical systemswith a coherent phase for all carriers, where a finite vector potential sets off a current irrespective of

2013


the history. The uniform phase was anticipated by Fritz and Heinz London already in 1935 without atheoretical justification. Also Ginzburg and Landau used a complex order parameter with a single-valuedphase to construct the Free-Energy functional but did not discuss its origin. It needed the Geistesblitz(flash of genius) of Bob Schrieffer to realize the necessity of a coherent state of fermions and to constructit in a fashion similar to that proposed by Schrodinger for bosons in 1926. So we have to understand thefull microscopic theory (see chapter 3) to properly explain the Meissner effect. A different approach ispossible via a thermodynamic argumentation which we shall discuss in chapter 4.

The full exclusion of magnetic flux is associated with a rapidly increasing energy E = V/2µ0B2 withV the sample volume. If we recall that the critical fields are moderate in elemental superconductors thenew phase will not survive sufficiently long to make the materials particularly useful for applications. Isthere a way around as insinuated in (iii)?

There are two qualitatively different answers. The first one is an effect of the demagnetization as de-scribed in Appendix 1. Around superconducting samples having shapes different from infinite slabsor cylinders oriented along H0 the field changes close to the surface once the Meissner state is estab-lished. This phenomenon originates in the usual electrodynamic relation ∇ ·B = 0 and the conditionn× (H0−Hi) = 0 describing the continuity of normal and, respectively, the tangential components ofthe fields around a superconductor. Whenever the demagnetization factor n is different from 0 the sur-face enhancement of the field around a generally shaped superconductor implies that the critical fieldis reached earlier in locations with enhanced field than in those without. In the case of a sphere theenhancement is 1/3 at the equator with the equatorial plane perpendicular to the homogeneous field withmagnitude B0. Consequently the field starts penetrating for B0 = 2/3Bc. In the range 2/3Bc < B0 < Bc,the sample is in the intermediate state with normal and superconducting regions coexisting on a macro-scopic scale. If a slab (for instance a thin-film sample) with area S ≈ a2 and thickness t much smallerthan a is aligned perpendicular to the field n is close to unity and the field becomes very large at the edgesrapidly exceeding the critical field, and flux penetrates even for very small applied fields B0 Bc. Forstability reasons the field in the normal regions must be Bc and vanishes in the superconducting parts.For satisfying the conservation of flux (∇ ·B = 0) everywhere the density of the superconducting regionsscales with the ratio of the applied to the critical field as ρsc = 1− (B0/Bc).

If the intermediate state is studied with a decoration experiment, with magneto-optics or neutrons onefinds a regular pattern of the normal and superconducting regions even in a perfect crystal. The regionsare aligned along high-symmetry lines of the crystal lattice. It is in fact the electronic structure whichdefines preferential orientations as will become plausible in chapter 3. The second part of the answerrefers to a new state which is sufficiently important to be discussed in a new paragraph.

2.1.3 Shubnikov phase (mixed state)

The energy needed to keep the magnetic field completely outside the superconducting volume increasesquadratically with the field. In turn, a complete penetration quenches superconductivity. Shubnikovshowed 1936 that a small amount of Tl in Pb leads to a new mixed state with some flux starting topenetrate the material long before superconductivity vanishes [?] The experimental magnetization curvesfor a Pb single crystal with 5%Pb replaced by Tl are shown in Fig. 2.2. For the cylindrical shape of thesample a demagnetization effect can be excluded. Later it turned out that practically all non-elementalsuperconductors and Nb share the property of normal regions encircled by superconducting screeningcurrents around the field lines [panel (b)]. Each vortex carries only the smallest amount of flux possible(see next paragraph). There is a lower critical field Bc1 at which the field starts to penetrate and an uppercritical field Bc2 at which the material becomes a normal metal. The creation of the first flux line at Bc1does not cost energy. Therefore the magnetization decays at an infinite rate as shown in panel (c) ofFig. 2.2. At Bc2 the magnetization vanishes at a finite rate.


Key Experiments SUPERCONDUCTIVITY 13

mixed state

Meissner state

Type II SC-0M

B0Bc1 Bc2

(a) (b)

(c)

Figure 2.2: Type II superconductor in a magnetic field. (a) Induction B vs. applied field of a cylindricalsample of Pb95Tl5 [?]. There is no jump of the induction at a critical field but rather a sharp onset at Bc1and a continuous approach to the normal state induction. At Bc2 the superconducting induction branchesof the normal one with a different slope. (b) Above Bc1 the partially penetrates the material. Instead ofrunning around the perimeter of the samples the screening currents encircle individual flux lines, eachcarrying Φ0, and form vortices. (c) Schematic plot of the magnetization −µ0M = Bi−B0 vs. B0. BelowBc1 Bi = 0. The rate of reduction of −µ0M is logarithmically divergent indicating that the energy needfor the first flux line is vanishingly small. At Bc2 the slope is finite.

The existence of the mixed state separates the superconductors in two classes. Those with completeMeissner effect are usually called type I while type II materials are characterized by a Meissner state atlow fields and a mixed state between Bc1 and Bc2. The infinite slope of −µ0M at Bc1 and the completereversibility of the magnetization indicate that the flux lines can move freely in the mixed state. Conse-quently, even though the field at which superconductivity vanishes is occasionally very high, also type IIsuperconductors still appear to be of little practical use since any current leads to a movement of theflux lines due to the Lorentz force and to dissipation. Only when the pinning of flux lines in disorderedalloys such as NbTi or by the introduction of artificial pinning center was achieved applications emergedrapidly. Materials with strong pinning are called magnetically hard or sometimes type III superconduc-tors. The applications aspects were among the reasons why Abrikosov earned the 2003 Nobel prize forhis theoretical description of the mixed state on the basis of the Ginzburg-Landau theory (chapter 4). Thestudy of the flux line lattice remains an important field of research into superconductors.

As briefly mentioned above the vortices carry a flux which, for energetic considerations, is small. Froma topological point of view a superconductor perforated by normal conducting regions is not simplyconnected any further. Hence, if a persistent current having a quantum mechanical nature encircles avortex quantization effects can be expected. Similar effects can be expected when a macroscopic sampleis not simply connected. This brings us finally to a discussion of question (iv) of paragraph 2.1.2.

2.1.4 Quantization of the flux

When electrons move freely they do not change their quantum state. Therefore, if they form a non-dissipative current in a ring one can expect the angular momentum to be quantized according to theSommerfeld condition. This was the reasoning following Fritz London’s prediction of 1950 when the

2013


B.S. Deaver and W.M. Fairbank, PRL 7, 43 (1961)R. Doll and M. Näbauer, PRL 7, 51 (1961)Figure 2.3: Experimental setup for (left) and results of flux quantization measurements [?, ?]. The mag-

netic flux trapped in the superconducting lead cylinder is measured with a resonance method. Once thecylinder is below Tc and the static field in y-direction B0 is switched off the field BM starts oscillat-ing at the resonance frequency of the cylinder. The oscillation amplitude as detected with a mirror isproportional to the magnetic moment of the cylinder, hence the included flux.

experiments on the flux quantization in multiply connected superconductors were discussed in the late1950ies and early 1960ies. The experiments were performed simultaneously by two groups which did nothave information about each other. Deaver and Fairbanks [?] in Stanford used a vibrating sample (Foner-type) magnetometer for determining the trapped flux in a small cylinder. In the experiment performedby Doll and Nabauer at the Bavarian Academy of Sciences and Humanities (shown in Fig. 2.3 l.h.s.) themagnitude of the magnetic field B0 stored in a hollow cylinder with cross section S was studied usinga resonance method. For freezing in the flux a homogeneous field B0 is applied along the axis of alead cylinder. Then the cylinder is cooled below Tc. The trapped flux and, consequently, the oscillationamplitude are found to increase in steps. The results demonstrate that the flux in the cylinder Φ = B0Scan assume only multiples of h/2e = 2.067833758(46)×10−15 Wb.

Similarly important as the observation of the quantization in itself is the magnitude of the flux quantum.In contrast to the expectation of London it is only h/2e rather than h/e. Onsager was probably the first toconsider this possibility soon after the publication of the BCS theory in 1957. The meaning can hardlybe overestimated: the wave function which imposes the quantization condition is that of electron pairs,exactly as derived by BCS, rather than single electrons.

2.1.5 Josephson effects

Another manifestation of the quantum nature of the superconducting state is the observation of coherencephenomena between two weakly connected superconductors. The setting is similar to that of a tunnelingcontact and the basic condition is that the wave functions of the superconducting condensates ratherthan those of the single particles overlap. Then one can observe all phenomena characteristic of twocoupled quantum mechanical oscillators as will be derived in detail in section ??. The main results are


Thermodynamics SUPERCONDUCTIVITY 15

two equations which describe the statics (dc) and dynamics of carriers across a weak link [?],

I(γ) = Ic sin(ϕ1−ϕ2) (2.1.7)∂γ

∂ t=

2π

Φ0U (2.1.8)

where ϕ1,2, γ , and Φ0 are, respectively, the phases in the weakly coupled superconductors 1 and 2,the phase difference and the flux quantum. U is the voltage across the junction. In the case of the dcJosephson effect there is a dissipationless supercurrent without a voltage drop across, e.g., an insulator.The second equation describes the dynamics if the current exceeds the critical current Ic which is theonly material dependent quantity. The Josephson frequency 1/Φ0 = 2e/h contains only constants andhas the value 483.597 870(11) MHz/µV.

2.1.6 Summary

Superconductivity was discovered in 1911 in Hg three years after the first liquefaction of He. Untilnow the resistivity in the superconducting state is zero to within the experimental precision of approx-imately 10−16 Ωm. The flux quantization and the Josephson effects demonstrate the quantum nature ofsuperconductivity. In addition, the magnitude of the flux quantum shows directly that the wave functionof the condensate must correspond to pairs of electrons. The perfect diamagnetism as discovered byMeissner and Ochsenfeld in 1933 demonstrates that superconductivity is a thermodynamic phase allow-ing us to derive relations for the condensation energy, the entropy, and the specific heat capacity in thesuperconducting state.

2.2 Thermodynamics

The thermodynamic properties provide us with a great deal of important information on a material inparticular on the energetics. Clearly, we learn something about the bulk while many spectroscopies(except for neutrons) suffer from surface sensitivities. However, it is not always trivial to isolate thedesired quantity out of a large variety of contributions. In addition, in the presence of magnetism thethermodynamic potentials are not uniquely defined (see Appendix 3) and the energy of the field B itselfneeds to be included in a way appropriate for the experimental circumstances. For a superconductor inthe Meissner state, for instance, the field energy stored in the sample volume equals the condensationenergy. For simplicity we use the magnitudes of B, H, and M rather than the vectorial quantities. Thiscorresponds to a specialization to cylindrical symmetry and homogeneous media.

2.2.1 Condensation Energy

Since superconductivity is a thermodynamic phase the energy difference between the normal (n) and thesuperconducting (s) state must be finite. Using the densities of the free energies fn,0 and fs,0, respectively,the energy difference for zero field (index 0) can be expressed as

fn,0− fs,0 =B2

s

2µ0. (2.2.9)

Here, the condensation energy is expressed in terms of the so far unknown field Bs. As long as theapplied field is smaller than the critical field Bc and completely excluded from the sample volume thecondensation energy does not change, hence fs,0 = fs,B. Now we assume that a cylindrical sample of

2013


a superconducting material is put in the homogeneous field B0 of a long solenoid. In this idealizedconfiguration there is no distortion of the field around the specimen since the demagnetization vanishes(n = 0). The total volume-integrated free energies F for the interior of the solenoid with the sample inthe normal and the superconducting states for a field B0 < Bc read, respectively,

Fn,B0 = V fn,0 +VB2

02µ0

+VextB2

02µ0

(2.2.10)

Fs,B0 = V fs,0 +VextB2

02µ0

, (2.2.11)

where V = A ·L is the volume of the sample having cross section A and length L and Vext is the volumeinside the coil surrounding the sample, Vext =Vcoil−V . For T < Tc(B0) (and B0 < Bc) the difference ofthe free energies becomes

Fn−Fs =V ( fn,0− fs,0)+VB2

02µ0

. (2.2.12)

If we increase B0 towards Bc from below and use Eq. (2.2.9) we obtain

Fn−Fs =VB2

s

2µ0+V

B2c

2µ0. (2.2.13)

Upon crossing the critical field superconductivity collapses and the field B0 = Bc enters V . Both thecondensation and the field energy induce an electromotive force E in the solenoid while the generatormaintains the constant current Ic required for the field Bc. If there are N windings of the solenoid over thelength L the field and the current are related as µ0NIc = BcL and E = −NΦ with Φ the time derivativeof the flux through one winding. The total electrical energy W generated at the transition is given by thetime integral over the power,

W =∫ b

a−NΦIcdt

=∫ a

b(NIc)dΦ

=BcL ·A

µ0

∫ Bc

0dB

= VB2

c

µ0. (2.2.14)

For the flux only the sample cross section is relevant since the field around the sample does not change.Since there is no dissipation (∆S = 0) W is equal to the free energy difference at Bc given in Eq. (2.2.13)and consequently Bs ≡ Bc. Hence the condensation energy is given by the critical field, and the freeenergy changes abruptly by twice the condensation energy upon crossing the Bc line.

While the (Helmholtz) free energy is useful for studying the electromagnetic energy released at thetransition it is less useful for practical purposes for depending on the volume V and, practically always,on the magnetization M. In an experiment it is next to impossible to control these variables. Therefore,a potential which depends on the pressure p and the applied magnetic field B0 = µ0H0 is desirable. TheGibbs potential (Gibbs free energy) G=U−T S− pV−BH0 having the differential dG=−SdT−V d p−V MdB0 (upon using the proper U) depends only on controllable parameters and, in addition, is the mostappropriate function for studying phase transition. Since G is the macroscopic version of the chemical



potential µ , G = Nµ , G1(T,Bc) and G2(T,Bc) must be equal at the transition between phase 1 and 2.Therefore, we replace now F by G. For the densities we can write g = f −BH0 and get

Gn,B0 = V fn,0−VB2

02µ0−Vext

B20

2µ0(2.2.15)

Gs,B0 = V fs,0 −VextB2

02µ0

. (2.2.16)

The second term on the r.h.s. of Eq. (2.2.15) is given only by the external field since M is usuallyvanishingly small in a superconducting material, and B = B0. The corresponding term in Eq. (2.2.16) iszero since M =−H0 hence B = 0 in the Meissner state. The difference

Gn−Gs =V ( fn− fs)−VB2

02µ0

(2.2.17)

obviously vanishes at B0 = Bc. For zero field the difference of both potentials yields the condensationenergy,

Gn−Gs =VB2

c

2µ0= Fn−Fs. (2.2.18)

Obviously, the field energy is irrelevant in this case and F can be replaced by F , which does not includethe vacuum field. F is preferable for theoretical considerations for containing only the magnetization asa microscopic quantity. Experimentally, F corresponds to the situation when a superconductor is firstcooled below Tc,B and then moved into the field. Then, if we recall the setting from above the generatorhas to supply only the energy of the excluded field equalling the condensation energy. What happens ifthe field is now cranked up with the sample inside until it reaches Bc(T )? One may have guessed, it isexactly the case described above and twice the condensation energy is released. This is no violation ofenergy conservation. Rather, the second half of the energy corresponds to the mechanical energy neededto move the sample into the filed from the field-free region. Using m =V M, the differentials of F and Gread,

dF(T,V,M) = −SdT − pdV +Bdm (2.2.19)

dG(T, p,B0) = −SdT +V d p−mdB0, (2.2.20)

where the index 0 at B0 is added to unambiguously denote that the field is controlled from outside. Thisis only possible in the case of G. In F , B depends on all other variables and on the location. However,for a homogeneous material in the Meissner state B = B0 is a good approximation. For constant T and pdGs can be integrated,∫ Bc

0dGs =−

∫ Bc

0mdB0 (2.2.21)

yielding

Gs(T,Bc)−Gs(T,0) =VB2

c

2µ0. (2.2.22)

In the normal state the magnetization is small and

Gn(T,Bc)−Gn(T,0)≈ 0. (2.2.23)

2013


At the the phase boundary Gs(T,Bc) = Gn(T,Bc), and the difference of Eqs. (2.2.23) and (2.2.22) yieldsagain the l.h.s. of Eq. (2.2.18). Now, without including of the field energy, F ≈G+V B0M yields F = Gat B0 = 0, and also the r.h.s. of Eq. (2.2.18) is recovered. At the critical field one finds

Fs(T,Bc) = Gs(T,Bc)+V BcMc = Gs(T,Bc)−V BcHc (2.2.24)

Fn(T,Bc) = Gn(T,Bc). (2.2.25)

Since the Gibbs potentials at Bc are equal the discontinuity of F is recovered. Finally, we calculate thevariation of Gs(T,B < Bc),

Gs(T,B)−Gs(T,0) =V

2µ0B2, (2.2.26)

Gs(T,B)−Gn(T,0)+V

2µ0B2

c =V

2µ0B2, (2.2.27)

Gs(T,B) = Gn(T,0)−V

2µ0

(B2

c−B2) (2.2.28)

which is sometimes called the Meissner parabola and shown in Fig. 2.4 (a).

Before deriving other quantities we note that B2c

2µ0is an energy density having the unit of a pressure. This

allows us to get a feeling for the order of magnitude of the condensation energy. For Nb with Tc = 9.2 K,Bc = 0.2 T, and a lattice constant a = 3.3 A we get

B2c

2µ0= 16.5

kJm3 (2.2.29)

= 16.5kPa (2.2.30)

corresponding to the pressure at 1.6 m under a water surface and a condensation energy of 2 µeV/atom.

2.2.2 Entropy

The entropy measures the degree of order in a system and can be derived from both thermodynamicpotentials (see Eqs. (2.2.19) and (2.2.20)). From an experimental point of view G is more convenient.For the consideration below, we can ignore the difference and write down the entropy S for either constantmagnetization and volume or constant applied field and pressure,

S(T ) =− ∂F∂T

∣∣∣∣M,V

=− ∂G∂T

∣∣∣∣B0,p

. (2.2.31)

In many cases one is interested in the temperature dependence of the entropy and other thermodynamicquantities. For the entropy change upon entering the superconducting state we obtain for F

∆S(T ) = Ss−Sn = − ∂

∂T(Fs−Fn)

=∂

∂T

(V

2µ0B2

c(T ))

=Vµ0

Bc(T )∂Bc(T )

∂T. (2.2.32)



Strictly speaking we can use only F here since the volume appears in the condensation energy. However,in a solid the differences between quantities measured for constant volume or constant pressure are onthe order of a few percent and will not be considered below. Apart from this subtlety Eq. (2.2.32) isextremely useful as soon as the temperature dependence of the critical field (or of F) is known. In factthe microscopic theory provides predictions which could be use. However, for getting qualitative insightwe use the phenomenological temperature dependence of Bc given in Eq. (2.1.6).

From Eq. (2.1.6) we see and from experiments we know that ∂Bc(T )∂T has no divergence at any temperature

and the following qualitative conclusions can be derived:

• Bc(T → Tc) = 0 implies that the entropy is continuous at Tc. In other words, there is no latent heatin zero field.

• In the limit T → 0 the entropy difference vanishes according to the Nernst theorem. SinceBc(0) 6= 0 the derivative ∂Bc(T→0)

∂T approaches zero. Obviously, Eq. (2.1.6) has the proper limitingbehavior for T → 0. However, systematic studies show that the curvature is not correct.

• In the range 0< T < Tc Bc(T ) decreases implying that ∂Bc(T )∂T < 0. As a consequence the entropy in

the superconducting state is smaller than in the normal state. This means that the superconductingstate has a higher degree of order and can transport heat less efficiently.

• For further considerations the derivatives of Eq. (2.1.6) are useful. We get

∂Bc(T )∂T

= −Bc(0)2TT 2

c= B′[= B′′T ] (2.2.33)

∂ 2Bc(T )∂T 2 = −Bc(0)

2T 2

c= B′′ = const. (2.2.34)

allowing us the calculate the latent heat as a function of temperature.

• The latent heat is defined as the entropy change times the temperature, L = ∆Q = T ∆S. UsingEq. (2.2.33) the density of the latent heat reads

L

V= 4

Bc(0)Bc(T )2µ0

(TTc

)2

(2.2.35)

Except for T = 0 and T = Tc there is a latent heat at the phase transition, and the transition becomesfirst order.

2.2.3 Specific heat capacity

The specific heat capacity cV,p at either constant volume or pressure is an extensively used quantitywhich reflects the bulk properties. In superconductors the low-temperature part is particularly importantfor characterizing the energy gap. Close to Tc the heat capacity has a discontinuity which reflects detailsof the transition and the coupling strength. There are various experimental complications such as defectsor the superposition of strong other contributions from, e.g., the lattice. At the moment we wish to focusonly on the concept.

2013


Usually the heat capacity is measured at constant pressure. However, cV is more desirable since allenergies depend on the Volume. Fortunately the difference |cp − cV | is small in a solid and can beignored here. The specific heat capacity is defined as

cV,p =1V

∆Q∆T

∣∣∣∣V,p

[J

m3K

](2.2.36)

where ∆Q is the thermal energy supplied in the interval between T and T +∆T . The molar heat capacitywhich is frequently used in earlier publications is then given by c Mmol

ρwith Mmol the molar mass and ρ

the mass density.

Using ∆Q = T ∆S and taking the limit ∆→ ∂ we get

cx =TV

(∂S∂T

)x,...

. (2.2.37)

It depends on the thermodynamic potential which of the variables x, . . . is taken constant (see above).For M and V being constant we obtain

cV =TV

(−∂ 2F

∂T 2

)M,V

. (2.2.38)

In the following the reference to constant variables is dropped for simplicity. The difference of thesuperconducting and the normal state heat capacities can be calculated right away from the entropydifference,

cs− cn =TV

∂ (Ss−Sn)

∂T=

TV

∂ 2

∂T 2 (Fs−Fn). (2.2.39)

From the condensation energy the famous Rutgers equation is obtained,

cs− cn =Tµ0

[(∂Bc(T )

∂T

)2

+Bc(T )(

∂ 2Bc(T )∂T 2

)]. (2.2.40)

This is a general result that can be used whenever Bc(T ) is known from a microscopic treatment. If theBCS predictions are used the data of Al, a prototypical weak-coupling superconductor, can be describedwell. Similarly well works the Eliashberg theory for strong coupling Pb.

For a qualitative visualization of the thermodynamic functions (Fig. 2.4) and for demonstrating a fewimportant limiting cases we use the parabolic approximation for Bc(T ) [Eq. (2.1.6)].

• In the limit T → Tc Bc(T ) has a finite slope and the heat capacity in the superconducting state isbigger than in the normal state,

cs = cn +8Tc

B2c(0)2µ0

. (2.2.41)

The discontinuity can be calculated on the basis of the low-temperature limes as will be donebelow. If we disregard the parabolic variation of Bc(T ) for a moment we realize that thediscontinuity depends on the rate of variation of the entropy below Tc with respect to that aboveor, sloppily speaking, on the sharpness of the kink in the entropy at Tc.



Thermodynamic functions

2

1)0()(c

cc TTBTB

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Tc,B

B

0.95 0.9 0.7 0.5 0.3 0.2 0.0

20[B

c(0)

]-2[g

s(T,

B)-

g n(T,

B)]

T/Tc,0

B/Bc(0)

F(T,0.3Bc)

(a)

0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0.0

0.2

(b) Tc,B

[ss-

s n]/T

c

T/Tc

0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

2.0 (c)

Tc,B

(cs-

c n)/c

n

T/Tc

Figure 2.4: Thermodynamic function calculated for the parabolic approximation of the critical field[Eq. (2.1.6)]. Shown are the respective differences of the superconducting and the normal state functionsfor the electronic part of (a) the Gibbs free energy gs(T,B)−gn(T,B), (b) the entropy ss(T,B)−sn(T,B),and (c) heat capacity cs(T,B)− cn(T,B) normalized to the volume and the condensation energy andrelated quantities. In (a) the Gibbs potential is plotted for various fields as indicated. For zero field theGibbs potential and the free energy F coincide (red line) and the slopes close to Tc,0 vanish indicatinga second order transition. For finite field the transition at Tc,B is first order. While |gs(0,B)− gn(0,B)|becomes smaller with increasing field for all temperatures and vanishes at B = Bc |F | starts to increasewith field as shown for B = 0.3Bc (dash-dotted green line). The free energy according to Eq. (2.2.19)stays constant at T = 0 (not shown). The entropy and the heat capacity are shown for zero field (full line)and for B/Bc = 0.5 (blue dashed line). For finite field the entropy has a discontinuity corresponding to alatent heat L [Eq. (2.2.35)] and a δ -like contribution to the heat capacity (thick blue vertical line).

• Since the entropy difference vanishes at T = 0 and T = Tc and since the superconducting state has ahigher order than the normal state ∆S is negative for 0< T < Tc and has a minimum. Consequently,∆c = cs− cn crosses zero or, in other words, cs and cn have an intersection point. For directlycomparing cs and cn we need an approximation for cn and take the Sommerfeld model whichdescribes quasi-free electrons in a solid (see, e.g., Ashcroft and Mermin). Here, cn = γT , and theconstant of proportionality γ is given by the electronic density of states for both spin projectionsat the Fermi energy N(EF),

cn = γT =π2

3k2

BN(EF)T. (2.2.42)

In a conventional metal with only the lattice and the electrons contributing to the specific heat,c(T ) = γT + bT 3, γ can be determined experimentally. If the total heat capacity c/T is plottedas a function of T 2 the intersection point at T = 0 yields γ . The discontinuity at Tc can now be

2013


expressed as

∆c(Tc)

cn(Tc)=

cs− cn

cn=

1γTc

8Tc

B2c(0)2µ0

. (2.2.43)

One could arrive at a complete phenomenology including a prediction for ∆c(Tc) if γ could bederived in “some way”. There is in fact a possibility opening up by further analyzing cs is in thelow-temperature limit [10].

• To this end we rewrite Eq. (2.2.40) using the results of Eqs. (2.2.33) and (2.2.34),

cn = γT = cs−Tµ0

[(B′′T

)2+Bc(0)

[1−(

TTc

)2]

B′′]. (2.2.44)

Since B′′ is a constant the last term approaches zero linearly for T → 0 while T (B′′T )2 vanishesrapidly. Also cs varies also faster than T as will be shown immediately and is found experimentallyfor basically all known superconductors3 hence the limit T → 0 yields

γ =4

T 2c

B2c(0)2µ0

[∝ N(EF)]. (2.2.45)

With γ plugged back in Eq. (2.2.44) the superconducting heat capacity at low temperature can beexpressed without a free parameter,

cs(T → 0) =12T 2

c

B2c(0)2µ0

(TTc

)3

. (2.2.46)

Using quite simple operations we derived several predictions demonstrating the importance of both thethermodynamic arguments and experiments. Eq. (2.2.45) demonstrates the relation between the densityof electronic states N(EF), Tc, and Bc(0) hence the condensation energy which is also found in themicroscopic theory. Even though the constant 4 is model dependent, i.e. depends here on the slope ofBc(T ) near Tc, Eq. (2.2.45) conveys the important message of an interrelation between the normal andthe superconducting state allowing one to check the plausibility and consistency of experimental results.According to Eq. (2.2.46) cs(T → 0) approaches zero as T 3. Trivially, this justifies our assumptioncs(T → 0) ∝ T 1+α with α > 0. Less trivially Eq. (2.2.46) predicts a power law for cs(T ) close to T = 0.For this reason all early data for the low-temperature specific of superconductors were compared withT 3. However the results for tin or vanadium [?] clearly show an exponential variation (see Fig.) whichwas immediately recognized as a manifestation of an energy gap in the electronic excitation spectrum.The discrepancy between the prediction of Eq. (2.2.46) and the observed variation or the BCS resultoriginates, as mentioned, in the small differences between the parabolic and the proper temperaturedependence of Bc(T → 0). From an experimental point of view affairs are even more complicated.Exponential variations are in fact rare and are found only in a few weak coupling superconductors suchas Al, Sn or V and some other elements. More recently, exponential temperature dependences werealso used for explaining cs(T ) in MgB2 or some iron-based systems. In either case multi-band effectsmake the analysis rather complicated and less stringent. In strong coupling conventional materials likeNb or Pb one finds in fact T 3 while in the copper-oxygen superconductors cs ∝ T 2 is established (seechapter 6). Different reasons are at the origin of these deviations: For Pb and Nb strong coupling effectslead to electronic states inside the gap even at T = 0 making the exponential temperature dependence tovanish. The T 3 power law can be derived in the framework of the Eliashberg theory [?, 3]. In contrast

3For a fully gapped superconductor like Al the T dependence is exponential, turning into T 3 in the strong coupling limit. Inthe case of a gap with line nodes cs ∝ T 2. Only in the case of osculating (kissing) nodes cs may vary linearly with T . Theseimportant new developments will be discussed in chapter 6.


Electrodynamics SUPERCONDUCTIVITY 23

to the conventional metals, the cuprates do not have a constant gap. Rather, the gap vanishes on lineson the Fermi surface (line nodes). Close to these nodes there are always thermal excitations except forT = 0. The density of those excitations and, hence, cs depends on the k-dependence of the gap close tothe nodes. For the cuprates the variation is linear, i.e. proportional to (ϕ−ϕ0), where ϕ is the angle onthe Fermi surface away from the position of the node at ϕ0.

Finally, we plug γ of Eq. (2.2.45) in Eq. (2.2.43) and obtain a universal number,

∆c(Tc)

cn(Tc)= 2 (2.2.47)

which is larger in comparison to the weak coupling BCS value of 1.43. Obviously, the approximation ofBc(T ) used for all our considerations here mimics strong coupling effects in general. This was showndirectly in detailed studies of the critical field [?] the main result of which is reproduced in Fig. In Alone finds 1.42(2) for the discontinuity in excellent agreement with the weak coupling result while 2.6and 2.1 are observed for Pb (Tc = 7.2 K) and Nb (Tc = 9.26 K), respectively. The BCS value is alsouniversal and is derived from the entropy of the quasiparticle excitations that reflect the gap. Largervalues for the discontinuity are strong coupling effects. The coupling strength in this context refers tothe interaction between the electrons and bosonic excitations such as phonons (considered by BCS) orcharge and spin fluctuations (believed to play a role in unconventional superconductors) for instancemediating the electron pairing below Tc. In the Landau-Fermi liquid picture the coupling is characterizedby a dimensionless parameter λ which measures the interaction related reduction of the Fermi velocityvF and the effective electron mass m∗ at the Fermi level (for an elementary discussion see, e.g., Ashcroftand Mermin). With increasing interaction vF is reduced as (1+λ )−1 while m∗ is enhanced as (1+λ )with λ varying between 0 (weak coupling) and values of order unity (strong coupling). For Al one findsλ ≈ 0.1. In the strong coupling elements Pb and Nb λ is close to 2.

We see that the discussion of thermodynamics provides deep insight into the microscopic origin of su-perconductivity without, however, explaining the origin of the ordered or condensed state. Nevertheless,the experimental observation of the entropy reduction below Tc (via the heat capacity around Tc) and ofthe existence of an energy gap in the electronic excitation spectrum (via the heat capacity close to zerotemperature) are hallmarks of research paving the way towards the microscopic explanation. In addition,the power laws found for the heat capacity of unconventional superconductors highlight the relevance ofthermodynamic studies in contemporary work.

Here, we have to pour some water on the vine. As already mentioned the heat capacity does not reveal thedesired isolated properties of the subsystem of interest which are the electrons here. Except for very lowtemperatures the contribution of the phonons dominates the heat capacity, and sophisticated methodsare necessary to isolate the electronic contribution (see Loram). In magnetic systems spin excitationsmay contribute enormously and in materials with disorder, even though very little, the contribution oftwo-level systems (Schottky anomaly) is very strong. For these reasons, various other thermodynamicquantities such as thermal conductivity or expansion are derived in order to bypass the problems inherentto the heat capacity. Concerning the copper and iron-base systems the study of the latter quantitiesrevealed qualitatively new insights (Zaini, Ando, Taillefer, Meingast, Hardy).

2.3 Electrodynamics

Naıvely, perfect conductivity occurs if the time between two collisions of an electron τ or the mean freepath ` diverge. It is a different story how this can be realized since even in an ideally clean materialperfect conductivity is a quantum mechanical effect and can be expected only at zero temperature. It isinstructive nevertheless to consider the transition to perfect conductivity before discussing the possible

2013


origin in a quantum mechanical context. In 1935, after the discovery of the Meissner-Ochsenfeld effectFritz and Heinz London considered electrodynamic consequences of a two fluid model consisting of amixture of two currents moving with and without dissipation. They did not further specify the origin ofthe superfluid current and used the limit τ→∞ but were aware that quantum mechanics must be at work.The Maxwell equations as well as some useful identities and the implications of gauge transformationsrequired here and in the following will be summarized in Appendix 4.

2.3.1 The Drude model in the limit τ → ∞

The first description of metallic conduction is due to Paul Drude [11] proposed shortly after the discoveryof the electron by Thomson in 1897. In this model the conduction electron having charge q move freelybetween the atoms of a metal with an average drift velocity vD. Between two collisions at times t andt + τ the electron is accelerated by the electric field E driving the current j which can be expressed as

j = nqvD =nqm

p, (2.3.48)

where n, m, and p are, respectively the electronic density, mass, and momentum. In an incremental timeinterval dt the electron’s momentum changes from p(t) to p(t +dt),

p(t +dt) =[p(t)+ f (t)dt +O(dt2)

](1− dt

τ

), (2.3.49)

driven by the force f (t) = qE. Contributions of higher than linear order in dt will be neglected asexpressed by O(dt2). Since the scattering time τ is finite the probability of an electron to not contributeto the average momentum change is dt/τ reducing the total acceleration by (1−dt/τ). Neglecting termsof order dt2 and higher Eq. 2.3.49 can be rewritten as

d pdt

=p(t)τ

+ f (t) (2.3.50)

which, after multiplication by nq/m and substitution of f yields(ddt

+1τ

)j =

nq2

mE. (2.3.51)

This result of Drude shows directly that the current increases linearly with time in the presence of a fieldE if τ becomes large constituting a slightly embarrassing statement. Therefore, before studying the limitτ→∞, we solve the differential equation (2.3.51). First we define the conductivity σ(ω) as j = σE andthen assume E to vary harmonically as E(t) = E0e−iωt ,(

−iω +1τ

)σE0e−iωt =

nq2

mE0e−iωt ,

yielding directly the final result for the conductivity,

σ(ω) =nq2

m1

τ−1− iω, (2.3.52)

or, upon separating real and imaginary part,

σ(ω) =nq2

mτ−1 + iωτ−2 +ω2 =

nq2

mτ

1+ iωτ

1+(ωτ)2 . (2.3.53)



The real part with q replaced by the electronic charge −e (e = 1.602 ·10−19 Cb) assumes the usual form

ℜσ(ω) =ne2

mτ

1+(ωτ)2 (2.3.54)

which is proportional to the lifetime τ . In the static limit the usual dc conductivity σ0 =ne2

m τ is recovered.The frequency dependence is given by a Lorentzian having a width τ−1. The integral over all frequenciesobeys the famous f -sum rule,

∫∞

−∞

dωℜσ(ω) = πne2

m(2.3.55)

which states here nothing else but charge conservation. Now we are prepared to derive the limit τ → 0.Eq. (2.3.52) shows directly that σ becomes purely imaginary,

limτ→∞

σ(ω) = ine2

ωm, (2.3.56)

and Eq. (2.3.55) can be interpreted as the Lorentz representation of the Dirac δ with spectral weight πne2

mand the width vanishing as τ−1. Since the conductivity is an analytic causal function, meaning that thesystem responds to a perturbation at time t0 only at t ≥ t0, real and imaginary part of σ are related byKramers-Kronig transformations, e.g. for the real part,

ℜσ(ω) =1π

℘

∫∞

−∞

dξℑσ(ξ )

ξ −ω, (2.3.57)

where ℘ means taking the principal value. Using the imaginary part of σ as derived in Eq. (2.3.56)the δ function centered at ω = 0 for the real part is recovered after some calculation and the complexconductivity in the collisionless limit becomes

σs(ω) =ne2

m

[πδ (ω)+ i℘

(1ω

)]. (2.3.58)

The preceding qualitative study of the Drude conductivity in the limit of diverging scattering time in thecontext of superconductivity is meaningful only for ω ≤ ∆/h where ∆ is the energy gap of the individualmaterial. ∆ is, as we remember, the energy reduction of the electronic system that drives the phasetransition and will be derived microscopically in the next chapter. We will see then that the infiniteconductivity at ω = 0 is a central result for superconductors. However, at finite frequency ℜσ vanishesonly for hω ≤ ∆/h and approaches the normal state conductivity asymptotically for ω > ∆/h.

The functional form of the imaginary part turns out to be very useful (once again for hω < ∆/h) if thecomplex conductivity can be determined experimentally. This is either possible by reflectivity measure-ments over wide energy ranges and exploitation of the analyticity of the response, here the reflectivity(Tanner, Basov), or by measuring reflection and transmission in thin samples or by ellipsometry experi-ments which directly return the real and imaginary part of the dielectric function ε = ε1 + iε2 and, aftersome simple algebra, the conductivity. Once σ is derived and ℑσ is divided by ω a constant is obtainedwhich directly reflects the plasma frequency

ω2pl =

ne2

ε0m(2.3.59)

and, as we will derive below, the London penetration depth λL for a magnetic field.

2013


The reasoning can be carried on a bit further by taking the curl of Ampere’s law [see Maxwell’s equations(problem 3 set 2 and problem 2 set 3)], and one obtains

(∇

2− 1c2

∂ 2

∂ t2

)B = −µ0∇× j (2.3.60)

= −µ0∇×σ(ω)E

= µ0σ(ω)∂

∂ tB. (2.3.61)

Note that we must use vectors here since derivatives with respect to r will be essential. If we solve thisdifferential equation by using the harmonic ansatz B(r, t) = B0(r)e−iωt as above for E we obtain

∇2B0(r) =

ω2pl−ω2

c2−iωτ

1− iωτB0(r). (2.3.62)

Here the focus is placed on small frequencies ω < ∆ ωpl since superconductivity is destroyed other-wise and ∆ is of order meV while ωpl is in the eV range. Therefore ω can be neglected, and the screeningequation is directly obtained

∇2B0(r) =

B0(r)[δ (ω)]2

, (2.3.63)

where δ (ω) is the frequency dependent skin depth of a normal metal,

δ (ω) =c

ωpl

√1+

iωτ

. (2.3.64)

Formally, the London penetration depth can be obtained in the limit τ → ∞. But wait a minute! This isonly possible as long as the frequency is finite. If we take the hydrodynamic or static limit ω→ 0 whichwe are interested in for a superconductor in a constant magnetic field the penetration depth diverges. Thisis exactly what happens in a normal metal where τ determines how fast the field penetrates. One couldargue that the limit τ → ∞ should be taken first. Then, however, the result is valid for all frequenciesincluding those well above ∆. Bottom line: A superconductor is not a perfect metal! While the resultsfor the conductivity are instructive and useful if the limitations are kept in mind they are not capableto provide a genuine understanding of the superconducting state. To this end a quantum mechanicaltreatment is necessary which yields the results also in the proper limits.

2.3.2 Generalized London theory

The quantum-mechanical nature of the condensate was anticipated by the London brothers already in1935 and was discussed in more detail in 1950 when, for instance, the possibility of flux quantizationwas mentioned. This implies that the condensate is described by a single wave function

ψ(r, t) = a(r, t)eiθ(r,t) (2.3.65)

having a real amplitude a(r, t) and a single rigid phase θ(r, t) 4 rather than O(1021) electronic wavefunctions in a macroscopic solid. ψ(r, t) will be inserted in the Schrodinger equation allowing us toderive an expression for a quantum mechanical current.

4Rigid does not mean constant. In the presence of fields and currents θ depends on position and time as does the amplitude.



Lorentz force and canonical momentum

With a magnetic field B present the kinematic momentum mv has to be replaced by the canonical mo-mentum p = mv+qA where q is the charge of a point-like particle (q =−e, with e = 1.602 ·10−19 Cb)and A is the vector potential. The Lorentz force Fq is directly related to this substitution as can be seenby replacing E and B by A,

Fq = m∂v∂ t

= q(E+v×B)

= q(−∇Φ− ∂A∂ t

+∇(v ·A)− (v ·∇)A). (2.3.66)

where all quantities depend on r and t. Using the total derivative of A

dAdt

=∂A∂ t

+(v ·∇)A

and reordering terms one finds a relation between the time derivative of the canonical momentum and anew scalar potential Φ∗ = Φ−v ·A at the position of the particle,

ddt(mv+qA) =−q∇(Φ−v ·A). (2.3.67)

Thus the Lorentz force is the force on a particle in the co-moving coordinate system and can be derivedfrom a generalization of Newton’s law [Eq. (2.3.67)] with the force −q∇Φ∗.

Quantum mechanical derivation of the supercurrent

In quantum mechanics we use the correspondence principle and replace the momentum by the momen-tum operator,

p → hi∇ canonical momentum

mv → hi∇−qA. kinematic momentum

Note that the kinematic momentum is not gauge invariant any further. For writing down the Schrodingerequation we substitute q and m for the charge and the mass to indicate that the formulation (here andlater in the Ginzburg-Landau theory, chapter 4) is more general and applicable also for electron pairs, forinstance,

ihψ =1

2m

(hi∇−qA

)2

ψ +qΦψ +Ξψ. (2.3.68)

The Schrodinger equation is, of course, gauge invariant since a gauge transformation has to be appliedall quantities including Φ and ψ . Eq. (2.3.68) describes a particle having charge q. In a neutral system(q = 0) only the potential Ξ = Ξ(r, t) survives which can be the chemical potential or a general potentialenergy. For q = 0 and Ξ = 0 Eq. (2.3.68) describes a free uncharged particle. However, it is useful tokeep track of possible interactions. We insert now the single-particle wave function Eq. (2.3.65) intoEq. (2.3.68) (Madelung transformation) and define the action S(r, t) = hθ(r, t). Using S and taking thereal part of Eq. (2.3.68) yields (see Problem 3 in set 4)

∂S∂ t

+(∇S−qA)2

2m+qΦ+Ξ =

h2∇2a

2ma. (2.3.69)

2013


The second term on the l.h.s. is the kinetic energy 12 mv2 and the sum of the second, third and forth terms

is the Hamilton function. The term on the r.h.s. of Eq.(2.3.69) indicates the quantum mechanical originof the equation and is proportional to the spatial variation of the amplitude of ψ . The quasi-classicallimit can be obtained by making the transition h2→ 0 directly leading to the classical Hamilton-Jacobiequation,

∂S∂ t

=−H. (2.3.70)

The imaginary part of Eq. (2.3.68) yields

∂a2

∂ t=−∇ ·

a2

m(∇S−qA)

(2.3.71)

which is a continuity equation for the probability-current density

jp =a2

m(h∇θ −qA) (2.3.72)

since a2 =ψ∗ψ is a probability density the multiplication of Eq. (2.3.72) with q or m yields the respectivecharge and mass densities a2q and a2m and current densities jpq and jpm. The result is, of course,equivalent to the usual quantum mechanical current density, which was already derived by Schrodingerand follows also from the Ginzburg-Landau equations,

jp =a2

m

ψ∗(

hi∇θ −qA

)ψ +ψ

(− h

i∇θ −qA

)ψ∗

(2.3.73)

and can be obtained by using Eq. (2.3.65) for ψ . The supercurrent in a charged system can be written as

js =a2qm

(h∇θ −qA)≡ nsq2

m(hq

∇θ −A) (2.3.74)

where ns = a2 is the density of superconducting “carriers” having mass m and charge q. This equationcan be considered the central result of the London theory. All London equations can be derived directlyfrom this quantum mechanical expression.

2.3.3 The London equations

As shown experimentally (see paragraph 2.1.4) and derived theoretically (see chapter 3) the charge ina superconducting condensate is 2e (for traditional reasons the positive sign is used) implying m→ 2mand ns→ n/2 for the mass and the density, respectively. These values will be used from now on.

