Unconventional Superconductors Experimental Investigation of the Order Parameter Symmetry

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Author: Gernot Goll

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Springer Tracts in Modern Physics Volume 214 Managing Editor: G. Höhler, Karlsruhe Editors: C. Varma, California F. Steiner, Ulm J. Kühn, Karlsruhe J. Trümper, Garching P. Wölfle, Karlsruhe Th. Müller, Karlsruhe

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Gernot Goll

Unconventional Superconductors Experimental Investigation of the Order-Parameter Symmetry

With 67 Figures

ABC

Gernot Goll Universität Karlsruhe Physikalisches Institut Wolfgang-Gaede-Str. 1 76128 Karlsruhe Germany E-mail: [email protected]

Library of Congress Control Number: 2005933767 Physics and Astronomy Classification Scheme (PACS): 74.70.Tx, 74.70.Pq, 74.20.Rp, 74.50.+r, 74.25.Bt, 74.25.Nf ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN-10 3-540-28985-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28985-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Cover concept: eStudio Calamar Steinen Cover production: design &production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11010715

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O Lord, how manifold are your works! In wisdom have you made them all; the earth is full of your creatures. (Psalm 104, 24)

To Claudia, Anne and Julia

S - a single word that still causes excitement, more than 90 years after its discovery

S - an amazing phenomenon of lossfree transport and levitation in a magnetic field

S - a challenging field of scientific activities between fundamental condensed-matter research and industrial applications

S - fascinating both experimentalists and theorists

S - sometimes unconventional

S

S

Preface

In the past decades a growing number of metals exhibited evidence for exotic types of superconductivity. The superconductivity is often called unconventional in order to classify superconductors with respect to the order-parameter symmetry. Many intriguing experiments have been carried out to elucidate the order-parameter symmetry. The characterization of the order parameter of superconducting materials needs information on (i) the pairing mechanism, (ii) the parity and spin state, (iii) the size and the nodal structure of the energy gap, and (iv) the macroscopic superconductive phase. Each property can be accessed by diﬀerent experiments. For example, the nodal structure can be investigated by measurements of the temperature dependence of thermodynamic and transport properties, by angular-resolved thermalconductivity measurements, and by the directional dependence of current–voltage characteristics in point-contact and tunnelling measurements, though the latter experiments have been mainly applied for the determination of gap size and the density of states around EF . On the other hand, the observation of a power law in the lowtemperature behaviour of one property, for example of the penetration depth, is not suﬃcient for the characterization of an unconventional superconductor and must be treated with some caution since there might be other, quite diﬀerent and rather conventional explanations of the data. Therefore, it is important to have corroborating evidence from a number of diﬀerent experiments before a conclusive picture of the order-parameter symmetry is obtained. As none of the experimental methods alone can definitely reveal the order-parameter symmetry, manifold experiments will be discussed, each of which accesses a certain aspect. In particular, some emphasis will be put on the investigation of the energy gap of unconventional superconductors by point-contact spectroscopy. The point-contact and tunnel spectra yield information on the size and the anisotropy of the superconductive energy gap, its temperature, and its field dependence. This book is subdivided into three parts. In part I, a brief overview on possible unconventional superconductors is followed by a short theoretical introduction of the main concepts of unconventional superconductivity combined with a general definition of often-used terms and notations. Part II presents several experimental methods that allow a characterization of the order-parameter symmetry. This include measurements of specific heat, thermal conductivity, penetration depth, and ultrasound attenuation for information on the nodal structure (Chap. 3), nuclear

X

Preface

magnetic resonance and muon spin rotation experiments for information on the parity and spin state (Chap. 4), point-contact and tunnelling spectroscopy for information about the energy gap (Chap. 5), and phase-sensitive probes, namely the Josephson eﬀect (Chap. 6). This part is concluded with an overview of experimental methods that probe the eﬀects of an unconventional order parameter on single vortices and the symmetry of the flux-line lattice (Chap. 7). Part III reviews three classes of unconventional superconductors, namely the Ce-based heavy-fermion superconductors (Chap. 8), the U-based heavy-fermion superconductors (Chap. 9), and the metal-oxide superconductors (Chap. 10), especially Sr2 RuO4 . This book is not intended to review the high-temperature superconductors. There exist a vast number of excellent review articles and monographs that give a quite complete overview of the state-of-the-art research on these materials. Nevertheless, the high-temperature superconductors will be mentioned, as the huge amount of research on these compounds has pushed the whole field, both the theoretical and experimental activities, and has shed light on the understanding of many other topics related to unconventional superconductivity. Finally, I thank all people who contributed to this work in many respects during the past years. First of all, I am grateful to Prof. H. v. L¨ohneysen for continuous support, encouragement, and many stimulating discussions. It is a pleasure to thank Prof. I. K. Yanson for initiating the interest in the experimental technique of pointcontact spectroscopy, and Dr. Y. Naidyuk for helpful discussions and correspondence. I thank Prof. L. Taillefer, Prof. Y. Maeno, Dr. Z. Q. Mao, Dr. F. Lichtenberg, Dr. V. Zapf, and Dr. E. Bauer for providing the samples necessary for these studies and for useful discussions. I would also like to acknowledge Prof. M. B. Maple, Prof. E. Dormann, Prof. C. Bruder, Prof. P. W¨olfle, Prof. J. Wosnitza, Dr. M. Eschrig, Dr. M. Fogelstr¨om, Dr. R. Werner, and Dr. O. Stockert for many fruitful and illuminating discussions. I thank Dr. C. Obermair, Dr. F. Laube, T. Brugger, S. Kontermann, and M. Marz for their contributions to the point-contact data reviewed here and all my colleagues for good collaboration. I thank L. Behrens for drawing, scanning, and editing many of the figures. Last, but not least, I thank my wife Claudia and my daughters Anne and Julia for their patience and sympathy during my weekend writing of this manuscript. This work was partly supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 195 “Lokalisierung von Elektronen in makroskopischen und mikroskopischen Systemen” and Graduiertenkolleg “Anwendungen der Supraleitung.”

Karlsruhe, October 2005

Gernot Goll

Contents

Part I Introduction to Unconventional Superconductivity 1

A Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 7

2

Basic Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Part II Experimental Methods 3

Probing the Nodal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Specific Heat and Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . 3.2 Magnetic Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Ultrasound Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 23 24 26

4

Probing the Parity and Spin State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hc2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nuclear Magnetic Resonance and Knight Shift . . . . . . . . . . . . . . . . . . 4.3 Muon Spin Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 30 32

5

Probing the Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Planar Tunnelling Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Scanning Tunnelling Spectroscopy (STS) . . . . . . . . . . . . . . . . . . . . . . 5.3 Point-Contact Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 37 39 53

6

Probing the Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Josephson Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Zero-Bias Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 59 59

XII

7

Contents

Probing the Vortices: Lattice Symmetry and Internal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Scanning Tunnelling Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Small-Angle Neutron Scattering (SANS) . . . . . . . . . . . . . . . . . . . . . . . 7.3 Muon Spin Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 62 63 63

Part III Possible Unconventional Superconductors 8

Ce-Based Heavy-Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . 8.1 CeCu2 Si2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 CeCoIn5 and CeIrIn5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 CePt3 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 72 81 84

9

U-Based Heavy-Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . 89 9.1 UPt3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9.2 UBe13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.3 URu2 Si2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.4 UNi2 Al3 and UPd2 Al3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10

Metal-Oxide Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.1 Sr2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.2 High-Temperature Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

A

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Part I

Introduction to Unconventional Superconductivity

1 A Brief Overview

The phenomenon of superconductivity is still one of the most exciting topics in solid state physics. In the history of superconductivity it was always a challenge for both experimentalists and theorists to reveal the nature of superconductivity. After the discovery of the ﬁrst superconductor by Kamerlingh-Onnes in 1911 [1] it took almost half a century until a pairing theory was developed which was able to account for many properties of superconductors known at that time. This famous pairing theory, formulated in 1957 by Bardeen, Cooper and Schrieﬀer (BCS) [2] points out that even a weak attraction between fermions will lead to simultaneous formation and condensation of so-called Cooper pairs. The Cooper pairs are bound electron pairs of opposite spin and momentum proposed in 1956 by Cooper [3]. In classical superconductors, ﬁrst considered by the BCS theory, the electron pairs have a total spin S = 0 (singlet pairing) and total orbital momentum L = 0. In analogy to quantum mechanics, the pairing is described by an isotropic wave function, and denoted as s-wave pairing. The weak attractive interaction arises from the virtual exchange of phonons and the screened Coulomb repulsion between electrons. A macroscopic quantum state is formed by the bosonic condensate which breaks the gauge invariance by setting a macroscopic phase relationship. S -wave pairing mediated by the electron-phonon interaction is the key ingredient of conventional superconductors like most element and alloy superconductors. In contrast, superconductivity is denoted unconventional, if below the transition temperature T c additional symmetries are broken besides the gauge symmetry, or if the pairing mechanism is mediated by non-phononic interactions. A generalized BCS theory [4, 5] can account for unconventional superconductivity, a fact that emphasizes the brilliance, universality and signiﬁcance of the concepts of the BCS pairing theory. A powerful example for the success of the generalized pairing theory is liquid 3 He, a Fermi system which was proposed as a candidate for a BCS transition as well. However, because of the hard-core nature of the interaction potential between 3 He atoms, the Cooper pairs should form a state of nonzero angular momentum. The basic properties of this superﬂuid system were calculated from generalized BCS theory [4, 5], actually before the superﬂuid phases of 3 He were ﬁnally discovered. Today, superﬂuid 3 He is a well-studied model system for unconventional superﬂuidity, where p-wave pairing with total spin S = 1 and total angular momentum L = 1 is mediated by spin ﬂuctuations [6]. The search for anomalous, i. e. non-s-wavetype superconductivity in solids, has been a stimulus for an abundance of research G. Goll: Unconventional Superconductors STMP 214, 3–9 (2006) c Springer-Verlag Berlin Heidelberg 2006

4

1 A Brief Overview

activities. Several classes of materials have been discussed since the late seventies of the last century as possible candidates for unconventional superconductivity, with the heavy-fermion superconductors and the high-T c cuprate superconductors as the most prominent and promising members. A straightforward application of the theory developed for superﬂuid 3 He to unconventional solid-state superconductors is possible, but meets serious limitations, as pointed out by W¨olﬂe [7] in a recent review on the similarities and diﬀerences in both systems. These limitations arise mainly from the reduced symmetry of electrons in a solid and from the existence of imperfections even in cleanest samples. The heavy-fermion superconductors are the class of solids, where hints at an unconventional nature of the superconductivity appeared at ﬁrst. Heavy-fermion compounds are mostly metals containing often rare-earth or actinides ions like cerium and uranium, where the f electrons of the incomplete inner f shell can couple to the conduction electrons. The electronic properties of these compounds can simply be described in a Fermi-liquid picture of “heavy fermions”, i. e. quasiparticle of immensely enhanced eﬀective masses of 100-1000 free-electron masses. Superconductivity in these systems arises from the pair formation of these heavy quasiparticles. CeCu2 Si2 [8] was the ﬁrst example, later several U-based compounds like UBe13 [9], UPt3 [10], URu2 Si2 [11], UPd2 Al3 [12], and UNi2 Al3 [13] were discovered with unusual superconducting properties and a coexistence with antiferromagnetic order. For a long time CeCu2 Si2 has been the only Ce-based heavy-fermion superconductor at ambient pressure, until in 2001 CeCoIn5 [14], and CeIrIn5 [15], and in 2003 the noncentrosymmetric CePt3 Si [16] have been synthesized. Most of the other Ce-based heavy-fermion superconductors discovered during the past few years become superconducting only under high pressure. CeRh2 Si2 [17], CePd2 Si2 [18], CeCu2 Ge2 [19], CeIn3 [20], and CeRhIn5 [21] can be tuned by pressure to show superconductivity close to the destruction of a magnetically ordered phase. In most of these “ancient” heavy-fermion superconductors (with the exception of CeCu2 Si2 and UBe13 , see below), superconductivity occurs out of a normal state which is well described by Landau’s Fermi liquid theory [22]. Within Fermi liquid theory it is assumed that the low-lying excitations of a strongly interacting Fermi system are fermionic quasiparticles which are assumed to be in one-to-one correspondence to the excitations of the non-interacting Fermi gas. As a consequence, the electronic heat-capacity coeﬃcient γ = C/T and the Pauli susceptibility χ are both temperature independent at low temperature and the resistivity follows a T 2 dependence. However, in UBe13 and some of the Ce-based superconductors superconductivity occurs in the vicinity of a quantum-critical point marking a phase transition at T = 0. Quantum criticality alters the interactions of the Fermi system, which leads to a breakdown of the Fermi-liquid behaviour, but superconductivity still appears out of this non-Fermi-liquid regime. The relationship between unconventional superconductivity and quantum criticality is emerging as an important issue in strongly correlated materials. A number of experiments point to the existence of a non-Fermi-liquid metallic state in some of these compounds. In CeCoIn5 , and CeIrIn5 , the key parameters of a Fermi liquid, namely the electronic heat-capacity

1 A Brief Overview

5

coeﬃcient γ = C/T and the Pauli susceptibility χ increase on cooling and show no sign of entering a temperature independent Fermi-liquid regime [23]. In addition, the resistivity ρ(T ) is far from the T 2 dependence and superconductivity appears out of this non-Fermi-liquid regime [24]. This behaviour can be understood in terms of magnetic ﬂuctuations near a zero-temperature critical point, where the theory predicts either γ ∼ − ln T or γ = γ0 −AT 1/2 , depending on the dimensionality and nature of the magnetism [25, 26]. The ﬂuctuations introduce new excitations which may as well mediate Cooper pairing that appears in the vicinity of a quantum-critical point. At present it is an open question what happens to such a quantum-critical system when it becomes superconducting. Recently, a new class of superconductors appeared, namely the class of superconducting ferromagnets, which are certainly good candidates for unconventional superconductivity due to the dominance of ferromagnetic interactions of the electrons which naturally favor a spin-parallel coupled superconducting state. Prominent members are the uniaxial ferromagnets UGe2 [27] and URhGe [28]. Both show coexistence of superconductivity and ferromagnetism. Superconductivity occurs deep inside the ferromagnetic state and pressure-dependent experiments on UGe2 suggest that superconductivity is tightly connected with the ferromagnetism. In these systems superconductivity emerges near a ferromagnetic quantum-critical point, i. e. when the ferromagnetic transition temperature is tunsed to T c = 0. At ﬁrst sight, a coexistence of superconductivity and magnetism was long believed to be impossible as they were regarded competing forms of electronic order. Twenty years ago, Fay and Appel [29] considered the possibility of an equal-spin-pairing superconducting state in itinerant ferromagnets, where the pairing is mediated by the exchange of longitudinal spin ﬂuctuations. They already proposed superconductivity in the weak band-ferromagnet ZrZn2 , however, the experimental observation of superconducitivity was hampered by metallurgical problems. Nevertheless, superconductivity was ﬁnally reported for ZrZn2 [30], but it seems to be caused by an impurity phase at the surface [31] and the proof for bulk superconductivity in this compound is still lacking. The high-temperature superconductors are the class of unconventional superconductors which had for sure the highest impact on the development of concepts and the understanding of the whole ﬁeld. The copper-oxide compounds (cuprates) with a perovskite-based layered structure are strongly two-dimensional electronic systems which become conductive under electron or hole doping of the copperoxide planes. The cuprates exhibit a number of intriguing properties. Superconductivity with highest T c (“optimally doped”) appears out of non-Fermi-liquid regime with a linear temperature dependence of the resistivity above T c . The reduction of the density of states in under-doped cuprates well above T c has been interpreted in terms of a pseudogap caused by preformed electron pairs. The material becomes superconducting when the quantum coherence becomes macroscopic and a gap opens at the Fermi energy. Although little is understood about the microscopic pairing mechanism, the order-parameter symmetry has unambiguously assigned to be of d-wave type, at least for the hole-doped cuprate superconductors [32]. Certainly,

6

1 A Brief Overview

the key experiments for the assignment have been the phase-sensitive experiments designed by Tsui and Kirtley which employed high-temperature superconductors in a thin ﬁlm tricrystal [33, 34]. In these experiments a scanning SQUID system has been used to show that there are half-integer ﬂux quanta in superconducting rings formed by three diﬀerently oriented YBa2 Cu3 O7 grains connected by grain boundary Josephson junctions. These results conﬁrm the d x2 −y2 symmetry [35] of the superconductive state. A quite complete review of the research on the hightemperature superconductors is given in a number of excellent review articles and monographs (see e. g. [36, 37, 38, 39, 40, 41]). Another metal-oxide superconductor is Sr2 RuO4 [42] which has the same layered perovskite structure, but behaves otherwise very diﬀerent: Sr2 RuO4 is metallic, displays Fermi-liquid behaviour and becomes superconducting at the rather low transition temperature T c ≈ 1.5 K [43]. Shortly after the discovery it has been suggested that it might form odd-parity (spin-triplet) Cooper pairs in contrast to evenparity pairing in cuprate systems [44, 45]. The basis for this claim was partially the analogy to 3 He and the presence of ferromagnetism in related compounds. The organic superconductors are believed to belong to the class of unconventional superconductors as well. Organic superconductors are the one-dimensional Bechgaard salts (TMTSF)2 X (Tetramethyltetraselenafulvalen) [46], which become superconducting mainly under high pressure only, and the quasi-two-dimensional systems based on the organic donor BEDT-TTF (bis(ethylenedithio)-tetrathiafulvalene or ET for short) and the stoichiometry (ET)2 X [47]. X denotes a monovalent anion. In the Bechgaard salts experiments hint at a triplet state. The strongest hints are provided by the Knight shift which is unchanged below T c with respect to the normal state [48], and a strong Pauli limiting of the upper critical ﬁeld [49, 50]. The quasi-two-dimensional materials show a variety of anomalous superconducting properties as well which in many respects resemble the behaviour seen in other layered superconductors. The proximity of magnetic phases close to the superconducting state as well as antiferromagnetic spin ﬂuctuations have certainly an inﬂuence on the superconducting properties. Although some experiments hint at unconventional superconductivity, there is still no general consensus, as no compelling evidence for this scenario has been given yet. Several short reviews on the experimental situation, which is interpreted either by the usual BCS theory or as an indication for unconventional superconductivity, were published recently [51, 52, 53]. The borocarbide superconductors RNi2 B2 C with R = Y, and the rare-earth elements, exhibit some properties which diﬀer from the standard behaviour of an isotropic superconductor [54]. Therefore, these compounds are sometimes mentioned in the context of unconventional superconductivity. A non-exponential temperature dependence of the speciﬁc heat [55], the absence of a Hebel-Slichter peak below T c in the nuclear relaxation rate [56], and the existence of a square ﬂux-line lattice in connection with a transition to a hexagonal vortex lattice [57, 58, 59] are the main reason for this suggestion. However, such non-standard behaviour is not necessarily inconsistent with s-wave pairing. For example, the square and hexagonal vortex lattices have very similar energies in Ginzburg-Landau theory and the

References

7

underlying tetragonal symmetry of the borocarbides might be amenable to favour a square symmetry at low magnetic ﬁelds. Therefore, it is commonly believed that these compounds are anisotropic superconductors perhaps even with zeros of the energy gap, but there is little evidence that they are unconventional in the sense of the deﬁnition given above.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

H.K. Onnes: Comm. Phys. Lab. Univ. Leiden 120b, 122b, 124c (1911) 3 J. Bardeen, L.N. Cooper, J.R. Schrieﬀer: Phys. Rev. 108, 1175 (1957) 3, 12 L.N. Cooper: Phys. Rev. 104(4), 1189 (1956) 3, 13 P.W. Anderson, P. Morel: Phys. Rev. 123(6), 1911 (1961) 3, 13 R. Balian, N.R. Werthamer: Phys. Rev. 131(4), 1553 (1963) 3, 13 D. Vollhardt, P. W¨olﬂe: The superﬂuid phases of Helium 3 (Taylor & Francis, London, 1990) 3 P. W¨olﬂe: Physica C 317-318, 55 (1999) 4, 75 F. Steglich, J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz, H. Sch¨afer: Phys. Rev. Lett. 43, 1892 (1979) 4, 67 H.R. Ott, H. Rudiger, Z. Fisk, J.L. Smith: Phys. Rev. Lett. 50, 1595 (1983) 4 G.R. Stewart, Z. Fisk, J.O. Willis, J.L. Smith: Phys. Rev. Lett. 52, 679 (1984) 4, 69 W. Schlabitz, J. Baumann, B. Pollit, U. Rauchschwalbe, H.M. Mayer, U. Ahlheim, C.D. Bredl: Z. Phys. B 62, 171 (1986) 4, 154 C. Geibel, C. Schank, S. Thies, H. Kitazawa, C.D. Bredl, A. Bohm, M. Rau, A. Grauel, R. Caspary, R. Helfrich, U. Ahlheim, F. Steglich: Z. Phys. B 84, 1 (1991) 4, 111 C. Geibel, S. Thies, D. Kaczorowski, A. Mehner, A. Grauel, B. Seidel, U. Ahlheim, R. Helfrich, U. Ahlheim, G. Weber, F. Steglich: Z. Phys. B 83, 305 (1991) 4, 111 C. Petrovic, P.G. Pagliuso, M.F. Hundley, R. Movshovich, J.L. Sarrao, J.D. Thompson, Z. Fisk, P. Monthoux: J. Phys.: Condens. Matter 13, L337 (2001) 4, 72, 74, 153 C. Petrovic, R. Movshovich, M. Jaime, P.G. Pagliuso, M.F. Hundley, J.L. Sarrao, Z. Fisk, J.D. Thompson: Europhys. Lett. 53(3), 354 (2001) 4, 72, 73, 74, 153, 154 E. Bauer, G. Hilscher, H. Michor, C. Paul, E.W. Scheidt, A. Gribanov, Y. Seropegin, H. No¨el, M. Sigrist, P. Rogl: Phys. Rev. Lett. 92(2), 027 003 (2004) 4, 81, 82 R. Movshovich, T. Graf, D. Mandrus, J.D. Thompson, J.L. Smith, Z. Fisk: Phys. Rev. B 53(13), 8241 (1996) 4 F.M. Grosche, S.R. Julian, N.D. Mathur, G.G. Lonzarich: Physica B 223-224, 50 (1996) 4 D. Jaccard, K. Behnia, J. Sierro: Phys. Lett. A 163, 475 (1992) 4 N.D. Mathur, F.M. Grosche, S.R. Julian, I.R. Walker, D.M. Freye, R.K.W. Haselwimmer, G.G. Lonzarich: Nature 394, 39 (1998) 4 H. Hegger, C. Petrovic, E.G. Moshopoulou, M.F. Hundley, J.L. Sarrao, Z. Fisk, J.D. Thompson: Phys. Rev. Lett. 84(21), 4986 (2000) 4, 72, 73 L.D. Landau: Sov. Phys. JETP 3(6), 920 (1957) 4 S. Ikeda, H. Shishido, M. Nakashima, R. Settai, D. Aoki, Y. Haga, H. Harima, Y. Aoki, ¯ T. Namiki, H. Sato, Y. Onuki: J. Phys. Soc. Jpn. 70(8), 2248 (2001) 5, 74, 153 V.A. Sidorov, M. Nicklas, P.G. Pagliuso, J.L. Sarrao, Y. Bang, A.V. Balatsky, J.D. Thompson: Phys. Rev. Lett. 89, 157 004 (2002) 5, 74 A.J. Millis: Phys. Rev. B 48, 7183 (1993) 5 T. Moriya, T. Takimoto: J. Phys. Soc. Jpn. 64(3), 960 (1995) 5

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1 A Brief Overview

27. S.S. Saxena, P. Agarwal, K. Ahilan, F.M. Grosche, R.K.W. Haselwimmer, M.J. Steiner, E. Pugh, I.R. Walker, S.R. Julian, P. Monthoux, G.G. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite, J. Flouquet: Nature 406, 587 (2000) 5 28. D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J.P. Brison, E. Lhotel, C. Paulsen: Nature 413, 613 (2001) 5 29. D. Fay, J. Appel: Phys. Rev. B 22(7), 3173 (1980) 5 30. C. Pﬂeiderer, M. Uhlarz, S.M. Hayden, R. Vollmer, H.v. L¨ohneysen, N.R. Bernhoeft, G.G. Lonzarich: Nature 412, 58 (2001). Erratum, ibid 412, 660 (2001) 5 31. E.A. Yelland, S.M. Hayden, S.J.C. Yates, C. Pﬂeiderer, M. Uhlarz, R. Vollmer, H.v. L¨ohneysen, N.R. Bernhoeft, R.P. Smith, S.S. Saxena, N. Kimura; condmat/0502341, submitted to Phys. Rev. B 5 32. C.C. Tsuei, J.R. Kirtley: Rev. Mod. Phys. 72(4), 969 (2000) 5, 136, 145, 146 33. J.R. Kirtley, C.C. Tsuei, J.Z. Sun, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, M. Rupp, M.B. Ketchen: Nature 373, 225 (1995) 6, 58 34. C.C. Tsuei, J.R. Kirtley, M. Rupp, J.Z. Sun, A. Gupta, M.B. Ketchen, C.A. Wang, Z.F. Ren, J.H. Wang, M. Bhushan: Science 271, 329 (1996) 6 35. D.J. Scalapino: Phys. Rep. 250, 329 (1995) 6, 141 36. A.V. Chubukov, D. Pines, J. Schmalian: A Spin Fluctuation Model for d-wave Superconductivity, Vol. 1: Conventional and High-Tc Superconductors (Springer, Berlin, 2003) 6, 140 37. E.W. Carlson, V.J. Emery, S.A. Kivelson, D. Orgad: Concepts in High Temperature Superconductivity, Vol. 1: Conventional and High-Tc Superconductors (Springer, Berlin, 2003) 6, 140 38. M.B. Maple: J. Magn. Magn. Mater. 177, 18 (1998) 6, 140 39. J. Klamut, B.W. Veal, B.M. Dabrowski, P.W. Klamut, M. Kazimierski: Recent Developments in High Temperature Superconductors (Springer, Berlin, 1996) 6, 140 40. N.M. Plakida: High-Temperature Superconductors (Springer, Berlin, 1995) 6, 140 41. C.P. Poole Jr., H.A. Farach, R.J. Creswick: Superconductivity (Academic Press, San Diego, 1995) 6, 140 42. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz, F. Lichtenberg: Nature 372, 532 (1994) 6 43. Y. Maeno: Physica C 282-287, 206 (1997) 6 44. T.M. Rice, M. Sigrist: J. Phys.: Condens. Matter 7(47), L643 (1995) 6, 122, 138 45. G. Basakaran: Physica B 223 & 224, 490 (1996) 6, 123 46. T. Ishiguro, K. Yamaji: Organic superconductors, Vol. 88 of Springer Series in SolidState Science (Springer, Berlin, 1990) 6 47. J.M. Williams, J.R. Ferrao, R.J. Thorn, K.D. Carlson, U. Geiser, H.H. Wang, A.M. Kini, M.H. Wangbo: Organic Superconductors: Synthesis, Structure, Properties, and Theory (Prentice Hall, Englewood Cliﬀs, 1992) 6 48. I.J. Lee, S.E. Brown, W.G. Clark, M.J. Strouse, M.J. Naughton, W. Kang, P.M. Chaikin: Phys. Rev. Lett. 88, 017 004 (2002) 6 49. I.J. Lee, M.J. Naughton, G.M. Danner, P. Chaikin: Phys. Rev. Lett. 78(18), 3555 (1997) 6 50. I.J. Lee, P.M. Chaikin, M.J. Naughton: Phys. Rev. B 62(22), R14 669 (2000) 6 51. J.F. Annett: Physica C 317-318, 1 (1999) 6 52. J. Wosnitza: Physica C 317-318, 98 (1999) 6 53. J. Singleton, C. Mielke: Contemporary Physics 43(2), 63 (2002) 6 54. S.L. Drechsler, S.V. Shulga, K.H. M¨uller, G. Fuchs, J. Freudenberger, G. Behr, H. Eschrig, L. Schultz, M.S. Golden, H. von Lips, J. Fink, V.N. Narozhnyi, H. Rosner, P. Zahn, A. Gladun, D. Lipp, A. Kreyssig, M. Loewenhaupt, K. Koepernik, K. Winzer, K. Krug: Physica C 317-318, 117 (1999) 6

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55. N.M. Hong, H. Michor, M. Vybornov, T. Holubar, P. Hundegger, W. Perthold, G. Hilscher, P. Rogl: Physica C 227 (1–2), 85 (1994) 6 56. Y. Iwamoto, T. Oda, K. Ueda, T. Kohara: Physica B 230-232, 886 (1997) 6 57. Y. Yaron: P.L. Gammel, A.P. Ramirez, D.A. Huse, D.J. Bishop, A.I. Goldman, C. Stassis, P.C. Canﬁeld, K. Movtensen, M.R. Eskildsen Nature 382, 236 (1996) 6 58. M.R. Eskildsen, P.L. Gammel, B.P. Barber, U. Yaron, A.P. Ramirez, D.A. Huse, D.J. Bishop, C. Bolle, C.M. Lieber, S. Oxx, S. Sridhar, N.H. Andersen, K. Mortensen, P.C. Canﬁeld: Phys. Rev. Lett. 78(10), 1968 (1997) 6, 137 59. M. Yethiraj, D.M. Paul, C.V. Tomy, E.M. Forgan: Phys. Rev. Lett. 78(25), 4849 (1997) 6

2 Basic Theoretical Concepts

Superconductivity is the phenomenon of dissipationless transport which occurs in many metals at suﬃciently low temperatures. Metals in the superconducting state reach a new ground state which allows the material to minimize its free energy. Superconductivity is the collective action of conducting electrons in a state of macroscopic quantum coherence formed by the Cooper pairs. The concept can account for many experimental ﬁndings. More generally, superconductivity is the formation of a new ground state of an ensemble of fermions which obey the Pauli exclusion principle. These fermions can be treated as a gas of fermions, the Fermi gas, if they are non- or only weakly-interacting, or as a Fermi liquid, if a strongly interacting Fermi systems is considered where the low-lying excitations are fermionic quasiparticles which are assumed to be in one-to-one correspondence to the excitations of the non-interacting Fermi gas. The theoretical concepts of superconductivity are not necessarily limited to a gas of fermions. The concepts are quite general and applicable for Fermi liquids as well as for non-Fermi-liquids – as indicated by many experiments –, and even other quantum liquids such as nuclear matter and neutron stars [1]. In the following a short introduction into the main concepts of superconductivity, especially unconventional superconductivity, is given, combined with a general deﬁnition of often used terms and notations. For a more detailed and theoretically based introduction the reader should refer textbooks [2, 3] and specialized review articles [4, 5, 6] which have also been used as a source for the following sections. Classiﬁcation of pair-correlated Fermi systems with respect to parity Noninteracting Fermi systems can be characterized by the particle density n, the energy ε k , the group velocity v k and the density of states N(EF ) at the Fermi energy EF . For an isotropic Fermi system these quantities are given by n=

kF3 (2mEF )3/2 = 3π2 3 3π2

(2.1)

εk = ξk + µ

(2.2)

v k = ∇ k ξ k /

(2.3)

3n 2EF

(2.4)

N(EF ) = G. Goll: Unconventional Superconductors STMP 214, 11–18 (2006) c Springer-Verlag Berlin Heidelberg 2006

12

2 Basic Theoretical Concepts

where ξk is the energy of the fermion with respect to the chemical potential µ and limT →0 µ = EF . The microscopic model by Bardeen, Cooper, and Schrieﬀer [7] was formulated only almost 50 years after the discovery of supeconductivity in mercury by Kamerlingh-Onnes. It postulates the formation of so-called Cooper pairs by pairing of a part of the fermions. Pair formation becomes energetically favored at low temperature. Originally, a variational method was used to calculate the condensation energy of the superconducting ground state relative to the normal state. In a more sophisticated modern approach the solution is obtained by canonical transformation. Then the pair formation in the momentum space is described by a pair amplitude g kσ1 σ2 ≡ ˆc−kσ1 cˆ kσ2

(2.5)

which is non-zero only at T < T c . Here, k = (k1 − k2 ) is the relative momentum of the pair, cˆ −kσ1 and cˆ kσ2 are the creation operators of a fermion with spin σ in the quantum state {−k, σ1 } and {k, σ2 }, respectively, and denotes the statistical average. The Pauli principle requires that the pair amplitude g kσ1 σ2 is asymmetric under spin σ1 , σ2 and momentum k1 , k2 exchange: g−kσ2 σ1 = −g kσ1 σ2 ,

(2.6)

which allows a classiﬁcation of the superconductors with respect to the spatial parity and spin rotational symmetry. For singlet pairing (total spin of the pair S = 0) the pair amplitude is given by 0 gk g kσ2 σ1 = = g k (iσ2 )σ1 σ2 (2.7) −g k 0 σ σ 1

2

where g k = − g k↑↓ ] and σ is one of the Pauli matrices1 . The Pauli principle expressed by (2.6) requires for g k even parity with respect to k, i. e. the spatial parity is (2.8) g−k = g k . 1 2 [g k↓↑

2

The k-dependence of g k can be expressed by a sum over spherical functions ˆ with Ylm ( k) ˆ the spherical harmonics with the orbital angug k = lm=−l alm Ylm ( k) lar momentum l and its z-projection m. Of course, this is useful only for isotropic Fermi surfaces. Accordingly, the pairing can be classiﬁed by the orbital quantum number l, which takes for singlet pairing the values 0, 2, 4, . . . , and the pair states are labeled by letters s, d, g, . . . . Consequently, the pairing is referred to as s-wave pairing (l = 0), d-wave pairing (l = 2), and so on. In the case of triplet pairing (S = 1) g k becomes g k↑↑ g k = 12 [g k↓↑ + g k↑↓ ] (2.9) g kσ2 σ1 = g k↓↓ g k = 12 [g k↓↑ + g k↑↓ ] σ1 σ2 = g k (σiσy )σ1 σ2 1

Pauli matrices are deﬁned as follows: 01 0 −i 1 0 σx = , σy = , σz = 10 i 0 0 −1

2 Basic Theoretical Concepts

13

The triplet components g kx = 12 [g k↓↓ − g k↑↑ ], g ky = 12 [g k↑↑ + g k↓↓ ], and g kz = 1 2 [g k↓↑ + g k↑↓ ] of the pair-amplitude vector g k are assigned to the magnetic quantum numbers m s = −1, 0, 1 and have odd parity with respect to k, i. e. g−k = −g k .

(2.10)

For triplet pairing the orbital quantum number l = 1, 3, . . . is odd and the pairing is referred to as p-wave pairing (l = 1), f -wave pairing (l = 3), and so on. Strictly speaking, this classiﬁcation is valid only for weak spin-orbit coupling. In the case of strong spin-orbit coupling, which is usually the case in materials containing chemical elements with large atomic numbers, the electron spin becomes a “bad” quantum number. Nonetheless, the electron states are doubly degenerated due to Kramers degeneracy, which allows a classiﬁcation in terms of “pseudospin” and the attributes “spin-singlet” and “spin-triplet” should be interpreted in terms of “pseudospin-singlet” and “pseudospin-triplet”. (s) . The The formation of Cooper pairs is mediated by an attractive interaction V kp basic idea was presented by Cooper in 1956 [8]. Cooper showed that the Fermi sea of electrons is unstable against the formation of bound pairs, regardless of how weak (s) combines the averaged pair amplithe interaction is, as long as it is attractive. V kp tudes g k and g k , respectively, with a new energy scale, the so-called pair potential: (0) (1) ∆(k) = − V kp g p ; d(k) = − V kp gp (2.11) p

p

(s) | 1) In the weak-coupling approximation the pair attraction is small (|N(EF )V kp and is attractive only close to the Fermi energy. The reason and mechanism for the (s) < 0) may vary. For classical superconductors lattice vibrations pair attraction (V kp (phonons) mediate the pair attraction between the electrons. In a simple picture, one electron interacts with the crystal lattice and perturbs it. The perturbed lattice interacts with another electron in such a way that there is an attraction between the two electrons that can exceed the Coulomb repulsion between them. The electronphonon interaction is not the only possibility for pair attraction. Other interactions could favor anisotropic pairing. Anderson and Morel [9] and Balian and Werthamer [10] investigated this general type of superconductivity with the prospect of superﬂuid 3 He, in which a spin-ﬂuctuation mechanism is assumed to be responsible for the creation of p-wave pairs. There is growing evidence that p-wave pairing is not limited to superﬂuidity only, but that there are superconductors where antiferromagnetic and ferromagnetic spin ﬂuctuations, respectively, give rise to the pair attraction. The scalar and vector pair amplitudes g k , g k , or equivalently the pair potential ∆(k) and d(k), respectively, are called order parameter of the superconducting phase of the pair-correlated Fermi system. The term order parameter originally was introduced in the phenomenological Ginzburg-Landau theory to describe ordering phenomena at a phase transition to an ordered phase where the thermodynamic properties can be derived from an expansion of the free energy in terms of the order parameter.

