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C. Noce A. VecchioneM. Cuoco A. Romano (Eds.)

Ruthenateand Rutheno-CuprateMaterialsUnconventional Superconductivity, Magnetismand Quantum Phase Transitions

1 3

EditorsC. NoceUniversita di SalernoDipartimento di Fisica‘‘E.R. Caianiello’’Facolta di ScienzeVia Salvator Allende84081 Baronissi (Salerno), Italy

A. VecchioneUniversita di SalernoDipartimento di Fisica‘‘E.R. Caianiello’’Facolta di ScienzeVia Salvator Allende84081 Baronissi (Salerno), Italy

M CuocoUniversita di SalernoDipartimento di Fisica‘‘E.R. Caianiello’’Facolta di ScienzeVia Salvator Allende84081 Baronissi (Salerno), Italy

A. RomanoUniversita di SalernoDipartimento di Fisica‘‘E.R. Caianiello’’Facolta di ScienzeVia Salvator Allende84081 Baronissi (Salerno), Italy

Cover Picture: (see contribution by Y. Maeno et al. in this volume)

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Preface

This volume is based on the lecture notes of the International Conference “Ruthen-ate and Rutheno-cuprate Materials: Theory and Experiments” held in Vietri sulMare (Salerno)–Italy from 25th to 27th October 2001.

One of the most important developments associated with the discovery ofhigh-temperature superconductivity in the cuprates has been a rapid growthin our understanding of related oxides. Oxides display all the ground states ofstrongly correlated electron physics, from many-body insulators to metals on theborder of applicability of the well-known Fermi liquid theory. The various formsof magnetism which also occur are linked to a host of interesting properties suchas colossal magnetoresistance and unconventional superconductivity. Recently,the class of ruthenate materials has been the focus of considerable work becauseof their interesting magnetic and superconducting properties.

Detailed studies of perovskite-like ruthenates belonging to the Ruddlesden-Popper series Srn+1RunO3n+1 have revealed an unexpectedly rich physics includ-ing itinerant 4d magnetism in SrRuO3, triplet superconductivity in Sr2RuO4,and quantum critical phenomena in the bilayer compound Sr3Ru2O7. Althoughmuch has been learned about these materials from a theoretical and experimentalpoint of view, there is a lot of interesting physics beyond this level.

The enthusiasm in the physics and phenomenology of the ruthenate oxideshas grown by the remarkable observation in hybrid rutheno-cuprate materials ofsuperconductivity arising up to at least Tc=35K in GdSr2RuCu2Oy, despite itsbeing ferromagnetic (FM) already at Tm=132K. In this respect, GdSr2RuCu2Oy

appears to be unique as a ferromagnet that becomes superconducting well withinthe FM phase. This compound can be derived from the YBCO high-Tc super-conductors by replacing the CuO chains by RuO2 layers and are characterizedby a sequence of CuO2 double layers carrying the superconductivity and RuO2layers responsible of the magnetism. Nevertheless, the coexistence of supercon-ductivity and long range magnetic order is intriguing and, in spite of extensiveinvestigation, a consistent picture of the magnetic structure is still lacking.

The volume includes articles on various topics in this field and are groupedin three main parts devoted to Sr2RuO4, to rutheno-cuprate materials, and toSrRuO3 and Sr3Ru2O7, respectively. However, the ordering of the papers islargely arbitrary, since the problems addressed overlap to a considerable extent.The authors are specialists in their respective fields and are actively engagedin the study of the problems touched upon by them. For this reason we are

VI Preface

confident that this book will attract the attention of the readers and will proveto be useful for researchers involved in Solid State Physics.

We would like to express our gratitude towards the eminent scientists whohave promptly and kindly accepted our invitation to give their lectures, and toall the participants who helped to create a warm and stimulating atmosphere,with their presence and interesting discussions.

This Conference has certainly summarized many of the recent theoreticaland experimental issues on ruthenate and rutheno-cuprate materials. A numberof factors, however, made it special: the non minor benefit coming from thewonderful and warm venue of Vietri sul Mare; the large number of young andenthusiastic people and the feeling of forming a community.

Salerno, Canio NoceJuly 2002 Antonio Vecchione

Mario CuocoAlfonso Romano

To our wivesRosangelaCaterina

Maria TeresaGiuliana

Acknowledgements

Organizing a conference is a real hard task, but a great honour too. So it has beenan honour when my colleagues Prof. Canio Noce, Dr. Antonio Vecchione, Dr.Mario Cuoco and Dr. Alfonso Romano asked me to join them in the organizationof the ”Ruthenate and Rutheno-Cuprate Materials: Theory and Experiments”conference. And an honour even greater is to have been asked to write a fewwords to thank, in the name of all the local organizing committee, those peoplewho have collaborated towards the good success of the conference.

These cultural initiatives, in fact, require the use of relevant financial re-sources that, in our case, have been provided by public as well as private in-stitutions and industries. Needless to say that, without their support, it wouldhave been really hard to maintain such a high standard for our conference, andto achieve the success the participants have so kindly recognized us, so we wantto acknowledge them all in an explicit form.

We would like to thank the ”Istituto Italiano per gli Studi Filosofici” inthe persons of its President Dr. Gerardo Marotta and of its General SecretaryProf. Antonio Gargano, that, with its prestigious support of experience andits important financial effort has significantly contributed to the success of ourinitiative. Noticeable financial assistance, as well as organizing contribute, havebeen provided by the Salerno’s research unit of the ”Istituto Nazionale di Fisicadella Materia” that we would like to thank in the person of its director Prof.Giovanni Costabile. Invaluable has been the contribute from our university, the”Universita degli Studi di Salerno”: we thank in particular the ”Dipartimento diFisica ’E.R. Caianiello’”, in the person of its director Prof. Ferdinando Mancini,that has given a financial and organizing support, and the ”Facolta di Scienze”, inthe person of its Headmaster Prof. Genoveffa Tortora, for financially contributingto this conference. A consistent financial support has been also provided by localinstitutions as the ”Provincia di Salerno”, that we thank in the person of itsPresident Dr. Alfonso Andria, and the ”Comune di Salerno”, that we thankin the persons of Aldermen Dr. Gianfranco Valiante and Dr. Ermanno Guerra.As regards private industry, we thank here ”Philips Analytical” for its valuablefinancial support, and in particular Dr. Gianfranco Brignoli.

Moreover, we acknowledge all those people who have contributed in personalform and mainly Prof. Attilio Immirzi, of the ”Dipartimento di Chimica, Uni-versita degli Studi di Salerno”, for scientific assistance and financial support, Dr.Sergio Marotta for its precious collaboration and advice, and Mr. Vincenzo DiMarino for assistance in graphics and designing.

Last but not least, we would like to thank all those people who have helped usin all backstage work, preparing bag kits, cutting and mounting badges, materialand personal transportation, technical assistance etc., it is a hard and sometimesboring work, but essential for the success of any such initiative: thanks again,you’ve been great!

Salerno, For the Local Organising CommitteeJuly 2002 Marcello Gombos

Table of Contents

Toward the Full Determinationof the Superconducting Order Parameter of Sr2RuO4

Y. Maeno, H. Yaguchi, K. Deguchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Evidence for Spin-Triplet Superconductivity . . . . . . . . . . . . . . . . . . . . . . . 33 Gap Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Multiple Superconducting Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Limiting of the Upper Critical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Magnetic Excitations in 214-RuthenatesM. Braden, O. Friedt, Y. Sidis, P. Bourges, P. Pfeuty, Y. Maeno . . . . . . . 151 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Inelastic Neutron Scattering:

Experiments and Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Magnetic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 RPA Analysis of the Magnetic Excitations . . . . . . . . . . . . . . . . . . . . 18

3 Magnetic Scattering in Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Incommensurate Fluctuations Due to Fermi-Surface Nesting . . . . . 193.2 Additional Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 SDW-Ordering in Sr2Ru1−xTixO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Magnetic Scattering in Ca2−xSrxRuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Unconventional Superconductivitywith Either Multi-component or Multi-band, or with ChiralityK. Machida, M. Ichioka, M. Takigawa, N. Nakai . . . . . . . . . . . . . . . . . . . . . . . 321 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Multi-component Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1 UPt3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2 UGe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Multi-band Superconductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Chiral Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Possible Pairing Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

XII Table of Contents

4.2 Quasi-classical Theory for Chiral Superconductivity . . . . . . . . . . . . 415 Conclusion and Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

The Behaviour of a Triplet Superconductorin a Spin Only Magnetic FieldB.J. Powell, J.F. Annett, B.L. Gyorffy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 A Microscopic Model for a Triplet Superconductor

in a Spin Only Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Ginzburg–Landau Theory of a Quasi–two Dimensional