For practical purposes it is more convenient to have a direct relationship between the current and themagnetic field. By applying the curl to Eq. (2.3.74) and remembering that ∇×∇θ = 0 the secondLondon equation

∇× j(r) =−ne2

mB0(r). (2.3.75)

follows immediately. It describes the existence of a current in the presence of a static field B0 andexplains the Meissner-Ochsenfeld effect phenomenologically. In the static limit and for an isolated pieceof a superconductor the charge density does not change and the continuity equation (2.3.71) yields ∇ · j =



−ρ = 0. Then the screening equations for the field B and the current density j can be derived directlyfrom Eq. (2.3.75) by either substituting j = µ

−10 ∇×B or taking the curl, respectively,

∇2B0(r) =

B0(r)λ 2

L, (2.3.76)

where we substituted m/µ0ne2 by λ 2L , the square of a length, and

∇2j(r) =

1λ 2

Lj(r). (2.3.77)

The expressions demonstrate that the field and the supracurrent exist only in a surface sheath of thicknessλL as will be discussed in detail in the following paragraph. Upon substituting B0 =∇×A in Eq. (2.3.75)the resulting equation

j(r) =−ne2

mA(r) (2.3.78)

is the not gauge invariant version of the second London equation. A closer look reveals that Eqs. (2.3.74)and (2.3.78) are equivalent modulo a gauge transformation. Using Eq. (2.3.78) one can derive a screeningequation also for A. Finally, by taking the time derivative of Eq. (2.3.78), one gets the accelerationequation

∂

∂ tj(r) =

ne2

mE(r), (2.3.79)

which was discussed earlier and is sometimes called the first London equation. Even if we finally arriveat equations, which are formally equivalent to those derived from the Drude model in the limit τ → ∞

the argumentation here is qualitatively different. Most importantly, the quantum mechanical nature isproperly taken into account allowing us to avoid artificial and unphysical arguments such as an infinitelifetime at finite temperature.5 In addition, the result is valid in the right limit of 0 ≤ ω ≤ ∆ whichparticularly includes the static limit. The upper limit ∆ will be derived in chapter 3.

2.3.4 Some conclusions

We discuss first consequences of the screening and then demonstrate the the main conclusions from thequantum mechanical nature of the Eq. (2.3.74) before returning to a special case of screening.

The penetration of field and current into a semi-infinite superconductor

Eq. (2.3.76) describes the screening of a magnetic field by a superconductor. Here we present a solutionfor the simplest possible geometry being a semi-infinite solid with the y− z plane at x = 0 separating thesuperconductor (x ≥ 0) from the vacuum. The homogeneous field B0 = Bz,0 is aligned along the z-axismaking the problem one-dimensional with Bz(x) only varying along x,

d2Bz(x)dx2 =

Bz(x)λ 2

L. (2.3.80)

5This is a point of view which became clear only after the formulation of the microscopic theory and is not intended tocriticize earlier proposals. However, it is the only sensible approach in these days.

2013


-2 -1 0 1 2-2

-1

0

1

A y(x)/

LBz(0

)

x/L

Ay(x<0) Ay(x>0)Bz,0

Bz,0

Bz(x)

Figure 2.5: Theoretical prediction for the penetration of the induction B = Bz(x), the current densityjs = j(s)y (x), and the vector potential A = Ay(x).

The usual ansatz for the differential equation (2.3.80) is an exponential, Bz(x) = beαx, yielding the char-acteristic equation α =±λ

−1L . The boundary conditions Bz(0) = Bz,0 and Bz(x→ ∞) = 0 determine the

solution completely,

Bz(x) = Bz,0e−x

λL . (2.3.81)

The current density follows from Ampere’s law, µ0js = ∇×B ≡ ∇× ezBz(x), and is oriented along they-axis represented by the unit vector ey. The vectorial relation reads

js(x) = eyBz,0

µ0λLe−

xλL . (2.3.82)

Eq. (2.3.78) gives one possible choice for the vector potential being anti-parallel to the current density,

A(x) =− mne2 js = ey

mµ0ne2

Bz,0

λLe−

xλL . (2.3.83)

B = ∇×A leads back to Eq. (2.3.81). The relationship between the various quantities is summarized inFig. 2.5. The characteristic length λL is called the London penetration depth and is given by

λL(0) =√

mµ0ne2 = c

√ε0mne2 =

cωpl

. (2.3.84)

The first equality in Eq. (2.3.76) is found frequently, probably because it follows directly from the screen-ing equation (2.3.76), but obscures the meaning a little. In fact, Eq. (2.3.84) is a remarkable result since itrelates a genuinely superconducting length scale with the velocity of light and a plasma frequency whichis orders of magnitude above the energies relevant for the condensate. In addition, ωpl is composed onlyof quantities characterizing the normal state. Note, however, that one can substitute the values of q, m,and ns without changing ωpl. Typical plasma frequencies for metals are in the visible range and above(1-5 eV) corresponding to approximately 2π(0.2 . . .1)1015s−1 and yield a range from 40 to 200 nm forλL in overall agreement with experiment (see Table 2.1).

Table 2.1: Experimental magnetic penetration depths in nm of selected superconductors.

Al Pb Nb3Sn YBa2Cu3O7

30±15 40 80 90±10



ωpl results from the broken gauge symmetry or the unique phase and phase stiffness of the condensateand is closely related to the Higgs mechanism which makes the photons to acquire a finite mass in thepresence of a gauge field being equivalent to the vector potential Ahere. [?, 12] Hence, the Meissnereffect is the Higgs mechanism in a superconductor and ωpl reflects the rigidity of the phase rather thanthe relevant energies. At Tc λL(T ) diverges, the field penetrates, and the photons lose the mass theyacquired due to fixed phase of the condensate.

λ−1L is proportional to

√ns which depends on temperature and vanishes at Tc. If there is a gap in the

electronic excitation spectrum ns saturates exponentially at low temperature while if the gap becomeszero for specific k-points ns can be described by characteristic power laws. For this reason precisemeasurements of λL(T ) play an important role for characterizing superconductors.

There are various methods which are particularly sensitive to changes rather than absolute magnitudes.Prominent examples are microwave, µ spin rotation, optical (see above) but also magnetometry tech-niques. For example if a sample is put in a maximum of the B field of a superconducting resonator thequality factor Q2 changes when the penetration depth of the sample changes as a function of tempera-ture. However, Q2 can be determined with very high precision using state of the art equipment. Theresults of this technique on very pure YBa2Cu3O7 samples (Hardy) finally convinced the majority of thepeople that superconductivity in the cuprates is unconventional. In a similar fashion by measuring thechanges of the penetration depth as a function of temperature in UPt3 and UBe13 with a magnetometerthe proposal of unconventional pairing was put forward for the first time [13].

Note that all previous equations here are strictly local in that the current in rdepends only on the field inr. Upon studying the penetration depth of PbIn alloys having various concentrations of defects and meanfree paths ` Pippard found the penetration depth to depend systematically on `. He concluded that theremust be another length scale characterizing the superconducting state and called it coherence length ξ0.Whenever ` and ξ0 are of the same order of magnitude the modification of λL turned out to be particularlystrong. in fact, ξ0 measures the volume around rover which one has to integrate to properly calculate theresponse to a perturbation in a point r. In other words, the response in a superconductor is not local butdepends on the neighborhood of a point. This purely experimental observation, to which we return whendiscussing the London vortex, was reproduced in an excellent fashion by the BCS theory [2].

Vanishing canonical momentum

If the velocity v in the canonical momentum Eq. (2.3.67) is expressed in terms of the supracurrent js =nevs one obtains

p =( m

nejs + eA

)(2.3.85)

and, upon using Eq. (2.3.78),

p = 0. (2.3.86)

This is another remarkable result that predicts the existence of superconducting charge carriers withvanishing total momentum and consequently anticipates Cooper pairing of electrons at momenta k and−k.

A simply connected superconductor

For an isolated bulk sample of a superconductor the charge density ρs is constant. This corresponds tothe absence of currents js flowing in or out,

0 =∂ρs

∂ t= ∇ · js = ∇ · e

2ns

m

(h2e

∇θ −A). (2.3.87)

2013

32 R. HACKL AND D. EINZEL Basic Experiments and UnderstandingFluxoid Quantisierung

AJ QM

QN ssQ

SS

sQLQ

JA 20

...,2,1,0;2),(

nntdS

rr

B0

Qh

n

nQ

d

d

S

sQL

S

sQL

0

0

20

20

'

2

JBS

JAr

Flux(oid) QuantFigure 2.6: Determination of the flux through a hollow cylinder. The homogeneous applied field B0is parallel to the cylinder axis and perpendicular to the surface S with boundary ∂S being inside thesuperconductor.

Except for the term ∇θ Eqs. (2.3.78) and (2.3.87) look similar. The question is therefore how they canbe reconciled. The problem was already realized by BCS, and they argue that ∇θ vanishes in the rightgauge. It is elucidating to argue the other way around and use the London gauge ∇A = 0. Then ∇2θ

follows immediately and ∇θ = const is a direct consequence. Since the starting point was an isolatedsuperconductor with no currents crossing the surface ∇θ has to vanish on the surface since the current isproportional to ∇θ and ∇θ |surface = 0. With ∇θ = const ∇θ ≡ 0 follows immediately demonstrating therigidity of the phase independent of statistical arguments.

It is interesting to study Eq. (2.3.87) without using the London gauge. To keep things simple we assumea harmonic spatial variation of all fields having the form f (r) = f0eikr yielding the replacement ∇→ ikand arriving at

js =q2ns

m(∇θ

∗−A) =q2ns

m(ikθ

∗−A) .

Taking the divergence in the static limit as above one finds ik ·A+ k2θ ∗ = 0 allowing one to eliminateθ ∗. The resulting current

js =q2ns

m

(k(k ·A)

k2 −A)

(2.3.88)

is now gauge invariant and equivalent to the second London equation [Eq. (2.3.75)]. k is the wave vectorcharacterizing position dependent fields on length scales of λL. k ·A is the longitudinal projection of thefield A on the direction of the fastest change of θ ∗, i.e. ∇θ ∗, and vanishes in the London gauge.

Quantization of the fluxoid

The quantum mechanical form of the supercurrent [Eq. (2.3.74)] provides us with the proper tool toexplain the flux quantization experiment described in paragraph 2.1.4 and to get additional insight intothe necessary experimental conditions. The setting, this time from a theoretical point of view, is sketchedin Fig. 2.6. We rearrange [Eq. (2.3.74)] and integrate over the entire boundary ∂S of the surface S (seeFig. 2.6) on both sides,

hq

∮∂S

∇θ ·d`=∮

∂S

A+

mnsq2 js

·d`. (2.3.89)



The line integral over a gradient of a field θ is given by the difference of the fields in the starting andend points. Since θ is the phase of the wave function Eq. (2.3.65), which has to be defined uniquely in aposition r, θ is defined only up to integer multiples of 2π ,

ψ(r) = a(r)eiθ(r) = a(r)ei(θ(r)+2πn) = ψ(r),

and the integral over the phase gradient yields∮∂S

∇θ ·d`= 2πn. n ∈ Z (2.3.90)

Using this result and Stokes’ theorem Eq. (2.3.89) can be recast,

hq

n =∫

S(∇×A) · ndS+µ0

∮∂S

λ2L js ·d`, (2.3.91)

where n is a unit vector perpendicular to S. The expression on the r.h.s. is the fluxoid since it dependsnot only on the flux through the cross section of the hollow cylinder but also on the screening currentsin the cylinder walls. The quantization of the fluxoid was demonstrated by Little and Parks [?] in 1963.Only if the walls are much thicker than λL one can find an integration path which is sufficiently buriedinside the wall so as to js→ 0 along ∂S and to make the second term on the r.h.s. of Eq. (2.3.91) vanish.Only in the case js = 0 the flux is quantized,

Φ =∫

SB · ndS =

hq

n. (2.3.92)

The London vortex

Finally we return to the screening equation in a mathematically more complicated geometry and calculatethe energy in a single vortex line for getting a first idea of how the mixed state (see section 2.1.3)develops. First we determine the asymptotic behavior of the field around the line as a function of thedistance r. The second step is a lengthy calculation (see Fossheim and Sudbø), and we write down onlythe result. A single vortex can penetrate a simply connected superconducting material for instance in atype II superconductor very close to Bc1. Then there is a normal filament with radius ξ λL (vortexcore) oriented along the field, e.g. ez, where the field is maximal and supercurrents circulating around thefilament which screen the field over a length scale of λL. By replacing ∇2B we can write the screeningequation as

B+λ2L ∇× (∇×B) = ezΦLδ (r) (2.3.93)

where δ (r) is the two dimensional Dirac δ function. The meaning of Eq. (2.3.93) becomes immediatelyclear if we integrate over circular surface S perpendicular to ez having a radius r λL,∫

SBdS+λ

2L

∮∂S(∇×B)d`= ezΦL. (2.3.94)

For the second term we used again Stokes’ theorem. With ∇×B = µ0js and js = 0 for r λL ΦL isthe total flux along the line. If, on the other hand, ξ < r < λL only a fraction r2/λ 2

L of the total fluxis screened, and we set

∫S B · ndS zero. The remaining part can be further simplified by exploiting the

cylindrical symmetry,

|∇×B|=−∂B∂ r

=ΦL

2πrλ 2L, (2.3.95)

2013


where 2πr results from the line integral, yielding

B(r) =ΦL

2πλ 2L

[c− ln

(r

λL

)]. (2.3.96)

Since this approximate expression diverges for r → 0 ξ has the role of a cut-off below which B(r)becomes constant. The full functional form is the solution of a Bessel differential equation resultingfrom ∇× (∇×B) in cylindrical spherical coordinates and B = ezBz,

1r

∂

∂ r

(r

∂Bz

∂ r

)− Bz

λ 2L= 0 (2.3.97)

that leads to a 0th-order Bessel or Hankel function K0,

B(r) =ΦL

2πλ 2L

K0

(r

λL

)(2.3.98)

which, for r < λL, has the asymptotic behavior described above and decays exponentially for r > λL. Ifone determines the energy of a vortex line by integrating over the field and the current density

L =1

2µ0

∫B2 +λ

2L (∇×B)2dr (2.3.99)

one finds for the energy per unit length L

L =Φ2

L

4πλ 2L µ0

[ε− ln

(ξ

λL

)]; ε ≈ 0.1 (2.3.100)

where the term ε originates in the condensation energy. For a qualitative argument (see Annett) onecan neglect the condensation energy and confine the integration to the region ξ ≤ r ≤ λL and substitute∇×B) by j = ΦL/(2πrλ 2

L )eφ )in Eq. (2.3.99). Since we are interested in the energy per unit length theintegration dr = d2r = 2πrdr yields Eq. (2.3.99) with ε = 0.

The significance of Eq. (2.3.100) is in the proportionality of the energy L to the square of the number offlux quanta, n2. This means that the costs of getting more than one flux quantum into a vortex increasemore rapidly than those for creating n vortices, nL1 < Ln. Eq. (2.3.100) but also the derivation of theasymptotic behavior highlight the existence of a second length scale ξ over which superconductivity isalmost completely suppressed while the supracurrents and the field can coexist between ξ and severalλL. ξ is essentially the same length scale as ξ0 derived by Pippard upon studying the penetration depth indisordered superconductors (see above). ξ ≈ ξ0 will turn out to be related to the length scale over whichcoherence in the condensed state can be maintained and which reflects the “diameter” of a Cooper pair.Further details will be discussed in the frameworks of the BCS and the Ginzburg-Landau theory.


Chapter 3

Microscopic Theory

The microscopic theory was finally presented in 1957. The last break-through was the derivation of acoherent wave function using a superposition of Fermions by Schrieffer. Cooper, on the other hand,showed that an infinitesimally small attractive interaction between Fermions having E > EF leads toa reduction in their energy by an amount proportional to the cut-off energy of the interaction. Thepossibility of an attraction between electrons mediated by phonons was already proposed by Frohlichand studied in detail by Pines and Bardeen. Under the supervision of John Bardeen the three ingredientswere combined. The plan of the chapter is as follows: We first prove the existence of the Cooperinstability. In the second paragraph we show that the electron-phonon interaction leads indeed to anattractive interaction. The third paragraph is devoted to the derivation of the coherent wave function.Finally, we show how the Hamiltonian of the interacting system can be minimized at zero and at finitetemperature. In the last part we calculate a few response functions in order to show how a superconductorreacts to external perturbations.

3.1 The Cooper Instability

We assume that the Fermi sea is completely filled at zero temperature. We now add two particles withmomenta k1,k2 > kF as shown in Fig. 3.1. If we construct a state of two Fermions the total wave functionhas to obey anti-commutation relations so as to satisfy the Pauli principle,

ψ(r1,σ1,r2,σ2) =−ψ(r2,σ2,r1,σ1). (3.1.1)

We can separate the wavefunction ψ into three factors if we introduce relative coordinates, r1 = R+r/2and r2 = R−r/2, with R and r describing the center of mass and the distance of the two electrons. Withspin part represented by χ(σ1,σ2) and the propagating part by plane waves ψ reads

ψ(r1,σ1,r2,σ2) =1√V

eiK·Reik·rχ(σ1,σ2), (3.1.2)

which can be reformulated as

ψ =1√V

eiK·RΦ(r1− r2)χ(σ1,σ2). (3.1.3)

35

36 R. HACKL AND D. EINZEL Microscopic TheoryCooper-Paare

,k

T = 0

EF

EF + D

,k

Triplett 2

1

Singulett 2

1

),,,(-

)(),,,(

spin,

spin,

1122

spin,212211

21

21

21

rr

rrrr RKie

Figure 3.1: Cooper instability of two particles having opposite momenta and spins in a shell with thick-ness hω0 above the Fermi energy EF . At T = 0 the Fermi sea is completely filled hence the additionalparticles have an energy above EF .

It was shown before that the London theory predicts vanishing canonical momentum of the supercon-ducting carriers [Eqs. (2.3.78) and (2.3.85)] on purely electrodynamic reasons. This momentum can nowbe identified as the center of mass of two electrons implying a state k2 = −k1. Consequently, the firstterm in Eq. 3.1.2 is essentially unity in equilibrium. The second term shows that the carriers can haveinternal relative momentum. The third term describes the spins. In a system with inversion symme-try where parity is a good good quantum number the two possible relative orientations of the spins areparallel and anti-parallel, and one arrives at triplet and singlet wave functions

χt(σ1,σ2) =

| ↑↑〉

1√2(| ↑↓〉+ | ↓↑〉)| ↓↓〉

(3.1.4)

χs(σ1,σ2) =

1√2(| ↑↓〉− | ↓↑〉) . (3.1.5)

If the two spins are exchanged χ t(σ1,σ2) and χs(σ1,σ2) conserve or change sign, respectively. Sincethe center-of-mass part is constant the product of the functions χ and Φ must be anti-symmetric tofulfill Eq. (3.1.1), Φ has to be symmetric and anti-symmetric for singlet and triplet spin wave functions,respectively. Possible representations are Φs = cos[k · (r1− r2)] and Φt = sin[k · (r1− r2)]. Recently,superconductivity was discovered in CePt3Si [?] and other systems without inversion symmetry. As aconsequence the distinction between singlet and triplet states is not possible any further and there arealways mixtures. In addition the Fermi surface splits up into two sheets. However, we won’t dwell onthis exotic though exciting and well studied case [?] but use it just as an appetizer for the part on novelsuperconductors.

Now, we derive the energy gain for the case of a small attractive interaction between the two extraelectrons outside the Fermi sea. If we leave out all complications this is a simple exercise in perturbationtheory. We assume that the unperturbed system has only kinetic energy T and count the energy fromthe chemical potential µ . From a statistical point of view we take the grand canonical potential with theparticle number being variable. Subtracting the energy of the Fermi sea means using the Landau-Fermiquasiparticle concept and excitation energies ξk = εk− µ rather than band energies εk. With V beingan attractive interaction between two electron states k and −k having magnitudes larger than the Fermimomentum, |k|> kF and energy ξk > 0, the eigen energies E of the perturbed Hamiltonian H = H0 +Vcan be found from the eigen states of the unperturbed system,

H0|k〉= 2ξk|k〉 (3.1.6)


The Cooper Instability SUPERCONDUCTIVITY 37

where the functions |k〉 are a complete set of eigen functions obeying 〈k′|k〉= δkk′ . Then

H|ψ〉 = E|ψ〉 (3.1.7)

|ψ〉 = ∑k

ϕk|k〉. (3.1.8)

Multiplication with 〈k′| from the left and collecting terms yields

〈k′|(H0 +V )∑k

ϕk|k〉 = 〈k′|E ∑k

ϕk|k〉

ϕk′(2ξk′−E) = −∑k

ϕk〈k|V |k〉. (3.1.9)

The calculation of the matrix element on the right hand side is usually complicated since the poten-tial between two electrons is unknown and since there are only approximations to the wave functions.Eliashberg was the first to present a realistic model [3] in terms of a material-specific electron-phononinteraction. Only recently, the wave functions could be approximated for a few conventional systems [?].Bardeen, Cooper and Schrieffer, although having a phenomenology for the attractive interaction, as-sumed simply that Vkk′ ≡ 〈k|V |k〉 is attractive for energies smaller than a typical phonon energy hω0 andzero otherwise,

Vkk′ =

−g2

eff |ξk| ≤ hω

0 otherwise

Now Eq. (3.1.9) can be solved right away by noting that, since |k〉 is a complete set, ∑k ϕk = ∑k′ ϕ′k,

ϕk′ = g2eff

∑k ϕk

2ξk′−E

∑k′

ϕk′ = g2eff ∑

k′

∑k ϕk

2ξk′−E

1 = g2eff ∑

k′

12ξk′−E

. (3.1.10)

The k-summation in Eq. (3.1.10) can be transformed into an energy integral as ∑k = NF∫ hω0

0 dξ whereNF is the density of states for both spin projections yielding

E =−2hω0exp− 2

NFg2eff

(3.1.11)

for the case NFg2eff 1. NFg2

eff is a dimensionless parameter which is often abbreviated by the couplingconstant λ . In the BCS weak coupling approximation λ is much smaller than 1. In the strong couplingcase discussed by Eliashberg λ can be of order 1 or even larger. Here we focus on the weak-couplinglimit and note that even in this case the energy gain E cannot be expanded in powers of λ . Via thecutoff energy hω0, which will be discussed in more detail in the next section, the energy gain dependson a material property. While there are various other possibilities, in the treatment of BCS hω0 was atypical phonon energy directly explaining the experimentally observed isotope effect Tc ∝ M−β where β

is expected to be 0.5 for a harmonic oscillator (Fig. 3.2).

2013

38 R. HACKL AND D. EINZEL Microscopic Theory

BCS-Theorie

Figure 3.2: Isotope effect in tin [10]. The different symbols refer to the results of different authors. Thetable shows the exponent β for various elements.

3.2 Origin of the interaction

Although suggested by the isotope effect (Fig. 3.2) an interaction depending on the lattice is not the mostobvious way towards an attraction between electrons. Rather, the electro-static force seems to prevail byfar. In order to understand the reasoning at the time of discovery and to get an idea of the relevant physicswe first have a closer look at the Coulomb interaction in a metal which is distinctly different from that ofpoint charges given by

VC =e2

4πε0r(3.2.12)

where r = |r2− r1| is the distance between two electrons at r1 and r2. The Fourier transform is given by

VC(q) =∫

drVCe−iq·r =e2

ε0q2 . (3.2.13)

In a metal, in contrast, the Coulomb interaction is screened. For instance, a charged impurity is hardlyvisible just a few lattice constants away. The exact distance depends on the charge density and is de-scribed ba the Thomas-Fermi theory. The screened potential was proposed by Yukawa in the context ofnuclear matter, reads

VC,TF(r) =e2

4πε0re−

rrT F , (3.2.14)

and has the Fourier transform

VC,TF(q) =e2

ε0(q2 + k2TF)

. (3.2.15)

If there is one electron per lattice site rT F ≈ a and kT F ≈ π

a close to the Fermi momentum. The potentialdoes not decay strictly exponentially but oscillates around the zero (Friedel oscillations). In any case


Origin of the interaction SUPERCONDUCTIVITY 39

Ei

tEi

kk-k

-k

-k-q-k-q

k+qk+q

-q q

(a) (b)

Figure 3.3: Feynman diagrams for the interaction between two electrons via the emission and reabsorp-tion of a bosonic particle, here a phonon.

the perturbation of the charge density is screened over distances of a few lattice constants, and otherinteractions can gain influence. In conventional metals the most important additional interaction is theelectron lattice interaction. Forces from charge and spin modulations are important in systems likecuprates or iron-based superconductors but will not be studied here at the moment. In most of the casesthe energy scales of the electronic degrees of freedom are much higher than those of the secondaryinteractions implying the existence of distinctly different time scales. In the case of the phonons thisdifference is quite obvious since the masses of the involved particles are roughly 4 orders of magnitudedifferent. While the electrons follow a perturbation practically immediately the ions react slowly butremember the perturbation for longer. This is called the retardation effect and can be understood asfollows: An electron with momentum k moves in the lattice and attracts the ions close to its trajectory.This distortion of the lattice survives on time scales much longer than the lifetime of an electron betweencollisions and provides an attractive potential for the other electrons especially (for phase space reasons)for those having opposite momenta−k. Pines and Bardeen have studied the resulting potentials betweenthe electrons in the framework of the jellium model (electrons on a homogeneous background of positivecharge) and found the result used for the BCS theory [?,?]. The derivation can be found in the textbooks[14] and will not be repeated here. The result for the full potential is expressed in terms of the dielectricconstant one obtains

Vkk′ =e2

ε0q2ε(q,ω), (3.2.16)

and if the Thomas-Fermi screening and the electron-ion interaction are included in the expression forε(q,ω) the full result can be written down as

Vkk′ =e2

ε0(q2 + k2T F)

(1+

ω2q

ω2−ω2q

). (3.2.17)

One can get an idea (not a rigorous derivation) of the quantum mechanical processes by consideringthe energy change due to emission and reabsorption of phonons by electrons as suggested by de Gennesand shown in Fig. 3.3. To this end we have to calculate in second order perturbation theory the matrixelement 〈k|Ve−p−e|k′〉 from two transitions of the form 〈k|He−p|i〉=W ∗q (i) and sum over all intermediate

2013


states i. Since there are two time-reversed processes we get

〈k|Ve−p−e|k′〉=12 ∑

i〈k|He−p|i〉

1

Ek′−Ei+

1Ek−Ei

〈i|He−p|k′〉. (3.2.18)

If we ignore the influence of the small differences between the states k and k′ on the electron phononmatrix elements the second element on the r.h.s. is just Wq(i). The initial and final energies are Ek = 2ξkand Ek′ = 2ξk′ , respectively, since ξk is a symmetric function of k as is the phonon energy hωq w.r.t. q.In the intermediate state the crystal momentum and the energy are conserved, yielding equal energies for

diagram (a) diagram (b)momentum energy momentum energy

electron 1 k′ = k+q ξk′ k ξk

electron 2 −k ξk −k′ =−(k+q) ξk′

phonon −q hωq +q hωq

both processes in the intermediate states, E(a)i =E(b)

i = ξk+ξk′+ hωq. After replacing±(ξk−ξk′) by hω

the energy denominators in Eq. (3.2.18) read ±hω− hωq. Since there is no reference to the intermediatestate any further the summation is only over the matrix elements and the total expression for the energychange due to electron-phonon coupling becomes

〈k|Ve−p−e|k′〉=|Wq|2

h12

1

ω−ωq− 1

ω +ωq

. (3.2.19)

For the full interaction including Coulomb part we use the Thomas-Fermi result [Eq. (3.2.15)] in additionto Eq. (3.2.19) and get

〈k|V |k′〉=VC,T F(q)+|Wq|2

hωq

ω2−ω2q. (3.2.20)

Although this result is anything else but quantitative it provides a feeling for the underlying physics:the energy dependence of the effective electron-electron interaction results from a phonon-mediatedperturbational correction to the electronic energies (Fig. 3.4). It is limited to a range of the order of theDebye energy hωD. At energies well above hωD the usual screened but repulsive Coulomb interactionprevails. For small energies the effect of the Coulomb interaction is overcompensated by the lattice,sometimes called overscreening effect, and can lead to a appreciable attractive potential that provides theinteraction needed for Cooper pairing.

3.3 The BCS wave function

As outlined in chapter 1 [Eq. (1.2.1)] the construction of a coherent state of bosons is relatively straight-forward. With bosons the macroscopic occupation of the ground state can be realized directly sinceall particles can be in the same state of vanishing momentum at sufficiently low temperatures. WithFermions the Pauli principle precludes a direct solution. Before we present Schrieffer’s solution to theproblem we have a brief look at the coherent state proposed by Schrodinger [?] since it has already allrelevant properties needed later and is easier tractable. We only summarize the results and leave thedetailed calculations as an exercise (problem 1 of set 1).


The BCS wave function SUPERCONDUCTIVITY 41

- 2 - 1 0 1 2- 4

- 2

0

2

4

6

e - p i n t e r a c t i o n B C S a p p r o x . V 0

(V C,TF)-1 <k

|V|k’>

ω/ ωq

Figure 3.4: Effective electron-phonon interaction for a mode hωq. For the plot Eq. (3.2.17) is used with asmall damping γ/ωq = 0.05. Both the energy and the potential are normalized. The red curve visualizesthe BCS approximation for the attractive potential using an arbitrary magnitude of V0 =−1.5.

3.3.1 Coherent states in a boson field

Bosonic and fermionic properties can be formulated in second quantization using creation (a†k) and an-

nihilation (ak) operators which usually depend on momentum k. The operators obey commutation oranti-commutation relations [a,b]∓ = ab∓ ba reflecting the symmetry properties of the wave functions.In the case of bosons they read,[

ak,a†k′

]= aka†

k′−a†k′ak (3.3.21)

= δk,k′ (3.3.22)

[ak,ak′ ] = 0 (3.3.23)[a†

k,a†k′

]= 0 (3.3.24)

where δk,k′ is the Kronecker δ . The simplest case of a coherent state is a superposition of an infinitenumber of harmonic oscillator wave functions without momentum dependence, |ψn〉 = 1/

√n!(a†)n|0〉

with |0〉 denoting the ground state,

|α〉= e−|α|2

2

∞

∑n=0

(α a†)n

n!|0〉= e−

|α|22 e(α a†)|0〉. (3.3.25)

After discussing the Madelung transformation of the Schrodinger equation and deriving the supracurrent[Eq. (2.3.74)] it becomes clear that writing the complex number α as α = |α|eiϕ means establishing anew wave function. Since α is an eigenvalue of the annihilator a, a|α〉 = α|α〉 the expectation valuesof the number operator N ≡ 〈n〉 = 〈α|a†a|α〉 and all its moments can be calculated right away withoutreferring to Eq. (3.3.25) yielding the following main results:

N = |α|2, (3.3.26)

∆NN

=

√〈n2〉−〈n〉2

N=

1√N, and (3.3.27)

∆N∆ϕ ≥ 12. (3.3.28)

2013


Kohärente Zustände

-1 0 1 2 3

0,0

0,1

0,2

0,3

0,4 <n> = 10 <n> = 25 <n> = 75

<n>1/

2 p(<n

>,n)

n/<n>

n<n>

0.4

0.3

0.2

0.1

0.0

Figure 3.5: Poisson distribution Pn(N) for different expectation values N = 〈n〉 as a function of theoccupation number n. The n-axis is normalized to N. For small N the asymmetry is still visible. Theordinate axis is normalized to

√N−1

since the maximum of Pn(N) scales as√

2πN−1

in the limit N→∞.In the sum O(

√N) distributions Pn(N) contribute for large N. In other words, in absolute units the width

increases while the relative width decreases.

Obviously, |α|2 corresponds to the average occupation number of the state |α〉. The distribution aroundN becomes increasingly sharp with increasing N and the amplitude and the phase are conjugate variables.The phase becomes infinitely rigid in the large N limit. Although all properties are describe in this wayit is instructive to write down the expectation value for the number operator explicitly using Eq. (3.3.25),

N = 〈n〉 =∞

∑n=0

n|α|2n

n!e−|α|

2(3.3.29)

= |α|2e−|α|2

∞

∑n=1

n|α|2(n−1)

n(n−1)!

= |α|2.

The r.h.s. of Eq. (3.3.29) is a Poisson distribution for n, 〈n〉 = ∑∞n=0 nPn(N) [see also Eq. (3.3.26)].

If Pn(N) is plotted for different values of N as a function of n Eq. (3.3.27) can be visualized directly(Fig. 3.5). If the k dependence is restored wave functions of atoms in a trap or laser fields can bedescribed,

|α〉= exp

(∑k

αka†k−

12|ak|2

). (3.3.30)

for instance, being, however, beyond our interest here. Rather we focus now on fermions.

3.3.2 Properties of fermions

Fermions are governed by the Pauli principle stating that each quantum state can only be occupied once.For this reason there is only the ground state |0〉 and a singly occupied state |1〉, and the wave functionsare anti-symmetric as formulated in Eq. (3.1.1). Using creators (c†

k) and annihilators (ck), the Pauli


The BCS wave function SUPERCONDUCTIVITY 43

principle corresponds to the following properties,

ck|0〉 = 0 (3.3.31)

c†k|0〉 = |1〉 (3.3.32)

ck|1〉 = |0〉 (3.3.33)

c†k|1〉 = 0. (3.3.34)

For the anti-commutation relations we use braces + in order to make affairs as clear as possible. Inaddition, we have to now take care of the spin σ ,

ckσ ,c†k′σ ′+ = ckσ c†

k′σ ′+ c†k′σ ′ckσ

= δkk′δσσ ′ (3.3.35)

ckσ ,ck′σ ′+ = 0 (3.3.36)

c†kσ,c†

k′σ ′+ = 0. (3.3.37)

We define now nkσ = c†kσ

ckσ and find 〈0|nkσ |0〉= 0, 〈1|nkσ |1〉= 1, etc. These properties are sufficientto derive a coherent state of fermions.

3.3.3 A coherent state of fermions

The target is to derive a fermion state similar to that of Eq: (3.3.30). To this end we first define pairannihilation and creation operators

Pk = c−k↓ck↑ and (3.3.38)

P†k = c†

k↑c†−k↓, (3.3.39)

respectively, and calculate their properties. P†k |0〉 corresponds to the creation of a state with an electron

at k having up-spin and simultaneously one at −k having down-spin. Since the two involved fermionshave opposite spin it is sensible to look at commutators rather than anti-commutators, yielding (as shownin problem 3 of set 5)

[Pk,Pk′ ]− = 0, (3.3.40)[P†

k ,P†k′

]−

= 0, and (3.3.41)[Pk,P

†k′

]−

= δkk′(1− nk↑− n−k↓). (3.3.42)

The latter relation is neither a commutator nor an anti-commutator in the usual sense for being differentfrom 1 or 0 in the general case. We finally need the powers of P†

k for calculating the exponential series,

P†k P†

k =(

P†k

)2

= c†k↑c

†−k↓c

†k↑c

†−k↓

= −c†k↑c

†k↑c

†−k↓c

†−k↓

= 0. (3.3.43)

2013


Note that we exchanged c†−k↓c

†k↑ in the second line producing a minus sign and a series of two equal

creators acting on the ground state thus making the whole operator to vanish. Using this and the anti-commutator in Eq. (3.3.41) two properties, Schrieffer’s ground state can now be written down as

|BCS〉 = a · exp

(∑k

αkP†k

)|0〉

= c ·∏k

exp(

αkP†k

)|0〉

= c ·∏k

(1+αkP†

k

)|0〉. (3.3.44)

For the last transformation we used that all powers of Pk beyond linear vanish. We set 〈BCS|BCS〉 = 1for determining the constant c and find

1 ≡ 〈BCS|BCS〉= c · 〈0|∏

k(1+α

∗kPk)(1+αkP†

k )|0〉.

(3.3.45)

The normalization condition can be satisfied if all factors are unity,

1 = c · 〈0|(1+α∗kPk)(1+αkP†

k )|0〉= c2(1+ |αk|2),

yielding the constant c. Using this normalization the BCS ground state can be written down as

|BCS〉 = ∏k(u∗k + v∗kP†

k )|0〉 with (3.3.46)

u∗k =1√

1+ |αk|2and

v∗k =αk√

1+ |αk|2where

|uk|2 + |vk|2 = 1. (3.3.47)

|BCS〉 is a wave function of a coherent state with the complex number αk having an amplitude and aphase. The same holds true for the “coherence factors” u∗k and v∗k which will be derived in detail later.Using the conjugate complex follows the convention in Annett’s book and is the most popular but not theonly one in the literature. The BCS wave function describes the coherent superposition of the vacuumand 2, 4, 6. . . electrons. Below we derive the average number of particles in the condensate and itsfluctuations and find full agreement with the coherent state Eq. (3.3.25).

3.4 Determination of the ground state

Using |BCS〉 we have to find the ground state of the Hamiltonian HBCS of the system by minimizing〈E〉 = 〈BCS|HBCS|BCS〉. There are various ways. BCS used real numbers, uk = sinθk and vk = cosθk,which obviously satisfy the normalization [Eq. (3.3.47)]. The most direct approach is the Bogoliubovtransformation. However, using Lagrange multiplicators helps visualizing several relevant details (An-nett). Therefore, we make this detour before deriving the Bogoliubov transformations. The problem wewish to solve is minimizing the ground state energy, E = 〈BCS|HBCS|BCS〉. To this end we switch tomomentum space which is equivalent to Fourier transform the Hamiltonian. The technique is namedsecond quantization.


Determination of the ground state SUPERCONDUCTIVITY 45

3.4.1 The BCS Hamilonian in second quantization

As already demonstrated in Problem 2 of set 5 the position dependent wave functions ψσ (r) can beexpanded in plan waves. The transformations and the corresponding inverse transformations read

ψσ (r) =1√V ∑

kckσ eik·r ckσ =

1√V

∫ψσ (r)e−ik·rdr (3.4.48)

ψ†σ (r) =

1√V ∑

kc†

kσe−ik·r c†

kσ=

1√V

∫ψ

†σ (r)e

ik·rdr. (3.4.49)

The Hamiltonian in configuration space is a sum over all N electrons and has a kinetic energy term, onefrom the external fields such as the potentials φ(r) or A(r), and the interaction between the electronswhich is supposed to lead to superconductivity,

H = ∑σ

N

∑i

[−h2

2m∇

2i −Vext(r)

]+

12 ∑

σ

N

∑i, j

V (ri− r j) (3.4.50)

The factor 1/2 in front of the interaction tern takes care of double counting. If the expansions Eqs. (3.4.48)and (3.4.49) are inserted the sum over the particles transforms into a k sum, and integrating over all spacecompletes the Fourier transformation,

H =∫

ψ†σ (r)

[−h2

2m∇

2−Vext(r)]

ψσ (r)dr+12

∫V (r−r′)ψ†

σ (r)ψσ (r)ψ†σ (r′)ψσ (r′)drdr′. (3.4.51)

While the evaluation of the kinetic energy term is straightforward that of the other contributions is moretime consuming and will be left as an exercise. The kinetic energy T works as follows:

T = −∫

ψ†σ (r)

[−h2

2m∇

2]

ψσ (r)dr

=1V ∑

σ

∑k,k′

∫ [c†

k′σ e−ik′·r h2k2

2mckσ eik·r

]dr

= ∑σ

∑k

h2k2

2mc†

kσckσ . (3.4.52)

After Fourier transforming the interaction term we can finally write

H = ∑k,σ

εkc†kσ

ckσ + ∑k1,k2,q,σ1,σ2

Vqc†k1+q,σ1

c†k2−q,σ2

ck2σ2 ck1σ1 . (3.4.53)

The interaction term can be simplified by using the assumptions for a Cooper pair, k2 = −k1 and, for asinglet state, σ2 =−σ1 and the resulting BCS Hamiltonian reads

HBCS = ∑k,σ

εkc†kσ

ckσ + ∑k,k′,σ

Vk,k′ c†k↑c

†−k↓c−k↓ck↑. (3.4.54)

The BCS Hamiltonian is a rather general and allows, for instance, for anisotropies in the interactionpotential. It does neither cover triplet pairing nor is it sufficient to deal with exotic cases such as non-centrosymmetric systems with mixtures of singlet and triplet pairing and spin-orbit splitting of the Fermisurface. However, for all our purposes below, including the discussion of the cuprates, the iron-based,and heavy fermion systems with nodes and sign changes of the energy gap ∆k. Before deriving theground state energy we determine a few expectation values for the BCS wave function.

2013


3.4.2 Some expectation values

We now calculate the expectation value for the momentum distribution function for one spin projection,

〈nk↑〉 = 〈BCS|c†k↑ck↑|BCS〉

= 〈0|(uk + vkc−k↓ck↑)c†k↑ck↑(u∗k + v∗kc†

k↑c†−k↓)|0〉

= |uk|2〈0|c†k↑ck↑|0〉+ukv∗k〈0|c

†k↑ck↑c

†k↑c

†−k↓|0〉+

+vku∗k〈0|c−k↓ck↑c†k↑ck↑|0〉+ |vk|2〈0|c−k↓ck↑c

†k↑ck↑c

†k↑c

†−k↓|0〉

= |uk|2 ·0+ukv∗k ·0+ vku∗k ·0+ |vk|2(1−〈0|c†−k↓c−k↓|0〉)

= |vk|2. (3.4.55)

The term with the prefactor |vk|2 can be broken down as 〈0|c−k↓ck↑c†k↑ck↑c

†k↑c

†−k↓|0〉 = 〈0|c−k↓(1−

c†k↑ck↑)(1 − c†

k↑ck↑)c†−k↓|0〉 = 〈0|c−k↓c

†−k↓|0〉 − 〈0|c−k↓c

†k↑ck↑c

†−k↓|0〉 − 〈0|c−k↓c

†k↑ck↑c

†−k↓|0〉 +

〈0|c−k↓c†k↑ck↑c

†k↑ck↑c

†−k↓|0〉. The first term is 1 either since the operators in this order yield 〈1|1〉

or can be rearranged as above. In all other cases the last two operators anti-commute, and . . .c|0〉 alwaysvanishes.

The total number of particles is just the sum over all k-points,

〈N〉= ∑k,σ|vk|2 = 2∑

k|vk|2 = ∑

k(1−|uk|2 + |vk|2). (3.4.56)

With the expectation value of the four-particle interaction operator given by 〈c†k↑c

†−k↓c−k′↓ck′↑〉 =

vkv∗k′uk′u∗k the energy expectation value of the BCS Hamilton operator can be written as

E ≡ 〈E〉= 2∑k

εk|vk|2 + ∑k,k′

Vk,k′vkv∗k′uk′u∗k. (3.4.57)

Before starting to determine the minimum of E we write down two other expectation values whichwill be useful below and are christened Gorkov amplitudes after one of the pioneers of the theory ofsuperconductivity,

〈c−k↓ck↑〉 = ukv∗k ≡ gk (3.4.58)

〈c†k↑c

†−k↓〉 = u∗kvk ≡ g†

k, (3.4.59)

which can easily be recognized as the expectation values of the pair annihilator and creator, respectively.

Finally, we look at the fluctuations ∆N of N. To this end we need quantitiessuch as ∆N =

√〈N2〉−〈N〉2 =

√2∑k(〈n2

k↑〉−〈nk↑〉2) =√

2∑k |vk|2−2∑k |vk|4 =√2∑k |vk|2−2∑k(1−|uk|2)|vk|2 =

√2∑k |uk|2|vk|2. This expression is proportional to

√N. So

we find immediately that ∆N/N =√

N−1

and realize that the relative fluctuations of the particle numbervanish in the limit N → ∞ (whereas the fluctuations ∆N diverge as

√N). In other words the particle

number has the statistical behavior of a coherent state [Eqs. (3.3.25)–(3.3.27)] meaning that order√

Nparticles fluctuate in an out of the unpaired part of the electrons. Where are they going to?

To find that out we determine the fluctuations of the condensate which have to be determined from〈c†

k↑c†−k↓〉± 〈c−k↓ck↑〉. The amplitude or particle number fluctuations are described by the + sign, and

∆N is proportional to√

N as above and the relative fluctuations vanish as√

N−1

. Correspondingly, theabsolute fluctuations of the phase to be derived from 〈c†

k↑c†−k↓〉− 〈c−k↓ck↑〉 vanish in limit N → ∞ as

expected for a coherent state. (I have to prove that mathematically but I believe it is true.)