14

2 Basic Theoretical Concepts

In the context of unconventional superconductivity sometimes the term “nonBCS” is used in order to account for properties which deviate from the standard s-wave behaviour. To be serious, for a classiﬁcation in terms of BCS and non-BCS behaviour, one should remind what happens at a BCS transition. A BCS transition is characterized by the fact that two important things appear simultaneously: 1. formation of bosons by the pairing of fermions and 2. condensation of these bosons into a ground state exhibiting macroscopic phase coherence. That sets superconductivity apart from the Bose-Einstein condensation (BEC). Bose-Einstein condensation is characterized by condensation of already existing bosons into a single-particle ground state [11]. In this sense, a non-BCS behaviour is observed when pair formation and phase coherence do not occur simultaneously for some reason. There are some hints that under-doped cuprate superconductors show non-BCS behaviour. The occurrence of a pseudogap well above T c was interpreted in terms of pair formation without macroscopic phase coherence which sets in below T c [12]. Classiﬁcation with Respect to Symmetry Real Fermi systems are classiﬁed be the symmetry of their pair potential. For this purpose ∆(k) and d(k), respectively, are decomposed in a temperature dependent maximum value ∆0 (T ) and a k-dependent orbital part f (k): ∆(k) = ∆0 (T ) f (k) ;

d(k) = ∆0 (T ) f (k)

(2.12)

Comparing the symmetry of the orbital part to the symmetry of the Fermi surface leads to a classiﬁcation into conventional and unconventional superconductors. Superconductors, where both symmetries are the same, are labelled conventional superconductors, whereas a lower symmetry of the pair potential characterizes an unconventional superconductor. This classiﬁcation is illustrated in Fig. 2.1. In general, a set of functions forms a basis of an irreducible representation of a certain spatial symmetry G if any function of this set is transformed under any symmetry operation of the group to a linear combination of the functions belonging to this set. The various types of Cooper pairing are in an one-to-one correspondence with the irreducible representations of the group of three-dimensional rotations. The classiﬁcation scheme of superconducting states in accordance with the irreducible representations of the symmetry group in the normal state is also valid in crystals where a normal state anisotropy is present. There exist a collection of articles listing the basis functions of diﬀerent irreducible representations Γ of a given point symmetry group G of the solid [3, 4, 5]. The functions g(k) and d(k) are expressed as linear combinations of these basis functions and the set of complex coeﬃcients ηi and ηi in the expansion acts as the order parameter in crystalline superconductors. Thus, superconducting states with either a one-component order parameter η = |η|eiϕ or multicomponent order parameters, e. g. η = (η1 , η2 ), are plausible. In addition to the point-symmetry operations, the full symmetry group G of the normal state also contains the operation of the time reversal R and gauge transformation U(1): G = U(1) × R × G (2.13)

2 Basic Theoretical Concepts

15

Fig. 2.1. Illustration of the classiﬁcation into conventional and unconventional superconductors by symmetry. Both panels show the Fermi surface cross-section (thick line) in the basal plane of a tetragonal crystal (symmetry group D4h ). The thin lines show: (a) amplitude of the order parameter of a conventional superconducting state; (b) amplitude of the order parameter of an unconventional superconducting state which lowers the crystal point symmetry to D2h (from [3])

Associated with the phase transition to superconductivity a spontaneous symmetry breaking occurs with respect to the local gauge invariance cˆ kσ → cˆ kσ eiϕ/2 , and the pair amplitude g kσ1 σ2 transforms into g kσ1 σ2 eiϕ . Therefore, an equivalent deﬁnition of conventional/unconventional superconductivity in terms of symmetry is that for conventional superconductors only gauge symmetry U(1) is broken at the phase transition, while additional symmetries are broken at the phase transition in unconventional superconductors. The conventional superconducting state has the full point symmetry of the crystal lattice, i. e. it belongs to the identity representation A1g . For the rest of superconducting states, which belong to non-identity representations the point-symmetry properties are broken. An important but not compulsory consequence of the broken point-group symmetry is the existence of zeroes in the order parameter, so called nodes, i. e. the order parameter vanishes at points or lines on the Fermi surface. This leads to a gapless excitation spectrum which alters the low-temperature behaviour of many physical properties in the superconductive state. In particular, power laws ∼ T n are observed instead of an exponential temperature dependence, and the exponent n is determined by the topology of the nodes. Some examples for the gap symmetry and the corresponding nodal structure are displayed in Fig. 2.2. Excitations of Paired Fermions In the BCS theory calculations of the spectrum of elementary excitations in a superconductor start from the mean-ﬁeld (mf) Hamilton operator of an ensemble of Fermi particles. Diagonalizing this Hamiltonian by applying the Bogoliubov transformation ﬁnally leads to a transformed Hamilton operator which is given by: Hmf = EBCS (0) + E k αˆ †kσ αˆ kσ (2.14) kσ

16

2 Basic Theoretical Concepts

Fig. 2.2. Some examples of the gap symmetry. The isotropic gap with A1g symmetry (no nodes), the polar gap (line node), the gap with E1u symmetry and the gap with E2g symmetry (both with point and line nodes) are shown (from left to right)

where EBCS (0) is the ground-state energy and E k the contribution of the thermal excitations, the so-called Bogoliubov quasiparticles, at ﬁnite temperatures. The energy spectrum of the Bogoliubov quasiparticles is given by E k = ξk2 + ∆2 (k) (2.15) After diagonalizing the Hamilton operator the pair amplitude in thermal equilibrium is given by ∆k Ek . (2.16) tanh gk = − 2Ek 2kb T Inserting (2.16) into (2.11) leads to two in the weak-coupling limit universal parameters at T c and T = 0, namely the relative size ∆C/CN of the jump in the speciﬁc heat at T c and the energy gap ∆0 (0) in units of kB T c at T = 0. For isotropic s-wave superconductors these parameters are ∆C/CN = 1.43 and 2∆0 (0)/kB T c = 3.53. For other order-parameter symmetries these values are altered, examples are given in Table 2.1. Thus, the pair potential ∆(k) plays the role of an energy gap in the spectrum of elementary excitations which can either have particle (ξk > 0) or hole character (ξk < 0). Table 2.1. BCS parameters ∆C/CN and 2∆0 (0)/kB T c of diﬀerent order-parameter symmetries (from [13]) isotropic

axial

polar

E1g

E2u

B1g B1g × Eu B2g × Eu

∆C/CN

1.43

1.19

0.79

1.00

0.97

0.95

2∆0 (0)/kB T c

3.53

4.06

4.92

4.22

4.26

4.28

2 Basic Theoretical Concepts

17

The excitation spectrum in the case of a superconductor with singlet pairing is deﬁned by Ek =

ξk2 + |∆(k)|2 .

(2.17)

iΦ which leads e. g. for the fully isotropic case (s-wave pairing) with g(k) = ∆0 e to

the well-known fully gapped superconductor with E k = case E k is determined by E k = ξk2 + |d(k)|2 ± |q(k)|

ξk2 + ∆20 . In the triplet

(2.18)

with q = i(d × d∗ ). The vector q(k) is only ﬁnite for d(k) d∗ (k). This case describes a so-called nonunitary pairing state where d(k) is not invariant under time reversal, i. e., it breaks time-reversal symmetry. Clearly, only triplet pairing states can be nonunitary. The physical meaning of a ﬁnite vector q(k) is that the structure of the pair correlation is diﬀerent for up- and down-spins in diﬀerent directions of k. In some cases this can give rise to a ﬁnite net magnetic moment associated with Cooper pairs which can couple to a magnetic ﬁeld. Furthermore, time-reversal symmetry can be broken, if the superconducting state has a degenerate representation, whereas it cannot be broken for non-degenerate representations (i. e. for all singlet states). When spin-orbit coupling is small, the pair wave function can be written as a product of the orbital part and the spin part. Non-zero angular momentum of either the orbital or the spin part could result in time-reversal symmetry breaking, although there are many such cases with conserved time-reversal symmetry, such as d-wave states in high-T c copper oxides. Time-reversal symmetry breaking is also possible in the case of strong spin-orbit coupling. In general, pairing states can be further classiﬁed in terms of gap functions attached to irreducible representations of a point group for a given crystal lattice symmetry of the system [5]. Time-reversal symmetry is broken for some unitary states and for all non-unitary states. For example, the B phase of 3 He has a unitary state which conserves time-reversal symmetry, the A phase is unitary but breaks timereversal symmetry, and the non-unitary A1 phase breaks time-reversal symmetry as well. In contrast, s-wave superconductors are unitary and conserve time-reversal symmetry. In the case of d(k) = zˆ(k x ± iky ) (A phase of 3 He) the Cooper pair has an orbital angular momentum parallel to the z-axis of the crystal. According to the symmetry classiﬁcation by Volovik and Gorkov [4] this pairing state belongs to the class of “ferromagnetic” time-reversal symmetry violating superconducting states and it is natural to expect that the presence of this angular moment appears in magnetic properties. In fact, however, the eﬀect of this angular momentum is invisible in the homogeneous superconducting phase due to Meissner screening. It only occurs where the superconducting state is disturbed in some way that screening eﬀects are insuﬃcient. This happens for example at the surface of samples, at domain walls between the phases zˆ(k x + iky ) and zˆ(k x − iky ), or defects of the crystal lattice, in particular impurities [5]. In such a state spontaneous supercurrents can be generated

18

2 Basic Theoretical Concepts

in the vicinity of an impurity which give rise to a spontaneous magnetic moment. This magnetic moment can be detected for example, by spin-polarized muons in zero magnetic ﬁeld.

References 1. D.H. Rischke, R.D. Pisarski: “Color superconductivity in cold, dense quark matter”, in Villefranche-sur-Mer 2000, Quantum chromodynamics (2000), pp. 220–23, nuclth/0004016 11 2. M. Tinkham: Introduction to Superconductivity (McGraw-Hill International Editions, Singapore, 1996) 11, 23, 24, 29, 56 3. V.P. Mineev, K.V. Samokhin: Introduction to Unconventional Superconductivity (Gordon and Breach Science Publishers, Amsterdam, 1999) 11, 14, 15, 28 4. G.E. Volovik, L.P. Gor‘kov: Sov. Phys. JETP 61(4), 843 (1985). (Zh. Eksp. Teor. Fiz. 88, 1412 (1985)) 11, 14, 17, 21 5. M. Sigrist, K. Ueda: Rev. Mod. Phys. 63(2), 239 (1991) 11, 14, 17, 29, 50, 61, 90, 100, 128, 137, 138 6. D. Einzel: “Supraleitung und Supraﬂuidit¨at”, in Lexikon der Physik Vol. 5 (Sc - Zz) (Spektrum Akademischer Verlag, Heidelberg, 2000), p. 228 11 7. J. Bardeen, L.N. Cooper, J.R. Schrieﬀer: Phys. Rev. 108, 1175 (1957) 3, 12 8. L.N. Cooper: Phys. Rev. 104(4), 1189 (1957) 3, 13 9. P.W. Anderson, P. Morel: Phys. Rev. 123(6), 1911 (1961) 3, 13 10. R. Balian, N.R. Werthamer: Phys. Rev. 131(4), 1553 (1963) 3, 13 11. J.M. Blatt: Theory of superconductivity (Academic Press, New York and London, 1964) 14 12. G. Deutscher: Nature 397, 410 (1999) 14 13. D. Einzel: J. Low Temp. Phys. 126(3/4), 867 (2002) 16

Part II

Experimental Methods

3 Probing the Nodal Structure

The topology and k-spatial symmetry of zeroes of the gap function provide useful information on the superconductive order parameter. Besides direct, i. e. spectroscopic probes of the nodal structure, bulk methods are usually used as an integral probe of the low-energetic excitation spectrum at low temperature. In the following sections frequently applied methods are discussed.

3.1 Speciﬁc Heat and Thermal Conductivity The determination of the heat capacity of a substance provides information about its internal energy. Speciﬁc heat is a bulk method which probes at low temperature the entire low-energy excitations of a solid close to the Fermi level, i. e., it is sensitive to the density of states at EF . The thermal conductivity is a transport property and a directional probe, which depends on the direction of the applied thermal gradient. Therefore, the thermal conductivity is capable to reveal the orientation and symmetry of nodes in the gap function. Thermal-conductivity measurements on metals are sensitive to the density of states at EF , as for metals at low temperatures primarily electrons near the Fermi surface are responsible for the transport of energy. In a standard s-wave superconductor with opening of an isotropic excitation gap at the Fermi surface below T c the speciﬁc heat C(T ) and the thermal conductivity κ(T ) vanish exponentially in the limit T → 0. The existence of line or point nodes alters this behaviour. The existence of quasiparticle excitations in the neighbourhood of these nodes gives rise to a non-exponential behaviour of both quantities, particularly at low temperatures where the node contributions are dominant. These low-energetic excitations are responsible for power laws ∼ T n in the temperature dependence of both properties for T → 0 and the exponent n of T hints at the nodal structure of the order parameter. For example, a T 2 behaviour of the speciﬁc heat at T T c hints at line nodes, while C(T ) ∼ T 3 indicates point nodes of the gap function. A list of exponents n of the low-temperature behaviour of C(T ) calculated under the aspect of order-parameter symmetry was published by Volovik and Gor’kov [1]. However, the hint at the nodal structure might be hidden by extrinsic origins of low-lying states within ∆. In particular, impurities are at the forefront of the extrinsic origins. On the other hand, if the impurity concentration is suﬃciently low, the dependence of the transport properties on the impurity concentration itself makes G. Goll: Unconventional Superconductors STMP 214, 21–26 (2006) c Springer-Verlag Berlin Heidelberg 2006

22

3 Probing the Nodal Structure

it feasible to uncover information on the order-parameter symmetry. It has been shown [2, 3] that the eﬀect of impurity scattering on the transport properties can only be addressed if the scattering is in the strong scattering (unitary) limit, i. e. a phase shift of δ0 = π/2 is involved in the scattering process. In this case, the temperature dependence of the thermodynamic properties is quantitatively related to the density of states and shows power laws depending on the position and form of the line or point nodes of the superconducting gap. If the scattering phase shift is near the Born limit δ = 0 no clear predictions can be made. Unitary scattering leads to virtual bound states on the impurity sites. For the typical concentration of impurities, these bound states overlap and lead to a small “normal state-like” contribution (linear in T ) to κ. Theoretically, it can be described by the development of a new energy scale γ, below which the density of states is nearly constant and in particular, ﬁnite at the Fermi level. The parameter γ is interpreted as the bandwidth of quasiparticle states bound to impurities [4] and provides a crossover energy scale as well. For energies larger than γ the transport properties of unconventional superconductors are determined by the quasiparticle excitations at the nodes, below γ the transport properties are dominated by the bound states. The energy scale and the zero-energy density of states depend on both the impurity concentration and the scattering phase shift δ0 . Graf et al. [5, 6] calculated the electronic contribution to the thermal conductivity for various order-parameter symmetries and found for some a universal value as the temperature approaches zero. Universal means that the thermal conductivity becomes independent of the impurity concentration and κ/T = const (see Fig. 3.1). Thus, experiments on unconventional superconductors with controlled impurity concentrations might allow to distinguish various order-parameter scenarios, depending on whether or not they approach a universal limit. A second test for the order-parameter symmetry arises from the magnetic-ﬁeld dependence of thermal conductivity and speciﬁc heat. As pointed out by Volovik [7] the density of states in a magnetic ﬁeld in superconductors with order-parameter nodes is dominated by contributions from extended quasiparticle states rather than the bound states associated with the vortex cores. The √remarkable consequence of this observation is a term in the speciﬁc heat varying as HT for a line node. K¨ubert and Hirschfeld [8] √ derived a scaling law for the quasiparticle transport properties in the variables T/ H, mixing ﬁeld and temperature dependence, which can be used as a test of the nodal structure, as no scaling is expected, e. g. for linear point nodes. A third experimental tool for probing the superconducting gap structure has been established recently by Izawa and coworkers [9]. They used angular-dependent thermal conductivity in the vortex state to study nodal superconductors. The most remarkable eﬀect for the understanding of the heat transport in the mixed state is the Doppler shift of the delocalized quasiparticle spectrum which is generated by the supercurrents around the vortices. This eﬀect gives rise to the ﬁnite density of states in the presence of nodes, at which the Doppler shift exceeds the local energy gap [10, 11, 12, 13].

3.2 Magnetic Penetration Depth

1.0

(a)

1.0

23

(b)

α=0.01 _ σ=1.0 κ/Τ [arb. units]

κ/κN

(unitary)

0.5

0.0 0.0

0.5

1.0

T/Tc

0.5 (i) (ii) (iii) (iv) BCS 0.0 0.0

0.5

1.0

T/Tc

Fig. 3.1. Thermal conductivity κ vs. temperature for unconventional superconductors in the unitary limit (σ = 1) with a dimensionless scattering rate α = 0.01. The diﬀerent pairing states are: (i) d x2 −y2 with B1g (D4h ) symmetry and 4 linear line nodes, (ii) polar state with A1u (D6h ) symmetry and 1 linear line node, (iii) hybrid I state with E1g (D6h ) symmetry and 2 linear point nodes and 1 linear line node, and (iv) hybrid II state with E2u (D6h ) symmetry and 2 quadratic point nodes and 1 linear line node. For comparison the result for an isotropic BCS superconductor is shown (by courtesy of M. Graf, LANL). As pointed out by Graf and coworkers κ/T has a ﬁnite intercept for the unconventional pairing states (i) – (iv)

3.2 Magnetic Penetration Depth A further probe for the nodal structure are measurements of the magnetic penetration depth λ. Two methods have been established to give access to this quantity, namely measurements of the surface impedance in a microwave resonator and transverseﬁeld µSR studies. The measurements of the surface impedance Z s have played a key role in expanding the understanding of superconductivity. Microwave measurements are made in cavity resonators applying resonance modes with frequencies f0 in the GHz range. Such measurements probe the complex conductivity σ = σ − iσ as a function of temperature and frequency, from which the superﬂuid density as well as the properties of the thermally excited quasiparticles can be deduced. The surface resistance and changes in surface reactance of the sample mounted inside the resonator are obtained from the full-width at half-maximum and the changes of f0 of the resonance curves. In the local limit of the two-ﬂuid model [14] the surface impedance and complex conductivity are given by iµ ω 1/2 0 Z s = R s + iX s = , (3.1) σ − iσ

24

3 Probing the Nodal Structure

where the real part R s is the surface resistance and the imaginary part X s is the surface reactance. The two-ﬂuid model gives a conductivity 2 e ns nn τ σ= + , (3.2) m∗ iω 1 + iωτ where n s = 1 − nn is the superﬂuid density, and nn is the normal ﬂuid density; in the superconducting state, τ is the scattering lifetime of the thermally excited quasiparticles and m∗ is their eﬀective mass. Both contributions to the surface impedance and complex conductivity reﬂect that in the two-ﬂuid model most of the current at frequencies in the microwave range will be carried as a lossless supercurrent, but there will be dissipation from the normal component for any nonzero frequency. The magnetic penetration depth λ and its temperature dependence is obtained from n s = λ(0)2 /λ(T )2 which is derived from σ . The second possibility for the determination of the magnetic penetration depth are transverse-ﬁeld µSR studies (see Sect. 4.3). In type-II superconductors the presence of the ﬂux-line lattice causes an additional line broadening of the µSR line which is assumed to be Gaussian. The muon depolarization rate then is given by 1/λ2 ∝ n s , where λ is the magnetic penetration depth and n s the superﬂuid density. Such studies yield information on the absolute value of λ, its anisotropy, and the temperature dependence λ(T ) which is altered in the presence of gap nodes. However, one has to be cautious against such experiments. As already pointed out by Tinkham [14] the magnetic penetration depth λ−2 cannot have a universal temperature dependence on T/T c even in the BCS theory because of the variation of the ratio ξ0 /λL (0) for diﬀerent metals. The temperature dependence slightly diﬀers in the pure local limit (l ξ, ξ λ), the pure anomalous limit (l ξ, ξ λ), and the dirty local limit (l ξ, ξ λ) compared to the empirical approximation of the two-ﬂuid model given by 1 λ(T ) = λ(0) [1 − (T/T c )4 ]1/2

(3.3)

i. e. an exponent n = 4 for λ−2 ∝ 1 − (T/T c )n . In a clean, local, weak-coupling BCS superconductor the exponent n is generally closer to n = 2 than to n = 4 [14]. For d-wave order parameter the temperature dependence is altered as well. The presence of line nodes gives rise to a continuum of low-lying excitations which results in a linear temperature dependence of λ−2 ∝ n s . Further, according to calculations by Hirschfeld and Goldenfeld [15] the presence of impurities leads to a crossover to a quadratic temperature dependence at low T .

3.3 Ultrasound Attenuation The last method which is introduced here is the use of ultrasound-attenuation measurements, a rarely utilized tool for probing the gap nodes. When a sound wave propagates through a metal the microscopic electric ﬁeld due to the displacement of

3.3 Ultrasound Attenuation

active

α/αn

1.0

25

0.5

inactive 0.0 0.0

0.5

1.0

1.5

T/TC

Fig. 3.2. Qualitative behavior of the ultrasound attenuation in the superconducting state. In the presence of “active” nodes the attenuation grows by a factor T 2 faster than in the presence of “inactive” nodes at low T (from [17])

the ions can impart energy to electrons near the Fermi level, thereby removing energy from the wave. In a superconductor well below T c the rate of attenuation α(T ) of sound waves with ω < 2∆ is markedly lower than in a normal metal. Therefore, measurements of the ultrasonic attenuation allow the determination of the temperature dependence and the anisotropy of the energy gap. Moreno and Coleman have developed a simple theory for the interpretation of transverse ultrasound attenuation coeﬃcients in systems with nodal gap anisotropy [16]. In their calculations, performed in the hydrodynamic limit where the electron mean free path is much shorter than the sound wavelength λ, the low temperature power-law behaviour of α(T ) is shown to depend strongly on the wave-vector direction qˆ and the polarization eˆ relative to the nodes. Nodes are “active” in attenuating sound, if neither of these vectors is perpendicular to the direction of the node, while nodes are “inactive” if either the vector qˆ or the polarization eˆ are perpendicular to the direction of the node. In this way ultrasound-attenuation experiments can locate nodes in the gap function. Based on the idea of “active” and “inactive” nodes Walker et al. worked out a more generalized theory by replacing the isotropic electron stress tensor in [16] by the electron-phonon matrix element. They derived an expression which is also applicable to anisotropic multisheet Fermi surfaces. If the matrix element is non-zero at the nodes for a particular phonon then the phonon can interact with the nodal quasiparticle (and thus attenuate a sound wave), i. e. the node is “active” for the particular phonon. If, on the other hand, the matrix element is zero at the nodes, then the coupling of the phonon to the quasiparticle precisely at the node is zero and grows as the distance from the node is increased. In this case, the node is “inactive” for the particular phonon [17]. In addition to the interpretation of ultrasound attenuation data, Moreno and Coleman have predicted a T 3.5 power law for the direction of inactive nodes and

26

3 Probing the Nodal Structure

a T 1.5 power law for the direction of active nodes for the case of a two-dimensional gap with k2x − ky2 symmetry. In general, the ultrasound attenuation at T T c exhibits a power law behaviour and the exponent n is larger by two in the case where only ”inactive“ nodes are present compared to the case when ”active“ nodes are present (see Fig. 3.2).

References 1. G.E. Volovik, L.P. Gor‘kov: Sov. Phys. JETP 61(4), 843 (1985). (Zh. Eksp. Teor. Fiz. 88, 1412 (1985)) 11, 14, 17, 21 2. P. Hirschfeld, D. Vollhardt, P. W¨olﬂe: Solid State Commun. 59(3), 111 (1986) 22, 50, 98 3. P.J. Hirschfeld, P. W¨olﬂe, D. Einzel: Phys. Rev. B 37, 83 (1988) 22, 50 4. L.J. Buchholtz, G. Zwicknagl: Phys. Rev. B 23(11), 5788 (1981) 22, 36 5. M.J. Graf, S.K. Yip, J.A. Sauls: J. Low Temp. Phys. 102, 367 (1996) 22, 74, 75, 91, 92, 93, 125 6. M.J. Graf, S.K. Yip, J.A. Sauls, D. Rainer: Phys. Rev. B 53(22), 15 147 (1996) 22 7. G.E. Volovik: JETP Lett. 58(6), 469 (1993). (Pis’ma Zh. Eksp. Teor. Fiz. 58, 457 (1993)) 22 8. C. K¨ubert, P.J. Hirschfeld: Phys. Rev. Lett. 80(22), 4963 (1998) 22 9. K. Izawa, Y. Matsuda: J. Low Temp. Phys. 131(3/4), 429 (2003) 22 10. P. Thalmeier, K. Maki: Europhys. Lett. 58(1), 119 (2002) 22 11. I. Vekhter, P.J. Hirschfeld, J.P. Carbotte, E.J. Nicol: Phys. Rev. B 59(14), R9023 (1999) 22, 124 12. K. Maki, G. Yang, H. Won: Physica C 341-348(3), 1647 (2000) 22 13. H. Won, K. Maki: cond-mat/0004105 (2000) 22, 124 14. M. Tinkham: Introduction to Superconductivity (McGraw-Hill International Editions, Singapore, 1996) 11, 23, 24, 29, 56 15. P.J. Hirschfeld, N. Goldenfeld: Phys. Rev. B 48(6), 4219 (1993) 24, 75, 125 16. J. Moreno, P. Coleman: Phys. Rev. B 53(6), R2995 (1996) 25 17. M.B. Walker, M.F. Smith, K.V. Samokhin: Phys. Rev. B 65, 014 517 (2002) 25, 95

4 Probing the Parity and Spin State

Classiﬁcation into classes of superconductors with singlet and triplet pairing, respectively, needs information on the parity and spin state of the pairing state. These information can be accessed through measurements in a magnetic ﬁeld due to the diﬀerent response of the pairs with S = 0 and S = 1, respectively, on an applied magnetic ﬁeld. In the following sections Hc2 measurements, the nuclear magnetic resonance (NMR), and the muon spin rotation (µSR) are introduced as frequently used probes of the parity and spin state.

4.1 H c2 Measurements Magnetic ﬁeld can suppress superconductivity via two eﬀects: orbital pair breaking of the superconducting pairs in the superconducting state and Pauli limiting due to the paramagnetism of the electron spins, which lowers the relative energy of the normal state. Keeping this in mind, measurements of the upper critical ﬁeld Hc2 can yield information on the parity of the superconducting state. A paramagnetic limitation of Hc2 arises in even parity superconductors due to the drop of the Pauli susceptibility in the superconducting state χS which tends to zero as T → 0. The limitation occurs at low temperatures when the increase in magnetic energy ∝ µ20 µ2B N0 H 2 becomes larger than the energy gain in the superconducting state ∝ N0 ∆20 /2, where N0 is the density of states per one spin projection. The superconducting state becomes unfavorable in a magnetic ﬁeld higher than the so called paramagnetic limit of superconductivity1 Hp given by ∆0 , Hp = √ 2µ0 µB

(4.1)

and the singlet state of Cooper pairs is destroyed. For the importance of Pauli limiting the diﬀerence χN − χS is decisive apart from the ratio χS /χN which is known for a given order parameter. χN is the susceptibility in the normal state. No Pauli limiting is expected in simple triplet states with equal spin pairing as χN = χS in 1

This limit is also known as the Clogston-Chandrasekhar paramagnetic limit named after Clogston and Chandrasekhar who ﬁrst pointed out that a ﬁrst-order transition to the normal state occurs at Hp due to the pair-breaking eﬀect of an external ﬁeld on the electronic spins [1, 2].

G. Goll: Unconventional Superconductors STMP 214, 27–33 (2006) c Springer-Verlag Berlin Heidelberg 2006

28

4 Probing the Parity and Spin State

these states. In heavy-fermion systems where the orbital limit is very large due to the large eﬀective mass, paramagnetic limitation of Hc2 can take place [3, 4]. Therefore, a simple Hc2 measurement already might probe the spin state of the Cooper pairs. So far, the considerations did only apply to isotropic systems without spin-orbit coupling. However, in systems with strong spin-orbit coupling the spin part of the order parameter cannot orientate itself freely with respect to the orbital part and χS can become smaller than χN for some orientations of the crystal relative to the magnetic ﬁeld. Therefore, anisotropic Pauli limiting can occur even in a triplet superconductor. The analysis of Hc2 curves is further complicated by impurity spin-orbit scattering which reduces the eﬀect of paramagnetic limiting [5, 6, 7]. Spin-orbit scattering leads to a ﬁnite susceptibility in conventional superconductors, and thus increases the Clogston limit Hp . In conclusion, the occurrence respectively the absence of Pauli limiting in itself does not allow deﬁnite conclusions on the nature of the superconducting state.

4.2 Nuclear Magnetic Resonance and Knight Shift The basic idea of nuclear magnetic resonance (NMR) is that radiofrequency signals can be used to measure resonant properties of nuclei in magnetic ﬁelds. In an external magnetic ﬁeld the magnetic dipole moments of nuclei are partially aligned, and the magnetization can be changed by irradiating with a radiofrequency. The important concepts of nuclear magnetization and resonance absorption are widely discussed in a number of textbooks [8] and therefore, will not be repeated here. However, two items will be focused on in the following, namely the Knight shift K = ∆ω/ω which is the shift ∆ω of the nuclear resonance at ω by polarized conduction electrons and the nuclear relaxation rate 1/T 1 which accounts for the spinlattice relaxation process in metals due to spin-ﬂip scattering of conduction electrons by the nucleus. The interaction responsible for both processes is the Fermi contact interaction given by 2 (4.2) Hhf = − µ0 γN γe |φ(0)|2 (S · I) 3 where µ0 = 4π·10−7 Vs/Am is the vacuum permeability, γN and γe are the gyromagnetic ratio of the nucleus and electron, respectively, and |φ(0)|2 is the density of the electrons at the nucleus normalized to one electron per unit volume. The diagonal terms contribute to the Knight shift, but the nondiagonal part is responsible for the spin relaxation. It follows that the Knight shift for s-electrons is given by K=

|ψ(0)|2 ∆ω 2 = χPauli ω 3 N

(4.3)

which is equal (apart from a factor 2/3) to the Pauli susceptibility χPauli multiplied by an ampliﬁcation factor |ψ(0)|2 /N, which gives the ratio of the electron density at the nucleus to the average electron density.

4.2 Nuclear Magnetic Resonance and Knight Shift

29

For the relaxation rate 1/T 1 one ﬁnds 1/T 1 =

4π 2 2 2 3 µ γ γ |φ(0)|4 D(EF )2 kB T 9 0 N e

(4.4)

with D(EF ) the density of states at the Fermi level and kB the Boltzmann constant. Both items are substantially inﬂuenced if a metal becomes superconducting. In an s-wave superconductor the spin susceptibility χspin drops to zero, while in equalspin-pairing states the spin susceptibility is unchanged in all ﬁeld directions as long as the order parameter d(k) is free to rotate2 . This is the case if the energy of the spin-orbit coupling is weaker than that of the applied magnetic ﬁeld. If the spinorbit coupling is strong enough to lock the order parameter to the crystal lattice, only χspin perpendicular to d is unchanged, whereas χspin parallel to d approaches zero for T → 0. Such a dependence of χspin on the direction and magnitude of the applied magnetic ﬁeld was actually observed in the spin-triplet superconductors UPt3 [12] and Sr2 RuO4 [13] (see Sects. 9.1 and 10.1). The relaxation process involves ﬂipping of spins so that the relevant matrix element for nuclear-spin relaxation by interaction with quasiparticles have the case II coherence factors [14]. This corresponds to constructive interference in the relevant low-energy scattering process and causes the relaxation rate 1/T 1 to rise above the normal value upon cooling through T c before it exponentially drop-oﬀ to zero with the freeze-out of quasiparticles at lower temperatures. This phenomenon is called the Hebel-Slichter peak after Hebel and Slichter who ﬁrst observed this peak in the nuclear relaxation rate of aluminum [15]. Its explanation was a great triumph for the BCS theory. For unconventional order parameters, case I and case II coherence factors are the same and do not lead to any enhancement3 [17]. The temperature dependence of the relaxation rate also acts as a probe for the nodal structure of the superconductor. Even though the coherence peak might be suppressed, s-wave superconductivity is evidenced by an exponential decrease of 1/T 1 below T c . In contrast, the relaxation rate of unconventional superconductors exhibits a T n power-law behaviour. For example, in the case of a line node 1/T 1 ∼ T 3 is observed. An overview over resonance experiments on heavy-fermion systems including heavy-fermion superconductors has been published by Kitaoka et al. [18]. 2

3

A constant Knight shift was also reported for the s-wave superconductor V [9]. The reason is that the orbital contribution to the Knight shift in V corresponding to the Van Vleck orbital susceptibility is the principal contribution, and this orbital shift is independent of the spin state of the conduction electrons. Other reasons for a constant spin susceptibility unrelated to pairing symmetry are strong spin-orbit coupling [10] or pecularities of the electronic structure [11]. Although the existence of this peak is one of the crucial tests for BCS superconductivity, one should keep in mind that indeed, this peak might be absent in the classical superconductors Nb and V [16] and other strong-coupling superconductors.

30

4 Probing the Parity and Spin State

4.3 Muon Spin Rotation The muon spin rotation (µSR) technique is a local-probe hyperﬁne method like nuclear magnetic resonance discussed above. Together with neutron scattering and NMR, it is one of the very few microscopic methods investigating the bulk of the material, as the muons penetrate tenth of a millimetre into the sample. In recent years, µSR has become a primary method for the study of type-II superconductors, because muons are an ideal tool to investigate weak-magnetism phenomena in zero external ﬁeld, and can be utilized to search for the occurrence of spontaneous magnetism below T c , which would signal a possible breakdown of the time-reversal symmetry invariance. Moreover, µSR transverse-ﬁeld measurements below T c can furnish valuable information on the possible anisotropies of the temperature dependence of the London penetration depth, which could indicate a crystal symmetry breaking. Due to the local character of the µ+ probe and its uniform implantation in a sample, µSR has been utilized to check whether the coexistence of magnetism and superconductivity in the heavy-fermion superconductors appears on a microscopic scale. In the following a brief introduction to the µSR technique is given, for more details the reader is referred to textbooks or specialized reviews (see e. g. [8, 19, 20] and refs. therein). Here, only µ+ spin rotation is considered since positive muons are used much more extensively than negative muons in solid state physics research. Muons belong to the lepton family and are produced via pion decay within the pion mean lifetime of τ = 26 ns. The muon is implanted in the solid sample and decays within 2.2 µs according to µ+ −→ e+ + νe + ν¯µ , where e+ is a positron and νe and ν¯µ are neutrinos. Two important properties make the muons suitable as a solid state probe: 1. muons produced in the above way are 100% spin-polarized in the pion rest frame and 2. the muon decay is anisotropic, i. e. the positrons are emitted preferentially in the direction of the muon spin. Hence, by measuring the positron distribution, it is possible to determine the original µ+ spin direction. Polarized muons are implanted into a sample where the polarization is aﬀected by the local magnetic ﬁeld until they decay. Because of its positive charge, the muon localizes at an interstitial site. If the implanted µ+ is subject to magnetic interactions it precesses about the local magnetic ﬁeld B(r) with a Larmor frequency ωµ = γµ B(r) ,

(4.5)

where γµ /2π = 135.5342 MHz/T is the muon gyromagnetic ratio. Consequently the polarization Pµ becomes time dependent with Pµ (t) = G(t)Pµ (0), where G(t) reﬂects the normalized µ+ -spin autocorrelation function which depends on the average value, distribution, and time evolution of the internal ﬁelds and therefore contains all physics of the magnetic interactions of the µ+ inside the sample. The envelope of G(t) is called the µ+ depolarization function and a fast Fourier transformation yields the µSR spectrum whose line shape and position can be further analyzed. Two techniques are important for the investigation of superconductors: the zeroﬁeld (ZF) µSR technique and the transverse-ﬁeld (TF) µSR technique. The zeroﬁeld (ZF) µSR technique monitors the time evolution of the muon ensemble under

4.3 Muon Spin Rotation

31

G(t)

B(r) (a)

t r G(t)

B(r) (b)

t r G(t)

B(r) (c)

t r

Fig. 4.1. The ﬁeld distribution inside a superconductor as a function of position and the corresponding muon-spin relaxation function in the normal state (a), the superconducting state (b), and in the superconducting state with a shorter penetration depth (c) (from [20])

the action of internal magnetic ﬁelds in zero external ﬁelds. The very large magnetic moment of the muon makes µSR sensitive to extremely small internal magnetic ﬁelds down to the order of 0.1 G. The zero-ﬁeld technique has been widely utilized to measure the spontaneous µ+ Larmor frequencies in magnetically ordered phases, providing valuable information about the values of the static moment and the magnetic structures. In superconductors with time-reversal symmetry-breaking ordered state an internal ﬁeld occurs which leads to an additional µ+ depolarization due to electronic magnetic moments. A broadening of the µSR line in zero ﬁeld can further be caused by a static distribution of internal ﬁelds or by ﬂuctuations arising from ﬂuctuating magnetic moments. For the latter process the depolarization rate σ1 = 1/T 1 describes the spin-lattice relaxation (1/T 1 process). The transverse-ﬁeld (TF) µSR technique gives access to the µ+ Knight shift. The + µ Knight shift originates from the magnetic-ﬁeld-induced polarization of the conduction electrons. Local electronic moments can also contribute to the frequency shift by producing an eﬀective dipolar ﬁeld and an additional hyperﬁne contact ﬁeld at the muon site. In the heavy-fermion compounds the µ+ Knight shift corresponds to contributions to Bint both from the polarization of conduction electrons and localized f moments induced by Hext . For this technique, an external ﬁeld Hext is applied perpendicular to the initial polarization Pµ (0). Pµ (t) precesses around the total ﬁeld Bµ at the µ+ site. From the oscillatory component of G(t) (see Fig. 4.1) the total ﬁeld Bµ can be extracted. After correction for the contribution of demagnetization and Lorentz ﬁeld one obtains the µ+ Knight shift Kµ =

|Bint | − |Hext | , |Hext |

where Bint are the internal ﬁelds induced by Hext .

(4.6)

32

4 Probing the Parity and Spin State

If an inhomogeneous ﬁeld distribution is present, the muons located at diﬀerent sites will feel slightly diﬀerent ﬁelds which will result in a loss of polarization by dephasing the muon ensemble and consequently to a line broadening in the µSR spectrum. A further origin of line broadening is the so-called 1/T 2 process which arises from the dephasing of the µ+ spins with depolarization rate σ2 = 1/T 2 . For transverse-ﬁeld µSR studies in type-II superconductors two eﬀects play a role: 1. the presence of the ﬂux-line lattice causes an additional ﬁeld distribution, and 2. below T c the µ+ Knight shift changes due to the formation of the Cooper pairs. For Hext Hc1 where Hc1 is the lower critical ﬁeld, a type-II superconductor is in the mixed state, where both superconducting and normal regions (vortex cores) coexist. The muons implanted close to the vortex core experience a larger magnetic ﬁeld than those implanted in the superconducting regions between vortices. The frequency shift is expected to be diﬀerent in the superconducting and in the normal regions, and consequently there is a spread in precession frequency. Therefore, the measured ﬁeld distribution is a convolution of the distribution due to the Knight shift and the ﬂux-line lattice [21]. The line broadening due to the presence of the ﬂux-line lattice is traditionally assumed to be Gaussian. The muon depolarization rate then is given by σ ∝ 1/λ2 ∝ n s , where λ is the magnetic penetration depth and n s the superﬂuid density. Therefore, such studies yield information on the absolute value of λ, its anisotropy, and the temperature dependence λ(T ) which gives information about the gap nodes (see Sect. 3.2). However, as pointed out by Sonier et al. in a recent review [22], a simple Gaussian ﬁt of the line shape is sometimes not suﬃcient and even yields false conclusions (see below in Sect. 7 and [22]).