Triplet Superconductor in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 504 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 d(k) Parallel to H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 d(k) Perpendicular to H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Coexistence of Spin-Triplet Superconductivityand Ferromagnetism Induced by the Hund’s Rule ExchangeJ. Spalek, P. Wrobel, W. Wojcik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Real-Space Pairing Induced by the Local Ferromagnetic Exchange . . . . 613 Spin-Triplet Superconducting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 Spin-Triplet Paired State Below the Stoner Threshold:

Phase Diagram and a Hidden Critical Point . . . . . . . . . . . . . . . . . . . . . . . 645 Spin-Triplet State in a Weak Ferromagnetic State:

Analytic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 Spin-Triplet Pairing for Strongly Correlated Electrons:

Role of Ferromagnetic Superexchange and Orbital Ordering . . . . . . . . . . 707 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Quasi-particle Spectra of Sr2RuO4A. Lichtenstein, A. Liebsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 Comparison with Photoemission Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 844 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Normal State Properties of Sr2RuO4M. Cuoco, C. Noce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912 Huckel-Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 Magneto-Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 Correlation Effects and Quantum Criticality . . . . . . . . . . . . . . . . . . . . . . . 995 Orbital Dependent Magnetic Correlations:

Dynamic Double Exchange vs Superexchange . . . . . . . . . . . . . . . . . . . . . . 1026 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table of Contents XIII

The Nature of the Superconducting Statein Rutheno-CupratesC.W. Chu, Y.Y. Xue, B. Lorenz, R.L. Meng . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Coexistence of Superconductivity and Weak-Ferromagnetismin Eu2−xCexRuSr2Cu2O10−δ

I. Felner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 X-ray Diffraction Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214 The Effect of Ce on TC and TM in Eu2−xCexRuSr2Cu2O10−δ . . . . . . . . 121

4.1 Superconductivity in Eu2−xCexRuSr2Cu2O10−δ:Results and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2 Superconductivity in the M-2122 System. . . . . . . . . . . . . . . . . . . . . . 1234.3 The Magnetic Properties of EuCeRuSr2Cu2O10 . . . . . . . . . . . . . . . . 1244.4 The Effect of Ce on the Magnetic Properties

of Eu2−xCexRuSr2Cu2O10−δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.5 Mossbauer Effect of 57Fe Doped in Gd1.4Ce0.6RuSr2Cu2O10−δ . . 129

5 The Magnetic Structure of Eu2−xCexRuSr2Cu2O10−δ . . . . . . . . . . . . . . . 1306 The Effect of Oxygen on the SC and Magnetic Behavior

of Eu1.5Ce0.5RuSr2Cu2O10−δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 The Effect of Hydrogen on the SC and Magnetic Behavior

of Ru-2122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358 The Mixed (Ru,Nb)-2122 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A Phase Diagram Approach to Superconductivityand Magnetism in Rutheno-CupratesH.F. Braun, L. Bauernfeind, O. Korf, T.P. Papageorgiou . . . . . . . . . . . . . . . 1421 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422 Structure and Properties of Rutheno-Cuprates . . . . . . . . . . . . . . . . . . . . . 1433 Phase Equilibria in the Sr–Gd–Ru–Cu–O System . . . . . . . . . . . . . . . . . . . 144

3.1 Pseudoternary Subsolidus Phase Diagrams . . . . . . . . . . . . . . . . . . . . 1463.2 The Section (XSr +XGd)/(XCu +XRu) = 1 . . . . . . . . . . . . . . . . . . . 150

4 The Precursor Route to Superconducting Ru-1212 . . . . . . . . . . . . . . . . . . 1515 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

The Synthesis, Structure and Physical Properties of the LayeredRuthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8ClA.C. Mclaughlin, J.P. Attfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1601 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1602 The Structure and Microstructure of RuSr2GdCu2O8 . . . . . . . . . . . . . . 1623 Doping Studies of RuSr2GdCu2O8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654 The Structure and Magnetic Properties of Pb2Sr2Cu2RuO8Cl . . . . . . . . 1685 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

XIV Table of Contents

Magnetism and Superconductivityin Ru1−xSr2RECu2+xO8−d (RE=Gd, Eu)and RuSr2Gd1−yCeyCu2O8 CompoundsP.W. Klamut, B. Dabrowski, S.M. Mini, S. Kolesnik, M. Maxwell,J. Mais, A. Shengelaya, R. Khazanov, I. Savic, H. Keller, C. Sulkowski,D. Wlosewicz, M. Matusiak, A. Wisniewski, R. Puzniak, I. Fita . . . . . . . . . 1761 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1762 Ru1−xSr2GdCu2+xO8−d (0 ≤ x ≤ 0.75) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1773 RuSr2Gd1−yCeyCu2O8 (0 ≤ y ≤ 0.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864 Superconducting and Non-superconducting Samples

of RuSr2RECu2O8 (RE=Gd, Eu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

ESR Studies of the Magnetism in Ru-1212 and Ru-2212O. Sigalov, A. Shames, S.D. Goren, H. Shaked, C. Korn, I. Felner,A. Vecchione . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1941 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1942 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

2.1 Ru-1212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1952.2 Ru-2212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1983.1 Ru-1212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2033.2 Ru-2212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Structure and Morphologyof NdSr2RuCu2Oy and GdSr2RuCu2Oz

L. Marchese, A. Vecchione, M. Gombos, C. Tedesco, A. Frache,H.O. Pastore, S. Pace, C. Noce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2051 Introduction: Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

1.1 From RE123 to RE1212 (RE = Nd, Gd) . . . . . . . . . . . . . . . . . . . . . . 2051.2 The Oxygen Problem

and the Synthesis of the First 1212 Phases . . . . . . . . . . . . . . . . . . . . 2071.3 The RE1212 Rutheno-Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2081.4 Coexistence of Superconductivity and Magnetic Ordering

in Gd1212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2092 NdSr2RuCu2Oy and GdSr2RuCu2Oz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

2.1 Synthesis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2102.2 Phase and Morphological Characterization . . . . . . . . . . . . . . . . . . . . 2112.3 Magnetisation and Susceptibility Measurements . . . . . . . . . . . . . . . 217

3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Synthesis Effects on Magnetic and Superconducting Propertiesof RuSr2GdCu2O8R. Masini, C. Artini, M.R. Cimberle, G.A. Costa, M. Carnasciali,M. Ferretti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2221 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Table of Contents XV

2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2232.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

3 Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2284 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2305 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Comparison of Electronic Structure, Magnetic Mechanism,and Symmetry of Pairing in Ruthenates and CupratesS.G. Ovchinnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2391 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2392 Generalized Tight-Binding Method for Quasiparticle Band Structure

in Strongly Correlated Electron Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2413 Evolution of the Electronic Structure with Doping in Cuprates . . . . . . . 2434 Comparison of Superconductivity in Cuprates and Ruthenates

in the t–J–I Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465 Electronic Structure of Ruthenates in the Multiband p–d Model . . . . . . 2496 Competition of Ferromagnetism and Antiferromagnetism

in Ruthenates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Magnetism, Spin Fluctuations and Superconductivityin Perovskite RuthenatesD.J. Singh and I.I. Mazin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2561 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2562 SrRuO3 and CaRuO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563 Band Structure of Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2594 Spin Fluctuations in Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615 Magnetism in (Sr,Ca)2RuO4 and Sr3Ru2O7 . . . . . . . . . . . . . . . . . . . . . . . . 2636 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Metamagnetic Quantum Criticality in Sr3Ru2O7A.J. Schofield, A.J. Millis, S.A. Grigera, G.G. Lonzarich . . . . . . . . . . . . . . . 2711 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2712 Deriving the Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2733 Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2754 Tree-Level Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775 One-Loop Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796 Integrating the RG Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2838 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2859 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Metamagnetic Transition and Low-Energy Spin DensityFluctuations in Sr3Ru2O7L. Capogna, E.M. Forgan, S.M. Hayden, G.J. McIntyre, A. Wildes,A.P. Mackenzie, J.A. Duffy, R.S. Perry, S. Ikeda, Y. Maeno . . . . . . . . . . . . 2901 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

XVI Table of Contents

2 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2922.1 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2922.2 Inelastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

3 The Metamagnetic Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2954 The Magnetic Fluctuation Spectrum

and the Dynamical Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2984.1 Energy Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2984.2 Momentum Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Decrease of Ferromagnetic Transition Temperaturein Nonstoichiometric SrRu1−vO3 PerovskitesB. Dabrowski, P.W. Klamut, O. Chmaissem, S. Kolesnik, M. Maxwell,J. Mais, C.W. Kimball, J.D. Jorgensen, S. Short . . . . . . . . . . . . . . . . . . . . . . 3031 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3032 Sample Preparation and Experimental Details . . . . . . . . . . . . . . . . . . . . . 3053 Magnetic and Resistive Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3054 Neutron Powder Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3065 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Strain Effects in SrRuO3 Thin Films and HeterostructuresG. Balestrino, P.G. Medaglia, P. Orgiani, A. Tebano . . . . . . . . . . . . . . . . . . . 3121 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3122 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