Determination of the ground state SUPERCONDUCTIVITY 47

3.4.3 Determination of the energy minimum at T = 0

We show here how the energy minimum for the BCS ground state at T = 0 can be found. The solutionwill be included in that of the following section but the derivation here is almost free of formalitiesand instructive. The minimum will be calculated with the constraints of (1) the conservation of thetotal number of particles and (2) of |uk|2 + |vk|2 = 1. The method used for this procedure is that ofthe Lagrangian multipliers. Since there are two constraints we have two factors λ1 and λ2 the meaningof which will become transparent during the derivation. Although returning a result identical to thatBardeen, Cooper, and Schrieffer obtained by using sine and cosine trigonometric functions for uk andvk, fixing the condition Eq. (3.3.47), the method used in the textbook by Annett has the advantage ofclarifying the physical meaning of the quantities µ and Ek and highlighting the quasiparticle concept.The two constraints are

ϕ1 = 〈N〉−∑k(1−|uk|2 + |vk|2) = 0

ϕ2 = uku∗k + vkv∗k +1 = 0,

and upon setting zero the partial derivatives of 〈E〉+λ1φ1 +λ2φ2 w.r.t. u∗k and v∗k one obtains the eigen-value equations

(εk +λ1)uk +∆kvk = λ2uk(3.4.60)

∆∗kuk− (εk +λ1)vk = λ2vk,

where

∆k = −∑k′

Vk,k′uk′v∗k′

≡ −∑k′

Vk,k′gk′ (3.4.61)

was used along with the definition in Eq. (3.4.58). ∆k is the gap parameter which has the full momentumdependence here and corresponds to the binding energy of the Cooper pairs and the energy gain due to thereduction of the electronic energy. The first multiplier shifts the energy, and we identify λ1 =−µ where µ

is the chemical potential. The second multiplier λ2 =±Ek corresponds to the energy of the quasiparticlesexcited out of the condensate across the gap (Bogolubov quasiparticles, see next paragraph), and uponsolving the eigenvalue equation we find

Ek =√

(εk−µ)2 + |∆k|2 =√

ξ 2k + |∆k|2 (3.4.62)

|uk|2 =12

[1+

ξk

Ek

](3.4.63)

|vk|2 =12

[1− ξk

Ek

](3.4.64)

ukv∗k =∆k

2Ek, (3.4.65)

where we have also defined the quasiparticle energy ξk = εk− µ in the normal state. Eqs. (3.4.63) and(3.4.64) are the coherence factors and describe the occupation probability of unpaired holes and electrons,

2013


respectively. Eq. (3.4.65) describes the occupation of the condensate. Upon inserting Eq. (3.4.65) intoEq. (3.4.61) the fully momentum dependent gap equation,

∆k =−∑k′

Vk,k′∆k′

2√

ξ 2k′+ |∆k′ |2

, (3.4.66)

is obtained. This equation cannot be solved analytically in the general case. If the gap and the interactionpotential are assumed to be k-independent one obtains an equation similar to (3.1.10). After transformingthe sum into an integral the solution for λ = NF |geff|2 (attractive interaction V =−|geff|2) and the cut-offof the attractive range hω0 ∆ reads

1 = λ lnhω0 +

√(hω0)2 + |∆|2|∆|

' lnhω0

[2+ 1

4

[|∆|hω0

]2]

|∆|

' ln2hω0

|∆|, yielding

∆ = 2hω0 exp(− 1

λ

). (3.4.67)

3.5 The general solution

In the preceding section it was shown how the ground state of a superconductor can be constructedfrom the coherent wave functions of the condensate [Eq. (3.3.46)]. For deriving the properties at finitetemperature we have to deal with the 4-operator term c†

k↑c†−k↓c−k↓ck↑ in the interaction energy.

3.5.1 Approximation of the four-operator term

Wick’s theorem states that four-operator terms can be approximated by a sum of pairs of operators timesthe expectation value of the corresponding pair, c†

k↑c†−k↓c−k↓ck↑ ≈ 〈c†

k↑c†−k↓〉c−k↓ck↑+ . . . . This can be

made plausible by using the identities

c−k↓ck↑ = gk + c−k↓ck↑−gk = gk +δgk

c†k↑c

†−k↓ = g†

k + c†k↑c

†−k↓−g†

k = g†k +δg†

k

with the Gorkov amplitudes defined in Eqs. (3.4.58) and (3.4.59). The interaction energy can now beexpressed as

Eint = ∑k,k′

Vk,k′(g†k +δg†

k)(gk′+δgk′)

≈ ∑k,k′

Vk,k′(g†kδgk′+gk′δg†

k +g†kgk′)

= ∑k,k′

Vk,k′(

g†kc−k′↓ck′↑+gk′ c

†k↑c

†−k↓−g†

kgk′)

= −

[∑k′

∆†k′ c−k′↓ck′↑+∑

k

(∆kc†

k↑c†−k↓−g†

k∆k

)]. (3.5.68)


The general solution SUPERCONDUCTIVITY 49

Here all terms higher than linear in δg(†)k were neglected, and Eq. (3.4.61) and its conjugate complexwere used. On noting that the quasiparticle energies are symmetric in momentum, ξ−k = ξk, the kineticenergy T can be rewritten as

T = ∑k,σ

ξkc†k,σ ck,σ

= ∑k

ξk

(c†

k↑ck↑+ c†−k↓c−k↓

)= ∑

k

(ξkc†

k↑ck↑−ξ−kc−k↓c†−k↓+ξ−k

), (3.5.69)

where Eq. (3.3.35) was applied in the last step. If the preceding two equations are added the mean fieldHamiltonian HMF−µN can be written in matrix form as proposed first by Nambu,

HMF−µN = ∑k

ξk +g†k∆k +

(c†

k↑, c−k↓

)︸︷︷︸

C†k

(ξk −∆k−∆

†k −ξk

)︸︷︷︸

Ξk

(ck↑

c†−k↓

)︸︷︷︸

Ck

. (3.5.70)

The operators C†k and Ck are vectors of two creators/annihilators and are called spinors. Ξk is the energy

matrix with the quasiparticle energies in the diagonal and the gap as off-diagonal elements. For find-ing the minimum of HMF− µN the energy matrix has to be diagonalized by a unitary transformation.The problem is similar to that of a two-level system with finite coupling between quantum mechanicaloscillators.

3.5.2 The Bogoliubov-Valatin transformation

For diagonalization we first insert the transformation matrices Bk and B†k into Eq. (3.5.70) and then

determine their coefficients,

HMF−µN = ∑k

[ξk +g†

k∆k +C†k(Bk︸︷︷︸B†

k)Ξk(Bk︸︷︷︸B†k)Ck︸︷︷︸

]. (3.5.71)

For a unitary transformation the product of Bk and B†k has to yield the unity matrix, BkB†

k = 1. Conse-quently one can write the transformation in terms of uk and vk etc. with |uk|2 + |vk|2 = 1 as above,

Bk =

(uk v∗k−vk u∗k

)(3.5.72)

B†k ≡ (B∗k)

′ =

(u∗k −v∗kvk uk

). (3.5.73)

This is the Bogoliubov-Valatin transformation which yields new operators B†k = C†

kBk and Bk = B†kCk

reading

β†k↑ = ukc†

k↑− vkc−k↓ (3.5.74)

β−k↓ = v∗kc†k↑+u∗kc−k↓ (3.5.75)

2013


and, respectively,

βk↑ = u∗kck↑− v∗kc†−k↓ (3.5.76)

β†−k↓ = vkck↑+ukc†

−k↓, (3.5.77)

and the new energy matrix

B†kΞkBk =

(Ek Dk−Dk −E∗k

). (3.5.78)

The coefficients uk and vk are found by the condition Dk = 0 and the normalization and lead to the sameresult as the minimization of the energy as displayed in Eqs. (3.4.62)–(3.4.65). The result is equivalentto the solution of the eigenvalue equation Eq. (3.4.60) and corresponds to finding the ground state. Herewe made an important additional step by deriving the new β operators. The two equations (3.5.74)and (3.5.76) define these new operators as a coherent superposition of creators and annihilators, henceelectrons and holes, at k and −k. It can be shown directly by insertion that the β operators obey fermionanti-commutation relations,

βkσ , β†k′σ ′+ = δkk′δσσ ′ (3.5.79)

βkσ , βk′σ ′+ = 0 (3.5.80)

β †kσ, β †

k′σ ′+ = 0. (3.5.81)

Consequently, in the superconductors there are no electrons or hole any further but, rather, compositefermions carrying the name Boguliubov quasiparticles and the condensate of Cooper pairs after their“inventors”. The probability of finding one of those objects in energy and momentum space is givenby the coherence factors uk and vk which can be visualized particularly easily by assuming a linearelectronic dispersion around the Fermi energy and momentum, ξk ≈ hvF · (k− kF) with vF and kF

the Fermi velocity and the Fermi momentum. In the state k,−k with particle-hole mixing every bandcrossing the chemical potential µ is reflected about µ . Since the particles in the reflected band are thoseof the original one the two bands mix in the usual quantum mechanical fashion and the levels repel eachother by virtue of the interaction energy 2∆k. At the intersection point kF |uk|2 and |vk|2 are both 1/2and the character of the Boguliubov quasiparticles is exactly 50% electronic and 50% hole-like. Awayfrom kF the character changes between 0 and 100%. Fig. 3.6 shows a theoretical result for particle-hole mixing in a strong coupling material, here the cuprate Bi2Sr2CaCu2O8. (From [15].) The band isreflected about µ and the color-coded occupation probability is given by the coherence factors shown inFig. 3.7.

If the new β operators [Eqs. (3.5.74) and (3.5.76)] and the coherence factors [Eqs. (3.4.63) and (3.4.64)]are plugged back into Eq. (3.5.71) HMF− µN can be separated into a ground state part and a part ofthermally excited quasiparticles,

HMF−µN = ∑k

[ξk +g†

k∆k +(

β†k↑, β−k↓

)( Ek 00 −Ek

)(βk↑

β†−k↓

)]= ∑

k

(ξk +g†

k∆k

)+∑

k

(Ekβ

†k↑βk↑−Ekβ−k↓β

†−k↓

)= ∑

k

(ξk +g†

k∆k

)+∑

k

(Ekβ

†k↑βk↑+Ekβ

†−k↓β−k↓−Ek

)= ∑

k

(ξk +g†

k∆k

)+∑

k,σ

(Ekβ

†k,σ βk,σ −Ek

)HMF−µN = ∑

k

(ξk−Ek +g†

k∆k

)+∑

k,σEkβ

†k,σ βk,σ . (3.5.82)



0.10

0.05

0.00

-0.05

-0.10

Ene

rgy k

(eV

)

= 0 meV = 20 meV

0 0a

a

Figure 3.6: Dispersion ξk along the (0,0)− (π,0) (Γ−X) direction, typical for a cuprate superconduc-tor, above (left) and below Tc. The chemical potential (yellow horizontal line) is at zero energy in thequasiparticle picture, the Fermi momentum is indicated in red. (The blue vertical line is an artifact.)Shown here is the result of a strong coupling calculation in false color representation with warm colorsindicating the maxima of the spectral functions. Above Tc the band crosses µ = 0 at k = kF . Below Tc

the band is reflected about µ . The two bands intersecting at kF interact similarly as states in a two-levelsystem. The spectral weight changes according to the coherence factors plotted in Fig. 3.7. For everyfixed k the integral over all energies is unity corresponding to one quasiparticle. From [15].

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0

|uk|2 , |

v k|2 , u*v

, f(

k)

Quasiparticle energy k ()

|u*u| |v*v| |u*v| f

0.0 0.5 1.0

0.0

0.5

1.0(0)

-1/kF = /t = 0.2

|uk|2 , |

v k|2 , u*v

k (/a)

Figure 3.7: Coherence factors |uk|2, |vk|2, and |ukv∗k| for electron, hole, and pair occupation, respectively.(a) shows the coherence factors as a function of momentum. The underlying band structure correspondsto the 2D tight-binding representation ξk =−t[cos(kxa)+ cos(kya)]−µ (a is the lattice constant) alongky = 0 at half filling, µ ≈ t. For better visualization the gap is as large as 10% of µ . For a conventionalmetallic superconductor ∆/µ = O(0.001) in the cuprates the maximal gap may reach 5–10% of µ . Thehalf width (FWHM) ∆k of |ukv∗k| defines the BCS coherence length, ξBCS ≈ ξ0 = 2π/∆k. (b) shows thecoherence factors as a function of energy. The Cooper pairs live around the Fermi energy but extendsubstantially into the band. As a result all electrons including the unpaired ones are “enslaved” and havethe same phase. The Fermi function at T = 1.76Tc, f (ξk,1.76Tc), has an energy dependence similar tothat of |vk|2.

The β operators obey fermionic anti-commutation relations [Eqs. (3.5.79)–(3.5.81)]. Hence the expec-tation values of the corresponding number operators are described by Fermi statistics,

〈β †k↑βk↑〉 = ν(Ek) ≡

1

exp(

EkkBT

)+1

(3.5.83)

〈β−k↓β†−k↓〉 = 1−ν(Ek). (3.5.84)

Here we have written the Fermi distribution function as ν(Ek) rather than f (Ek) as usual in order to

2013


highlight that the energy of a Bogoliubov quasiparticle is in the argument. These equations show that thesecond term on the r.h.s. of Eq. (3.5.82) describes the kinetic energy of free Fermions having dispersionEk. It vanishes in the limit T → 0 and in the limit ∆→ 0. (Note that for correctly determining thetransition into the normal state the limit is lim∆→0 Ek = |ξk| at any temperature.) Since the second termvanishes in the limit T → 0 the first term is the ground state energy UBCS. Using these limits appropriatelyEq. (3.5.82) is the basis for determining the energy gain in the superconducting state (see Problem 2 inset 7) yielding

〈∆E〉T=0 = 〈HMF−µN− Hn〉

= −14

NF∆2 (3.5.85)

for a constant gap ∆ and NF the density of states at the Fermi energy EF for both spin projections (whichis twice the number of k points).

From Eq. (3.5.83) the expectation value of the Gorkov amplitude at finite temperature can be derived(see problem 1, set 7),

〈c−k↓ck↑〉 = 1−2ν(Ek)ukv∗k

= tanh(

Ek

2kBT

)ukv∗k (3.5.86)

with ukv∗k given by Eq. (3.4.65). This result enables us to write down the gap equation at finite tempera-ture.

3.5.3 Solution of the gap equation for T ≥ 0

Together with Eq. (3.5.86) the gap equation (3.4.61) with the definition Eq. (3.4.58) can be written downas a major result of this section,

∆k = −∑k,k′

Vk,k′〈c−k′↓ck′↑〉

= ∑k,k′

Vk,k′uk′v∗k′ tanh(

Ek

2kBT

)∆k = −∑

k,k′Vk,k′

∆k′

2Ek′tanh

(Ek

2kBT

). (3.5.87)

This equation reproduces the result of Eq. (3.4.67) of section 3.4.3 if the limit T = 0 is taken. At finitetemperature one can calculate the limit ∆→ 0 and derive the transition temperature Tc (where the gapvanishes). To simplify we use the BCS approximation, as above, and assume that ∆k = ∆ is constant, thedispersion is parabolic and isotropic, and that the interaction is attractive and also isotropic yielding

1NF |geff|2

≡ 1λ

=∫ hω0

0

dξ

Etanh

(E

2kBT

). (3.5.88)

The limit E→ |ξ | and the substitution x = ξ/2kBTc yield

1λ

=∫ hω0/2kBTc

0dx

tanhxx

= ln(

2eγ

π

hω0

2kBTc

), (3.5.89)



where γ = 0.5772 is the Euler constant. Combining this result and that of Eq. (3.4.67),

kBTc =eγ

π2hω0e−

1λ (3.5.90)

∆ = 2hω0e−1λ

returns the famous universal BCS relationship between the gap and the transition temperature,

∆

kBTc=

π

eγ= 1.764. (3.5.91)

In the majority of cases the ratio of twice the gap to Tc is given, 2∆/kBTc = 3.528. Note that anydependence on materials properties, in particular the cutoff energy hω0, corresponding to the Debyefrequency hωD in the case of electron-phonon coupling, and the electron-phonon coupling parameter λ

drop out entirely. While this is typical for a mean field theory as here it is not realistic for most of thematerials. Only a few of the elements such as Al, Sn or V are close to the BCS prediction. Elements likePb or Nb and most of the alloys have gap ratios well above 3.53 and reach 5.6 in Nb3Sn. This meansthat the influence of the cutoff or, more physically, of details of the interaction cannot be ignored anyfurther. Eliashberg solved that problem in 1960 [3]. In essence the solution takes into account the fullenergy dependence of the interaction potential leading to energy dependent gap functions as well. Thetreatment requires the knowledge of Green’s function techniques, and the discussion is far beyond thescope of these lecture notes.

If the interaction potential Vk,k′ is either repulsive or strongly momentum dependent the gap function∆k may become momentum dependent as well and may even change sign on the Fermi surface [16, 17].What means a sign change of the gap? Formally, this is a simple question since we can write down thegap in terms of k-dependent functions such as

∆k =∆0

2[cos(kxa)− cos(kya)] , (3.5.92)

where a is the lattice constant of a square lattice. Eq. (3.5.92) reproduces the dx2−y2 gap in the copper-oxygen compounds, for instance. Fossheim and Sodbø describe in some detail how this gap followsfrom a specific choice of Vk,k′ . The problem with visualizing such a gap lies in the way it is measuredexperimentally. In the typical spectroscopic experiments such as tunneling, photoemission, optical orRaman spectroscopy or thermodynamic experiments such as heat capacity, heat transport or penetrationdepth only the magnitude of the gap |∆k| enters. In addition, it is hard to imagine how an energy reductionbelow the chemical potential can be both positive and negative. The solution is that the gap has a phasederiving from the wave function of the coherent state. This phase enters whenever the relative phases oftwo coupled condensates are probed. Also remember that the expression for the current contains a phasegradient. Consequently, whenever experiments with coupled superconductors are performed the phaseand, hence, the sign of ∆k becomes intuitively clearer. The seminal phase-sensitive experiments in thecuprates [18, 19] will be discussed in more detail in chapter 6.

A momentum-dependent gap raises additional questions as to the possible outcome of thermodynamicexperiments or the condensation energy. In the latter case Eq. (3.5.87) has to be solved for the respec-tive momentum-dependent gap. In the case of dx2−y2 gap in Eq. (3.5.92) the mean field condensationenergy leads to a maximal gap larger than the canonical BCS value, and 2∆0 = 4.28. In the cupratesone finds 2∆ ≈ 8 indicating very strong coupling. If there is a multiband system the smallest gap canbe well below 3.53 while the largest one always exceeds 3.53 since otherwise the condensation energyis not reached. Concerning thermodynamics the answer is similarly complicated. For a system with afinite gap everywhere on one of the Fermi surfaces one typically finds activated behavior, i.e. a responsebeing characterized by a gap. Then the smallest gap dominates (but does not exclusively determines) the

2013


activation energy since, with increasing temperature, the branches with the smaller Bogolyubov energyEk are populated first. If the gap has nodes, ∆(k0) = 0, there is no activated behavior any further becauseunpaired particles exist whenever T 6= 0, hence always. Then the specific heat, the penetration depthor the thermal conductivity are described by temperature power laws rather than exponentials. Promi-nent examples are UPt3, UBe13 [13] and YBa2Cu3O7 [20]. For predictions the explicit variation withtemperature of the gap and of the condensate density is required.

Neither the temperature dependence of the gap nor that of other thermodynamic quantities can be writtendown explicitly. Rather the implicit Eq. (3.5.88) needs to be solved. Muhlschlegel derived numericalvalues which are tabulated [21]. In addition, interpolation schemes can be derived [22] which providean analytic expression for 0 < T ≤ Tc which agrees with the exact numerical solution to better than afraction of a percent and is sufficient for most of the practical purposes,

∆0(T )∆0(0)

= tanh

(πkBTc

∆0(0)

√23

cs− cn

cn

∣∣∣∣Tc

[∆0(0)]2

〈[∆k(0)]2〉FS

[Tc

T−1+0.191

TTc

]). (3.5.93)

Note that the Tc/T term is not defined at T = 0 while the limit T → 0 exits. The result of Eq. (3.5.93),as plotted in Fig. 3.8, has the right asymptotic behavior and can be used for estimates of other prop-erties. It is derived for the weak-coupling BCS limit but it is also valid approximately for the strongcoupling case. In particular, it is also valid for superconductors with anisotropic gaps. Then the gap ratio

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

∆(T)/∆

(0)

T / T c

Figure 3.8: Temperature dependence of the superconducting energy gap according to Eq. (3.5.93). Thedeviations from the numerical solution [21] is of the order of a percent.

∆0(0)/kBTc (see above), the discontinuity of the heat capacity cs−cncn

at Tc, and the Fermi surface averageof [∆k/∆0(0)]2 have to be used accordingly. In the case of an isotropic gap one finds 1.764, 1.426, and1, respectively, for the so-called BCS-Muhlschlegel parameters. For the case of dx2−y2 gap [Eq. (3.5.92)]in a tetragonal system, which, cum grano salis, applies for the cuprates, the weak-coupling values are,respectively, 2.14, 0.951 and 0.5 [22]. The influence of replacing the BCS-Muhlschlegel parameters on∆0(T ) is mild. Other quantities may change substantially.

3.6 Connection to experiments

In this section we derive a few quantities which follow from the BCS theory and can be observed experi-mentally including thermodynamic properties, the tunneling density of states, and two-particle propertiesfor which the coherence factors play an important role.


Connection to experiments SUPERCONDUCTIVITY 55

3.6.1 Thermodynamic properties

The solution of the gap equation for all temperatures enables us to determine the energy o the Bogoliubovquasiparticles Ek(T ) = [ξ 2

k + |∆k(T )|2]1/2 and their thermal occupation ν(Ek(T )) leading directly to thethermodynamic functions. For transforming the k sums into integrals the momentum dependence willbe dropped occasionally. Since the Bogoliubov quasiparticles are a system of free fermions and sincethe condensate does not contribute for being a fully ordered state the density of electronic entropy in thesuperconducting state comes only from the quasiparticle and can be written in terms of ν(Ek(T )),

ss =−2kB ∑k[νk lnνk +(1−νk) ln(1−νk)] . (3.6.94)

From Eq. (3.6.94) the heat capacity cs follows directly as

cs = Tdss

dT. (3.6.95)

Some writing can be saved if one takes the derivative w.r.t. β = (kBT )−1,

cs = −βdss

dβ

= 2βkB ∑k− ∂νk

∂Ek

[E2

k +12

βd

dβ|∆k(T )|2

]. (3.6.96)

Here the derivative of the occupation number w.r.t. the quasiparticle energy leads directly to the Yosidafunction Y (T ) which enters many thermodynamic properties,

NFY (T ) = ∑k− ∂νk

∂Ek=

NF

4kBTc

∫∞

−µ

dξ[cosh

(E

kBT

)]2 (3.6.97)

which is exponentially small at low temperature for an isotropic gap and shows directly that the heatcapacity show activated behavior then. At Tc the heat capacity of the normal state is also needed and canbe derived from Eq. (3.6.96) by setting the gap zero reproducing the result (2.2.42),

cn = γT =π2

3k2

BN(EF)T.

From that and Eq. (3.6.96) one finds for an isotropic gap,

∆c(Tc) = N(EF)

(d

dT|∆(T )|2

)and for the ratio to the normal-state value another universal number,

∆ccn

= 1.43. (3.6.98)

Using the result in Eq. (3.6.96) the internal energy and finally the free energy density in the normal andthe superconducting state and, thereof, the critical field as a function of temperature can be found. Now,by using the BCS expressions, the variation with T is slightly different from the parabola as shown inFig. 3.9. As mentioned earlier weak coupling superconductors are described well by the BCS predictionwhile those with strong coupling first get closer to the parabola and then even go beyond. Fig. 3.10 showsthe temperature dependence of the thermodynamic functions as derived above.

2013


Figure 3.9: Temperature dependence of the critical field. The lower curve follows from the BCS predic-tion the upper one represents the parabola from the two fluid model proposed first by Gorter and Casimir.From Ref. [2].

Figure 3.10: Temperature dependence of the thermodynamic function according to the BCS model. FromTinkham.

3.6.2 Single particle response

Since the condensate is a coterie only pairs of electrons can be added or removed (Andreev). Therefore,if single particle properties are being studied, only the quasiparticle contribute and the response fromthe condensate can be disregarded. This holds for (single particle as opposed to Josephson) tunnelingand photoemission spectroscopy for instance whenever the final state is outside the superconductor. Oneof the most important quantities in this is the single particle density of states N(E) which is a momen-tum integrated function. Since the number of particles above and below Tc is conserved on can simplyequate N(ξ )dξ = N(E)dE and, using the approximation N(ξ )≈ N(0) in the normal state, finds for the


Connection to experiments SUPERCONDUCTIVITY 57

superconducting density of states,

N(E)N(0)

=dξ

dE=

E√

E2+∆2 E > ∆

0 E < ∆,

(3.6.99)

exhibiting the typical square root singularity at the gap edge. This result is well reproduced in weak-coupling isotropic superconductors.

2013

Chapter 4

Ginzburg-Landau Theory

The model of Ginzburg and Landau (GL) proposed 1950 is an extension of the London theory and allowsthe treatment of spatially inhomogeneous mixtures of normal and superconducting regions in a material.In this way all configurations of currents and fields in a superconductor can be treated. Therefore theGL theory has a wide range of applications in practice. Gorkov showed in the beginning of the 1960iesthat the BCS theory and the GL model are equivalent in the temperature range around Tc where the GLtheory is valid.

4.1 Phase transitions

Before discussing the results for superconductors we briefly summarize the theory for second order phasetransition originally proposed for magnets by L. D. Landau in 1937. Landau expanded the density of thefree energy in powers of the magnetization M being a natural order parameter different from zero inthe ordered state (T < Tc) and identical zero above. Hence the description of magnetism is a clear-cutproblem. The purpose of the theory, formulated a long time before numerical methods were successfullyintroduced to the problem of magnetism, was essentially to derive thermodynamical properties in theordered state and later also above in the fluctuation regime from the simplest possible assumptions. Thenthe density of the free energy can be written as

fM = fn +α(p,T )M(r)2 +β (p,T )

2M(r)4. (4.1.1)

Without external fields only the magnitude of M(r) = |M(r)| enters. In the simplest case (no position de-pendence) the free energy becomes minimal if the first derivative of f vanishes and the second derivativeis positive,

0 =∂ fM

∂M= 2α(p,T )M+2β (p,T )M3 (4.1.2)

0 <∂ 2 fM

∂M2 = 2α(p,T )+6β (p,T )M2. (4.1.3)

A non-trivial solution of Eq. (4.1.2) is

M2 =−α

β(4.1.4)

which has a real solution only if α and β have opposite sign and β 6= 0. This is fulfilled at T < Tc for

α(T ) = α0 (T −Tc) (4.1.5)

59

60 R. HACKL AND D. EINZEL Ginzburg-Landau Theory

with α0 and β positive constants. If M2 from Eq. (4.1.4) is inserted into Eq. (4.1.3) −4α is obtainedbeing positive in the ordered phase, and the extremum of fM is a minimum if and only if α < 0 andβ > 0. The evolution of the free energy landscape for various temperatures is shown in Fig. 4.1. The

- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

0 . 0 0

0 . 0 5

0 . 1 0

f(p,T,

M)

M / M 0

α = 0 . 1 α = 0 . 0 3 α = 0 α = - 0 . 0 3 α = - 0 . 1

Figure 4.1: Free energy density in a magnetic system as a function of the temperature. The temperatureis expressed in terms of α . For negative α the minimum of fM is found at finite magnetization M.

difference

fM− fn =−α2

2β(4.1.6)

corresponds to a reduction of the free energy in the ordered state and opens the way towards the derivationof all thermodynamic properties as in section 2.2. With α given in Eq. (4.1.5) the transition is of secondorder, implying that there is no volume and entropy change and, consequently, no latent heat at Tc. Thenthe Ehrenfest relations [?] between the discontinuities in the heat capacity ∆cp, the thermal expansion,∆αT , and the compressibility, ∆κT can be derived (Landau-Lifshitz, Statistical Physics),

∆αTc = − d pdT

∣∣∣∣Tc

∆κTc (4.1.7)

∆cp

Tc=

d pdT

∣∣∣∣Tc

∆αTc . (4.1.8)

The pressure dependence of the transition temperature Tc, dTc/d p is closely (but certainly not trivially)related to microscopic mechanisms at the origin of a phase transition, as can be seen for instance fromthe Eq. (3.5.90) in the case of superconductivity, and is therefore a highly desirable quantity. Since ∆cp

and ∆αT are experimentally accessible Eq. (4.1.8) can be used to indirectly determine dTc/d p. This issometimes simpler than a direct measurement of Tc in a pressure cell. In any case, one obtains an internalthermodynamic consistency check and can get the compressibility from Eq. (4.1.7). Obviously, this typeof approach proves very useful already in its simplest version. Ginzburg and Landau showed that it canbe applied to superconductivity and even well beyond [12].

4.2 Application to superconductivity

A major problem is the proper definition of an order parameter. One could argue that the temperaturedependent condensate density of the London theory would be the right choice. However, it becomes im-mediately clear that one would not even get a current where the phase gradient plays a crucial role. This


Application to superconductivity SUPERCONDUCTIVITY 61

was the motivation for Ginzburg to suggest an order parameter in the spirit of the Madelung substitutionfor the wave function, Eq. (2.3.65),

ψ(r, t) = a(r, t)eiθ(r,t),

where a(r, t) and θ(r, t) are real functions of position and time. In the GL theory the function ψ dependson also temperature T is normalized as

2|ψ(r, t,T )|2 = 2a(r, t,T )2 = ns(T ) = n fs(T ). (4.2.9)

where n is the constant density of electrons in a material, and ns(T ) is the number of condensed electronsthat is twice the number of Cooper pairs. fs(T ) is a dimensionless function that vanishes at Tc and reachesunity if all electrons are condensed in the limit T → 0. Note that the GL model is valid only for T . Tc.

4.2.1 Density of the free energy

The density of the free energy of a superconductor can be derived in the spirit of Eq. (4.2.16) as long asthere are no external fields or currents,

fs = fn +α(p,T )|ψ(r, t)|2 + β (p,T )2|ψ(r, t)|4 (4.2.10)

≡ fn +α(p,T )|a(r, t)|2 + β (p,T )2|a(r, t)|4, (4.2.11)

which has solutions equivalent to those of Eq. (4.2.16). Here, only the amplitude is important for findingthe minimum, and the phase can still be chosen arbitrarily as shown in Fig. 4.2. In the figure this

fs()

Im

Re

Figure 4.2: Free energy density in a superconductor with out field and currents. The energy minimumdepends only on the amplitude of ψ but is independent of the phase θ .

corresponds to moving the order parameter around the bottom of the valley. Moving out of the valley isan amplitude fluctuation for which energy must be invested.

As soon as there are currents and fields the kinetic energy of the electrons and the field energy becomesimportant. In addition there is an interrelation. The kinetic and field energy are given by

12M v2→ 1

2M| hi∇ψ(r, t)−QAψ(r, t)|2 (4.2.12)

and, respectively, by

W =1

2µ0

∫B2

MdV =1

2µ0

∫[B(r)−B0]

2 dV (4.2.13)

2013


neglecting demagnetization effects. The magnitude is taken for arriving always at real numbers for theenergies. Nowadays one would call that an educated guess since the result is correct but there is nojustification otherwise. Since all functions depend now on position only appropriate integrals over thesample and over all space lead to the proper energy minimum corresponding to deriving and minimizingthe functional of the free energy.

4.2.2 Functional of the free energy

To this end the energies related to the order parameter are integrated over the sample volume while themagnetic energies have to be taken in the entire space yielding

Fs = Fs +∫

sampleαψ

∗ψ +

β

2(ψ∗ψ)2 +

h2

2M︸︷︷︸γ

∣∣∣∣∇ψ +Qih

Aψ

∣∣∣∣2 dV +1

2µ0

∫∞

−∞

[B(r)−B0]2 dV. (4.2.14)

Originally the parameter γ was used in order to indicate that the third term in the first integral is anotherexpansion coefficient. It will turn out, as expected, that M and Q are the mass and the charge of aCooper pair having being unknown at the time the GL theory was proposed. For finally deriving the GLequations the variation of Eq. (4.2.14) has to be determined. Since the absolute value is inconvenient inthis context we rewrite the γ term,

h2

2M

∫sample

∣∣∣∣∇ψ +Qih

Aψ

∣∣∣∣2 dV =h2

2M

∫sample

ψ∗(

∇+Qih

A)

ψdV +ih

2M

∫S

ψ∗ (ih∇+QA)ψ · ndS,

(4.2.15)

where S is the sample surface and n the surface normal in point r. In addition we will need the variationof the magnetization reading

δ1

2µ0

∫B2

MdV =1µ0

∫BM ·δBMdV

=1µ0

∫BM · (∇×δAM)dV

=1µ0

∫(∇×BM) ·δAM−∇ · (BM×δAM)dV

=1µ0

∫(∇×BM) ·δAMdV (4.2.16)

4.2.3 The Ginzburg-Landau equations

For the deriving the GL differential equations the variational derivative of F has to be evaluated. Thismeans finding the change δF if ψ or A are replaced by ψ +δψ and A+δA. If F is minimal

∂F∂ψ∗

δψ∗ = 0;

∂F∂A

δA = 0

where for traditional reasons the variation of ψ∗ is considered, and the subscript M referring to themagnetization has been dropped for simplicity. The variation of ψ∗ is obviously calculated only inside


Two new length scales SUPERCONDUCTIVITY 63

the first integral which vanishes only if the integrand vanishes for all possible δψ∗ yielding,

12M

(hi∇−QA

)2

ψ +(α +βψ∗ψ)ψ = 0 (4.2.17)(

hi∇n−QAn

)ψ = 0 (4.2.18)(

hi∇n−QAn

)ψ = bψ. (4.2.19)

Eq. (4.2.17) is the first Ginzburg-Landau equation (GL1) from which most of the properties can bederived. Eq. (4.2.18) results from the second integral of Eq. (4.2.15) over the sample surface and saysthat the gradient of the wave function perpendicular to the surface vanishes in zero field meaning thatno current flows in or out and that there are no variations of the pair density close to the surface aswe have seen already before. If one attaches a metal to the surface of a superconductor one arrivesat Eq. (4.2.19). Note that deriving such an equation implies that Eq. (4.2.15) is valid also outside thesuperconductor, at least in a small distance. This observation is called proximity effect and is consistentwith the microscopic theory in that the coherent wave function of the pairs “leaks” out of the materialover a length scale given by the coherence length. Along with the proximity effect, which means thatsuperconductivity can be induced in a normal metal attached to a superconductor, the Josephson effect isanother important consequence.

The variation of the vector potential δA leads to the second Ginzburg-Landau equation (GL2) whichdescribes the current density in the presence of superconductivity,

js =hi

Q2M

(ψ∗∇ψ−ψ∇ψ∗)− Q2

MAψ

∗ψ (4.2.20)

≡ Q2M

[ψ∗(

hi∇−QA

)ψ + c.c.

](4.2.21)

where c.c. means complex conjugate and indicates that the current density is a real quantity. If we insertagain the Madelung wave function we obtain

js =a2Q2

M

(hQ

∇θ −A)

(4.2.22)

and with the usual replacement Q→ 2e (traditionally no minus sign here), M → 2m, and 2a2→ n,

js =ne2

m

(h2e

∇θ −A)

(4.2.23)

reproducing the London result Eq. (2.3.74).

4.3 Two new length scales

The equations for the current (4.3.25) and (4.2.22) and the first GL equation (4.2.17) without field leadto two new length scales tha turn out to be similar to the scales λL and ξ0. However, though being relatedsome care has to be taken to keep the definitions clear and to understand the meaning properly. We shalloutline the relation without derivation later.

2013


4.3.1 Screening

For a moment we go back to Eq. (4.2.22) and take the curl,

∇× js =−a2Q2

M(∇×A) =−a2Q2

MB (4.3.24)

and find upon using Ampere’s law

∇× (∇×B) = µ0∇× js

∇2B = µ0

a2Q2

MB. (4.3.25)

Eq. (4.3.25) is formally equal to Eq. (2.3.76), and if a2 =−α

β, Q = 2e, M = 2m are substituted the GL

penetration depth is obtained,

λ2GL =

mβ

2µ0|α|e2 . (4.3.26)

It is tempting to use 2a2 = n as above to recover the London penetration depth but then we get the zerotemperature limit while the GL theory is valid only close to Tc. However, we can insert the temperaturedependence of α and find λGL to diverge as

√Tc−T −1 at Tc and, in the clean limit,

λGL(T, `→ ∞) =λL(0)√2 Tc−T

Tc

. (4.3.27)

4.3.2 Ginzburg-Landau coherence length

If A = 0 in the first GL equation (4.2.17) and a20 = −α/β is substituted for a real equilibrium order

parameter GL1 can be made dimensionless by dividing the entire equation by a0,

h2

4m|α|∂ 2 f∂x2 + f + f 3 = 0. (4.3.28)

Here the problem is confined to one dimension and f = a/a0. For dimensional reasons the prefactor ofthe second derivative w.r.t. to x must by a length squared and defines already the GL coherence lengthξGL,

ξ2GL(T ) =

h2

4m|α0(T −Tc)|(4.3.29)

the meaning of which becomes transparent by looking at small deviations from the equilibrium, δ f =f −1. By neglecting all terms of order (δ f )2 and higher one finds a differential equation for δ f ,

∂ 2

∂x2 (δ f ) =2

ξ 2GL

(δ f ), (4.3.30)

which identifies ξGL as the “healing” length of a perturbation to the equilibrium order parameter. Withthis length scale we have a phenomenological argument in favor of Eq. (4.2.19) and observe that theorder parameter ψ does not change abruptly but rather varies continuously over a characteristic distanceξGL. However, similarly as in the case of the penetration depth, ξGL is different from the BCS ξ0 inboth magnitude and meaning. While the magnitudes of ξGL and ξ0 differ only by a factor of order 1


Two new length scales SUPERCONDUCTIVITY 65

the interpretation of ξGL is purely phenomenological. It has neither a connection to Cooper pairs nor tonon-locality: the GL theory is strictly local. However, in contrast to the London limit with the coherencelength exceeding the penetration depth ξGL is the smallest length scale in those cases where the GLtheory develops its full power. Hence, effects of non-locality can be ignored safely.

The ratio of the penetration depth and the coherence length is an important quantity and we define thedimension less GL parameter,

κ =λGL

ξGL=

√β

2µ0

2mhe

=

√β

2µ0

1µB

. (4.3.31)

where µB is the Bohr magneton. κ depends only weakly on temperature through β , so in a first approx-imation it can be considered constant close to Tc. However, Eq. (4.3.31) and, similarly, the equations(4.3.29) and (4.3.27) conceal that ξGL and λGL are not solely given by constants of nature but, rather,depend in subtle ways on the BCS coherence length ξ0 and the mean free path ` of the electrons. Thefunctional dependence is obtained by comparing the results of BCS and GL close to Tc. While the de-pendences of ξGL(ξ0, `) and λGL(ξ0, `) are very important whenever real materials are to be analyzedquantitatively they will not particularly further the basic understanding here. We study now the conse-quences of varying κ . In practice this means looking at different (clean) materials covering a range ofapproximately 0.1 < κ . 100.

4.3.3 Energy of the normal-superconductor interface

We consider now an infinite superconductor. At x < 0 the magnetic field is homogeneous, points alongthe z-axis, and just reaches the critical field Bc. For x≥ 0 the field decays, hence the surface x = 0 sepa-rates the normal from the superconducting part. Then superconductivity is almost completely suppressedat x = 0 implying that the order parameter, i.e. the density of pairs a2, at the surface is vanishingly smalland negligible in comparison to the equilibrium value a2

0 deep inside the superconductor for x→ ∞. Onthe other hand, the magnetic field is screened over a typical length scale of a few λGL.

What is energetically more favorable, small or large κ? Small κ means that the field is pushed out over ashort length scale while condensation energy is lost in the range ξGL λGL needed for a/a0 to recoverto unity. In the opposite case the field can penetrate over a much larger range than that needed for a toapproach a0. Obviously, condensation energy is lost here in a smaller volume than in the previous case,and less energy has to be invested to keep the field out. This situation is energetically more favorable andcan be written down by considering the free energy density for x≥ 0,∫

fsdx =∫ (

fn−B2

c

2µ0

)dx+(ξGL−λGL)

B2c

2µ0. (4.3.32)

The last term is an estimate for the trade-off between condensation and field energy. The energy is readilywritten down since the condensation energy has exactly the same magnitude as the the energy stored inthe field. The signs of ξGL and λGL are determined in a way that the free energy increases with ξGLand decreases with λGL. In this way a surface energy σ can be defined. The sign of σ determines as towhether or not the field penetrates,

σ = ∆ f δ ≈ B2c

2µ0(ξGL−λGL) (4.3.33)

where a negative sign corresponds to an energy gain. For calculating numbers the functional form of thefield penetration and the recovery of a/a0 cannot be neglected but will not be considered here since it

2013


will be derived in a very simple way below. The result yields a limiting value for κ for which σ changessign,

κ <1√2

typeI (4.3.34)

κ >1√2

typeII (4.3.35)

dividing the world into type I superconductors which exclude the field always completely except for thethin surface layer carrying the screening current and type II superconductors into which the field canpenetrate. We have seen earlier that vortices are formed which carry one flux quantum each. Since κ

reaches values in the range of 100 for copper-oxygen superconductors the upper critical field up to whichsuperconductivity survives can be orders of magnitude higher than the field Bc1 at which the first fluxline penetrates and the thermodynamical critical field Bth

c (to be defined later).

4.4 States with internal flux

For κ > 1 it becomes increasingly favorable for the superconductor to let the field penetrate as observedfirst by Shubnikov and coworkers (see Fig. 2.2). The consequences are far-reaching and the basis ofmany applications such as coils for generating high magnetic fields. The way the field penetrates firstwas discussed in section (2.3.4), and the field penetration depth λL was found to be the characteristicdimension of the vortex while a “mysterious” cutoff ξ λL was postulated to keep the problem tractableby avoiding a divergence in the center of the vortex. It is intuitively clear now that the scale ξ ≈ ξ0introduced there is related to (but not equal to) the GL coherence length ξGL. We show now that ξGL isintimately related to the upper critical field Bc2 at which superconductivity collapses.

4.4.1 The upper critical field Bc2

To this end the first GL equation (4.2.17) has to be analyzed. Assuming that the applied field B0 isonly slightly below Bc2 Eq. (4.2.17) can be linearized since the order parameter ψ∗ψ is suppressedsubstantially below its equilibrium value |ψ0|2 = −α/β and βψ∗ψ |α|. The linearized equationreads

h2

2M(−i∇+QA)2

ψ = |α|ψ. (4.4.36)

Eq. (4.4.36) is formally equal to a Schrodinger equation for a particle with mass M and charge Q in afield B = ∇×A, and the problem can be mapped on a known one. For doing so, the field is assumed topoint along the z-axis, B = B0ez, and a gauge is chosen for which A = eyB0x. With the abbreviations

ωc =QB0

M≡ eB0

m(e > 0); (4.4.37)

x0 = −hky

M ωc

[=

kyΦ0

2πB0

]and the ansatz (4.4.38)

ψ = exp(i[kyy+ kzz]) f (x) (4.4.39)

where ωc is the cyclotron frequency, the equation of a one-dimensional harmonic oscillator,

− h2

2M

∂ 2 f (x)∂x2 +

M ωc

2(x− x0)

2 f (x) =

(|α|−

h2k2z

2M

)f (x)≡ E f (x), (4.4.40)


States with internal flux SUPERCONDUCTIVITY 67

is found which is entirely equivalent to finding the Landau levels of an electron in a normal metal. x0 isthe equilibrium position, and ωc is the eigenfrequency. The harmonic oscillator has the spectrum

E =

(n+

12

)hωc (4.4.41)

where n is an integer. The final step is to find the largest field B0 for which Eq. (4.4.41) has a solution.With the usual substitutions for M and Q, and E given in Eq. (4.4.40) one obtains(

n+12

)h

eB0

m+

h2k2z

4m= α0(Tc−T ). (4.4.42)

The field becomes maximal for n = 0 and kz = 0, hence

Bc2 =4mα0(Tc−T )

h2h2e

(4.4.43)

=Φ0

2πξ 2GL(T )

. (4.4.44)

The second equation shows immediately that the vortex lines are essentially at a distance of ξGL(T )before superconductivity collapses. As a consequence, for κ & 1, the field and the Cooper pair densitya2 oscillate weakly on similar length scales. For large κ the field penetrates the superconductor almosthomogeneously and a2 oscillates between zero in the vortex core and a value much smaller than a2

0. Thetemperature dependence of Bc2 is linear close to Tc, as can be seen directly from Eq. (4.4.43), henceis consistent with Eq. (2.1.6). To make it consistent with the BCS prediction the prefactors becomeimportant. This problem has been addressed by Gorkov and will be discussed in a later version of thelecture notes.