References 1. A.M. Clogston: Phys. Rev. Lett. 9, 266 (1962) 27 2. B.S. Chandrasekhar: Appl. Phys. Lett. 1, 7 (1962) 27 3. V.P. Mineev, K.V. Samokhin: Introduction to Unconventional Superconductivity (Gordon and Breach Science Publishers, Amsterdam, 1999) 11, 14, 15, 28 4. U. Rauchschwalbe: Physica B+C 147, 1 (1987) 28, 69 5. P. Fulde, K. Maki: Phys. Rev. 141, 275–280 (1966). (E: Phys. Rev. 147, 414 (1966)) 28 6. N.R. Werthamer, E. Helfand, P.C. Hohenberg: Phys. Rev. 147, 295–302 (1966) 28 7. K. Maki: Phys. Rev. 148, 362–369 (1966) 28 8. G. Schatz, A. Weidinger: Nuclear condensed matter physics: nuclear methods and applications (John Wiley & Sons, Ltd., England, 1992) 28, 30 9. R.J. Noer, W.D. Knight: Rev. Mod. Phys. 36(1), 177–185 (1964) 29 10. P.W. Anderson: Phys. Rev. Lett. 3(7), 325–326 (1959) 29 11. D.E. MacLaughlin: Solid State Physics: Advances in research and applications 31, 1 (1976) 29 12. H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura, Y. Onuki, E. Yamamoto, Y. Haga, K. Maezawa: Phys. Rev. Lett. 80(14), 3129 (1998) 29, 91, 96, 101, 156 13. H. Murakawa, K. Ishida, K. Kitagawa, Z.Q. Mao, Y. Maeno: Phys. Rev. Lett. 93(16), 167 004 (2004) 29, 128, 157

References

33

14. M. Tinkham: Introduction to Superconductivity (McGraw-Hill International Editions, Singapore, 1996) 11, 23, 24, 29, 56 15. L.C. Hebel, C.P. Slichter: Phys. Rev. 113(6), 1504–1519 (1959) 29 16. D.M. Ginsberg, L.C. Hebel: Nonequilibrium properties: Comparison of Experimental Results with Predictions of the BCS Theory (Marcel Dekker, Inc., New York, 1969), p. 193 29 17. M. Sigrist, K. Ueda: Rev. Mod. Phys. 63, 239 (1991) 11, 14, 17, 29, 50, 61, 90, 100, 128, 137, 138 18. Y. Kitaoka, S. Kawasaki, T. Mito, Y. Kawasaki: J. Phys. Soc. Jpn. 74(1), 186–199 (2005). (see also cond-mat/0412288v1) 29 19. A. Amato: Rev. Mod. Phys. 69(4), 1119 (1997) 30, 94, 109 20. S.J. Blundell: Contemporary Physics 40(3), 175 (1999). (see also cond-mat/0207699) 30, 31 21. R. Feyerherm, A. Amato, F.N. Gygax, A. Schenck, C. Geibel, F. Steglich, N. Sato, T. Komatsubara: Phys. Rev. Lett. 73(13), 1849 (1994) 32, 113, 157 22. J.E. Sonier, J.H. Brewer, R.F. Kieﬂ: Rev. Mod. Phys. 72(3), 769 (2000) 32, 63

5 Probing the Energy Gap

The most direct tool for the determination of the excitation gap of superconductors is spectroscopy. Spectroscopy probes the basic excitation of a system by absorption and emission of a well known amount of energy. In general, diﬀerent kind of probes can be used such as photons, neutrons and electrons, as long as the initial and ﬁnal states with respect to energy and momentum can be determined. In the following sections tunnelling spectroscopy and the related methods of scanning tunnelling spectroscopy and point-contact spectroscopy will be discussed, three methods which have been frequently used for the investigation of conventional and unconventional superconductors. Of course, there are further methods, e. g. the photoemission spectroscopy which has played an important role for the investigation of the high-T c material. However, most of the superconducting compounds discussed in this book have T c ’s of the order of 1 K and therefore, an excitation gap below 1 meV, where these methods have serious drawbacks.

5.1 Planar Tunnelling Spectroscopy The concept of tunnelling is a direct consequence of quantum mechanics, following from the nature of the solution ψ(x) of Schr¨odinger’s equation and the probability interpretation of ψ∗ ψ. The rate at which such processes occur is dominated by the exponential decay of ψ(x) in the classically forbidden barrier region where the potential energy V exceeds the kinetic energy E. In particular, at a metal-vacuum interface the wavefunction for an electron at the Fermi energy does not vanish outside the metal but rather decays exponentially as exp(−κx) where κ (2m/2 )1/2 Φ1/2 , x is the distance into the vacuum, and Φ is the metal work function. This exponential dependence on x is the basic mechanism for the scanning tunnelling microscope which will be considered in Sect. 5.2. For metal-insulator-metal junctions the electronic mass m and the work function Φ are modiﬁed reﬂecting the properties of the solid-state insulator. If such a junction is biased by an external voltage V, electrons in a well-deﬁned energy range 0 ≤ E ≤ eV between the two shifted Fermi energies may elastically tunnel from one side to empty states opposite. This feature makes possible several forms of spectroscopy including a spectroscopy of the superconducting state, which probes both the details of the energy gap structure and of the boson spectrum which produces the paired-electron system. There exist G. Goll: Unconventional Superconductors STMP 214, 35–54 (2006) c Springer-Verlag Berlin Heidelberg 2006

36

5 Probing the Energy Gap

several monographs which give a general insight into the wide range of tunnelling spectroscopy especially on ordinary superconductors [1, 2, 3]. In a simple phenomenological model, the net current ﬂowing across a normal metal-superconductor junction at applied bias V is calculated by considering the transition probability of electrons crossing from the normal metal into the superconductor and vice versa. The transition probability is proportional to the number of ﬁlled states on one side times the number of empty electronic states on the other side of the junction at the same energy. The net current can then be written as

∞ ρN (E)ρS (E + eV) f (E) − f (E + eV) dE , (5.1) INS = GNN −∞

where ρS /ρN = |E|/ (E 2 − ∆2 ) is the normalized BCS density of states, ρN is the electronic density of states of the normal metal at the Fermi level, GNN = R−1 NN is the junction conductance when both electrodes are in the normal state and f (E) is the Fermi-Dirac function. A justiﬁcation of this approach was given by Bardeen [4] and Cohen et al. [5] using a transfer Hamiltonian approach. The quantum-mechanical tunnelling current ﬂowing through a thin insulating layer between metal electrodes was ﬁrst investigated by Fisher and Giaever [6, 7, 8]. Soon, it became clear that the tunnelling technique enables the study of the quasiparticle density of states. From such studies the size of the energy gap could be derived, and its temperature dependence which could be compared to the predictions of the BCS theory. Further, depairing eﬀects in magnetic ﬁelds could be veriﬁed and tunnelling studies of strong-coupling superconductors like Pb reveal direct information about the Eliashberg function α(ω)2 F(ω). Already since the early days of research on unconventional superconductors it has been proposed that tunnelling spectroscopy might be suitable to identify the pairing symmetry. This has mainly to do with the signiﬁcantly diﬀerent behaviour of the order parameter at the tunnel-junction interface in junctions with unconventional superconductors. One important example is the formation of Andreev bound states at zero energy conﬁned to the surface for an order parameter exhibiting a sign change for diﬀerent k-directions. In general, bound states occur at energies for which the phases of Andreev-reﬂected particlelike and holelike excitations interfere constructively. This eﬀect is pronounced if the scattering of the quasiparticles at the surface is connected with a sign change of the order parameter along the classical trajectory. In this case a zero-energy bound state forms. Surface bound states of this origin were discussed in the context of tunnelling into p-wave superconductors by Buchholtz and Zwicknagl [9] and more recently for the d x2 −y2 pairing model of the cuprates by Hu [10], Tanaka and Kashiwaya [11], and Fogelstr¨om et al. [12]. The bound states can carry current and therefore are observed as a zero-bias conductance peak in the tunnelling conductance dI/dV vs. V. The identiﬁcation of a zero-bias conductance peak with the surface bound states is important because the surface bound states is associated with an unconventional order-parameter symmetry and the origin of the zero-bias conductance peak therefore is the same as that of the π shift in Josephson interference experiments (see Sect. 6.1). The occurrence of a

5.2 Scanning Tunnelling Spectroscopy (STS)

37

zero-bias conductance peak has been investigated in great detail in tunnelling experiments on the high-temperature superconductors. A summary of these experiments can be found in two reviews by Alﬀ et al. [13] and by Cucolo [14]. A further distinction arises from the diﬀerent behaviour of the order parameter components at the surface. Buchholtz and Zwicknagl [9] showed for p-wave superconductors that, in general, the perpendicular component of the order parameter is suppressed to zero and the parallel components are slightly enhanced as they approach the surface. Consequently, bound states appear ﬁlling up the excitation gap. In contrast, s-wave pairing is not disrupted by a (nonmagnetic) surface, i. e. a surface does not act as a pair-breaking scatterer. Consequently, the order parameter will not be diminished which is the crucial distinction.

5.2 Scanning Tunnelling Spectroscopy (STS) The enormous potential of the scanning tunnelling microscope (STM) is the ability to measure the tunnel spectrum at any desired position of the STM tip on the sample. Therefore, the STM provides access to both, the topography of a surface and its local electronic structure. Both properties are of course not independent of each other and the correct interpretation of the topographic information needs the knowledge of the local density of states of sample and tip. This information can be derived from measurements of the tunnelling current as a function of applied voltage. As already noticed in the previous Sect. 5.1 the key for vacuum tunnelling is the exponential decay of the electron wavefunction outside the metal (see Fig. 5.1). In order to describe the mechanism of imaging on an atomic scale the total threedimensional topographic and electronic structure of the sample surface and tip has to be taken into account for the calculation of the tunnel current. In a ﬁrst semiquantitative approach Tersoﬀ and Hamann [15, 16] derived an expression for the tunnelling current I under the assumption of small bias voltage V, low temperature and s-wave character of the tip: I ∼ VρT (EF )ρS (r0 , EF )

(5.2)

ρT (EF ) is the (constant) density of states of the tip at the Fermi energy EF and ρS (r0 , EF ) the local density of states at r0 . The result allows the interpretation of constant-current-mode (topography mode) STM pictures in terms of surfaces of constant density of states at EF . In many cases, however, the assumptions of the Tersoﬀ-Hamann model are not fulﬁlled. Many materials used as tip material are transition metals where the assumption of an s-wave character of the tip wave function is not a good approximation. A generalization of the Tersoﬀ-Hamann model for other tip wave functions was proposed by Chen [17] who approximated the vacuum eigenstates of the tip by the radial solutions of the vacuum Schr¨odinger equation. Another case is tunnelling spectroscopy where the assumption of small applied voltage is not valid and the energy dependence of the matrix elements and the potential shift of the wave function at the barrier has to be considered. In an extension of the

38

5 Probing the Energy Gap

Fig. 5.1. Basic sketch of a scanning tunnelling microscope (STM)(a), simple picture of the 1D electron wavefunction (b), and potential at the sample-tip interface (c). This sketch qualitatively explains the mechanism of operating a STM based on the tunnelling eﬀect

Tersoﬀ-Hamann model for not too large bias V < Φ, where Φ is the mean work function of tip and sample, the tunnelling current is generally given as a folding of the local energy-dependent density of states ρS (E) and ρT (E) of sample and tip, respectively, expressed by

∞ I(s, V) ∼ ρS (E)ρT (E − eV)T (s, V, E) f (E) − f (E + eV) dE , (5.3) −∞

which is under certain conditions equivalent to the ”Golden-Rule” tunnelling current given in (5.1). A more detailed discussion and appropriate references can be found in [18]. The tunnelling current I(s, V) in (5.3) depends on the local density of states, their temperature-dependent occupation determined by the Fermi-Dirac distribution f (E), and on the transmission probability T (s, V, E) through the (vacuum) tunnelling barrier. In the simple case of a planar (vacuum) tunnel contact the transmission probability is given by [19]       2m eV   − E Φ + (5.4) T (s, V, E) ∼ exp  −2s    2   2

5.3 Point-Contact Spectroscopy

39

where s is the distance between tip and sample and Φ = (ΦS + ΦT )/2 is the mean work function of tip and sample. As expected, T (s, V. E) depends exponentially on the distance s. The exponential dependence reﬂects the main principle of a scanning tunnelling microscope and explains the high sensitivity of the tunnelling current on the atomic surface roughness. In a simple estimate one can verify that a change of 1 Å in the surface-tip distance leads to a change of almost one order of magnitude in the tunnelling current. Equation (5.3) already shows the main result of the present consideration, namely that the tunnel current depends on the density of states of the sample and I − V measurements give access to the local density of states at the surface. Usually however, for experimental reasons in order to make use of the Lock-In technique, the diﬀerential conductance dI/dV is regarded which is more sensitive on ρS (E). Under the above given conditions and the further assumptions that the density of states of the tip is energy-independent and that the transmission probability behaves monotonic as given by (5.4), one obtains for the diﬀerential conductance at low temperatures (kB T eV): dI ∼ ρS (E)ρT (E − eV)T (s, V, E) E=eV dV

(5.5)

Equation (5.5) allows the interpretation of dI/dV vs. V spectra in terms of spectroscopy of the local density of states comparable to the information obtained from planar tunnel junctions, but locally. Spectroscopy, not topography, was also the initial intention of G. Binnig and H. Rohrer for the invention of the STM, as they note in their Nobel prize lecture [20]: “Our ﬁrst setup was designed to work at low temperatures . . . because our thoughts were ﬁxed on spectroscopy.”

5.3 Point-Contact Spectroscopy Point-contact spectroscopy is a method which probes the excitations of a solid by the use of ballistic electrons moving through a small metallic contact. These electrons are accelerated within the potential drop V along a microcontact between two electrodes gaining the energy eV. The contact is called point contact if its lateral dimensions are smaller than the characteristic electronic length scales of the electrode material. In a metal the length scale is deﬁned by the elastic and inelastic mean free path le and li , respectively, on which the electron is scattered elastically, if it changes its momentum, or inelastically, if it changes its well deﬁned energy eV. Point-contact spectroscopy studies the non-linearities of the current-voltage characteristics resulting from the scattering of the electrons. Several review articles have appeared since this method has been developed in the early 1970’s [21, 22, 23, 24, 25], a comprehensive survey has been published recently [26].

40

5 Probing the Energy Gap

There is an important diﬀerence between tunnelling and point-contact spectroscopy1 , namely the existence of a direct metallic conductivity in the latter case. Nonlinearities of the current-voltage characteristics of a point contact are primarily caused by scattering of the electrons while the availability of free current-carrying states in which the electrons can tunnel, determine the current-voltage characteristic of tunnel junctions. Transport through a point contact is determined on one hand by the possible electron transfer processes through the contact, where the electronic surface states play an important role, and on the other hand by backscattering processes in a region at the contact on a scale of the order of contact size. In this sense, point-contact experiments are sensitive to the surface properties of the material under investigation. Historically, shorted thin-ﬁlm tunnel junctions were the ﬁrst structures which were investigated by this method [27]. A few examples of present-day structures are schematically depicted in Fig. 5.2. The “needle-anvil” geometry of bulk electrodes (a) and the ”edge-to-edge” contact of sharp edges of bulk metals (b) are the easiest ways to realize point contacts between two metals and either a homocontact between electrodes of the same metal or a heterocontact between electrodes of diﬀerent metals can be established. The stability of the point contact can be substantially improved if a mechanically controllable break junction (c) or a nanolithographic point contact through a Si3+x N4−x membrane (d) is used. A sophistication of the break-junction technique has enabled even the study of the transport channels through metallic one-atom point contacts, so called metallic quantum point contacts [28, 29]. Contacts of type (d) have the advantage that the contact size can be controlled by the preparation in a deﬁned way, while for types (a) – (c) the contact resistance is the only control parameter, and an appropriate modelling of the contact is required to derive the contact dimensions from the measured contact resistance. Normal Metal - Normal Metal Contacts Several models have been derived to describe a point contact and to estimate the lateral dimensions of the constriction. For simplicity only homocontacts are considered in the following. The most commonly used theoretical model considers a point contact as a circular constriction of length L and diameter d connecting two bulk metallic half-spaces. The two extreme cases are the oriﬁce (L d) in a non-transparent inﬁnitely thin partition and the one-dimensional channel (L d). Depending on the relationship between the mean free path of the electrons and the point-contact diameter d, various regimes of electron ﬂow have been √ established. These regimes are called ballistic (le , li d), diﬀusive (le d le li ), and thermal (le , li d). A summary of basic point-contact relations for diﬀerent geometries and regimes can be found in [30, 31]. In contrast to the tunnelling spectroscopy, there exists always a metallic conductance without a tunnel barrier, but the intermediate regime between the purely metallic constriction and the pure tunnel junction can be modelled by a 1

Yet another diﬀerence is the displaying of data which sometimes causes confusion: for experimental reasons tunnel spectra show the diﬀerential conductance dI/dV, point-contact spectra usually the diﬀerential resistance dV/dI.

5.3 Point-Contact Spectroscopy

(a)

41

(b)

S Si 3+x N4−x N (c)

(d)

Fig. 5.2. Experimental realisation of point-contacts: (a) “edge-to-edge” contact of sharp edges of bulk metals [33], (b) “needle-anvil” geometry of bulk electrodes [34], (c) mechanically controllable break junction [35], (d) nanolithographic point contact through Si3+x N4−x membrane [36]

probability D of electrons passing through the constriction [32]. In the ﬁrst regime, the ballistic regime, electrons in the presence of an applied voltage V are transmitted ballistically through the contact yielding energy-resolved, i. e. spectroscopic information on basic excitations of the metal like phonons, magnons, etc. The conductance of the contact in the ballistic regime can be calculated with the help of the Landauer formula [37] for the conductance 2e2 2e2 Tr{t † t} = Tα . (5.6) G= h h α t is the transmission matrix which describes the transmission of N electrons through a contact from mode i of the wave function into the mode j with probability ti j . T α is the transmission probability of the α-th scattering channel after diagonalizing the transmission matrix. If the contact is modelled as an oriﬁce with cross-section area S = πd2 /4 in an impenetrable partition, the total number of transverse channels is given by S kF2 /4π and one obtains the Sharvin conductance GS , or the Sharvin resistance RS = G−1 S :

42

5 Probing the Energy Gap

RS =

16 h 16ρl · = , (kF d)2 2e2 3πd2

(5.7)

where h/e2 = 25.8 kΩ is the inverse conductance quantum, ρl = kF /ne2 , kF is the Fermi wave number, n is the density of electrons, e is the electron charge, and ρ is the electrical resistivity of the metal in the contact region. The factor of 2 in (5.7) arises because of spin degeneracy. In the thermal regime energy-resolved spectroscopy is not possible and Joule heating occurs in the contact volume, i. e. in a sphere of radius r around the contact. The contact resistance R0 is dominated by the Maxwell resistance RM = ρ/d. Joule heating of the contact region yields a hot spot with maximum temperature T max given by the Kohlrausch relation (5.8) T max = T B2 + V 2 /4L with T B the bath temperature, V the applied voltage and L the Lorenz number. As an example, the diﬀerential resistance dV/dI vs voltage V of an UPt3 -Pt contact in the thermal regime is shown in Fig. 5.3. Choosing L = 2L0 in qualitative agreement with the experiment [38], where L0 = 2.45·10−8 WΩ/K2 , the voltage axis of Fig. 5.3 can be rescaled into a T axis and all spectra collapse together onto one “master” curve determined by the local resistivity of the contact area, which means that the voltage-dependent contact resistance resembles the temperature dependent bulk resistivity ρbulk (T ) (see Fig. 5.4). In the heating model [32] the current through the contact is given by 2.5

(1/R0(7.5K)) dV/dI

UPt3-Pt

2.0

1.5

T (K) 35 20 11.5

1.0

7.5

-20

-10

0

10

20

V (mV)

Fig. 5.3. Diﬀerential resistance dV/dI vs voltage V at diﬀerent temperatures T for a lowohmic point contact with R0 = 0.1 Ω. The solid lines are calculated from the bulk resistivity using the heating model [32] (from [39])

5.3 Point-Contact Spectroscopy

43

2.5

(1/R0(7.5K)) dV/dI

UPt3-Pt 2.0

TB (K) 35 20

1.5

11.5 7.5 1.0

R 0(TB) 0

10

20

30

40

50

60

T = (TB2+V2/4L)1/2 (K)

Fig. 5.4. Scaling behaviour of the spectra in the thermal regime. The voltage dependence of dV/dI is replaced by a temperature dependence via the Kohlrausch relation T = (T B2 + V 2 /4L)1/2 (from [39])

I(V) = Vd 0

1

dx ρ(T max (1 − x2 )1/2 )

(5.9)

2 2 where x = ((1 − T 2 /T max )/(1 − T B2 /T max ))1/2 parametrizes the temperature distribution in the contact. ρ denotes the local resistivity of the contact area and has been approximated by ρ(T ) = ρbulk (T ) + ρi . ρi takes into account that the local resistivity might be enhanced by structural defects, boundary scattering or impurities. The solid lines in Fig. 5.3 are calculated with ρi = 6 µΩ cm and L = 2L0 for all T by numerical diﬀerentiation of the above equation for I(V), evidentially proving that thermal eﬀects dominate the point-contact spectra in this T range. In the diﬀusive regime it is still possible to obtain direct energy-resolved information about the inelastic scattering of the electrons as in the ballistic regime. The same holds true for the intermediate tunnel regime. However, in both cases the nonlinearities in the I–V curves become considerably weaker, the relative intensity vanishes proportionally to li /d and D, respectively [32]. The crossover between the ballistic and nonballistic regimes takes place at suﬃciently small values of li or D, respectively. A simple interpolation formula for the total resistance was derived by Wexler [40]: 16ρl ρ(T ) . (5.10) R0 = +γ d 3πd2 γ is a slowly variing monotonic function which varies between γ(0) = 1 for d and γ(∞) ≈ 0.7 for d. Therefore, the predominance of one or another term in R0 depends on the relation between d and ρ. For high-resistive materials and/or large contacts, the Maxwell contribution dominates the current-voltage characteristic, while the Sharvin resistance dominates for clean metals and small constrictions.

44

5 Probing the Energy Gap

Normal Metal - Superconductor Contacts An additional mechanism of transport through the contact occurs when the temperature is lowered below the superconducting transition temperature T c of one electrode of a metal-metal point contact: Andreev reﬂection at the normal metalsuperconductor interface. As a conventional superconductor is characterized by a gap ∆ in its single-particle spectrum, single electrons cannot be injected into the superconductor for energy |eV| < ∆, but two electrons forming a Cooper pair can be injected. This scattering process in which an electron is injected and a hole is retroreﬂected with probability A is called Andreev reﬂection. Andreev reﬂection leads to a characteristic increase in the diﬀerential conductance dI/dV (see Fig. 5.5a) of a point contact at bias voltages of the order of the gap energy ∆/e [41, 42], i. e. a minimum in the diﬀerential resistance dV/dI vs V (see Fig. 5.5b). A small fraction of normally reﬂected electrons due to an interface barrier increases the zero-bias resistance and leads to the characteristic double-minimum feature in dV/dI.

Fig. 5.5. Diﬀerential conductance dI/dV vs V (a), and diﬀerential resistance dV/dI vs V (b) for a N-S contact with an isotropic energy gap ∆(k) = ∆0 and arbitrary interface barrier strength Z. Calculated after [41]

To describe the point-contact experiments theoretically, the dynamics of quasiparticles in an inhomogeneous superconductor have to be known. There is a wellestablished method to solve this problem, viz. the Andreev equations which are a quasiclassical version of the Bogoliubov - de Gennes equations d ˆ z)¯v(z) E u¯ (z) = −ivF ( kˆ · zˆ ) dz u¯ (z) + ∆( k, ˆ z)¯u(z) . E v¯ (z) = ivF ( kˆ · zˆ ) d v¯ (z) + ∆∗ ( k, dz

(5.11)

Here u¯ and v¯ are the particle and hole components of the quasiparticle wave functions, zˆ is a unit vector in the z direction, and it was assumed that the situation is one-dimensional, i.e., that the order parameter is homogeneous in the x and y directions. In these equations, the pair potential ∆ acts as an oﬀ-diagonal potential coupling particle and hole degrees of freedom. In general, the Andreev equations

5.3 Point-Contact Spectroscopy

45

(5.11) contain a k-dependent and spatially dependent pair potential ∆(k, r) which has to be determined self-consistently, because a boundary of an unconventional superconductor will be pair-breaking [43]. Once an order-parameter symmetry has been chosen and the self-consistency problem has been solved, the Andreev equations can be integrated numerically. From the solution the energy-dependent Andreev reﬂection coeﬃcient, the normal (specular) reﬂection coeﬃcient, and, ﬁnally, the diﬀerential resistance dV/dI can be determined. Blonder, Tinkham and Klapwijk [41] and - in a more general way - Zaitsev [42] have done this for a N-S constriction with an interface barrier of arbitrary strength Z separating the normal metal (N) and a superconductor (S) which has an isotropic energy gap ∆(k) = ∆0 = const. (conventional s-wave superconductor). Both derived an expression for the current

∞ [ f0 (E − eV) − f0 (E)]T (E)dE

I(V) ∼

(5.12)

−∞

and computed a family of I-V curves ranging from the tunnel junction (Z 1) to the metallic limit (Z = 0) (see Fig. 5.5). Here, the electrochemical potential µ of the pairs in the superconductor was choosen as a reference level for the energy E, and it was assumed that the distribution function of electrons in N and quasiparticles in S is given by the Fermi functions f0 (E) = 1/(exp(E/kB T ) + 1) ≡ 12 (1 − tanh(E/2kB T )) and f0 (E − eV) shifted relatively to each other by eV, where V is the voltage across the contact. The transmission coeﬃcient for the current T (E) which accounts for both normal reﬂection and Andreev reﬂection is given by [42]:  2∆2    2 2 2 2 2 |E| < ∆    E +(∆ −E )(2Z +1) (5.13) T (E) =    2|E|   √ |E| > ∆  2 2 2 |E|+ E −∆ (2Z +1)

Using the above model the diﬀerential resistance dV/dI as a function of applied voltage V can be calculated, and the gap value ∆0 can be derived from the position of the minima in the spectrum, V = ±∆/e. As an example, the point-contact spectra for a nanolithographic Al-Cu contact (see sketch in Fig. 5.2d) at diﬀerent temperatures together with the curves calculated according to (5.12) and (5.13) are shown in Fig. 5.6. The ﬁt curves are hardly visible because of the perfect agreement with the data. Minima in the diﬀerential resistance are expected for V ≤ ∆/e for both conventional superconductors [41] and unconventional superconductors [43]. However, for the latter, information on the gap symmetry and width cannot be obtained so straightforwardly as in the case of ordinary superconductors by following reasons: (1) Because of the self-consistency problem (see above) ∆ in (5.13) cannot simply be replaced by ∆(k). (2) The gap value ∆/e cannot directly be read from the position of the minima in dV/dI. The position of the minima depends on the choice of the orderparameter symmetry, the microscopic details of the contact and the barrier strength.

46

5 Probing the Energy Gap

2

G / GN

1.8 1.6

Cu / Al τ = 0.782 Γ / ∆ = 0.024 ∆ = 207 µeV

T (mK) 95 212 327 497 646 796 920 1040

1.4 1.2 1 -0.8

-0.4

0 U (mV)

0.4

0.8

Fig. 5.6. Andreev spectra for a nanolithographic point contact between Cu and Al (T c = 1.12 K) (symbols) and the corresponding theoretical curves according to Blonder-TinkhamKlapwijk theory (lines). The ﬁt parameters are ∆0 = 207 µeV, and τ = 1/(1 + Z 2 ) = 0.782 corresponding to Z = 0.528 at given temperature T . In order to improve the ﬁt quality, a small inelastic broadening Γ = 5 µeV has been taken into account (from [44])

In order to illustrate this in more detail the following example is considered: The diﬀerential resistance dV/dI as a function of applied voltage V was calculated for the following order-parameter symmetries [45]: isotropic order parameter: ∆(k) = ∆0 = const. E1u :

∆(k) ∼ η1 |kz |k x + η2 |kz |ky η = (1, ±i) η = (1, 0) η = (1, 0) with belt nodes

E2g :

∆(k) ∼ η1 |kz |(k2x − ky2 ) + 2η2 |kz |ky k x η = (1, ±i) η = (1, 0)

For η = (1, ±i), both the E1u and E2g phases deﬁned above have a hybrid gap with both a line of nodes in the basal plane and point nodes along the c axis (see Fig. 2.2.). Although these order-parameter symmetries have been considered to account for point-contact data on UPt3 [45], the result is quite general and is valid for other choices as well. Since the microscopic structure of point contacts usually is not known, it has to be parameterized by its normal reﬂectivity properties (reﬂection coeﬃcient R) and its injection characteristics (preferentially in forward direction with an angular spread described by a cone with an opening angle θ). The resulting point-contact

5.3 Point-Contact Spectroscopy

47

Fig. 5.7. Calculated dV/dI vs. eV curves for diﬀerent order parameter symmetries for current injection kz within a cone with maximum angle Θ = 18◦ . dV/dI has been normalized to the normal resistance RN , the energy scale has been normalized to ∆0 . The reﬂection coeﬃcient at the interface was chosen to be R = 0.025 (a), 0.15 (b) and 0.30 (c). (from [45])

48

5 Probing the Energy Gap

Fig. 5.8. Calculated dV/dI vs. eV curves under the same assumptions as in Fig. 5.7, but with line and belt nodes. (from [45])

5.3 Point-Contact Spectroscopy

49

spectra normalized to the normal resistance RN for diﬀerent reﬂection coeﬃcients R are displayed in Fig. 5.7a–c. All spectra in Figs. 5.7, 5.8, and 5.10 have been calculated for a ﬁnite temperature T/T c = 0.08. ∆0 is always the maximum value of the energy gap. The microscopic structure of the point contact has been modelled by an injection of the electrons preferentially in forward direction kz (direction of point nodes) with an angular spread described by a cone with a maximum angle Θ = 18◦ and reﬂection coeﬃcients R = 0.025, 0.15, and 0.30, corresponding to Z = (R/(1 − R))1/2 = 0.16, 0.42, and 0.65, respectively. The occurrence of a double-minimum structure in dV/dI depends on three parameters: (1) on the maximum angle Θ, (2) on the reﬂection coeﬃcient R, and (3) on the choice of the order-parameter symmetry. For ﬁxed Θ and R (see Fig. 5.7) the order parameter with E2g symmetry always leads to smaller structures in dV/dI than the order parameter with E1u or the isotropic order parameter. This can be explained by a diﬀerence in the excitation spectrum near the point nodes. While both 2D-representations have the same topology of nodes, namely a line of nodes in the basal plane and point nodes at the poles, in the E1u state the gap opens up linearly with polar angle, whereas it opens up quadratically in the E2g state. This diﬀerence explains why one has to choose a cone with larger maximum angle for electron injection to “see” the same energy gap as in the E1u state. This diﬀerence is also important for the occurrence of the characteristic double-minimum structure and the position of the minima. For low R (and/or small Θ) only the isotropic order parameter leads to a distinct double-minimum structure, while a weak structure is obtained for E1u symmetry and only a single minimum for E2g symmetry. The double-minimum structure can be reproduced by increasing R and/or Θ for the E2g order parameter, too (Fig. 5.7 c). The voltage at which the double minimum occurs is expected to be ±∆/e for the isotropic order parameter [41] and to be only weakly dependent on R. In the anisotropic case the position of the minimum also depends only weakly on R, but is strongly inﬂuenced by the choice of Θ. For larger opening angles Θ the minima occur at higher voltages (not shown) and therefore the structures are much wider. This might explain the large variety of the shape of dV/dI curves observed experimentally. The last point which should be considered in the analysis of Andreev spectra is the resistance change r = (RN − R0 )/RN , where R0 is the zero-bias resistance. For unconventional superconductors it has been notoriously diﬃcult to describe the absolute size of the resistance change caused by Andreev scattering, where experimentally r typically amounts to a few percent only. For an isotropic order parameter, Andreev reﬂection should reduce the diﬀerential resistance by a factor of two for Z = 0 (or at least at eV ≈ ∆0 in other cases) i. e. r = 1/2. For an anisotropic order parameter, however, the resistance change depends on the direction of current ﬂow. Figure 5.7a shows that in the isotropic case and for low R, the ratio r is almost 1/2. If R is increased, r decreases and at least becomes negative for large R as already shown by BTK [41]. Nevertheless the ratio (RN − Rmin )/RN , where Rmin is the minimum diﬀerential resistance, remains nearly unchanged at 1/2. For the anisotropic

50

5 Probing the Energy Gap

order parameters r is of the order of 1/2 only for low R and large Θ (Fig. 5.7a), for higher R it is reduced signiﬁcantly (Fig. 5.7c). A strong reduction of r is obtained for a current ﬂow within a direction which contains a line of nodes, i. e. either for a preferred current direction in the basal plane, where both E1u and E2g exhibit a line of nodes or for current ﬂow kz in the η=(1,0) state of E1u and E2g , when an additional line of nodes appears in the plane k x = 0 or two additional lines appear in the plane k x = ±ky , respectively. Figure 5.8 shows the diﬀerential resistance calculated under the same assumptions as in Fig. 5.7, but for the η=(1,0) state of E1u and E2g . The resistance change is only about 1/2 to 1/3 of that obtained in Fig. 5.7, i. e. for current ﬂow within a direction which contains only point nodes, but it is still larger than the often experimentally found resistance change which is of the order of a few percent. Thus the nodal structure of an anisotropic order parameter alone cannot explain the small values of r, and an additional mechanism is required. For current ﬂow in a direction which contains a line of nodes, a possible explanation might be at hand. Whereas point nodes in the gap are stable to the addition of resonantly scattering impurities to the superconductor [46, 47] (the density of states N(F ) at the Fermi energy becoming ﬁnite only once a certain threshold in the impurity concentration has been reached), resonant impurity scattering leads to a ﬁnite quasiparticle density of states at the chemical potential in a system with a line of nodes. This means that the line of nodes is broadened to a “belt” of nodes, i.e., the system has a normal spectrum for a two-dimensional manifold of points on the Fermi surface. Andreev reﬂection will be suppressed in these directions and completely disappear if the electrons are injected in a polar angle less than the angle subtended by the belt. Figure 5.8 includes calculated dV/dI curves with belt angle 16◦ and with the same injection cone Θ = 18◦ as for the other calculations. Another possible reason for the often observed smallness of r is the presence of a normal-state surface layer due to a suppression of the order parameter at the surface [48]. Recently, it was shown that such a normal-state surface layer has a natural explanation in a p-wave triplet scenario [49], if a region near the interface is assumed in which scattering is enhanced. As shown in Fig. 5.9, such a diﬀusive region leads eﬀectively to a normal-conducting layer near the interface in a p-wave superconductor. The reason is that the p-wave order parameter is very sensitive to scattering. Thus, an enhanced scattering near the surface leads to a suppression of the order parameter near the surface. The presence of a normal-state layer aﬀects strongly point-contact spectra and tunnelling spectra. For sake of completeness, a possible but purely phenomenological mechanism should be mentioned that has been invoked to describe a broadening and a reduced absolute magnitude of the spectra. This mechanism is inelastic broadening of the quasiparticle density-of-states [50]:       E − iΓ   N(E, Γ) = Re  (5.14)

    (E − iΓ)2 − ∆2 

5.3 Point-Contact Spectroscopy

specular surface diffusive surface

dI/dV*R0

∆p−wave/Tc

0.5

2 1.5 1 0.5 0 2

kx+iky

Tunnel

lmfp=15ξ0

1.5 1 0.5

Andreev −2

0 eV/∆max

lmfp=0.3ξ0

1

dI/dV*R0

2

1.5

51

2 kx−iky

0 −10

−8

−6

−4

−2

0

x/ξ0 Fig. 5.9. Creation of a normal-conducting surface layer in a p-wave superconductor due to an increased scattering rate near the surface. For comparison, dashed lines indicate the behaviour of a clean surface. The two p-wave order-parameter components are the bulk k x + iky order parameter, and the subdominant k x − iky order parameter which is stabilized only within a few coherence lengths (ξ0 = v f /2πT c ) near the surface. The p-wave order parameter is calculated self-consistently together with impurity self-energies within the framework of quasiclassical Green’s functions formalism. The results for a mean free path of 0.3 coherence lengths in the shaded region, and of 15 coherence lengths elsewhere are shown. Both orderparameter components are suppressed in a layer with increased scattering, leading eﬀectively to a normal-conducting surface layer. The calculations are for T = 0.05T c . The insets show the corresponding point-contact spectra (bottom) and tunnelling spectra (top) (from [49])

Here Γ is the broadening parameter which accounts for the ﬁnite quasiparticle lifetime τ with Γ = /τ. This will lead to a broadening of any structure in dV/dI and to a decrease in its absolute size. The eﬀect of an energy-independent inelastic width is shown in Fig. 5.10. Clearly, the absolute size of the structures in dV/dI can be diminished by an order of magnitude or more and the minimum in dV/dI can be broadened. The most remarkable consequence of Andreev reﬂection is the occurrence of an extra current Iexc in addition to the normal-state current I = V/RN , where RN is the normal-state resistance of the contact. For bias larger than the gap energy this current is constant if strong-coupling, inelastic and nonequilibrium processes are disregarded [51]. It has been shown that the excess current in s-wave superconductors is proportional to the superconductive energy gap [41, 42] and consequently contains further information on the superconducting state. For a clean metallic N-S contact the excess current is given by

52

5 Probing the Energy Gap

Fig. 5.10. Calculated dV/dI vs. eV curves for the E1u state under the same assumptions as in Fig. 5.7a and b, respectively, but with inelastic broadening Γ/∆0 = 0 (long-dashed line), Γ/∆0 = 0.1 (short-dashed line), Γ/∆0 = 0.2 (solid line) (from [45])

Iexc =

4∆ , 3eRN

(5.15)

but it decreases rapidly with increasing barrier strength Z. The proportionality between ∆ and Iexc requires that both quantities exhibit the same functional dependence on temperature and magnetic ﬁeld, respectively. For unconventional superconductors, however, this relation is altered and the excess current is not necessarily proportional to the order parameter [49]. In general, one obtains a scaling relation near Tc Iexc = constant × ∆1/ν , (5.16) where ν is deﬁned by the order-parameter symmetry. In summary, the current-voltage characteristics between a superconductor and a normal metal yield a lot of information about the energy gap via the mechanism of Andreev reﬂection, and they are suited for directly probing the order parameter of unconventional superconductors. A vast number of experiments have already been performed on these materials and the results will be discussed in Part III. In addition, for the heavy-fermion superconductors there exist two excellent review articles [52, 53] which should furthermore be referred to. In the case of heavy-fermion superconductors there is some controversy on the interpretation [54] of the minimum in dV/dI as caused by Andreev reﬂection or by a Maxwell contribution to the contact resistance. The latter might also lead to a resistance change δR below T c with a scaling behaviour δR ∼ a−1 ∼ R0 . Gloos argues that in those cases the Andreev-reﬂected hole current is suppressed and the point-contact spectra do not contain spectroscopic information about the energy gap in superconductors. An argument in favor of Andreev reﬂection is the simultaneous occurrence of an excess current which is concatenated with the Andreev-reﬂected hole current. Further, if the zero electrical resistance is fully established within a few tenth K below T c , one would expect, for |eV| < ∆, the diﬀerential resistance to

References

53

decrease sharply at T c , reaching its minimum just below T c . In that case there would be no obvious reason why the zero-bias resistance should further decrease substantially with decreasing temperature, as is observed very often. A last phenomenon unexplained in the Maxwell scenario is the occurrence of a zero-bias conductance anomaly for some superconductors. This is a surface resonance phenomenon which is caused by the presence of supgap Andreev bound states, and which is totally unaffected by a Maxwell contribution. Nevertheless, heating eﬀects as well as the inﬂuence of a high current density and the magnetic ﬁeld connected with these currents should be taken into consideration in each case, before dealing with more exotic phenomena in point contacts. It should be mentioned that besides the information on the superconductive energy gap further information might be derived from nonlinearities of the currentvoltage characteristics of N-S contacts at energies well above the gap energy. One can expect those nonlinearities caused by the electron-boson interaction responsible for the formation of Cooper pairs. Unfortunately, such phenomena have up-to-now mainly observed for point contacts on ordinary superconductors. These results have recently been reviewed by Yanson [51].