2.1 SrRuO3 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3142.2 SrRuO3 / SrTiO3 Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . 316

3 Structural Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3164 Electrical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3205 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

List of Contributors

J.F. AnnettH.H. Wills Physics LaboratoryUniversity of BristolTyndall AvenueBristol BS8 1TL, UKJames.Annett@bristol.ac.uk

C. ArtiniINFM and DCCIUniversity of GenoaVia Dodecaneso 3116146 Genova, Italyartini@chimica.unige.it

J.P. AttfieldDepartment of Chemistry,University of CambridgeLensfield RoadCambridge CB2 1EWandInterdisciplinary Research Centre inSuperconductivityDepartment of PhysicsUniversity of CambridgeMadingley RoadCambridge CB3 0HE, UKjpa14@cam.ac.uk

G. BalestrinoDipartimento di Scienze e TecnologieFisiche ed EnergeticheUniversita di Roma “Tor Vergata”Via di Tor Vergata 11000133 Roma, Italybalestrino@uniroma2.it

L. BauernfeindPhysikalisches InstitutUniversitat BayreuthD-95440 Bayreuth, Germany

P. BourgesLaboratoire Leon BrillouinC.E.A./C.N.R.S.F-91191-Gif-sur-Yvette CEDEX,France

M. BradenII. Physikalisches InstitutUniversitat zu KolnZulpicher Str. 77D-50937 Koln, Germanybraden@ph2.uni-koeln.de

H.F. BraunPhysikalisches InstitutUniversitat BayreuthD-95440 Bayreuth, GermanyHans.Braun@uni-bayreuth.de

L. CapognaMax Planck Institute for Solid StateResearchStuttgart, D-70569, GermanyandSchool of Physics and AstronomyUniversity of BirminghamBirmingham B15 2TT, U.K.capognal@isdux1.bham.ac.uk

XVIII List of Contributors

M. CarnascialiINFM and DCCIUniversity of GenoaVia Dodecaneso 3116146 Genova, Italypani@chimica.unige.it

O. ChmaissemPhysics DepartmentNorthern Illinois UniversityDeKalb, IL 60115, U.S.A.

C.W. ChuDepartment of Physics and TCSUHUniversity of Houston 202Houston Science CenterHouston TX 77204-5002, USAandLawrence Berkeley National Labora-tory1 Cyclotron RoadBerkeley CA 94720, USAandHong Kong University of Science andTechnologyClear Water BayKowloon, Hong Kongcwchu@uh.edu

M.R. CimberleCNR – IMEM, Sezione di GenovaVia Dodecaneso 33, 16146 Genova,Italy

G.A. CostaINFM and DCCIUniversity of GenoaVia Dodecaneso 31, 16146 GenovaItalycosta@chimica.unige.it

M. CuocoDipartimento di Fisica“E.R. Caianiello”Universita di Salernovia S. Allende

I-84081 Baronissi (Salerno), Italymarcuo@sa.infn.it

P. DabrowskiPhysics DepartmentNorthern Illinois UniversityDeKalb, IL 60115, U.S.A.dabrowski@anl.gov

K. DeguchiKyoto University InternationalInnovation CenterKyoto 606-8501Japan

J.A. DuffyH.H. Wills Physics LaboratoryUniversity of BristolBristol BS8 1TL, U.K.andDepartment of PhysicsUniversity of WarwickCoventry CV4 7AL, U.K.

I. FelnerInstitute of PhysicsThe Hebrew UniversityJerusalem, Israel 91904israela@vms.huji.ac.il

M. FerrettiINFM and DCCIUniversity of GenoaVia Dodecaneso 3116146 Genova, Italyferretti@chimica.unige.it

I. FitaInstitute of Physics of Polish Academyof Sciences02-668 Warszawa, Poland

E.M. ForganSchool of Physics and AstronomyUniversity of BirminghamBirmingham B15 2TT, U.K.

List of Contributors XIX

A. FracheDipartimento di Scienze e TecnologieAvanzateUniversita del Piemonte Orientale“A. Avogadro”C.so BorsalinoI-15100 Alessandria, Italia

O. FriedtII. Physikalisches InstitutUniversitat zu KolnZulpicher Str. 77D-50937 Koln, GermanyandLaboratoire Leon BrillouinC.E.A./C.N.R.S.F-91191-Gif-sur-Yvette CEDEXFrance

M. GombosDipartimento di Fisica“E.R. Caianiello”Universita di SalernoVia S. AllendeI-84081 Baronissi (SA), Italiagombos@sa.infn.it

S.D. GorenDepartment of PhysicsBen Gurion UniversityBeer Sheva, Israelshaulg@bgumail.bgu.ac.il

S.A. GrigeraSchool of Physics and AstronomyUniversity of St. AndrewsNorth Haugh, St Andrews, Fife, KY169SS, United Kingdom

B.L. GyorffyH.H. Wills Physics LaboratoryUniversity of BristolTyndall Avenue, BS8 1TL, UK

S.M. HaydenH.H. Wills Physics LaboratoryUniversity of BristolBristol BS8 1TL, U.K.

M. IchiokaDepartment of PhysicsOkayama UniversityOkayama 700-8530, Japan

S. IkedaDepartment of PhysicsKyoto UniversityKyoto 606-8502, Japan

J.D. JorgensenMaterials Science DivisionArgonne National LaboratoryArgonne, IL 60439, U.S.A.

R. KazhanovPhysik-InstitutUniversitat ZurichCH-8057 Zurich, Switzerlandand Laboratory forMuon-Spin SpectroscopyPaul Scherrer InstitutCH-5232 Villigen PSI, Switzerland

H. KellerPhysik-InstitutUniversitat ZurichCH-8057 Zurich, Switzerland

C.W. KimballPhysics DepartmentNorthern Illinois UniversityDeKalb, IL 60115, U.S.A.

P.W. KlamutDepartment of PhysicsNorthern Illinois UniversityDeKalb, Illinois 60115, USAandInstitute of Low Temperature andStructure Research of Polish

XX List of Contributors

Academy of Sciences50-950 Wroclaw, Polandth0pwk1@corn.cso.niu.edu

S. KolesnikDepartment of PhysicsNorthern Illinois UniversityDeKalb, Illinois 60115, USA

O. KorfUniversitat BayreuthPhysikalisches InstitutD-95440 Bayreuth, Germany

C. KornDepartment of PhysicsBen Gurion UniversityBeer Sheva, Israel

F. LichtenbergExperimentalphysik VICenter for Electronic Correlations andMagnetism (EKM)Institute of PhysicsAugsburg UniversityD - 86135 Augsburg, GermanyFrank.Lichtenberg@physik.uni-augsburg.de

A. LichtensteinUniversity of Nijmegen6525 ED Nijmegen, The NetherlandsA.Lichtenstein@sci.kun.nl

A. LiebschInstitut fur FestkorperforschungForschungszentrum52425 Julich, Germany

G.G. LonzarichCavendish LaboratoryMadingley RoadCambridge, CB3 0HE, UnitedKingdom

B. LorenzDepartment of Physics and TCSUH202 Houston Science CenterUniversity of HoustonHouston TX 77204-5002, USA

K. MachidaDepartment of PhysicsOkayama UniversityOkayama 700-8530, Japanmachida@mp.okayama-u.ac.jp

A.P. MackenzieSchool of Physics and AstronomyUniversity of BirminghamBirmingham B15 2TT, U.K.andSchool of Physics and AstronomyUniversity of St. AndrewsSt. Andrews KY16 9SS, U.K.pm9@st-andrews.ac.uk

Y. MaenoDepartment of PhysicsKyoto UniversityKyoto 606-8502, JapanandCRESTJapan Science and TechnologyCorporationKawaguchi, Saitama 332-0012, Japanmaeno@scphys.kyoto-u.ac.jp

J. MaisDepartment of PhysicsNorthern Illinois UniversityDeKalb, Illinois 60115, USA

L. MarcheseDipartimento di Scienze e TecnologieAvanzateUniversita del Piemonte Orientale“A. Avogadro”C.so BorsalinoI-15100 Alessandria, Italia

List of Contributors XXI

R. MasiniCNR – IENI, Sezione di MilanoVia Cozzi 5320125, Milano, Italy

M. MatusiakInstitute of Low Temperature andStructure ResearchPolish Academy of Sciences50-950 Wroclaw, Poland

I.I. MazinNaval Research LaboratoryWashington, DC 20375 U.S.A.