In a finite field B < Bc2 there is no phase transition at Tc but only at a lower temperature Tc(B) which canbe obtained by inverting Eq. (4.4.43),

Tc(B) = Tc(0)[

1−2πξ(0)GL

BΦ0

], (4.4.45)

where ξ(0)GL = h2/4mα0Tc(0) was used. Not unexpectedly, the transition temperature into the supercon-

ducting state decreases linearly with increasing field. It should be noted, however, that some care shouldbe taken again upon extrapolating the result to low temperatures and identify Φ0/2πξ

(0)GL with the critical

field at zero temperature. In other words, mind the range of validity!

4.4.2 The nucleation field Bc3 on the surface

So far only an infinite superconductor was considered. If there is a surface, for instance at x = 0, theboundary conditions (4.2.18) or (4.2.19) have to be taken into account depending on whether the su-perconductor is either in vacuum or covered by an insulator or, respectively, by a metal. In the case ofvacuum, which is usually the case for a typical experimental situation, (or in the case of an insulator) thegauge invariant current across the surface must vanish,(

hi

∂

∂x−2eA(x)

)ψ(x)

∣∣∣∣x=0

= 0. (4.4.46)

2013


In the above paragraph we found that the maximal field corresponds to vanishing momentum along thez-axis, kz = 0, and the lowest level of the harmonic oscillator, n = 0. The corresponding eigenfunctionreads

f (x) = C exp

(−[

x− x0√2ξGL

]2), (4.4.47)

which, however, satisfies the boundary condition only for x0 = 0 and x0 = ∞ giving the eigenvalue above.The question is as to whether or not eigenfunctions and -values can be found in the intervall 0 < x0 < ∞

for which the field is different from Bc2. Saint-James and de Gennes [?] found a solution in 1963 andshowed that for x0 = 0.59ξGL

0.59heB0

2m= α0(Tc−T ) (4.4.48)

replaces Eq. (4.4.42) (in the case of kz = 0 and n = 0) leading to the surface critical field Bc3 = 1.695Bc2.The exact solution is complicated and requires numerics. In particular, the eigenfunction must have avanishing derivative at x = 0 which is not the case for Eq. (4.4.47) at x0 6= 0. Kittel and de Gennessuggested variational approaches using analytical functions which are described briefly in the textbooksby Tinkham and de Gennes. Fossheim and Sudbø give a quite detailed derivation.

Physically speaking the enhanced surface field corresponds to an enhanced Cooper pair density at thesurface being enforced by the boundary condition Eq.(4.4.46). Without the boundary condition the orderparameter would have a negative slope at x = 0 whenever x0 > 0 while it is pushed up otherwise. Theboundary condition can be satisfied with any function which is symmetric about x = 0. The simplestground-state wave function in agreement with this requirement is a Gaussian centered at x = 0. Forx0 > 0 symmetry can be imposed by a superposition of two Gaussians at ±x0 or, simpler, by a trialwave function f (x) = (1+ ax2)exp(−bx2). If the potential energy in Eq. (4.4.40) is also symmetrizedabout x = 0 one arrives at a double well potential. For appropriate values of x0 the maximum at x = 0is low and the potential is wide enough to facilitate a lower eigenvalue. The modulus of correspondingeigenfunction has then a small depression at x = 0 but maxima at x≈ x0 just as the trial wave function fora < 0. This symmetry considerations make the reasoning behind the surface effects more plausible andshow how boundary conditions can be imposed. Needless to say that all parts for x < 0 have no physicalmeaning and are constructed in a similar fashion as mirror charges for instance.

The existence of the surface critical field Bc3 shows directly that the transition at Bc2 cannot be sharpunder realistic conditions but, rather, must be a crossover. In particular, if the surface is rough on theorder of the coherence length the maximal value of Bc3 is not reached and, in addition, vortices arepinned to the surface making affairs even more complicated. In other words, for determining the rangeof superconductivity in the presence of magnetic fields and, similarly important, currents, which wehave disregarded here for the sake of simplicity, become very important. Hence, for characterizingsuperconductors and determining their intrinsic properties the conditions of the experiments have tobe controlled. What we discuss here are the intrinsic properties of clean materials in the limit of fullreversibility in order to understand the thermodynamic and microscopic properties at the origin of thecondensed state. Deviations from the described behavior are equally important since they are the basisof applications and will be discussed there.

4.4.3 The thermodynamic critical field Bc

While the thermodynamic critical field is intuitively clear in a type I superconductor the definition is lessobvious when the flux can penetrate. However, the GL approach provides a direct interpretation since



the difference between the free energy densities in the superconducting and the normal state is given by

fs− fn =−α2

2β=− B2

c

2µ0. (4.4.49)

Recalling the expressions for ξGL and λGL in Eqs. (4.3.29) and (4.3.27) we find that the ratio |α|/β canbe derived from λGL while ξGL depends on |α| alone and obtain

|α| =h2

4mξ 2GL

(4.4.50)

|α|β

=m

2µ0λ 2GLe2 (4.4.51)

B2c =

h2

2(2e)2ξ 2GLλ 2

GL

=Φ2

0

2(2π)2ξ 2GLλ 2

GL

Bc =Φ0√

22πξGLλGL. (4.4.52)

Along with the definition of κ in Eq. (4.3.31) and the expression for the upper critical field in Eq. (4.4.44)we find,

Bc2 =√

2κBc, (4.4.53)

immediately proving the dichotomy spelled out in Eqs. (4.3.34) and (4.3.35): Whenever√

2κ < 1 thenucleation field Bc2 is smaller than the condensation field making it energetically unfavorable for the fluxto penetrate into the material. Rather the field is completely expelled at Bc. We may ask as to whetherBc2 has still a physical meaning. In fact, neglecting surface effects, Bc2 is now the “supercooling” field,and (in very clean samples) an applied field decreasing from B0 > Bc cannot be expelled before reachingBc2 < Bc . Once the field is expelled Bc can be approached from below. In optimal conditions, includingnearly atomically flat surfaces and the suppression of edge effects, the field can be cranked up furtherand penetrates only at Bc1 [?]. Consequently, as expected for a first order phase transition, there is ahysteresis. It is also clear from the discussion of thermodynamics that the integral over the magnetizationcorresponds to the condensation energy.

4.4.4 The lower critical field Bc1

In the opposite case the nucleation takes place at Bc2 > Bc but what happens at Bc? Unfortunatelynothing, making the determination of the condensation energy particularly complicated in all type IIsuperconductors with high Bc2. For this reason there is still a debate in the case of the cuprates aftermore than two decades. However, since the thermodynamical critical field is still determined by theintegral over the magnetization there must be a compensation of the contributions to the integral at highfields in the low-field range in the spirit of a Maxwell construction. In other words, the field is completelyexcluded up to a critical field Bc1 and then starts to penetrate. The first flux line penetrates without energysince the Gibbs potentials with and without flux should be the same in this point of the phase diagram.Then, as a first guess, one would expect B2

c ≈ Bc1Bc2 and Bc ≈√

2κBc1. In reality and for κ 1, thecore enhances the energy stored in a single vortex line by a factor of order lnκ + ε [see Eq. (2.3.100)],and the better estimate for Bc1 is given by (see textbooks)

Bc1 =Bc

4πλ 2GL

(lnκ +0.08) κ 1 (4.4.54)

yielding Bc ≈√

Bc1Bc2O(lnκ). The relation between the various bulk fields is sketched in Fig. 4.3.

2013


BBc2Bc1 Bc

-0M

Figure 4.3: Magnetization of type I and type II superconductors as a function of the applied field B forvarious GL parameters κ (sketch). For κ < 1/

√2 the field is excluded completely (black). The red and

the blue curves correspond to κ ≈ 0.8 and 2, respectively. As indicated for κ ≈ 2 the shaded areas aboveand below the thermodynamical field Bc are equal (Maxwell construction). The infinite slope at Bc1 isonly observed in ideally clean samples and indicates that the first flux line can penetrate free of energy.

4.4.5 The Abrikosov lattice (1957)

For κ 1 and parallel flux lines along, e.g. ez, it can be shown relatively easily (Tinkham) that the forcedensity f on one flux line from all other flux lines and a potential transport current is given by

f = js× ezΦ0 (4.4.55)

where js is the sum of all contributions to the supercurrent density at the location of the flux line, and Φ0 isthe flux quantum. f is repulsive for two neighboring flux lines in equilibrium. If there is a transport currentjtr perpendicular to the flux lines the screening currents around the vortices have components paralleland anti-parallel to jtr leading to a Lorentz force on the vortices and making them move perpendicularto jtr and ez. This movement results in a voltage drop parallel to jtr hence a finite resistance and ispotentially detrimental if large currents have to transported in a magnetic field such as in a solenoid.Only if the movement can be suppressed efficiently type II superconductors will be useful for this kindof applications. As will be discussed in chapter 6 the flux lines can be pinned in various ways, and thetype II materials with pinning (sometimes called type III superconductors) can in fact be used for highcurrent and high field applications.

Here we focus on the problem of having forces on a flux line whenever the line is not in a fully symmetricenvironment where all screening currents from neighboring lines cancel out by symmetry. Abrikosovfound a solution to this problem by deriving a general solution ψL for the linearized first GL equationwhich is strictly valid only in the limit B0→Bc2. With the field in ez direction we found Bc2 by evaluatingthe lowest eigenvalue of Eq. (4.4.40) at n = 0 and kz = 0. Hence, only ky has to be considered inEq. (4.4.39) and will be abbreviated by k in the following. Eq. (4.4.38) shows that for each k thereexists a slab in the yz-plane centered at x0 carrying one flux quantum. Since we need a symmetric, i.e. aperiodic solution, we set

k = nq (4.4.56)

with a fixed q yielding a real space periodicity ∆y = 2π/q and

xn =nqΦ0

2πB0(4.4.57)

showing directly that the spacing in x-direction is

∆x =qΦ0

2πB0(4.4.58)



and that B0∆x∆y = Φ0 as to be expected. Since the functions

ψn = exp(inqy)exp

(−[

x− xn√2ξGL

]2)

(4.4.59)

are orthogonal for different n because of the factor exp(inqy) the general solution is a superposition ofall functions ψn,

ψL = ∑n

Cn exp(inqy)exp

(−[

x− xn√2ξGL

]2)

(4.4.60)

with coefficients Cn to be determined. While periodicity in y-direction is built in via Eq. (4.4.56) theperiodicity in x-direction is recovered only if the coefficients Cn are periodic, Cn+m =Cn. From the dis-cussion above a square lattice follows immediately for index-independent coefficients Cn or, equivalently,m = 1. For C1 = iC0 and m = 2 a triangular lattice is found. For deciding as to which of the possiblelattices is being realized Abrikosov found that the linearized GL equation is insufficient and observedthat the normalization independent parameter

βA =〈ψ4

L〉〈ψ2

L〉2(4.4.61)

should be as close to 1 as possible. βA = 1 obviously holds for constant ψL corresponding to the fieldfree case. All other variations of ψL (dictated by flux penetrating the material) lead to larger values of βA.For the square lattice βA = 1.18 for the triangular lattice βA = 1.16. Hence the triangular lattice wins bya very small margin. Since numerical calculations are necessary it is not that surprising that Abrikosov,in his original paper in 1957, found the square lattice to prevail. Kleiner, Roth, and Autler showed in1964 that the triangular lattice out of all periodic solutions has in fact the optimal value of βA. Tinkhamnotices that, for the same density of flux lines, the distance ai between the flux lines is slightly larger inthe triangular lattice (closest packing) than in the square lattice,

atriangle =

(43

) 14

≈ 1.075asquare (4.4.62)

favoring the first one since it reduces the positive repulsive energy between the lines. The flux line latticehas been visualized first by Essmann and Trauble in 1967 by decoration and, more recently, by scanningtechniques as shown in Fig. 4.4.

The difference between the possible periodic solutions is in fact sufficiently small to allow solutions otherthan triangular in many cases were either the underlying lattice or the symmetry of the order parameterintroduce additional anisotropies. For instance in Nb one finds various transitions between different fluxline configurations as a function of temperature and field [?]. In YBa2Cu3O7 the dx2−y2 symmetry of theorder parameter is sufficient to make the square lattice more favorable [?].

2013


Abricosov vortex lattice

2

2

02 )(21exp q

Bnx

TeC

cn

inqyn

(a) (b) (c)

Figure 4.4: Flux line lattice. (a) and (b) show the different filling for the square (a) and the triangular (b)flux line lattice. From Tinkham [?]. (c) Scanning tunneling map of NbSe2 in the Shubnikov phase [23].The dark areas correspond to the vortex cores where the superconducting gap and, hence, the orderparameter vanishes.


Chapter 5

The Josephson Effect

Elementary optics and quantum mechanics show that waves can tunnel through regions of space evenin the case of a potential barrier. For instance, light can travel across a sufficiently narrow gap betweentwo dielectrica even if the incoming wave is totally reflected from the inner surface of one of the ma-terials. Another example are electrons tunneling through an insulator sandwiched between two metals.In all cases the wavelength and the height of the barrier govern the transmitted intensity. Typically,for distances larger than a few wavelengths the transmitted intensity vanishes. What happens in thesecircumstances with the Cooper pairs of a condensate, and is it worthwhile to bother?

If we could couple two condensates weakly it appears very intriguing to utilize the rigid phases on eitherside of the junction so as to generate beat frequencies in the MHz or GHz range having unprecedentedstability and watch quantum mechanics at work on a macroscopic scale. When Josephson considered thispossibility in 1962 [?] the experts were skeptical. John Bardeen for instance argued that the tunnelingprobability would be orders of magnitude too small for being the square of the tunneling efficiency ofa single electron. It turned out that the rules for condensates are different and that the single particleprobability still determines the order of magnitude of the matrix element.

5.1 Weakly coupled superconductors

The Ginzburg-Landau model gives us an idea of how we should think of pair tunneling at least in the caseof a metallic interface between two superconductors. Eq. (4.2.19) explicitly describes how far the wavefunction extends into the normal metal via the proximity effect, and Eq. (4.3.29) shows ξGL(T ) to bethe approximate length scale. A schematic view of the experimental setup for observing the Josephsoneffects is shown in Fig. 5.1. The two superconductors which may be made of different materials areseparated by a thin insulating or metallic layer (grey). Other realizations of weak links are either narrowsuperconducting bridges with lateral dimensions close to ξGL ≈ ξ0 or boundaries between differentlyoriented single crystals or intrinsic contacts in natural very anisotropic materials [?] or hetero structures.Most importantly, the critical supercurrent density across the weak link must be orders of magnitudesmaller than in the bulk and coherence between the two condensates must be maintained. The junctionscan be either voltage or current biased. In most of the cases the current I is determined from out side andthe voltage U is the dependent quantity. Nevertheless, the current-voltage characteristics will be plottedthe other way around partially for traditional reasons. However, since the energy is given eU this is alsomore instructive as we will see below.

73

74 R. HACKL AND D. EINZEL The Josephson Effect

superconductor superconductorinsulator/metal

~0

U

I

Figure 5.1: Schematics of a weak link. The grey slab separating the superconductors on the left andthe right can be either an insulator or a semi conductor or a metal. If the cross section is reduced todimensions of the coherence length ξ0 even a superconductor can be used a weak link. The thickness ofthe layer should be on the order of ξ0 or less. Usually the contact is current biased and the voltage dropU is the dependent quantity.

5.2 The Josephson equations

The dynamics of the Cooper pairs can be described by time-dependent Schrodinger equations for thecondensate wave functions ψ1,2(r, t) for superconductors 1 and 2 having stationary energy eigenvaluesE1,2. Then the most physical approach to a weak link is to introduce an interaction W12 E1,2 betweenthe the two superconductors leading to an equation system of two coupled harmonic oscillators [?, ?],

ihψ1(r, t) = E1ψ1(r, t)+W12ψ2(r, t)ihψ2(r, t) = E2ψ1(r, t)+W21ψ1(r, t),

where ψ is the partial derivative of ψ w.r.t. time. For solving these equations we use again the Madelungrepresentation of the wave function [Eq. (2.3.65)]. In contrast to the chapter on GL theory we substitutehere a2

1,2(r, t) = Qnpair(r, t)≡ ρQ(r, t) for the amplitude. After separating imaginary and real part we get,respectively,

a21 =

2W12a1a2

hsin(θ2−θ1) =−a2

2 and (5.2.1)

−ha21θ1 = E1a2

1 +W12a1a2 cos(θ2−θ1) (5.2.2)

ha22θ2 = −E2a2

2−W21a1a2 cos(θ1−θ2). (5.2.3)

(5.2.4)

From the first equation we obtain that a21 =−a2

2 meaning that the rate of change of the charge density onthe left side is equal but opposite to that on the right side of the junction. In addition, the time derivativeof the density is related to a current density by virtue of the continuity equation. However, it is clear thatan imbalance of charges is not possible. Hence, so long as no current is supplied from outside there won’tbe a current across the junction, and the phase difference vanishes. In turn, if an external current from asource is supplied Cooper pairs may move from one side to the other side, and θ2−θ1 will be differentfrom zero. The maximal current is proportional to the matrix element W12 ≈W21 and is expected to besmall. These considerations allow us to write down the first Josephson equation,

j = jc sin(θ2−θ1)≡ jc sin(∆θ), (5.2.5)


The Josephson equations SUPERCONDUCTIVITY 75

where jc is the critical Josephson current density, which states that there is a finite supercurrent across,e.g., an insulator without a voltage drop. The current is proportional to the sine of the relative phase ofthe condensates in superconductor 1 and 2 or directly to ∆θ for small currents.

An estimate of the critical current can be obtained from GL theory or from microscopic considerations.The latter was achieved by Ambegaokar and Baratoff [?] who found the full temperature dependence ofthe product of the critical current Ic and the normal resistance Rn of the junction which can be approxi-mated as Rn(Tc),

IcRn =π∆

2etanh

(∆

2kBT

). (5.2.6)

This important result is universal, i.e. independent of the type of junction. In the limit T → 0it reduces to IcRn = π∆(0)/2e while close to Tc the temperature dependence becomes linear andIcRn = 2.34πkB/e(Tc−T ) with the constant being 635 µV/K.

The phase difference ∆θ is not gauge invariant implying that the current would depend on the selectedgauge of the vector potential A in an applied magnetic field. Therefore, before proceeding, we introducethe gauge-invariant phase difference γ ,

γ = ∆θ − 2π

Φ0

∫ 2

1A ·ds, (5.2.7)

where the integration limits correspond to the two sides of the weak link. Applying the gauge transfor-mations A→ A+∇χ and θ → θ + 2e

h χ immediately proves Eq. (5.2.7). γ is generally valid with andwithout field and has to used later when we derive properties of junctions in a field and study quantuminterference effects. The gauge invariant first Josephson equation the reads

j = jc sinγ. (5.2.8)

For the analysis of Eqs. (5.2.2) and (5.2.3) we assume the same material on each side and get a1 = a2and W12 = W21. Since the cosine is an even function, the sum of the two expressions yields the secondJosephson equation,

ddt

γ =2eh

U, (5.2.9)

where U =E1−E2 and Q= 2e. Obviously, upon exceeding jc there is a finite voltage across the junction,and the phase difference becomes time dependent. The differential equation can be integrated right awayyielding

γ(t) = γ(0)+∫ t

0

2π

Φ0Udt ′ = γ(0)+2π

UΦ0

t. (5.2.10)

If we insert γ(t) into the first Josephson equation we obtain a current

j(t) = jc sin(

γ(0)+2πUΦ0

t). (5.2.11)

that oscillates at the frequency U/Φ0 where Φ0 is the flux quantum. The inverse of the flux quantum isthe celebrated Josephson frequency

1Φ0

= 483.597870(11)MHzµV

, (5.2.12)

meaning that for a voltage drop of 1 µV a frequency of 483.597870(11) MHz can be observed that de-pends only on constants of nature. Along with the resistance normal given by the von-Klitzing constanth/e2 = RK = 25812.8074434(84)Ω the Josephson frequency plays an important role in metrology sincethe voltage can be linked to elementary constant.

2013


5.3 The RCSJ model

While the fundamental properties are easily derived the details important for the design and understand-ing of sensors and other devices require the study of individual junctions. For that we note that a weaklink is not only a highly non-linear resistance but has an Ohmic and a capacitive shunt which are of cru-cial importance for the dynamics. Therefore we analyze the properties of the device sketched in Fig. 5.2.The bias current Ib is distributed between the three branches across the resistor, the weak link, and the ca-

Figure 5.2: Equivalent circuit of a resistively and capacitively shunted junction (RCSJ). (a) The weaklink is symbolized by a cross. The total bias current Ib is distributed between the three branches asIR = U/R, IJ = Ic sinγ , and IC = CU . (b) and (c) The dynamics of the RSCJ model can be visualizedby a washboard potential where U(δ ) = −EJ cosδ − Ib/Ic〈δ 〉 where 〈δ 〉 is the coordinate. (b) If theIb < Tc the phase is time independent and a supercurrent flows across the junction. (c) Above the criticalcurrent 〈δ 〉 becomes time dependent but does not vary linearly as expected for a harmonic oscillator.From Clarke.

pacity as IR =U/R, IJ = Ic sinγ , and IC =CU . We use the second Josephson equation (5.2.9) to eliminateU and U and get

Ib

Ic= sinγ +

Φ0

2πIcRγ +

Φ0C2πIc

γ. (5.3.13)

Eq. (5.3.13) is equivalent to the second order differential equation of a driven pendulum. The termIc sinγ replaces the linear term of the harmonic oscillator equation with small amplitude and allows forfull rotations. The mechanical analogue reads

D = mgl sinϕ +Γϕ +Θϕ (5.3.14)

where D is the driving torque, mgl sinϕ is the gravitational energy if the pendulum is rotated out ofequilibrium, Γ is the damping, and Θ is the moment of inertia. In the case of the RCSJ model it iscustomary (and convenient) to introduce the parameters

τc =Φ0

2πIcR, (5.3.15)

βc =2πIcR2C

Φ0Stewart−McCumber parameter (5.3.16)

=RCτc≡ τRC

τc∼ Q2 quality factor,

τ =tτc, and dt = τcdτ, (5.3.17)


The RCSJ model SUPERCONDUCTIVITY 77

and obtain the dimensionless sine-Gordon equation1

i = sinγ +∂γ

∂τ+βc

∂ 2γ

∂τ2 . (5.3.18)

The Stewart-McCumber parameter βc corresponds to a quality factor and indicates how freely the systemcan oscillate. However, independent of βc the time dependences of i(τ) and γ(τ) will never be harmonic.Rather, as we will see below, βc indicates as to whether or not the I−V characteristics is hysteretic.

For βc = 0 we find

i = sinγ +∂γ

∂τ(5.3.19)

which can be integrated for i > 0 over one period allowing us to derive an average voltage from (5.2.9).Separation of the variables and integration between 0 and 2π and, respectively, 0 and the period durationτ0 yields

2π√i2−1

= τ0 (5.3.20)

and after restoring units and inserting the average oscillation frequency 2π/T = 2π(τcτ0)−1 in Eq. (5.2.9)

we find

U = R√

I2b − I2

c . (5.3.21)

This result shows that the voltage is zero for Ib = Ic and then asymptotically approaches Ohmic behaviorfor Ib Ic. For Ib < Ic integral (5.3.20) diverges implying that the phase γ does not depend on time andU vanishes.

It is not possible to solve Eq. (5.3.18) analytically for βc > 0, and the I−V characteristics can only beobtained numerically. For βc > 0.5 the I−V curves become increasingly hysteretic as shown in Fig. 5.3.This means that the voltage assumes a finite value immediately at Ic jumping directly to the Ohmic linefor βc 1. On reducing Ib the voltage follows Ohm’s law down to I< where U finally jumps back tozero and the current flows free of losses. If we integrate over one period we find that

R. KleinerDissertation TUM

Figure 5.3: I−V curves for differently damped Josephson contacts. The numerical simulation is accord-ing to the model of a resistively and capacitively shunted junction (RCSJ) for βc as indicated. From R.Kleiner (PhD thesis, TUM 1092).

1The equation is the non-linear analogue of the one-dimensional Klein-Gordon equation, being the relativistic form of theSchrodinger equation. Goldstein, Poole, and Safko (Addison Wesley 2002) speculate that the name is a frivolous pun.

2013


∫ T

0Udt = Φ0 (5.3.22)

meaning that the phase changes exactly by one flux quantum per cycle. The energy is radiated off thejunction with a frequency on the order of the Josephson frequeny times the voltage.

5.4 Josephson contact in a microwave field

We go now the other way around and irradiate the junction with a microwave field UHF = U0 +U1 cos(ω1t) with U1U0. Then the phase varies as

γ(t) = γ0 +2eh

U0t +2eh

U1

ω1sin(ω1t) (5.4.23)

which has to be inserted into Eq. (5.2.8). The general solution is given by Bessel functions of order n, Jn

as

j(t) = jc∞

∑n=0

Jn

(2eU1

hω1

)· sin

(2eU0

ht±nω1t + γ0

). (5.4.24)

We search now for the dc contributions to j(t) and observe that the time integral over the sum is differentfrom zero only if

ω1 =2eh

U0 (5.4.25)

and integer multiples thereof. Hence the distance on the voltage axis Un between different steps is aninteger multiple of U0,

Un =nhω

2e= nU0 (5.4.26)

The way the calculation works can be visualized relatively easily for the first step (see Problem 2 in set10).

5.5 Josephson effect in a magnetic field

In a magnetic field the gauge invariant phase difference (5.2.7) across the junction becomes relevant asone may anticipate already from the supercurrent (2.3.74).

5.5.1 Ring with a single weak link

We consider now the setup sketched in Fig. (5.4). There is a ring having a central hole. The total diameterof the ring is at least an order of magnitude larger than the penetration depth λeff of the magnetic fieldB = ∇×A implying that a contour Γ can be found along which the screening currents vanish. The fieldcan penetrate the weak link between points 1 and 2 and the superconductor above and below up to λeff.The distance of 1 and 2 from the weak link is larger than λeff. Then Eq. (2.3.74) yields∫ 1

2∇θ ·ds =

2eh

∫ 1

2A ·ds → θ1−θ2 =

2eh

∫ 1

2A ·ds (5.5.27)


Josephson effect in a magnetic field SUPERCONDUCTIVITY 79

Js = 0

1

2

(ds)

Figure 5.4: Ring with a weak link in a magnetic field. The field B = ∇×A points out of the plane. Thering walls are much thicker than the penetration depth λeff of the field. Therefore the current Js alongthe contour Γ (integration variable ds) vanishes. The field penetrates the weak link and the adjacentsuperconductor as long as the distance from the weak link is smaller than λeff. Points 1 and 2 are in thefield free part.

and

θ2−θ1 = γ +2eh

∫ 2

1A ·ds. (5.5.28)

Upon adding Eq. (5.5.27) and (5.5.28) one obtains

2πn = γ +2eh

∮Γ

A ·ds

= γ +2eh

∫B ·dS (5.5.29)

where S is the total are inclosed by Γ. Since the phase is unique only modulo 2πn the factor 2πn, wheren is an integer, has to be added. The last integral is the magnetic flux Φ through the ring. If we insert theflux quantum Φ0 we find that the gauge invariant phase

γ = 2π

(n− Φ

Φ0

)(5.5.30)

changes proportional to the flux through the ring. The rate of change is given by Φ0 which is, as weknow, a very small number. Hence, a ring as shown in Fig. (5.4) can in principle be a very sensitivesensor if the γ can be measured.

5.5.2 Ring with two weak links: Quantum interference

To this end we consider a ring having two weak links as shown in Fig. 5.5 (a). With this setup the totalflux penetrating the ring is proportional to the phase differences γ1 and γ2 of links 1 and 2,

γ1− γ2 = 2πΦ

Φ0. (5.5.31)

2013


Figure 5.5: Superconducting quantum interference device (SQUID). (a) Schematic view. (b) I−V curveas a function of the applied field. (c) Modulation of the maximal critical current as a function of the field.Note that the Josephson junctions have to have a relatively small βc so as to avoid hysteresis.

If the two contacts have exactly the same properties the total current across the two links is simply thesum of the individual currents, Itot = Ic(sinγ1 + sinγ2). With the substitution

γ1 = γ0 +πΦ

Φ0

γ2 = γ0−πΦ

Φ0

where γ0 takes care of all problems with screening currents (which are a complicated issue in practice)we find immediately

Imax(B) = 2Ic sinγ0

∣∣∣∣cosπΦ

Φ0

∣∣∣∣ . (5.5.32)

Eq. (5.5.32) states that the current is maximal for vanishing field. This is not surprising since the twocurrents add symmetrically. In all other cases there is a superposition of the bias and the screening currentwhich lead to a pattern of Imax(B) which is equivalent to that of an optical interference experiment witha double slit. Therefore the ring with two weak links is named superconducting quantum interferencedevice (SQUID)2. If βc of the two contacts is in the right range Imax(B) oscillates around a mean valuebut does not reach zero. In addition all hysteresis effects vanish. Then the voltage in the resistive state isa unique function of B as long as 2Φ < Φ0. If the field through the SQUID is compensated in the mostsensitive part of the I−V curve [Fig. 5.5 (b)] one has a very sensitive field detector (10−6Φ0 typically).If not the voltage oscillates [Fig. 5.5 (c)].

5.5.3 Quantum interference in a long junction

So far we assumed that the current is constant across the entire junction. However, the width of thesuperconductor perpendicular to the field B and the current I is usually as large as several penetrationdepths λ as shown in Fig. 5.6. Hence, the gauge invariant phase may vary substantially in the directionperpendicular to the current and the field. The width of the contact a is much larger than the penetrationdepth λ , and the field points into the positive z-direction. Note that d . ξ can be larger than λ in type Isuperconductors. Then the flux through the area enclosed by the curve Γ (blue) is given by

Φ(x) = Bx(d +2λ ) (5.5.33)

2John Clarke who invented the SQUID in 1966 is a gourmet.


Josephson effect in a magnetic field SUPERCONDUCTIVITY 81

yielding the gauge invariant phase,

γ(x) = γ(0)+2πΦ(x)Φ0

. (5.5.34)

For getting the maximal supercurrent Imax in y direction Eq. (5.5.34) will be inserted in the first Josephson

0

a/2

-a/2

x

y

B

x

d

Figure 5.6: Long (wide) weak link. The width a in x-direction is much larger than the penetration depthλ . The current I is along y, and the field B points along the z-direction. For getting the flux Φ(x) one hasto integrate in the x− y plane inside the curve Γ (blue). The screening currents jscreen(λ ) are indicatedschematically (red). Here they are larger than the critical current across the weak link. in y direction thesample is larger than shown.

equation (5.2.8) that is then integrated in x and z direction in the limits [±a/2] and [0,c] where c is thethickness of the sample:

Imax = Ic sin(γ(0))

∣∣∣∣∣∣sin(

πΦ(x)Φ0

)πΦ(x)

Φ0

∣∣∣∣∣∣ . (5.5.35)

Eq. (5.5.35) is equivalent to the Fraunhofer pattern in optics describing the intensity distribution of thediffracted light after a single slit while the SQUID corresponds to the double slit experiments. In practicethe flux dependence of the maximal current across a SQUID is a superposition of a sin and cos modulationsince the legs of the SQUID have also a finite width. If Eq. (5.5.34) is inserted in the Eq. (5.2.8) an x-dependent current density as shown in Fig. 5.7 is found which explains the zeros in the maximal currentphysically.

If the field is reduced it can be expelled from the junction. However, since the critical current across thejunction is very small in comparison to that in the bulk the Josephson penetration depth λJ ,

λJ =

(Φ0

2πµ0 jcd

) 12

, (5.5.36)

is usually much larger than the London penetration depth λL. Eq. (5.5.36) is derived from the Ferrell-Prange equation [?]. Typical values for λJ exceed those for λL by at least one order of magnitude.

2013


e

f

Magnetic field B = 0 Magnetic field B = B0/2

Magnetic field B = B0 Magnetic field B = 3B0/2

-4 -3 -2 -1 0 1 2 3 40.0

0.5

1.0

I max

/I c/0

Figure 5.7: Current in a long junction. (a)–(d) Current distribution for different field/flux values whereB0 corresponds to one flux quantum. (e) Maximal current (in units of the critical current) as a functionof the flux penetrating the junction according to Eq. (5.5.35). Φ0 is the flux quantum. (f) Experimentalresults from a Pb/YBa2Cu3O7 junction. From [?].


Chapter 6

Unconventional Materials

The entire field of superconductivity is driven by the discovery and development of new compoundsbringing the materials scientists into the center of the game. As opposed to semiconductor physicsit is extremely hard to predict or at least guess the tendency towards superconductivity. In addition,any prediction of transition temperatures from first principles, even for known materials, is difficultand was successful only for a few elements and simple compounds with conventional electron-phononcoupling [9]. To our knowledge BKBO is the only material having a Tc close to 30 K which was predictedon the basis of the isostructural low-Tc sister compound BaPbBiO3 [?]. In all other cases empirical rulessuch as those of Bernd Matthias [?] or highly complex quantum chemistry considerations have little orno predictive power yet meaning that our understanding is still insufficient and requires new and brightideas. Hence, the purpose of this chapter is twofold in that the zest for understanding on the theoreticalside is highlighted and the inventions of the materials scientist are celebrated by describing some of thefascinating properties of unconventional superconductors.

6.1 Classification

Electron-phonon coupling was considered the only mechanism leading to superconductivity for a longtime although magnetic polarizability was recognized as possible alternative [24]. However, only thediscovery of systems with heavy electrons [25] and superconductivity therein [26] opened new vistas.1

It was pointed out from the beginning that the heavy mass (large heat capacity) may originate in theKondo-like interaction of the conduction electrons with the spins of the 4 f -electrons of cerium. The dis-covery of superconductivity at approximately 0.5 K in CeCu2Si2 in 1979 [26] was a new paradigm andopened a completely new field of research that remained active and vibrant until now [27]. The heavyfermion systems were certainly the first examples of materials belonging to the class of unconventionalsuperconductors which include now also the copper-oxygen [6] and the iron-based compounds [7] ascompiled in Table 6.1. What means unconventional? The distinction between conventional and uncon-ventional superconductors is not clear-cut. Sometimes all electron-phonon superconductors carry thename conventional while all others are unconventional. In the terminology of Anderson [?] dividing ruleis given by the Fermi surface average of the gap ∆k: Whenever ∑k ∆k = 0 superconductivity is uncon-ventional. Considering the modern developments it makes also sense to ask as to whether or not there

1The name “heavy electrons” (or fermions) derives from measurements of the electronic heat capacity cp = γT with the γ

the Sommerfeld constant. As shown first for CeAl3 cp is approximately three orders of magnitude larger than that of a usualmetal. Since γ is proportional to the density of states at the Fermi energy NF and, hence, to the effective mass m∗ of theelectrons a large γ was associated with a large m∗. Maurice Rice pointed out early that, by definition, NF ∝ |vF |−1 implying alarge γ to be more indicative of slow rather than heavy electrons. He added that slow electrons might sound less exciting andimportant.

83

84 R. HACKL AND D. EINZEL Unconventional Materials

Table 6.1: The main catagories of superconductors. Conventional includes all metallic compounds withelectron-phonon coupling where a lattice instability driven by strong coupling is the only additionalphase. MgB2 is listed separately for holding the record of the conventional systems due to a mixtureof intra- and interband pairing. CuO2 represents all superconductors on copper-oxygen basis. Fe-basedsystems include also those with Fe replaced by Rh. f -electron systems are typically those having ahigh electronic specific heat at low temperature and are also known as heavy fermion compounds. Thefollowing abbreviations are used: electron (el.), instability (inst.), antiferromagnetism (AF), density wave(DW), spin density wave (SDW), ferromagnetism (FM), and cylinder (cyl.). The missing entry in the lastcolumn reflects the existence of various types of pairing states in f -electron systems. The years inbrackets indicate the first chance for observing superconductivity (see text). References can be found inthe text.

conventional MgB2 CuO2 Fe-based f -electron

structural anisotropy isotropic 10 . . .100 10 . . .105 5 . . .100 ∼isotropicnormal state metal metal strange metal metal heavy el.competing phases (lattice inst.) – AF, DW SDW AF/FMFermi surface multi-band cyl. + sphere cyl. 1 band ∼cyl. 5 bands multi-bandTc,max (K) 30 39 160 56 18.5Hc2,max (T) 50 50 . . .100 100 . . .250 100 27+ξ0 (nm) 10 . . .1000 1 . . .5 ∼ 3 ∼ 5 3 . . .30κGL (nm) 0.01 . . .50 ∼ 100 ∼ 100 ∼ 50 ∼ 100superconductivity isotropic s anisotropic s dx2−y2 s±year of discovery 1911 2002(1952) 1986(1978) 2006(1995) 1979

is one or more other phases competing with superconductivity. The heavy fermions, the cuprates, theFe-based materials, and organic metals share this property. In the vast majority of the cases magnetismcompetes with superconductivity. Less frequently, charge order plays a role. Very often the competingphase is suppressed continuously as a function of a parameter r other than temperature such as appliedpressure or magnetic field or doping. If the transition temperature to the ordered state To(r) approacheszero for a specific rc the material belongs to the large class of quantum critical systems [?, 28]. Here,the physics is dominated by the fluctuations of the spin or charge density in a wide range of temperatureabove rc which are considered to contribute substantially to superconductivity in both the cuprates andthe Fe-based compounds [29, 30]. In fact, since the electron-phonon coupling is too small to explaintransition temperatures in the range of 100 K other types of exchange bosons have to take over at least solong as a the interaction is retarded (meaning that the timescale is much longer than that of the electronicscreening).

Although the definitions are not generally accepted and/or used one has at least an intuitive idea alongwhich lines the divide runs. Nevertheless, when reading the literature one should be aware of the slightlyfuzzy definitions. For the present purposes it is enough to know that there are systems having propertiesfundamentally different from those of the usual metallic superconductors. In most of the cases they arecharacterized by rich phase diagrams and the proximity of magnetism and superconductivity. In thefollowing we provide an overview of the Fe-based and copper-oxygen systems and a brief summary ofthe heavy fermions and the organic metals.


The iron-age of superconductivity SUPERCONDUCTIVITY 85

6.2 The iron-age of superconductivity

High-temperature superconductivity in compounds with iron2 was not considered seriously beforeYoichi Kamihara and coauthors in Hideo Hosono’s group discovered a transition at Tc = 26 K inLa(O1−xFx)FeAs. [31] It is interesting to note that LaFePO having Tc = 6 K was synthesized alreadyin 1995 by Barbara Zimmer in the group of Wolfgang Jeitschko3. The observation of superconductivitywas only mentioned in Zimmer’s thesis. Since the publication of Kamihara’s results in 2008 severalten thousand papers have appeared, and the highest Tc so far exceeds 50 K. Even though the materialscontain arsenic or other rather toxic elements many laboratories started with the preparation of poly-and single-crystalline samples. Generally the Fe atoms form two-dimensional layers and are coordinatedwith atoms from the fifth column in most of the cases (with Se being an exception) as shown in Fig. 6.1.The family name pnictides (Pn) comes from Greek πν iχτıκ oς (asphyxiant, suffocative) for the columnof nitrogen.

In particular Chinese scientists contributed a lot of important results and found Nd(O1−xFx)FeAs withthe so far highest transition at 55 K [?]. While in the beginning many people believed the iron pnic-tides to be another class of oxides Dirk Johrendt and his group [8] demonstrated that high transitiontemperatures can also be obtained in purely intermetallic compounds. At optimal doping with x ≈ 0.4Ba1−xKxFe2As2 reaches a Tc of 38 K. In contrast to the oxifluorides (see Fig. 6.1) large single crystalscan be grown although the homogeneity and quality is not in all cases satisfactory. However, crystalsof BaFe2(As1−xPx)2 and LaFePO are already sufficiently clean for the observation of quantum oscilla-tions [32, 33].

Figure 6.1: Structures of iron pnictide compounds (by courtesy of D. Johrendt). LaFeAsO1−xFx (1111;T max

c = 28 K) was the first pnictide superconductor with high Tc [31]. With Pr, Nd or Sm replacingLa Tc exceeds 50 K [?]. LaFeAsO1−xFx is isostructural to LaFePO having the transition at 6 K at thestoichiometric composition. BaFe2As2 (122) develops a spin-density wave (SDW). When doped with Kfor Ba [8], Co for Fe [34] or P for As the SDW is suppressed and superconductivity appears. LiFeAs(111) has a maximal Tc of 18 K. The simplest of the materials, FeSe (11), with the same structuralelements but Se for As has a maximal Tc of 8 K with 9% Se deficiency at ambient pressure and reaches27 K at 1.5 GPa [35, 36].

The question as to the origin of superconductivity arose immediately. Are the iron pnictides similar tothe cuprates or to MgB2 with Tc = 39 K due to electron-phonon coupling or are they a material class ontheir own? The spin-density-wave (SDW) order of the parent compound indeed suggests a proximity tothe cuprates, where the superconducting phase emerges from a Mott insulator. With doping p away from

2This section is partially copied from a contribution of one of us (R.H.) to the Annual Report 2009 of the WMI. The title isborrowed from a Viewpoint by Michelle Johannes in Physics 1, 28 (2008).

3Note that the undoped parent compounds of both the cuprates and the pnictides were well known quite some time ahead ofthe first observation of superconductivity in doped variants.

2013


half filling spin and charge fluctuations as well as superconductivity follow antiferromagnetic long rangeorder. However, in contrast to the cuprates there is no universal phase diagram in the pnictides (Fig. 6.2).In the essentially hole doped oxifluorides there is an abrupt transition from a magnetically ordered phaseto a superconducting one with Tc only weakly depending on doping. The electron-doped intermetalliccompounds have a smooth transition, and SDW order and superconductivity may even coexist [37]. Thephase boundary of superconductivity is dome shaped.

June 16, 2009; 4

BaFe2As2

Chu et al. PRB 79, 014506 (2009)

Mandrus, Canfield, Büchner, Klauss, Dai,….

Ba(Fe1-xCox)2As2

Figure 6.2: Phase diagrams of LaFeAsO1−xF−x (left) [38] and Ba(Fe1−xCox)2As2 (right) [37]. WhileSDW und superconductivity (SC) overlap in 122 there is a strict separation in La-1111. The structuraltransition at Tα always precedes spin density wave order at Tβ .

The differences in the phase diagrams of the pnictides are surprising since the electronic structures areremarkably similar. There are 5 bands derived from the Fe 3d orbitals. Two (α1,2) form concentric hole-like Fermi cylinders around the center of the Brillouin zone (BZ), two (β1,2) have FSs which encircle thecorner of the small BZ derived from the 2Fe crystallographic unit cell (Fig. 6.3). Since the cross sectionsof the resulting Fermi surfaces are nearly equal the α and β sheets are nested with the vector Q' (π,π).Consequently, the electronic susceptibility as described by the Lindhard function is strongly momentumdependent [30] and is believed to be at the origin of the SDW. The pronounced peaks in the susceptibilitymake the strong variations of the properties upon small changes of the electronic and lattice structures atleast plausible.