References 1. E.L. Wolf: Principles of Electron Tunneling Spectroscopy (Oxford University Press, New York, 1985) 36 2. L. Solymar: Superconductive Tunneling and Applications (Chapman and Hall, London, 1972) 36 3. W.L. McMillan, J.M. Rowell: Tunneling and strong-coupling superconductivity (Marcel Dekker, Inc., New York, 1969), p. 561 36, 129 4. J. Bardeen: Phys. Rev. Lett. 6(2), 57 (1961) 36 5. M.H. Cohen, L.M. Falicov, J.C. Phillips: Phys. Rev. Lett. 8(8), 316 (1962) 36 6. I. Giaever: Phys. Rev. Lett. 5, 147 (1960) 36 7. I. Giaever: Phys. Rev. Lett. 5, 464 (1960) 36 8. J.C. Fisher, I. Giaever: J. Appl. Phys. 32(2), 172 (1961) 36 9. L.J. Buchholtz, G. Zwicknagl: Phys. Rev. B 23(11), 5788 (1981) 22, 36 10. C.R. Hu: Phys. Rev. Lett. 72(10), 1526 (1994) 36, 59 11. Y. Tanaka, S. Kashiwaya: Phys. Rev. Lett. 74(17), 3451 (1995) 36, 59 12. M. Fogelstr¨om, D. Rainer, J.A. Sauls: Phys. Rev. Lett. 79(2), 281 (1997). E: Phys. Rev. Lett. 79, 2754 (1997) 36, 59, 79, 144 13. L. Alﬀ, S. Kleeﬁsch, U. Schoop, M. Zittartz, T. Kemen, T. Bauch, A. Marx, R. Gross: Eur. Phys. J. B 5, 423 (1998) 37 14. A.M. Cucolo: Physica C 305, 85 (1998) 37 15. J. Tersoﬀ, D.R. Hamann: Phys. Rev. Lett. 50(25), 1998–2001 (1983) 37 16. J. Tersoﬀ, D.R. Hamann: Phys. Rev. B 31(2), 805–813 (1985) 37 17. C.J. Chen: J. Vac. Sci. Technol. A 6(2), 319 (1988) 37 18. T.E. Feuchtwang, P.H. Cutler, N.M. Miskovsky: Phys. Lett. A 99, 167 (1983) 38 19. A. Messiah: Quantenmechanik (de Gruyter, Berlin, 1990) 38 20. G. Binnig, H. Rohrer: Rev. Mod. Phys. 59(3), 615 (1987) 39 21. A.G.M. Jansen, A.P. van Gelder, P. Wyder: J. Phys. C 13(33), 6073 (1980) 39

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5 Probing the Energy Gap

22. I.K. Yanson: Sov. J. Low Temp. Phys. 9, 343 (1983). (Fiz. Nizk. Temp. 9, 676 (1983)) 39 23. I.K. Yanson, O.I. Shklyarevskii: Sov. J. Low Temp. Phys. 12(9), 509 (1986). (Fiz. Nizk. Temp. 12, 899 (1986)) 39 24. I.K. Yanson: Physica Scripta T23, 88 (1988) 39 25. A.M. Duif, A.G.M. Jansen, P. Wyder: J. Phys.: Condens. Matter 1(20), 3157 (1989) 39 26. Y.G. Naidyuk, I.K. Yanson: Point-contact spectroscopy, Vol. 145 of Springer Series in Solid-State Sciences (Springer, 2004) 39 27. I.K. Yanson: Sov. Phys. JETP 39, 506 (1974) 40 28. J.M. van Ruitenbeek, A. Alvarez, I. Pi˜neyro, C. Graham, P. Joyez, M.H. Devoret, D. Esteve, C. Urbina: Rev. Sci. Instrum. 67(1), 108 (1996) 40 29. E. Scheer, N. Agra¨ıt, J.C. Cuevas, A.L. Yeyati, B. Ludolph, A. Mart´ın-Rodero, G.R. Bollinger, J.M. van Ruitenbeek, C. Urbina: Nature 394, 154 (1998) 40 30. I.O. Kulik, A.N. Omelyanchouk, R.I. Shekhter: Sov. J. Low Temp. Phys. 3(12), 740 (1977). Fiz. Niz. Temp. 3, 1543 (1977). 40 31. I.O. Kulik, R.I. Shekhter, A.G. Shkorbatov: Sov. Phys.- JETP 54(6), 1130 (1981) 40 32. I.O. Kulik: Sov. J. Low Temp. Phys. 18, 302 (1992) 41, 42, 43, 131 33. P.N. Chubov, I.K. Yanson, A.I. Akimenko: Sov. J. Low Temp. 8, 32 (1982) 41 34. A.G.M. Jansen, F.M. Mueller, P. Wyder: Phys. Rev. B 16, 1325 (1977) 41 35. C.J. Muller, J.M. Ruitenbeek, L.J. de Jongh: Physica C 191, 485 (1992) 41 36. K.S. Ralls, R.A. Buhrmann, R.C. Tiberio: Appl. Phys. Lett. 55, 2459 (1989) 41 37. Y. Imry: Directions in Condensed Matter Physics (World Scientiﬁc, Singapore, 1986), p. 101 41 38. J.J.M. Franse, A. Menovsky, A. de Visser, C.D. Bredl, U. Gottwick, W. Lieke, H.M. Mayer, U. Rauchschwalbe, G. Sparn, F. Steglich: Z. Phys. B 59, 15–19 (1985) 42, 69, 153, 154 39. G. Goll, H. v. L¨ohneysen: Physica C 317-318, 82 (1999) 42, 43 40. A. Wexler: Proc. Phys. Soc. 89(4), 927 (1966) 43 41. G.E. Blonder, M. Tinkham, T.M. Klapwijk: Phys. Rev. B 25, 4515 (1982) 44, 45, 49, 51, 78, 99, 133 42. A.V. Zaitsev: JETP 59, 1015 (1984) 44, 45, 51 43. C. Bruder: Phys. Rev. B 41, 4017 (1990) 45, 78 44. F. Perez-Willard: (2003), Ph.D. thesis, Universit¨at Karlsruhe 46 45. G. Goll, C. Bruder, H. v. L¨ohneysen: Phys. Rev. B 52, 6801 (1995) 46, 47, 48, 52, 97, 101, 158 46. P. Hirschfeld, D. Vollhardt, P. W¨olﬂe: Solid State Commun. 59(3), 111 (1986) 22, 50, 98 47. P.J. Hirschfeld, P. W¨olﬂe, D. Einzel: Phys. Rev. B 37, 83 (1988) 22, 50 48. M. Sigrist, K. Ueda: Rev. Mod. Phys. 63(2), 239 (1991) 11, 14, 17, 29, 50, 61, 90, 100, 128, 137, 138 49. F. Laube, G. Goll, M. Eschrig, M. Fogelstr¨om, R. Werner: Phys. Rev. B 69, 014 516 (2004). (see also cond-mat/0301221) 50, 51, 52, 133, 159 50. R.C. Dynes, V. Narayanamuri, J.P. Garno: Phys. Rev. Lett. 41, 1509 (1978) 50, 115 51. I.K. Yanson: “Point-contact spectroscopy of superconductors”, in Quantum Mesoscopic Phenomena and Mesoscopic Devises in Microelectronics, ed. by I.O. Kulik, R. Ellialtioglu (Kluwer Academic Publisher, Dordrecht, Netherlands, 2000), p. 61 51, 53 52. H. v. L¨ohneysen: Physica B 218, 148 (1996) 52 53. Y.G. Naidyuk, I.K. Yanson: J. Phys.: Condens. Matter 10(40), 8905 (1998) 52 54. K. Gloos, F.B. Anders, B. Buschinger, C. Geibel, K. Heuser, F. J¨arling, J.S. Kim, R. Klemens, R. M¨uller-Reisener, C. Schank, G.R. Stewart: J. Low Temp. Phys. 105(1/2), 37 (1996) 52, 71, 97, 114, 158, 159

6 Probing the Phase

Condensation into a macroscopic quantum state ﬁxes the phase of the superconductor and makes it macroscopically accessible. Besides the size of the energy gap, information on the superconductive phase is a second important property which helps to assign the superconductive order parameter. Phase information is usually probed by the Josephson eﬀect which is based on pair tunnelling. However, unconventional superconductors even provide phase information via quasiparticle tunnelling. A sign change of the superconductive phase leads to the formation of an Andreev bound state which can be probed by the tunnelling eﬀect. Both eﬀects will be discussed in the following sections.

6.1 Josephson Eﬀect The Josephson eﬀect is certainly one of the most intriguing phenomena in superconductivity. It is a consequence of coherent tunnelling between two superconducting condensates, each of which is represented by a complex macroscopic wave function. Two superconductors separated by a thin insulating barrier form a junction which is called a Josephson junction, based on the prediction in 1962 by B. Josephson that Cooper pairs could tunnel across such a junction from one superconductor to the other with no resistance [1]. The tunnelling of Cooper pairs constitutes a current, which is observed even when there is no voltage drop across the junction. The current depends on the diﬀerence in phase of the wave functions that describe the Cooper pairs. Let ϕL be the phase for the wave function of a Cooper pair in one superconductor. All the Cooper pairs act coherently, so they all have the same phase. If ϕR is the phase constant for the Cooper pairs in the second superconductor, the current across the junction is given by I = Imax sin(ϕL − ϕR )

(6.1)

where Imax is the maximum current (Josephson critical current), which depends on the coupling strength of the two pair condensates, and can be modiﬁed, e. g. by the thickness of the barrier. The prediction of pair tunnelling is known as the dc Josephson eﬀect and was almost immediately veriﬁed by experiment [2]. Josephson tunnelling is not just limited to superconductor-insulator-superconductor (SIS) junctions, but has been generalized to all weak-link structures consisting of two not G. Goll: Unconventional Superconductors STMP 214, 55–60 (2006) c Springer-Verlag Berlin Heidelberg 2006

56

6 Probing the Phase

Fig. 6.1. The Josephson eﬀect can take place in diﬀerent types of structures. Structures (b) to (h) are diﬀerent weak links with a direct (non-tunnel-type) conductivity. (a) tunnel junction (SIS sandwich), (b) sandwich, (c) proximity eﬀect bridge, (d) ion-implanted bridge, (e) Dayem bridge, (f) variable-thickness bridge, (g) point contact, (h) blob-type junction. The conducting regions of the two last types of weak links are shown schematically in the circle on the bottom. S stands for superconductor, S for the superconductor with reduced critical parameters, N for normal metal or alloy, SE for semiconductor (usually heavily doped), and I stands for insulator (from [3])

necessarily identical superconductors coupled by a small region of depressed order parameter [3, 4]. Some examples are shown in Fig. 6.1. Josephson weak links include SIS, SNS, and SCS, where N and C stand for normal metal and constriction, respectively. Initially, Pals et al. [5] suggested that under quite general conditions there is no Josephson coupling between a spin-singlet (even parity) and -triplet (odd parity) superconductor up to the second order in the transition-matrix element. This remains true even with paramagnetic impurities in the tunnel barrier. By coupling a particular superconductor weakly to a well-known spin-singlet pairing superconductor it should be possible to investigate experimentally whether the particular supercon-

6.1 Josephson Eﬀect

57

ductor has spin-triplet pair. However, it was subsequently pointed out that a Josephson coupling could arise from spin-orbit coupling [6]. In the presence of spin-orbit coupling, the Cooper pairs of diﬀerent parities will be mixed at the interface between the s- and p-wave superconductor, resulting in a direct Josephson coupling between them. A second property of the Josephson eﬀect is that it contains information about the magnitude and phase of the order parameter of the superconductors on both sides of the tunnel barrier. In the simplest treatment of an isotropic and uniform tunnelling probability and identical isotropic s-wave superconductors on both sides the Josephson current as a function of temperature is [7] ∆(T ) π∆(T ) tanh (6.2) Imax (T ) = 2eRn 2kB T where Rn is the normal-state tunnelling resistance. This is the Ambegaokar-Baratoﬀ formula for the temperature dependence of the Josephson critical current which has been conﬁrmed by numerous experiments. The concept of Josephson tunnelling implies that the Josephson eﬀect is a direction-sensitive phenomenon connected with the orientation of the junction and with the crystal axes of the superconductor on each side. This fact is of minor importance for conventional s-wave superconductors with an essentially isotropic pair wave function. However, in the case of non-s-wave superconductivity, where the pair wave functions have an internal angular structure, this property can lead to intriguing new eﬀects. The combination of two Josephson contacts in a parallel arrangement forming a superconducting loop is the main constituent of a superconducting quantum interferometer or dc SQUID (see Fig. 6.2). The supercurrent through this arrangement depends on the critical currents Ic1 and Ic2 through the two junctions and the phase jumps ϕ1 and ϕ2 across the junctions: I = Ic1 sin ∆ϕ1 + Ic2 sin ∆ϕ2

(6.3)

The diﬀerence between the phase jumps, i. e. the total change in the order parameter phase after the round trip over the entire loop, is composed of the phase diﬀerence due to the applied magnetic ﬁeld, Adr = φ/φ0 , and the ’internal’ phase jump at the interface between both junctions: ∆ϕ1 − ∆ϕ2 =

2πφ + δ12 φ0

(6.4)

The extra term δ12 which accounts for the intrinsic phase shift is absent in conventional superconductors, i. e. δ12 = 0, for all orientations of the junctions. However, it becomes non-zero for unconventional superconductors where the order parameter exhibits a sign change for certain directions. δ12 is the key term for all phasesensitive experiments identifying the order-parameter symmetry. For a symmetric dc SQUID with Ic1 = Ic2 = I0 the maximum supercurrent modulates with applied ﬂux according to

58

6 Probing the Phase

Fig. 6.2. Design of a corner SQUID experiment used to determine the relative phase between orthogonal directions and the resulting modulation of the critical current as a function of applied magnetic ﬂux (from [8])

Imax

πφ δ12 = 2I0 cos + φ0 2

(6.5)

For δ12 = 0 the supercurrent reaches its maximal value Imax = 2I0 for zero external ﬂux and all integer multiples of φ0 (left panel of Fig. 6.2). If the superconductor exhibits an additional phase shift π due to the order-parameter symmetry, the interference pattern is shifted by half a ﬂux quantum and the maxima of Imax are now at φ = φ0 (2n + 1)/2 (right panel of Fig. 6.2). A further consequence of an internal phase shift due to the order-parameter symmetry is the occurrence of a half-integer ﬂux quantum enclosed in a superconductive ring containing an odd number of π shifts. This can be realized if the ring is ﬁt together from several segments in such a way that due to diﬀerent orientations of the segments a π shift is achieved at the junction between two segments. Therefore, for an even number of π junctions the quantization condition for the ﬂux enclosed by the loop is unaltered from the standard φ = φ0 n, but for an odd number of π shifts one obtains 2n + 1 φ = φ0 . (6.6) 2 Pairing symmetry tests using the half-ﬂux quantum eﬀect were ﬁrst proposed for the heavy-fermion superconductors by Geshkenbein and coworkers [9, 10], but the existence of the sign change in the order-parameter phase has ﬁrst been demonstrated for the high-temperature superconductors. The SQUID interferometry experiments by Wollman et al. [11] on a YBa2 Cu3 O7 -Pb corner SQUID and the half-integer ﬂux quantum experiments by Tsui et al. [12, 13] of YBa2 Cu3 O7 rings on a tricrystal substrate have unambiguously revealed the d x2 −y2 order-parameter symmetry of

References

59

YBa2 Cu3 O7 . These experiments have become a hallmark of similar pairing symmetry tests on high-temperature superconductors and will be reviewed in Sect. 10.2.

6.2 Zero-Bias Anomalies As already mentioned in Sect. 5.1 under certain conditions quasiparticle tunnelling through a normal metal-superconductor junction may reveal information on the superconductive phase as well. This is the case if a zero-energy Andreev bound state is formed in the potential well at the superconductors surface. The Andreev bound state is a result of constructive interference of electron- and holelike quasiparticles. It appears at zero energy if the pair potential exhibits a sign change. A well examined example is the Andreev bound state formed at the [110] surface of the hightemperature superconductors with d x2 −y2 symmetry of the order parameter and a sign change under 90◦ rotation. The appearance of the surface bound state in the highT c ’s has been treated by several theoretical groups [14, 15, 16]. Experimentally, a zero-bias conductance peak is observed in the tunnel spectra [17, 18, 19]. It has also been shown that for any d x2 −y2 superconductor the presence of the zero-energy bound state allows the emergence of a phase with spontaneously broken time-reversal symmetry [20, 21] which can lead to a spontaneous splitting of the zero-energy conductance peak with decreasing temperature [16]. In a magnetic ﬁeld, the bound state spectrum is shifted by screening currents which leads to a splitting of the zero-bias conductance peak. The splitting is linear in H and saturates at the pair-breaking critical ﬁeld [16]. These ﬁndings give additional tools at hand for an identiﬁcation of the order-parameter symmetry. The formation of the zero-energy bound states does not only inﬂuence the tunnelling properties. Many important eﬀects peculiar to d-wave superconductivity can be related to this state, e. g. paramagnetic currents at surfaces and defects, unusual Josephson current-phase relation, π-junction behaviour, etc.. An overview over these eﬀects and a general discussion of Andreev bound states in high-T c superconducting junctions is found in [22].

References 1. 2. 3. 4. 5. 6. 7. 8.

B.D. Josephson: Phys. Lett. 1(7), 251 (1962) 55 P.W. Anderson, J.M. Rowell: Phys. Rev. Lett. 10, 230 (1963) 55 K.K. Likharev: Rev. Mod. Phys. 51, 101 (1979) 56 M. Tinkham: Introduction to Superconductivity (McGraw-Hill International Editions, Singapore, 1996) 11, 23, 24, 29, 56 J.A. Pals, W. van Haeringen, M.H. van Maaren: Phys. Rev. B 15, 2592 (1977) 56, 72 E.W. Fenton: Solid State Commun. 54, 709 (1985) 57 V. Ambegaokar, A. Baratoﬀ: Phys. Rev. Lett. 10, 486 (1963). E: Phys. Rev. Lett. 11, 104 (1963). 57 D.J. Van Harlingen: Rev. Mod. Phys. 67(2), 515 (1995) 58, 136, 145

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9. V.B. Geshkenbein, A.I. Larkin: JETP Lett. 43, 395 (1986). (Zh. Eksp. Teor. Fiz. 43, 306 (1986)) 58 10. V.B. Geshkenbein, A.I. Larkin, A. Barone: Phys. Rev. B 36, 235 (1987) 58 11. D.A. Wollman, D.J. Van Harlingen, W.C. Lee, D.M. Ginsberg, A.J. Leggett: Phys. Rev. Lett. 71(13), 2134 (1993) 58, 144, 145 12. C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T. Shaw, J.Z. Sun, M.B. Ketchen: Phys. Rev. Lett. 73(4), 593 (1994) 58, 144, 146 13. J.R. Kirtley, C.C. Tsuei, J.Z. Sun, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, M. Rupp, M.B. Ketchen: Nature 373, 225 (1995) 6, 58 14. C.R. Hu: Phys. Rev. Lett. 72(10), 1526 (1994) 36, 59 15. Y. Tanaka, S. Kashiwaya: Phys. Rev. Lett. 74(17), 3451 (1995) 36, 59 16. M. Fogelstr¨om, D. Rainer, J.A. Sauls: Phys. Rev. Lett. 79(2), 281 (1997). E: Phys. Rev. Lett. 79, 2754 (1997) 36, 59, 79, 144 17. J. Geerk, X.X. Xi, G. Linker: Z. Phys. B 73, 329 (1988) 59 18. J. Lesueur, L.H. Greene, W.L. Feldmann, A. Inam: Physica C 191(3–4), 325 (1992) 59 19. M. Covington, M. Aprili, E. Paraoanu, L.H. Greene, F. Xu, J. Zhu, C.A. Mirkin: Phys. Rev. Lett. 79, 277 (1997) 59 20. L.J. Buchholtz, M. Palumbo, D. Rainer, J.A. Sauls: J. Low Temp. Phys. 101(5/6), 1079 (1995) 59 21. L.J. Buchholtz, M. Palumbo, D. Rainer, J.A. Sauls: J. Low Temp. Phys. 101(5/6), 1099 (1995) 59 22. T. L¨ofwander, V.S. Shumeiko, G. Wendin: Supercond. Sci. Technol. 14(5), R53–R77 (2001) 59

7 Probing the Vortices: Lattice Symmetry and Internal Structure

A spatial variation of the superconducting properties occurs when a magnetic ﬁeld is present perpendicular to the surface of a type-II superconductor. For applied magnetic ﬁelds larger than the lower critical ﬁeld Hc1 and below the upper critical ﬁeld Hc2 of the superconductor the magnetic ﬁeld penetrates the superconductor in quantized magnetic ﬂux lines called vortices. The vortex core contains a single ﬂux quantum φ0 and the material is in the normal state. In a conventional s-wave superconductor, shielding currents circulate around the individual vortices, but the ﬁeld decays outside the core region over the length scale λ. Within the core region, the superconducting order parameter is strongly suppressed. With increasing ﬁeld strength, the vortex density increases and repulsive forces between the vortices become more and more important. The interaction leads to the formation of a triangular ﬂux-line lattice, the so-called Abrikosov ﬂux-line lattices, in order to minimize its free energy. The diﬀerence in the free energy for the triangular lattice compared to square lattice is only about 2% [1], and therefore, it is understandable that originally, Abrikosov found that the square lattice is the most stable arrangement [2]. Later this (numerical) error has been rectiﬁed and it was shown that the triangular array is the most favorable one [1]. A consequence of the small diﬀerence is that a relatively weak anisotropy is capable of changing this balance, leading to a distorted triangular or a square ﬂux line lattice. In general, if superconductivity is described by a multicomponent order parameter various types of vortex structures are expected. The problem has been treated theoretically by several groups [3, 4, 5, 6]. It has been shown that the multiple components of the order parameter allow an internal structure of the vortex without generating a normal core region, i. e. a zero of the total order parameter. In addition, fractionally quantized vortices can be observed under certain conditions. Experiments that probe the vortex state can be divided into three categories, those that measure thermal and transport properties (thermal conductivity, resistivity, speciﬁc heat, etc.), those that measure the electronic structure (scanning tunnelling spectroscopy) and those that measure the inhomogeneous magnetic ﬁeld (small-angle neutron scattering, magnetic imaging with SQUIDs [7], nuclear magnetic resonance, muon spin rotation). A fourth, and historically very important method is the decoration method, however, this method will not be considered here.

G. Goll: Unconventional Superconductors STMP 214, 61–64 (2006) c Springer-Verlag Berlin Heidelberg 2006

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7 Probing the Vortices: Lattice Symmetry and Internal Structure

7.1 Scanning Tunnelling Spectroscopy The ability of the scanning tunnelling microscope (STM) to perform local spectroscopic measurements makes it an excellent tool for locating the vortices and studying their electronic structure. By measuring the spatial variation of dI/dV at a bias voltage where a considerable diﬀerence between the normal and superconducting state exists, images of the vortex lattice can be obtained at diﬀerent values of applied magnetic ﬁeld. Furthermore, information on the internal electronic structure and the spatial symmetry of the single vortex are revealed by measurements of the density of states in and near the vortex core. One early example of the ability of this method has been the investigation by Hess et al. on 2H-NbSe2 [8, 9] who imaged ﬁrst the triangular Abrikosov lattice in a ﬁeld of B = 1 T. Surprisingly, they observed a zerobias enhancement in the vortex core instead of a constant conductivity due to a ﬂat normal state density of states. This ﬁnding was explained by Andreev reﬂection of an electron-like quasiparticle contained in the potential well around the vortex [10]. Further experiments have been done on the high-temperature superconductors [11] and the borocarbide superconductors [12].

Fig. 7.1. Abrikosov ﬂux lattice in NbSe2 at 1.8 K and B = 1 T (left panel) and diﬀerential conductance spectra dI/dV vs. V along a line that intersects a vortex (right panel) (from [8])

7.2 Small-Angle Neutron Scattering (SANS) The most often used method to investigate the vortex lattice structure has become neutron scattering. Neutrons are spin-1/2 particles and therefore sensitive to magnetic scatterers within a sample. The vortex lattice consists of an arrangement of vortices each carrying one ﬂux quantum φ0 and therefore can be regarded as a periodic arrangement of magnetic scatterers. Small-angle neutron scattering provides a direct determination of the geometry and orientation of the ﬂux-line lattice. The scattered intensities give further quantitative information about the spatial form of

References

63

the magnetic ﬁeld distribution in the sample. The lattice constant a of the vortex lattice depends on the applied magnetic ﬁeld and is given by φ0 (7.1) a=c B with φ0 = 2 · 10−15 Tm2 and c2 = 1 for the square array and c = 1.075 for the Abrikosov triangular array. For B = 1 T the ﬂux lattice has a triangular lattice constant a = 48 nm which is much larger than the usual lattice constants of crystals. Therefore, the scattering angles at which the Bragg peaks occur are quite small. A comprehensive introduction into the measurement technique and its application to vortex lattices in superconductors is given in [13].

7.3 Muon Spin Rotation The muon spin rotation (µSR) technique provides a sensitive local probe of the spatially inhomogeneous magnetic ﬁeld associated with the vortex state. For the case of a regular vortex lattice (Abrikosov lattice [2]), the magnetic penetration depth λ and the coherence length ξ can be extracted from the measured internal ﬁeld distribution. Furthermore, as shown by Kogan [14], the angle α characterizing the vortex lattice depends only on the penetration length ratio: √ (7.2) tan α = 3(λ x /λy ) If the penetration length is isotropic, α equals 60◦ . Therefore, although α is most naturally measured by small-angle neutron scattering (SANS), one can determine this angle by µSR as well. Certainly, the main advantage of this technique is that it yields information on the internal structure of the vortices. For this purpose the autocorrelation function G(t) (see Sect. 4.3) has to be analyzed carefully. As pointed out by Sonier et al. [15], a simple Gaussian ﬁt of the line shape is not suﬃcient and even yields false conclusion regarding the symmetry of the pairing state. For this purpose the complete line shape has to be taken into account which enables investigations of the change in the spatial arrangement of vortices and the structure and symmetry of the vortex cores. More details on µSR studies of the vortex state in type-II superconductors can be found in [15].

References 1. 2. 3. 4. 5. 6.

W.H. Kleiner, L.M. Roth, S.H. Autler: Phys. Rev. 133(5), A1226 (1964) 61 A.A. Abrikosov: Sov. Phys. JETP 5(6), 1174 (1957) 61, 63 T.A. Tokuyasu, D.W. Hess, J.A. Sauls: Phys. Rev. B 41(13), 8891 (1990) 61, 103 M. Sigrist, K. Ueda: Rev. Mod. Phys. 63(2), 239 (1991) 11, 14, 17, 29, 50, 61, 90, 100, 128, 137, 138 N. Ogawa, M.E. Zhitomirsky: Physica B 281-282, 947 (2000) 61 T. Champel, V.P. Mineev: Phys. Rev. Lett. 86(21), 4903 (2001) 61, 103

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7 Probing the Vortices: Lattice Symmetry and Internal Structure

7. C. Veauvy, K. Hasselbach, D. Mailly: Rev. Sci. Inst. 73(11), 3825 (2002) 61 8. H.F. Hess, R.B. Robinson, R.C. Dynes, J.M. Valles, Jr., J.V. Waszczak: Phys. Rev. Lett. 62(2), 214–216 (1989) 62 9. H.F. Hess, R.B. Robinson, J.V. Waszczak: Phys. Rev. Lett. 64(22), 2711 (1990) 62 10. J.D. Shore, M. Huang, A.T. Dorsey, J.P. Sethna: Phys. Rev. Lett. 62(26), 3089 (1989) 62 11. I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, Ø. Fischer: Phys. Rev. Lett. 75(14), 2754–2757 (1995) 62, 142, 146 12. Y. De Wilde, M. Iavarone, U. Welp, V. Metlushko, A.E. Koshelev, I. Aranson, G.W. Crabtree, P.C. Cauﬁeld: Phys. Rev. Lett. 78(22), 4273–4276 (1997) 62 13. A. Huxley: Neutron scattering from vortex lattices in superconductors, Solid State Sciences (Springer, 2002), Chap. 18, pp. 301–320 63 14. V.G. Kogan: Phys. Lett. A 85(5), 298 (1981) 63 15. J.E. Sonier, J.H. Brewer, R.F. Kieﬂ: Rev. Mod. Phys. 72(3), 769 (2000) 32, 63

Part III

Possible Unconventional Superconductors

8 Ce-Based Heavy-Fermion Superconductors

Cerium-based heavy-fermion superconductors are one class of unconventional superconductors. Most of the members exhibit superconductivity in the vicinity of a quantum critical point which is often reached only under hydrostatic pressure. In the following sections only those Cerium compounds are reviewed where superconductivity appears already at ambient pressure. In those compounds the superconducting state has been investigated and characterized by many diﬀerent probes which results in a detailed knowledge of the superconducting properties.

8.1 CeCu2 Si2 The discovery of superconductivity in CeCu2 Si2 by Steglich et al. [1] marks the beginning of the study of heavy-fermion superconductivity. The transition into the superconducting state occurs at T c ≈ 0.6 K, but the exact value depends on the sample stoichiometry and quality. CeCu2 Si2 is a heavy-fermion with a coeﬃcient of the linear term in speciﬁc heat γ = C/T ≈ 1000 mJ/molK2 . The initial experimental observation that ∆C/γT c ≈ 1.4, was taken as an indication that the superconducting state is carried by Cooper pairs formed by the heavy-fermion quasiparticles. However, since the Debye temperature is larger than the typical electronic energy scales in CeCu2 Si2 , it appears unlikely that superconductivity is described by the conventional phonon-mediated theory. Although CeCu2 Si2 was the ﬁrst heavy-fermion superconductor discovered [1], many questions concerning its properties still remain unresolved. Several problems are certainly connected to metallurgical diﬃculties arising from its incongruent melting. Hence, large variations in the properties of CeCu2 Si2 have been reported, depending on the sample preparation, exact stoichiometry, and annealing conditions. For example, the presence of magnetism and/or superconductivity in this system is known to be extremely sensitive to the stoichiometry even in monocrystalline samples. In the past few years more light was shed on the complex phase diagram which led to a closer understanding of the system [2]. Within a narrow homogeneity range of the primary 1:2:2 phase, several groundstate properties were realized. The ground state can be either the so-called A phase, sometimes coexisting with superconductivity (S), A/S where superconductivity expels the A phase, or only superconducting. The A phase is of magnetic origin [3], G. Goll: Unconventional Superconductors STMP 214, 67–87 (2006) c Springer-Verlag Berlin Heidelberg 2006

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8 Ce-Based Heavy-Fermion Superconductors

and there exists a subtle interplay between magnetism and superconductivity. Doping experiments wih Ge on the Si site as well as experiments under hydrostatic pressure strongly suggest that CeCu2 Si2 is located very close to a quantum critical point connected with the disappearance of the A phase. Non-Fermi-liquid behaviour is observed in the vicinity of the quantum critical point, e. g. in the speciﬁc heat and the electrical resistivity [4]. From the direction-dependent resistivity measurements it was concluded that the A phase is a conventional spin-density wave with very small ordered moment. The magnetic origin of the A phase was proved by NMR [5] and muon spin relaxation [6] measurements as well. The observation of long-range incommensurate antiferromagnetic order in neutron diﬀraction experiments ﬁnally reveals the nature of the A phase, and suggests that a spin-density-wave instability is the origin of the quantum critical point [7].

Fig. 8.1. Crystal structure of CeCu2 Si2 which is a ThCr2 Si2 type structure with T=Cu and X=Si belonging to the tetragonal space group I4/mmm

A further puzzle arises from the pressure dependence of the superconductivity. It is commonly believed that the superconducting properties of CeCu2 Si2 are stabilized through a slight Cu excess, which by some mechanism produces a source of positive internal pressure. On the other hand, Ge doping on the Si site expands the lattice, i. e., it is equivalent with the application of negative pressure. As a consequence, superconductivity is suppressed and a more detailed study of the A phase is enabled. Subsequent application of hydrostatic pressure reverses the eﬀect of lattice expansion, only the eﬀect of impurity scattering at the dopants remains. Yuan et al. recently investigated the p-T phase diagram of Ge-doped CeCu2 Si2 (x = 0.1) and found two distinct superconducting phases as a function of pressure p [8]. One superconducting phase is observed on the threshold of antiferromagnetic order, a second at higher p which seemingly coincides with a weak instability of the Ce

8.1 CeCu2 Si2

69

valence [9]. It was speculated the superconductivity may be caused by diﬀerent pairing mechanisms with both of them of an unconventional nature, namely, superconductivity mediated by the exchange of virtual ﬂuctuations of the spin density (at low p) and of the charge density (at high p), respectively [10]. The apparent sensitivity of the physical properties of CeCu2 Si2 on the exact stoichiometry and the crystal-growth conditions makes it diﬃcult to compare the results and to reconcile with possible implication for the order-parameter symmetry. Therefore, more recent results on samples with better deﬁned stoichiometry deserve closer attention. Nodal Structure The nodal structure of CeCu2 Si2 has been investigated by diﬀerent probes. While early speciﬁc-heat measurements on polycrystalline samples point to a T 3 dependence at low T [11, 12], more recent measurements indicate a T 2 dependence below 0.2 K [13, 14], consistent with line nodes of the order parameter. Measurements on a single crystal show a T 2 dependence as well (see Fig. 8.2) [15]. A T 2 dependence was also reported from measurements of the thermal conductivity [11] and the penetration depth [16].

Fig. 8.2. Speciﬁc heat of single-crystalline CeCu2 Si2 (a) and linear thermal expansion (b) measured along [100] (•) and [001] () (from [15])

Spin State and Parity Hints at an unconventional nature of the superconducting order parameter are given as well by measurements of the upper critical ﬁeld [17, 18, 19, 20]. A ﬂattening at

70

8 Ce-Based Heavy-Fermion Superconductors

low temperatures of the Bc2 (T ) curve was ascribed to Pauli limiting eﬀects [18]. An immediate conclusion was that CeCu2 Si2 probably has an even-parity order parameter. Early NMR measurements [21] on a polycrystal with T c = 0.72 K reported the absence of a Hebel-Slichter peak in the relaxation rate 1/T 1 and a T 3 dependence of 1/T 1 down to 0.1 K. The rapid decrease of the 29 Si and 63 Cu Knight shifts in a powdered polycrystalline sample with T c (H) = 0.66 K has been taken as a support of even-parity pairing without signiﬁcant spin-orbit scattering by impurities [22]. Ishida et al. [23, 24] reported for samples with exact stoichiometry Ce1.00 Cu2.05 Si2 and Ce1.025 Cu2 Si2 the absence of a Hebel-Slichter peak in the relaxation rate 1/T 1 and T 3 dependence in a T range of 0.1-0.6 K. These ﬁndings have been interpreted in terms of d-wave superconductivity. Further support has come from NQR measurements under hydrostatic pressure on Ge-doped samples [25, 26] which allowed a detailed study of the crossover from the antiferromagnetic (T A ) to the superconducting phase (T c ) as well. While a peak at T c and the large enhancement of 1/T 1 T in samples with T c < T A point to the presence of ﬂuctuating antiferromagnetic waves even in the superconducting state, the observation of a 1/T 1 ∝ T 3 -like dependence in a sample with T c ∼ T A was taken as evidence for a typical heavy-fermion superconductor as a consequence of the suppression of ﬂuctuating antiferromagnetic waves. Investigations of CeCu2 Si2 by the µSR technique under a weak transverse ﬁeld gave strong evidence against a microscopic coexistence between magnetism and superconductivity [27, 28]. The conclusion is based on the observation of the occurrence of a clear two-component structure in the µSR spectra when lowering the temperature below T c . However, the presence of magnetically ordered and superconducting domains seriously hampers the characterization of the superconducting state by µSR. On the other hand, experiments under high transverse ﬁelds provide evidence that the observed decrease of the Knight shift below ∼ 1 K at 0.5T is due to superconductivity coexisting within the magnetic A phase microscopically [29]. Energy Gap The energy gap of CeCu2 Si2 was investigated by several groups using point-contact, tunnelling, and vacuum tunnelling spectroscopy. From the point-contact experiments by de Wilde et al. [30] (see Fig. 8.3(2)) a gap size of ≈ 50 µeV was derived which corresponds to a rather small value 2∆/kB T c ≈ 1.9. Iguchi et al. [31] determined from tunnelling experiments on CeCu2 Si2 -I-Pb junctions a much larger gap with 2∆(0)/kB T c ≈ 6.4 and interpreted their data in terms of singlet d-wave superconductivity in CeCu2 Si2 . Using vacuum-tunnelling spectroscopy with a lowtemperature STM only V-shaped structures of a typical width of 5-10 meV below T c have been reported [32]. The measurements did not reveal any superconducting anomalies down to 300 mK. Either surface degradation due to oxygen or water molecules on the surface or an intrinsic property of the heavy-fermion surface might be possible reasons. Hence it follows that the experimental situation concerning the question of the gap size in CeCu2 Si2 remains unclear. As pointed out by Gloos et

8.1 CeCu2 Si2

71

Fig. 8.3. Normalized diﬀerential resistance R(V)/R0 of contacts between Ag and the heavyfermion superconductors UPt3 (1, R0 = 1.4 Ω), CeCu2 Si2 (2, R0 = 0.7 Ω), and URu2 Si2 (3, R0 = 0.1 Ω). Curves a are measured at T ≈ 50 mK in zero magnetic ﬁeld, curves b are the corresponding curves in a magnetic ﬁeld B > Bc2 (from [30])

al. [33] this might partially by caused by the suppression of the Maxwell contribution to the contact resistance which can occur in non-ballistic contacts. In addition, thermoelectric eﬀects and local heating have been reported for CeCu2 Si2 -Mo point contacts [34]. Phase-Sensitive Measurements First investigations of the Josephson eﬀect date back to the mid 1980’s when Poppe [35] investigated CeCu2 Si2 -Al point-contact junctions. He observed a crit-

72

8 Ce-Based Heavy-Fermion Superconductors

ical current and some oscillating behaviour in small magnetic ﬁeld. Concerning the magnitude of the critical current he reached the upper limit estimated from the Ambegaokar-Baratoﬀ formula. Therefore, he concluded that it is very unlikely that the observed Josephson current is a fourth-order eﬀect as proposed by Pals et al. [36] for triplet superconductors. Poppe claimed that CeCu2 Si2 behaves like a spinsinglet superconductor. The observation of a Josephson current through SNS -type junctions has been reported by Koyama et al. [37] and Sumiyama et al. [38] as well, but in both cases without a deﬁnite conclusion about the superconducting order parameter.