M. MaxwellDepartment of PhysicsNorthern Illinois UniversityDeKalb, Illinois 60115, USA

G.J. McIntyreInstitut Laue-Langevin6 Rue Jules HorowitzF38042 Grenoble Cedex, France

A.C. MclaughlinDepartment of ChemistryUniversity of CambridgeLensfield RoadCambridge CB2 1EWandInterdisciplinary Research Centre inSuperconductivityDepartment of PhysicsUniversity of CambridgeMadingley RoadCambridge CB3 0HE, UKacm43@cus.cam.ac.uk

P.G. MedagliaDipartimento di Scienze e TecnologieFisiche ed EnergeticheUniversita di Roma “Tor Vergata”Via di Tor Vergata 11000133 Roma, Italy

R.L. MengDepartment of Physics and TCSUH202 Houston Science CenterUniversity of HoustonHouston TX 77204-5002, USA

A.J. MillisDepartment of PhysicsColumbia University538 W 120th StNew York, NY 10027 USA

S.M. MiniDepartment of PhysicsNorthern Illinois UniversityDeKalb, Illinois 60115, USA

N. NakaiDepartment of PhysicsOkayama UniversityOkayama 700-8530, Japan

C. NoceDipartimento di Fisica“E.R. Caianiello”Universita di Salernovia S. AllendeI-84081 Baronissi (Salerno), ITALYcanio@sa.infn.it

S. PaceDipartimento di Fisica“E.R. Caianiello”Universita di SalernoVia S. AllendeI-84081 Baronissi (SA), Italiapace@sa.infn.it

T.P. PapageorgiouPhysikalisches InstitutUniversitat Bayreuth,D-95440 Bayreuth, Germany

XXII List of Contributors

R.S. PerrySchool of Physics and AstronomyUniversity of BirminghamBirmingham B15 2TT, U.K.andDepartment of PhysicsKyoto UniversityKyoto 606-8502, Japan

B.J. PowellH.H. Wills Physics LaboratoryUniversity of BristolTyndall AvenueBS8 1TL, UKben.powell@bristol.ac.uk

R. PuzniakInstitute of Physics of Polish Academyof Sciences02-668 Warszawa, Poland

P. OrgianiDipartimento di Scienze e TecnologieFisiche ed EnergeticheUniversita di Roma “Tor Vergata”Via di Tor Vergata 11000133 Roma, Italypasquale.orgiani@na.infn.it

S.G. OvchinnikovL.V. Kirensky Institute of PhysicsSiberian Branch of RASandUNESCO Chair of New Materials andTechnologyKrasnoyarsk State Technical Univer-sityKrasnoyarsk, 660036, Russiasgo@iph.krasn.ru

H.O. PastoreInstituto de QuımicaUniversitade Estadual de CampinasCP6154, CEP 13083-970, Campinas,SP, Brazil

P. PfeutyLaboratoire Leon BrillouinC.E.A./C.N.R.S.F-91191-Gif-sur-Yvette CEDEX,France

I. SavicPhysik-InstitutUniversitat ZurichCH-8057 Zurich, SwitzerlandandFaculty of PhysicsUniversity of Belgrade11001 Belgrade, Yugoslavia

A.J. SchofieldSchool of Physics and AstronomyUniversity of BirminghamEdgbaston, Birmingham, B15 2AD,United Kingdomajs@th.ph.bham.ac.uk

H. ShakedDepartment of PhysicsBen Gurion UniversityBeer Sheva, Israel

A. ShamesDepartment of PhysicsBen Gurion UniversityBeer Sheva, Israel

A. ShengelayaPhysik-InstitutUniversitat ZurichCH-8057 Zurich, Switzerland

S. ShortMaterials Science DivisionArgonne National LaboratoryArgonne, IL 60439, U.S.A.

O. SigalovDepartment of PhysicsBen Gurion UniversityBeer Sheva, Israel

List of Contributors XXIII

D.J. SinghNaval Research LaboratoryWashington, DC 20375 U.S.A.

C. SulkowskiInstitute of Low TemperatureandStructure Research of Polish Academyof Sciences50-950 Wroclaw, Poland

J. SpalekMarian Smoluchowski Institute ofPhysicsJagiellonian Universityulica Reymonta 4, 30-059 Krakow,Polandufspalek@if.uj.edu.pl

M. TakigawaDepartment of PhysicsOkayama UniversityOkayama 700-8530, Japan

A. TebanoDipartimento di Scienze e TecnologieFisiche ed EnergeticheUniversita di Roma “Tor Vergata”Via di Tor Vergata 11000133 Roma, Italy

C. TedescoDipartimento di ChimicaUniversita di SalernoVia S. AllendeI-84081 Baronissi (SA), Italiatedesco@chem.unisa.it

A. VecchioneDipartimento di Fisica“E.R. Caianiello”Universita di SalernoVia S. AllendeI-84081 Baronissi (SA), Italiavecchione@sa.infn.it

A. WildesInstitut Laue-Langevin6 Rue Jules HorowitzF38042 Grenoble Cedex, France

A. WisniewskiInstitute of PhysicsPolish Academy of Sciences02-668 Warszawa, Poland

D. WlosewiczInstitute of Low TemperatureandStructure Research of Polish Academyof Sciences50-950 Wroclaw, Poland

W. WojcikInstitute of PhysicsTadeusz Kosciuszko TechnicalUniversityulica Podchorazych 130-084 Krakow, Poland

P. WrobelMarian Smoluchowski Institute ofPhysicsJagiellonian Universityulica Reymonta 430-059 Krakow, Poland

Y.Y. XueDepartment of Physics and TCSUH,University of Houston202 Houston Science CenterHouston TX 77204-5002, USA

H. YaguchiKyoto University InternationalInnovation CenterKyoto 606-8501, Japan

Toward the Full Determinationof the Superconducting Order Parameterof Sr2RuO4

Y. Maeno1,2, H. Yaguchi2, and K. Deguchi2

1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan2 Kyoto University International Innovation Center, Kyoto 606-8501, Japan

Abstract. The layered perovskite superconductor Sr2RuO4 (Tc = 1.5 K) has attractedmuch research interest, particularly because of its unconventional superconductivity.From the NMR Knight shift, as well as from the polarized neutron scattering, the spinwave function is most likely triplet with a vector order parameter pointing to the zdirection (with paired parallel spins lying within the RuO2 plane). However, the orbitalwave function, which determines the gap symmetry, has not been so well establishedyet. The remaining problems towards the full characterization of the superconductingorder parameter of Sr2RuO4 will be presented and possible means of resolving themwill be discussed. In view of these, we will highlight the following problems: (1) locationof the line nodes; (2) mechanism of the strong suppression of the upper critical field;(3) origin of the two superconducting phases.

1 Introduction

Extensive investigation triggered by the discovery of high-temperature super-conductivity in 1986 [1] soon made it clear that the structural essence of thehigh-Tc superconductors is the quasi-two-dimensional planes consisting of cop-per and oxygen. Worldwide searches for superconductivity in such layered struc-ture without copper also started and finally yielded the discovery in Sr2RuO4 in1994 [2]. Sr2RuO4 shares the same layered perovskite structure with the high-Tc

superconductor La2−xBaxCuO4 as illustrated in Fig. 1. The experimental searchfor superconductivity in ruthenium oxides (ruthenates) in J.G. Bednorz’s groupat IBM Zurich Laboratory started in 1987 and eventually lead to the discoveryat Hiroshima University in collaboration with the IBM group. Their initial moti-vation to investigate the ruthenates was the similarity of the electronic structureto that of the cuprates. The electronic states of the high-Tc cuprates are basedon the 3d9 configuration of Cu2+ with one hole in the eg orbits. Those of theruthenates are based on the 4d5 or 4d4 configuration of Ru3+ or Ru4+ with oneor two holes in the t2g orbits (compounds with Ru5+ are also well known).

The hybridization between the transition-metal d -electrons and the oxygenp-electrons are strong in both systems. In contrast with the cuprates based onthe spin S =1/2 configuration, however, the superconductivity in the ruthenatewas found unexpectedly in the Ru4+ compound with S = 1 configuration.