The real part of the susceptibility is also considered a possible origin of superconductivity [30, 39–41]while the electron-phonon coupling is probably weak [42]. From this point of view, the pnictides and thecuprates appear to be cousins in the same family even if the strong metallicity of the parent phases of theFePn compounds may argue otherwise. But how can the coordinates of the pnictides be determined? Ina recent optical transport study the authors conclude from the reduced band width that the pnictides arehalf way between normal metals and the cuprates [43]. However, the related spectral redistribution to beexpected upon doping is not observed by angle-resolved photoemission (ARPES) and x-ray absorption(XAS) [44, 45]. Perhaps one of the most telling similarities would be if the pnictides had the signatureproperty of all cuprates – an energy gap ∆k having nodes and a sign change along the Fermi surface [19].

In the 6 years after the discovery the understanding and the experimental knowledge has advanced sub-stantially. The results from the usual spectroscopic methods such as tunneling and photoemission startto converge. In most of the compounds the modulation of the energy on the Fermi surface is not sup-portive of gap nodes on individual Fermi surfaces [47]. However, it becomes more likely that the signchange between the hole and the electron surfaces leaves imprints in the tunneling spectra in an appliedmagnetic field [?] supporting the early suggestion of Mazin and coworkers [30] of an s± gap. Here thephases of the gaps on the hole- and electron-like Fermi surfaces differ by π . This would imply that, for


The iron-age of superconductivity SUPERCONDUCTIVITY 87Generic band structure and Fermi surfaces

1Fe

ky

1Fe

2F

2Fe

M

XQ

kx

SPP 1458- Iron Pnictides page 14November 30, 2009

Figure 6.3: Iron plane (left), Brillouin zone (BZ), and real part of the non-interacting susceptibilityχ0(q,ω) [30] (right) of FePn materials. The cell relevant for the electronic structure contains 1 Fe atom(dashes) and is smaller by a factor of 2 and rotated by 45 with respect to the crystal cell (full line andaxes a and b). The BZ of the unit cell (full line) and the first quadrant of the Fe plane (dashed line) areshown along with the FS cross sections at π/c = 0 (adopted from Ref. [46]). The dotted FSs are obtainedby downfolding the 1 Fe BZ. Even with the ellipsoidal elongation of the M barrels the α and β bands areapproximately nested. Reχ0(q,ω) controls the pairing strength [30].

ideal conditions of equal cross sections of the respective Fermi surfaces and magnitudes of the gaps, theJosephson current in c-direction perpendicular to the Fe planes would vanish for a conventional counterelectrode. While this Gedankenexperiment is good for visualizing the consequences of the s± gap (andalso other gaps) the technical realization is hampered by the real shape of the Fermi surfaces and theremaining variation of the gaps. Here the coherence factors come into play.

As was shown in problem 2 of set 5 external perturbations having the potentials φ or A are either in-dependent of the particle momentum or proportional to the momentum or its square. This and the spinintroduce a sensitivity to the sign of k in the response since the Bogoliubov particles in a superconductorare coherent superpositions of electrons and holes and the amplitudes of the perturbation matrix elementshave to be added before being squared. (The problem is outlined in the books of Tinkham and de Gennesand will be discussed in detail in a later version of chapter 3 of this manuscript.) The bottom line is thattwo particle response functions given by a superposition of occupied and unoccupied states such as thespin susceptibility, the ultrasound absorption, the NMR properties, the optical conductivity or the lightscattering response assume characteristic energy and temperature dependences depending on the relatedperturbation operator. In the case of the spin susceptibility and of the longitudinal ultrasound absorptionthere is no momentum involved. For NMR and optical absorption (one dipole transition) k enters lin-early via k ·A and for light scattering the perturbation is proportional to k2 (two dipole transitions) viaA2. Since the full momentum dependence of the gap enters the coherence factors the different types ofresponses assume energy and temperature dependences characteristic for individual ∆k.

In the Fe-based superconductors a resonance in the spin correlation has been observed by neutron scatter-ing [48] which may in fact be indicative of gaps having opposite sign on the electron and hole bands. TheNMR and optical experiments are not conclusive yet. Also the angle-resolved photoemission (ARPES)is more complicated than in the cuprates due to the relatively strong electronic kz dispersion followingfrom the weak anisotropy (see Table 6.1). In fact ARPES and Andreev tunneling favor large, essentiallyconstant gaps on all Fermi surfaces [49, 50]. In some compounds such as Ba1−xKxFe2As2 the modu-lation of the gap is stronger along kz than in the kx− ky-plane [?]. Nevertheless, some of the materialshave very small gaps on parts of the Fermi surface in addition to the possible sign change [47, 51, 52].Some experiments indicate a strong modulation of ∆k and even true nodal behavior [53, 54]. The strongmaterial dependence inferred from the experiments is also expected from theoretical consideration thatindicate the close proximity of various ground states [?].

2013


The possible anisotropies in the superconducting state are accompanied by band dependent carrier dy-namics in the normal state as observed by quantum oscillatory phenomena [32, 33] and the analysis ofHall data [55]. For the clarification of these fundamental questions at the heart of the physics of thepnictides bulk sensitive spectroscopies with band and momentum resolution will be instrumental.

Recently, the critical current density was determined in magnetic fields up to 45 T and found to be inthe range of 104 A cm−2 up to at least 30 T (Fig. 6.4). These properties open the possibility to useBa(Fe1−xCox)2As2 for static fields in the range above 21 T the limiting value of Nb3Sn. It became

Figure 6.4: Critical currents jc and pinning forces of Ba(Fe1−xCox)2As2 in a magnetic field. The trans-port current is in the a− b plane. (a) The field B that points in z-direction. (b) The field B is in thea−b-plane. From Ref. [?].

apparent relatively early that the pnictides are much harder a problem to solve than, e.g., MgB2. In fact,electron-spin or direct electron-electron interactions moved into the main focus of research. Therefore,the people working on the CuO2 compounds were naturally attracted. The hope is that the results inthe pnictides pave the way also towards a better understanding of the cuprates and of high-temperaturesuperconductivity in general.

6.3 Copper-oxygen compounds

When superconductivity was discovered in LaBaCuO [6] in 1986 and the proper phase La2−xBaxCuO4isolated soon thereafter [?] an unprecedented goldrush started. Now almost 3 decades and some 300,000publications later the puzzle is still among the most important issues of solid state research but the knowl-edge of superconductivity, competing phases, strange metals, Mott and charge transfer insulators and thetheoretical description thereof has grown tremendously. In addition, several experimental methods werepushed close to perfection and opened completely new insights. Here, we just give a brief summary ofwhat has been revealed in this still exciting field.4

4The section is partially copied from the article in Zeitschrift fur Kristallographie published by one of us (R.H.) [56].


Copper-oxygen compounds SUPERCONDUCTIVITY 89

6.3.1 History

It is enlightening to look at the activities preceding the final successful synthesis of BaxLa5−xCu5O5(3−y)[6]. As in many other cases there was a program where to search for higher transition temperatures. Thefamous empirical rules of Bernd Matthias, who succeeded to discover more than 1000 new supercon-ducting materials, are probably the best documented example. They include, among other statements,“Stay away from oxides” which had its origin in the unsuccessful attempt to find an oxide with a Tc sub-stantially above 10 K. Therefore, the search for new superconductors was focused on cubic intermetalliccompounds with the hope to possibly reach a Tc close to 40 K, the theoretically expected upper limit forisotropic metals [57]. In fact, such a compound would have been very advantageous for applications withliquid hydrogen sufficing as a cryogen.

Beyond the very successful mainstream research on intermetallic superconductors there were also variousinnovative ideas on the market. Little, picking up a comment of F. London, theoretically studied the pos-sibility of superconductivity in organic materials with the conducting and polarizable structural elementsbeing separated in space [58]. Allender, Bray, and Bardeen continued partially in this direction [59] butconsidered also metal-semiconductor hetero-structures [60]. The basic idea is that electron-phonon cou-pling is expected to be stronger in highly polarizable poorly conducting materials. Then, the electrons inan adjacent thin metallic layer are being coupled to Cooper pairs via the polarizability of the insulator orsemiconductor over a length scale on the order of the superconducting coherence length ξ0 as defined inthe microscopic theory of Bardeen, Cooper, and Schrieffer (BCS) [2]. Alternatively, the role of polaronswas explored for essentially homogeneous materials with low carrier density [61]. On this substrate andwith a sound background in ferroelectricity, Bednorz and Muller (Fig. 6.5) started their search for strongcoupling superconductors in the early 1980ies which lead to the discovery of the cuprates.

Figure 6.5: J. Georg Bednorz (left) and K. Alex Muller. By courtesy of the Nobel Foundation.

6.3.2 Materials

Within a few years almost ten families of cuprates have been discovered. They have rather differentcrystal structures with occasionally huge unit cells accommodating between 8 and over 100 atoms. Themost popular families are compiled in Table 6.2 along with their maximal superconducting transitiontemperatures T max

c .

2013


Table 6.2: Important families of copper oxygen superconductors. R is for Nd, Sm or even La in the caseof thin films. n is the number of adjacent CuO2 planes; three seems to be best for Tc [62]. Hg-1223 hasthe highest Tc so far. With applied pressure, 150 K can be reached [63].

number of CuO2 planes n 1 2 3 4 5

chemical formula nickname T maxc (n) [K] Ref.

La2−xBaxCuO4 LBCO 30 [?, 6]La2−xSrxCuO4 LSCO 39 [64, 65]R2−xCexCuO4 RCCO 29 [66, 67]YBa2Cu3O6+x Y-123 93 [68]

Bi2Sr2Can−1CunO2n+4+δ Bi-22(n-1)(n) 39 98 110 [?]Tl2Ba2Can−1CunO2n+4+δ Tl-22(n-1)(n) 95 118 123 112 105 [69]TlBa2Can−1CunO2n+3+δ Tl-12(n-1)(n) 70 103 125 112 107 [69]HgBa2Can−1CunO2n+2+δ Hg-12(n-1)(n) 96 128 136 123 100 [62, 69]

Structure and chemistry

The copper-oxygen compounds have nearly tetragonal crystal structures with b/a ratios between 1.00and 1.02. The parent compounds La2CuO4 and Nd2CuO4 (see insets of Fig. 6.8) have been known fora long time [70–72] and have K2NiF4 structure [73] with 3 approximately cubic perovskite-like blocksper unit cell stacked along the crystallographic c-axis. The resulting c/a ratio is close to 3. According tothe valence count (and also to band structure calculations) the compounds should be metals with a halffilled conduction band (1 electron per unit cell) but the strong electronic correlations block the transport(for details see section 6.3.3). Therefore, the number of free carriers is zero at half filling (n = p = 0).If Nd is partially replaced by Ce or La by Sr(Ba) the materials are doped away from half filling, 1+n or1− p, respectively, and become conductors with a small number n or p of mobile carriers directly givenby x. The solubility limits for Ce and Sr(Ba) are x' 0.18 and 0.33, respectively. Both compounds haveCuO2 planes at distances c/2 which are offset by (1/2,1/2) in tetragonal lattice units a. In La2CuO4 theCu atom is in the center of an oxygen octahedron, in Nd2CuO4 there is no oxygen in apex position aboveand below the CuO2 plane. In contrast to K2NiF4, La2CuO4 is tetragonal only at temperatures above530 K [74]. Below, the structure is slightly orthorhombic with the octahedra tilted about their basal axiswhich corresponds to the orthorhombic a∗-axis. The transition temperature to the orthorhombic structuredecreases with Sr doping and vanishes at x' 0.20. Nd2CuO4 remains tetragonal for all temperatures. Indoped Nd2−xCexCuO4 (NCCO) and in La2CuO4 there is excess oxygen in the structures which has tobe removed for optimal physical properties. La2−xSrxCuO4 (LSCO) for x > 0.05 has an oxygen deficitafter preparation. It is not clear yet whether fully stoichiometry La2CuO4 and Nd2CuO4 exist at all.Nevertheless, La2−xSrxCuO4 is one of the most intensively studied cuprates, since the entire dopingrange is accessible with a single relatively well ordered compound.

YBa2Cu3O6+x (Y-123) is the only material which exists at the stoichiometric composition. Crystals withthe exact cation ratio 1:2:3 can be grown from the flux. When prepared in BaZrO3 crucibles they are vir-tually free of defects [75–77]. The structures for the limiting doping levels YBa2Cu3O6 and YBa2Cu3O7are shown in Fig. 6.6. YBa2Cu3O6 has a half filled conduction band but is an antiferromagnetic (AF)insulator for the same reasons as La2CuO4 and Nd2CuO4. The copper atoms in the chains have valence1+, and holes on the CuO2 planes are generated only when one chain Cu has two oxygen neighbors. Forcertain intermediate compositions in the range 0 < x < 1 highly ordered phases can be prepared with do-main sizes of several 100 A in all 3 crystallographic directions [78–80]. Within small limits doping canalso be achieved by replacing Y3+ by Ca2+ [81]. Y-123 has two neighboring CuO2 planes at a distance



c

ab

YBa2Cu3O6 YBa2Cu3O7a

Figure 6.6: Structure of YBa2Cu3O6+x (Y-123). (left) YBa2Cu3O6 and (right) fully oxygenatedYBa2Cu3O7. The small (blue) sphere in the center is Y, all other small (red) spheres are Cu. Ba andO are represented as large (yellow) spheres and open circles, respectively. In spite of a half filled con-duction band (1 electron per unit cell) YBa2Cu3O6 is insulating due to electronic correlations and has nomobile carriers (p = 0). Doping in Y-123 is achieved by adding O to the CuO chains along the crystallo-graphic b-axis. In this way electrons are removed from the CuO2 planes. YBa2Cu3O7 has 0.82 electronsor 0.18 mobile holes per CuO2 formula unit (p = 0.18). This is already slightly above the doping leveloptimal for superconductivity (see Fig. 6.8). The CuO2 planes (shaded, see Fig. 6.9) are common to allcuprates with Y-123 having two of them at a distance c/3. The maximal Tc is reached in triple-layercompounds (see Table 6.2). Details about most of the crystal structures of the cuprates can be found inShaked et al. [69].

c/3 ' a which share one octahedron and are in symmetric positions with respect to the central Y atomof the conventional unit cell.

Bi-based cuprates, in particular Bi2Sr2CaCu2O8+δ (Bi-2212) [?], are very popular although they growonly far off stoichiometry and have complex modulated crystal structure according to Ref. [82]. However,since the two adjacent BiO2 layers are bound by van der Waals interaction, they can be cleaved easily.The surfaces obtained in this way are stable, electrically neutral, and typically atomically flat over mmranges. This makes Bi-2212 the workhorse of surface sensitive experimental methods such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). The dopinglevel can be changed via the oxygen content in the Bi−O layer and by replacing Ca2+ by Y3+. Thehighest T max

c at p ' 0.16 is obtained with 8 % Y doping and reaches typically 96 K [83]. By annealingthe crystals at oxygen partial pressures in excess of 1000 bar [84] p = 0.23 can be obtained. In order toreach the underdoped range below p = 0.16 annealing protocols with low oxygen partial pressure andlow temperatures are required in contrast to what is expected from the phase diagram [85].

In addition to these three extremely well studied families, the thallium and mercury-based cupratesare also important since the highest transition temperatures and the most extreme doping levels canbe reached here in materials with T max

c above 90 K. In either case, compounds with up to five neigh-boring CuO2 planes exist [69] all of them becoming superconducting above 90 K. The record breakingHgBa2Ca2Cu3O8+δ has three adjacent CuO2 planes and reaches a Tc close to 150 K at 25 GPa appliedpressure [63]. Three CuO2 planes are apparently optimal for Tc, for four and more layers Tc decreasesagain. Very recently, HgBa2CuO4+δ (Hg-1201) was demonstrated to be another model system in that thedoping can be tuned in the range 0.07 ≤ p ≤ 0.24 while a very high degree of order can be maintainedmanifesting itself in a Meissner effect of almost 100 % [86].

Single layer Tl2Ba2CuO6+δ (Tl-2201) can be driven non-superconducting metallic on the overdopedside by high-pressure oxygen annealing [87]. Since optimal doping is slightly below the accessible

2013


range the maximal transition temperature at p = 0.16 can only be extrapolated to be close to 95 K in thelatest generation of crystals [88]. Hence, Tl-2201 is expected to display the properties of a true high-Tc

compound in contrast to LSCO.

Growth of crystals

All materials were first prepared in crucibles. The main complications are that the liquid phases andfluxes corroded all standard materials and that the desired compositions do not normally correspond tostable points in the high-temperature phase diagrams. Y-123 alone can be grown stoichiometrically fromthe flux. Instead of Y other rare earth metals can be used but only crystals with Y are free of site defects.In all other cases a finite number of rare earth atoms change position with Ba thus reducing the degreeof order. The only stable crucible material is BaZrO3 which had to be developed first [75–77]. All othermaterials get partially dissolved and contaminate the crystals. Crystals grown in BaZrO3 are shown inFig. 6.7.

Figure 6.7: Y-123 single crystals in BaZrO3 crucibles. BaZrO3 is not corroded by the flux, which isaggressive to all conventional crucible materials, and had to be developed first [75]. After decantingthe flux freestanding crystals, many of them with atomically flat surfaces, are left behind on the walls.From [76] with permission.

Single crystals of NCCO, LSCO and of the Bi family were relatively soon grown in optical furnaces usingthe [traveling solvent] floating zone technique ([TS]FZ) with a small volume of appropriate flux betweenthe feed and the seed rod . When the flux is liquefied the material in the feed rod is dissolved and diffusesto the colder seed where it crystallizes. Normally, several tricks are necessary to obtain a single crystalor at least sufficiently large grains. The temperature of the melt, the external pressure of the atmosphereand the oxygen partial pressure are all very critical for a successful growth and have to be determinedsystematically [83,89,90]. In this way large high-quality single crystals can be prepared in several placesof the world having very similar properties. However, in contrast to Y-123 full reproducibility cannot beachieved.

For the high vapor pressures of Hg and Tl at elevated temperatures the TSFZ technique cannot be appliedfor the preparation of single crystal of the respective families. Using sealed crucibles and a secondcontainment the volatility and also the toxicity can be controlled reasonably well at the price of a reducedparameter space for the growth conditions. In spite of these challenges, single crystals of Tl-2201 andHg-1201 have now a reproducibly high quality and reach sizes in the mm range [86–88,91]. The potentialof these materials was already demonstrated and can hardly be overestimated.



Thin films

Along with these bulk methods thin-film techniques were developed and applied successfully. With afew exceptions all applications rely on thin films. There are various methods, (i) chemical vapor depo-sition (CVD), (ii) sputtering, (iii) pulsed laser deposition (PLD), (iv) molecular beam epitaxy (MBE),(v) thermal co-evaporation, and (vi) jet printing. All have been known before the advent of the cuprates,(ii)-(vi) are still used for the cuprates. In a PLD machine a high-energy UV laser evaporates materialfrom a target with the desired composition. The substrate on which the film is supposed to be grown isin the center of the plume of the evaporating atoms. PLD is very efficient and fast and is probably themost widely used technique. In contrast to PLD, the MBE technique allows full access to the ratio ofthe cations. The regulation of the respective ion currents is practically instantaneous facilitating layer-by-layer growth and an extremely high film perfection [92]. Artificial multi-layers with digital interfacescan be produced.

Thermal co-evaporation is a comparably simple and cheap approach which turned out to be very suc-cessful for the production of large-scale films of Y-123. Since the crucibles with the starting materialsare heated resistively the regulation is only important for the long-term control of the cation ratio. In allcases the substrates have to be heated to the appropriate temperature. Usually one uses mono-crystallinematerial with lattice parameters close to those of the deposited films. Within certain limits the atmo-sphere in the preparation chamber can be adjusted. In most of the cases the deposited cuprate films haveoxygen deficits which are fixed by in-situ post-annealing protocols [93]. The quality of MBE films cancome very close to those of single crystals although the structures are not normally fully relaxed. On theother hand, materials can be prepared which do not exist in equilibrium. This holds particularly true forelectron-doped cuprates [66].

6.3.3 Physical properties

As mentioned above the purity of the samples has crucial influence on the properties. In particular, theintegrity of the CuO2 planes is essential to reach the maximal transition temperature of a material class.In turn, the effect of impurities in the CuO2 plane can be studied systematically [94] and demonstratesthat the majority of the properties derives from the planes in accordance with band structure calculations.Here, I try to focus on the properties of virtually clean materials and widely ignore defect-related andpreparation problems.

Phase diagram

The most remarkable property of the cuprates, in my opinion right after the high Tc, is the universalityof the phase diagrams as shown in Fig. 6.8. Here, the phases are only described separately; possibleinterrelations will be postponed to section 6.3.3.

At least on the hole-doped side, the shape of the superconducting dome is almost independent of thematerial class. In the doping range 0.05≤ p≤ 0.27 the transition temperature is well reproduced by [98]

Tc

T maxc

= 1−82.6(p−0.16)2. (6.3.1)

The maximum is always close to 16 % doping but T maxc can vary between 38 K for La1.84Sr0.16CuO4 [64]

and approximately 150 K in HgBa2Ca2Cu3O8+δ at 25 GPa [63]. A variety of materials following theuniversal curve are compiled in Ref. [95]. The relation is valid only in clean materials. In disorderedsamples the dome shrinks in that superconductivity occurs in a narrower doping range and at lower

2013


insulating metallicmetallic

La2-xSrxCuO4Nd2-xCexCuO4

300insulating metallicmetallic

2-x x 42-x x 4

T*(K) Cu

OT*200

atur

e( Ln

AFTO100em

pera

Te

SCSC00.30

00.30

Doping pn

Figure 6.8: Generic phase diagrams of electron (left) and hole-doped (right) cuprates. The insets showthe tetragonal unit cells of the prototypical compounds Nd2CuO4 and La2CuO4, respectively. The shadedrange in the center of the diagram indicates long-ranged antiferromagnetism (AF). The superconductingdome on the hole-doped side reaches from 0.05 to 0.27 in all clean cuprates [95] independent of themaximal transition temperature T max

c close to 0.16. T maxc varies between 30 K in La2−xBaxCuO4 [?] and

150 K in HgBa2Ca2Cu3O8+δ at an applied pressure of 25 GPa [63]. T ∗ and T0 schematically indicate theonset temperatures of the pseudogap range [96] and of charge and spin ordering [97].

temperatures. This effect can be studied systematically by substituting Zn for Cu, for instance, or byirradiation [99–102]. Also single-layer Bi2Sr1.61La0.39CuO6+δ has a narrower dome and is not super-conducting for p < 0.10 [103]. In the following we discuss only the clean limit, where all correlationlengths of ordering phenomena are much smaller than the electronic mean free path `. As has been shownfor Y-123 ` depends more on the sample quality than on the doping and can be in the micrometer rangein YBa2Cu3O6.53 with alternating completely filled and empty chains [104].

Given that zero doping (half filling) can be reached, the antiferromagnetism is similarly universal as su-perconductivity. The maximal Neel temperatures TN range between 280 and 420 K. As-grown La2CuO4has TN ' 280 K and only post-annealing in Ar yields TN = 325 K [74, 105]. In Y-123 TN reaches420 K [97]. There is no direct scaling between TN and Tc.

Long-ranged AF disappears rapidly on the hole-doped side and somewhat slower for electron-doping, inany case much faster than one would expect from the percolation limit [106]. However, spin correlationswithout long-ranged order can be observed well above TN and up to very high doping levels. In LSCO,Wakimoto and coworkers find them to disappear along with superconductivity above psc2 = 0.27 [107].Presently, there is not enough experimental material available to decide whether or not spin fluctuationsand superconductivity generally coexist up to psc2.

In the range up to p ' 0.20 there is another transition or crossover at a doping dependent temperatureT ∗(p) which is usually referred to as the pseudogap line [96, 108]. The nomenclature pseudogap isrelated to the observation of a reduced spectral weight in the ARPES spectra measured below T ∗(p) on



fractions of the Fermi surface [109, 110]. This gap in the single-particle properties leaves imprints onpractically all types of responses. In some material classes, and here individual behavior sets in, othercrossover phenomena can be observed below T ∗(p). Particularly in the La-based compounds, static spinand charge textures develop [97,111–113]. More recently, indications of fluctuating ordering phenomenawere also observed in Bi-2212 and Y-123 [114–120]. Finally, in some compounds a spin glass phase atlow doping can be observed below some 10 K.

For the rich variety of materials available, the hole-doped side is more exhaustively studied than electron-doped cuprates. Nevertheless, it is well established that the AF range is broader and the superconductingdome is narrower for n-doping [66, 90]. A pseudogap was observed by various methods [66, 121–123]and may be a signature of the back-folding of the conduction band at the AF Brillouin zone [124]. Thetemperature range of the results from different methods is not yet consistent. While ARPES indicatesthe opening of the pseudogap above Tc [125], tunneling reveals the pseudogap only below Tc whensuperconductivity is suppressed by a high magnetic field [121]. Systematic transport studies on thinfilms uncover a quantum critical point at n' 0.165 [123] which could be the end point of the T ∗(n) line.Similarly, a crossover from a small to a large Fermi surface close to n = 0.17 may be interpreted as aclosing of the SDW-like gap [126].

Electronic structure

In conventional superconductors, the band width, the Fermi, phonon, and gap energies are well sepa-rated, and superconductivity can be treated as a small perturbation of the normal state. In the cuprates,all energy scales are in close proximity including the magnetic exchange coupling J. The correlationeffects originate in the large Coulomb repulsion U , lead to a substantial incoherent part of the electronicspectral functions, and k is not a good quantum number any further. Consequently, interaction effectsare interrelated and cannot be observed independently, a fact which still creates confusion.

Instead of dealing with these complications it appears more fruitful to search for the origin of thevariation of T max

c in an otherwise rather universal phase diagram. A natural starting point seems theelectronic structure of the CuO2 plane as the basic building unit in the individual environment of a givenmaterial class.

High energies The most transparent access to the electronic structure of the CuO2 plane is throughLa2CuO4 since with the valences of La and O, given as 3+ and 2–, respectively, Cu, residing only in theplane, is in a 2+ state. The 4s orbital is not relevant for the plane and, on an atomic level, we are dealingwith oxygen 2p and copper 3d states. In the tetragonal environment of the cuprates the degeneracy of thenine 3d electrons is lifted and the dx2−y2 orbital happens to be the highest occupied one hybridizing withthe oxygen px,y orbitals as shown in Fig. 6.9 (a). The resulting conduction band is half filled and therefore,on this level of sophistication, La2CuO4 should be a metal. This is also predicted by band structurecalculation in local density approximation (LDA). However, already a Hartree-Fock calculation showsthat the exchange energy is higher than the kinetic energy and blocks the metallic transport. This effectis usually referred to as a Mott metal-insulator transition. The appropriate description is the Hubbardmodel which sets the kinetic and the Coulomb energy in relation. If next-nearest neighbor hopping t ′ isincluded in addition to the nearest neighbor integral t for a more realistic description of the cuprates [seeEq. (6.3.3)] the one-band Hubbard Hamilonian reads [127],

H = ∑〈i, j〉σ

t(c†iσ c jσ )+ t ′(c†

iσ c jσ )+U ∑i

ni↑ni↓, (6.3.2)

where c†iσ and ciσ creates and, respectively, annihilates an electron with spin σ on site i and niσ = c†

iσ ciσ

is the density operator. 〈i, j〉 indicates that the sum is restricted to nearest- and next nearest neighborhopping in the case of t and t ′, respectively.

2013


CuO Mπ(b)(a)

ϕ

XΓky

ϕ

XΓy

kxa π−π

−π

Figure 6.9: Electronic structure at EF = µ(T → 0) of an idealized quadratic CuO2 plane. Panel (a)shows the orbital character of Cu and O (without phases) at low quasiparticle energy ξk = εk− µ ' 0.(b) Brillouin zone and Fermi surface of a single CuO2 plane. The Fermi surface encircles the emptystates around the M points at (±π,±π) for unit lattice spacing a. The diagonal (Γ−M) is usually calledthe nodal direction since the superconducting gap ∆k crosses 0 here (see section 6.3.4). Correspondingly,the neighborhood of X is called antinode. The number of carriers, electrons or holes, correspond to theimbalance of the areas around M and Γ. At half filling (1 electron per CuO2 ) the areas equal. The planeshould be metallic but the correlations make it insulating.

If one considers a situation with one hole per copper site the hole tries to hop from site i via the bridgingoxygen to one of the four nearest-neighbor sites j. The kinetic energy which can be expressed in termsof the transfer integral t is much smaller than the Coulomb repulsion U for double occupancy. In thisway the transport is blocked. In addition, since the Pauli principle is also at work, hopping is onlypossible when the spins on sites i and j are anti-parallel hence antiferromagnetically ordered. For tUthe magnetic coupling between the neighboring spins is given as J = t2/U in second order perturbationtheory. If one constructs a tight-binding Fermi surface with only nearest-neighbor hopping (t ′ = 0) andhalf filling in the idealized quadratic Brillouin zone (BZ) of the CuO2 plane a square results covering halfof the BZ area and being rotated by 45 [see Fig. 6.9 (b)] which coincides with the magnetic BZ. Due tothe extended parallel parts, the configuration is unstable and a gap would open at the Fermi energy suchas in charge or spin density wave (CDW/SDW) systems [5].

It is instructive to have a closer look at the effect of the Hubbard U on the electronic structure as shown inFig. 6.10. On an atomic level, the Cu 3d and the O 2p orbitals at εd and εp, respectively, are split by some2 eV. Crystal field splitting and hybridization broaden the atomic levels considerably, and the Cu 3dx2−y2

and O 2px,y orbitals are mixed covalently. The Fermi energy EF is in the middle of the anti-bonding(AB) band indicating metallicity. The dispersionless non-bonding (NB) and the bonding (B) bands areapproximately 3 eV below EF . The correlation energy U opens a gap at EF , which was first introducedby Mott, and the system becomes an insulator. In the case of the cuprates U is approximately 8 eV hencemuch larger than |εp− εd |, and all three bands are needed for a proper description.

Now the system will be doped by replacing La3+ by Sr2+. In a one-band picture, part of the copper isnominally transformed into Cu3+ for compensation, and hopping becomes possible into empty dx2−y2

orbitals thus opening a channel for transport. This picture is quite useful but according to the precedingparagraph not quite true. In fact the first hole is created on oxygen as demonstrated experimentally veryearly [128,130] since the charge-transfer energy |εp−εd | is smaller than U [?]. More precisely, the extrapositive charge forms a cloud around the central Cu2+ which compensates also the copper spin and istherefore called a Zhang-Rice singlet (ZRS) [129]. With minor modification, it still obeys the dynamicsof the one-band Hubbard model as if the charge would reside on the copper.

The fact that |εp − εd | < U can only be taken into account properly - rather than approximately asin the one-band Hubbard model sketched in the preceding paragraph - if the Cu 3dx2−y2 , O 2px, and



εUHB

ABEF

ε

εd |εp – εd| UNB ZRS

AB

E

EF

εpB

EF

LHBcrystal fieldcovalency

t U

LHBcovalency

t U

Figure 6.10: Schematic band structure of the CuO2 plane at eV energies in electron representation. Thelevels εp and εd correspond, respectively, to the O 2p and Cu 3d atomic levels. The bonding band (B) atapproximately 6 eV below the Fermi level EF (long dashes), the dispersionless non-bonding (NB) andthe anti-bonding (AB) band originate from the covalently overlapping Cu 3dx2−y2 and O 2px,y orbitals.All other orbitals are neglected. Since the hopping integral t is much smaller than the Coulomb energyU the configuration is insulating at half filling, and the AB conduction band is split by U into the upper(UHB) and lower (LHB) Hubbard band separated by the Mott gap U . For small U the first hole goes oncopper. If U exceeds |εp− εd | the material becomes a charge transfer insulator [?] and the first dopedhole is created on the oxygen [128]. The new state is called a Zhang-Rice singlet (ZRS) [129]. (Thefigure is inspired by lecture notes of D. Einzel.)

O 2py orbitals are included in a three-band Hubbard model [131, 132]. It has been shown via numericalsolutions of the two models that both give rather similar results for the phase diagrams at T = 0 includingsuperconductivity [133], since the physics apparently does not change qualitatively if the role of theHubbard U is taken over by |εp− εd |. Beyond the phase diagrams on either side of half filling also theelectron-hole asymmetry [see Fig. 6.8] can be captured. The holes go on the oxygen atoms and quenchthe superexchange coupling between the antiferromagnetically ordered Cu spins while the electronsappear first on the Cu site and just dilute the spins. Consequently, the antiferromagnetism survives muchlonger on the electron-doped side. To which extent the substantial differences between the materialclasses in T max

c and other ordering phenomena can be explained through the material dependent finetuning of |εp− εd | is a matter of present research [133].

Energies in the range kBT

In order to arrive at energies in the range of a few kBT around EF , where the relevant physics such assuperconductivity occurs, the high-energy degrees of freedom have to be integrated out [133, 134]. Thisprocedure is a further idealization and is only qualitative in that individual properties cannot be captured.Nevertheless, a thorough understanding of the coherent part of the electrons’ spectral properties is at theorigin of the explanation of the relevant interactions and of superconductivity.

In this spirit, a downfolded LDA band structure [135] can be derived which reproduces the experimentalresults from ARPES [136,137]. This holds particularly true for the shape of the Fermi surface which canbe obtained to within a few percent from the tight binding band structure (now in momentum space),

ξk =−2t(cos(kx)+ cos(ky))+4t ′ cos(kx)cos(ky)−µ. (6.3.3)

Data from slightly overdoped Bi-2212 can be fitted satisfactorily using t = 250 meV, t ′/t = 0.35 andneglecting band-splitting effects. With µ/t = 1.1 one arrives at a filling close to p = 0.16. With minorchanges in the parameters the CuO2 Fermi surfaces of all other cuprates can be reproduced. The Brillouinzone with the Fermi surface for a single CuO2 plane is shown in Fig. 6.9. The parametrization of the

2013


band structure in terms of hopping integrals is frequently used as a basis for phenomenological modeling[138–141] and microscopic calculations [133, 142, 143].

The experimental dispersion yields a Fermi velocity vF ' 2 · 107 cm/s [144] substantially smaller thanthe LDA prediction indicating strong interaction effects. A hallmark is the kink-like change of theslope vk = h−1

∂εk/∂k at approximately 70 meV observed in the nodal ARPES spectra [145,146] whichturns out to be quite universal [144]. In the presence of conventional electron-phonon interaction akink is expected in the strong coupling limit [14]. In the cuprates, the origin of the interactions is stillcontroversial. The usual electron-phonon coupling [146–150], the coupling to spin excitations [151,152],fluctuations of the charge density [142, 153, 154] and orbital currents [155, 156] have been proposed.

The strong renormalisation is in accordance with the short lifetime τk(ω) of the electrons which de-creases rapidly away from the Fermi surface [145, 157]. For a while it appeared that the imaginary partΣ′′k(ω) = 2[τk(ω)]−1 of the electronic self energy Σ = Σ′+ iΣ′′ is scale free and varies linearly with en-ergy ω and temperature T [145] as expected when the electrons scatter on critical fluctuations of anyorigin (such as spin or charge density or orbital currents). As a consequence, the quasiparticle weightZk = [1−Σ′k(ω)]−1 at the Fermi energy EF vanishes and the electron’s spectral function is distributedover very large energy scales in contrast to the properties of a Landau Fermi liquid with 0 < Zk < 1. Thisphenomenology is called marginal Fermi liquid (mFL) [158] and has a big share in the discussion of thecuprates.

With the continuously improved resolution of ARPES experiments, various substructures were found atlow energies and analyzed in terms of phonons [147, 148] and spin fluctuations [151, 157]. In all cases,Σ′′k(ω) varies faster than linear at low energies and crosses over to the more linear behavior in the 50 meVrange. This phenomenology is expected if the electrons couple to a dispersionless (Einstein) mode whichmust not necessarily be a lattice vibration. These recent observations let it appear very likely that severalinteractions contribute to the coupling spectrum.

At a given energy ω , τk(ω) depends strongly on the position on the Fermi surface. Along the diagonalΓ−M line of the BZ (nodal direction) Σ′′k(ω) is relatively small and does not depend substantially ondoping [136, 137]. In the vicinity of the X point (antinode) things are more complicated. At veryhigh doping as in Tl-2201, the lifetime of antinodal quasiparticles may even be longer than that ofnodal ones [159]. With decreasing doping the lifetime and the weight of the antinodal quasiparticlesdecreases continuously. Slightly below optimal doping the interactions along the principle directionsbecome strong enough to completely suppress coherence even in the superconducting state [160, 161].At elevated temperatures no quasiparticle develops any more for p < 0.18 indicating a loss of coherencein the pseudogap regime on parts of the Fermi surface [137].

Bridging the gap At first glance one would expect that the renormalized conduction band with a widthin the range of an eV and the UHB and LHB split by U ' 8 eV do not overlap. However, the strongcorrelations leave only a small fraction Zk 1 of the quasiparticles’ spectral weight close to EF anddistribute 1−Zk over large energy scales. This incoherent part can be observed experimentally betweenthe LHB and the conduction band as a faint structure with almost vertical dispersion [162–164] and isreproduced theoretically using Monte-Carlo techniques for solving the Hubbard model [165,166]. Thus,there is one more indication that the Hubbard model is a reasonable starting point for the description ofthe CuO2 planes in the cuprates.

Two-particle dynamics

The CuO2 planes determine the majority of the physical properties and, in particular, carry the currents inthe normal and in the superconducting states. Owing to the layered structure, the cuprates are electrically



highly anisotropic [167]. The anisotropies of the resistivities ρc/ρab, with the subscripts indicating thecrystallographic directions, range from 30 to 105 [168, 169].

Transport At optimal doping all clean cuprates have an essentially linear resistivity in the a− b planedown to Tc, ρab(T ) ∝ T , which saturates only at very low temperature, as demonstrated for LSCO inmagnetic fields [170]. Below a doping dependent temperature T ∗ρ (p) ' T ∗(p) there is a reduction ofρab(T ) below the linear variation which is associated with a pseudogap [108, 168, 169, 171, 172]. If thehigh-temperature part is extrapolated to T = 0 the residual resistivity is very small and approaches 0 forthe cleanest optimally doped crystals. If superconductivity is suppressed by high magnetic fields ρab(T )saturates at a finite value [170]. For p > 0.16, also the out-of-plane resistivity becomes purely metallicand ρc(T ) ∝ ρab(T ). The ratio ρc/ρab(T ' 1.5Tc) at optimal doping is approximately 30 in Y-123 [168]and 5000 in Bi-2212 [169] having the same T max

c and close to 500 in LSCO with T maxc = 38 K [170].

There is a dichotomy between the under- and the over-doped ranges on the hole-doped side which be-comes particularly clear at low temperatures when superconductivity is suppressed by magnetic fields.Various systematic studies have been carried out recently by Ando and coworkers [64]. The results forLSCO and Y-123 are shown in Fig. 6.11. In both Y-123 and LSCO the resistivity generally turns in-

(d)(c)

(a)(b)

Figure 6.11: Resistivity vs T for different doping levels p for LSCO (a,b) and Y-123 (c,d). The compari-son indicates the similarity of the different classes as long as the crystals are sufficiently clean. Note thedeviation from linearity below a doping dependent temperature T ∗(p) which is particularly clearly seenin Y-123 at 0.11 . p . 0.15 corresponding to 6.60 . y . 6.85. From Ref. [64] with permission.

sulating, corresponding to dρab(T )/dT < 0, at low temperatures and doping before superconductivityappears at psc1 ' 0.05. For p > psc1 dρab(T )/dT becomes essentially positive for T > Tc. If a magneticfield is applied which is high enough to suppress superconductivity completely an upturn is observedwith a logarithmic divergence towards zero temperature. With increasing p the minimum shifts to lowertemperature and approaches T = 0 close to p = 0.17 [170]. For p > 0.17 the resistivity remains metallicand exhibits a T α variation over extended temperature ranges with α ' 1.5. Apparently, full metal-licity develops above optimal doping. In strongly over-doped Tl-2201 with Tc ' 15 K, 1.5 < α < 2 isfound [173]. If the resistivity is fitted to ρ(T ) = ρ0 +AT +BT 2 the coefficient A of the linear termapproaches zero as |psc2− p| for p≤ psc2 [174, 175].

The details become more transparent when the dynamics is studied as a function of the electron mo-mentum k as discussed already briefly in the context of the electronic structure and single-particle life-times in paragraph 6.3.3. Concerning transport, two-particle properties have to be considered where anelectron is scattered from an occupied into an empty state leading to the usual restrictions and cor-rections. In an early nuclear magnetic resonance (NMR) experiment, deviations from the Korringalaw, (T1T )−1 = const, with T1 the spin lattice relaxation time and T the temperature was found wellabove Tc for p ≤ 0.17 [176], indicating the loss of a relaxation channel for the electrons. The NMR

2013


BB2000 T = 200K 2gB1gB

ΓΓ

opt.doping

(K)

Bi-2212T = 200 K

1000 X MX MΓ (

μ0 X

Γ-M(a)0

2

K/K)

(a)

Γ-M

2

0

Γ ∂/

T (K

μ0 X

0.05 0.10 0.15 0.20 0.25-4

-2∂Γ (b)

doping p

Figure 6.12: In-plane anisotropy of the electronic relaxation Γµ = [τµ ]−1 at 200 K as seen by electronic

Raman scattering [183]. (11.6K = 8cm−1 = 1meV) Γµ is closely related to a resistivity ρ , and τ−1 =ne2

m ρ in a Drude model. As an additional information from Raman scattering, the regions around the Xpoints (diamonds) and along the Γ−M line (squares) of the Brillouin zone (see Fig. 6.9) can be projectedindependently with different light polarizations µ [181]. (a) There is little doping dependence along thenodal directions (Γ−M). At X there is an abrupt change at p = 0.21±0.01. The crossover seems to beuniversal since it is also seen in LSCO and in Tl-2201 [184] and by NMR [185]. It is predicted by theHubbard-Holstein model [143]. (b) The temperature dependence of Γµ , ∂Γµ/∂T |200K is isotropic abovep = 0.21±0.01. Below the crossover, the antinodal derivative decreases continuously and changes signclose to p = 0.16.

form factors suggest that particles close to (π,0) may experience a gap which was directly observed byARPES [109, 110]. Similarly, optical transport (IR) results in Y-123 show that the electrons with mo-menta along the diagonal relax differently from those at the X points of the BZ. The distinction is pos-sible for the specific crystal and band structure of Y-123 which facilitates to project diagonal and (π,0)momenta for in-plane and out-of-plane polarizations, respectively [135]. For this reason, the pseudo-gap as an anti-nodal property was discovered first by c-axis polarized IR spectroscopy [177]. However,the projection in optical spectroscopy with in-plane polarizations is incomplete with finite sensitivityeverywhere in the Brillouin zone, and the pseudogap is clearly visible below optimal doping also forE‖a,b [178, 179].

The electronic Raman response measures a quantity similar to the conductivity [180, 181] but has in-plane selection rules which facilitate independent access to nodal and antinodal electrons by appropri-ately selecting the light polarizations [181, 182]. Results for Raman relaxation rates Γµ (µ is for thepolarizations corresponding to symmetry projections [181]) of differently doped Bi-2212 at 200 K areshown in Fig. 6.12. In the nodal configuration the doping dependence of the spectra and of the corre-sponding carrier relaxation is weak for psc1 < p < psc2 [183, 186, 187]. Above p' 0.21 no polarizationdependence corresponding to a relaxation anisotropy can be observed. The relaxation rate for antinodalelectrons increases abruptly below p' 0.21, and approximately 30 % of the spectral weight is lost in theenergy range up to 250 meV. This was traced back to a doping and momentum dependent correlation gap



extrapolating to 2∆C ' 200 meV at p = 0 [183]. A similar phenomenology emerges with a progressiveloss of quasiparticle coherence starting at X and proceeding to the node upon reducing p as observedby ARPES [109, 110, 137, 188] and studied also in the context of light scattering [186, 189, 190]. Thecorrelation gap, the pseudogap and, finally, the superconducting gap (see section 6.3.4), have differentenergy scales and their interrelation has to be determined yet.