8.2 CeCoIn5 and CeIrIn5 The ternary rare-earth compounds CeCoIn5 and CeIrIn5 belong to a class of Cebased heavy-fermion superconductors which have recently been discovered [39, 40]. This new family with common stochiometry CeTIn5 , where T is one of the transition metals Co, Rh, and Ir, crystallizes in the tetragonal HoCoGa5 structure type which can be regarded as alternating layers of CeIn3 and TIn2 stacked sequentially along the tetragonal c-axis (see Fig. 8.4). All members of this family exhibit heavy-fermion behaviour as indicated by their large Sommerfeld speciﬁc-heat coeﬃcients γ. The linear coeﬃcient of the speciﬁc heat increases through the transition metal series from γ(T c ) = 350 mJ/molK2 for T=Co [41] over γ(T N ) > 420 mJ/molK2 for T=Rh [39] to γ(T c ) = 720 mJ/molK2 for T=Ir [40]. In contrast, the La analogues, which do not contain a 4 f electron, are Pauli paramagnets with coeﬃcient γ of about 5 mJ/molK2 [40]. Two members of this family, CeIrIn5 and CeCoIn5 , become superconducting at ambient pressure. Superconducting transition temperatures T c = 0.4 K [42] for CeIrIn5 and T c = 2.3 K [40] for CeCoIn5 have been reported. For the latter, this is

Fig. 8.4. Crystal structure of CeTIn5 with T=Co, Rh, Ir. The structure belongs to the tetragonal space group P4/mmm (No. 123, D14h ) and consists of alternating layers of CeIn3 and TIn2 stacked along the [001] direction

8.2 CeCoIn5 and CeIrIn5

73

Fig. 8.5. Speciﬁc heat divided by temperature C/T (circles, left ordinate), ac susceptibility χac (triangles, arbitrary units), and electrical resistivity ρ (squares, right ordinate) of CeIrIn5 as a function of temperature (from [40])

the highest transition temperature among the heavy-fermion superconductors. The Cooper pairs are formed by the heavy quasiparticles as indicated by the sizable jump ∆C in the speciﬁc heat at T c . For CeIrIn5 ∆C/γT c = 0.76 was estimated from the data, while ∆C/γT c = 1.43 is predicted in weak-coupling BCS theory. The enormous jump ∆C in the speciﬁc heat at T c with ∆C/γT c = 4.5 reported for CeCoIn5 probably hints at an unconventional pairing mechanism in these materials. A further peculiarity is observed for CeIrIn5 . As shown in Fig. 8.5 the resistivity already drops to zero at about T 0 = 3T c = 1.2 K, while the thermodynamic and magnetic signatures of superconductivity appear at much lower T . It was speculated [40] that the resistivity drop might be caused by the ”preformation” of electron pairs in analogy with the cuprates [43], but intrinsic ﬁlamentary superconductivity cannot be ruled out [44]. The third member of this family is CeRhIn5 which is an incommensurate heavyfermion antiferromagnet with ordering temperature T N = 3.8 K [39, 45]. Small magnetic Ce moments (0.26µB at 1.4 K) form a helical spiral along the c axis and are antiparallel for nearest-neighbour pairs in the tetragonal basal plane [45]. The observation of superconductivity at a very high T c = 2.1 K under hydrostatic pressure above 16.3 kbar [39] has led to speculations regarding to its magnetic ﬂuctuation spectrum [39, 40]. The enhancement of T c in layered CeTIn5 over CeIn3 has been suggested to be due to quasi-2D structure taking advantage of favorable coupling of 2-D antiferromagnetic ﬂuctuations. However, the presence of three-dimensional antiferromagnetic ﬂuctuations up to 7 K as reported by Bao et al. [46] demonstrate that this hypothesis seems not applicable in its simplest form.

74

8 Ce-Based Heavy-Fermion Superconductors

Recently it has been discovered that isostructural Pu intermetallic compounds become superconducting as well. Superconductivity with a transition temperature above 18 K has been reported for single crystals of PuCoGa5 [47, 48], while the sister compound PuRhGa5 has still remarkable T c = 8.6 K [49]. Beside the structural similarity with the Ce-based compounds, PuCoGa5 exhibits as well a local-moment behaviour close to that expected for Pu3+ , but only a modest quasiparticle mass enhancement is indicated by heat-capacity measurements [50]. Further, in both Co compounds the superconductivity evolves out of a non-Fermi-liquid state where the temperature dependence of the resistivity follows a T 1.35 power law for PuCoGa5 and a nearly linear behaviour for CeCoIn5 . The origin of superconductivity with relatively high transition temperature in this compounds is still unsettled, however, it has been argued that the high T c might be attributed to increased hybridization of the Pu 5 f electrons compared to the more localized Ce 4 f electrons, and that the superconductivity is unconventional, and most likely spin-ﬂuctuation mediated [47]. Spin-ﬂuctuation mediated superconductivity is also possible for the Ce compounds because the normal phase shows a proximity to quantum criticality. Superconductivity in CeCoIn5 is built out of a normal state displaying non-Fermi-liquid behavior as evidenced by a resistivity almost linear in T [40, 51], a logarithmically diverging speciﬁc-heat coeﬃcient [40, 52], and a power-law behavior in acsusceptibility [40, 52] and nuclear spin-lattice relaxation rate [53]. On the other hand, under pressure the ground state evolves into a conventional Fermi liquid [54]. Both points to the existence of a pressure-tuned quantum critical point close to ambient pressure [51]. For the T dependence of C/T in magnetic ﬁelds, some discrep√ ancy arises from the fact that for CeCoIn5 either a ln T behaviour [52] or a T [55] has been reported, which inﬂuences the interpretation of the data in view of the quantum critical ﬂuctuation spectrum. There is strong experimental evidence that superconductivity in this material is unconventional. Regarding the question of the order-parameter symmetry, the experimental situation for the ambient-pressure superconductors CeCoIn5 and CeIrIn5 is the following: Nodal Structure Speciﬁc-heat data unambiguously and unanimously reveal C(T ) ∝ T 2 for both compounds, if the data are evaluated at low enough temperature (below T c /3 − T c /2) [40, 42, 55, 56]. The T 2 electronic speciﬁc heat is an indication of the presence of line nodes in the superconducting gap. In addition, a linear-T contribution in CeIrIn5 and CeCoIn5 was attributed to a impurity band that forms in the line nodes of the superconducting energy gap. Contributions that can be attributed to an impurity band have been seen in thermal-conductivity measurements as well. Both compounds reveal a residual κ/T for T → 0. However, although the residual term in CeIrIn5 is rather large, it has been shown [56] that it is consistent with universal limit estimates [57] (see Sect. 3.1), i. e. the residual contribution κ/T |T →0 is independent of the concentration of scattering impurities. For CeCoIn5 , similar estimates suggest that the sam-

8.2 CeCoIn5 and CeIrIn5

75

ple is close to universal limit as well. An upper limit estimate of the impurity concentration is nimp = 20 ppm in the unitary scattering limit [57]. However, the thermal-conductivity data yield a T 3.37 low-temperature behaviour which slightly deviates from the T 3 behaviour predicted for an unconventional superconductor with line nodes in the clean limit. For CeIrIn5 , a T 2 dependence of κ below T < 0.2 K has been reported [56]. Further information on the nodal structure are obtained from thermal-conductivity measurements of CeCoIn5 with in-plane applied ﬁeld [58]. These measurements show fourfold symmetry, consistent with nodes along the (±π, ±π) positions. Field-angle dependent measurements of the speciﬁc heat support a fourfold symmetry of the nodal structure in the c-plane but disagree on the location of the nodes [59]. Although both quantities, κ(Θ) and C(Θ), respectively, show a fourfold oscillation with minima along the [010] and [100] directions (for C(Θ) see Fig. 8.6), the interpretation of the thermal-conductivity data in terms of quasiparticle scattering lead to the conclusion that the superconducting order parameter has d x2 −y2 symmetry, while the speciﬁc-heat data which probe directly the zero-energy density-of-states reveal a d xy symmetry, i. e. the same nodal structure but rotated by π/4. Measurements of the penetration depth on both compounds have been performed by several groups either by µSR [60] or microwave measurements [61, 62, 63]. The experiments revealed a power-law dependence of λ(T ). From transverseﬁeld µSR measurements with ﬁeld applied parallel to the c axis the magnetic penetration depth λab and its temperature dependence was deduced. For CeIrIn5 a magnetic penetration depth λab 6700 Å was estimated, quite similar to that for CeCoIn5 with λab 5500 Å. The temperature dependence deviates from the T 4 −2 −2 behaviour of ∆λ−2 expected for an isotropic s-wave superconab = λab (0) − λab (T ) ductor. However the observed exponents n = 3.0 ± 0.4 for CeIrIn5 and n = 3.3 ± 0.4 for CeCoIn5 are far from the linear temperature dependence expected for an order parameter with line nodes in the clean and local limits [64]. It was argued that impurity scattering [65] or nonlocal eﬀects [66, 67] can cause the deviations from the linear dependence of the superﬂuid density at low temperature although both eﬀects change the linear-T dependence only into a T 2 dependence. The inﬂuence of impurity scattering on the temperature dependence of λ(T ) was considered in the high-frequency experiments as well. For a sample with slightly lower T c = 2.17 K [61] a linear-T behaviour was reported, though the data clearly show some curvature. In another experiment with a sample with higher T c = 2.3 K [62, 63] the in-plane penetration depth λ exhibits a linear-T behaviour as well, but only for high temperature. For low T a crossover to a quadratic behaviour with crossover temperature T ∗ ≈ 0.32 K is observed. Such behaviour can arise in a superconductor with nodes in the gap either in a dirty d-wave model [68] or from nonlocal electrodynamics [67]. The authors argue that the linear-T dependence of the c-axis penetration depth λ⊥ strongly favors the nonlocal model with line nodes parallel to the c axis. Further, T ∗ is about ﬁve times larger than the upper limit obtained in the dirty d-wave model with unitary scattering, suggesting that the sample is too clean for the dirty d-wave model to be applicable.

76

8 Ce-Based Heavy-Fermion Superconductors

Fig. 8.6. Field-angle dependence of the speciﬁc heat of CeCoIn5 . Θ = 0 is the [100] direction, open symbols show the result without sample. The solid curves are ﬁts to C(H, Θ)/T = (C0 + C H (1 + A4 cos 4Θ))/T with C0 = 0.0361, 0.0631 and 0.4042 J mol−1 K−1 at T = 0.29, 0.38 and 1 K, and A4 = −0.0217, −0.0157 and −0.0061 for (2 T, 0.29 K), (5 T, 0.38 K) and (2 T, 1 K) (from [59])

8.2 CeCoIn5 and CeIrIn5

77

Parity and Spin State As pointed out in Sect. 4.1 Hc2 measurements can already probe the spin state of the Cooper pairs. The upper critical ﬁeld of CeCoIn5 has been determined by measurements of the speciﬁc heat and thermal expansion. Both indicate a change from a second-order transition at Hc2 at ﬁelds below 4.7 T to a ﬁrst-order transition [69]. The ﬁrst-order superconducting phase transition has been interpreted as caused by Pauli limiting. The fact that Hc2 is determined by the Pauli paramagnetic limit is a direct evidence for a spin singlet pairing.

Fig. 8.7. Temperature dependence of the 115 In nuclear spin-lattice relaxation rate of CeIrIn5 . The solid line is a calculation assuming a line-node gap ∆(φ) = ∆0 cos φ with 2∆0 = 5.0kB T c and the BCS temperature dependence of the gap (from [70])

Nuclear quadrupolar resonance (NQR) measurements show that there is no Hebel-Slichter peak just below T c for both compounds [53, 70, 71]. Below T c the spin susceptibility is suppressed, indicating singlet pairing. For CeIrIn5 the nuclear relaxation rate 1/T 1 follows a T 3 dependence down to 50 mK [53]. Zheng et al. [70] showed that this behaviour is compatible with a line-node gap ∆(φ) = ∆0 cos φ. The linear increase in E of the density of states N(E) gives rise to the T 3 variation of 1/T 1 at low T . The data (see Fig. 8.7) have been described assuming a BCS tem-

78

8 Ce-Based Heavy-Fermion Superconductors

perature dependence of the gap and 2∆0 = 5.0kB T c . However, NQR measurements on the sister compound CeCoIn5 [53] did not show the T 3 low-temperature behaviour of 1/T 1 that is expected for a line-node gap; instead 1/T 1 saturates below 0.3 K probably due to paramagnetic impurities, which mask the intrinsic superconducting quasiparticle contribution. Up to now µSR studies on CeCoIn5 and CeIrIn5 have been performed only by one group [60]. From zero-ﬁeld µSR measurements a quasi-static and temperatureindependent internal ﬁeld was observed which indicates the absence of long-range magnetic order in both compounds. Furthermore, the results indicate no enhancement of the spin relaxation rate below T c in either CeCoIn5 or CeIrIn5 , which demonstrates the absence of additional spontaneous magnetic ﬁeld in the superconducting phase which is expected only in superconductors with breakdown of time-reversal symmetry (e. g. a d x2 −y2 + iηd xy state). Hence, this experiment is consistent with spin-singlet pairing. Energy Gap Point-contact spectroscopy has been performed to study the order-parameter symmetry through the mechanism of Andreev reﬂection [72, 73, 74]. In [72] heterocontacts between CeCoIn5 and Pt as normal-metal counterelectrode were established in a 4 He bath cryostat at lowest T = 1.5 K by means of a mechanical diﬀerential screw mechanism. The typical contact resistances range between 1 and 130 Ω. The main direction of current ﬂow was within the basal plane of CeCoIn5 , however, the current direction with respect to the crystallographic axes could not be determined. Figure 8.8 gives an overview of the point-contact spectra at T ≈ 1.5 K at diﬀerent points on the surface. For comparison, the spectra have been normalized to the diﬀerential resistance at V = −5 mV and shifted with respect to each other. The main result is that two distinctly diﬀerent types of spectra are observed. Curves displayed in panel (a) represent spectra which exhibit a double-minimum structure, while curves displayed in panel (b) represent spectra which show a single minimum in dV/dI centered at V = 0, and shoulders symmetric in V at higher bias. In both cases, the features are related to superconductivity as they become weaker with increasing T and vanish near T c (see Figs. 8.9a and b). As already discussed in Sect. 5.3 minima at ﬁnite V in the diﬀerential resistance are expected for normal metal-superconductor point contacts due to Andreev reﬂection at the interface for both conventional [75] and unconventional [76] superconductors. This scattering process, where an electron is injected and a hole is retro-reﬂected with probability A, leads in isotropic fully gapped superconductors to a reduction of dV/dI for voltages < ∆/e. A ﬁnite but small probability 1 − A of normal reﬂection due to an interface barrier increases the zero-bias resistance and leads to the characteristic double-minimum feature with minima at ±∆/e. Qualitatively, the same general behaviour is expected for unconventional superconductors with a k-dependent gap function ∆ = ∆(k), although the “transparency” of the junction for Andreev reﬂection has to be determined self-consistently to take into account that the interface itself might be pairbreaking for some order-parameter symmetries. In

8.2 CeCoIn5 and CeIrIn5

1.4

1.10 Pt/CeColn5 1.05

Pt/CeColn5

1.3 3 2

1.00 1 0.95

(dV/dI)/(dV/dI|-5mV)

(dV/dl)/(dV/dl|-5mV)

79

3

1.2 1.1

2

1.0

1

0.9 0.8

(a) -4

-2

2 0 V (mV)

4

(b) 0.7

-4

-2

0 2 V (mV)

4

Fig. 8.8. Diﬀerential resistance dV/dI vs voltage V of point contacts between CeCoIn5 and Pt at T = 1.5 K. The contact resistance is R0 = 3.3 Ω (1), 2.1 Ω (2), and 3.6 Ω (3) for curves panel (a) and R0 = 4.2 Ω (1), 1.2 Ω (2), and 1.6 Ω (3) for curves in panel (b)

this sense, the curves displayed in Fig. 8.8a are in line with both conventional and unconventional order-parameter symmetry. In principle, from the width of the structure and the position of the minima, the gap value ∆ can be inferred by ﬁtting the curves, although the exact value depends on the choice of the order-parameter symmetry. A rough estimate from the position of the minima shows that ∆ is well above the weak-coupling limit 2∆ = 3.5kB T c = 0.69 meV. In contrast, the observation of a zero-bias conductance anomaly (ZBA) is expected only if the order parameter exhibits a sign change as a function of k. Andreev reﬂection of the quasiparticles at the surface leads to a surface bound state which is detected in the dV/dI vs V spectra. It was shown [77] that for the d x2 −y2 symmetry, the occurrence of a ZBA depends on the direction of current ﬂow with respect to the crystallographic axis. For a ﬂat surface and specular reﬂection, no zero-bias anomaly is detected for current injection along the [100] direction. In contrast, along [110], the order parameter exhibits a sign change and, therefore, a surface bound state at zero energy is formed which is seen in the spectra as a enhanced conductance at zero bias, i. e., a single minimum in dV/dI. However, ZBAs with approximately equal spectral weight appear for both direction if surface roughness is taken into account. In this sense, the result displayed in Fig. 8.8b strongly support an unconventional pairing state. The undeﬁned direction of current ﬂow and the limited temperature range T > 0.66T c of the experiment, however, prevents a deﬁnite assignment of the order-parameter symmetry. Measurements at lower T have been performed by [73] down to 0.4 K and by [74] to 0.15 K. Point-contact spectra with two types of features were also reported by [74] for low-ohmic contacts (R < 1 Ω) with I c but the features occur within

80

8 Ce-Based Heavy-Fermion Superconductors

Fig. 8.9. Temperature dependence of both types of structures in the diﬀerential resistance dV/dI vs V

the same spectrum and depend on the point-contact resistance. The data have been interpreted in terms of d-wave superconductivity involving multiple bands with two coexisting order parameters ∆1 = 0.95 meV and ∆2 = 2.4 meV which evolve differently with temperature. There are some doubts concerning this interpretation because the spectra might be obtained from non-ballistic contacts close to the thermal regime [78, 79]. The results reported by [73] for contacts with I c exhibit only the double-minimum feature which is a clear hint at bulk Andreev reﬂection in ballistic contacts albeit suppressed in magnitude. A ﬁt to the spectra assuming a d-wave order parameter indicates strong coupling with 2∆/kB T c = 4.6. Phase-Sensitive Measurements The Josephson eﬀect has been investigated on an SNS -type weak link with S=CeIrIn5 , N=Cu and S =Nb [80] in order to shed more light on the peculiar behaviour of the resistivity in comparison with bulk superconductivity (see Fig. 8.5). Indeed, a Josephson critical current Ic was observed below the temperature T 0 , where the resistivity of the sample drops to zero, i. e., well above bulk T c as indicated by ac-susceptibility measurements on the same sample. The result conﬁrms the presence of a superconducting state above T c and possible phase coherence between CeIrIn5 and Nb, at least at the surface of CeIrIn5 . In an applied magnetic ﬁeld a decrease of Ic is observed instead of a Fraunhofer diﬀraction pattern in the ﬁeld dependence of Ic which hints at a spatial ﬂuctuating local critical current density in a non-uniform junction. The non-uniform junction prevents also from observing further information on a possible intrinsic π phase shift of the order parameter, in analogy to experiments on the cuprates [81].

8.3 CePt3 Si

81

Flux Line Lattice The ﬂux-line lattice of CeCoIn5 has been imaged using small-angle neutron scattering [82]. At low magnetic ﬁelds Eskildsen et al. found a hexagonal ﬂux-line lattice. With increasing applied ﬁeld the ﬂux-line lattice undergoes a transition to square symmetry at about 0.6 T with the nearest neighbours oriented along [110] direction. This orientation is consistent with theoretical predictions based on the d-wave order-parameter with d x2 −y2 symmetry. Theoretically, d-wave pairing is expected to stabilize a square ﬂux-line lattice [83, 84], which was indeed observed in the highT c superconductors [85]. The transition is believed to be very likely a ﬁrst-order transition. However, it has been argued that the orientation of the square vortex lattice in tetragonal crystals may not serve as a conclusive test for the position of gap nodes [59], and that the low-ﬁeld vortex lattice is consistent with d xy symmetry as well. In summary, there is little doubt that superconductivity in CeCoIn5 and CeIrIn5 can be described by a spin-singlet pairing state. All experiments reviewed above point in the same direction. An order-parameter with d x2 −y2 or d xy symmetry is the most prominent candidate for CeCoIn5 , namely an order parameter with four linear line nodes. The angular dependence of the thermal conductivity, the power laws in various properties and the point-contact data give support to this order-parameter symmetry. For CeIrIn5 , the experimental situation points at a polar gap dz2 with one linear line node in the basal plane. The size of the speciﬁc-heat jump and the temperature dependence of the nuclear relaxation rate in connection with the estimated gap size deliver the strongest arguments.

8.3 CePt3 Si The heavy-fermion superconductor CePt3 Si is the youngest member of the heavyfermion family, but it is highlighted by an exceptional crystal structure [86]. It crystallizes in the tetragonal space group P4mm (see Fig. 8.10), and is therefore the ﬁrst heavy-fermion superconductor without a center of symmetry. The lack of an inversion center has certain implications for the physical properties, and that’s why it has attracted a widespread attention. CePt3 Si exhibits long-range antiferromagnetic order below T N = 2.2 K and becomes superconducting at T c = 0.75 K. Neutron scattering experiments [87] and zero-ﬁeld muon-spin relaxation investigations [88] both point to a microscopic coexistence of magnetism and superconductivity. The magnetic structure consists of two interleaved ferromagnetic sublattices of local Ce 4f moments (see Fig. 8.10) and has inversion symmetry under simultaneous space-time reversal. The magnetic moment of 0.16 µB /Ce is reduced compared to the free Ce 4f moment, but sizable compared to that observed in other Ce-based compounds. Bulk superconductivity below T c = 0.75 K has been proved by speciﬁc-heat [86], susceptibility [89], resistivity [86] and NMR measurements [90, 91]. From

82

8 Ce-Based Heavy-Fermion Superconductors

Fig. 8.10. Crystal and magnetic structure of CePt3 Si. The structure belongs to the space group P4mm and does not have centrosymmetry or an inversion center. The arrows on the Ce atoms indicate the magnetic moment lying in the basal plane (from [87])

a careful extrapolation of the normal state speciﬁc heat invoking the T 3 dependence associated with antiferromagnetic ordering a Sommerfeld coeﬃcient γn ≈ 0.39 J/(molK2 ) has been determined [86]. The jump ∆C = 0.075 J/molK at T c implies Cooper pairs formed from the heavy quasiparticles, however the reduced BCS parameter ∆C/γn T c = 0.25 instead of 1.43 has lead to the speculation that not all electrons condense into Cooper pairs and therefore, a signiﬁcant portion of the Fermi surface is not involved in the superconducting condensate [92]. However, this scenario has to be revised in the light of recent ﬁndings that a double transition occurs in the speciﬁc heat of a sample annealed at slightly higher temperatures [93]. Up to now, experimental hints at an unconventional nature of the superconductity come from measurements of the upper critical ﬁeld Hc2 and the unusual temperature dependence of the NMR relaxation rate (see Fig. 8.11). From the theoretical point of view the lack of an inversion center has certain implications for the order-parameter symmetry of CePt3 Si. According to Anderson’s theorem [94] Cooper pairing in the spin-singlet channel relies on the presence of time-reversal symmetry and additionally inversion symmetry is required for spin-triplet pairing to obtain the necessary degenerate electron states [95]. Therefore, it is commonly believed that a material without inversion symmetry would be an unlikely candidate for spin-triplet pairing. However, the upper critical ﬁeld of CePt3 Si exceeds the usual paramagnetic limiting ﬁeld which might indicate spin-triplet pairing, although the lack of inversion symmetry reduces the eﬀect of paramagnetic limiting for spinsinglet pairing, too [96]. In addition, Frigeri et al. [96] have shown by an analysis

8.3 CePt3 Si

83

1.2 1.0

8.9 MHz ~ 1 T 18.1 MHz ~ 2 T

(1/T1T)/(1/T1T)TC

CePt3Si TC

0.8 0.6

BW (full gap) 2 /kBTC = 4

0.4

ABM (point-node) 2 /kBTC = 3.6

0.2

polar (line-node) 2 /kBTC = 5.1

0.0 0.0

0.2

0.4

0.6

0.8 T/TC

1.0

1.2

1.4

1.6

Fig. 8.11. Normalized relaxation rate (1/T 1 T )/(1/T 1 T )Tc vs. T/T c of CePt3 Si in comparison with three models: the Balian-Werthamer model (isotropic spin-triplet state, dashed line), a point-node model (dotted line), and a line-node model (dashed-dotted line)(from [92])

of the BCS gap equations with respect to the point-group symmetry of CePt3 Si that spin-triplet pairing with p-wave symmetry is not entirely excluded in this system. A symmetry classiﬁcation of possible pairing states was also given by Samokhin et al. [97]. They conclude that a possible gap structure of CePt3 Si has line nodes if the order parameter belongs to a one-dimensional representation, but ﬁnally, the gap structure depends on the dimensionality of the order parameter which is not known so far. For a pairing state with line nodes the temperature dependence of the NMR relaxation rate should exhibit a T 3 power-law behaviour. However, as shown in Fig. 8.11, the relaxation rate of CePt3 Si shows an exceptional behaviour among the heavy-fermion superconductors: (i) it has a coherence peak at T c , although with signiﬁcantly smaller peak height than for conventional BCS superconductors, and (2) neither an exponential law nor a T 3 behaviour is observed down to T = 0.2 K [90]. A comparison with three diﬀerent models gives no satisfactory description over the entire temperature range [92]. While the Balian-Werthamer (BW) model for an isotropic spin-triplet state and the Anderson-Brinkman-Morell (ABM) model for a state with point-nodes can account for the reduced coherence peak and the temperature dependence close to T c , the favored line-node model agrees only with the data points at lowest T . The failure of above models opens the door for other, more sophisticated orderparameter scenarios including a mixing of spin-singlet and spin-triplet components

84

8 Ce-Based Heavy-Fermion Superconductors

of the superconducting order parameter [98] or a pairing model with rotating spins around the momentum-space Fermi surface [99]. Such states might be realized in the absence of inversion symmetry. Future experiments certainly will provide more data to discriminate among such exotic scenarios.

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9 U-Based Heavy-Fermion Superconductors

9.1 UPt3 The low-temperature normal state of the binary compound UPt3 presents a strongly renormalized Fermi liquid evidenced by the large coeﬃcient of the linear term in speciﬁc heat, γ =430 mJ/molK2 [1] and the equally enhanced Pauli susceptibility. It orders antiferromagnetically at T N ≈ 5 K [2], but the ordered moment m = 0.02µB /U is unusually small and is directed along the a∗ axis in the hexagonal plane. Below T c ≈ 0.5 K UPt3 becomes superconducting. The evolution of magnetism and its interplay with superconductivity has been studied by doping with Pd on the Pt site. Neutron diﬀraction [3] and µSR experiments [4, 5] shows that UPt3 is close to an antiferromagnetic instability. A long-range antiferromagnetic phase with large moments exists in U(Pt1−x Pd x )3 for x > 0.006 and it becomes instable for smaller Pd concentrations at the critical concentration xc,a f ≈ 0.006. At the quantum critical point superconductivity appears together with a small-moment antiferromagnetic phase. This has been interpreted in terms of a competition between large-moment antiferromagnetism and superconductivity and has lead to the suggestion that superconductivity in UPt3 is not mediated by antiferromagnetic interactions, but rather by ferromagnetic spin ﬂuctuations, which cannot coexist with long-range antiferromagnetic order [6]. Hints at ferromagnetic ﬂuctuations have been found by inelastic neutron scattering [7, 8] in the magnetic ﬂuctuation spectrum which is complex and consists of both antiferromagnetic and ferromagnetic contributions.

U Pt

Fig. 9.1. Crystal structure of UPt3 belonging to the hexagonal space group P63 /mmc

G. Goll: Unconventional Superconductors STMP 214, 89–120 (2006) c Springer-Verlag Berlin Heidelberg 2006

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The small-moment antiferromagnetic order which exists for x < 0.006 is rather robust on Pd doping and the small ordered moment m varies from m = 0.018µB /U for x = 0 to m = 0.048µB /U for x = 0.005. The increase of m(x) correlates with the splitting ∆T c of the superconducting transition (see below) which provides evidence for the idea that small-moment antiferromagnetism acts as the symmetry-breaking ﬁeld in a Ginzburg-Landau scenario for unconventional superconductivity. Among the heavy-fermion superconductors UPt3 is the best candidate for unconventional superconductivity based on the observation of at least three superconducting phases and strong evidence for an anisotropic order parameter. Several reviews appeared over the years [9, 10, 11, 12], quite recently, a comprehensive review of the superconducting phases of UPt3 was given by Joynt and Taillefer [13]. The phenomenology of the phase diagram (see Fig. 9.2) has been studied extensively using Ginzburg-Landau theory, where the free energy functional is derived exclusively by symmetry arguments. Several models [14] have been proposed to explain the exceptional phase diagram by (i) a two-dimensional (2D) order parameter

Fig. 9.2. (B, T ) phase diagram of UPt3 for a ﬁeld B applied perpendicular (a) and parallel (b) to the c direction as determined from the speciﬁc heat (•) and the magnetocaloric eﬀect () (from [10])

9.1 UPt3

91

with a degeneracy of the orbital part of the pairing function and either even [15] or odd parity [12], (ii) a one-dimensional (1D) order parameter with a degeneracy of the spin part of the pairing function [16], or (iii) two accidentally degenerate 1D representations [17]. Although, the latter model can explain the apparent tetracritical point for all ﬁeld orientations [18, 19], it provides no explanation for the observed correlation between the T c splitting and the antiferromagnetic order parameter [20, 21]. In the so-called E-representation model the splitting ∆T c is caused by the lifting of the degeneracy of a two-component superconducting order parameter by a symmetry-breaking ﬁeld. Substantial evidence is at hand that the small-moment antiferromagnetic phase acts as a symmetry-breaking ﬁeld [20, 22]. In the spin degenerate model the order parameter is characterized by a nonunitary spin-triplet pairing with an orbital part belonging to a 1D representation such as A2u , B1u , or B2u . NMR experiments have been explained successfully within this model [23]. Thermodynamic and transport measurements which mainly probe the nodal structure and the anisotropy of the order parameter favor the 2D scenario with a degeneracy of the orbital part. Measurements of surface superconductivity in UPt3 have been analyzed in terms of a superconducting order parameter with E2u symmetry, i. e. a 2D odd-parity order parameter with point and line nodes [24]. The suppression of T c with increasing residual resistivity is also in line with this orderparameter symmetry [25]. Good agreement is obtained in an analysis which follows from a generalization of the Abrikosov-Gor’kov formula for pair-breaking including anisotropic scattering, unconventional pairing and Fermi surface anisotropy. Further support for a 2D-order parameter with a line of nodes in the basal plane and point nodes along c axis comes from measurements of the thermal conductivity [26, 27] along b (κb ) and c axis (κc ) at very low temperatures T , i. e. in the B phase of UPt3 . The T dependence of the anisotropy κc /κb and its behaviour for T → 0 should be able to discriminate possible order-parameter scenarios [28, 29]. The data down to T = 16 mK favor an order parameter with the above-mentioned nodal structure, however, they preclude an unambiguous distinction between the E1g and E2u models. Additional restriction to possible order parameters is obtained by µSR experiments [30] which detect the appearance of a spontaneous magnetic ﬁeld within the superconducting B phase. The result hints at a broken time-reversal symmetry of the B phase which is valid for all nonunitary and some unitary states. A superconducting state which breaks time-reversal symmetry is also favored by investigations of the ﬂux dynamics in the vortex state of UPt3 [31]. Experimentally, such a strong pinning was observed for vortices in the B phase that no creep could be detected for those vortices. This behaviour has been attributed to the trapping and strong pinning of fractional vortices on domain walls between domains of degenerate superconducting phases which inhibits also the motion of ordinary vortices due to the repulsive vortex-vortex interaction. Results on the magnetization in the superconducting mixed state are rather in favor of an odd-parity pairing with an appreciable anisotropy in the pair-spin correlation [32].

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Nodal Structure Speciﬁc-heat measurements do not allow a deﬁnite statement on the low-temperature T dependence due to the existence of a speciﬁc-heat anomaly at very low T with a maximum at 18 mK [33]. Up to now, no microscopic explanation of the anomaly is at hand, and the anomaly might even be caused by stress due to clamping the sample for the measurements. Therefore, the T 2 dependence and the residual linear-T term which was deduced for T > 0.1 K and reported by several authors [1, 34, 35, 36], has to be handled with caution. An unambiguous hint at a two-dimensional order parameter comes from the thermal-conductivity measurements, where the low-T anomaly is absent. The thermal conductivity of UPt3 in absence of a magnetic ﬁeld was measured by several groups. Lussier et al. [38, 26] found a strong anisotropy between the heat ﬂow parallel and perpendicular to the basal plane. Furthermore, the anisotropy evolves only below the lower transition T c− . This was taken as an indication for a diﬀerent nodal structure of the A (between T c+ and T c− ) and B phase (below T c− ). Two potential candidates for the symmetry of the order parameter were considered, the E1g gap parameter with lines of nodes along the basal plane and linear point nodes along the c-axis and the E2u gap parameter with lines of nodes along the basal plane and quadratic point nodes along the c-axis. Both order parameter show a hybrid gap structure in the B phase and are compatible with the observed T dependence and anisotropy of κ. In order to discriminate further between the E1g gap and the E2u

Fig. 9.3. Thermal conductivity κ/T of UPt3 in the low-temperature regime (between 16 mK and 70 mK) as a function of T 2 with ﬁt curves based on the theoretical predictions of Graf et al. [29] (from [37])

9.1 UPt3

93

gap, measurements at very low T on samples with low impurity concentration are necessary. Such measurements have been performed by Suderow et al. [37] down to 16 mK resp. T c /30 (see Fig. 9.3). As already discussed in Sect. 3.1 the thermal conductivity should remain ﬁnite due to the creation of a band of impurity bound states. Graf et al. [29] showed for unitary scattering, that this residual κ/T term should be independent of the number of defects (universal limit) for certain order-parameter symmetries. Calculations for plausible order-parameter symmetries of UPt3 found that for E2u , κb /T and κc /T should be both independent of the number of defects, and for E1g only κb /T should have a universal value. Suderow et al. observed for both directions of heat ﬂow a T 3 dependence at T < 0.1 K in line with the predictions in the clean limit. However, no conclusive picture could be drawn for the residual term, as for both directions the residual term is much lower than expected. Suderow et al. [39] also investigated the ﬁeld dependence of the thermal conductivity of UPt3 single crystals. At very low temperatures down to 16 mK κ(B, T ) scales as a function of x = (T/T c )(Bc2 /B)1/2 as theoretically predicted for a superconductor with lines of nodes [40, 41]. Measurements of the penetration depth deduced from the transverse-ﬁeld depolarization rate in the superconducting state have been carried out as well [43]. The observed rise of the depolarization rate results from the magnetic penetration depth λc (0) = 7200 Å and λa (0) = 6900 Å. From the temperature dependence of the depolarization rate the magnetic penetration depth λc (T ) and λa (T ) were extracted which appears to be consistent with a superconducting gap with a line of nodes in the basal plane. This was ascribed to an even-parity superconducting state belonging to the

6

hybrid II hybrid I polar

λ−2 (µm−2)

5 4 3 2 1 / λ2c

1

1 / λ2a

0 0.0

0.1

0.2

0.3

0.4

Temperature (K) Fig. 9.4. Temperature dependence of the penetration depth perpendicular (λc ) and within the basal plane (λa ) of UPt3 (from [42])

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9 U-Based Heavy-Fermion Superconductors

Fig. 9.5. Temperature dependence of the Gaussian transverse-ﬁeld µ+ SR depolarization rate in UPt3 obtained in ﬁeld-cooled (FC) and zero-ﬁeld-cooled (ZFC) runs in comparison to the two superconducting transition determined by speciﬁc heat on the same sample (from [42])

2D scenario with representation E1g . However, the order parameter belonging to the 2D E2u representation has a similar topology of nodes, but form an odd-parity state. Hence, no deﬁnitive conclusions have been drawn from these results. In addition, these results have later been put into question when it was noticed [30] that bulk measurements indicated much larger values of λ. More recent µSR measurements by Yaouanc et al. [42] on a single crystal with a clear double transition conﬁrmed the earlier data. As shown in Fig. 9.4 both gap symmetries ﬁt the data quite reasonably. Peculiarly, the increase of the depolarization rate is conﬁned to the lower superconducting phase below T c− (see Fig. 9.5). For comparison speciﬁc-heat data on the same sample [44] are shown together with the µSR data. The origin of the increase of the depolarization rate in the transverse ﬁeld is still open. Possibly, it might be due to the drastic change of the ﬂux-line lattice. The temperature dependence of the nuclear magnetic relaxation rate indicates the presence of a line node of the gap function as well. Below T c no Hebel-Slichter peak is seen and at low temperature, between 0.1 and 0.3 K, a T 3 dependence is observed (see Fig. 9.6) which is compatible with a quasiparticle density of states that grows linearly with energy. For the measurements of the longitudinal ultrasound attenuation either a T 2 [46, 47, 48] or a T dependence [49] have been reported, while for measurements of the transverse ultrasound attenuation one gets a T od T 2 dependence as well but depending on the sound direction. From this measurements no conclusive statement

9.1 UPt3

95

Fig. 9.6. Temperature dependence of the nuclear magnetic relaxation rate (1/T 1 ) measured on a powdered sample of UPt3 (from [45])

can be made regarding any possible gap structure. Other experiments [50, 51] clearly support the E2u scenario and are in agreement with calculations by Graf et al. [52] for E2u symmetry of the order parameter. Both groups unisonously report a linear T dependence of the attenuation for transverse sound waves propagating in the basal plane with polarization also in the basal plane, but a T 3 power law for polarization parallel to c. Walker et al. pointed out that this result can be interpreted in favor of the existence of horizontal line nodes which are “active” for transverse sound waves with qˆ a (qˆ b) and eˆ b (ˆe a), but “inactive” for eˆ c [53].