In the very early stage of the study of superconductivity in Sr2RuO4, thismaterial was widely recognized as a reference to the high Tc cuprates because

C. Noce et al. (Eds.): LNP 603, pp. 1–14, 2002.c© Springer-Verlag Berlin Heidelberg 2002

2 Y. Maeno, H. Yaguchi, and K. Deguchi

Fig. 1. Layered perovskite structure common to the cuprate and ruthenate supercon-ductors.

of the structural similarity. However, since the possibility of spin triplet pairingwas pointed out on theoretical ground [3], the superconductivity of Sr2RuO4 it-self has been of great interest despite its superconducting transition temperaturebeing rather low (Tc = 1.5 K) [4]. This article describes unconventional natureof the superconductivity of Sr2RuO4, now firmly established by a large numberof experiments [5], and addresses the current issues toward the full determina-tion of its superconducting order parameter. Conventional superconductivity ischaracterized by electron pairs in the singlet s-wave channel. High-Tc supercon-ductivity is also carried by spin-singlet pairs, although the orbital symmetry isd -wave. In principle, fermions like electrons can pair in the spin triplet state.In fact, superfluidity of 3He is carried by atomic pairs of 3He for which the nu-clear spins are paired in the triplet channel [6]. The important question thenarises whether or not superconductivity with spin-triplet pairs is ever realizedand if so, what novel properties they may exhibit. In recent years, several super-conductors have been considered as candidates of such long-sought spin-tripletsuperconductors. To list some of the better known examples:

(1) Heavy Fermion superconductors.Probably the best-known candidate for the spin-triplet superconductor is theheavy fermion compound UPt3. The existence of multiple superconducting pha-ses of UPt3 is well established [7]. Recently, NMR Knight shift experiments havegiven strong evidence in favor of spin-triplet pairing [8]. In spite of the accumula-tion of the detailed data over the past years, however, the final consensus has not

Full Determination 3

been reached over the identification of the order parameter for each phase. Be-cause of the strong electron correlations among the f -electrons, unconventionalsuperconductivity with non-s-wave pairing often emerges in the heavy fermioncompounds. In addition to UPt3, UNi2Al3 is also considered as a possible can-didate of a spin-triplet superconductor on the basis of the recent NMR Knightshift experiment [9], although a relative compound UPd2Al3 is clearly a spinsinglet superconductor [10].

(2) Ruthenate superconductor Sr2RuO4.This superconductor, of course, is the topic of this article.

(3) Ferromagnetic superconductors UGe2 [11], URhGe [12], and ZrZn2 [13].In these compounds, ferromagnetic ordering of itinerant electrons with tiny or-dered moment occurs, and unconventional superconductivity emerges below theCurie temperature. Although the spin state of the Cooper pairs are probablytriplet, there has not been any detailed experimental information at present.

(4) Quasi-one-dimensional organic superconductor (TMTSF)2PF6.Spin triplet pairing has been discussed on the basis of the NMR Knight shiftexperiments [14].

Among these candidate superconductors of spin-triplet pairing, Sr2RuO4stands as a unique compound. First, it is probably the only oxide supercon-ductors for which the possibility of spin-triplet state is discussed in any depth.Second, the simplicity of the crystal and electronic structures allows the anal-ysis based on the realistic material parameters. The details of the electronicstructure with quasi-two-dimensional Fermi surface (Fig. 2) have been preciselycharacterized [15,16], and the Fermi-liquid behavior of its normal state is de-scribed quantitatively in terms of the Fermi-surface parameters. Third, largesingle crystals of extremely high quality can be grown by a floating-zone methodusing an infrared furnace as shown in Fig. 3 [4]. The residual resistivity reachesas low as 50 nΩ cm, corresponding to the quasiparticle mean-free-path of l =2 µm and the ratio to the coherence length of l/ ξ = 30, making it in the ex-tremely clean limit. Another advantage to the experimental investigation is thatthe superconductivity occurs at ambient pressure. Because of these properties,there is a real hope that the physics of spin-triplet superconductivity may bematured through the investigation of Sr2RuO4 in a level comparable to that ofsuperfluid 3He.

For spin-triplet pairing, it is expected that multiple superconducting phasescould be induced under certain conditions because of the internal degree offreedom possessed by Cooper pairs. This is actually exemplified in superfluid3He and the heavy fermion superconductor UPt3. Our recent studies indeedsuggest that multiple superconducting phases emerge in Sr2RuO4 in magneticfields accurately parallel to the RuO2 plane.

2 Evidence for Spin-Triplet Superconductivity

In this section we will briefly introduce the expression of the order parameterof spin-triplet superconductivity in terms of the d -vector, and then discuss the

4 Y. Maeno, H. Yaguchi, and K. Deguchi

Fig. 2. Fermi surface of Sr2RuO4 based on the measurements of quantum oscilla-tions [16].

Fig. 3. Growth of a single crystal of Sr2RuO4 in a floating-zone furnace. Polycrystallinesample rod fed from above is melted at the center at approximately 2100 C. A singlecrystal is extracted from below, typically at the very rapid rate of 40 mm/hr.

most appropriate form of the d -vector for Sr2RuO4 based on the experimentalfacts. For spin-singlet superconductivity, the order parameter can be expressedby a single complex parameter.

|Ψ〉=∆s(|↑↓〉−|↓↑〉) . (1)

For the spin-triplet superconductivity, it has three spin bases:

|Ψ〉=∆↑↑|↑↑〉 + ∆↓↓|↓↓〉 + ∆↑↓(|↑↓〉 + |↓↑〉) . (2)

Full Determination 5

Fig. 4. Schematic of spin-triplet Cooper pairs represented by the d-vector d =z∆0(kx + iky). The thin arrows depict the spin state of equal spin pairs in the Sz

= 0 state; the thick (vertical) arrows illustrate the orbital wave function with Lz =1.

By introducing a new set of spin bases,

x =1√2

(−|↑↑〉 + |↓↓〉) ,

y =i√2

(|↑↑〉 + |↓↓〉) ,

z = |Sz = 0〉 =1√2

(|↑↓〉 + |↓↑〉) ,

one can express the order parameter in terms of a vector in the spin space, calledthe d -vector: d = dxx + dyy + dzz = (dx(k), dy(k), dz(k)), where k is the unitvector specifying the direction on the Fermi surface. Thus,

|Ψ〉=√

2d=√

2 (dxx + dyy + dzz) . (3)

Just as z corresponds to the |S z = 0〉 state, x (y) corresponds to the |S x = 0〉(|Sy = 0〉) state.

As we see below, the d -vector appropriate for Sr2RuO4 has the spin z compo-nent only: d = d z (k)z with the orbital part of the wave function d z (k) = ∆0 (kx± iky). The spin and orbital states of this superconducting state is illustratedin Fig. 4.

In order to obtain decisive evidence for spin-triplet superconductivity, a mostdirect way is to determine the spin susceptibility of the Cooper pairs. Figure 5(a) and (b) represent the spin susceptibilities, normalized by the normal statevalues, χS/χN of spin-singlet and spin triplet states. For spin singlet states withS = 0 pairing, χS/χN decays to zero toward zero temperature irrespective of thedirection of the applied field (Fig. 5 (a)). The temperature dependence reflects

6 Y. Maeno, H. Yaguchi, and K. Deguchi

Fig. 5. Spin susceptibility of superconducting states with (a) spin-singlet pairs and (b)spin triplet pairs, normalized by the normal-state susceptibility. It is isotropic for singletstates; the temperature dependence reflects the difference in gap structures associatedwith different orbital symmetry (s or d-wave). It depends on the field direction fortriplet states; it remains invariant for the field perpendicular to the d-vector. (c) Theobserved NMR Knight shift of Sr2RuO4 with the field parallel to the RuO2 plane. Thedotted curve represents the expectation for a singlet pairing.

the contribution from thermally excited quasiparticles across the superconduct-ing gap. At temperatures much below Tc, it is exponential for s-wave pairingand T -linear for d -wave pairing with lines of nodes in the superconducting gap.In contrast, for the spin-triplet pairing χS/χN depends on the direction of theapplied magnetic field. In particular, it remains unity for the field perpendicularto the d -vector. For d = d z (k)z this should occur for the field along the basalplane.

Measurements of the spin susceptibility in the superconducting state needs tobe performed by a microscopic means since the bulk susceptibility is dominated

Full Determination 7

by the diamagnetism due to the Meissner effect. The NMR Knight shift, theshift of the nuclear magnetic resonance frequency in the presence of an externalmagnetic field, is a powerful technique for this purpose. In fact, the separation ofthe spin and orbital contribution to the Knight shift is relatively straightforwardfor Sr2RuO4, and reveals the dominance of the term proportional to the spinsusceptibility. Such property of the d -electron system makes it more reliableto identify the spin state of the Cooper pairs, compared with many of the f -electron systems. As shown in Fig. 5, NMR measurements performed by Ishidaet al. [17] yielded that the Knight shift remains unchanged upon entering itssuperconducting state, providing a definitive identification of spin-triplet pairing.We should add that polarized neutron scattering experiments also indicated thatthe magnetization remains invariant across Tc for the field parallel to the RuO2plane [18].