The onset of anomalies in the doping range around p = 0.16 are also seen in the Hall effect [191] and,particularly clearly in the Nernst signal [175, 192–194] constituting, respectively, a transverse voltagein response to a charge and heat current in a perpendicular magnetic field. The Hall effect exhibits amaximum close to T ∗ [175, 194] but only the quantum oscillations of the Hall resistivity at very lowtemperature indicate that the anomaly may be related to a reconstruction of the Fermi surface [117].Very recently, indications of a Fermi surface reconstruction were also discovered on the electron-dopedside in NCCO [126].

The Nernst effect is sensitive to superconducting vortices [195] and density-wave order [196]. For a longtime the Nernst signal observed between T ∗ and Tc in LSCO was considered a signature of vortex motionabove the coherence temperature Tc in the spirit of a 2D Kosterlitz-Thouless transition [192]. Only recentresults in Eu-doped LSCO showed that the onset of the Nernst voltage coincides with the charge-orderingtemperature [194,197] found in various other experiments [?, 113]. In Y-123 the onset of the anisotropicNernst signal [175, 192, 193] coincides with various other indications of broken rotational symmetrysuch as Kerr rotation [198] or incommensurable peaks in the dynamic spin susceptibility [118, 119].New frequencies in the quantum oscillations indicate that a partial reconstruction of the Fermi surfacegoes along with the ordering phenomena. Hence, the superstructures found first in the the spin channelin LSCO [199, 200] and Nd-doped LSCO [111] seem to be a generic phenomenon of all cuprates that isaccompanied by charge order. Since the superstructures are static only in exceptional cases signaturesof them in the transport escaped observation for a long time, in particular in the compounds with highTc. Their importance is being unveiled only slowly. Further details will be discussed at the end of thefollowing subsection and in sections 6.3.3 and 6.3.4.

Spin dynamics Homogeneous magnetism exists in wide doping ranges. At 0 ≤ p ≤ 0.03 the antiferro-magnetism is long-ranged in LSCO. In NCCO three-dimensional (3D) antiferromagnetism exists belown = 0.13. The exchange coupling J ' 130 meV is among the largest ones existing. The order is truly 3Dbut the coupling along the c-axis is orders of magnitude smaller than along a.

The dynamics at high energy was studied early by Raman scattering. The photon flips essentially twoneighboring spins breaking six bonds with energy J [201]. More accurately, a two-magnon density ofstates is measured and projects the flat parts of the dispersion in the vicinity (π,0). The maximal energyobserved is therefore at E2M ' 6Js. In a spin 1/2 system the peak is close to 3J (2.7J for quantumcorrections) [?, 201, 202]. In LSCO, Y-123 and Bi-2212 spin correlations can be observed by Ramanscattering up to approximately p = 0.20 [?, 187, 203]. In LSCO and NCCO magnetic short-range orderwas observed by inelastic neutron scattering up to p ' 0.27 [107] and n ' 0.17 [204], respectively.For the large magnitude of J the full dispersion of the spin excitations was studied with neutrons onlyrecently. The spectrum extends beyond J well above the energy of thermal neutrons. Results for variouscompounds up to approximately 200 meV are shown in Fig. 6.13 [97]. If the energy axis is normalized tothe exchange coupling J the dispersions collapse on top of each other lending evidence to the universalityof the spin excitations.

At intermediate and low energies spin excitations were studied in detail by neutron scattering and NMR.The decrease of the Knight shift below Tc indicated spin singlet pairing [207] (see section 6.3.4). ForT > Tc, the spin-lattice relaxation rate T−1

1 is proportional to T compatible with Fermi liquid-like carriers[176,207] only close to optimal doping and above. At low doping a spin gap is found below Tc by neutronscattering [208] and magnetic resonance [176,209] putting magnetism and superconductivity in relation.It was conjectured early that most of the spin susceptibility results from itinerant electrons rather than

2013


-0.2 -0.1 0 0.1 0.20

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

(0.5+h, 0.5) (rlu)

E /

J

La1.90Sr0.10CuO4La1.875Ba0.125CuO4La1.84Sr0.16CuO4YBa2Cu3O6.5YBa2Cu3O6.6

Figure 6.13: Dispersion of the spin excitations of La-based cuprates and in Y-123. If the energies are nor-malized to the exchange coupling J the spectra are universal. Low and high-energy parts are compatiblewith fluctuating charge and spin order [205, 206]. From [97] with permission.

localized Cu moments. This inspired the model of a nearly antiferromagnetic Fermi liquid [210]. Theinteraction of carriers and spin fluctuations can be studied systematically in the fluctuation exchange(FLEX) scheme [?, ?]. However, in which way spin fluctuations participate in the Cooper pairing is stillan open question (see section 6.3.4).

With polarized neutrons an intriguing narrow mode with wavevector Q = (π,π) was found at low tem-peratures [208,211–215] which is usually referred to as the π-resonance. For p≥ 0.16 the energy of theresonance ER is proportional to Tc. In the underdoped range the mode appears already between T ∗ andTc when the pseudogap opens up. The spectral weight of the mode is between 1 and 6 % of the integratedspectral weight of the spin susceptibility [216] and its origin is controversial. Its role as a mediator ofsuperconductivity has been explored in various studies [152, 217–219]. The results, however, did notgenerate general agreement yet. In spite of that the π-mode is characteristic of the cuprates and possiblyother superconductors in close proximity to a magnetic phase [?].

In the La-based compounds incommensurate peaks shifted by δ (0,π) and δ (π,0) (in the square unit cellof the CuO2 planes) away from the AF reflex at (π/2,π/2) were discovered early in the inelastic channelindicating a dynamic superstructure on top of the antiferromagnetic order for p = 0.075 and 0.14 [199].If part of the La is replaced by Nd the superstructure becomes static [111]. Charge order accompaniedby a lattice distortion with a periodicity of four unit cells appear before the eight unit cell superstructureof the spins is established. Below the onset point of superconductivity, p ≤ psc1 ' 0.05, static diagonalstripe order is observed in LSCO. At psc1 the stripes rotate by 45 and start to fluctuate meaning theycan only be observed at finite energy [199, 220]. Generally, charge order precedes spin order uponcooling [111, 113, 221]. Fluctuating modulations of the charge density cannot normally be observedby neutron scattering but can be visualized by tunneling spectroscopy due to interference effects [114].Fig. 6.14 shows that charge and spin order in Bi-2212 and LSCO have the same orientation above psc1.Recently, equally oriented nematic order was also observed in underdoped superconducting Y-123 in thedynamic spin susceptibility [118, 119]. Assuming a stripe-like superstructure of the spins the dispersion(Fig. 6.13) can be predicted quantitatively [205, 206, 222]. Hence, evidence mounts that dynamic phase



Bi2Sr2CaCu2O8+La2-xSrxCuO

(c)

Figure 6.14: Spin and charge ordering in LSCO (a,b) and Bi-2212 (c), respectively. Parts (a,b) displayincommensurate neutron reflexes at (h,k) = (0.5± ε,0.5± ε) below psc1 ' 0.05 and at (0.5± δ ,0.5)and (0.5,0.5±δ ) above psc1. They originate from a static and dynamic spin superstructure, respectively,corresponding to static and fluctuating stripes [111,199,220]. Above psc1 the spin and charge superstruc-tures are observed to have the same orientation. Superstructures with similar orientations including therotation [see panel (b)] are also seen in Y-123 [116, 118, 119, 175]. From [220] with permission. The re-sult on Bi-2212 (c) is a filtered STS image and shows a modulation of the charge density which becomesvisible due to interferences. The arrows indicate the orientation of the CuO2 planes. From [114] withpermission.

separation and the related ordering phenomena are generic properties of the cuprates contributing toanomalies such as critical fluctuations, Fermi surface reconstruction and the pseudogap in the electronicexcitation spectrum.

Competing phases

The pseudogap range is one of the most intensively studied areas of the phase diagram being observedbelow the T ∗ line (see Fig. 6.8) and for doping levels below approximately p = 0.21 [96, 108, 172,223]. Various properties discussed above indicate a gap in the quasiparticle excitation spectrum and,hence, an instability above the transition to superconductivity. It is clear from ARPES [109, 110] thatthe Fermi surface is not fully gapped above Tc. Concomitantly, the materials remain metallic, and theresistivity even decreases slightly below T ∗ since the strongly interacting quasiparticles are gapped out(see Fig. 6.11).

To some extent the resistivity of the cuprates (Fig. 6.11) indicates similarities to CDW or SDW systemssuch as the recently discovered FeAs superconductors [7,8,31], where the resistivity also drops upon en-tering the ordered phase in the undoped parent compounds [37]. Similarly, in several f -electron systemsa magnetically ordered phase is suppressed as a function of either doping or pressure giving room forsuperconductivity [27]. In either case, superconductivity is in close proximity to other ordered phases, incomplete contrast to conventional materials [?]. Clearly, there is another instability above the supercon-ducting phase, and it is of pivotal importance to understand the relationship of the phases as to whetherthey compete or cooperate [96].

In most of the cases, in particular in the compounds with high transition temperatures, T ∗ is a crossoverrather than a phase transition. There are various indications of a broken symmetry [114, 142, 223–228]

2013


with the fluctuations of incipient order widely considered important and responsible for many of theanomalous properties of the cuprates including superconductivity. Very early Varma and coworkers in-troduced the concept of a quantum critical point above which temperature is the only energy scale ratherthan collective excitations such as spin waves or phonons. The related fluctuations lead to an almost com-plete collapse of Landau’s Fermi liquid model and to the marginal Fermi liquid phenomenology [158],where an electron has vanishingly small coherent weight even at EF . The quantum critical point (QCP), atwhich the fluctuations suppress any phase transition above absolute zero, is somewhere buried below thesuperconducting dome in the range 0.15 < p < 0.22 on the hole-doped side [175] and close to n = 0.165for electron-doped systems [123]. Order or incipient order (for reviews see Ref. [28] or the book bySachdev [229]) are expected below T ∗(p) and between n = p = 0 and the critical doping. Similarly asin many other systems the QCP cannot be accessed directly since it is “protected” by superconductiv-ity. The appearance of superconductivity above a QCP is one of the reasons why the fluctuations areconsidered a possibility to mediate Cooper pairing.

It is controversial which types of fluctuations dominate. Inspired by the NMR results Anderson proposedthe resonating valence bond (RVB) model where fluctuating spin singlets are formed at high temperatureand condense below Tc [230]. A similar phenomenology follows if polarons condense into bi-polarons[61]. Ong and coworkers interpreted the onset of the Nernst signal between Tc and up to maximally 3 Tc'T ∗ at p ' 0.10 in terms of superconducting fluctuations which survive even below psc1 [192]. Recentstudies in La2−x−yEuySrxCuO4 (LEuSCO) show that the Nernst signal sets in along with the formationof a CDW-like superstructure and may originate from the related charge ordering [194, 197] rather thanfrom vortex motion [196]. While the superstructure is static in LEuSCO, LNdSCO [111], LBCO andLa2−x(Ba1−ySry)xCuO4 [221, 231] below T0 (Fig. 6.8) fluctuating order is observed in LSCO [220] andalso in Y-123, at least at specific doping levels [116, 117, 119, 191]. Fermi surface reconstruction hasalso been observed recently in NCCO. It can be described in terms of band folding resulting from theAF order [89, 126]. Yet, the details and the origin behind the reconstruction remain important problemsto solve on either side of zero doping.

As already mentioned, to some extent there is a similarity to CDW and SDW materials with dimensiond greater than one, where only part of the Fermi surface is gapped while the rest sustains metallicity[232, 233] or even superconductivity such as in 2H-NbSe2 [234]. Beyond these similarities the type oforder in the cuprates has many new features. In particular in the high-Tc compounds, an ordered phase isnot established, and it is probably sensible to speak of nematic order [114, 228], including spontaneousdeformations of the Fermi surface [235], with only the rotational symmetry broken.

The lattice appears to play a crucial role in stabilizing the order. It has been shown for La-based com-pounds that the tilting angle θt of the copper-oxygen octahedra may be a parameter to quantify theproximity to static order [112]. If θt exceeds a critical value static order is established by kind of alock-in transition and superconductivity disappears. Hence, in realistic models electron-phonon interac-tion should be included in the Hubbard model [142] to bring the derived phase diagrams closer to theexperiments [143, 185].

There is no evidence whether and which fluctuations contribute to superconductivity (see also sec-tion 6.3.4). However, it was shown for LBCO at 1/8 doping and for LEuSCO that static order quenchesthe 3D phase transition [112, 236]. In the case of LBCO the CuO2 planes decouple and the phase transi-tion to 2D superconductivity is suppressed by fluctuations [236]. Upon applied pressure the static orderbecomes nematic and superconductivity is restored [237].

There are two conclusions. (i) The pseudogap phase has many signatures of a broken symmetry otherthan the gauge symmetry of superconductivity. There are many experimental indications that the rota-tional symmetry is broken and that the electronic states partially reconstruct due to incipient charge orderdriven by the strong correlations. The contribution of superconducting fluctuations to the pseudogap is



small as can be seen independently from the spectral weight redistribution in the optical conductiv-ity [238]. (ii) As soon as the order becomes static superconductivity is quenched. In this sense thereis a competition between the two phases, and the possible coexistence is clearly different from that inconventional CDW and SDW systems [234, 239]. To which extent the critical fluctuations of incipientorder contribute to or drive superconductivity needs to be clarified [96].

6.3.4 Superconductivity

Superconductivity in the cuprates mesmerized nearly all solid state scientists for its unprecedented ro-bustness. The essential parameters of Y-123 are summarized in Table 6.3 and compared with those of Aland Nb3Sn. Applications on a large scale were expected to be realized within a few years. The transitiontemperatures were high enough to make cooling with liquid nitrogen an option. Upper critical fields inthe 100 T range and critical current densities of jc ' 107 A cm2 close to those of the best metallic alloysand three to four orders of magnitude above the maximal capacity of Cu triggered expectations of com-pletely replacing power transmission lines, storing energy or constructing magnets with fields in excessof 30 or even 40 T virtually free of energy consumption. However, a brief look at Table 6.3 shows werethe problems are buried. Nevertheless, substantial progress could be made since 1986 and several of theideas have become commercial products (see chapter 7).

Table 6.3: Superconducting parameters of Al, Nb3Sn, and Y-123. The data for Al and Nb3Sn are takenfrom Ref. [10]. The references for Y-123 are indicated in the last column. Some entries are estimatedusing the relations ξBCS = hvF(π∆0)

−1 and Bc2 = Φ0(√

2πξGL)−2 with vF the Fermi velocity and Φ0 =

2.07 ·10−15 Wb the flux quantum. The critical current for Al is determined via the Silsbee criterion [10]for a wire with 1 mm diameter. All derived quantities should be considered order of magnitude estimates.Note that ξBCS and ξGL are related but different quantities. λ has no index since it can be measured withsome accuracy (see, e.g., Ref. [104]) Also in the case of λ , the London and the Ginzburg-Landau (GL)definitions should be distinguished.

quantity unit Al Nb3Sn Y-123 comment Ref.

T maxc K 1.19 18 93 [81]∆0 meV 0.18 4.3 35 [240]∆0 kBTc 1.76 2.7 4.3

ξ abBCS A 2 ·104 100 20Bc

c1 T 0.01 0.1 0.05 B‖cBc

c2 T 24 130 B‖c [241]Bab

c2 T 240 B‖ab [10]ξ ab

GL A 15ξ c

GL A 40 1.5λ ab A 500 800 900±100 B⊥ ab [104]κGL 0.5 20 80

jc(5K,10T) A/cm2 1.6 ·103 106 5 ·107 [242]

2013


Experiments

One can spot the short coherence length ξ0 (used if specialization to ξBCS or ξGL is not necessary) tobe among the major problems to deal with, since it prevents effectively the pinning of flux-lines. Theresulting flux flow goes along with energy dissipation and kills all applications in high fields and withlarge currents. Typically, one pinning center per coherence volume ξ 2

0,abξ0,c is needed. Hence, in con-ventional alloys flux flow can be suppressed by a moderate density of defects (see Table 6.3). In addition,(non-magnetic) impurities have little impact on superconductivity in an s-wave superconductor [243]. Incontrast, Tc is rapidly reduced in the cuprates, and a high density of pinning centers is required. Fortu-nately, the structure helps, but one had to learn to keep the pinning centers away from the CuO2 planes.For instance, oxygen clusters on the CuO chain sites of Y-123 pin effectively [?] and have only mildinfluence on Tc [78]. For practical purposes the proper distribution of pinning centers is among the majorchallenges, and the dynamics of flux lines remains an important field of research [?,?]. What is the originof the short coherence length and of the sensitivity to disorder?

One could phrase it this way: you have the choice between Skylla and Charybdis. The high transitiontemperature goes along with a large energy gap ∆ which results in a short coherence length, ξBCS =hvF(π∆)−1. The high transition temperatures in turn, come from an exotic coupling mechanism whichmakes superconductivity in the cuprates unconventional. Following the definition proposed by Pitaevski[17] and Brueckner et al. [16] unconventional means that

∑k

∆k

2√

ξ 2k + |∆k|2

= 0. (6.3.4)

Hence the gap ∆k is strongly anisotropic, changes sign and has nodes on the Fermi surface making Tc

highly susceptible to defects. The sign change is topologically different from a strongly anisotropic butgenerally positive gap with vanishingly small minima and goes along with a discontinuous transitionfrom a four-fold to a two-fold rotational symmetry which implies a change in the phase of the gapsimilar to the structure of atomic orbitals with l ≥ 1. Since spin singlet pairing was identified early byNMR [207], odd internal angular momentum of the Cooper pairs going along with spin triplet states canbe excluded. Hence, l = 2 is the lowest possible angular momentum and dx2−y2 is realized.

Experimentally, the sign change of the gap can be demonstrated only in a phase-sensitive experiment[18,19] and not by spectroscopy probing the magnitude of the gap |∆k|. The dx2−y2 character was pinneddown by Wollman and coworkers [18] and consecutively corroborated in various ways for both electronand hole-doped cuprates [19].

On the Fermi surface k = kF the d-wave gap is simply given by ∆ϕ = ∆0 cos(2ϕ) with ∆0 the gapmaximum and ϕ the azimuthal angle which is zero on the M-X line [see Fig. 6.9] with the origin in M.On the tight-binding band structure, as given in Eq. (6.3.3), the gap is parameterized as

∆k =∆0

2[cos(kxa)− cos(kya)] (6.3.5)

for a quadratic unit cell with lattice parameter a.

It is an enchanting coincidence that the paper on the unconventional gap in UBe13 [13] directly precedesthe article on superconductivity in La-Ba-Cu-O [6]. There, the magnetic penetration depth λ (T ) wasused as a diagnostic tool which also brought the break-through for the cuprates [20]. Although therewere very early indications that the gap is strongly anisotropic and may even have nodes [244–247], onlythe experiment of Hardy and coworkers on high-quality Y-123 single crystals [20] triggered an avalancheof activities including the first phase sensitive experiment by Wollman et al. [18].



To map out the magnitude of the gap in the cuprates spectroscopically, resolution in k-space is needed.For this reason ARPES became particularly important since one can map electronic single-particle ener-gies with a resolution ∆E . 2 meV as a function of the in-plane momentum k‖ with a resolution of theFermi surface angle ϕ of better than one degree. The cuprates and ARPES profited mutually from eachother, in kind of a symbiosis, since Bi-2212, due to its extremely two-dimensional structure and the ex-ceptional cleaving plane between the Bi-O layers, facilitated deep insights into the physics of the cupratesby photoemission and thus enormously fueled the method itself [137]. The most important results arethe observation of the Fermi surface and of the momentum dependence of |∆k| [248, 249] as shown inFig. 6.15, of the pseudogap [109, 110] having the same momentum dependence as |∆k|, the doping de-pendence of the dispersion [136, 188], and various renormalization effects on the band structure whichare believed to be in close but not yet understood relationship with the Cooper pairing [146, 151, 250].

Fermi-surface angle (deg)

Figure 6.15: The magnitude of the gap |∆k| as a function of the Fermi surface angle as defined in Fig. 6.9.Note that the labels on the Brillouin zone in the inset correspond to the reciprocal lattice of Bi-2212. Theinset shows experimental points for the Fermi surface (circles) and the tight-binding fit (full line). Thehairlines indicate the replica originating from the superstructure of the Bi-O layers. From [249] withpermission.

Electronic Raman scattering [251] is among the few other possibilities to see the gap anisotropy directlysince different parts of the Fermi surface are projected independently by appropriately adjusting thepolarizations of the incoming and outgoing photons [181, 182]. Since light scattering is a two-particlemethod both the gap in the excitation spectrum and the condensate are seen. At optimal doping the resultsagree with those from ARPES. Additional information is obtained predominantly at more extreme dopinglevels closer to the onset points of superconductivity. It turns out that the gap close to the nodal directionscales with Tc in very wide doping ranges [?,190,240,252–256] in qualitative agreement with low-energytunneling [257] and recent ARPES results [258]. In the latter experiment the particle-hole mixing typicalfor (k,-k) pairing can be seen below but close to Tc at E > EF . This identifies the observed gap as thesuperconducting one.

On the electron-doped side ARPES [125] and Raman scattering [259, 260] reveal gap magnitudes∆0/kBTc in the range 2 to 2.5, much smaller than for hole-doping. Phase sensitive experiments showthat the gap changes sign similarly as on the hole-doped side [19, 261]. The gap appears to vary non-monotonically [125,260]. However, the interference with the pseudogap may influence the magnitude of

2013


the superconducting gap and needs to be clarified further. The relatively small gap ratios signal interme-diate to weak coupling, and it is not overly surprising that ∆0 approximately follows Tc in the relativelysmall doping range where superconductivity exists [262].

For p-doping, the approximate scaling of the gap extracted from extended portions of the Fermi surfacearound the nodes as ∆0 ' 4.5kBTc in the under- and over-doped ranges [181, 190, 254, 257] is rathersurprising as one would expect the gap to reflect the supposedly doping dependent coupling strength. Infact, most of the electron-electron interactions such as those from spin fluctuations or Coulomb repulsionincrease towards p = 0. However, all these coupling potentials are not isotropic but dominate along theprinciple axes so that the nodal part could be less influenced. It has been argued that the increase ofthe gap may be compensated by the loss of major parts of the Fermi surface due to the interactionitself [137, 189, 190, 263–265]. The reduction of the superfluid density towards low doping may be a afingerprint of this phenomenon [172, 266, 267] but there is no quantitative understanding yet.

One would expect that this problem could be clarified by looking at the environment of the X points (anti-node) where the strong interactions prevail. However, the loss of coherent quasi-particles and the openingof the pseudogap for p≤ 0.19 progressively shroud the pairing dynamics. This is further complicated bythe emergence of inhomogeneities which can be observed by scanning tunneling spectroscopy (STS) asshown in Fig. 6.16. In the spectra of Bi-2212 large and small gaps are spatially separated, and the largegaps line up with oxygen defects where the doping level is expected to be reduced [268]. The pseudogapand the superconducting gap may even mix in some doping ranges [223, 269, 270].

Sample Bias (mV)

dI/d

V (a

rb)

05001-

(a) 1

234

5 6

-50 0

(b)

Figure 6.16: Inhomogeneity of the tunneling spectra of slightly underdoped (p = 0.15± 0.01) Bi-2212as seen by STS. The spectra in panel (a) are measured at the spots in panel (b) having the same color.Note that the slope close to zero bias depends only little on the position. The asymmetry for positive andnegative bias results from the charge order. From [268] with permission.

Affairs do not simplify on the overdoped side. While the condensation energy has a maximum close top = 0.19 and the superfluid density tends to saturate [172, 267] the nodal and anti-nodal gaps continueto develop independently. In addition, the interpretation of the coherence peaks remains controversial.There are particularly enlightening experiments. (i) Electronic Raman scattering with applied pressuredemonstrates that the superconductivity-induced anti-nodal structures decouple from Tc already at op-timal doping [271]. (ii) In STS the energy of the coherence peaks in the range of ∆0 is lower thanone would expect from the slope close to zero bias [257, 268, 270] and depend on the location on thesample [268].

The π-mode at energy ER (see section 6.3.3) follows Tc(p) and ER(p)' 1.3∆0(p) on the overdoped side[?,211,213–215,272]. On the underdoped side, data on Y-123 [213,272] show that the scaling with Tc isnot valid any more. As to whether or not the proportionality to ∆0 takes over depends on the definition ofthe gap which, in my opinion, remains problematic, in particular in the presence of the pseudogap. Hence,the π-resonance proves to be in close relationship to unconventional superconductivity [?, 216, 219],while the more stringent question as to its relationship with the magnitude of the gap and with the originof the Cooper pairing is not settled.



In summary, not all microscopic properties in the superconducting state are clarified. While the symmetryof the energy gap is found to be universally dx2−y2 [19] the magnitude, which could be instrumental toidentify the origin of Cooper pairing, is hard to pin down. Close to the node scaling with Tc seems likelybut around X discrepancies as large as ±50 % between different probes are typical. The condensationenergy seems to peak at p = 0.19 [172] making superconductivity most robust slightly above optimaldoping.

Origin of superconductivity

The basic notions of the superconducting state are a d-wave gap and a universal dome-shaped dependenceof the transition temperature Tc on doping p. Whenever p = 0 is accessible the cuprates are AF insulatorshighlighting the importance of strong electronic correlations. In the range 0 < p < 0.20 a competitionof ordering phenomena is observed. These facts should be captured at least qualitatively by a theoreticalapproach towards superconductivity.

In conventional superconductors the condensation of electrons into Cooper pairs is an instability of thenormal metallic state. Cooper showed that an infinitesimally weak attractive potential, −V0, which isnon-zero only for energies ξk ≤ hω0 with hω0 the energy of the coupling boson (phonon in conventionalmetals), makes two electrons with opposite momenta and spins living above a filled Fermis sphere,ξk = εk−EF > 0, to pair and to reduce their energy by ∆ [273]. Bardeen, Cooper and Schrieffer (BCS)derived how N = O(1023) electrons can exploit this energy gain by forming a condensate which ischaracterized by a single wave function similar to that of an electron in an isolated atom or an infiniteplane wave [2]. The energy gain and the transition temperature Tc depend linearly on the cutoff hω0and exponentially on the coupling strength λ = NFV0 with NF the density of electronic states at EF .In the BCS approximation, λ 1, there is no direct relationship to real materials. This open problemwas solved by Eliashberg who showed how λ can be derived from the phonon spectrum and reachvalues in excess of 1 [3, 274]. Coupling spectra α2(ω)F(ω) with α2(ω) the energy dependent electron-phonon interaction and F(ω) the phonon density of states have been derived for elements and alloysfrom electron tunneling spectra [275]. Since F(ω) can be measured directly by neutron scattering α

and F can be derived independently and serve as a basis for a quantitative comparison with theoreticalpredictions [276] and, therefore, provide key information for a microscopic understanding of the pairing.Generalizations including momentum dependent coupling and bosonic excitations other than phononshave been put forward but require various approximations (similary as Eliashberg’s original approach)[151, 218, 250, 277, 278]. In all cases the characteristic bosonic energy must be much smaller than theelectronic energies, hω0 EF , [279] implying that the interaction is retarded. This means that one dealswith two fairly different time scales. The electronic one reacts instantaneously to a perturbation. Theother one maintains the polarization field created by one electron sufficiently long so as to allow a secondelectron to experience it. This condition holds excellently in conventional metals having hω0/EF =O(10−2).

In the beginning (see section 6.3.1) strong electron phonon coupling with λ > 1 was considered to leadto sufficiently stable Cooper pairing in the cuprates. The limiting case is the formation of polaronswhich, at low temperature, condense into bi-polarons [61]. Polaronic behavior was indeed observed atlow doping [280, 281] but it is hard to pin down at optimal doping and beyond. This does not implythat electron-phonon coupling can be disregarded. Actually, there are experimental indications such asdoping dependent shifts in the phonon spectra [282, 283], isotope effects at low doping [284], kinks inthe electronic dispersion in the entire doping range [146, 147, 149] or strong coupling effects of specificphonons [285, 286]. However, the derived overall coupling constants are considered to be too small tosupport superconductivity in the 100 K range [62,250], and the way the electron-lattice interaction entersis probably different from the situation in conventional superconductors [62, 143].

2013


Coupling mechanisms other than phononic can arise from low-energy spin [?,?,210,287–290] or chargefluctuations [153] or from high-energy instantaneous interactions such as the Coulomb repulsion U orthe exchange coupling J [230, 291, 292]. In all cases the Hubbard model is a useful starting point thatpredicts antiferromagnetism, phase separation and superconductivity [29, 133] as shown in Fig. 6.17.

ter antiferromagnetic

superconductingaram

et superconductingrder pa

AF+SC pseudogap SC

or

doping p

Figure 6.17: Phases predicted by the 3-band Hubbard model on the p-doped side. Similar to the 1-bandversion (oxygen orbitals not explicitly taken into account) antiferromagnetic (AF) and superconducting(SC) correlations corresponding to finite order parameters are found. In the overlap region a pseudogapappears in the derived spectral functions. SC vanishes only at p= 0. Additional crossover lines are foundat higher doping if electron-phonon interaction is included [142, 143]. From [133] with permission.

For t < U ∞ superconductivity can be obtained even though the interaction is repulsive, since the d-wave gap changes sign and facilitates a solution of the gap equation. In a real-space argument one wouldsay that the two electrons avoid the repulsive part of U by arranging in a d-wave pair function whichvanishes when the potential is repulsive [29, 133] as already pointed out by Pitaevskii and Brueckneret al. [16, 17]. This type of interaction is instantaneous since it is purely electronic and on a very highenergy or short time scale [292].

In the limit U → ∞ the Coulomb repulsion is integrated out and J ' 130meV becomes the highest en-ergy scale right after the band width. For p . 0.03 the nearest-neighbor coupling J leads to the usualHeisenberg-type long-ranged AF order (given that there is finite coupling in c-direction). At higherdoping only short range order and paramagnetism survive. There are indications that the coupling be-tween 2 spins survives beyond optimal doping [107]. On this basis Anderson formulated the RVB ap-proach [230], where local singlet pairs start to couple well above Tc via the exchange energy J andcondense into Cooper pairs below Tc. Then, the pseudogap is the energy reduction in the RVB state andphase fluctuations prevent the singlets from condensing above Tc.

Upon proceeding to lower energies the pairing becomes more conventional in the sense that the inter-action is retarded. Then, the Eliashberg theory can be applied [3, 274, 277], and a coupling spectrumshould be derivable from any type of electronic response or, turning the argument around, observableby an appropriate independent experimental method such as neutron scattering in the case of spin fluc-tuations. There are quantitative studies of how the spin spectrum could provide enough coupling forthe cuprates [152] but there is no consensus yet since the interaction between a spin fluctuation and anelectron can be treated only phenomenologically. Alternatively, the electrons can also interact via fluc-tuations of orbital currents [155, 156] or of the charge density [153, 154]. Small magnetic moments aspossible indications of orbital currents were discovered recently below T ∗ [?] but it is as complicated asin the case of spins to determine the coupling. Traces of charge fluctuations are even harder to pin downsince there is no independent probe for the related excitations [114]. Caprara et al. propose to study the



related Aslamazov-Larkin fluctuations by light scattering [293]. Since charge fluctuations couple to thelattice, the phonons may be back in the game in an indirect fashion [142].

Presently, spectra measured with different methods are analyzed in order to find fingerprints of therelevant bosons. These include ARPES [146, 147, 151, 152], neutron scattering [152], STS [138],IR [218, 294–296], and Raman spectroscopy [?, 139, 184]. There are indications for both electron-phonon [146, 147] and electron-spin [151] interaction in the nodal ARPES spectra. With adjusted cou-pling constants the nodal electron dispersion can be reproduced by and large on the basis of the spin sus-ceptibility [152]. Away from the node high-resolution ARPES data do not exist yet. The optical and Ra-man spectra always show two well separated energy scales at approximately 50 and 200 meV [184, 296]which cannot a priori be identified with specific excitation. Possibly the strong polarization dependencein the Raman results [184] may help to assign the modes via the selection rules. Then charge and spinfluctuations would dominate at low and high energies, respectively [?] making the exchange couplingJ an important player, as suggested by Anderson [292], along with coupled charge-phonon excitations.However, as in the other cases the experiments have been done above Tc and the coupling constants canat best be guessed. Is there any independent criterion to foster a decision?

Poilblanc and Scalapino derived a partial sum rule for the complex Eliashberg gap function Φ(k,ω),I(k,Ω) = fk(Φ)|(ω≤Ω), varying between 0 for Ω = 0 and approximately 1 for Ω→ ∞ which measuresthe contributions to the pairing interaction at a given momentum k as a function of the cut-off energyΩ [297]. When all the coupling (attractive or repulsive) is exhausted at high energies I(k,Ω) approaches1. In Pb for instance, I(Ω) increases most rapidly at the transversal and longitudinal phonon frequencies,exceeds unity above the energy range of the phonons due to the unretarded repulsion in normal metalsand approaches 1 asymptotically in the high energy limit [298]. This type of analysis requires highresolution data for the gap function Φ(k,ω) which do not exist for the cuprates yet. However, theoreticalmodels can be studied, and in the Hubbard model 80 % of the coupling occurs in the energy range ofthe spin fluctuations. In contrast to Pb, I(Ω) does not exceed unity in the Hubbard model indicatinginstantaneous pairing interactions at higher energies such as J and U . Their relative weight has still to beclarified [133, 298–300].

It will be an important step forward if the relevant interactions or energy scales in the cuprates can beidentified. In spite of enormous progress both experimental and theoretical, the main question as to therelative contribution of the various possible pairing mechanisms is not yet answered. Probably, it is theright mixture which allows one to explain not only the phase diagram but also the material dependence.

It has been noticed that the maximal Tc and the ratio t ′/t depend in a systematic way on the distance of theapical oxygen from the CuO2 plane reflecting the individual electronic structure of the compounds [301].Johnston and coworkers derived the corresponding electron-phonon coupling λph and find that Tc can betuned substantially by varying the ratio λs/λph with λs the coupling via spin fluctuations [62]. Similarly,since the three-band Hubbard model includes p− d charge fluctuations which depend sensitively ondetails of the materials it may supplement the “plain vanilla” one-band model [302] by adding a channelfor tuning T max

c [133].

In any case, we have to further sharpen our diagnostic tools to finally tackle the proper origin(s) ofsuperconductivity in the cuprates and, maybe, get ideas towards novel materials. Although new su-perconductors have usually been found through the intuition of the materials scientists, the search wasalways guided by concepts.

6.3.5 Summary and perspectives

The highest transition temperatures to superconductivity so far are observed in copper-oxygen com-pounds with CuO2 planes as the common building elements. The planes are separated by perovskite-like

2013


blocks. Tc can reach 135 K under normal conditions and exceed 150 K with 25 GPa applied pressure.The upper critical field Bc2(T → 0) reaches values well above 100 T in compounds with Tc ' 100 K.

The cuprates share the proximity of superconductivity and other ordered phases or states with incipientorder with various materials such as f -electron systems, organic metals and the recently discovered iron-based superconductors. It is a crucial question as to which extent superconductivity gets fueled by theneighboring instabilities and their fluctuations. Apparently, the dimensionality plays a role since lowdimensions favor fluctuations and reduce the screening [303].

At present the Hubbard model seems a viable way towards a microscopic description since it capturesantiferromagnetism, competing phases and superconductivity. However, it is an open question whichof the interactions emerging from the model, i.e. spin fluctuations, exchange coupling and Coulombrepulsion, dominate in driving superconductivity. Even phonons may enter in a couple of ways yetdifferent from those in conventional systems.

So far neither theory nor experiment are in a state to suggest search strategies for new superconductingmaterials. However, from what we know from the comparison of the cuprates, iron-pnictides and f -electron systems, nearly planar systems at the brink of stability of a magnetic or charge-ordered phaseseem to be favorable.

After almost three decades of research into cuprates various applications emerge (see next chapter).The most popular passive devices are frequency filters, fault current limiters, power transmission lines,dynamical capacitors, high field magnets, efficient eddy current heaters, and low-noise pick-up coils.Presently, only SQUID magnetometers are among the cuprate-based applications using the dynamicproperties (Josephson effect). Although cycle frequencies in the THz range would be possible computingwith Josephson junctions appears very unlikely an application of the cuprates at the moment for theenormous and continuous progress of semiconductor devices relying on established technology.


Chapter 7

An overview of applications

Materials and applications have a strong impact on the development of a field since they generate fundingat a high rate and from many sources. The potential for applications of superconductivity was recognizedalready by Kamerlingh-Onnes but it was a long way to go before the first solenoid was presented in 1955by Yntema [304], providing a field of 0.71 T. The conductor was Nb, a weak type II superconductorhaving κ ≈ 0.9. The development of the Ginzburg-Landau theory was one major step towards the un-derstanding of the kind of problems that had to be solved for this application: high currents in highfields. In addition, the metallurgy of superconducting materials, connections between superconductorsand between superconductors and normal metals, and efficient cooling at the temperature of liquid he-lium were on the agenda. NbTi was a big leap since it has an upper critical field of approximately 16 Twhile maintaining a high ductility. It is still the material that dominates high field applications. Only forfields in excess of 12 T Nb3Sn [305] and YBa2Cu3O7 conductors 1 are used and developed, respectively.Apart from big magnetic coils high-power applications on a large scale are still at their infancy. On theother hand active devices for detecting small voltages and magnetic fields are widely used and commer-cially available. These include SQUID magnetometers for research, exploration and military, filters formobile telecommunication, and Josephson elements for metrology. To get superconducting technologystarted the advantages must be striking, and other solutions have to be outperformed technically andeconomically.

7.1 Potential areas

7.1.1 Economic considerations

By and large2 new technologies prevail only if the production and operation costs are substantiallysmaller and if the reliability is comparable or better in comparison to an established solution. In addition,the inertia of running systems can hardly be overestimated. While the operation costs can be estimatedand controlled reasonably well the production costs may include a completely new infrastructure. Awell known example is hydrogen technology. Similarly, for power transmission over larger distanceswith superconductors requiring cooling with liquid He the costs He recovery and liquefaction systemsare prohibitively high. Although operative cables were and are available there was no progress in instal-lation before the cuprate superconductors permitted cooling with liquid nitrogen. Here, the liquefactionand distribution is orders of magnitude simpler and cheaper than in the case of He. Consequently, also the

1http://www.magnet.fsu.edu/usershub/publications/sciencehighlights/2010/28/0654118 Boebinger ASC1 coil final.pdf2For details search for the Peter principle.

113

114 R. HACKL AND D. EINZEL An overview of applications

reliability is much higher. For the grid, the requirements are approximately 1/365 days for maintenanceas can be realized with nitrogen technology but not with He.

If, on the other hand, the operating costs exceed the those for installation new approaches are desirable.As an example we compare the costs for a conventional and a superconducting magnet for a laboratory.Similar considerations apply for a magnet used for magnetic resonance imaging (MRI) in medical ap-plications or chemical analysis. Table 7.1 shows a comparison. This example shows that experiments in

Table 7.1: Comparison of the approximated overall cost of a laboratory magnet providing a field of 10 Tin a bore of 50 mm. ke is 1,000e, LHe is for liquid helium. In modern MRI machines there is noLHe refill any more since the evaporating gas is reliquefied with a closed-cycle refrigerator consumingapproximately 5 kW corresponding to 40 e/d for electricity.

NbTi solenoid Cu solenoid

installation 100 ke 30 keinfrastructure 0 10 kecooling 3 l LHe/d 1 m3 water/min

30 e/d 3 keenergy 0 5,000 kW

0 40,000 ke/dmaintenance LHe refill electricity/waterreliability 1/365 d ?conductor 1 e(kA m)−1 0.2 e (kA m)−1

high fields were out of reach for standard laboratories before the advent of superconducting coils sim-ilarly as medical examination using MRI. There is just one exception for which resistive magnets arestill needed: the generation of continuous fields above 21 T. The power consumption of the 45 T magnetin the National High Field Laboratory in Tallahassee (FL, USA) or of the 33 T magnet in the CNRSHigh Field Laboratory in Grenoble (France) is accordingly high and reaches values in the 30 MW rangerequiring cooling facilities with capacities on the order of 1,000 m3 per hour.

We may conclude that superconductivity will preferably be used for local applications since cooling ofextended installations is complicated and costly. However, with the advent of the cuprates the restrictionsbecome more relaxed since LHe can be replaced by liquid nitrogen (LN2) entailing a cost reduction byapproximately a factor of 100. For instance, cooling with LN2 costs approximately 0.5e (m d)−1 for ahigh power cable. Nevertheless, the production costs for the conductors remain high.

7.1.2 Areas of application

It is clear that the so far available superconductors will not completely replace conventional materials.Even if a material with a Tc above room temperature will successfully be synthesized in the future prob-lems such as flux pinning (see below), AC losses, connection with other materials, and, last but not least,production costs will be an issue. On the basis of current technology the following applications arerealized or feasible. Some will be discussed in more detail below.

• High current applicationsThis includes high-field magnets for research, medicine, fusion, and accelerators, the improve-ment of local grids, the interconnection of independent grids, fault current limiters, dynamic syn-chronous condensers, motors, and generators.


Passive applications SUPERCONDUCTIVITY 115

• Filter technologyFilter characteristics can be improved substantially with an enhanced conductivity. Here the dy-namic conductivity in the MHz and predominantly GHz range is relevant. Using cuprates at40 . . .77 K at least an order of magnitude improvement in performance and size can be reachedin comparison to copper. Here mobile telecommunication and defense prevail.

• SensorsSQUID magnetometers are widely used in research and development but also in Geology for ex-ploration. The Josephson effect is used for metrology for instance.

• Superconducting computersIn the beginning one was thinking of superconducting logic circuits switching between zero andfinite resistance. This concept was outperformed by Si technology before any realization. Therapid single flux quantum logics (RSFQ) is still competitive concerning frequency but is not pur-sued any further. Currently SQUIDS are being studied as to whether they can be used as Qbits ina quantum computer.

Prior to all applications many physical concepts had to be developed and technical challenges had to besolved. We shall discuss the important aspects before providing an overview of realized applications ina field. A broad discussion will follow in the lecture “Applied Superconductivity” in the Summer Term.

7.2 Passive applications

In most passive applications one wants to exploit the dissipation free transport of high currents sometimesin a magnetic field. In type I superconductors the critical currents and fields are very small limiting therange of applications to the screening of small static fields and electromagnetic noise in the sub GHzrange (hω < ∆ where ∆ is the gap). In type II materials the fields destroying superconductivity are muchhigher but in the presence of a transport current the vortices may move giving rise to dissipation. Wediscuss no the origin of the dissipation and possible remedies.

7.2.1 Physical and technical challenges

If a transport current jtr flows in a superconductor in the presence of a magnetic field as shown in Fig. 7.1there is a force on the vortices originating from the Lorentz force. We saw in the discussion of theAbrikosov lattice that the vortices assume a symmetric arrangement without a transport current since thenet forces vanish. For a finite transport current perpendicular to the field B the symmetry is broken sincethe transport and the screening currents add vectorially, j(r)total = j(r)tr + j(r)screen resulting in a netprojection of the currents around the flux lines on the y-direction and a force FL in positive x-direction.The symmetry breaking can also be seen in a different way: the applied current jtr creates a field aroundthe superconductor which enhances the field on the l.h.s. and leads to a reduction of B on the oppositeside. As a consequence the vortex density starts to become location dependent thus establishing a forceto restore equilibrium. The density of the Lorentz force reads

f =FV

= nevtr×B≡ jtr×B. (7.2.1)

If we integrate over the volume of one flux line, cπλ 2, we get

F1 ≈ c∫

πλ 2|jtr||B| (7.2.2)

2013


x

z

ya

b

c

jtr

B

FL

Figure 7.1: Superconductor in a magnetic field. The field B that points in z-direction is in the rangeBc1 < B < Bc2. The transport current jtr and the screening currents add resulting in an imbalance of thescreening currents around the flux lines. The net force on the flux lines originates in this asymmetry.

since most of the flux is in a cylinder having a radius of order λ , and since the current is perpendicularto the field. Inside the cylinder the force is approximately constant, and we arrive at the result derivedalready earlier,

F1 = c|jtr|∫

πλ 2|B|d2r. (7.2.3)

or, more generally, F1/c = jtr× ezΦ0 since a flux line carries one flux quantum. The power dissipated byone flux line is then

dQ1

dt=

ddt

∫ x

0F1dx′ ≈ F1

dxdt

= F1|vFL| (7.2.4)

where vFL is the (unknown) drift velocity of the flux lines in x-direction. The last part of the derivationis counting the number of flux lines N, and the total dissipation is P = NQ1

P =UI =|B|ab

Φ0F1|vFL| (7.2.5)

with U the voltage drop along y and I = ac · |jtr|. Using Eq. (7.2.3) the voltage drop over the length b ofthe superconductor can be expressed as

U = B|vFL|b, (7.2.6)

meaning that, for homogeneous conditions, the electric field is E = B|vFL|. The drift velocity vFL can bemeasured by virtue of Eq. (7.2.6). We can estimate |vFL| by defining the maximally acceptable voltagedrop. Usually one defines the onset of the resistive state by occurrence of one µV along the direction ofthe transport current implying that in a field of 1 T and and b = 10−2 m the drift velocity reaches only|vFL|= 10−4 m/s.