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9 U-Based Heavy-Fermion Superconductors

Spin State and Parity The spin state of UPt3 was revealed by 195 Pt Knight-shift measurements. Early measurements [54] already indicated that the Knight shift is unchanged through the transition in the superconducting state. Intriguing results have been reported by Tou et al. [23, 55, 56]. For applied ﬁelds above 0.23 T, the Knight shift is temperature independent through T c , but small changes at T c are observed for lower applied ﬁelds along some crystallographic directions. These results are interpreted by many people as conclusive evidence in favour of odd-parity triplet superconductivity in UPt3 , with the higher ﬁeld experiments showing that the complex vector order parameter is depinned from the crystal lattice. From the detailed dependence of the Knight shift on temperature, magnetic ﬁeld and crystal orientation Tou et al. concluded that the A and C phase belong to a class of unitary triplet-pairing states while the B phase is nonunitary. In addition, they claimed that the spin-orbit coupling felt by the Cooper pair is weak. From µSR measurements on UPt3 no conclusive picture about the superconducting state can be drawn. Hints at unconventional superconductivity came from the remarkable observation in zero-ﬁeld experiments that below the superconducting transition T c− a supplementary very weak spontaneous magnetization occurs [30]. The results are shown in Fig. 9.7. In the discussion of possible origins the increased depolarization ﬁrst has been interpreted by a reorientation of the ordered moments in the basal plane. However, such a reorientation of all magnetic moments in an ideal crystal would lead to a cancellation of the internal ﬁelds, and therefore occurs only as a side eﬀect in nonperfect crystals. In this respect, one has to mention that the µSR

Fig. 9.7. Temperature dependence of the zero-ﬁeld (ZF) exponential µ+ SR depolarization rate on a monocrystalline UPt3 sample with the initial µ+ SR polarization along the basal plane (from [30])

9.1 UPt3

97

data obtained on a high-quality single crystal [57] do not indicate a change of the depolarization below T c . As a second origin of the increased depolarization the occurrence of a superconducting state with no time-reversal symmetry was discussed, which creates an additional ﬁeld at the µ+ side. The magnitude of this supplementary ﬁeld (0.1 G) has been taken as evidence for a lifting of the degeneracy in the spin space rather than in the orbital space. On the basis of this interpretation a spin-triplet odd-parity superconducting state belonging to the 1D scenario has been proposed [58]. However, the non-observation of an increase of the depolarization rate below T c− on a high-quality single crystal [57] seriously questions this latter interpretation. Energy Gap While planar-tunnel-junction experiments failed because of diﬃculties in growing superconducting UPt3 ﬁlms, Andreev reﬂection in point contacts has been investigated by several groups [59, 60, 61, 62]. For UPt3 with T c ≈ 0.5 K one has to go to very low T to investigate the energy gap and its nodal structure via Andreev reﬂection. A minimum structure in dV/dI vs V of about the same width was observed by all groups. While early measurement by Nowack et al. showed the existence of a minimum, de Wilde et al. [61] attributed the point-contact spectra (an example is shown in Fig. 8.3(1)) to the presence of nodes in the gap function. They interpreted their data in terms of Andreev reﬂection in a 1D-order parameter scenario, however, without investigation of the angular dependence and the magnetic-ﬁeld dependence. Gloos et al. [62] examined W-UPt3 contacts and attributed the observed resistance change as a function of bias to changes of the Maxwell contribution to the contact resistance and not to Andreev reﬂection (see Sect.5.3). The most detailed investigation studied the directional, temperature, and magnetic-ﬁeld dependence of Pt-UPt3 point contacts [60] and compared the results quantitatively with various order-parameter models [63]. In addition, the dependence of the point-contact spectra on the surface properties has been examined [64]. Figure 9.8 shows representative spectra at T < 50 mK for preferred current ﬂow parallel to the a, b, and c directions of hexagonal UPt3 on a single crystal with a double transition at T c+ = 521 mK and T c− = 460 mK as determined from the speciﬁc heat [66]. The directional dependence of Andreev reﬂection is one of the clearest indications of an anisotropic superconducting order parameter [60]. While for I c (curves 3a and b) almost always a pronounced Andreev reﬂection feature is observed - quite often even a double-minimum structure -, the absence of such a feature for I a and I b was the rule (curves 1a and 2a). The rare observation of a shallow minimum is attributed to contributions to the current oﬀ the basal plane due to the rough UPt3 /Pt interface. In a comparison to various order-parameter models it was shown [63] that an anisotropy of point-contact spectra is caused by the nodal structure of the order parameter. Following the expositions of Sect. 5.3 for current ﬂow parallel to a direction with line nodes, the resistance change is only about 10 % and it is further reduced by broadening of the line nodes into belt nodes which may be caused by

98

9 U-Based Heavy-Fermion Superconductors 1.04

UPt3-Pt

1.02

1a

1.00

I || a

1.02

1b

(1/R0) dV/dI

1.00 1.04 1.02

2a

1.00

I || b

1.02

2b

1.00 1.04

3a

1.02

I || c

1.00

3b

1.02 1.00 0.98 -0.4

-0.2

0

0.2

0.4

V (mV)

Fig. 9.8. Diﬀerential resistance dV/dI vs voltage V for point contacts on UPt3 for diﬀerent directions of current ﬂow. 1a and b: I a; 2a and b: I b; 3a and b: I c. The spectra were taken at T = 37 mK (1a), 37 mK (1b), 50 mK (2a), 53 mK (2b), 37 mK (3a), and 42 mK (3b) (from [65])

impurities [67]. Thus the observed anisotropy can be understood qualitatively. However, the experimental data always show resistance changes of a few percent only for I c. Systematic investigations of the inﬂuence of the surface condition on Andreev reﬂection should help clarify this question [64]. Figure 9.9 gives an overview over point-contact spectra at T ≈ 50 mK for different surface treatments of UPt3 . Representative spectra are shown for the polished surface with n c (curve 1), the polished and lightly etched surface with n c (curve 2), for a surface with n c after several etching steps denoted as ”heavily etched” (curve 3), and a freshly cleaved surface (curves 4). The common feature of all spectra is the minimum of dV/dI for |V| ≤ ∆/e which is less pronounced for the polished surface (curve 1) and most pronounced for the heavily etched (curve 3) and the cleaved surface (curve 4). Although the surface quality strongly aﬀects the intensity of the structures in dV/dI, even for a freshly cleaved surface the

9.1 UPt3

99

1.04

UPt3-Pt

1.02

1

1.00

(1/R0) dV/dI

1.04 1.02

2

1.00 1.04

3

1.02 1.00 1.04

4

1.02 1.00 -0.4

-0.2

0

0.2

0.4

V (mV)

Fig. 9.9. Diﬀerential resistance dV/dI vs voltage V for point contacts on UPt3 with diﬀerent surface treatments. 1: polished, I c; 2: polished and lightly etched; 3: polished and heavily etched, I c; 4: freshly cleaved, Θ(I, c) ≈ 40◦ . The spectra were taken at T = 53 mK (1), 51 mK (2), 42 mK (3), 44 mK (4) (from [65])

resistance change remains of the order of a few percent only which suggests an intrinsic suppression of the order parameter at the surface. With increasing T or B, the minimum in dV/dI vs V weakens and ﬁnally Andreev reﬂection vanishes. For a detailed analysis the area F between dI/dV vs V curves in the superconducting state and the corresponding normal-state background was calculated according to   V∆/e V∆/e

 

dI dI   |S dV − |N dV  F = 2    dV dV 0

0

and the ﬁeld and temperature value, respectively, was determined, where F becomes zero. In a simple isotropic model of Andreev reﬂection, F corresponds to the excess current and should vanish at Bc2 and T c , respectively [68]. The values of T or B where Andreev reﬂection vanishes can be compared to the (B, T ) phase diagram obtained from measurements of the speciﬁc heat and magnetocaloric eﬀect for B ⊥ c [66]. Independent of surface conditions for I c, Andreev reﬂection occurs only in the low-T , low-B phase, i. e. the B phase of UPt3 . For predominant current ﬂow oblique with respect to the c direction Andreev reﬂection occurs in all three superconducting phases (see Fig. 9.10).

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9 U-Based Heavy-Fermion Superconductors

B⊥c I || c, polished I || c, polished, lightly etched I || c, heavily etched

2

o

B (T)

Θ(I, c) ~40 , freshly cleaved

C 1

N

A

B 0 0

0.2

0.4

0.6

T (K) Fig. 9.10. (B, T ) phase diagram for Andreev reﬂection on UPt3 . The solid phase lines are obtained from measurements of the speciﬁc heat and the magnetocaloric eﬀect on a sample cut from the same crystal [66] (from [65])

Several order-parameter scenarios for unconventional superconductivity in UPt3 have been proposed in the past decade [9, 14]. A 2D order parameter with orbital degeneracy lifted by a symmetry-breaking ﬁeld [12] and a spin-triplet order parameter with 1D orbital part [69, 70] are considered as the most promising candidates. In the 2D model the order parameter is a vector η = (η1 , η2 ) with basic functions either with E1 or E2 symmetry. The B phase is described by η = (1, ı˙), a state with broken time-reversal symmetry, while the A and C phase, respectively, are described by η = (1, 0) and η = (0, 1), respectively. The splitting of the T c ’s for the A and B phase may arise from the antiferromagnetic ordering which occurs [20] below T N = 5 K and which acts as a symmetry-breaking ﬁeld. The nodal structure is characterized by point nodes in c direction and a line node in the basal plane for the B phase, and at least one additional line node in a plane perpendicular to the basal plane in the A and C phase, i. e. even in c direction. In the previous paragraphs it was shown that point-contact spectroscopy has revealed the following properties of Andreev reﬂection on UPt3 : (i) Andreev reﬂection occurs only in the B phase, and (ii) Andreev reﬂection is directionally dependent in the B phase. Both ﬁndings consistently can be explained with a 2D order parameter with point and line nodes, provided that Andreev reﬂection is suppressed for current ﬂow in a direction with line nodes and only little aﬀected by point nodes. Line nodes occur for current ﬂow in the basal plane for all three phases, for current ﬂow I c in the A and C phases only. Self-consistent calculations of the diﬀerential resistance of a contact between a normal metal and an unconventional superconductor with a k-dependent pair poten-

dV/dI (arb. units)

9.1 UPt3

101

data isotropic E2g

E1u

-0.2

-0.1

0.0

0.1

0.2

V (mV) Fig. 9.11. Diﬀerential resistance dV/dI vs voltage V calculated for diﬀerent order-parameter symmetries in comparison with the data of a point contact with R0 = 1.62 Ω (from [65])

tial ∆(k) indeed support such behaviour [63]. Furthermore, the shape of the spectra is reproduced with an order parameter of above nodal structure. Theoretically however, no double-minimum structure was obtained if ∆(k) ∼ kz , because there is always a bound state at zero energy leading to a minimum in dV/dI at eV = 0. However, for ∆(k) ∼ |kz | the pair potential contains a bound state at energy ±∆ which leads to the characteristic minima in dV/dI at eV = ±∆. Figure 9.11 shows a spectrum for I c in comparison with calculations for diﬀerent order parameter symmetries: isotropic order parameter: ∆(k) = ∆0 = const. ∆(k) ∼ η1 |kz |(k2x − ky2 ) + 2η2 |kz |ky k x E2g : E1u : ∆(k) ∼ η1 |kz |k x + η2 |kz |ky The choice of the E representations is diﬀerent only with respect to parity, the nodal structure is identical to the usually considered gap functions. Both 2D order parameter symmetries are barely distinguishable and follow the data quite well, while an isotropic BCS gap fails to describe the data. The spin-space degenerate scenario [69, 70] has not been considered, although strongest support for odd-parity pairing with parallel spin pairing (triplet pairing) have come from NMR studies [23] where the 195 Pt Knight shift does not change for both ﬁeld directions H ⊥ c and H c for H > 5 kOe. In this model the degeneracy of the spin part of the pairing function causes the T c splitting and the diﬀerent

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superconducting phases rather than the orbital part. In more detail, superconductivity in UPt3 is characterized by an order parameter ∆(k) = ηλ f (k)τλ λ

with λ = x, y, z, τλ = ı˙σy σλ , and η = (η x , ηy , ηz ) the spin part and f (k) the orbital part belonging to a one-dimensional representation. In the A and C phase a unitary state with η x 0, ηy = ηz = 0 (A phase) and η x = 0, ηy 0, ηz 0 (C phase), respectively, is realized. By entering the B phase the spin part becomes complex (η = η x xˆ + ı˙ηz zˆ) and therefore ∆(k) nonunitary. This implies that the degeneracy of the dispersion function E(k) is lifted and spin-up and spin-down particles have diﬀerent dispersion relations. In this sense the three diﬀerent phases of UPt3 clearly have diﬀerent order parameters ∆(k), however with the same orbital part f (k) which determines the nodal structure. In other words, all three phases have the same nodal structure. Therefore this model in the present form cannot account for the pointcontact data which hints at diﬀerent nodal structures, although the nodal structure in the B phase might be the same (e. g. for A2u ). Phase-Sensitive Measurements Early experiments of the Josephson eﬀect between an ordinary superconductor and UPt3 did not detect any Josephson current [71]. Although the absence of a pair current due to pair breaking at the surface might be explainable if UPt3 is a triplet superconductor, a suppression of superconductivity by mechanical stress could not be ruled out. More than a decade later, Sumiyama et al. ﬁnally succeeded in the observation of a Josephson current in SNS -type and SS -type junctions [72, 73]. Although the critical current as a function of applied ﬁeld shows a falling envelope, no Fraunhofer diﬀraction pattern was observed. Ic oscillates with no deﬁnite period, which suggests that the junction is not uniform and the local critical current density ﬂuctuates spatially. Sumiyama et al. observed an anisotropy of the supercurrent and attributed this to the unconventional order parameter of UPt3 . The temperature of the critical current is quite remarkable as well. For I c only a weak increase is observed below T c+ , but below T c− the critical current increases steeply. In contrast to I c the increase becomes rather slow below T c− for I b. However, the junction quality prevents from any conclusion about the order parameter. Flux Line Lattice The only direct measurement on the vortex structure is low-angle neutron scattering [74, 75] for an applied magnetic ﬁeld in the basal plane. A conventional centered rectangular lattice is observed in the B phase and for B ⊥ c, i. e. a triangular lattice compressed in the c direction. The opening angle is 2α = 38◦ (at low ﬁeld) instead of the standard 2α = 60◦ for a perfect triangular lattice. As α is governed by anisotropy in both the Fermi velocity and the gap, and the observed anisotropy m⊥ /m = (3 tan2 α)−1 = 2.8 is close to the normal-state mass ratio, the authors [74]

9.2 UBe13

103

claim that no additional anisotropy from the gap structure seems to show up in the B phase. This is in contrast to theoretical expections [76, 77]. Based on µSR results an unusual form of the ﬂux-line lattice has been proposed for time-reversal-symmetry breaking superconducting states. For B c neutron scattering was used to look at the vortex lattice at B = 0.19 T as a function of temperature [78], in order to investigate the A-B transition. It was found that the lattice is accurately hexagonal in both phases. In the B phase, the nearest-neighbour vector is parallel to the a∗ direction, while in the A phase it rotates by ±15◦ , i. e. each phase shows a distinctly diﬀerent orientation of the ﬂux-line lattice, which can be directly related to the diﬀerence in anisotropy of the superconducting gap function. The result was interpreted as an alignment of the lattice to a gap of E2u form rather than of E1g [78, 79]. A full Ginzburg-Landau analysis of the problem [80] supports this interpretation. The authors conclude that the data favor E2u . In summary, there remain several important mysteries, which have to be solved. Examples are the smallness of the changes in the Knight shift at low ﬁelds, the implication of small spin-orbit coupling in a material for which it would be expected to be very large, and the role of antiferromagnetism. These puzzles will need to be resolved before a universal agreement on the superconducting order parameter in UPt3 will be achieved.

9.2 UBe13 In contrast to UPt3 , which is in a well-developed Fermi-liquid state when it becomes superconducting, and which orders antiferromagnetically, UBe13 is very diﬀerent. Strangely, no clear evidence for magnetic ordering has been observed in UBe13 despite of the large paramagnetic Curie temperature. Furthermore, the magnitude of the electrical resistivity at low temperatures is 200 µΩcm, which is considerable larger than those of other compounds, suggesting that the spin-disorder scattering is dominant even at 1 K. In addition, UBe13 is not in a Fermi-liquid regime when superconductivity appears below ≈ 1 K as seen from the speciﬁc-heat and resistivity measurements (see Fig. 9.13). There is strong experimental evidence that non-Fermi-liquid behaviour is present in UBe13 due to the vicinity of a quantumcritical point [81]. A clear picture of the experimental situation is hampered by the fact that two variants of UBe13 exist with markedly diﬀerent superconducting and normal-state properties [82]: “H-type” (T c ≈ 0.9 K) and “L-type” (T c ≈ 0.75 K) UBe13 . It is believed that the diﬀerence originates from slightly diﬀerent actual compositions due to the preparation procedure [82]. With regard to the superconducting state, no clear picture of the symmetry of the order parameter has emerged, and apparent inconsistencies still remain. The power-law temperature dependence of all thermodynamic or transport properties early have suggested the presence of nodes in the superconducting gap of UBe13 , but a precise identiﬁcation of the order-parameter symmetry is still lacking. In the following, the experimental situation will be shortly sketched.

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9 U-Based Heavy-Fermion Superconductors

¯ Fig. 9.12. Crystal structure of UBe13 (cubic space group Fm3c)

Nodal Structure On entering the superconducting state the speciﬁc heat exhibits a jump ∆C with an height ∆C = 1.43γT c for an isotropic s-wave superconductor. In UBe13 however, this jump ∆C = 2.5γT c , is quite large (see Fig. 9.13) and points to the presence of strong-coupling eﬀects. Strong-coupling eﬀects are manifested in the unusual temperature dependence of the upper critical ﬁeld as well [83]. The speciﬁc heat below T c 0.86 K follows closely a T 3 law, compatible with the presence of points on

Fig. 9.13. Speciﬁc heat (C − CN )/T vs. T of UBe13 at B = 0 and 12 T after subtraction of a nuclear hyperﬁne contribution CN due to the external magnetic ﬁeld B (from [87])

9.2 UBe13

105

the Fermi surface with vanishing superconducting gap [84]. Further substantial evidence for the existence of point nodes is provided by the magnetic-ﬁeld dependence of the speciﬁc heat. W¨alti et al. [85] showed that for ﬁelds B > 2 T, C(T, B) exhibits a scaling behaviour with respect to T B1/2 . As already mentioned in Sect. 3.1 this scaling behaviour is a consequence of the ”Volovik” eﬀect, i. e., a nonzero density of states which depends on the topology of nodes and is caused by a Doppler shift of the energy scale due to the supercurrents circulating around vortices in the mixed state. A gap function with point nodes like the axial state of 3 He allows a fairly good ﬁt to the penetration depth data as well. Einzel et al. [86] observed a T 2 dependence of ∆λ and showed that this behaviour is consistent with an axial p-wave state. Thermal-conductivity measurements on UBe13 in the superconducting state have been reported by Jaccard et al. [88] on a single-crystalline sample and by Ravex et al. [89] on a polycrystal. While κ(T ) of the single crystal unambigously exhibits a T 2 behaviour over the whole T range between 0.15 and 0.8 K, the data on the polycrystal show a diﬀerent T dependence. Only after subtraction of a residual contribution linear in T which appears below 100 mK, the remaining thermal conductivity follows an exponential BCS behaviour. The NMR relaxation rate do not indicate an Hebel-Slichter peak and an exponential temperature dependence below T c , but follows a power law with exponent n = 3 [90], which, in contrast to the speciﬁc-heat and penetration-depth data, is compatible with the presence of lines with vanishing superconducting gap. At T ≈ 0.2 K the 9 Be spin-lattice relaxation rate 1/T 1 shows a crossover from a T 3 to a linear-T dependence [91, 92]. Spin State and Parity Subsequent transverse-ﬁeld µSR studies found that the µ+ Knight shift is virtually unchanged in the superconducting state [93, 94]. This behaviour is compatible with odd-parity pairing and consistent with the observed power-law behaviour of the speciﬁc heat. However, a more detailed study reveals the existence of two magnetically inequivalent µ+ stopping sites produced by the dipolar ﬁelds of the uranium 5felectrons and subsequently diﬀerent Knight shifts on each site. The temperature dependence is diﬀerent as well. While the magnitude of the Knight shift decreases with decreasing T at one of the sites, it anomalously increases at the other site below T c until both become constant below T ≈ 0.2 K [95]. This change coincides with the temperature at which the 9 Be spin-lattice relaxation rate 1/T 1 shows a cross over from a T 3 to a linear-T dependence. Obviously, the formation of an equal-spin pairing state which has a T -independent electronic spin susceptibility, does not explain the behaviour below T c . Zero-ﬁeld µSR studies [96, 97] checked the possible occurrence of static magnetism in the superconducting phase. The depolarization rate does not exhibit any detectable change, implying a complete absence of magnetic correlations with moments ≥ 10−3 µB /U-atom down to at least 50 mK. The temperature-independent zero-ﬁeld depolarization rate arises solely from the spread in dipolar ﬁelds due to the 9 Be nuclear magnetic moments.

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9 U-Based Heavy-Fermion Superconductors

Energy Gap Early point-contact experiments by Nowack et al. on UBe13 -W point contacts [59] showed either a V-shaped minimum or a double-peak structure in the dV/dI spectra. Although both features are related to the superconductivity in UBe13 and vanish close to T c , they are unexpected for normal conductor-s-wave-superconductor contacts. For lack of an appropriate quantitative description in terms of non-s-wave superconductivity, the authors took the full width at half maximum as a measure for the energy gap 2∆ and deduced a ratio 2∆/kB T c closely to the BCS value. Later Gloos et al. [98] repeated these measurements and obtained similar features, the V-shaped minimum for high-ohmic contacts of the order of tens of ohms and the doublepeak structure for low-ohmic contacts of the order of 1 Ω, which he interpreted in terms of local heating. The width was about twice of that obtained by Nowack et al.. Even wider features have been found for UBe13 -Au contacts by W¨alti et al. [99]. A value of 2∆/kB T c > 6.7 has been extracted from the data and interpreted in terms of strong-coupling eﬀects. In addition, a huge maximum in dI/dV at zero-bias was reported. The maximum hints at the existence of low-energy Andreev surface bound states and provides strong evidence for an unconventional energy-gap function. The interpretation, however, was put into question by Gloos [100, 101] who argued that local heating in the contact region might aﬀect the results. The three examples show that the experimental situation for UBe13 is still contradictory.

Fig. 9.14. Josephson eﬀect between UBe13 and the s-wave superconductor Nb. Panel (a) shows the sample arrangement for an SNS -type junction, panel (b) the temperature dependence of the critical supercurrent density Jc and junction resistance R close to T c of an SS type contact (from [102])

9.3 URu2 Si2

107

Phase-Sensitive Measurements Josephson-eﬀect studies [103] in a contact between UBe13 and the s-wave superconductor Ta with T c (Ta) > T c (UBe13 ) indicate that the s-wave order parameter induced by the proximity eﬀect is strongly suppressed by the occurrence of the UBe13 order parameter, suggesting that the latter might have odd parity. Josephson eﬀect related to the bulk superconductivity of UBe13 was probably observed for SNS type and SS -type contacts between UBe13 and Nb (see Fig. 9.14) [102]. For the dc Josephson eﬀect an increase of the critical current Ic was observed for decreasing temperature below T c , and in the ac Josephson eﬀect conventional Shapiro steps have been found. However, the junctions have not been uniform and therefore no Fraunhofer pattern has been observed for Ic (B).

9.3 URu2 Si2 URu2 Si2 was the ﬁrst heavy-fermion system for which the coexistence between magnetism occurring below T 0 17.5 K and superconductivity occurring below T c 1.3 K was reported [104]. Additional interest was created by the apparent contradiction between the large entropy loss at T 0 and the extremely small value of the ordered moment of µS 0.03µB detected by neutron scattering [105] which has been attributed to the development of an enigmatic hidden order. A quadrupolar model of the 17.5 K phase transition has been developed [106, 107] where the order parameter of the phase transition is not the tiny staggered magnetic moment, but the ordering of the localized f quadrupoles. Chandra et al. showed that incommensurate orbital antiferromagnetism, associated with circulating currents between the U ions,

Fig. 9.15. Crystal structure of URu2 Si2 which is a ThCr2 Si2 type structure with T=Ru and X=Si belonging to the space group I4/mmm

108

9 U-Based Heavy-Fermion Superconductors

can account for the local ﬁelds and entropy loss at the transition [108]. Nevertheless a purely local picture cannot provide a fully consistent explanation and the clear signatures of Fermi-liquid behaviour above T 0 suggest that itinerant behaviour is involved. Thus URu2 Si2 demands a description which accounts for both its itinerant and its local aspects. A discussion of possible models was given recently by Mydosh et al. [109]. Possible unconventional superconductivity is supported mainly by the observation of power laws in the temperature dependence of various physical properties. In detail, following results have been obtained: Nodal Structure Early speciﬁc-heat measurements achieved contradictory results on the superconducting transition. Some observed a double transition reminiscent that in UPt3 [110], others only a single jump [111, 112, 113]. Ramirez et al. ﬁnally showed that the multiple phase transitions originate from spatially separated regions of structurally indistinguishable material [114]. Nevertheless, a common feature of all measurements was the observation of a T 2 behaviour of the speciﬁc heat in the superconducting state which hints at a line node in the superconducting energy gap. Hasselbach et al. [115] analyzed all the gap functions allowed by the tetragonal crystal symmetry to derive the full temperature dependence of the speciﬁc heat using weak-coupling BCS theory. Although their theoretical curves do not unequivocally reproduce the data, they conclude that the analysis favors either B1g symmetry with a possible gap function ∆(k) ∝ k2x − ky2 [115] or B2g symmetry with a possible gap function ∆(k) ∝ k x ky [116]. Both gap functions, which have the same nodal structure, but turned around the kz axis by 45◦ , are an indication for d-wave (even-parity) superconductivity in URu2 Si2 . However, Brison et al. have shown that this interpretation is not unambiguous [117], since for an even-parity s-wave pairing in a clean superconductor, a line node can be generated by the presence of a large antiferromagnetic molecular ﬁeld and a gapless region appears on a equatorial line perpendicular to the wave vector q when the molecular-exchange ﬁeld exceeds T c . Within this model, the calculated temperature dependence of the speciﬁc heat is in better agreement with the experimental observations than the d-wave calculations. Thermal-conductivity measurements by Behnia et al. [118] down to 40 mK did not allow any statement on the nodal structure owing to the presence of large nonelectronic contributions to κ below T c . Spin State and Parity Nuclear-resonance experiments in the superconducting state have been done by the Kohara group in zero and applied magnetic ﬁelds. For both, 101 Ru NQR measurements [119] and 29 Si NMR measurements [120], they report the absence of a HebelSlichter peak just below T c and a decrease of the inverse nuclear spin-lattice relaxation time T 1−1 proportional to T 3 . However, while the NMR results show deviations from the power-law dependence at lower temperatures, no deviation was found in

9.3 URu2 Si2

109

the NQR results down to 0.2 K. The deviations result from the coupling to paramagnetic impurities which is more eﬀective for the 29 Si nuclei than for 101 Ru nuclei. The T 3 behaviour indicates the existence of line nodes in the superconducting gap. The 29 Si Knight shift, however, shows no decrease within experimental errors in contradiction with the existence of Pauli limiting of the upper critical ﬁeld [121] and the decrease of the muon Knight shift (see below) which were considered as the evidence of spin-singlet superconductivity in URu2 Si2 . Zero-ﬁeld µSR studies [122] reveal the presence of antiferromagnetism below T N 17.5 K by an increase of the µ+ depolarization rate. However, no change of the zero-ﬁeld depolarization rate is detectable at the superconducting transition T c [44] which suggests that the coupling between the superconducting and antiferromagnetic order parameter must be weak. Transverse-ﬁeld µSR measurements below T c show a second increase of the depolarization rate in addition to the increase below T N . This additional increase is caused by the formation of the ﬂux-line lattice, which generates a supplementary ﬁeld distribution at the µ+ site. The key information furnished by µSR is that down to the lowest temperature magnetism and superconductivity microscopically coexist. Further, the µ+ depolarization-rate increase below T c is only observed for Hext c, whereas forHext ⊥ c the depolarization appears to be constant [93, 123]. This implies that the magnetic penetration depth is much smaller in the basal plane than along the c axis and λ⊥ (0) = 9500 Å was calculated from the data. The most interesting eﬀect is the increase of the µ+ frequency with lowering the temperature, signaling a decrease in absolute value of the µ+ Knight shift, which tracks the conduction-electron spin susceptibility. Such reduction is reminiscent of a Yosidatype decrease of the spin susceptibility below T c and is often taken as an indication for even-parity pairing. This appears in line with the observation of a T 2 behaviour of the speciﬁc heat in the superconducting state [111]. Energy Gap Direct information on the superconducting order parameter was accessible by means of point-contact spectroscopy. However, the data reported are contradictory and not entirely conclusive. Hasselbach et al. [125] and Samuely et al. [126] found that their point-contact data were described slightly better with a d-wave gap function ∆(k) ∝ k x ky , while de Wilde et al. [61] from their data suggest the absence of zeros in the gap (see Fig. 8.3(3)). Naidyuk et al. [124] investigated in detail the directional dependence of the superconducting order parameter and found a pronounced anisotropy of the dV/dI characteristics for two directions, along the c axis and along the a axis, i. e. perpendicular and parallel to the tetragonal basal plane (see Figs. 9.16 and 9.17). For point contacts with current ﬂow predominantly along the c axis, the superconducting double-minimum structure is broader for almost a factor three than for the perpendicular direction. The strong anisotropy might be caused by a suppression of superconductivity in the basal plane, and gives support to the model proposed by Brison et al. [117] that the exchange ﬁeld due to ferromagnetically aligned layers of the U magnetic moments in the basal plane may lead to line of zeros for ∆,

110

9 U-Based Heavy-Fermion Superconductors

Fig. 9.16. Temperature (a) and magnetic-ﬁeld dependence at T = 50 mK (b) of point-contact data of URu2 Si2 for I c. The data have been symmetrized. Insets: measured dV/dI curves below and above T c (a), dV/dI (solid line) and its symmetric part (dashed line) on a larger scale (from [124])

independent of the pairing mechanism (see above). For all data reported, the gap width is rather large, 2∆/kB T c ranges from 5 to 12. Phase-Sensitive Measurements A strong anisotropy of the dc Josephson current below the transition temperature of URu2 Si2 was reported for point contacts between URu2 Si2 and Nb, namely, the Josephson current is found only, if the contacts are aligned within the basal plane, whereas it is absent in contacts aligned along the c direction [127]. A straightforward explanation of this extreme anisotropy of the Josephson current was done in terms of an unconventional order parameter in URu2 Si2 with a symmetry leading to destructive interference for Josephson currents along the c direction. Among the possible order-parameter states Wasser et al. [127] found best agreement with their results for the even-parity B1g symmetry which shows two line nodes and maximum gap values along the two a axes. The result however, completely disagrees with the model suggested by Brison et al. (see above), which yields maximum gap values for the c direction. A Josephson current of the same order of magnitude for both c- and a-axis direction was reported by Tachibana et al. [128], who investigated rf sputtered SNS -type and SS -type Josephson junctions between URu2 Si2 and Nb (see Fig. 9.14a for a scheme of the junction). A corner SQUID experiment designed to measure the phase of the order parameter has been performed by Nowack et al. [129]. A closed-loop setup with two NbTi

9.4 UNi2 Al3 and UPd2 Al3

111

Fig. 9.17. Temperature (a) and magnetic-ﬁeld dependence at T = 80 mK (b) of point-contact data of URu2 Si2 for I a. The data have not been symmetrized. Insets: T dependence of R(0) (a), symmetric part of dV/dI below and above T c (b) (from [124])

contacts mechanically ﬁxed on URu2 Si2 showed SQUID oscillations in a small magnetic ﬁeld below T c of URu2 Si2 . However, no further conclusion concerning the order-parameter symmetry could be drawn because of the ill-deﬁned geometry of the mechanical setup and the incomplete magnetic shielding of the experiment.

9.4 UNi2 Al3 and UPd2 Al3 The heavy-fermion superconductors UNi2 Al3 [130] and UPd2 Al3 [131] exhibit coexistence between superconductivity and a magnetic state with relatively large ordered moments. The detection of large static moments contradicted the picture that small values of these moments µS ≤ 10−3 µB were a prerequisite for a coexistence of both types of ground states. Both systems crystallize in a rather simple hexagonal structure P6/mmm (D16h ), which is shown in Fig. 9.18. Whereas UNi2 Al3 forms peritectally, UPd2 Al3 has a congruent melting, resulting in the availibility of large single crystals for the latter compound. UPd2 Al3 was found to exhibit a simple antiferromagnetic structure [wave vector q = (0, 0, 12 )] [132] below T N 14.5 K and the static magnetic moments of U lying in the basal plane. The neutron-scattering data are consistent with an ordered magnetic moment µS 0.85µB , reduced compared to the eﬀective moment obtained from the high-temperature susceptibility, but exceeding by up to two orders of magnitude the small moments found, for example, in UPt3 . Hence, in contrast to UPt3 , a picture of local-moment magnetism seems to describe the magnetic state in

112

9 U-Based Heavy-Fermion Superconductors

Fig. 9.18. Crystal structure of UPd2 Al3 (and UNi2 Al3 ) belonging to the hexagonal space group P6/mmm (D16h )

UPd2 Al3 . The simultaneous occurrence of local moment and heavy-mass itinerant behaviour was attributed to a dual nature of the electrons in the 5f shell in UPd2 Al3 [133] which has an average occupation of slightly less than three: two electrons are “localized” 5f electrons in the U4+ state, while the remaining 5f electrons are “itinerant” due to the hybridization with the conduction electrons leading to a Sommerfeld coeﬃcient γ = 140 mJ/(mol K2 ) in the speciﬁc heat. Superconductivity due to these heavy quasiparticles occurs below T c ≈ 2 K as evidenced by a jump in the speciﬁc heat with ∆C = 1.2γT c . UNi2 Al3 has similar properties. It exhibits coexistence of long-range antiferromagnetic order below T N 4.5 K and superconductivity below T c ≈ 1 K as well. However, the ordered moment is much smaller, only about µS 0.2µB have been reported [134]. There exists only a rare number of experiments on UNi2 Al3 compared to the Pd homologue due to the metallurgical diﬃculties. Nodal Structure The nodal structure of UPd2 Al3 was investigated by several probes. Speciﬁc-heat measurements [135] found a T 3 power law and the data have been interpreted in terms of an octahedral d-wave state with eight zeros on points on the Fermi surface. The temperature dependence of the thermal conductivity of UPd2 Al3 single crystals is quasiquadratic [136] and contains a cubic term, indicating that the lowtemperature dependence is dominated by the electron contribution. The T 2 dependence is contradictory to the exponential one expected for conventional s-wave superconductors and in particular, the speciﬁc-heat results on this compound. A ﬁnite linear term found in measurements on a polycrystal [137] has been interpreted in terms of a residual density of low-energy quasiparticle states in qualitative agreement with resonant impurity scattering theories applied to unconventional superconductors. From these measurements a d-wave state characterized by lines of nodes has been proposed. The position of these line nodes have been probed by angleresolved magnetothermal transport measurements [138]. A distinct twofold oscilla-

9.4 UNi2 Al3 and UPd2 Al3

113

tion of the thermal conductivity is reported for H rotated in the plane orthogonal to the basal ab plane, while no oscillations are observed when H is rotated within the basal plane. These results have been interpreted in terms of a superconducting pairing function with d-wave symmetry of the form ∆(k) = ∆0 cos (kz c), i. e. a gap function with a single line node orthogonal to the c axis and isotropy within the basal plane. Measurements of the nuclear relaxation rate are in line with line nodes. Independently of the local microscopic probe 27 Al [139, 140] or 105 Pd [141] a T 3 power law has been observed down to 0.2 K without any indication of a coherence peak. Spin State and Parity A strong paramagnetic limitation of the upper critical ﬁeld of UPd2 Al3 was observed by Amato et al. [142] which points to a spin-singlet pair state. Furthermore, from the analysis of 27 Al Knight shift below T c [140] it was shown that the anisotropic residual Knight shift originates from the f -electron susceptibility in the antiferromagnetic state, whereas the isotropic reduction of the shift below T c is due to a singlet pairing nature of the superconductivity. In contrast to UPd2 Al3 , the sister compound UNi2 Al3 shows evidence for spintriplet superconductivity. The evidence arises from 27 Al Knight-shift measurements which do not change down to 50 mK [143]. This result is in line with earlier measurements of the upper critical ﬁeld which showed the absence of paramagnetic limitation [144, 145]. However, there is also experimental evidence for a spin-singlet state realized in UNi2 Al3 . From measurements of Hc2 performed on thin ﬁlms of UNi2 Al3 a much lower upper critical ﬁeld Hc2 (0) ≈ 1.6 T was estimated [146] which provides evidence for non-negligible paramagnetic pair-breaking contributions. Zero-ﬁeld µSR studies on polycrystals of UPd2 Al3 [142] indicate the absence of a precessing component in the zero-ﬁeld µSR data even below T N , indicating that the internal ﬁeld at the µ+ site has an average value of zero and must therefore be symmetric with respect to the antiferromagnetic sublattices. Furthermore, the depolarization is unchanged below T c , indicating that the magnetic state is not aﬀected by the onset of superconductivity. Additional transverse-ﬁeld µSR measurements on monocrystalline samples in the magnetic and superconducting phase [147] yield an isotropic character of the penetration depth with λ⊥ (0) = 4800 ± 500 Å and λ (0) = 4500 ± 500 Å. Furthermore, the µ+ Knight shift was found to manifest a peculiar behaviour below T c as it is only partially reduced and the derived reduction is isotropic. The partial reduction of the µ+ frequency shift is explained within a physical picture of two rather independent electron subsystems: One is associated with the 5 f electron subsystem that forms the heavy quasiparticle condensing into Cooper pairs below T c , whereas the second is ascribed to the electron subsystem that is associated with the local antiferromagnetism, which is uneﬀected in the superconducting state. Additional support for the coexistence of two distinct electron subsets was provided by speciﬁc-heat measurements under pressure [135], which demonstrated that the itinerant subsystem accounts for only 80 % of the linear coef-

114

9 U-Based Heavy-Fermion Superconductors

ﬁcient of the speciﬁc heat. As a result the observations of µSR studies on UPd2 Al3 are compatible with singlet pairing of the superconducting state. For the related system UNi2 Al3 µSR studies have been performed with the aim of unravelling the magnetic structure [148]. The zero-ﬁeld spectra below T N 4.3 K show a complex signal, in sharp contrast with the situation observed in UPd2 Al3 . Unfortunately, such a complex signal prohibits a µSR study of the superconducting state, and therefore only information about the magnetic phase have been provided which is beyond the scope of this review. Energy Gap Using the vacuum-tunnelling spectroscopy on UPd2 Al3 with a low-temperature STM only V-shaped structures of a typical width of 5-10 meV below T c have been reported [150]. The measurements did not reveal any superconducting anomalies down to 300 mK. Either surface degradation due to oxygen or water molecules on the surface or an intrinsic property of the heavy-fermion surface might be the reason for this. Point-contact data on both UPd2 Al3 and UNi2 Al3 [62, 151] show a minimum in the diﬀerential resistance as a function of applied voltage, however, the data has been interpreted in terms of non-spectroscopic eﬀects, e. g. local heating in the contact region. UPd2 Al3 has the great advantage that it has been possible to grow epitaxial thin ﬁlms with only slightly reduced T c . Therefore, UPd2 Al3 -AlO x -Pb tunnel junctions have been fabricated to study the superconducting properties [149]. The left panel in Fig. 9.19 shows a sketch of the junction. At T = 0.3 K, well below T c of both electrodes, the diﬀerential conductance shows conductance peaks related to the sum and diﬀerence of the energy gaps of both electrodes. Additionally, the phononic strongcoupling modulations typical for Pb are observed (not shown in Fig. 9.19). Applying a magnetic ﬁeld large enough to suppress superconductivity in the Pb electrode, but

Fig. 9.19. Left panel: Sketch of a UPd2 Al3 -AlO x -Pb tunnel junction; Right panel: Normalized diﬀerential conductivity of a UPd2 Al3 -AlO x -Pb tunnel junction at T = 0.3 K and B = 0.3 T (black lines) together with a ﬁt curve with ∆ = 235 µeV and Γ = 35 µeV. The inset shows spin-ﬂuctuation strong-coupling structures of UPd2 Al3 (from [149])

References

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well below the upper critical ﬁeld of UPd2 Al3 allows the study of the tunnelling density of states of the heavy-fermion superconductor. From a ﬁt according to Dynes formula [152] with a broadening parameter Γ = 35 µeV (see Sect. 5.3) a gap value ∆ = 235 µeV was determined which correspondes to 2∆/kB T c ≈ 3.4. Surprisingly, a modulation of the conductivity was observed at V ≈ 1 meV. With the help of inelastic neutron scattering data [153, 154, 155] the modulation straightforwardly has been assigned to strong-coupling eﬀect between the charge carriers and magnetic excitations that mediate the superconducting pairing interaction between the heavy quasiparticles. However, the question whether these excitations are spin ﬂuctuations [154] or magnetic excitons [155, 156] is still a matter of ongoing debate. In order to explain their inelastic neutron scattering spectra Sato et al. [155] assumed a two-component 5f-electron character with a strong exchange coupling between the quasiparticles and the localized electrons that couples the magnetic excitons to the particle-hole excitations. This model yields a strong-coupling constant between the two components which is consistent with the strong-coupling feature observed in the tunnelling experiment. They showed that within the framework of the (conventional) Eliashberg equation both inelastic neutron scattering and tunnelling spectra are consistent with each other. Phase-Sensitive Measurements Phase-sensitve measurements have been performed on UPd2 Al3 by two groups [157, 158]. Both succeeded in obtaining a Josephson current in their experiments. However, the only conclusions which have been drawn from these experiments, are, that the reduction of the Josephson current might arise from a unconventional supercondcuting state in UPd2 Al3 .