What is more, µSR experiments observed that the spontaneous magneticmoments arise upon entering the superconducting state, which indicates bro-ken time reversal symmetry [19]. Since the spin state deduced from the NMRstudy does not break time reversal symmetry, the spontaneous internal field isattributable to the orbital moment of the Cooper pairs. Thus the existing ex-perimental results allow us to understand that the superconducting symmetryis represented by the degenerate two-component order parameter, d = z ∆0 (kx+ iky) [3]. Here, z and (kx + iky) are the spin part and the orbital part of thevector order parameter, respectively. The two-component order parameter, con-sisting of kx and ky components , is required also to account for the observed fielddistribution in the vortex state [20]. This superconducting symmetry is schemat-ically depicted in Fig. 4. The spin of Cooper pairs lies within the RuO2 planeand the orbital function corresponds to Lz = 1. The superconducting energy gapis

|∆(k)| = (d · d∗)1/2 = ∆0(k2x + k2

y)1/2 , (4)

and is isotropic on a cylindrical Fermi surface.

3 Gap Structure

Whereas the two-dimensional order parameter deduced mainly from NMR andµSR leads to the superconducting gap being isotropic without nodes, there areseveral experimental results suggestive of a nodal structure in the gap [21,22,23,24,25].As shown in Fig. 6, the specific heat divided by temperature, C/T, decreases lin-early in T, rather than exponentially, thus suggesting a line-node gap. It shouldbe noted that C/T at low temperatures is even greater than the expectation forthe simple line node. Such a large entropy release implies that a large numberof quasiparticles are thermally excited on a part of the Fermi surface at lowtemperatures. This behavior is accounted for by extending the original modelof orbital dependent superconductivity (ODS) [26]. The superconducting gapon one or two of the three Fermi surface cylinders needs to have line-node-likeanisotropy and, moreover, characteristic energy smaller than Tc.

8 Y. Maeno, H. Yaguchi, and K. Deguchi

Fig. 6. Temperature dependence of the specific heat of Sr2RuO4. It deviates sharplyfrom the expectation for an isotropic gap, and suggests the presence of line nodes inthe superconducting gap.

Recent studies of the thermal conductivity down to 0.3 K under orientedmagnetic fields suggest that there is no in-plane node, implying a circular lineof nodes running around the Fermi-surface cylinder [27,28]. The attenuation ofthe ultrasound also supports this picture, although the in-plane anisotropy inthe ultrasound attenuation at 50 mK is substantial [25]. Regarding the nodalstructure of the gap, recent theory by Zhitomirsky and Rice [29] satisfies experi-mental constraints as a whole. They have extended the idea of orbital-dependentsuperconductivity [26] and proposed circular line nodes around one or two of thethree Fermi-surface cylinders, as depicted in Fig. 7. Because of the difference inthe inversion symmetry between the dxy orbit and the dyz -d zx orbits, Cooperpair hopping between γ and α, β Fermi surfaces is expected to be mainly viainterlayer processes. Combination with the direct in-plane processes would re-sult in the formation of a node at a particular kz location. A direct experimentalidentification of such a circular line node is needed for the full characterizationof the gap structure.

4 Multiple Superconducting Phases

Before describing experimental aspects of the multiple superconducting phasesin Sr2RuO4, we briefly introduce Agterberg’s theoretical prediction that theelectron system in Sr2RuO4 will undergo a second superconducting transitionwithin its superconducting state [30]. The theory assumes that the supercon-ducting state is represented by the degenerate two-component order parameterd = z ∆0 (kx + iky). This is the most probable superconducting wave function(in zero magnetic field) although some modification such as circular line nodes isrequired. The superconducting state d = z ∆0 (kx + iky) relies on the tetragonal

Full Determination 9

Fig. 7. Model of the superconducting gap of Sr2RuO4 [29]. A large nodeless gap openson the gamma Fermi surface, while circular line nodes are anticipated in the subdom-inant gap on the other Fermi surfaces.

Fig. 8. Proposed theoretical field-temperature phase diagram of Sr2RuO4. Order pa-rameters and the corresponding gap structures of the expected multiple superconduct-ing phases are illustrated.

symmetry. Therefore, the application of a field that lowers the symmetry will liftthe twofold degeneracy in energy, causing a second superconducting transition tooccur [3]. Possible symmetry lowering fields are a magnetic field and a uniaxialpressure parallel to the RuO2 plane. Agterberg [30] discussed the superconduct-ing state in magnetic fields parallel to the ab plane (RuO2 plane) based on theGinzburg-Landau formulation. Figure 8 illustrates the expected gap symmetryas well as the field-temperature (H-T ) phase diagram. The d -vector in the par-allel magnetic field is expressed as d = z ∆0 (kx ′ + iεky′) (0 < ε <1), where x ’is the field direction, and ε approaches zero with increasing field. At a certain

10 Y. Maeno, H. Yaguchi, and K. Deguchi

Fig. 9. Field-temperature phase diagram of Sr2RuO4 for H ‖ [100] direction, basedon specific heat measurements. The upper critical field Hc2 and the second transitionline H2 appear to merge at a bicritical point close to H = 1.2 T, T = 0.8 K. Inset:temperature dependence of the specific heat divided by T at 1.4 T, indicating thedouble transitions.

field H2 (H2 < Hc2), a second superconducting transition to the one-componentstate d = z ∆0 kx ′ will occur. While the two-component state is a fully gappedstate (no nodes), aside from the circular line nodes discussed above, the one-component state has nodes along the direction normal to the applied field. Thenodes in the gap in the latter state rotate with the in-plane direction of theapplied field.

The above theoretical prediction indeed stimulated experimental studies forexploring multiple superconducting phases. In fact, we carried out three kindsof measurements (ac susceptibility [31,32,33], specific heat [21,34] and thermalconductivity [24]) on single crystals of high quality in magnetic fields accuratelyparallel to the ab plane. These three kinds of experiments all obtained directand/or indirect evidence for the second superconducting transition.

Here we will mainly focus on the results for the specific heat as representedin Figs. 9 and 10. The temperature sweep curves of the electronic specific heatC e divided by temperature start to exhibit an unusual upturn near Tc for fieldsabove 1.2 T. Furthermore, as shown in the inset of Fig. 9, the peak associatedwith the transition splits into two with nearly the same intensity above 1.4T. Field sweep measurements with fixed T also indicate two superconducting

Full Determination 11

Fig. 10. Polar-angle dependence of the upper critical field Hc2 of Sr2RuO4. The curvesare the fitting based on an anisotropic mass model for 0 < q < 88o. The inset clarifiesa strong suppression of Hc2 for the field nearly parallel to the RuO2 plane, for whichthe second superconducting phase concurrently emerges.

transitions near Hc2. These observations provide thermodynamic evidence forthe double superconducting transitions.

The field-temperature phase diagram, mapping the region of the double tran-sitions, is shown in the main frame of Fig. 9. The second transition line H2(T )appears to merge into Hc2 line at a point (a bicritical point) µ0H = 1.2 T andT = 0.8 K. Let us note two characteristic points here. (1) Measurements of allthree quantities, susceptibility, specific heat, and thermal conductivity, suggestthat very accurate alignment of the applied magnetic field to the ab plane isessential for inducing the second superconducting transition. (2) The existenceof a bicritical point is inconsistent with the Agterberg theory: the theory sug-gests that the H2 line merges into the Hc2 line at H = 0, T = Tc [30]. Thus,while the presence of the two superconducting phases is attributable to the two-component order parameter, the mechanism of the second transition appears tobe different from that originally proposed.

5 Limiting of the Upper Critical Field

Another important aspect to discuss is that the second superconducting transi-tion is probably accompanied by the strong suppression of the low-temperatureupper critical field [34]. We point out that the suppression manifests itself atleast in the following three ways:

12 Y. Maeno, H. Yaguchi, and K. Deguchi

(a) Polar angle θ dependence of Hc2.Figure 10 shows the polar angle θ dependence of the upper critical field Hc2determined from the specific heat. Hc2 is clearly suppressed for θ∼ 90 (θ =90 corresponds to H // ab). This may be quantified by fitting experimentaldata for a limited range of θ to the Ginzburg-Landau anisotropic effective massapproximation. For example, fitting the data for 0 < θ < 88 yields µ0Hc2//ab=1.84 T; experimentally µ0Hc2//ab= 1.50 T at 0.1 K. The inset of Fig. 10 indi-cates that the suppression sets in rather abruptly at about two degrees awayfrom the exact alignment. It is in this region of suppressed Hc2 that the secondsuperconducting transition emerges.