Whenever vFL 6= 0 energy is dissipated. For understanding how the energy is dissipated we can followtwo ways. Qualitatively we observe that Cooper pair are broken up in the center of the vortex and turninto normal particles. As long as the vortex is at rest the normal electrons do so as well making the mixedstate dissipationless after the relaxation time of the normal electrons. However, if the vortices move theunpaired electrons are dragged on with velocity vFL and dissipate energy via collisions. Finally, vorticesappear on one side and get annihilated on the opposite side giving rise to an ac component in U . If thespacing of the vortices is d the average voltage drop over the distance d is Ud = B|vFL|d, and a frequency

ω = 2π|vFL|

d(7.2.7)


Passive applications SUPERCONDUCTIVITY 117

can be found. If we solve this for |vFL| and substitute it into Ud we find

Ud =ωBd2

2π. (7.2.8)

Bd2 is the flux through one vortex that is just the flux quantum Φ0, and we find

Ud =hω

2e(7.2.9)

and recover the second Josephson equation. Very sloppily one can say that the current trajectory getsinterrupted by a vortex when it moves through and causes a phase slip of 2π that gives rise to a voltagepulse.

The important question is as to whether or not the vortices can be prevented from moving since oth-erwise the mixed state (Shubnikov phase) is next to useless for applications. We now discuss how the“pinning” of the vortices can in fact be accomplished. The most direct way of describing pinning (seeV. V. Schmidt) is by realizing that a vortex has a normal core inside which the condensation energy van-ishes implying that the energy density inside the core is enhanced by approximately B2

c/2µ0 where Bc isthe thermodynamic critical field or by

B2c

2µ0πξ

2 (7.2.10)

per unit length over that of the superconducting material at distances larger than ξ from the center ofthe core. The same loss of condensation energy occurs if a hole with diameter 2ξ is drilled into thesuperconductor. This can be realized experimentally by shooting accelerated heavy atoms on the super-conductor. Now, if the vortex core is centered around this columnar defect, no additional condensationenergy is lost. If, however, the vortex is moved away from the hole the free energy of the material ispushed up again by the same amount. Therefore a restoring force fp (per unit length) exists trying to pullthe vortex back into the cylindrical cavity. At the edge of the cavity fpξ should be approximately equalto the condensation energy,

fp ≈B2

c

2µ0πξ , (7.2.11)

yielding

fp,c ≈B2

c

2µ0πξ c, (7.2.12)

for the slab of thickness c (see Fig. 7.1). Whenever fp,c < F1 (see above) the restoring force is big enoughto the vortex pinned. If we equate fp,c and F1 we find for the current density

jtr =B2

c

2µ0Φ0πξ , (7.2.13)

or, by using the GL expression (4.4.52) for the thermodynamic critical field,

jtr =Bcξ

4√

2µ0λ, (7.2.14)

which is comparable to the pair-breaking critical current (not derived yet from the GL theory). In otherwords, given efficient pinning a superconductor in in field can carry a high current. However, the totalfield is substantially enhanced by the current. Therefore, the critical current is reduced substantially uponapproaching Bc2.

2013


7.2.2 Examples

Conventional materials

The most popular application with the biggest market are solenoids for MRI, and many of us had anopportunity to experience one of the machines. A field of up to 3 T with well-defined small gradients ina volume of approximately 1 m3 is provided by a system of superconducting coils. The multifilamentarywires are made of NbTi embedded in a matrix of alloyed Cu. They can be fabricated by establishedextrusion techniques and wound up [10]. Hydrogen is the only nucleus being probed here and the spatialresolution is approximately 0.5 mm. Coils for 4–8 T are in the experimental phase since the resolutioncan be improved and other nuclei such as phosphorus can be used as a probe.

The other big market is laboratory magnets. As mentioned above relatively inexpensive NbTi standardlaboratory magnets are in wide use. For chemical analysis high-field NMR has an increasing share.After a very long experimental phase Nb3Sn coils providing fields in excess of 21 T became availablein 1980ies and are standard for high-field NMR magnets only since the nineties [306]. Here, hybridcoils are used. The outer coil is a NbTi solenoid providing some 10 T the inner part is a Nb3Sn solenoidadding another 10 or 11 T, and a field 21 T can be reached in a bore of 50 mm having a homogeneityof better than 10−6 in a volume of 1 cm3. Similarly, the 45 T magnet in Tallahassee is a hybrid havinga NbTi, a Nb3Sn, and a resistive coil with the latter one providing an additional field of some 25 T atthe price of 30 MW electrical power and 1440 m3 per hour cooling water. On of the biggest installationsof superconducting magnets is the accelerator of the CERN. In Karlsruhe coils for fusion reactors weredeveloped and built. The coils have complicated shapes and are as big as a two-story building. Since awhile prototypes for toroidal coils for energy storage are being developed.

Beyond the conventional metallic systems there is an increasing number of applications of high-Tc

cuprates but none of the established techniques can be used for the cuprates.3

Cuprates with high transition temperature

It is not only the higher transition temperatures which relax the requirements for cooling but also theenhanced robustness of superconductivity as quantified by the condensation energy ∆F ∝ B2

c ∝ T 2c with

Bc the thermodynamical critical field. In Y-123 for instance, the upper critical field Bc2(T ), at whichsuperconductivity collapses, is in the range 140 T in the low-T limit and still some 40 T at 77 K [241]. Thecritical current densities exceed 100 A mm2 and 10,000 A mm2 at 77 and 4.2 K, respectively. For activedevices a switching frequency in the THz range can be reached for the large energy gap, τ−1 ' ∆/h.Given these advantages, why took it more than 20 years until the first application was commercialized?

It is rather complicated to get sufficiently homogeneous large-scale products at competitive costs sincethe materials have to be synthesized at 600-800 C and are brittle. Y-123, the workhorse in applications,is a quaternary compound and small deviations from stoichiometry reduce Tc substantially. On the otherhand, defects at average distances close to the coherence length ξ0 ' 20 A are necessary for high criticalfields and currents. The enormous a− c-anisotropy and the critical current of in-plane grain boundarieswhich decreases exponentially with the misalignment angle [307] require essentially mono-crystallinespecimens. For coils or for power transmission lines the quality must be maintained over hundreds ofmeters.

In spite of the difficulties various products are in the test phase now (for recent overviews see, e.g.,Ref. [308–310] or the web-links in the references). Wires with Bi-2223 can be bought from the shelfwith lengths up to 1.5 km [311]. For this product Bi-2223 powder is filled in metal tubes, mostly Ag, and

3The following section is an almost literal quote from a publication of one of us (R.H.) [56]


Active devices SUPERCONDUCTIVITY 119

then rolled and reheated several times to get the platelet-like micro-crystals aligned. Very good results areobtained with oriented films of Y-123, which are deposited on strained metallic ribbons on top of variousbuffer layers [312, 313]. There are various techniques to grow thin films.For industrial production costsplay a central role. Thermal co-evaporation of the metallic elements followed by in-situ post-annealingin oxygen [93] is among the promising methods. It was developed in a spin-off company of the TechnicalUniversity Munich which now sells superconductors as well as production and test equipment [312]. Forsome applications jet-printing is used to deposit liquid precursors which are processed afterwards [314].In this way complicated geometries can be realized, possibly at the price of a slightly reduced criticalcurrent. On the other hand, the technique is extremely simple and cost effective.

In wider collaborations fault current limiters have been developed. In contrast to conventional technologythe device can reversible interrupt the connection between the generator and the grid in the case of anoverload. Inductive fault current limiters can cut spikes without fully interrupting the transmission. Thefirst device produced by Zenergy Power was delivered in 2010 to CE Electric in the United Kingdom.In the United States dynamic synchronous condensers are used to compensate variable inductive andcapacitive loads resulting from and enhancing rapid fluctuations of the power consumption in the grid.Filters for base stations of mobile communication have a much better selectivity and are substantiallysmaller. The superconducting filter element has an area of only a few quare centimeters. Thousands ofthese filters which fit into a 19′′ rack have been installed already [315]. Motors and generators havingrotors with superconducting coils have a slightly better efficiency and substantially reduced size andweight. This helps to reduce the material consumption and makes the technology favorable for, e.g.,ships or upcoming applications such as wind turbines. It has been demonstrated that thin-film Y-123receiver coils improve the signal-to-noise ratio of MRI by a factors between 2 and 9 compared to thatachievable with copper [316] as shown in Fig. 7.2 At the National High Field Laboratory in Tallahassesolenoids for the 30 to 40 T range are under development. For many experiments these magnets canreplace the 45 T Hybrid magnet with a superconducting outer and a conventional inner coil at a price ofapproximately 20 MW power consumption.

All these applications are local, and cryogen-free cooling is possible and is actually used widely. Thedevelopment and optimization of pulse-tube cryo-coolers in the last decade simplified the refrigerationsubstantially and improved the reliability. The maintenance intervals are years and the the base tem-perature of a single stage engine is close to 30 K. For the filters miniature Stirling coolers have beendeveloped with a power consumption in the 100 W range [315].

For non-local application such as power transmission cooling is still an issue. It is very much relaxedby the use of liquid nitrogen but still complicated and subject to failure [310]. Therefore, supercon-ductors are mainly considered for specialized applications, where conventional techniques cannot carrythe increased load, although the development of cables is very advanced . Nevertheless, using a ca-ble manufactured by AmSC, the Long Island Power Authority started to transmit electricity for 300,000households in April 2008 as shown in Fig. 7.3 [313]. Demonstration projects have been started in variousother places of the world.

7.3 Active devices

The sensing technique with SQUIDS on the basis of conventional superconductors was already matureat the end of the 1980ies while the use of cuprates operated at 77 K has still to overcome some prob-lems [308]. Due to low pinning potentials the noise is the main problem to fix. While the noise ofapproximately 50 fT Hz−1/2at 1 Hz is still too high for magneto-encephalography the study of the heartis feasible and has enough resolution [317]. There exist small start-up companies producing integratedsolutions [318]. SQUIDs are now used mainly in the laboratory but also in military applications, in ge-ology, and in quality control. Bits for quantum computing are so far only made of conventional metals.

2013


tissue

0 5 mB

0.5 m

c-shaped open magnet plasticp p g plastictube

He compressorHe compressor

Figure 7.2: Schematic view of a setup for magnetic resonance imaging (MRI). The c-shaped open magnetis fabricated with Bi-2223 tape wire. The sensor has an Y-123 receiver. Magnet and receiver are cooledcryogen free with closed-cycle He refrigerators. The lower image on the r.h.s. shows the MRI image ofthe little finger with a conventional Cu receiver. The image on top is the result obtained with the Y-123receiver. The improvement of the signal to noise ratio for the same scanning time is approximately afactor of 5. Reproduced from [316].

Figure 7.3: Superconducting power transmission line (front) operated by the Long Island Power Author-ity (LIPA). The cables have a core of Y-123 conductor including Cu for protection, a channel for liquidnitrogen and thermal insulation. The superconductors carry the same energy as the over-head lines in thebackground. The system is operative since April 2008. Courtesy of American Superconductors [313].


Bibliography

[1] H. Kamerlingh-Onnes, , Comm. Leiden 120b, (1911).

[2] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Superconductivity, Phys. Rev. 108, 1175(1957).

[3] G. M. Eliashberg, , Zh. Eksp. Teor. Fiz. 38, 966 (1960).

[4] A. J. Leggett, A theoretical description of the new phases of liquid 3He, Rev. Mod. Phys. 47, 331(1975).

[5] G. Gruner, in Density waves in Solids, edited by D. Pines (Addison-Wesley, ADDRESS, 1994).

[6] J. G. Bednorz and K. A. Muller, Possible high Tc superconductivity in the Ba-La-Cu-O System,Z. Phys. B 64, 189 (1986).

[7] Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, and H. Hosono,Iron-Based Layered Superconductor: LaOFeP, J. Am. Chem. Soc.. 128, 10012 (2006).

[8] M. Rotter, M. Tegel, and D. Johrendt, Superconductivity at 38 K in the Iron Arsenide(Ba1−xKx)Fe2As2, Phys. Rev. Lett. 101, 107006 (2008).

[9] M. Luders, M. A. L. Marques, N. N. Lathiotakis, A. Floris, G. Profeta, L. Fast, A. Continenza,S. Massidda, and E. K. U. Gross, Ab initio theory of superconductivity. I. Density functionalformalism and approximate functionals, Phys. Rev. B 72, 024545 (2005).

[10] W. Buckel and R. Kleiner, Superconductivity (Wiley, Weinheim 2004, Weinheim, 2004).

[11] P. Drude, Zur Elektronentheorie der Metalle, Annalen der Physik 306, 566 (1900).

[12] P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13, 508(1964).

[13] F. Gross, B. S. Chandrasekhar, D. Einzel, K. Andres, P. J. Hirschfeld, H. R. Ott, J. Beuers, Z. Fisk,and J. L. Smith, Anomalous temperature dependence of the magnetic field penetration depth insuperconducting UBe13, Z. Phys. B: Condens. Matter 64, 175 (1986).

[14] N. Ashcroft and N. Mermin, Solid State Physics (Saunder College, Philadelphia, ADDRESS,1976), p. 848.

[15] W. Prestel, Study of the Interaction Processes in Cuprate Superconductors by a Quantitative Com-parison of Spectroscopic Experiments, Ph.d. thesis, Technische Universitat Munchen, 2012.

[16] K. A. Brueckner, T. Soda, P. W. Anderson, and P. Morel, Level Structure of Nuclear Matter andLiquid 3He, Phys. Rev. 118, 1442 (1960).

121

122 R. HACKL AND D. EINZEL BIBLIOGRAPHY

[17] L. P. Pitaevskii, , Sov. Phys. JETP 10, 1267 (1960).

[18] D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett, Experimentaldetermination of the superconducting pairing state in YBCO from the phase coherence of YBCO-Pb dc SQUIDs, Phys. Rev. Lett. 71, 2134 (1993).

[19] C. C. Tsuei and J. R. Kirtley, Pairing symmetry in cuprate superconductors, Rev. Mod. Phys. 72,969 (2000).

[20] W. N. Hardy, D. A. Bonn, D. C. Morgan, R. Liang, and K. Zhang, Precision measurements of thetemperature dependence of λ in YBa2Cu3O6.95: Strong evidence for nodes in the gap function,Phys. Rev. Lett. 70, 3999 (1993).

[21] B. Muhlschlegel, Die thermodynamischen Funktionen des Supraleiters, Z. Phys. 155, 313 (1959).

[22] D. Einzel, Supraleitung und Superfluiditat, Lexikon der Physik (2000).

[23] H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, and J. V. Waszczak, Scanning-Tunneling-Microscope Observation of the Abrikosov Flux Lattice and the Density of States near and insidea Fluxoid, Phys. Rev. Lett. 62, 214 (1989).

[24] P. G. de Gennes, Superconductivity of Metals and Alloys (W. A. Benjamin, ADDRESS, 1964),iSBN 0-7382-0101-4.

[25] K. Andres, J. E. Graebner, and H. R. Ott, 4 f -Virtual-Bound-State Formation in CeAl3 at LowTemperatures, Phys. Rev. Lett. 35, 1779 (1975).

[26] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer, Supercon-ductivity in the Presence of Strong Pauli Paramagnetism: CeCu2Si2, Phys. Rev. Lett. 43, 1892(1979).

[27] C. Pfleiderer, Superconducting phases of f -electron compounds, Rev. Mod. Phys. 81, 1551(2009).

[28] M. Vojta, Quantum phase transitions, Rep. Prog. Phys. 66, 2069 (2003).

[29] D. J. Scalapino, The case for dx2−y2 pairing in the cuprate superconductors, Physics Reports 250,329 (1995).

[30] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Unconventional Superconductivity with aSign Reversal in the Order Parameter of LaFeAsO1−xFx, Phys. Rev. Lett. 101, 057003 (2008).

[31] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Communication Iron-Based Layered Su-perconductor La[O1−xFx]FeAs (x = 0.05-0.12) with Tc = 26 K, J. Am. Chem. Soc. 130, 3296(2008).

[32] A. I. Coldea, J. D. Fletcher, A. Carrington, J. G. Analytis, A. F. Bangura, J.-H. Chu, A. S. Erickson,I. R. Fisher, N. E. Hussey, and R. D. McDonald, Fermi Surface of Superconducting LaFePODetermined from Quantum Oscillations, Phys. Rev. Lett. 101, 216402 (2008).

[33] J. G. Analytis, C. M. J. Andrew, A. I. Coldea, A. McCollam, J.-H. Chu, R. D. McDonald, I. R.Fisher, and A. Carrington, Fermi Surface of SrFe2P2 Determined by the de Haas–van AlphenEffect, Phys. Rev. Lett. 103, 076401 (2009).

[34] A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D. J. Singh, and D. Mandrus, Superconductivityat 22 K in Co-Doped BaFe2As2 Crystals, Phys. Rev. Lett. 101, 117004 (2008).


BIBLIOGRAPHY SUPERCONDUCTIVITY 123

[35] F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. Huang, P. M. Wu, Y.-C. Lee, Y.-L. Huang,Y.-Y. Chu, D.-C. Yan, and M.-K. Wu, Superconductivity in the PbO-type structure α-FeSe, PNAS105, 14262 (2008).

[36] Y. Mizuguchi, F. Tomioka, S. Tsuda, T. Yamaguchi, and Y. Takano, Superconductivity at 27 K intetragonal FeSe under high pressure, Appl. Phys. Lett. 93, 152505 (2008).

[37] J.-H. Chu, J. G. Analytis, C. Kucharczyk, and I. R. Fisher, Determination of the phase diagram ofthe electron-doped superconductor Ba(Fe1−xCox)2As2, Phys. Rev. B 79, 014506 (2009).

[38] H. Luetkens, H.-H. Klauss, M. Kraken, F. J. Litterst, T. Dellmann, R. Klingeler, C. Hess, R.Khasanov, A. Amato, C. Baines, M. Kosmala, O. J. Schumann, M. Braden, J. Hamann-Borrero,N. Leps, A. Kondrat, G. Behr, J. Werner, and B. Buchner, The electronic phase diagram of theLaO1−xFxFeAs superconductor, Nat. Mater. 8, 305 (2009).

[39] A. V. Chubukov, D. V. Efremov, and I. Eremin, Magnetism, superconductivity, and pairing sym-metry in iron-based superconductors, Phys. Rev. B 78, 134512 (2008).

[40] S. Graser, T. Maier, P. Hirschfeld, and D. Scalapino, Near-degeneracy of several pairing channelsin multiorbital models for the Fe pnictides, New J. Phys. 11, 025016 (2009).

[41] F. Wang, H. Zhai, and D.-H. Lee, Antiferromagnetic correlation and the pairing mechanism of thecuprates and iron pnictides: A view from the functional renormalization group studies, Europhys.Lett. 85, 37005 (2009).

[42] L. Boeri, O. V. Dolgov, and A. A. Golubov, Is LaFeAsO1−xFx an Electron-Phonon Superconduc-tor?, Phys. Rev. Lett. 101, 026403 (2008).

[43] M. M. Qazilbash, J. J. Hamlin, R. E. Baumbach, L. Zhang, D. J. Singh, M. B. Maple, and D. N.Basov, Electronic correlations in the iron pnictides, Nat. Phys. 5, 647 (2009).

[44] D. H. Lu, M. Yi, S.-K. Mo, A. S. Erickson, J. Analytis, J.-H. Chu, D. J. Singh, Z. Hussain,T. H. Geballe, I. R. Fisher, and Z.-X. Shen, Electronic structure of the iron-based superconductorLaOFeP, Nature 455, 81 (2008).

[45] E. Z. Kurmaev, R. G. Wilks, A. Moewes, N. A. Skorikov, Y. A. Izyumov, L. D. Finkelstein, R. H.Li, and X. H. Chen, X-ray spectra and electronic structures of the iron arsenide superconductorsRFeAsO1−xFx (R = La,Sm), Phys. Rev. B 78, 220503 (2008).

[46] K. Terashima, Y. Sekiba, J. H. Bowen, K. Nakayama, T. Kawahara, T. Sato, P. Richard, Y.-M. Xu,L. J. Li, G. H. Cao, Z.-A. Xu, H. Ding, and T. Takahashi, Fermi surface nesting induced strongpairing in iron-based superconductors, PNAS 106, 7330 (2009).

[47] P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Gap symmetry and structure of Fe-basedsuperconductors, Rep. Prog. Phys. 74, 125508 (2011).

[48] A. D. Christianson, E. A. Goremychkin, R. Osborn, S. Rosenkranz, M. D. Lumsden, C. D. Malli-akas, I. S. Todorov, H. Claus, D. Y. Chung, M. G. Kanatzidis, R. I. Bewley, and T. Guidi, Un-conventional superconductivity in Ba0.6K0.4Fe2As2 from inelastic neutron scattering, Nature 456,930 (2008).

[49] D. V. Evtushinsky, D. S. Inosov, V. B. Zabolotnyy, A. Koitzsch, M. Knupfer, B. Buchner, M. S.Viazovska, G. L. Sun, V. Hinkov, A. V. Boris, C. T. Lin, B. Keimer, A. Varykhalov, A. A. Kordyuk,and S. V. Borisenko, Momentum dependence of the superconducting gap in Ba1−xKxFe2As2, Phys.Rev. B 79, 054517 (2009).

2013


[50] P. Samuely, Z. Pribulova, P. Szabo, G. Pristas, S. L. Bud’ko, and C. P.C., Point contact Andreev re-flection spectroscopy of superconducting energy gaps in 122-type family of iron pnictides, PhysicaC 469, 507 (2009).

[51] B. Muschler, W. Prestel, R. Hackl, T. P. Devereaux, J. G. Analytis, J.-H. Chu, and I. R. Fisher,Band- and momentum-dependent electron dynamics in superconducting Ba(Fe1−xCox)2As2 asseen via electronic Raman scattering, Phys. Rev. B 80, 180510 (2009).

[52] M. A. Tanatar, N. Ni, C. Martin, R. T. Gordon, H. Kim, V. G. Kogan, G. D. Samolyuk,S. L. Bud’ko, P. C. Canfield, and R. Prozorov, Anisotropy of the iron pnictide superconductorBa(Fe1−xCox)2As2 (x = 0.074, Tc = 23 K), Phys. Rev. B 79, 094507 (2009).

[53] J. D. Fletcher, A. Serafin, L. Malone, J. G. Analytis, J.-H. Chu, A. S. Erickson, I. R. Fisher, and A.Carrington, Evidence for a Nodal-Line Superconducting State in LaFePO, Phys. Rev. Lett. 102,147001 (2009).

[54] C. W. Hicks, T. M. Lippman, M. E. Huber, J. G. Analytis, J.-H. Chu, A. S. Erickson, I. R. Fisher,and K. A. Moler, Evidence for a Nodal Energy Gap in the Iron-Pnictide Superconductor LaFePOfrom Penetration Depth Measurements by Scanning SQUID Susceptometry, Phys. Rev. Lett. 103,127003 (2009).

[55] L. Fang, H. Luo, P. Cheng, Z. Wang, Y. Jia, G. Mu, B. Shen, I. I. Mazin, L. Shan, C. Ren, andH.-H. Wen, Roles of multiband effects and electron-hole asymmetry in the superconductivity andnormal-state properties of Ba(Fe1−xCox)2As2, Phys. Rev. B 80, 140508 (2009).

[56] R. Hackl, Superconductivity in copper-oxygen compounds, Z. Kristallogr. 226, 323 (2011).

[57] C. Varma, in Superconductivity in d- and f-Band Metals, edited by W. Buckel and W. Weber(Kernforschungszentrum Karlsruhe, ADDRESS, 1982), p. 500.

[58] W. A. Little, Possibility of Synthesizing an Organic Superconductor, Phys. Rev. 134, A1416(1964).

[59] D. Allender, J. W. Bray, and J. Bardeen, Theory of fluctuation superconductivity from electron-phonon interactions in pseudo-one-dimensional systems, Phys. Rev. B 9, 119 (1974).

[60] D. Allender, J. Bray, and J. Bardeen, Model for an Exciton Mechanism of Superconductivity,Phys. Rev. B 7, 1020 (1973).

[61] A. Alexandrov and J. Ranninger, Bipolaronic superconductivity, Phys. Rev. B 24, 1164 (1981).

[62] D. Johnston, The Puzzle of High Temperature Superconductivity in Layered Iron Pnictides andChalcogenides , Adv. Phys. 59, 803 (2010).

[63] D. T. Jover, R. J. Wijngaarden, H. Wilhelm, R. Griessen, S. M. Loureiro, J.-J. Capponi, A.Schilling, and H. R. Ott, Pressure dependence of the superconducting critical temperature ofHgBa2Ca2Cu3O8+y and HgBa2Ca3Cu4O10+y up to 30 GPa, Phys. Rev. B 54, 4265 (1996).

[64] Y. Ando, S. Komiya, K. Segawa, S. Ono, and Y. Kurita, Electronic Phase Diagram of High-Tc

Cuprate Superconductors from a Mapping of the In-Plane Resistivity Curvature, Phys. Rev. Lett.93, 267001 (2004).

[65] R. J. Cava, B. Batlogg, R. B. van Dover, D. W. Murphy, S. Sunshine, T. Siegrist, J. P. Remeika,E. A. Rietman, S. Zahurak, and G. P. Espinosa, Bulk superconductivity at 91 K in single-phaseoxygen-deficient perovskite Ba2YCu3O9−δ , Phys. Rev. Lett. 58, 1676 (1987).



[66] N. P. Armitage, P. Fournier, and R. L. Greene, Progress and perspectives on electron-dopedcuprates, Rev. Mod. Phys. 82, 2421 (2010).

[67] Y. Tokura, H. Takagi, and S. Uchida, A superconducting copper oxide compound with electronsas the charge carriers, Nature 337, 345 (1989).

[68] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang,and C. W. Chu, Superconductivity at 93 K in a new mixed-phase Y-Ba-Cu-O compound system atambient pressure, Phys. Rev. Lett. 58, 908 (1987).

[69] H. Shaked, P. M. Kean, J. C. Rodriguez, F. F. Owen, R. L. Hitterman, and J. D. Jorgensen, CrystalStructures of the High-Tc Superconducting Copper-Oxides (Elsevier Science B.V., ADDRESS,1994), p. 71.

[70] J. Longo and P. Raccah, The structure of La2CuO4 and LaSrVO4, J. Solid State Chem. 6, 526(1973).

[71] B. Grande, H. Muller-Buschbaum, and M. Schweizer, Uber Oxocuprate. XV Zur Kristallstrukturvon Seltenerdmetalloxocupraten: La2CuO4, Gd2CuO4, ZAAC 428, 120 (1977).

[72] H. Muller-Buschbaum and Wollschlager, Uber ternare Oxocuprate. VII. Zur Kristallstruktur vonNd2CuO4, Z. Anorg. Allg. Chem. 414, 76 (1975).

[73] H. Muller-Buschbaum, Zur Kristallchemie der oxidischen Hochtemperatur-Supraleiter und derenkristallchemischen Verwandten, Angew. Chem. 101, 1503 (1989).

[74] M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. Endoh, Magnetic, transport, and optical prop-erties of monolayer copper oxides, Rev. Mod. Phys. 70, 897 (1998).

[75] A. Erb, E. Walker, and R. Flukiger, BaZrO3: The solution for the crucible corrosion problemduring the single crystal growth of high-Tc superconductors REBa2Cu3O7−δ ; RE = Y, Pr, PhysicaC 245, 245 (1995).

[76] A. Erb, E. Walker, and R. Flukiger, The use of BaZrO3 crucibles in crystal growth of the high-Tcsuperconductors. Progress in crystal growth as well as in sample quality, Physica C 258, 9 (1996).

[77] R. Liang, D. A. Bonn, and W. N. Hardy, Growth of high quality YBCO single crystals usingBaZrO3 crucibles, Physica C 304, 105 (1998).

[78] R. Liang, D. A. Bonn, and W. N. Hardy, Preparation and X-ray characterization of highly orderedortho-II phase YBa2Cu3O6.50 single crystals, Physica C 336, 57 (2000).

[79] R. Liang, D. A. Bonn, W. N. Hardy, J. C. Wynn, K. A. Moler, L. Lu, S. Larochelle, L. Zhou, M.Greven, L. Lurio, and S. G. J. Mochrie, Preparation and characterization of homogeneous YBCOsingle crystals with doping level near the SC-AFM boundary, Physica C 383, 1 (2002).

[80] R. Liang, D. A. Bonn, and W. N. Hardy, Evaluation of CuO2 plane hole doping in YBa2Cu3O6+xsingle crystals, Phys. Rev. B 73, 180505 (2006).

[81] M. Klaser, YBa2Cu3Ox-Einkristalle dotiert mit Kalzium: Zchtung und Charakterisierung, Ph.D.thesis, Universitt Karlsruhe, 1999.

[82] R. E. Gladyshevskii and R. Flukiger, Modulated structure of Bi2Sr2CaCu2O8+δ , a high-Tc super-conductor with monoclinic symmetry, Acta Crystallogr., Sect. B: Struct. Sci 52, 38 (1996).

2013


[83] H. Eisaki, N. Kaneko, D. L. Feng, A. Damascelli, P. K. Mang, K. M. Shen, Z.-X. Shen, and M.Greven, Effect of chemical inhomogeneity in bismuth-based copper oxide superconductors, Phys.Rev. B 69, 064512 (2004).

[84] C. Kendziora, M. C. Martin, J. Hartge, L. Mihaly, and L. Forro, Wide-range oxygen doping ofBi2Sr2CaCu2O8+δ , Phys. Rev. B 48, 3531 (1993).

[85] G. Triscone, J. Y. Genoud, T. Graf, A. Junod, and J. Muller, Variation of the superconductingproperties of Bi2Sr2CaCu2O8+x with oxygen content, Physica C 176, 247 (1991).

[86] N. Barisic, Y. Li, X. Zhao, Y.-C. Cho, G. Chabot-Couture, G. Yu, and M. Greven, Demonstratingthe model nature of the high-temperature superconductor HgBa2CuO4+δ , Phys. Rev. B 78, 054518(2008).

[87] A. P. Mackenzie, S. R. Julian, G. G. Lonzarich, A. Carrington, S. D. Hughes, R. S. Liu, andD. S. Sinclair, Resistive upper critical field of T l2Ba2CuO6 at low temperatures and high magneticfields, Phys. Rev. Lett. 71, 1238 (1993).

[88] D. Peets, R. Liang, M. Raudsepp, W. Hardy, and D. Bonn, Encapsulated single crystal growth andannealing of the high-temperature superconductor Tl-2201, J. Cryst. Growth 312, 344 (2010).

[89] M. Lambacher, Crystal growth and normal state transport of electron doped high temperaturesuperconductors, Dissertation, Technische Universitat Munchen, 2008.

[90] M. Lambacher, T. Helm, M. Kartsovnik, and A. Erb, Advances in single crystal growth and an-nealing treatment of electron-doped HTSC, Eur. Phys. J. Special Topics 188, 61 (2010).

[91] X. Zhao, G. Yu, Y.-C. Cho, G. Chabot-Couture, N. Barisic, P. Bourges, N. Kaneko, Y. Li, L.Lu, E. Motoyama, O. Vajk, and M. Greven, Crystal Growth and Characterization of the ModelHigh-Temperature Superconductor HgBa2CuO4+δ , Adv. Mater. 18, 3243 (2006).

[92] I. Bozovic, Atomic-layer engineering of superconducting oxides: yesterday, today, tomorrow,Appl. Supercond. 11, 2686 (2001).

[93] P. Berberich, W. Assmann, W. Prusseit, B. Utz, and H. Kinder, Large area deposition ofYBa2Cu3O7 films by thermal co-evaporation, J. Alloys Compd. 195, 271 (1993).

[94] G. Logvenov, A. Gozar, and I. Bozovic, High-Temperature Superconductivity in a Single Copper-Oxygen Plane, Science 326, 699 (2009).

[95] J. L. Tallon, C. Bernhard, H. Shaked, R. L. Hitterman, and J. D. Jorgensen, Generic superconduct-ing phase behavior in high-Tc cuprates: Tc variation with hole concentration in Y Ba2Cu3O7−δ ,Phys. Rev. B 51, 12911 (1995).

[96] M. R. Norman, D. Pines, and C. Kallin, The pseudogap: friend or foe of high Tc?, Adv. Phys. 54,715 (2005).

[97] J. Tranquada, in Handbook of High-Temperature Superconductivity Theory and Experiment,edited by J. Schrieffer and J. Brooks (Springer, Berlin-Heidelberg, 2007).

[98] R. Presland, M. L. Tallon, J. G. Buckley, R. S. Liu, R. and E. Flower, N. General trends inoxygen stoichiometry effects on Tc in Bi and Tl superconductors, Physica C 176, 95 (1991).

[99] S. H. Naqib, J. R. Cooper, J. L. Tallon, R. S. Islam, and R. A. Chakalov, Doping phase diagram ofY1−xCaxBa2(Cu1−yZny)3O7−δ from transport measurements: Tracking the pseudogap below Tc ,Phys. Rev. B 71, 054502 (2005).



[100] F. Rullier-Albenque, H. Alloul, F. Balakirev, and C. Proust, Disorder, metal-insulator crossoverand phase diagram in high-Tc cuprates, Europhys. Lett. 81, 37008 (2008).

[101] H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld, Defects in correlated metals and supercon-ductors, Rev. Mod. Phys. 81, 45 (2009).

[102] R. S. Islam, J. R. Cooper, J. W. Loram, and S. H. Naqib, Pseudogap and doping-dependent mag-netic properties of La2−xSrxCu1−yZnyO4 , Phys. Rev. B 81, 054511 (2010).

[103] S. Ono and Y. Ando, Evolution of the resistivity anisotropy in Bi2Sr2−xLaxCuO6+δ single crystalsfor a wide range of hole doping, Phys. Rev. B 67, 104512 (2003).

[104] R. Harris, P. J. Turner, S. Kamal, A. R. Hosseini, P. Dosanjh, G. K. Mullins, J. S. Bobowski, C. P.Bidinosti, D. M. Broun, R. Liang, W. N. Hardy, and D. A. Bonn, Phenomenology of a[over] -axisand b[over] -axis charge dynamics from microwave spectroscopy of highly ordered Y Ba2Cu3O6.50and Y Ba2Cu3O6.993, Phys. Rev. B 74, 104508 (2006).

[105] A. Erb, private communication (PUBLISHER, ADDRESS, 2008).

[106] O. P. Vajk, P. K. Mang, M. Greven, P. M. Gehring, and J. W. Lynn, Quantum Impurities in theTwo-Dimensional Spin One-Half Heisenberg Antiferromagnet, Science 295, 1691 (2002).

[107] S. Wakimoto, H. Zhang, K. Yamada, I. Swainson, H. Kim, and R. J. Birgeneau, Direct Relationbetween the Low-Energy Spin Excitations and Superconductivity of Overdoped High-Tc Super-conductors, Phys. Rev. Lett. 92, 217004 (2004).

[108] T. Timusk and B. Statt, The pseudogap in high-temperature superconductors: an experimentalsurvey, Rep. Prog. Phys. 62, 61 (1999).

[109] A. G. Loeser, Z.-X. Shen, D. S. Dessau, D. S. Marshall, C. H. Park, P. Fournier, and A. Kapitulnik,Excitation Gap in the Normal State of Underdoped Bi2Sr2CaCu2O8+, Science 273, 325 (1996).

[110] H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria, M. R. Norman, T. Mochiku,K. Kadowaki, and J. Giapintzakis, Spectroscopic evidence for a pseudogap in the normal state ofunderdoped high-Tc superconductors, Nature 382, 51 (1996).

[111] J. Tranquada, B. Sternlieb, J. Axe, Y. Nakamura, and S. Uchida, Evidence for stripe correlationsof spins and holes in copper oxide superconductors, Nature 375, 561 (1995).

[112] H.-H. Klauss, W. Wagener, M. Hillberg, W. Kopmann, H. Walf, F. J. Litterst, M. Hucker, andB. Buchner, From Antiferromagnetic Order to Static Magnetic Stripes: The Phase Diagram of(La,Eu)2− xSrxCuO4, Phys. Rev. Lett. 85, 4590 (2000).

[113] J. Fink, E. Schierle, E. Weschke, J. Geck, D. Hawthorn, V. Soltwisch, H. Wadati, H.-H. Wu,H. A. Durr, N. Wizent, B. Buchner, and G. A. Sawatzky, Charge ordering in La1.8−xEu0.2SrxCuO4studied by resonant soft x-ray diffraction, Phys. Rev. B 79, 100502 (2009).

[114] S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, andC. Howald, How to detect fluctuating stripes in the high-temperature superconductors, Rev. Mod.Phys. 75, 1201 (2003).

[115] M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Local Ordering in the PseudogapState of the High-Tc Superconductor Bi2Sr2CaCu2O8+δ , Science 303, 1995 (2004).

[116] L. Tassini, W. Prestel, A. Erb, M. Lambacher, and R. Hackl, First-order-type effects inYBa2Cu3O6+x at the onset of superconductivity, Phys. Rev. B 78, 020511 (2008).

2013


[117] N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J. Bonnemaison, R. Liang, D. A. Bonn,W. N. Hardy, and L. Taillefer, Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor, Nature 447, 05872 (2007).

[118] V. Hinkov, D. Haug, B. Fauque, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. Lin, and B.Keimer, Electronic liquid crystal state in the high-temperature superconductor YBa2Cu3O6.45,Science 319, 597 (2008).

[119] V. Hinkov, C. Lin, M. Raichle, B. Keimer, Y. Sidis, P. Bourges, S. Pailhes, and A. Ivanov, Su-perconductivity and electronic liquid-crystal states in twin-free YBa2Cu3O6+x studied by neutronscattering, Eur. Phys. J. Special Topics 188, 113 (2010).

[120] R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choiniere, F. Laliberte, N. Doiron-Leyraud, B. J.Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Broken rotational symmetryin the pseudogap phase of a high-Tc superconductor, Nature 463, 519 (2010).

[121] L. Alff, Y. Krockenberger, B. Welter, M. Schonecke, R. Gross, D. Manske, and M. Naito, A hiddenpseudogap under the dome of superconductivity in electron-doped high-temperature superconduc-tors, Nature 422, 698 (2003).

[122] N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Damascelli, K. M. Shen, D. L. Feng, H. Eisaki,Z.-X. Shen, P. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y. Tokura, Doping De-pendence of an n-Type Cuprate Superconductor Investigated by Angle-Resolved PhotoemissionSpectroscopy, Phys. Rev. Lett. 88, 257001 (2002).

[123] Y. Dagan, M. M. Qazilbash, C. P. Hill, V. N. Kulkarni, and R. L. Greene, Evidence for a QuantumPhase Transition in Pr2−xCexCuO4−δ from Transport Measurements, Phys. Rev. Lett. 92, 167001(2004).

[124] C. Kusko, R. S. Markiewicz, M. Lindroos, and A. Bansil, Fermi surface evolution and collapse ofthe Mott pseudogap in Nd2−xCexCuO4±δ , Phys. Rev. B 66, 140513 (2002).

[125] H. Matsui, K. Terashima, T. Sato, T. Takahashi, M. Fujita, and K. Yamada, Direct Observationof a Nonmonotonic dx2−y2-Wave Superconducting Gap in the Electron-Doped High-Tc Supercon-ductor Pr0.89LaCe0.11CuO4, Phys. Rev. Lett. 95, 017003 (2005).

[126] T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, andR. Gross, Evolution of the Fermi Surface of the Electron-Doped High-Temperature Superconduc-tor Nd2−xCexCuO4 Revealed by Shubnikov–de Haas Oscillations, Phys. Rev. Lett. 103, 157002(2009).

[127] F. Gebhard, The Mott Metal-Insulator Transition - Models and Methods (Springer Tracts in Mod-ern Physics, ADDRESS, 1997), Vol. 137.

[128] N. Nucker, J. Fink, J. C. Fuggle, P. J. Durham, and W. M. Temmerman, Evidence for holes onoxygen sites in the high-Tc superconductors La2−xSrxCuO4 and Y Ba2Cu3O7−y, Phys. Rev. B 37,5158 (1988).

[129] F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev.B 37, 3759 (1988).

[130] J. M. Tranquada, S. M. Heald, and A. R. Moodenbaugh, X-ray-absorption near-edge-structurestudy of La2−x(Ba,Sr)xCuO4−y superconductors, Phys. Rev. B 36, 5263 (1987).

[131] V. J. Emery, Theory of high-Tc superconductivity in oxides, Phys. Rev. Lett. 58, 2794 (1987).



[132] C. M. Varma, S. Schmitt-Rink, and E. Abrahams, Charge transfer excitations and superconductiv-ity in ”ionic metalls, Solid State Commun. 62, 681 (1987).

[133] W. Hanke, M. Kiesel, M. Aichhorn, S. Brehm, and E. Arrigoni, The 3-band Hubbard-model versusthe 1-band model for the high-Tc cuprates: Pairing dynamics, superconductivity and the ground-state phase diagram, Eur. Phys. J. Special Topics 188, 15 (2010).

[134] C. Honerkamp, Iron pncitide superconductors studied by the functional renormalization group,Eur. Phys. J. Special Topics 188, 33 (2010).

[135] O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F. Paulsen, LDA energy bands, low-energyhamiltonians, t ′, t ′′, t⊥(k), and J⊥, J. Phys. and Chem. Solids 56, 1573 (1995).

[136] A. A. Kordyuk, S. V. Borisenko, M. S. Golden, S. Legner, K. A. Nenkov, M. Knupfer,J. Fink, H. Berger, L. Forro, and R. Follath, Doping dependence of the Fermi surface in(Bi,Pb)2Sr2CaCu2O8+δ , Phys. Rev. B 66, 014502 (2002).

[137] A. Damascelli, Z. Hussain, and Z.-X. Shen, Angle-resolved photoemission studies of the cupratesuperconductors, Rev. Mod. Phys. 75, 473 (2003).

[138] B. W. Hoogenboom, C. Berthod, M. Peter, O. Fischer, and A. A. Kordyuk, Modeling scanningtunneling spectra of Bi2Sr2CaCu2O8+δ , Phys. Rev. B 67, 224502 (2003).

[139] F. Venturini, R. Hackl, and U. Michelucci, Comment on ıNonmonotonic dx2−y2 SuperconductingOrder Parameter in Nd2−xCexCuO4 , Phys. Rev. Lett. 90, 149701 (2003).

[140] D. S. Inosov, R. Schuster, A. A. Kordyuk, J. Fink, S. V. Borisenko, V. B. Zabolotnyy, D. V.Evtushinsky, M. Knupfer, B. Buchner, R. Follath, and H. Berger, Excitation energy map of high-energy dispersion anomalies in cuprates, Phys. Rev. B 77, 212504 (2008).