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10 Metal-Oxide Superconductors

Metal-oxide superconductors commonly base on a perovskite-type structure. The standard perovskite BaTiO3 forms a body-centered cubic structure with Ba at the body-centered position, Ti at the corners of a cube and O at the edge-centered positions. This structure can be regarded as stack of Ti-O and Ca-O layers as well. Starting from this structure a vast of superconductors have been synthesized or even designed by replacing the d metal ion, doping on the Ba site, doubling the unit cell in one direction, inserting additional d metal ion-oxygen layers, tuning the chargecarrier concentration, e. g. by changing the oxygen stoichiometry, or all changes together. In this way, supercells with ordered stacking sequences along the c direction have created which become superconducting at temperatures up to 133 K, as for example in HgBa2 Ca2 Cu3 O8+δ [1]. HgBa2 Ca2 Cu3 O8+δ has a sandwich structure consisting of three CuO2 layers separated by Ca layers and embedded in a roll made of a Hg/Ba-O double layers. A common feature of most metal-oxide superconductors is the d metal ion-oxygen layer as a building block and its subtle inﬂuence on the electronic properties. Very often a strongly two-dimensional metal is observed with conductivity (and superconductivity) mainly within these layers.

Fig. 10.1. Crystal structure of the perovskite CaTiO3 , the main structure element of the hightemperature superconductors (from [2]). It corresponds to the standard perovskite BaTiO3 unit cell shifted along the [111] direction by (1/2, 1/2, 1/2) G. Goll: Unconventional Superconductors STMP 214, 121–151 (2006) c Springer-Verlag Berlin Heidelberg 2006

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10 Metal-Oxide Superconductors

10.1 Sr2 RuO4 Sr2 RuO4 belongs to a perovskite-related layered series, the so-called RuddlesdenPopper series Srm+2 Rum+1 O3m+4 where the d metal site is occupied by Ru. Sr2 RuO4 is one example of this general structure with m = 0 and one RuO2 layer. There exist two further members of this series Sr3 Ru2 O7 (m = 1) and SrRuO3 (m = ∞), which are non-superconducting, but exhibit strongly ferromagnetic correlations. SrRuO3 is an itinerant 4d ferromagnet with a Curie temperature T c ≈ 160 K [3, 4] and Sr3 Ru2 O7 a metamagnet with strong critical ﬂuctuations [5]. The structure of Sr2 RuO4 is closely related to the high-temperature superconductor (La,Sr)2 CuO4 with Cu replaced by Ru (see Fig. 10.2), but superconductivity appears only below T c ≤ 1.5 K.

Fig. 10.2. Crystal structure of Sr2 RuO4 which crystallizes in a K2 NiF4 -type structure belonging the tetragonal space group I4/mmm

The discovery of superconductivity in Sr2 RuO4 has quickly triggered a large amount of interest because of the unconventional properties [6] and the initially proposed analogy [7] to 3 He. The normal state of Sr2 RuO4 is well described by Fermi liquid theory with strong correlation eﬀects which enhance the eﬀective mass seen in quantum oscillation [8, 9], speciﬁc heat and Pauli spin susceptibility measurements1 . The quasi-two-dimensional Fermi surface consists of three weakly corrugated sheets, the hole-like α band, and the electron-like β and γ bands, which can be mainly associated with three t2g orbitals of Ru [9]. A major source of complications arises from the presence of four electrons in three nearly degenerate Ru orbitals 1

For a review of the normal-state properties see [10].

10.1 Sr2 RuO4

123

4d xy , 4dyz , and 4d xz and the competing eﬀects of crystal ﬁeld, spin-orbit interaction and Hund’s coupling. In the normal state orbital dependent, predominantly ferromagnetic spin correlations have been reported by NMR experiments [11], but neutron scattering did not detect ferromagnetic correlations so far, which would help to explain a spin-triplet superconducting state in a simple picture. In fact, incommensurable antiferromagnetic ﬂuctuations above [12] and below [13] T c are reported by neutron scattering with a strong ﬂuctuation amplitude at q ≈ (0.6π, 0.6π). The ﬂuctuations are due to nesting of the α and β sheets which had been predicted independently by Mazin and Singh [14]. This suggests a subtle competition between ferro- and antiferromagnetic spin correlations and the role of these spin ﬂuctuations for the pairing mechanism is still a matter of an ongoing debate. The superconducting state is extremely sensitive to impurities and disorder. Systematic doping of nonmagnetic Ti on the Ru site increases the residual resistivity ρab above ≈ 1 µΩ cm for x > 10−3 and immediately suppresses superconductivity [15, 16]. This is in line with earlier observations [17, 18] that defects or impurities both rapidly suppress superconductivity, if the residual resistivity is increased above ≈ 1 µΩcm. The extreme sensitivity of T c to nonmagnetic impurities is consistent with pair-breaking in unconventional superconductors and has been observed in the unconventional superconductor UPt3 [19] and the high-T c cuprates [20] as well. The suppression can be modelled by a generalized Abrikosov-Gor’kov pairbreaking theory [21] adapted for nonmagnetic impurities in a superconductor with an unconventional gap symmetry [22, 23]. The exact symmetry of the superconducting order parameter of Sr2 RuO4 [24, 25, 26, 27, 28, 29] and notably the pairing mechanism [12, 14, 30, 31, 32] are still controversial as well. On the base of the later discussed experiments several models for the superconductivity in Sr2 RuO4 have been developed and adapted according to the experimental progress. These models will be discussed at the end of the section after a short description of the experimental situation2 . Nodal Structure Early measurements [34] showed a large residual speciﬁc heat for T → 0 which prevent from a deﬁnite assignment of the low-T temperature dependence. Due to an improved sample quality the residual speciﬁc-heat contributions were signiﬁcantly reduced. From measurements on samples with T c ranging from 0.43 K to 1.17 K [35] it is expected , that no residual contribution remains for samples with optimal T c , i. e. that the residual speciﬁc heat is not intrinsic. The speciﬁc heat data on the best crystals reveal a T 2 dependence at T < T c /2 probably caused by line nodes of the energy gap. In addition, the jump ∆C at T c is much smaller than that expected from an isotropic gap. One ﬁnds ∆C/γT c = 0.74 in line with the low-temperature behaviour. The angular dependence of the speciﬁc heat on the applied magnetic ﬁeld orientation was used as a probe for the quasiparticle density of states induced by the 2

A more extended review of the superconducting properties was given by Mackenzie and Maeno [33] (on the basis of the knowledge in spring 2002)

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Fig. 10.3. Field-orientation dependent speciﬁc heat Ce /T measured by rotating the applied magnetic ﬁeld on a cone with polar angle Θ. φ is the azimuth angle and the measurements have been done at ﬁxed T = 0.12 K and reduced magnetic ﬁeld H/Hc2 = 0.4. For details see text (from [36])

magnetic ﬁeld. An oscillatory modulation of the thermodynamic quantities has theoretically been predicted by Vekhter et al. [37] and Won and Maki [38, 39] for the vortex state of d-wave superconductors. The observed in-plane anisotropy (see Fig. 10.3 for Θ = 90◦ (H ⊥ c)) below 0.3 K which is absent in the normal state has been interpreted as a modulation of the superconducting gap on the γ band with a minimum along the [100] direction [40]. For the ﬁeld range 0.15 T < µ0 H < 1.2 T the non-sinusoidal 4-fold angular variation can be approximated by Ce (φ) = C0 + C4 · f4 (φ)

(10.1)

with f4 (φ) = 2| sin 2φ| − 1. Further studies extended to smaller polar angles show a steeply decrease of C4 (φ) with decreasing Θ from 90◦ (i. e. towards H c). No angular modulation of Ce (φ) was detected below Θ = 80◦ (see Fig. 10.3). Deguchi et al. concluded that the origin of this behaviour is the compensation by gap anisotropy in

10.1 Sr2 RuO4

125

the α and β bands which most probably have line nodes or gap minima along [110], i. e. with antiphase with that of the γ band [36]3 . Further hints at the nodal structure were obtained from thermal conductivity measurements. Zero-ﬁeld experiments on single crystal with T c = 1.44 K show a quadratic temperature dependence of κ below T c down to ≈ 0.3 K and a residual κ/T which is independent of T c , i. e. of the impurity concentration [41, 42, 43, 44]. Furthermore, below T ≈ 0.3 K samples with a high T c show an crossover to a T 3 dependence of κ [43] signalling the crossover from heat transport dominated by the quasiparticle excitations at the nodes to heat transport dominated by bound states. Both, the residual and the T 3 contributions, have been discussed in the context of various models and it appears that the residual term is compatible with the universal limit of κ/T [45]. The T 3 dependence has been predicted by Zhitormirsky and Walker [46] and Graf and Balatsky [26] for the orbital dependent gap structure and scattering in the unitary limit. In an applied magnetic ﬁeld the thermal conductivity increases linearly with ﬁeld [41, 47]. It has been shown that this is consistent with κ(H) in very clean superconductors with line nodes. In addition, Izawa et al. [41] rotated the magnetic ﬁeld within the RuO2 plane and observed contributions with two- and fourfold symmetry of the in-plane thermal conductivity. In comparison with various proposals for the gap function they conclude that both contributions are incompatible with any model with vertical line nodes, but strongly indicate the presence of horizontal line nodes. Hints at line nodes of the gap function came from the temperature dependence of the penetration depth λ as well. Bonalde et al. [48] reported on penetration-depth measurements on Sr2 RuO4 single crystals of diﬀerent quality utilizing a 28 MHz tunnel diode oscillator. Samples with a rather high T c close to 1.4 K unanimously exhibit a T 2 dependence below 0.8 K down to 40 mK. In contrast, a dirty sample with T c = 0.82 K shows ∆λ(T ) = λ(T ) − λ(0.04 K) ∝ T 3 below 0.6 K. These power laws have been discussed in the framework of several models including the orbitaldependent superconductivity model [49], an anisotropic p-wave model [50] and a weak-coupling d-wave model in order to account for possible line nodes. Only the gap function with line nodes reproduces the quadratic temperature dependence at low temperatures satisfactory, if, in addition, the presence of impurities is taken into account. However, no crossover to a linear behaviour has been found as expected from the local-limit calculations by Hirschfeld and Goldenfeld [51]. The observation of a T 3 dependence of ∆λ in the sample with lower T c instead of a more pronounced T 2 dependence lead to the conclusion, that nonlocal eﬀects might be responsible for the observed exponents. According to Kosztin and Leggett [52] nonlocal electrodynamics in superconductors with line nodes of the order parameter can cause a quadratic T dependence instead of a linear T dependence even in the clean limit. In the presence of impurities (dirty nonlocal limit) a T 3 dependence would be expected [52] at low temperatures. Recently, Kusunose and Sigrist [28] showed that the orbital-dependent superconductivity model in the clean nonlocal limit allows a consistent description of the experimental data of Bonalde et al. [48] as well. 3

The underlying order-parameter scenario is discussed later at the end of this section.

126

10 Metal-Oxide Superconductors

Penetration-depth measurements by µSR experiments in the transverse ﬁeld have been performed only on samples with rather low T c 0.7 K [53] and T c 1.15 K [54]. From the real part of the Fourier transform of the time evolution of the muon polarization the probability distribution of the local ﬁelds and its width ∆B2 1/2 ∝ λ−2 was determined. A closer inspection of the temperature dependence yields −2 α (10.2) λ−2 ab (T ) = λab (0)(1 − (T/T c ) ) with λab (0) = 200 ± 20 nm [53] and λab (0) = 190 nm [54]. Both experiments yield n = 2.5 [53] and n = 2.78 [54] for the exponent α which is close to the BCS expectation (see discussion in Sect. 4.3). This would imply a nodeless energy gap of s- or p-wave symmetry. However, at low temperature the data in [53] are also consistent with a linear temperature dependence which is usually taken as an indication of the presence of nodes. For a deﬁnite conclusion one has to await measurements on samples with higher T c where impurities do not mask the intrinsic temperature dependence. A non-exponential T dependence has also been reported from ultrasound attenuation measurements [57, 56]. While Matsui and coworkers reported a crossover from T 2 to T 3 behaviour above ≈ 0.4 K for the transverse sound attenuation (see also comment by Gavenda [58]), Lupien et al. [56] investigated the longitudinal and transverse ultrasound attenuation in the normal and superconducting states with sound propagating along the 100 and 110 direction in the basal plane and found both a signiﬁcant anisotropy and power-law behaviour down to T c /30. The data have been analyzed by Walker et al. in a model where the ultrasound attenuation is determined by the electron-phonon matrix element [55] (see Sect. 3.3). They showed that the strong anisotropy of the ultrasound attenuation is connected with the layered square-lattice structure of Sr2 RuO4 and occurs only in the interaction of phonons with electrons in the γ band, but not with electrons in the α and β bands. Therefore, the attenuation of the most strongly attenuated modes is associated with their interaction with the electrons in the γ band and thus gives information about the nodal structure of this band, while the attenuation of the most weakly attenuated modes probes the nodal structure of the α and β bands. Under the assumption of an order parameter that transforms as the Eu irreducible representation of the point group D4h and parametrized by dni eıkRn (10.3) di (k) = n

with dix (k)

ky a kx a kz c = δ sin (k x a) + sin cos cos 2 2 2 i

and dyi (k) = δi sin (ky a) + cos

ky a kx a kz c sin cos 2 2 2

(10.4)

(10.5)

where i = α, β, γ denotes the band index, they achieved a quantitative description of the data (see Fig. 10.4). The best ﬁts were obtained with ∆γ0 = 0.7 meV and

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127

Fig. 10.4. Numerical ﬁts of the temperature dependence of the in-plane ultrasound attenuation modes of Sr2 RuO4 (from [55]). The free parameters are δγ and ∆γ0 (for explanation see text). The exerimental data have been published by Lupien et al. [56]

δγ = 0.35. The non-zero δ indicates that the horizontal line node at kz = ±π/c (δi = 0) is removed and point nodes appear instead. Parity and Spin State One of the early hints at unconventional superconductivity in Sr2 RuO4 came from zero-ﬁeld muon-spin relaxation measurements that reveal the spontaneous appearance of an internal magnetic ﬁeld below the transition temperature [59]. As already discussed, the appearance of such a ﬁeld indicates that the superconducting state is

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characterized by the breaking of time-reversal symmetry. This spontaneous internal magnetic ﬁeld can be caused either by a ﬁnite hyperﬁne ﬁeld or spontaneous supercurrents in the vicinity of inhomogeneities in the order parameter near impurities, surfaces and/or domain walls [60, 61]. The knowledge of the origin would help to discriminate between unitary and non-unitary states. However, this distinction cannot be made from the µSR results. From the increase in the exponential relaxation below T c a characteristic ﬁeld strength of 0.5 G was estimated. This is about the same as observed in the B phase of UPt3 [62]. NMR/NQR measurements on Sr2 RuO4 beneﬁt from the presence of three possible local probes within the RuO2 plane, namely the natural abundant Ru isotopes 99 Ru and 101 Ru both with nuclear spin I = 5/2 and the O isotope 17 O with I = 1/2 which has to replace the natural 16 O without nuclear spin. Therefore, the Knight shift and the nuclear relaxation rate can be measured on diﬀerent local sites. As already mentioned in Sect. 4.2 the Knight shift reveals information on the spin state in the superconducting state. Both 17 O [63] and 99 Ru [64] Knight shift measurements with magnetic ﬁeld applied perpendicular to the c axis, i. e. within the RuO2 planes, indicate an unchanged spin susceptibility below T c which identiﬁes Sr2 RuO4 as a spin-triplet superconductor. The result reinforces that the electrons in Sr2 RuO4 are bound together in parallel-spin pairs parallel to the RuO2 plane, consistent with a vector order parameter d(k) pointing along the c axis. Measurements of the 101 RuKnight shift in a magnetic ﬁeld parallel to the c axis (see Fig. 10.5) also ﬁnd an invariance of the Knight shift with respect to the ﬁeld and temperature on passing through the upper critical ﬁeld and T c , respectively, if the applied ﬁeld is larger than 200 Oe [65]. This ﬁnding was interpreted in a scenario where the d vector pointing to the c axis in zero ﬁeld can ﬂip perpendicular to the c axis by application of a small magnetic ﬁeld along c. The absence of a Hebel-Slichter peak in the nuclear relaxation rate 1/T 1 on both sites was taken as a ﬁrst sign of non-s-wave superconductivity in Sr2 RuO4 . Further, the temperature dependence of the nuclear relaxation rate yields information on the nodal structure. While early measurements of the 101 Ru relaxation rate on a sample with rather low T c = 0.7 K found a linear T dependence [66], subsequent experi101 Ru down ments on samples with T c close to 1.5 K showed a T 3 power law of (1/T 1 )NQR to 0.15 K. The T 3 dependence is an indication of the gap function with line nodes. Consequently, the T -linear behaviour was ascribed to impurity eﬀects. The observation of a T 3 dependence has been used for a comparison of various models for the superconductivity in Sr2 RuO4 . The result favors a p-wave model with a strongly anisotropic gap [50] or with a line node [67]. In contrast, the nuclear relaxation rate 17 O on the 17 O site shows a T -linear behaviour even for high-T c samples in (1/T 1 )NQR zero ﬁeld [68]. Application of a magnetic ﬁeld parallel to the planes completely suppresses the linear T dependence at low T . This was interpreted as an emergence of gapless thermal spin ﬂuctuations for the out-of-plane component along the c axis which possibly originate from the thermal motion of the domain structure of a superconducting k x ± iky state.

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129

Fig. 10.5. Magnetic-ﬁeld (a) and temperature dependence (b) of the 101 Ru-Knight shift in a ﬁeld applied parallel to the c axis of Sr2 RuO4 (by courtesy of K. Ishida, U Kyoto)

Energy Gap Information on the energy-gap size and symmetry have been obtained from several experiments. While Laube et al. [69] investigated point contacts between Sr2 RuO4 and a sharpened Pt needle as a normal metal counterelectrode, Jin et al. [70] used planar Pb-Sr2 RuO4 tunnel junctions, and Upward et al. [71] vacuum tunnelling spectroscopy between a Pt/Ir tip and a cleaved Sr2 RuO4 surface. The planar tunnelling measurements reveal the presence of an insulating surface layer of unknown origin. The tunnelling characteristics of the Pb-Sr2 RuO4 junction show a clear gap feature, but resembles very much the tunnelling characteristics of superconducting Pb with ∆Pb = 1.4 meV in accordance with the literature value of 2∆Pb /kB T c = 4.3 [72]. No further gap structure is observed (see Fig. 10.6), although, features from both superconductors should be present in the tunnelling characteristic of a superconductor-superconductor tunnel junction. This hints at a strong surface degradation of Sr2 RuO4 during the preparation process. The vacuum tunnelling data reveal a much smaller width of the relevant features in the tunnelling spectra (see Fig. 10.7). Similar tunnelling characteristics have been obtained irrespectively of the position of the Pt/Ir tip on diﬀerent crystals or at diﬀerent locations of the same Sr2 RuO4 crystal and the characteristic features disappear above T c . The authors interpreted the maxima at ﬁnite bias as a manifestation of the singularity in the density of states expected at ∆max and determined ∆max from the position of this maxima. They derived a BCS parameter 2∆max /kB T c = 6.2 − 8 and accounted the enhanced value to a large gap anisotropy as arising e. g. from vertical line nodes.

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Fig. 10.6. Tunneling characteristic for a Pb-Sr2 RuO4 tunnel junction along the c axis at T = 0.5 K (from [70])

Fig. 10.7. Tunneling spectra on diﬀerent crystal or at diﬀerent locations of the same Sr2 RuO4 crystal at T = 0.08 K (from [71])

The point-contact measurements were performed on two single crystals, grown both by a ﬂoating zone technique in diﬀerent groups4 . Determination of T c was carried out via bulk resistivity measurements. A total of about 150 contacts have been investigated with contact resistances typically ranging from R0 = 0.5 to 25 Ω. 4

Crystal #5 was provided by F. Lichtenberg at Augsburg University and shows a midpoint transition temperature T c50% = 1.02 K with a transition width ∆T c90%−10% = 0.035 K, crystal #C85B5 was grown by Z. Q. Mao and Y. Maeno at Kyoto University and has T c50% = 1.54 K and ∆T c90%−10% = 0.15 K.

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131

The measurements were performed in diﬀerent conﬁgurations of the predominant current injection relative to the crystallographic axis of Sr2 RuO4 and the applied magnetic ﬁeld. In addition, the inﬂuence of the surface treatment has been investigated. Spectroscopic information was obtained only from contacts with current injection parallel to the ab-planes because the transport through such contacts was in the ballistic limit. In the spectra with current perpendicular to the ab-planes evidence for non-spectroscopic eﬀects have been found [73]. These have been explained in terms of the heating model [74] for contacts in the thermal limit (see Sect. 5.3). For both crystals two distinctly diﬀerent types of structures in dV/dI vs V spectra are observed in the superconducting state: either a double-minimum structure (curve 1 and 2 in Fig. 10.8) or a zero-bias anomaly, i. e. a single-minimum structure centered

Fig. 10.8. Diﬀerential resistance dV/dI vs bias voltage V normalized to the zero-bias resistance R0 measured at low T = 0.2 K T c with R0 = 4.4 (curve 1), 5.0 (2), 5.3 (3), and 3.6 Ω (4). The inset shows a dV/dI curve with R0 = 7.6 Ω to higher bias. Note that the diﬀerential resistance dV/dI and not the diﬀerential conductance dI/dV is shown, therefore, the zero-bias anomaly in curve 3 and 4 appears as a minimum and not as a maximum

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at V = 0 (curve 3 and 4 in Fig. 10.8). These structures appear independently of the contact resistance and the in-plane orientation of the current. The occurrence of a zero-bias anomaly is believed to have its origin in an Andreev-bound surface state hinting at an unconventional order parameter of the superconducting phase which changes sign as function of the k-vector. The shape of the spectra were satisfactorily reproduced by an analysis of the data in terms of a p-wave pairing state with order parameter d(k) = zˆ (k x ± iky ). The current transport across the contact was modelled allowing a tunable transparency and a phenomenological acceptance cone. Within this model, amongst other things, information about the transmittivity of the contact has been extracted [69]. Contacts showing the zero-bias anomaly are described by a low transmission probability, the ones with double-minimum structures are described by a high transmission probability. The temperature dependence of both types of spectra is shown in Fig. 10.9a and b, together with the calculated spectra. In both cases, the structures related to superconductivity become weaker with increasing temperature and vanish near T c . The temperature dependence of the theoretical curves were calculated without additional parameter once ∆0 has been determined by ﬁtting to the spectrum at lowest T . Identical values for ∆0 = 1.1 meV have been extracted from both types of spectra, which is about 5 times the value expected from a weak-coupling theory.

Fig. 10.9. Temperature dependence of the spectra with a double-minimum structure (a) or with a single-minimum (b). Closed (open) symbols denote the measured (calculated) spectra. For clarity, the curves at higher T are shifted with respect to the curve at lowest T . D0 is the transmission probability for quasiparticles along the contact normal and λ is a measure for the opening angle of the acceptance cone

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133

Point contacts of high transmission exhibit a direct metallic conductivity, and, in contrast to the tunnelling limit, excess current Iexc occurs. This additional current through the constriction is a consequence of the Andreev scattering process and carries further information on the order parameter (see Sect. 5.3). For classical superconductors with an isotropic energy gap ∆ the excess current is proportional to ∆ [75]. From the proportionality it follows that both quantities exhibit the same functional dependence on temperature and magnetic ﬁeld, respectively. For unconventional superconductors, however, this relation is altered in the presence of impurities and the excess current is not necessarily proportional to the order parameter. Experimentally, a strikingly linear dependence of the excess current as a function of temperature and applied magnetic ﬁelds over a surprisingly wide range of the phase diagram has been obtained (see Fig. 10.10). Each symbol in the lower panel represents one point contact either on sample #5 (open symbols) or #C85B5 (ﬁlled symbols). The excess current was determined by numerical integration of dI/dV vs. V after subtraction of a normalconducting background. The normalization values ﬁt (H = 0) and H ﬁt (Iexc = 0) have been determined from linear regression of the Iexc Iexc (H) vs. H data for each point contact separately. The inset shows typical dI/dV curves from which the excess current was derived. Note that the result is quite universal since it is found in both samples in spite of their diﬀerent value of T c . As outlined by Laube et al., the observed equivalence of the linearity in the ﬁeld and temperature dependence implies a well deﬁned functional dependence of the excess current Iexc on the superconducting gap ∆. In general, one obtains a scaling relation Iexc = constant × ∆1/ν close to T c , where ν is deﬁned by the order-parameter symmetry [76]. These observations have been discussed in the framework of the p-wave triplet pairing state [24, 25] in the presence of impurities. The pairing state can qualitatively account for the linear behaviour of the excess current. The p-wave analysis to the point-contact spectra [69] yields the BCS temperature dependence for the superconducting gap in Sr2 RuO4 , i.e., ν = 0.5 (thin full line in Fig. 10.10 upper panel). The resulting temperature dependence of the excess current determined in the framework of the p-wave analysis is shown in Fig. 10.10 as Iexc,1 (diamonds). The calculations are performed for a mean free path of 15 coherence lengths (ξ0 = v f /2πT c ) and for a diﬀusively scattering surface. It is clear that this model is insuﬃcient to describe the experimental data (squares). This is also true for the overall magnitude of the bulk gap ∆(0) = 1.1 meV = 5.6 × 1.76kB T c . The general conclusion, however, is that unlike in the s-wave case in unconventional superconductors the excess current is not necessarily proportional to the order parameter. This is a result of the fact that impurities and disorder strongly aﬀect the surface properties of unconventional superconductors. To reconcile the measured ∆(0) and Iexc (T ) with a p-wave order parameter an additional pair-breaking channel has to be considered. It was shown by Millis et. al. [22] that a low-frequency bosonic mode at a characteristic frequency ω p described by an Einstein spectrum A p (ω) = π2 J p ω p δ(ω − ω p ) leads to a temperature dependent pair-breaking parameter ωp (1 − g) J p ω p coth( ) , (10.6) Γin (T ) = 4 2T

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Iexc(T)/Iexc(0) and ∆/∆0

1

0.5

Iexc(T) (experiment) ∆Bulk,1(T) (theory: d=z[kx+iky]) Iexc,1(T) ∆Bulk,2(T) (theory: d=z[kx+iky] & Γin(T)) ∆Bulk,2(T) Iexc,2(T)

0 0

0.2

1/ν

with ν=1/3, ∆(0)=5.6 ∆BCS(0)

0.4

0.6

T/Tc

0.8

1

Fig. 10.10. Upper panel: Temperature dependence of the normalized excess current across a point contact in Sr2 RuO4 . Experimental results (squares) and the results of the calculation (open symbols) for the excess current from a p-wave analysis are shown, without (diamonds) and with (circles) the eﬀect of an inelastic scattering channel Γin (T ). The dashed thick curve illustrates the scaling relation Iexc,2 (T ) ∼ ∆bulk,2 (T )1/ν for the inelastic scattering Lower panel: Field dependence of the normalized excess current across several point contacts in Sr2 RuO4 . The magnetic ﬁeld H is aligned almost parallel to the c-axis, and the current accross the point contact is applied in the ab-plane. Each symbol represents one point contact on one of the two studied samples. The full line is a guide to the eye. For explanation of ﬁt (H = 0) and H ﬁt (Iexc = 0) see text. The inset shows for one point contact typical dI/dV Iexc curves from which the excess current was determined as a function of magnetic ﬁeld

where g is the coupling-constant appearing in the gap equation. For the excess current this model gives an excellent agreement with experimental data, as shown by Iexc,2 (circles) in Fig. 10.10 for ω p = 0.5T c and (1−g) 4 J p = 2π × 0.25.

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135

Field-dependent calculations near Hc2 are complicated due to the presence of vortices near the interface, and have not been performed within the p-wave approach so far. However, from experiment it is obvious that with increasing magnetic ﬁeld both types of structures are weakened until they vanish close to Bc2 . In addition to the structures attributed to superconductivity, a maximum in the spectra centered at V = 0 starts to develop - already below Bc2 - and becomes more pronounced with increasing ﬁeld. The maximum is observed in a wide range of the (B, T ) diagram and persists even above T c up to several K. The maximum in dV/dI signals insulating behaviour, but its origin is not yet understood. Several scenarios, including magnetic or Kondo-type origin, a lattice distortion at the surface or a structural instability of Sr2 RuO4 , have been discussed [77]. A ﬁeld-induced structural instability at the surface seems the most promising candidate, but further work is necessary for a deﬁnite assignment. For example, scanning tunnelling microscopy in an applied magnetic ﬁeld can be helpful for a clariﬁcation. Phase-Sensitive Measurements Several experiments have been performed on Sr2 RuO4 by diﬀerent groups [70, 78, 79, 80, 81] in order to study the pairing symmetry and to obtain a doubtless conﬁrmation of the proposed order-parameter symmetry. However, despite of the experimental eﬀorts, the situation is still contradictory. This has mainly to do with diﬃculties in preparing reliable and uniform Josephson junctions. Although pair tunnelling was achieved in most junctions, none of the junctions show a clear Fraunhofer pattern of the critical current as a function of applied magnetic ﬁeld which hints at spatial ﬂuctuation of the critical supercurrent density. A reduction of the critical current between two Pb electrodes was observed in a Pb/Sr2 RuO4 /Pb junction (see Fig. 10.11a) when it was cooled below the critical temperature of Sr2 RuO4 [70]. In SN S type junctions a weakening of the coupling between two s-wave electrodes is expected, if the interlayer N becomes superconducting with non-s-wave pairing symmetry [82]. However, this experiment both beneﬁts and suﬀers from the presence of Ru lamellas in the junction region

Fig. 10.11. Several examples of junctions on Sr2 RuO4 for studies of the pairing symmetry (from [81])

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(see Fig. 10.11c). It beneﬁts, because proximity-induced coupling between Pb and Sr2 RuO4 occurs probably only due to the presence of these inclusions, which open a “window” through the insulating surface layer on Sr2 RuO4 and it suﬀers from it, because the inﬂuence of the lamellas on the pairing symmetry in the junction region is elusive. Nevertheless, the result at least points to a non-s-wave pairing symmetry of Sr2 RuO4 . Jin et al. [79] prepared a SS type Josephson junction by pressing an Indium wire against the sample surface and observed a orientation-dependent Josephson current. A supercurrent was present for in-plane junctions or absent for junctions along the c axis (see Fig. 10.11b). This result was interpreted as an indication for an order parameter with either p- or d- wave symmetry. The directional dependence of the supercurrent arises from a directional dependence of spin-orbit coupling strength which is responsible for a mixing between s- and p-wave pairs and thereby enables Josephson coupling (see Sect. 6.1). Sumiyama et al. [78] prepared SS type junctions by evaporating Nb or Sn through a shadow mask on a pre-structured Sr2 RuO4 surface (see Fig. 10.12). No directional dependence of the supercurrent was observed, for both orientations of the crystal a Josephson coupling was reported. The reason for the discrepancy between both experiments is not clear. As possible origins the surface roughness or the presence of Ru lamellas are discussed which avoid direction-sensitive measurements. A nonconducting surface barrier might account for the absence of Josephson coupling along c-axis oriented junctions as well.

Fig. 10.12. Experimental realization of SS Josephson junctions on Sr2 RuO4 (from [78])

Early SQUID experiments (see Fig. 10.11d) following the pioneering work by Kirtley and Tsuei [83] and Wollman et al. [84] on the high-temperature superconductors suﬀer from the reduced T c of the junctions and the obvious surface degradation during junction preparation [80, 81]. More recently, Nelson et al. found a clear diﬀerence between the critical current in the limit of zero magnetic ﬂux of SQUID devices with junctions on the same side of a Sr2 RuO4 crystal (c-axis junction, see Fig. 10.11d) and junctions on opposite sides (in-plane junctions) [85]. They reported a maximum of Ic (Φ = 0) for the c-axis junction and a minimum of Ic (Φ) at Φ = 0 for in-plane junctions. The result indicates that the superconducting phase changes by π under inversion and veriﬁes odd-parity, spin-triplet pairing in Sr2 RuO4 . The position of the minimum, however, slightly deviates from Φ = 0. The asymmetry of the critical current with respect to Φ was interpreted as an additional conﬁrmation of

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137

the chiral p-wave symmetry by Asano et al. who calculated the response of the critical Josephson current to applied magnetic ﬁelds for diﬀerent types of ring topology and pairing symmetries [86]. Flux Line Lattice Information about the vortex lattice have been obtained both by µSR measurements in the transverse ﬁeld [53, 54] and by small-angle neutron scattering (SANS) [87, 88]. In the µSR experiment the time evolution of the muon polarization was measured and from the real part of the Fourier transform the probability distribution of the local ﬁelds has been obtained. The line shape of this distribution function is in good agreement with vortices forming a square lattice, at least at the applied ﬁeld of 6 mT [53] and 15 mT [54]. The observation of a square lattice is not by itself a proof for non-s-wave superconductivity. The lattice structure may change as a function of ﬁeld in conventional s-wave superconductors as well and a square vortex lattice can arise from non-local eﬀects [89]. Indeed, this was experimentally observed in members of the borocarbide family RNi2 B2 C [90, 91, 92]. The presence of a square vortex lattice was conﬁrmed by the SANS experiments [87, 88] in magnetic ﬁelds of B = 10, 20, and 30 mT parallel to the c-axis of Sr2 RuO4 , i. e. perpendicular to the layers. The orientation of the ﬂux line lattice is such that the nearest-neighbour direction is at 45◦ to the Ru-O-Ru direction in the crystal lattice (compare with Fig. 10.2). The result was interpreted in terms of the two-component Ginzburg-Landau model for the superconductivity of Sr2 RuO4 in presence of a magnetic ﬁeld as proposed by Agterberg [93, 94]. In this model the order parameter has two components (η1 , η2 ) and belongs to the two-dimensional representations Γ5u of the tetragonal point group D4h [60]. A time-reversal symmetry breaking state would be realized for (η1 , η2 ) ∝ (1, i) and the orbital symmetry properties are reproduced by k x + iky . The measured intensities of diﬀerent ﬂux lattice Bragg reﬂections have been shown to be in qualitative agreement with this model which describes a p-wave state. On the base of this Heeb and Agterberg calculated the vortex core structure and proposed a fourfold deformation of the vortex core which should be observable in scanning tunnelling measurements. A certainly intriguing method was used to study the ﬁeld penetration in a magnetic ﬁeld applied almost within the ab plane of Sr2 RuO4 , namely µSQUID force microscopy [95]. The µSQUID force microscope consists of a piezoelectric tuning fork carrying the SQUID. By detecting the interaction forces between the SQUID tip and sample surface the magnetic ﬂux penetrating the superconductor in the vortex state has been imaged. The images taken at low ﬁelds show the magnetic ﬂux starting to penetrate the crystal in form of elongated ﬂux chains, oriented along the ﬁeld direction. The formation of vortex chains is driven by the transverse magnetization of the tilted vortices. The eﬀect has already been observed for other superconductors, e. g. the high-T c ’s (see Sect. 10.2).