(b) Shape of the H -T phase diagram.The Hc2-vs-T curve in Fig. 9 is notably distorted in the region close to thenarrow second superconducting phase. The upper critical field Hc2 at low tem-peratures appears to be strongly suppressed compared with that expected fromorbital depairing. The application of the Werthamer-Helfand-Hohenber (WHH)formula or a corresponding formula for orbital depairing in p-wave superconduc-tor, Hc2(0) = 0.7 K (dHc2/dT )T=T c results in an overestimation of Hc2 by afactor of nearly two.

(c) Field dependence of the specific heat and thermal conductivity.The second superconducting transition is accompanied by a very steep change inboth the specific heat C e and thermal conductivity κ. With increasing H bothquantities exhibit rapid approaches to their respective normal-state values inthe narrow region near Hc2 indicating that the superconducting state becomesunstable. In other words, a smooth extrapolation of the C e(H )/T or κ(H )/Tcurve from below H2 to higher fields would yield a much higher Hc2.

Because of the extreme sensitivity of the limiting behavior of Hc2 to the ex-act field alignment, it is difficult to interpret the behavior in terms of the spindepairing. In addition, according to our current understanding of the supercon-ducting state, the spin of Cooper pairs is parallel to the ab plane, so that spindepairing is irrelevant in fields parallel to the ab plane. Although we are notaware of any theoretical proposal for the mechanism of Hc2 limiting, orbital mo-tion on the quasi-two-dimensional Fermi surface under in-plane field, combinedwith the particular orbital wave function of the Cooper pairs, kx + iky , shouldboth be relevant in accounting for the limiting behavior.

6 Conclusions

We described experimental aspects of multiple superconducting phases inSr2RuO4. It is now established that a second superconducting transition oc-curs in magnetic fields precisely parallel to the RuO2 plane. We discussed thepossible origin of the second superconducting transition in terms of the modelbased on the Ginzburg-Landau formulation. However, the theory only receivespartial support from experimental observations; apparent disagreement such asthe bicritical point is significant. We also point out that the suppression of the

Full Determination 13

low-temperature upper critical field is closely related to the second supercon-ducting transition. Apparently neither conventional orbital depairing nor spindepairing can explain the Hc2 suppression. This suggests that our understand-ing of the second (high-field low-temperature) superconducting state needs tobe reexamined with a new mechanism of depairing specific to spin triplet super-conductors or possibly to Sr2RuO4.

Acknowledgements

We are grateful to Canio Noce, Mario Cuoco, and other members of the orga-nizing committee of the international workshop on the ruthenates and cupro-ruthenates. We thank many who have contributed in the development of theactive field of ruthenate superconductivity. We are particularly grateful to M.Sigrist for his invaluable theoretical contributions. The first section of this arti-cle has been inspired by the presentation by L. Taillefer. This work has been inpart supported by the Grant-in-Aid for Scientific Research (S) from the JapanSociety for Promotion of Science and by the Grant-in-Aid for Scientific Researchon Priority Area Novel Quantum Phenomena in Transition Metal Oxides fromthe Ministry of Education, Culture, Sports, Science and Technology (MEXT),as well as by a grant for Core Research for Evolutional Science and Technol-ogy from Japan Science and Technology Corporation. K.D acknowledges JapanSociety for Promotion of Science for its support.

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Magnetic Excitations in 214-Ruthenates

M. Braden1, O. Friedt1,2, Y. Sidis2, P. Bourges2, P. Pfeuty2, and Y. Maeno3,4

1 II. Physikalisches Institut, Universitat zu Koln,Zulpicher Str. 77, D-50937 Koln, Germany

2 Laboratoire Leon Brillouin, C.E.A./C.N.R.S.,F-91191-Gif-sur-Yvette CEDEX, France

3 Department of Physics, Kyoto University,Kyoto 606-8502, Japan

4 CREST, Japan Science and Technology Corporation,Kawaguchi, Saitama 332-0012, Japan

Abstract. We discuss the magnetic excitations in several 214-ruthenates as observedby inelastic neutron scattering. In the spin-triplet superconductor Sr2RuO4 the mag-netic excitation spectrum is dominated by incommensurate peaks arising from strongnesting between quasi-one-dimensional bands. These excitations are found to freezeout through minor substitution of Ru by Ti. One may conclude that Sr2RuO4 itselfis already very close to spin density wave ordering. Sr2RuO4 seems to exhibit alsoadditional, much weaker magnetic excitations; these are still incommensurate but situ-ated much closer to the zone-center. Similar almost ferromagnetic fluctuations becomestrongly enhanced and dominant by replacing Sr through Ca.

1 Introduction

The current interest in the ruthenates goes far beyond the spin-triplet supercon-ductivity in pure Sr2RuO4 [1]. Besides superconductivity, newly studied ruthen-ates show a variety of magnetic phenomena: a metal insulator transition inCa2RuO4 which ends in an antiferromagnetic Mott-insulator [2,3,4,5], strong ma-gneto-elastic coupling in the phase diagram of Ca2−xSrxRuO4 [6], ferromagneticorder appearing at very low temperatures in Ca1.5Sr0.5RuO4 [7], a metallic spin-density wave (SDW) ordered phase in Sr2Ru1−xTixO4 [8], and the coexistence ofsuperconductivity with weak ferromagnetism in the rutheno-cuprates [9,10]. Wepresent the magnetic excitations in the ruthenates with K2NiF4-type structure(214) as they are observed by inelastic neutron scattering (INS) and discuss theirrelation to the different types of magnetic ordering.

Sr2RuO4 exhibits unconventional superconductivity where Cooper pairs arebound into triplets with p-wave symmetry [11]. This has been demonstratedin Knight-shift experiments [12] which indicate no change in the spin suscep-tibility when crossing the superconducting transition. The loss of time rever-sal symmetry was deduced from the occurrence of spontaneous fields reportedby µSR-experiments [13]. There is general agreement about the unconventionalcharacter of the superconductivity in Sr2RuO4 [11]; however, the mechanism re-sponsible for spin-triplet pairing is still under debate. Rice and Sigrist proposedSr2RuO4 to be an analogue of superfluid He3 [14]; they argue that coupling

C. Noce et al. (Eds.): LNP 603, pp. 15–31, 2002.c© Springer-Verlag Berlin Heidelberg 2002

16 M. Braden et al.

is mediated through the interaction with ferromagnetic fluctuations. The ev-idence for such ferromagnetic fluctuations was deduced from the comparisonwith the Ru-perovskites [15] which order ferromagnetically or are close to ferro-magnetism. Also, a NMR experiment by Imai et al. seemed to indicate dominantferromagnetic fluctuations as a similar signal was observed on the Ru- and on theO-sites [16]. However, the rich variety of magnetic phenomena in 214-ruthenatescasts already some doubt on a unique role of ferromagnetic fluctuations in thepairing mechanism in Sr2RuO4.

When replacing Sr through the smaller but isovalent Ca one finds an extraor-dinarily rich phase diagram [3,6]. The smaller ionic radius is the key element tounderstand the sequence of the structural phases, which imply rather differentmagnetic properties due to strong magneto-elastic coupling. Short range ferro-magnetic order occurs near the concentration Ca1.5Sr0.5RuO4 [7] and this obser-vation might be relevant for the superconducting mechanism in Sr2RuO4. Thereexists also another layered ruthenate which exhibits ferromagnetic order or isclose to that depending on sample quality, Sr3Ru2O7 [17]. In all these ferromag-netic or nearly ferromagnetic layered ruthenates, the magnetic susceptibility isabout two orders of magnitude higher than that in Sr2RuO4 and strongly tem-perature dependent, whereas the susceptibility in Sr2RuO4 is only weakly tem-perature dependent [18]. Complete or almost complete substitution of Sr by Caleads to the metal insulator transition and, in particular, to antiferromagneticorder with rather large ordered moments [19]. In spite of the fact that the insulat-ing phase exhibits quite different structural parameters, this observation clearlyshows that also antiferromagnetic interactions play a role in the 214-ruthenates.Mazin and Singh were inspired by this finding when they analyzed the spin-susceptibility in Sr2RuO4 by LDA-band structure calculations [20]. These cal-culations yielded nesting induced peaks as discussed in detail in Sect. 3.1.

Several substitutions on the Ru-site in Sr2RuO4 have been analyzed, moststudies used non-magnetic impurities. The common result concerns the rapidsuppression of the superconducting transition temperature which in general isused as a strong argument against conventional s-wave pairing [21]. In the case ofTi, Minakata et al. [22] have found a weak ferromagnetic signal in susceptibilitymeasurements which clearly indicates some static ordering.