[141] W. Prestel, F. Venturini, B. Muschler, I. Tutto, R. Hackl, M. Lambacher, A. Erb, S. Komiya, S.Ono, Y. Ando, D. Inosov, B. Zabolotnyy, V. and V. Borisenko, S. Quantitative comparison ofsingle- and two-particle properties in the cuprates, Eur. Phys. J. Special Topics 188, 163 (2010).

[142] C. Castellani, C. Di Castro, and M. Grilli, Singular Quasiparticle Scattering in the Proximity ofCharge Instabilities, Phys. Rev. Lett. 75, 4650 (1995).

[143] S. Andergassen, S. Caprara, C. Di Castro, and M. Grilli, Anomalous Isotopic Effect Near theCharge-Ordering Quantum Criticality, Phys. Rev. Lett. 87, 056401 (2001).

[144] X. J. Zhou, T. Yoshida, A. Lanzara, P. V. Bogdanov, S. A. Kellar, K. M. Shen, W. L. Yang, F.Ronning, T. Sasagawa, T. Kakeshita, T. Noda, H. Eisaki, S. Uchida, C. T. Lin, F. Zhou, J. W.Xiong, W. X. Ti, Z. X. Zhao, A. Fujimori, Z. Hussain, and Z.-X. Shen, High-temperature super-conductors: Universal nodal Fermi velocity, Nature 423, 398 (2003).

[145] T. Valla, A. V. Fedorov, P. D. Johnson, B. O. Wells, S. L. Hulbert, Q. Li, G. D. Gu,and N. Koshizuka, Evidence for Quantum Critical Behavior in the Optimally Doped CuprateBi2Sr2CaCu2O8+, Science 285, 2110 (1999).

[146] A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H.Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, and Z.-X. Shen,Evidence for ubiquitous strong electron-phonon coupling in high-temperature superconductors,Nature 412, 510 (2001).

2013


[147] X. J. Zhou, J. Shi, T. Yoshida, T. Cuk, W. L. Yang, V. Brouet, J. Nakamura, N. Mannella, S.Komiya, Y. Ando, F. Zhou, W. X. Ti, J. W. Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, H.Eisaki, S. Uchida, A. Fujimori, Z. Zhang, E. W. Plummer, R. B. Laughlin, Z. Hussain, and Z.-X.Shen, Multiple Bosonic Mode Coupling in the Electron Self-Energy of (La2−xSrx)CuO4, Phys.Rev. Lett. 95, 117001 (2005).

[148] T. Cuk, D. Lu, X. Zhou, Z.-X. Shen, T. Devereaux, and N. Nagaosa, A review of electron-phononcoupling seen in the high-Tc superconductors by angle-resolved photoemission studies (ARPES),PSS 242, 11 (2005).

[149] W. Meevasana, N. J. C. Ingle, D. H. Lu, J. R. Shi, F. Baumberger, K. M. Shen, W. S. Lee, T.Cuk, H. Eisaki, T. P. Devereaux, N. Nagaosa, J. Zaanen, and Z.-X. Shen, Doping Dependence ofthe Coupling of Electrons to Bosonic Modes in the Single-Layer High-Temperature Bi2Sr2CuO6Superconductor, Phys. Rev. Lett. 96, 157003 (2006).

[150] W. Meevasana, X. J. Zhou, S. Sahrakorpi, W. S. Lee, W. L. Yang, K. Tanaka, N. Mannella, T.Yoshida, D. H. Lu, Y. L. Chen, R. H. He, H. Lin, S. Komiya, Y. Ando, F. Zhou, W. X. Ti, J. W.Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, K. Fujita, S. Uchida, H. Eisaki, A. Fujimori, Z.Hussain, R. S. Markiewicz, A. Bansil, N. Nagaosa, J. Zaanen, T. P. Devereaux, and Z.-X. Shen,Hierarchy of multiple many-body interaction scales in high-temperature superconductors, Phys.Rev. B 75, 174506 (2007).

[151] A. A. Kordyuk, S. V. Borisenko, V. B. Zabolotnyy, J. Geck, M. Knupfer, J. Fink, B. Buchner, C. T.Lin, B. Keimer, H. Berger, A. V. Pan, S. Komiya, and Y. Ando, Constituents of the QuasiparticleSpectrum Along the Nodal Direction of High-Tc Cuprates, Phys. Rev. Lett. 97, 017002 (2006).

[152] T. Dahm, V. Hinkov, S. V. Borisenko, A. A. Kordyuk, V. B. Zabolotnyy, J. Fink, B. Buchner, D. J.Scalapino, W. Hanke, and B. Keimer, Strength of the spin-fluctuation-mediated pairing interactionin a high-temperature superconductor, Nat. Phys. 5, 217 (2009).

[153] A. Perali, C. Castellani, C. Di Castro, and M. Grilli, d-wave superconductivity near charge insta-bilities, Phys. Rev. B 54, 16216 (1996).

[154] S. A. Kivelson, E. Fradkin, and V. J. Emery, Electronic liquid-crystal phases of a doped Mottinsulator, Nature 393, 550 (1998).

[155] C. M. Varma, Non-Fermi-liquid states and pairing instability of a general model of copper oxidemetals , Phys. Rev. B 55, 14554 (1997).

[156] V. Aji and C. M. Varma, Theory of the Quantum Critical Fluctuations in Cuprate Superconductors,Phys. Rev. Lett. 99, 067003 (2007).

[157] A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, J. Fink, M. Knupfer, B. Buchner, H. Berger, G. Mar-garitondo, C. T. Lin, B. Keimer, S. Ono, and Y. Ando, Manifestation of the Magnetic ResonanceMode in the Nodal Quasiparticle Lifetime of the Superconducting Cuprates, Phys. Rev. Lett. 92,257006 (2004).

[158] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phe-nomenology of the normal state of Cu-O high-temperature superconductors, Phys. Rev. Lett. 63,1996 (1989).

[159] M. Plate, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, R. Liang, D. A. Bonn, W. N. Hardy,S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Fermi Surface and QuasiparticleExcitations of Overdoped Tl2Ba2CuO6+δ , Phys. Rev. Lett. 95, 077001 (2005).



[160] X. J. Zhou, T. Yoshida, D.-H. Lee, W. L. Yang, V. Brouet, F. Zhou, W. X. Ti, J. W. Xiong,Z. X. Zhao, T. Sasagawa, T. Kakeshita, H. Eisaki, S. Uchida, A. Fujimori, Z. Hussain, and Z.-X.Shen, Dichotomy between Nodal and Antinodal Quasiparticles in Underdoped (La2−xSrx)CuO4Superconductors, Phys. Rev. Lett. 92, 187001 (2004).

[161] J. Chang, M. Shi, S. Pailhes, M. Mansson, T. Claesson, O. Tjernberg, A. Bendounan, Y. Sassa, L.Patthey, N. Momono, M. Oda, M. Ido, S. Guerrero, C. Mudry, and J. Mesot, Anisotropic quasipar-ticle scattering rates in slightly underdoped to optimally doped high-temperature La2−xSrxCuO4superconductors, Phys. Rev. B 78, 205103 (2008).

[162] J. Graf, G.-H. Gweon, K. McElroy, S. Y. Zhou, C. Jozwiak, E. Rotenberg, A. Bill, T. Sasagawa,H. Eisaki, S. Uchida, H. Takagi, D.-H. Lee, and A. Lanzara, Universal High Energy Anomalyin the Angle-Resolved Photoemission Spectra of High Temperature Superconductors: PossibleEvidence of Spinon and Holon Branches, Phys. Rev. Lett. 98, 067004 (2007).

[163] J. Chang, S. Pailhes, M. Shi, M. Mansson, T. Claesson, O. Tjernberg, J. Voigt, V. Perez, L. Patthey,N. Momono, M. Oda, M. Ido, A. Schnyder, C. Mudry, and J. Mesot, When low- and high-energyelectronic responses meet in cuprate superconductors, Phys. Rev. B 75, 224508 (2007).

[164] D. S. Inosov, S. V. Borisenko, I. Eremin, A. A. Kordyuk, V. B. Zabolotnyy, J. Geck, A. Koitzsch,J. Fink, M. Knupfer, B. Buchner, H. Berger, and R. Follath, Relation between the one-particlespectral function and dynamic spin susceptibility of superconducting Bi2Sr2CaCu2O8−δ , Phys.Rev. B 75, 172505 (2007).

[165] R. Preuss, W. Hanke, and W. von der Linden, Quasiparticle Dispersion of the 2D Hubbard Model:From an Insulator to a Metal, Phys. Rev. Lett. 75, 1344 (1995).

[166] B. Moritz, F. Schmitt, W. Meevasana, S. Johnston, E. Motoyama, M.Greven, D. Lu, C. Kim, R.Scalettar, Z.-X. Shen, and T. Devereaux, Effect of strong correlations on the high energy anomalyin hole- and electron-doped high-Tc superconductors, New J. Phys. 11, 093020 (2009).

[167] T. Ito, H. Takagi, S. Ishibashi, T. Ido, and S. Uchida, Normal-state conductivity between CuO2planes in copper oxide superconductors, Nature 350, 596 (1991).

[168] K. Takenaka, K. Mizuhashi, H. Takagi, and S. Uchida, Interplane charge transport inY Ba2Cu3O7−y: Spin-gap effect on in-plane and out-of-plane resistivity, Phys. Rev. B 50, 6534(1994).

[169] L. Forro, Out-of-plane resistivity of Bi2Sr2CaCu2O8+x high temperature superconductor, Phys.Lett. A 179, 140 (1993).

[170] G. S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K.Tamasaku, N. Ichikawa, and S. Uchida, Insulator-to-Metal Crossover in the Normal State ofLa2−xSrxCuO4 Near Optimum Doping, Phys. Rev. Lett. 77, 5417 (1996).

[171] T. Ito, K. Takenaka, and S. Uchida, Systematic deviation from T-linear behavior in the in-planeresistivity of Y Ba2Cu3O7−y: Evidence for dominant spin scattering, Phys. Rev. Lett. 70, 3995(1993).

[172] J. L. Tallon and J. W. Loram, The doping dependence of T* - what is the real high-Tc phasediagram?, Physica C 349, 53 (2001).

[173] M. Abdel-Jawad, J. G. Analytis, L. Balicas, A. Carrington, J. P. H. Charmant, M. M. J. French, andN. E. Hussey, Correlation between the Superconducting Transition Temperature and AnisotropicQuasiparticle Scattering in Tl2Ba2CuO6+δ , Phys. Rev. Lett. 99, 107002 (2007).

2013


[174] R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y.Koike, M. Nohara, H. Takagi, C. Proust, and N. E. Hussey, Anomalous Criticality in the ElectricalResistivity of La2−xSrxCuO4, Science 323, 603 (2009).

[175] L. Taillefer, Scattering and Pairing in Cuprate Superconductors, Annu. Rev. Cond. Mat. Phys. 1,51 (2010).

[176] H. Alloul, T. Ohno, and P. Mendels, 89Y NMR evidence for a fermi-liquid behavior inYBa2Cu3O6+x, Phys. Rev. Lett. 63, 1700 (1989).

[177] C. C. Homes, T. Timusk, R. Liang, D. A. Bonn, and W. N. Hardy, Optical conductivity of c axisoriented Y Ba2Cu3O6.70: Evidence for a pseudogap, Phys. Rev. Lett. 71, 1645 (1993).

[178] A. V. Puchkov, D. N. Basov, and T. Timusk, The pseudogap state in high- superconductors: aninfrared study, J. Phys. Condens. Matter 8, 10049 (1996).

[179] D. N. Basov and T. Timusk, Electrodynamics of high- Tc superconductors, Rev. Mod. Phys. 77,721 (2005).

[180] B. S. Shastry and B. I. Shraiman, Theory of Raman scattering in Mott-Hubbard systems, Phys.Rev. Lett. 65, 1068 (1990).

[181] T. P. Devereaux and R. Hackl, Inelastic light scattering from correlated electrons, Rev. Mod. Phys.79, 175 (2007).

[182] T. P. Devereaux, D. Einzel, B. Stadlober, R. Hackl, D. H. Leach, and J. J. Neumeier, ElectronicRaman scattering in high-T c superconductors: A probe of dx2-y2 pairing, Phys. Rev. Lett. 72,396 (1994).

[183] F. Venturini, M. Opel, T. P. Devereaux, J. K. Freericks, I. Tutto, B. Revaz, E. Walker, H. Berger, L.Forro, and R. Hackl, Observation of an Unconventional Metal-Insulator Transition in OverdopedCuO2 Compounds, Phys. Rev. Lett. 89, 107003 (2002).

[184] B. Muschler, W. Prestel, L. Tassini, R. Hackl, M. Lambacher, A. Erb, S. Komiya, Y. Ando, D.Peets, W. Hardy, R. Liang, and D. Bonn, Electron interactions and charge ordering in CuO2 com-pounds, Eur. Phys. J. Special Topics 188, 131 (2010).

[185] S. Billinge, M. Gutmann, and E. Bozin, Structural Response to Local Charge Order in Underdopedbut Superconducting La2−x(Sr,Ba)xCuO4, Int. J. Mod. Phys. B 17, 3640 (2003).

[186] S. Blanc, Y. Gallais, A. Sacuto, M. Cazayous, M. A. Measson, G. D. Gu, J. S. Wen, and Z. J.Xu, Quantitative Raman measurement of the evolution of the Cooper-pair density with doping inBi2Sr2CaCu2O8+δ superconductors, Phys. Rev. B 80, 140502 (2009).

[187] B. Muschler, F. Kretzschmar, R. Hackl, J.-H. Chu, J. G. Analytis, and I. R. Fisher, Doping depen-dence of the electronic properties of Ba(Fe1−xCox)2As2, 49 (2010).

[188] T. Yoshida, X. J. Zhou, K. Tanaka, W. L. Yang, Z. Hussain, Z.-X. Shen, A. Fujimori, S. Sahrakorpi,M. Lindroos, R. S. Markiewicz, A. Bansil, S. Komiya, Y. Ando, H. Eisaki, T. Kakeshita, and S.Uchida, Systematic doping evolution of the underlying Fermi surface of La2−xSrxCuO4, Phys.Rev. B 74, 224510 (2006).

[189] X. K. Chen, J. G. Naeini, K. C. Hewitt, J. C. Irwin, R. Liang, and W. N. Hardy, Electronic Ramanscattering in underdoped Y Ba2Cu3O6.5, Phys. Rev. B 56, R513 (1997).



[190] M. Le Tacon, A. Sacuto, A. Georges, G. Kotliar, Y. Gallais, D. Colson, and A. Forget, Twoenergy scales and two distinct quasiparticle dynamics in the superconducting state of underdopedcuprates, Nat. Phys. 2, 537 (2006).

[191] D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou, J.-B. Bonnemaison, N. E. Hussey, L.Balicas, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, S. Adachi, C. Proust, and L. Taillefer,Electron pockets in the Fermi surface of hole-doped high-Tc superconductors, Nature 450, 533(2007).

[192] Y. Wang, L. Li, and N. P. Ong, Nernst effect in high- Tc superconductors, Phys. Rev. B 73, 024510(2006).

[193] F. Rullier-Albenque, R. Tourbot, H. Alloul, P. Lejay, D. Colson, and A. Forget, Nernst Effect andDisorder in the Normal State of High-Tc Cuprates, Phys. Rev. Lett. 96, 067002 (2006).

[194] O. Cyr-Choiniere, R. Daou, F. Laliberte, D. LeBoeuf, N. Doiron-Leyraud, J. Chang, J.-Q. Yan,J.-G. Cheng, J.-S. Zhou, J. B. Goodenough, S. Pyon, T. Takayama, H. Takagi, Y. Tanaka, and L.Taillefer, Enhancement of the Nernst effect by stripe order in a high-Tc superconductor, Nature458, 743 (2009).

[195] H.-C. Ri, R. Gross, F. Gollnik, A. Beck, R. P. Huebener, P. Wagner, and H. Adrian, Nernst,Seebeck, and Hall effects in the mixed state of Y Ba2Cu3O7−δ and Bi2Sr2CaCu2O8+x thin films:A comparative study, Phys. Rev. B 50, 3312 (1994).

[196] K. Behnia, The Nernst effect and the boundaries of the Fermi liquid picture, J. Phys. Condens.Matter 21, 113101 (2009).

[197] C. Hess, E. Ahmed, U. Ammerahl, A. Revcolevschi, and B. Buchner, Nernst effect of stripeordering La1.8−xEu0.2SrxCuO4, Eur. Phys. J. Special Topics 188, 103 (2010).

[198] J. Xia, E. Schemm, G. Deutscher, S. A. Kivelson, D. A. Bonn, W. N. Hardy, R. Liang, W.Siemons, G. Koster, M. M. Fejer, and A. Kapitulnik, Polar Kerr-Effect Measurements of the High-Temperature Y Ba2Cu3O6+x Superconductor: Evidence for Broken Symmetry near the PseudogapTemperature, Phys. Rev. Lett. 100, 127002 (2008).

[199] S.-W. Cheong, G. Aeppli, T. E. Mason, H. Mook, S. M. Hayden, P. C. Canfield, Z. Fisk, K. N.Clausen, and J. L. Martinez, Incommensurate magnetic fluctuations in La2−xSrxCuO4, Phys. Rev.Lett. 67, 1791 (1991).

[200] A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka,and T. Ito, Determination of the Local Lattice Distortions in the CuO2 Plane of La1.85Sr0.15CuO4,Phys. Rev. Lett. 76, 3412 (1996).

[201] K. B. Lyons, P. A. Fleury, J. P. Remeika, A. S. Cooper, and T. J. Negran, Dynamics of spinfluctuations in lanthanum cuprate, Phys. Rev. B 37, 2353 (1988).

[202] P. E. Sulewski, P. A. Fleury, K. B. Lyons, S.-W. Cheong, and Z. Fisk, Light scattering fromquantum spin fluctuations in R2CuO4 (R=La, Nd, Sm), Phys. Rev. B 41, 225 (1990).

[203] S. Sugai, H. Suzuki, Y. Takayanagi, T. Hosokawa, and N. Hayamizu, Carrier-density-dependentmomentum shift of the coherent peak and the LO phonon mode in p-type high-Tc superconductors,Phys. Rev. B 68, 184504 (2003).

[204] E. M. Motoyama, G. Yu, I. M. Vishik, O. P. Vajk, P. K. Mang, and M. Greven, Spin correlationsin the electron-doped high-transition-temperature superconductor Nd2−xCexCuO4±δ , Nature 445,05437 (2007).

2013


[205] M. Vojta and T. Ulbricht, Magnetic Excitations in a Bond-Centered Stripe Phase: Spin Waves Farfrom the Semiclassical Limit, Phys. Rev. Lett. 93, 127002 (2004).

[206] G. S. Uhrig, K. P. Schmidt, and M. Gruninger, Unifying Magnons and Triplons in Stripe-OrderedCuprate Superconductors, Phys. Rev. Lett. 93, 267003 (2004).

[207] S. E. Barrett, D. J. Durand, C. H. Pennington, C. P. Slichter, T. A. Friedmann, J. P. Rice, and D. M.Ginsberg, 63Cu Knight shifts in the superconducting state of Y Ba2Cu3O7−δ (Tc=90 K), Phys. Rev.B 41, 6283 (1990).

[208] J. Rossat-Mignod, L. Regnault, C. Vettier, P. Bourges, P. Burlet, J. Bossy, J. Henry, and G. Laper-tot, Neutron scattering study of the YBa2Cu3O6+x system, Physica C 185, 86 (1991).

[209] P. M. Singer and T. Imai, Systematic 63Cu NQR and 89Y NMR Study of Spin Dynamics inY1−zCazBa2Cu3Oy across the Superconductor-Insulator Boundary, Phys. Rev. Lett. 88, 187601(2002).

[210] A. J. Millis, H. Monien, and D. Pines, Phenomenological model of nuclear relaxation in the normalstate of Y Ba2Cu3O7, Phys. Rev. B 42, 167 (1990).

[211] H. A. Mook, M. Yethiraj, G. Aeppli, T. E. Mason, and T. Armstrong, Polarized neutron determi-nation of the magnetic excitations in Y Ba2Cu3O7, Phys. Rev. Lett. 70, 3490 (1993).

[212] H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, A. Ivanov, G. D. Gu, N. Koshizuka, and B. Keimer,Neutron scattering from magnetic excitations in Bi2Sr2CaCu2O8+δ , Nature 398, 588 (1999).

[213] H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, J. Bossy, A. Ivanov, D. L. Milius, I. A. Aksay, andB. Keimer, Spin susceptibility in underdoped YBa2Cu3O6+x , Phys. Rev. B 61, 14773 (2000).

[214] H. He, P. Bourges, Y. Sidis, C. Ulrich, L. P. Regnault, S. Pailhs, N. S. Berzigiarova, N. N.Kolesnikov, and B. Keimer, Magnetic Resonant Mode in the Single-Layer High-Temperature Su-perconductor Tl2Ba2CuO6+, Science 295, 1045 (2002).

[215] G. Yu, Y. Li, E. M. Motoyama, X. Zhao, N. Barisic, Y. Cho, P. Bourges, K. Hradil, R. A. Mole, andM. Greven, Magnetic resonance in the model high-temperature superconductor HgBa2CuO4+δ ,Phys. Rev. B 81, 064518 (2010).

[216] H.-Y. Kee, S. A. Kivelson, and G. Aeppli, Spin–1 Neutron Resonance Peak Cannot Account forElectronic Anomalies in the Cuprate Superconductors, Phys. Rev. Lett. 88, 257002 (2002).

[217] D. Munzar, C. Bernhard, and M. Cardona, Does the peak in the magnetic susceptibility determinethe in-plane infrared conductivity of YBCO? A theoretical study, Physica C 312, 121 (1999).

[218] J. P. Carbotte, SchachingerE., and D. N. Basov, Coupling strength of charge carriers to spin fluc-tuations in high-temperature superconductors, Nature 401, 354 (1999).

[219] A. Abanov, A. V. Chubukov, M. Eschrig, M. R. Norman, and J. Schmalian, Neutron Resonance inthe Cuprates and its Effect on Fermionic Excitations, Phys. Rev. Lett. 89, 177002 (2002).

[220] M. Fujita, K. Yamada, H. Hiraka, P. M. Gehring, S. H. Lee, S. Wakimoto, and G. Shirane, Staticmagnetic correlations near the insulating-superconducting phase boundary in La2−xSrxCuO4,Phys. Rev. B 65, 064505 (2002).

[221] J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada,Quantum magnetic excitations from stripes in copper oxide superconductors, Nature 429, 534(2004).



[222] M. Vojta, Tendencies toward nematic order in YBa2Cu3O6+δ distortion vs. incipient chargestripes, Eur. Phys. J. Special Topics 188, 49 (2010).

[223] L. Benfatto, S. Caprara, and C. D. Castro, Gap and pseudogap evolution within the charge-ordering scenario for superconducting cuprates, Eur. Phys. J. B 17, 95 (2000).

[224] J. Zaanen and O. Gunnarsson, Charged magnetic domain lines and the magnetism of high-Tc

oxides, Phys. Rev. B 40, 7391 (1989).

[225] K. Machida, Magnetism in La2CuO4 based compounds, Physica C 158, 192 (1989).

[226] V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phase separation in the t-J model, Phys. Rev. Lett. 64,475 (1990).

[227] E. Demler, W. Hanke, and S.-C. Zhang, SO(5) theory of antiferromagnetism and superconductiv-ity, Rev. Mod. Phys. 76, 909 (2004).

[228] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P. Mackenzie, Nematic FermiFluids in Condensed Matter Physics, Annu. Rev. Cond. Mat. Phys. 1, 153 (2010).

[229] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, ADDRESS, 1999).

[230] P. W. Anderson, The Resonating Valence Bond State in La2CuO4 and Superconductivity, Science235, 1196 (1987).

[231] M. Fujita, H. Goka, K. Yamada, J. M. Tranquada, and L. P. Regnault, Stripe order, depinning,and fluctuations in La1.875Ba0.125CuO4 and La1.875Ba0.075Sr0.050CuO4 , Phys. Rev. B 70, 104517(2004).

[232] S. V. Borisenko, A. A. Kordyuk, A. N. Yaresko, V. B. Zabolotnyy, D. S. Inosov, R. Schuster, B.Buchner, R. Weber, R. Follath, L. Patthey, and H. Berger, Pseudogap and Charge Density Wavesin Two Dimensions, Phys. Rev. Lett. 100, 196402 (2008).

[233] V. Brouet, W. L. Yang, X. J. Zhou, Z. Hussain, R. G. Moore, R. He, D. H. Lu, Z. X. Shen, J.Laverock, S. B. Dugdale, N. Ru, and I. R. Fisher, Angle-resolved photoemission study of theevolution of band structure and charge density wave properties in RTe3 ( R = Y , La, Ce, Sm, Gd,Tb, and Dy), Phys. Rev. B 77, 235104 (2008).

[234] P. Trey, S. Gygax, and J. P. Jan, Anisotropy of the Ginzburg-Landau parameter κ in NbSe2, J. LowTemp. Phys. 11, 421 (1973).

[235] W. Metzner, D. Rohe, and S. Andergassen, Soft Fermi Surfaces and Breakdown of Fermi-LiquidBehavior, Phys. Rev. Lett. 91, 066402 (2003).

[236] Q. Li, M. Hucker, G. D. Gu, A. M. Tsvelik, and J. M. Tranquada, Two-Dimensional Supercon-ducting Fluctuations in Stripe-Ordered La1.875Ba0.125CuO4, Phys. Rev. Lett. 99, 067001 (2007).

[237] M. Hucker, M. v. Zimmermann, M. Debessai, J. S. Schilling, J. M. Tranquada, and G. D. Gu,Spontaneous Symmetry Breaking by Charge Stripes in the High Pressure Phase of Superconduct-ing La1.875Ba0.125CuO4, Phys. Rev. Lett. 104, 057004 (2010).

[238] A. Dubroka, L. Yu, D. Munzar, K. Kim, M. Rossle, V. Malik, C. Lin, B. Keimer, T. Wolf, andC. Bernhard, Pseudogap and precursor superconductivity in underdoped cuprate high temperaturesuperconductors: A far-infrared ellipsometry study, Eur. Phys. J. Special Topics 188, 73 (2010).

[239] N. Toyota, M. Lang, and J. Muller, Low-Dimensional Molecular Metals (Springer, Berlin-Heidelberg, 2007).

2013


[240] R. Nemetschek, M. Opel, C. Hoffmann, P. F. Muller, R. Hackl, H. Berger, L. Forro, A. Erb, andE. Walker, Pseudogap and Superconducting Gap in the Electronic Raman Spectra of UnderdopedCuprates, Phys. Rev. Lett. 78, 4837 (1997).

[241] J. L. Smith, J. S. Brooks, C. M. Fowler, B. L. Freeman, J. D. Goettee, W. L. Hults, J. C. King, P. M.Mankiewich, E. I. Obaldia, M. L. O’Malley, D. G. Rickel, and W. J. Skocpol, Low-temperaturecritical field of YBCO, J. Supercond. 7, 269 (1994).

[242] B. Roas, L. Schultz, and G. Saemann-Ischenko, Anisotropy of the critical current density in epi-taxial Y Ba2Cu3Ox films, Phys. Rev. Lett. 64, 479 (1990).

[243] P. Anderson, Theory of dirty superconductors, J. Phys. and Chem. Solids 11, 26 (1959).

[244] R. Hackl, W. Glaser, P. Muller, D. Einzel, and K. Andres, Light-scattering study of the supercon-ducting energy gap in YBa2Cu3O7 single crystals, Phys. Rev. B 38, 7133 (1988).

[245] S. L. Cooper, F. Slakey, M. V. Klein, J. P. Rice, E. D. Bukowski, and D. M. Ginsberg, Gapanisotropy and phonon self-energy effects in single-crystal YBa2Cu3O−7−δ , Phys. Rev. B 38,11934 (1988).

[246] A. Yamanaka, T. Kimura, F. Minami, K. Inoue, and S. Takekawa, Superconducting Gap Excita-tions in Bi-Sr-Ca-Cu-O Superconductor Observed by Raman Scattering, Jpn. J. Appl. Phys. 27,L1902 (1988).

[247] H. Monien and A. Zawadowski, Theory of interband electron Raman scattering in YBa2Cu3O7:A probe of unconventional superconductivity, Phys. Rev. Lett. 63, 911 (1989).

[248] Z.-X. Shen, D. S. Dessau, B. O. Wells, D. M. King, W. E. Spicer, A. J. Arko, D. Marshall,L. W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, T. Loeser, and C. H. Park,Anomalously large gap anisotropy in the a-b plane of B2Sr2CaCu2O8+δ , Phys. Rev. Lett. 70, 1553(1993).

[249] H. Ding, M. R. Norman, J. C. Campuzano, M. Randeria, A. F. Bellman, T. Yokoya, T. Taka-hashi, T. Mochiku, and K. Kadowaki, Angle-resolved photoemission spectroscopy study of thesuperconducting gap anisotropy in Bi2Sr2CaCu2O8+x, Phys. Rev. B 54, R9678 (1996).

[250] T. Cuk, F. Baumberger, D. H. Lu, N. Ingle, X. J. Zhou, H. Eisaki, N. Kaneko, Z. Hussain, T. P.Devereaux, N. Nagaosa, and Z.-X. Shen, Coupling of the B1g Phonon to the Antinodal ElectronicStates of Bi2Sr2Ca0.92Y0.08Cu2O8+δ , Phys. Rev. Lett. 93, 117003 (2004).

[251] M. V. Klein and S. B. Dierker, Theory of Raman scattering in superconductors, Phys. Rev. B 29,4976 (1984).

[252] C. Kendziora and A. Rosenberg, a-b plane anisotropy of the superconducting gap inBi2Sr2CaCu2O8+δ , Phys. Rev. B 52, 9867 (1995).

[253] L. V. Gasparov, P. Lemmens, N. N. Kolesnikov, and G. Guntherodt, Electronic Raman scatteringin Tl2Ba2CuO6+δ Symmetry of the order parameter, oxygen doping effects, and normal-statescattering, Phys. Rev. B 58, 11753 (1998).

[254] M. Opel, R. Nemetschek, C. Hoffmann, R. Philipp, P. F. Muller, R. Hackl, I. Tutto, A. Erb, B.Revaz, E. Walker, H. Berger, and L. Forro, Carrier relaxation, pseudogap, and superconductinggap in high-Tc cuprates: A Raman scattering study, Phys. Rev. B 61, 9752 (2000).

[255] S. Sugai and T. Hosokawa, Relation between the Superconducting Gap Energy and the Two-Magnon Raman Peak Energy in Bi2Sr2Ca1−xYxCu2O8+δ , Phys. Rev. Lett. 85, 1112 (2000).



[256] F. Venturini, M. Opel, R. Hackl, H. Berger, L. Forr, and B. Revaz, Doping dependence of theelectronic Raman spectra in cuprates, J. Phys. and Chem. Solids 63, 2345 (2002).

[257] C. Panagopoulos and T. Xiang, Relationship between the Superconducting Energy Gap and theCritical Temperature in High- Tc Superconductors, Phys. Rev. Lett. 81, 2336 (1998).

[258] W. S. Lee, I. M. Vishik, K. Tanaka, D. H. Lu, T. Sasagawa, N. Nagaosa, T. P. Devereaux, Z.Hussain, and Z.-X. Shen, Abrupt onset of a second energy gap at the superconducting transitionof underdoped Bi2212, Nature 450, 81 (2007).

[259] B. Stadlober, G. Krug, R. Nemetschek, R. Hackl, J. L. Cobb, and J. T. Markert, Is Nd 2−xCe xCuO4a High-Temperature Superconductor?, Phys. Rev. Lett. 74, 4911 (1995).

[260] G. Blumberg, A. Koitzsch, A. Gozar, B. S. Dennis, C. A. Kendziora, P. Fournier, and R. L. Greene,Nonmonotonic dx2−y2 Superconducting Order Parameter in Nd2−xCexCuO4, Phys. Rev. Lett. 88,107002 (2002).

[261] B. Chesca, K. Ehrhardt, M. Moßle, R. Straub, D. Koelle, R. Kleiner, and A. Tsukada, Magnetic-Field Dependence of the Maximum Supercurrent of La2−xCexCuO4−y Interferometers: Evidencefor a Predominant dx2−y2 Superconducting Order Parameter, Phys. Rev. Lett. 90, 057004 (2003).

[262] M. M. Qazilbash, A. Koitzsch, B. S. Dennis, A. Gozar, H. Balci, C. A. Kendziora, R. L. Greene,and G. Blumberg, Evolution of superconductivity in electron-doped cuprates: Magneto-Ramanspectroscopy, Phys. Rev. B 72, 214510 (2005).

[263] J. Orenstein and A. J. Millis, Advances in the Physics of High-Temperature Superconductivity,Science 88, 468 (2000).

[264] A. Kaminski, H. M. Fretwell, M. R. Norman, M. Randeria, S. Rosenkranz, U. Chatterjee, J. C.Campuzano, J. Mesot, T. Sato, T. Takahashi, T. Terashima, M. Takano, K. Kadowaki, Z. Z. Li,and H. Raffy, Momentum anisotropy of the scattering rate in cuprate superconductors, Phys. Rev.B 71, 014517 (2005).

[265] M. R. Norman, A. Kanigel, M. Randeria, U. Chatterjee, and J. C. Campuzano, Modeling the Fermiarc in underdoped cuprates, Phys. Rev. B 76, 174501 (2007).

[266] Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer, J. F. Carolan, W. N. Hardy, R. Kadono,J. R. Kempton, R. F. Kiefl, S. R. Kreitzman, P. Mulhern, T. M. Riseman, D. L. Williams, B. X.Yang, S. Uchida, H. Takagi, J. Gopalakrishnan, A. W. Sleight, M. A. Subramanian, C. L. Chien,M. Z. Cieplak, G. Xiao, V. Y. Lee, B. W. Statt, C. E. Stronach, W. J. Kossler, and X. H. Yu, Uni-versal Correlations between Tc and ns

m∗ (Carrier Density over Effective Mass) in High-Tc CuprateSuperconductors, Phys. Rev. Lett. 62, 2317 (1989).

[267] D. L. Feng, D. H. Lu, K. M. Shen, C. Kim, H. Eisaki, A. Damascelli, R. Yoshizaki, J.-i. Shi-moyama, K. Kishio, G. D. Gu, S. Oh, A. Andrus, J. O’Donnell, J. N. Eckstein, and Z.-X. Shen,Signature of Superfluid Density in the Single-Particle Excitation Spectrum of Bi2Sr2CaCu2O8+,Science 289, 277 (2000).

[268] K. McElroy, D.-H. Lee, J. E. Hoffman, K. M. Lang, J. Lee, E. W. Hudson, H. Eisaki, S. Uchida,and J. C. Davis, Coincidence of Checkerboard Charge Order and Antinodal State Decoherence inStrongly Underdoped Superconducting Bi2Sr2CaCu2O8+δ , Phys. Rev. Lett. 94, 197005 (2005).

[269] R. Zeyher and A. Greco, Influence of Collective Effects and the d Charge-Density Wave on Elec-tronic Raman Scattering in High-Tc Superconductors, Phys. Rev. Lett. 89, 177004 (2002).

2013


[270] A. N. Pasupathy, A. Pushp, K. K. Gomes, C. V. Parker, J. Wen, Z. Xu, G. Gu, S. Ono, Y. Ando,and A. Yazdani, Electronic Origin of the Inhomogeneous Pairing Interaction in the High-Tc Su-perconductor Bi2Sr2CaCu2O8+, Science 320, 196 (2008).

[271] A. Goncharov and V. Struzhkin, Raman spectroscopy of metals, high-temperature superconduc-tors and related materials under high pressure, J. Raman Spectrosc. 34, 532 (2003).

[272] S. Pailhes, C. Ulrich, B. Fauque, V. Hinkov, Y. Sidis, A. Ivanov, C. T. Lin, B. Keimer, and P.Bourges, Doping Dependence of Bilayer Resonant Spin Excitations in (Y,Ca)Ba2Cu3O6+x, Phys.Rev. Lett. 96, 257001 (2006).

[273] L. N. Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Phys. Rev. 104, 1189 (1956).

[274] D. J. Scalapino, in Superconductivity - An Introduction To Fluid, Heat And Mass TransportProcesses, edited by R. Parks (Marcel Dekker, ADDRESS, 1969), p. 1456, iSBN-13: 978-0824705961.

[275] W. L. McMillan and J. M. Rowell, Lead Phonon Spectrum Calculated from SuperconductingDensity of States, Phys. Rev. Lett. 14, 108 (1965).

[276] W. Weber, The phonons in high Tc A15 compounds, Physica B 126, 217 (1984).

[277] J. P. Carbotte, Properties of boson-exchange superconductors, Rev. Mod. Phys. 62, 1027 (1990).

[278] J. Carbotte and F. Marsiglio, in Handbook of High-Temperature Superconductivity Theory andExperiment, edited by J. Schrieffer and J. Brooks (Springer, ADDRESS, 2007).

[279] A. Migdal, , Sov. Phys. JETP 7, 996 (1958).

[280] S. Lupi, P. Maselli, M. Capizzi, P. Calvani, P. Giura, and P. Roy, Evolution of a Polaron Bandthrough the Phase Diagram of Nd2−xCexCuO4−y, Phys. Rev. Lett. 83, 4852 (1999).

[281] P. Calvani, Optical properties of polarons, La Rivista del Nuovo Cimento 24, 1 (2001).

[282] B. Renker, F. Gompf, E. Gering, N. Ncker, D. Ewert, W. Reichardt, and H. Rietschel, Phonondensity-of-states for the high-Tc superconductor La1.85Sr0.15CuO4 and its non-superconductingreference La2CuO4, Z. Phys. B: Condens. Matter 67, 15 (1987).

[283] B. Renker, F. Gompf, E. Gering, G. Roth, W. Reichardt, D. Ewert, H. Rietschel, and H. Mutka,Phonon density-of-states for high-Tc (Y,RE)Ba2Cu3O7 superconductors and non-superconductingreference systems, Z. Phys. B: Condens. Matter 71, 437 (1988).

[284] R. Khasanov, A. Shengelaya, K. Conder, E. Morenzoni, I. Savic, and H. Keller, The oxygen-isotope effect on the in-plane penetration depth in underdoped Y1−xPrxBa2Cu3O7−δ as revealedby muon-spin rotation, J. Phys. Condens. Matter 15, L17 (2003).

[285] B. Friedl, C. Thomsen, and M. Cardona, Determination of the superconducting gap inRBa2Cu3O7−δ , Phys. Rev. Lett. 65, 915 (1990).

[286] A. Pashkin, M. Porer, M. Beyer, K. W. Kim, A. Dubroka, C. Bernhard, X. Yao, Y. Dagan, R. Hackl,A. Erb, J. Demsar, R. Huber, and A. Leitenstorfer, Femtosecond Response of Quasiparticles andPhonons in Superconducting Y Ba2Cu3O7−δ Studied by Wideband Terahertz Spectroscopy, Phys.Rev. Lett. 105, 067001 (2010).

[287] K. Miyake, S. Schmitt-Rink, and C. M. Varma, Spin-fluctuation-mediated even-parity pairing inheavy-fermion superconductors, Phys. Rev. B 34, 6554 (1986).



[288] D. J. Scalapino, E. Loh, and J. E. Hirsch, d-wave pairing near a spin-density-wave instability,Phys. Rev. B 34, 8190 (1986).

[289] A. Kampf and J. R. Schrieffer, Pseudogaps and the spin-bag approach to high-Tc superconductiv-ity, Phys. Rev. B 41, 6399 (1990).

[290] P. Monthoux and D. Pines, Y Ba2Cu3O7: A nearly antiferromagnetic Fermi liquid, Phys. Rev. B47, 6069 (1993).

[291] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperaturesuperconductivity, Rev. Mod. Phys. 78, 17 (2006).

[292] P. W. Anderson, Is There Glue in Cuprate Superconductors?, Science 316, 1705 (2007).

[293] S. Caprara, C. Di Castro, M. Grilli, and D. Suppa, Charge-Fluctuation Contribution to the RamanResponse in Superconducting Cuprates, Phys. Rev. Lett. 95, 117004 (2005).

[294] S. V. Dordevic, C. C. Homes, J. J. Tu, T. Valla, M. Strongin, P. D. Johnson, G. D. Gu, and D. N.Basov, Extracting the electron-boson spectral function α2F(ω) from infrared and photoemissiondata using inverse theory, Phys. Rev. B 71, 104529 (2005).

[295] J. Hwang, T. Timusk, E. Schachinger, and J. P. Carbotte, Evolution of the bosonic spectral densityof the high-temperature superconductor Bi2Sr2CaCu2O8+δ , Phys. Rev. B 75, 144508 (2007).

[296] E. van Heumen, E. Muhlethaler, A. B. Kuzmenko, H. Eisaki, W. Meevasana, M. Greven, and D.van der Marel, Optical determination of the relation between the electron-boson coupling functionand the critical temperature in high- Tc cuprates, Phys. Rev. B 79, 184512 (2009).

[297] D. Poilblanc and D. J. Scalapino, Gap function φ(k,ω) for a two-leg t− J ladder and the pairinginteraction, Phys. Rev. B 71, 174403 (2005).

[298] T. A. Maier, D. Poilblanc, and D. J. Scalapino, Dynamics of the Pairing Interaction in the Hubbardand t− J Models of High-Temperature Superconductors, Phys. Rev. Lett. 100, 237001 (2008).

[299] P. Prelovsek and A. Ramsak, Spin-fluctuation mechanism of superconductivity in cuprates, Phys.Rev. B 72, 012510 (2005).

[300] N. M. Plakida, Theory of antiferromagnetic pairing in cuprate superconductors (Review article),Low Temp. Phys. 32, 363 (2006).

[301] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen, Band-Structure Trendin Hole-Doped Cuprates and Correlation with Tcmax, Phys. Rev. Lett. 87, 047003 (2001).

[302] P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N. Trivedi, and F. C. Zhang, The physicsbehind high-temperature superconducting cuprates: the ’plain vanilla’ version of RVB, J. Phys.Condens. Matter 16, R755 (2004).

[303] P. Monthoux, D. Pines, and G. G. Lonzarich, Superconductivity without phonons, Nature 450,1177 (2007).

[304] G. B. Yntema, Minutes of the 1955 Annual Meeting Held at New York City, January 27-29, p.1197, Phys. Rev. 98, 1144 (1955).

[305] J. E. Kunzler, E. Buehler, F. S. L. Hsu, and J. H. Wernick, Superconductivity in Nb3Sn at HighCurrent Density in a Magnetic Field of 88 kgauss, Phys. Rev. Lett. 6, 89 (1961).

[306] B. BioSpin, http://www.bruker-biospin.com/.

2013


[307] H. Hilgenkamp and J. Mannhart, Grain boundaries in high-Tc superconductors, Rev. Mod. Phys.74, 485 (2002).

[308] D. Koelle, R. Kleiner, F. Ludwig, E. Dantsker, and J. Clarke, High-transition-temperature super-conducting quantum interference devices, Rev. Mod. Phys. 71, 631 (1999).

[309] A. P. Malozemoff, J. Mannhart, and D. Scalapino, High-Temperature Cuprate SuperconductorsGet to Work, Physics Today 58, 41 (2005).

[310] J. Bray, Superconductors in Applications; Some Practical Aspects, Appl. Supercond. 19, 2533(2009).

[311] Sumitomo Electric Industries Ltd., http://global-sei.com/super/index.en.html.

[312] Theva, http://www.theva.de/.

[313] American Superconductor, http://www.amsc.com/.

[314] Zenergy Power, http://www.zenergypower.com/.

[315] Superconductor Technologies Inc., http://www.suptech.com/home.htm.

[316] Siemens, http://siemens.de.

[317] D. Drung, F. Ludwig, W. Muller, U. Steinhoff, L. Trahms, H. Koch, Y. Q. Shen, M. B. Jensen, P.Vase, T. Holst, T. Freltoft, and G. Curio, Integrated YBa2Cu3O7−x magnetometer for biomagneticmeasurements, Appl. Phys. Lett. 68, 1421 (1996).

[318] I. Tristan Technologies, http://www.tristantech.com/.


i/sc_20… · contents 1 introduction 1 1.1 a brief history of low-temperature physics . . . . . ....

Documents