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Superconductivity Models One possibility to deliberate on the question of the order-parameter symmetry is to look at the possible pairing states allowed under the aspects of the group theory. The possible pairing states of Sr2 RuO4 can be classiﬁed according to the irreducible representations of the D4h point group [60]. The only state which breaks time-reversal symmetry is the odd parity Eu state which can be expressed by the vector function d(k) = zˆ (k x ± iky ). Other possible time-reversal symmetry breaking states would require complex combinations of states belonging to diﬀerent representations which would in general lead to double superconducting transitions, with broken time reversal only below the lower transition. As there is no evidence to date of multiple superconducting transitions, the Eu state is the only one compatible with the experimental data. Based on the proposal by Rice and Sigrist [7] that modest ferromagnetic enhancement in the Fermi liquid state favors p-wave superconductivity the earliest model assumed a p-wave order parameter in analogy to the A phase of 3 He, namely the unitary spin-triplet d(k) = zˆ (k x ± iky ) order parameter. However, the pairing state is isotropic in the basal plane and fully gapped, which contradicts the observation of line nodes in several experiments. As shown by Blount [96] symmetry requires spin-triplet superconductors to be nodeless or have point nodes but not line nodes. Line nodes in a spin-triplet state can appear only accidentally if spin-orbit coupling is present. The proposed non-unitary p-wave model [97, 98] is incompatible with the observation of line nodes as well. In order to account for the NMR results Miyake and Narikiyo [50] showed that starting from a pairing interaction mediated by short-range ferromagnetic spin ﬂuctuations a modiﬁed order parameter d(k) = ∆0 zˆ (sin k x +i sin ky ) is obtained instead of the isotropic d(k) = ∆0 zˆ (k x +iky ). Furthermore, the amplitude of the gap is described approximately by a fourfold-symmetry gap function of |d(k)| = ∆0 [1 − r cos(4θk )] with r = 0.692. This gap function has vertical line nodes. Hasegawa et al. [67] have proposed a phenomenological model with several possible f -wave pairing states which have been constructed from the product of two-dimensional odd-parity Eu representations with diﬀerent even-parity representations which are chosen to have zeros. The stability of such mixed-parity states has been investigated by Eschrig et al. [27]. Among all states allowed by symmetry the following states are the most promising candidates for an explanation of the experimental ﬁndings of µSR, NMR and speciﬁc heat: d(k) = ∆0 zˆ (k x + iky )k x ky d(k) = ∆0 zˆ (k x + iky )(k2x − ky2 ) d(k) = ∆0 zˆ (k x + iky )(cos(ckz ) + a0 )

(10.7) (10.8) (10.9)

These are time-reversal symmetry-broken states which have line nodes running either vertically (10.7 and 10.8) or horizontally (10.9). However, only d(k) = zˆ ∆0 (k x + iky )(cos(ckz ) + a0 ) with |a0 | < 1 and real, can account for the θ dependence of the thermal conductivity under rotation of the magnetic ﬁeld [41]. Graf

10.1 Sr2 RuO4

139

and Balatsky [26] analyzed the speciﬁc-heat data and conclude that pairing states similar to those above in 10.7 are possible. On the other hand, Wu and Joynt [99] analyzed ultrasonic-absorption and thermal-conductivity measurements and came to the conclusion that only the state in 10.8 can qualitatively account for the transport data. Other models take into account the multi-sheet Fermi surface of Sr2 RuO4 . Won and Maki [100] assumed that the order parameter is the same for all three bands and considered a f -wave order parameter with d(k) = ∆0 zˆ kz (k x + iky )2 . This order parameter has horizontal line nodes and can account for the observed power laws in the thermodynamic properties. The orbital-dependent superconductivity model proposed by Agterberg et al. [49] assumes that interband pairing interaction of quasiparticles is weak between the γ band characterized by the Ru 4d xy orbitals and the α and β bands by the Ru 4d xz and 4dyz orbitals so that a large gap occurs on the γ band and only small proximity-induced gaps on the α and β bands. As a consequence, the ungapped Fermi surfaces can account for the residual density of states observed in early speciﬁc-heat measurements, at least until a second phase transition appears at very low T . However, the residual density of states observed in early speciﬁc-heat measurements seems not to be intrinsic and, up to now, no second phase transition has been found in zero ﬁeld. Zhitomirsky and Rice [101] extended this model in order to account for the experimental ﬁndings. They proposed that line nodes exist only on two of the three Fermi surfaces: a fully-gapped Fermi surface exists in the active γ band, which drives the superconducting transition, while line nodes develop in passive α and β bands by the interband proximity eﬀect. They found that a nodeless axial order parameter d(k) ∝ (k x + iky ) in the active band can induce superconducting gaps d(k) ∝ (k x + iky ) cos(kz /2) with horizontal line nodes in the passive bands and they showed that this model almost perfectly describes the speciﬁc-heat data. A similar approach was chosen by Nomura and Yamada [103, 104]. By solving the linearized Eliashberg equation on the basis of the two-dimensional three-band Hubbard model they determined the superconducting gap structure of Sr2 RuO4 . They showed that the momentum dependence of the eﬀective pairing interaction on the γ band favors p-wave pairing on this band, while the pairing on the α and β

Fig. 10.13. Schematic representation of the vector order parameter d(k) = zˆ ∆0 (k x + ıky ) for Sr2 RuO4 . Small arrows depict the spins and the large arrows depict the orbital angular moments (from [102])

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10 Metal-Oxide Superconductors

bands is induced through pair scattering from the γ band. Therefore, spin-triplet superconductivity in Sr2 RuO4 is the natural result of electron correlations, and cannot considered as a result of strong magnetic correlations. Under the assumption that the orbital symmetry of the Cooper pairs is p-wave with k x + iky symmetry they found that the induced gap function on the β band has vertical line nodes at k x = ±ky . With this model the speciﬁc-heat data [105] have been explained successfully. They calculated that the speciﬁc-heat jump is dominated by that of the γ band but the low-temperature thermal excitations in the superconducting state are dominated by those of the β band due to the vertical line-node-like structure. In addition, on the basis of this model the angular modulated speciﬁc-heat data have been interpreted [36, 40]. In conclusion, there is still a lot of controversy on a deﬁnite assignment of the order-parameter symmetry. But further discussion of the various models and additional excellent experiments will add more pieces to that puzzle and ﬁnally reveal a consistent picture of the pairing state of Sr2 RuO4 .

10.2 High-Temperature Superconductors The discovery of superconductivity in the La-Ba-Cu-O system in 1986 by Bednorz and M¨uller [106] was like a big drumbeat in the hall of superconductivity and it opened a run of research activities into the new ﬁeld of metal-oxide compounds. As already mentioned above, a common structural element of high-temperature superconductors are copper-oxide planes which dominate the superconducting properties (see Fig. 10.14). In addition, the “123” class of materials contains CuO chains which are thought to serve as a charge-carrier reservoir to control the electron density in the planes. The mobile charge carriers which reside primarily within the CuO2 planes can be electrons, but are usually holes. Many of the cuprates can be doped with charge carriers and rendered superconducting by substitution of appropriate elements into an antiferromagnetic insulating parent compound. One well-investigated example is YBa2 Cu3 O7−δ with T c = 93 K at optimal doping. The undoped parent compound YBa2 Cu3 O6 is an antiferromagnetic insulator. There is still no consensus on the mechanism causing the high T c in these materials. However, most properties can be well described within the concepts of Bardeen-Cooper-Schrieﬀer theory and Ginzburg-Landau theory. As already stated in the beginnning, this book is not dedicated to review the high-temperature superconductors. There exist a vast of excellent review articles and monographs [107, 108, 109, 110, 111, 112, 113] which give a quite complete overview on the state-of-art of research on this material and which should be referred to. However, for sake of completeness and especially, to account for the importance of the research on the high-temperature superconductors for the understanding of the many properties of other unconventional superconductors, a short ﬂash on the order-parameter symmetry and the concomitant physical properties will be given. Nodal Structure A very large number of experiments that probed the nodal structure showed that there exists a ﬁnite density of states all the way down to zero energy. However,

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141

Fig. 10.14. Schematic representation of the unit cells of the crystal structure of YBa2 Cu3 O7−δ for δ = 0 and δ = 1 (from [2])

there is limited quantitative consistency between diﬀerent samples and diﬀerent experimental techniques and it is not clear what is the intrinsic property. Penetrationdepth measurements, for example, showed a linear temperature dependence of λ−2 in line with theoretical predictions for the d x2 −y2 gap. Measurements of the thermal conductivity at low temperature T < 10−3 T c (see Fig. 10.15) report a T 3 power law and a residual linear term in the thermal conductivity of both optimally doped YBa2 Cu3 O7−δ and Bi2 Sr2 CaCu2 O8 [114]. The magnitude of the linear term is in excellent agreement with the value expected from Fermi-liquid theory and the d-wave energy spectrum. This was taken as a hint that thermally excited quasiparticles are a signiﬁcant mechanism in suppressing the superﬂuid density in cuprate superconductors. Spin State and Parity NMR measurements on the high-temperature superconductors showed the absence of a Hebel-Slichter peak in the relaxation rate 1/T 1 . At low temperature the relaxation rate exhibits a power law with an exponent between 3 and 4.5. Knight shift measurements involving nuclei on diﬀerent lattice sites of YBa2 Cu3 O7−δ show a strong reduction of K in the superconducting state which clearly reﬂects spin singlet pairing [116, 117]. Theoretical model calculations gave better agreement, if d-wave rather than s-wave pairing was assumed [118]. For further information on NMR measurements the review by Pennington and Slichter should be referred to [119].

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Fig. 10.15. Low-temperature thermal conductivity of YBa2 Cu3 O6.9 and Bi2 Sr2 CaCu2 O8 at optimum doping. The lines are linear ﬁts to the data below 130 mK (from [114])

Energy Gap There have been many experimental data reported of point-contact and tunnelling spectroscopy on high-T c superconductors. The experimental methods that have been applied to study the electronic states range from scanning tunnelling microscopy/spectroscopy, thin-ﬁlm junctions to point contacts and break junctions. A review of the tunnelling spectroscopy on high-T c superconductors was given recently by Kashiwaya and Tanaka [120]. Most measurements of the gap width suggest considerably larger values for 2∆/kB T c than the BCS weak-coupling value of 3.5. For example, STM measurements by Maggio-Apprile et al. [121] found a gap width of YBa2 Cu3 O7−δ of about 20 meV corresponding to 2∆/kB T c = 5.1. Other values range between 4 and 7. A certainly exceptional test of the pairing symmetry has been reported by Wei et al. [115]. They measured tunnelling and point-contact spectra for diﬀerent orientations of a YBa2 Cu3 O7−δ crystal using a low-temperature scanning tunnelling microscope. For measurements on tunnel junctions with tunnelling current along the [110] direction a zero-bias anomaly has been reported

10.2 High-Temperature Superconductors

143

10 Mixed symmetry models

8

8

Normalized dI/dV

6

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d+is 4

6

2

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4

{110} STM junction

10

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d-wave fit: Z=3.3, β=π/8 ∆0=27 meV

2

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100

V (mV) 3 Mixed symmetry models

{001} STM junction

4 3 2

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-6 -4 -2

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1

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d-wave fit: Z=4.5, β =2π ∆0=19 meV 0

-150

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Mixed symmetry models 3

Normalized dI/dV

2

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1

d-wave fit: Z=0.4, β =2π ∆0=28 meV -150

-100

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0

50

100

150

V (mV)

Fig. 10.16. Tunnelling spectroscopy on YBa2 Cu3 O7−δ for diﬀerent orientation of the crystal and diﬀerent transparency of the interface. For measurements on tunnel junctions with tunnelling current along the [110] direction a zero-bias anomaly has been reported which is absent for tunnelling current along the [100] direction and for the point contact with high transparency (by courtesy of J. Wei, U Toronto)

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which is absent for tunnelling current along the [100] direction and for the point contact [115]. Theoretical results for the ab plane tunnelling conductance by Fogelstr¨om et al. [122] have shown that a zero-bias conductance anomaly is expected if the order parameter exhibits a sign change (see Sect. 6.2). Therefore, the measurements directly prove the d x2 −y2 symmetry of the order parameter. Phase-Sensitive Experiments Although supported by many experiments, the d x2 −y2 symmetry of the order parameter has deﬁnitely assigned only by phase-sensitive experiments as previously suggested by Sigrist and Rice [123]. The pioneering experiments were carried out by Wollman et al. [124], who compared the SQUID characteristics of two conﬁgurations either involving a π junction either without a π junction in the SQUID loop (see Figs. 10.17 and 6.2). They found the expected shift by φ0 /2 in the response to an externally applied ﬂux (see Fig. 10.18). However, the SQUID experiments had a number of complicating factors in their interpretation. For example, twinning eﬀects, ﬂux trapping, demagnetization, and ﬁeld-focusing eﬀects can strongly effect the data and there has been some controversy on the interpretation of the data [126, 127, 128].

Fig. 10.17. Scheme of a SQUID-type loop for a phase-sensitive test of the order-parameter symmetry (from [123])

The possibility of errors from undetected stray trapped ﬂux was addressed in subsequent experiments by Tsuei et al. [129] and Mathai et al. [126]. Both experiments used a scanning SQUID microscope to measure trapped ﬂux in a ring. Mathai et al. found half-quanta of trapped ﬂux in rings where both junctions were on diﬀerent faces of a YBa2 Cu3 O7−δ crystal, while there was an integral number of quanta for both junctions on the same face of the crystal. The experiment of Tsuei et al. compared three-junction and two-junction rings in a YBa2 Cu3 O7−δ ﬁlm (see Fig. 10.19, upper panel), ﬁnding half-integer numbers of ﬂux quanta in the three-junction conﬁguration and integer numbers in the two-junction rings (see Fig. 10.19, lower

10.2 High-Temperature Superconductors

145

Fig. 10.18. Summary of the experimental result of Wollman et al. [124, 125]: (a) Extrapolation of the measured SQUID resistance minimum vs ﬂux to zero-bias current for a corner SQUID and an edge SQUID on the same sample; (b) Measured critical current vs applied magnetic ﬁeld for an edge junction and (c) for a corner junction (from [83])

panel). All of these results provide strong support for the correctness of d-wave pairing. The tricrystal experiment has also applied to other high-temperature superconductors. A review on the phase-sensitive experiments can be found in [83, 84]. All these experiments indicate dominant d-wave pairing. Flux Line Lattice Various techniques have been applied to high-temperature superconductors to reveal information on the ﬂux line lattice. For ﬁelds applied parallel to the c axis a weakly distorted Abrikosov ﬂux lattice has been reported [130, 131]. The distortion arises from a small in-plane eﬀective mass anisotropy. Besides the classical Bitter decoration studies the scanning tunnelling microscope has been the main tool for such

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10 Metal-Oxide Superconductors

Fig. 10.19. Upper panel: Tricrystal geometry and ring orientation of the Scanning-SQUID experiment by Tsuei et al.. The central, three-junction ring should be a π ring for d x2 −y2 pairing (from [129]). Lower panel: Three-dimensional image of a thin-ﬁlm YBa2 Cu3 O7−δ tricrystal ring sample, cooled and imaged in nominally zero ﬁeld. The outer rings have no ﬂux, the central three-junction ring has half of a ﬂux quantum spontaneously generated (from [83])

investigations. Both methods revealed a vortex chain state [132, 133] in tilted ﬁelds. For YBa2 Cu3 O7−δ the vortex lattice is transformed into a pinstripe array of vortex chains, oriented to lie in the plane deﬁned by B and c [132]. As the applied ﬁeld is increased, the chains merge smoothly into an isotropic vortex lattice, still oriented in the same direction. Most attention have attracted studies of the electronic structure of the vortex cores. Bound quasiparticle states can exist inside the vortex cores with lowest energy given approximately by E ∝ ∆2 /2EF , where EF is the Fermi energy and ∆ is the superconducting gap. Vortex core states have been identiﬁed for YBa2 Cu3 O7−δ by Maggio-Aprile et al. [121] and for Bi2 Sr2 CaCu2 O8 by Pan et al. [20]. However, a fourfold symmetry sometimes predicted for d-wave vortices was not seen in the latter experiment. This was explained by quasiparticle states bound in the vortex cores whose order parameter is locally nodeless due to the presence of a magnetic ﬁeld.

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A Appendix

Table A.1. A survey on experiments probing the nodal structure of unconventional superconductors by specific heat (sp), thermal conductivity (tc), penetration depth (pd) measurements, and ultrasound attenuation (us) Compound

Sample

Technique

C ∼ T n, κ ∼ T n

Reference

pc pc pc sc pc pc pc

sp sp sp sp sp tc pd

n = 3 (< 0.2 K) n = 2.43 (0.1-0.6 K) n=3 n=2 n=2 n = 2(< 0.25 K) n = 2(< 0.25 K)

Steglich et al. [1] Bredl et al. [2] Franse et al. [3] Lang et al. [4] Ishikawa et al. [5, 6] Steglich et al. [1] Gross et al. [7]

sc sc sc sc sc sc sc sc sc sc sc sc

n = 3 (0.9-1.9 K) Ikeda et al. [8] n = 2 (0.2-0.6 K) Ikeda et al. [8] sp n = 2 (0.1-0.7 K) Movshovich et al. [9] sp n = 3.3 (0.6-1.8 K) Zapf et al. [10] sp n = 2 (0.033-0.1 K) Petrovic et al. [11] sp C(Θ) Aoki et al. [12] tc – Izawa et al. [13] tc n = 3.37 (0.033-0.1 K) Movshovich et al. [9] Ormeno et al. [14] pd (GHz) ∆λ ∼ T 1 (0.3-0.8 K) 3.3±0.4 Higemoto et al. [15] pd (µSR) ∆λ−2 ab ∼ T 1.5 ¨ Ozcan et al. [16] pd (MHz,GHz) ∆λ ∼ T (0.1-0.9 K) pd (MHz) ∆λ ∼ T 1.5 (0.14-1.13 K) Chia et al. [17] Chia et al. [17] pd (MHz) ∆λ⊥ ∼ T 1 (0.14-1 K)

sc

sp

CeCu2 Si2

CeCoIn5

sp

CeIrIn5

G. Goll: Unconventional Superconductors STMP 214, 153–168 (2006) c Springer-Verlag Berlin Heidelberg 2006

n = 2 (0.085-0.2 K)

Movshovich et al. [9] Petrovic et al. [18] continued on next page

154

A Appendix

continued from previous page Compound Sample

Technique C ∼ T n , κ ∼ T n n = 2 (< 0.2 K)

Reference

sc

tc

sc

3.0±0.4 pd (µSR) ∆λ−2 ab ∼ T

pc, annealed pc, annealed pc, annealed pc, annealed sc pc sc sc sc sc sc sc

sp sp sp sp sp tc tc tc tc tc pd pd(µSR)

sc

pd(µSR)

sc sc sc sc sc

us us us us us

pc pc pc pc pc sc

sp sp sp sp tc tc pd pd

Ott et al. [39] Mayer et al. [40] Ravex et al. [41] W¨alti et al. [42] Ravex et al. [41] n = 2(0.15-0.8 K) Jaccard et al. [43] ∆λ ∼ T 2 (0.06-0.5 K) Einzel et al. [44] Gross et al. [45] ∆λ ∼ T 2

pc sc sc sc sc sc

sp sp sp sp sp sp

n = 2(0.2-0.7 K) n = 3(0.5-0.9 K) n = 2(0.2-0.7 K) n = 2(0.5-0.9 K) n = 2(0.5-1 K) n = 2(0.15-1.1 K)

UPt3

UBe13

URu2 Si2

n=2 n = 2(0.15-0.5 K) n = 2(0.07-0.2 K) n = 2(0.2-0.4 K) n = 2(> 0.1 K) n=1−2 n = 2(0.07-0.25 K) n ≈ 3 (0.03-0.1 K) n = 3(0.03-0.05 K) ∆λ ∼ T 2 λ−2 ab ∼ (1 − (T/T c )) (0.05-0.3 K) λ−2 ab ∼ (1 − (T/T c )) (0.04-0.4 K) n = 1 (qˆ b, eˆ a) n = 3 (qˆ b, eˆ c) n = 2 (qˆ b) n = 1 (qˆ a, eˆ b) n = 3 (qˆ a, eˆ c)

Movshovich et al. [9] Petrovic et al. [18] Higemoto et al. [15]

Franse et al. [3] Sulpice et al. [19] Ott et al. [20] Fisher et al. [21] Hasselbach et al. [22] Franse et al. [3] Behnia et al. [23] Huxley et al. [24] Lussier et al. [25, 26] Suderow et al. [27, 28, 29, 30, 31, 32] Gross-Alltag et al. [33] Broholm et al. [34] Yaouanc et al. [35] Shivaram et al. [36] Shivaram et al. [36] Maurer et al. [37] Ellman et al. [38] Ellman et al. [38]

n=3 n = 2.9(0.2-0.85 K) n = 2.9(0.2-0.7 K) n = 3(0.07-0.2 K)

Schlabitz et al. [46] Maple et al. [47] Hasselbach et al. [48] Ramirez et al. [49] Knetsch et al. [50] Brison et al. [51] continued on next page

A Appendix

155

continued from previous page Compound Sample

Technique

sc

tc

sc sc pc sc

sp sp tc tc tc

UPd2 Al3

Sr2 RuO4

sc, T c = 1.13 K sp sc, T c = 1.48 K sp sc sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c sc, T c

= 1.52 K = 1.48 K < 1.1 K < 1.45 K < 1.44 K < 1.44 K < 1.44 K = 1.44 K = 0.82 K = 0.7 K = 1.15 K = 1.37 K = 1.42 K

C ∼ T n, κ ∼ T n

Reference Behnia et al. [52]

n = 3 (0.5-1 K) n = 2 (0.3-0.5 K) n = 2 (0.35-1 K)

Caspary et al. [53] Sakon et al. [54] Hiroi et al. [55] Chiao et al. [56] Watanabe et al. [57]

n=2 n=2 n=2 n=2 n = 2 (< 0.7 K) C(φ, Θ) n=2 n=2 n = 2 (0.3-0.6 K) n = 3 (0.1-0.3 K) n = 2 (> 0.3 K) ∆λ ∼ T 2 (0.04-0.85 K) ∆λ ∼ T 3 (0.04-0.6 K) 2.5 λ−2 ab ∼ (1 − (T/T c ) ) −2 λab ∼ (1 − (T/T c )2.78 )

Nishizaki et al. [58] Nishizaki et al. [59, 60] Maeno et al. [61] Langhammer et al. [62] Yoshida et al. [63] Deguchi et al. [64, 65] Suderow et al. [30] Izawa et al. [66, 67] Suzuki et al. [68] Suzuki et al. [68] Tanatar et al. [69, 70] Bonalde et al. [71, 72] Bonalde et al. [71] Aegerter et al. [73] Luke et al. [74] Lupien et al. [75] Matsui et al. [76]

sp sp sp tc tc tc tc tc pd (MHz) pd (MHz) pd (µSR) pd (µSR) us (long/trans) us (trans) n = 2 (0.17-0.4 K) n = 3 (0.3-0.8 K)

156

A Appendix

Table A.2. A survey on experiments probing the parity of unconventional superconductors by NMR/NQR and µSR studies Compound

Sample

Technique

Reference

pc, sc pc, powdered

63

Cu NMR Si,63 Cu NMR 63 Cu NMR 63 Cu NQR 63 Cu NQR µSR -ZFµSR -ZF-,-LF-,-WTFµSR -ZF-,-WTFµSR -HTF-

Kitaoka et al. [77] Ueda et al. [78] Asayama et al. [79] Ishida et al. [80] Kawasaki et al. [81, 82] Uemura et al. [83] Luke et al. [84] Amato et al. [85, 86] Koda et al. [87]

115

CeCu2 Si2

pc

pc pc

29

CeCoIn5 sc sc powdered sc powdered sc

In, 59 Co NMR In, 59 Co NMR 115 In NQR µSR -ZF,TF-

Curro et al. [88] Kohori et al. [89] Kohori et al. [89] Higemoto et al. [15]

sc powdered sc powdered sc powdered sc

115

In, 59 Co NMR In NQR 115 In NQR µSR -ZF,TF-

Kohori et al. [89] Kohori et al. [89] Zheng et al. [90] Higemoto et al. [15]

pc, sc pc

195

Pt NMR µSR -ZF-

Yogi et al. [91] Amato et al. [92]

sc

µSR -TFµSR -ZF,TFµSR -ZFµSR µSR -TFµSR -ZF,TF195 Pt NMR 195 Pt NMR

Broholm et al. [34] Luke et al. [93, 94] Dalmas de R´eotier et al. [95] Higemoto et al. [96] Yaouanc et al. [35] Dalmas de R´eotier et al. [97] Kohori et al. [98, 99] Tou et al. [100, 101]

115

CeIrIn5 115

CePt3 Si

UPt3

sc powder sc

continued on next page

A Appendix

157

continued from previous page Compound

Sample

Technique

Reference

9

MacLaughlin et al. [102, 103], Tien et al. [104] Heﬀner et al. [105] Heﬀner et al. [106] Luke et al. [107] Sonier et al. [108] Dalmas de R´eotier et al. [109]

UBe13

pc

sc sc

Be NMR

µSR -ZF,TFµSR -TFµSR -TFµSR -TFµSR -ZF,TF-

URu2 Si2 pc mc, aligned

29

Si Ru NQR µSR -ZFµSR -TFµSR -TF-

Kohori et al. [110] Matsuda et al. [111] Knetsch et al. [112] Heﬀner et al. [106] Luke et al. [113]

pc pc pc pc sc mc

27

Al NMR/NQR Al NQR 105 Pd NMR/NQR µSR -ZF,TFµSR -TFµSR -TF-

Kyogaku et al. [114] Tou et al. [115] Matsuda et al. [116] Amato et al. [117] Amato et al. [118] Feyerherm et al. [119]

pc sc pc

27 27

Al NMR/NQR Al NMR µSR -ZF-

Kyogaku et al. [114] Ishida et al. [120] Amato et al. [121]

sc sc sc sc sc sc sc sc

µSR -ZFµSR -ZF,TF17 O NMR 101 Ru NMR 101 Ru NQR 99 Ru NMR 17 O, 101 Ru NQR 101 Ru NQR

Luke et al. [122] Luke et al. [74] Ishida et al. [123] Ishida et al. [124] Ishida et al. [125] Ishida et al. [126] Mukuda et al. [127] Murakawa et al. [128]

101

UPd2 Al3 27

UNi2 Al3

Sr2 RuO4

158

A Appendix

Table A.3. A survey on experiments probing the energy gap of unconventional superconductors. The experiments have been performed either on polycrystals (pc) or single crystals (sc) by classical point-contact technique (PC), mechanically controllable break-junctions (MCBJ), planar tunnel junctions (TJ) technique, and scanning tunneling spectroscopy (STS) Compound

Sample

Counterelectrode

Technique

Reference

pc pc sc pc pc sc sc

Nb Al Ag W, Cu, Ag Pb CeCu2 Si2 PtIr

PC PC PC PC TJ MCBJ STS

Han et al. [129] Poppe [130] de Wilde et al. [131] Gloos et al. [132, 133, 134] Iguchi et al. [135] Gloos et al. [136] Goschke et al. [137]

sc sc sc

Pt Au Pt-Ir

PC PC PC

Goll et al. [138] Park et al. [139] Rourke et al. [140, 141, 142]

sc pc sc pc pc sc pc sc

Al, Nb, UPt3 Pt Pt Ag W Zn, Pb, NbTi UPt3 Pt

PC PC PC PC PC PC MCBJ PC

Poppe [130] Nowack et al. [143] Goll et al. [144, 145, 146, 147] de Wilde et al. [131, 148] Gloos et al. [133, 134] Naidyuk et al. [149] Gloos et al. [136] Obermair et al. [150]

pc pc pc pc pc pc pc

Nb, Ta W UBe13 W Cu, Pt, Ir, Ta, UBe13 UBe13 Au

PC PC PC PC PC MCBJ PC

Han et al. [129, 151] Nowack et al. [143, 152] Moreland et al. [153] Gloos et al. [133, 154] Kvitnitskaya et al. [155] Gloos et al. [136] W¨alti et al. [156, 157, 158]

sc sc sc

Ag Ag, URu2 Si2 Mo, W

PC PC PC

Naidyuk et al. [159] Nowack et al. [160] Hasselbach et al. [161] continued on next page

CeCu2 Si2

CeCoIn5

UPt3

UBe13

URu2 Si2

A Appendix

continued from previous page Compound

Sample

Counterelectrode

Technique

Reference

sc sc

Ag Zn, NbTi

PC PC

pc pc sc sc sc sc

Ag W Pt URu2 Si2 Pt, URu2 Si2 URu2 Si2

PC PC PC MCBJ PC MCBJ

de Wilde et al. [131] Nowack et al. [162], Naidyuk et al. [163] Samuely et al. [164] Gloos et al. [133, 134] Naidyuk et al. [165] Naidyuk et al. [166] Rodrigo et al. [167] Gloos et al. [136]

pc sc sc sc film film

Nb W UPd2 Al3 PtIr Au/Ag Pb

PC PC MCBJ STS TJ TJ

He et al. [168] Gloos et al. [133, 134] Gloos et al. [136] Goschke et al. [137] Jourdan et al. [169] Jourdan et al. [170]

pc pc

W UNi2 Al3

PC MCBJ

Gloos et al. [133, 134] Gloos et al. [171, 136]

sc sc sc sc

Pt Pb In Pt/Ir

PC TJ TJ STS

Laube et al. [172, 173, 174] Jin et al. [175] Liu et al. [176] Upward et al. [177]

UPd2 Al3

UNi2 Al3

Sr2 RuO4

159

160

A Appendix

Table A.4. A survey on experiments probing the phase of unconventional superconductors by the Josephson eﬀect Compound CeCu2 Si2

CeIrIn5

UPt3

UBe13

URu2 Si2

UPd2 Al3

Sr2 RuO4

Junction

Reference

CeCu2.2 Si2 -Al (SS ) CeCu2.2 Si2 -Cu-Nb (SNS ) CeCu2.2 Si2 -Cu-Nb (SNS ) CeCu2 Si2 (MCBJ)

Poppe et al. [130] Sumiyama et al. [178] Koyama et al. [179] Gloos et al. [180]

CeIrIn5 -Cu-Nb (SNS )

Sumiyama et al. [181]

UPt3 -Al (SS ) UPt3 -Cu-Nb (SNS )

Poppe et al. [130] Sumiyama et al. [182, 178]

UBe13 -Ta (SS ) UBe13 -Cu-Nb (SNS ) UBe13 -Nb (SS )

Han et al. [151] Shibata et al. [183] Shibata et al. [183]

URu2 Si2 -NbTi (SS ) URu2 Si2 -Nb (SS ) URu2 Si2 -Nb (SS ) URu2 Si2 -Cu-Nb (SNS )

Naidyuk et al. [163, 162] Wasser et al. [184] Tachibana et al. [185] Tachibana et al. [185]

UPd2 Al3 -Cu-Nb (SNS ) UPd2 Al3 -Nb (SS )

Koyama et al. [179] He et al. [168]

Pb-Sr2 RuO4 -Pb (SN S) Sr2 RuO4 -In (SS ) Sr2 RuO4 -Au0.5 In0.5 (SS ) Sr2 RuO4 -Sn (SS ) Sr2 RuO4 -Nb (SS ) Sr2 RuO4 -Au0.5 In0.5 (SQUID)

Jin et al. [175] Jin et al. [186], Liu et al. [176] Nelson et al. [187] Sumiyama et al. [188] Sumiyama et al. [188] Nelson et al. [189]

References

161

Table A.5. A survey on experiments probing the vortex lattice/structure of unconventional superconductors Compound

Sample

Technique

Reference

sc

neutron diﬀraction

Eskildsen et al. [190]

neutron diﬀraction neutron diﬀraction neutron diﬀraction

Kleiman et al. [191] Yaron et al. [192] Huxley et al. [193]

µSR µSR neutron diﬀraction neutron diﬀraction µSQUID force microscope

Aegerter et al. [73] Luke et al. [74] Kealey et al. [194] Riseman et al. [195] Dolocan et al. [196]

CeCoIn5

UPt3

Sr2 RuO4 sc, T c sc, T c sc, T c sc, T c sc, T c

= 0.95 K = 1.45 K = 1.39 K = 1.28 K = 1.31 K

References 1. F. Steglich, C.D. Bredl, W. Lieke, U. Rauchschwalbe, G. Sparn: Physica B+C 126, 82 (1984) 2. C.D. Bredl, H. Spille, U. Rauchschwalbe, W. Liedke, F. Steglich, G. Cordier, W. Assmus, M. Herrmann, J. Aarts: J. Magn. Magn. Mater. 31-34, 373 (1983) 3. J.J.M. Franse, A. Menovsky, A. de Visser, C.D. Bredl, U. Gottwick, W. Lieke, H.M. Mayer, U. Rauchschwalbe, G. Sparn, F. Steglich: Z. Phys. B 59, 15–19 (1985) 4. M. Lang, R. Modler, U. Ahlheim, R. Helfrich, P.H.P. Reinders, F. Steglich, W. Assmus, W. Sun, G. Bruls, D. Weber, B. L¨uthi: Physica Scripta T39, 135 (1991) 5. M. Ishikawa, N. Takeda, M. Koeda, M. Hedo, Y. Uwatoko: Phys. Rev. B 68, 024522 (2003) 6. M. Ishikawa, N. Takeda, P. Ahmet, Y. Karaki, H. Ishimoto: J. Phys.: Condens. Matter 13, L25–L31 (2001) 7. F. Gross, B.S. Chandrasekhar, K. Andres, U. Rauchschwalbe, E. Bucher, B. L¨uthi: Physica C 153-155(1), 439–440 (1988) 8. S. Ikeda, H. Shishido, M. Nakashima, R. Settai, D. Aoki, Y. Haga, H. Harima, Y. Aoki, ¯ T. Namiki, H. Sato, Y. Onuki: J. Phys. Soc. Jpn. 70(8), 2248 (2001) 9. R. Movshovich, M. Jaime, J.D. Thompson, C. Petrovic, Z. Fisk, P.D. Pagliuso, J.L. Sarao: Phys. Rev. Lett. 86, 5152 (2001) 10. V.S. Zapf, E.J. Freeman, E.D. Bauer, J. Petricka, C. Sirvent, N.A. Frederick, R.P. Dickey, M.B. Maple: Phys. Rev. B 65(01), 014 506 (2001)

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Index

Abrikosov lattice 63 Ambegaokar-Baratoﬀ formula 57 Anderson-Brinkman-Morell (ABM) model 83 Andreev bound states 36, 59, 132 Andreev equations 44 Andreev reflection 44 Balian-Werthamer (BW) model 83 Bardeen-Cooper-Schrieﬀer attractive interaction 3 generalized theory 3 model of 12 non-BCS behaviour 14 pair attraction 13 parameter 17 theory of 3 weak-coupling 13 Blonder-Tinkham-Klapwijk excess current 51, 133 resistance change 49 theory 45 transmission coeﬃcient 45 borocarbides 6 Bose-Einstein condensation 14 CeCoIn5 72 CeCu2 Si2 67 CeIrIn5 72 CePt3 Si 81 classical superconductor 3 Clogston-Chandrasekhar paramagnetic limit 27 conventional superconductor 3 Cooper pairs 3, 12 crystal structure CeCoIn5 72 CeCu2 Si2 68

CeIrIn5 72 CePt3 Si 82 perovskite 121 Sr2 RuO4 122 UBe13 104 UNi2 Al3 112 UPd2 Al3 112 UPt3 89 URu2 Si2 107 YBa2 Cu3 O7−δ 141 cuprates 5 electron-phonon interaction excess current 51, 133

3

Fermi liquid theory 4 ferromagnets itinerant 5 superconducting 5 flux quantum 62 half-integer 58 flux-line lattice 61 hexagonal 6 squared 6 transition in 6 Hc2 measurements 27 CeCoIn5 77 CeCu2 Si2 69 CePt3 Si 82 3 He 3 heat capacity 4 heating model 42 heavy-fermion compounds 4 heavy-fermion superconductors 4 high-temperature superconductor 5, 140 impurity scattering

22, 50

170

Index

inelastic broadening

50

Josephson eﬀect 55 CeCu2 Si2 71 CeIrIn5 80 dc Josephson eﬀect 55 Sr2 RuO4 135 UBe13 107 UPd2 Al3 115 UPt3 102 URu2 Si2 110 Josephson junction 55 critical current 55 Knight shift µ+ Knight shift 31 nuclear magnetic resonance Kohlrausch relation 42

28

Landauer formula 41 London penetration depth UBe13 105 UPd2 Al3 113 UPt3 93 URu2 Si2 109 magnetic penetration depth 23 Maxwell resistance 42 Meissner screening 17 metal-oxide superconductor 121 Sr2 RuO4 122 muon spin rotation 30 µ+ Knight shift 31 CeCoIn5 78 CeCu2 Si2 70 CeIrIn5 78 CePt3 Si 81 Sr2 RuO4 127, 137 study of vortices 63 UBe13 105 UNi2 Al3 114 UPd2 Al3 113 UPt3 96 URu2 Si2 109 NMR/NQR measurements CeCoIn5 77 CeCu2 Si2 70 CeIrIn5 77

CePt3 Si 83 Sr2 RuO4 128 UBe13 105 UNi2 Al3 113 UPd2 Al3 113 UPt3 96 URu2 Si2 108 nodal structure CeCoIn5 74 CeCu2 Si2 69 CeIrIn5 74 CePt3 Si 83 Sr2 RuO4 123 UBe13 104 UPd2 Al3 112 UPt3 92 URu2 Si2 108 non-Fermi-liquid 4, 5 CeCoIn5 74 nuclear magnetic resonance UPd2 Al3 113 order parameter 13 nodes 15 organic superconductor

28

6

pair amplitude 12 pairing nonunitary 17 singlet 3, 6, 12 triplet 6, 12 pairing symmetry d-wave 6 p-wave 3, 132 s-wave 3, 6 paramagnetic limit 27 Pauli matrices 12 Pauli principle 12 Pauli susceptibility 4 penetration depth Sr2 RuO4 125 phase diagram UPt3 90 phase-sensitive experiments 6 point-contact spectroscopy 39 break junction 40 CeCoIn5 78 CeCu2 Si2 70 edge-to-edge contact 40

Index N-N contact 40 N-S contact 44 nanolithographic contact 40 needle-anvil contact 40 regimes of ballistic 41 diﬀusive 43 thermal 42 Sr2 RuO4 129 UBe13 106 UNi2 Al3 114 UPd2 Al3 114 UPt3 97 URu2 Si2 109 PuCoGa5 74 PuRhGa5 74 Quantum criticality 4 quantum-critical point 4 CeCoIn5 74 CeCu2 Si2 68 ruthenates

6

scanning tunnelling spectroscopy 37 study of vortices 62 Sharvin resistance 41 small-angle neutron scattering 62 CeCoIn5 81 Sr2 RuO4 137 UPt3 102 specific heat 21 CeCu2 Si2 69 Sr2 RuO4 123 UBe13 104 UPd2 Al3 112 UPt3 92 URu2 Si2 108 spin fluctuation 3 spin-lattice relaxation 28 spin-orbit coupling 17 Sr2 RuO4 122 superconductor borocarbide 6 conventional 3 cuprate 5 ferromagnetic 5

heavy-fermion 4 high-temperature 5, 140 organic 6 ruthenate 6 unconventional 3 supercurrents, spontaneous 17 Tersoﬀ-Hamann model 37 thermal conductivity 21 Bi2 Sr2 CaCu2 O8 142 influence of impurities 22 Sr2 RuO4 125 UBe13 105 UPd2 Al3 113 UPt3 92 URu2 Si2 108 YBa2 Cu3 O6.9 142 time-reversal symmetry 17 breaking of 17 tricrystal SQUID experiment YBa2 Cu3 O7−δ 146 tunnelling spectroscopy 35, 40 Sr2 RuO4 129 UPd2 Al3 114 UPt3 97 YBa2 Cu3 O7−δ 143 UBe13 103 ultrasound attenuation 24 Sr2 RuO4 126 UPt3 94 unconventional superconductor UNi2 Al3 111 UPd2 Al3 111 UPt3 89 URu2 Si2 107

3, 14

vacuum tunnelling 37 vacuum-tunnelling spectroscopy UPd2 Al3 114 vortex lattice 6, 61 vortices 61, 102 Wexler formula

43

zero-bias anomaly 59 zero-bias conductance peak

36, 59

171