In this contribution we will discuss the magnetic excitation spectra of theSr2−xCaxRu1−yTiyO4-compounds which cover various types of magnetic phe-nomena. The excitations were studied by inelastic neutron scattering on singlecrystals using a thermal triple axis spectrometer. In all compounds, we findsome characteristic features in the magnetic excitation spectra which reflect thedifferent schemes of magnetic order observed in the 214-ruthenates. The exactcomposition seems to drive the balance between these excitations and betweenthe distinct types of magnetic ordering finally occurring.

Magnetic Excitations in 214-Ruthenates 17

2 Inelastic Neutron Scattering:Experiments and Theoretical Background

2.1 Experimental Setup

Inelastic neutron scattering experiments have been performed with samples ofthe following compositions: Sr2RuO4, Ca2−xSrxRuO4 with x close to 0.5 andSr2Ru1−xTixO4with x=0.09. All the used samples were grown by a floating zonemethod in an image furnace. For the pure compound up to three crystals werecoaligned in order to gain statistics, yielding a total sample volume of about1350 mm3. For the Ti-substituted composition only a small crystal with about100 mm3 volume was available rendering inelastic studies extremely difficult.Also, the crystal growth of Ca2−xSrxRuO4 is quite difficult and did not yieldlarge crystals up to now. We have analyzed the compositions Ca2−xSrxRuO4with x=0.62 and 0.52 on a sample with 350 mm3 and on a set of two crystalswith a total volume of 140 mm3, respectively.

The INS results reported here were obtained on the thermal triple axis spec-trometer 1 T at the Orphee reactor in Saclay, France, using double-focusing py-rolithic graphite monochromator and analyzer crystals. In order to gain in statis-tics, the diaphragms determining the divergences of the beams were opened morethan usually. This procedure turned out not to reduce the data-quality since themagnetic scattering was always found to be broadened in comparison to the ex-perimental resolution, but it increased the intensity significantly. More detailsabout the experimental setup can be found in [23,24].

2.2 Magnetic Neutron Scattering

The magnetic neutron scattering cross section per formula unit can be describedby the Fourier transform of the spin correlation function, Sαβ(Q, ω) (labels α, βcorrespond to x,y,z) as [25];

d2σ

dΩdω=

kf

kir20F

2(Q)∑

α,β

(δα,β − QαQβ

|Q|2 )Sαβ(Q, ω) , (1)

here ki and kf are the incident and final neutron wave vectors, r20=0.292 barn,

F (Q) is the magnetic form factor. We use the common notation to split thescattering-vector Q into Q =q+G, where q lies in the first Brillouin-zone andG is a reciprocal lattice vector. All reciprocal space coordinates (Qx, Qy, Ql) aregiven in reduced lattice units of 2π/a or 2π/c.

According to the fluctuation-dissipation theorem [25], the spin correlationfunction is related to the imaginary part of the dynamical magnetic susceptibilitytimes (n(ω) + 1), with n(ω) the Bose-factor and g the Lande-factor:

Sαβ(Q, ω) =1

π(gµB)2χαβ”(Q, ω)

1 − exp(−ω/kBT ). (2)

18 M. Braden et al.

In case of weak anisotropy, which is usually observed in a paramagnetic state,χαβ”(Q, ω) reduces to χ”(Q, ω)δα,β . For itinerant magnets, anisotropy of thesusceptibility can already occur due to spin-orbit coupling. Due to the tetrag-onal symmetry the generalizedspin susceptibility in Sr2RuO4 separates into:χ”± := χ”xx = χ”yy = χ”zz. INS can distinguish the components even withouta polarization analysis since only the component perpendicular to the scatteringvector contributes. The measured intensity is then given by:

d2σ

dΩdω∝ F 2(Q)[(1 − Q2

l

|Q|2 )χ”zz + (1 +Q2

l

|Q|2 )χ”±] . (3)

Experiments at high Ql will mainly measure the in-plane component of thesusceptibility. For a quantitative analysis, one has to compare with the formfactor F(Q), which can be described by the Ru+ magnetic form factor in firstapproximation. Once determined, the magnetic scattering is converted into thedynamical susceptibility, and absolute units may be obtained by comparisonwith acoustic phonons, using a standard procedure depicted in [23,24].

2.3 RPA Analysis of the Magnetic Excitations

At low temperatures Sr2RuO4exhibits well defined Fermi-liquid properties andhigh metallic conductivity. Furthermore, the Fermi-surface has been studied indetail both by experiment and by theory yielding good agreement [20,26,27,28].Therefore, it seems appropriate to analyze themagnetic excitations in Sr2RuO4wi-thin the RPA approach basing on the calculated electronic band structure.

The four electrons of the nominally four-valent Ru occupy the t2g-levels ina low spin configuration. Bands reflecting the character of all three t2g-orbitals,dxy, dxz and dyz, contribute to the Fermi-surface [20,27]. Here one has to dis-tinguish between the dxy-band on one side and the dxz and dyz-orbitals on theother side. The dxy-band is characterized by strong hybridization in the xy-plane and, therefore, it is two-dimensional in nature. The dxz- and dyz-orbitalshybridize only along the x and y-direction, respectively. The bands correspond-ing to these orbitals are, therefore, quasi-one-dimensional and form flat sheetsof the Fermi-surface, α-sheet and β-sheet. The latter parts of the Fermi-surfaceget slightly modified by weak hybridization. In contrast, the sheet arising fromthe two-dimensional band is cylindrical, γ-sheet.

The barenon-interacting susceptibility, χ0(q), can be obtained by summingthe matrix elements for an electron-hole excitation [25]:

χ0(q, ω) = −∑

k,i,j

Mki,k+qjf(εk+q,j) − f(εk,i)

εk+q,j − εk,i − ω + iε, (4)

where ε →0, f is the Fermi distribution function, and εk the quasiparticle dis-persion relation. This was first calculated by Mazin and Singh [20,27] under theassumption that only excitations within the same orbital character are relevant(the matrix-elements Mki,k+qj are equal one or zero). Mazin and Singh pre-dicted the existence of peaks in the real part of the bare susceptibility at ω=0

Magnetic Excitations in 214-Ruthenates 19

x

y

(0 0 0) (1 0 0)

(0.3 0.3 0)

Fig. 1. Scheme of the (hk0)-plane in reciprocal space. Thin lines show the boundariesof the Brillouin-zones and small filled and open circles the zone-centers and the Z-pointsof the body centered lattice, respectively. Large filled circles indicate the position ofthe incommensurate peaks and thick lines connecting four of them correspond to thewalls of enhanced susceptibility also suggested in [20]

due to the pronounced nesting of the α-bands and β-bands. These peaks werecalculated at (0.33,0.33,0) and experimentally confirmed very close to this posi-tion at qi=(0.3,0.3,0), see Fig. 1. In addition to the peaks at qi, this study findsridges of high susceptibility at (0.3, qy,0) for 0.3 < qy < 0.5 and some shoulderfor 0 < qy < 0.3.

The susceptibility gets enhanced through the Stoner-like interaction which istreated in RPA by:

χ(q) =χ0(q)

1 − I(q)χ0(q), (5)

with the q-dependent interaction I(q). For the nesting positions Mazin andSingh get I(q)χ0(q)=1.02, which already implies a diverging susceptibility anda magnetic instability. A more detailed description of the calculation of thegeneralized susceptibility in Sr2RuO4 may be found in the contribution of D.Singh et al. in this book.

In Fig. 1 we show a scheme of the (hk0)-plane in reciprocal space, where thepositions of the nesting peaks calculated by Mazin and Singh as well as the ridgesof enhanced susceptibility are included. Note that due to the I-centering of thestructure in Sr2RuO4 (100) for instance is not a zone center, but a Z-point.

3 Magnetic Scattering in Sr2RuO4

3.1 Incommensurate Fluctuations Due to Fermi-Surface Nesting

The prediction of the incommensurate signal due to Fermi-surface nesting byMazin et al. [20] was confirmed by our first INS experiment on a single sample-crystal [23]. Figure 2 shows the result of a constant energy scan across the

20 M. Braden et al.

Fig. 2. Scans across the nesting position at low and high temperature revealing theincommensurate signal, from [23]. Lines are fits with Gaussians

incommensurate position at low and high temperatures. The clear peak in the lowtemperature scan gets washed out by heating, although any phononic scatteringshould become enhanced through the Bose-factor. This temperature dependence,the analysis of many different positions in reciprocal space, and the comparisonwith the Ru-form factor has unambiguously established the magnetic origin ofthis scattering. We do not observe any shift in the incommensurate position asfunction of energy and also the broadening in Q-space is not found to increasesensitively with energy.

In the meanwhile we have crystals of larger volume at our disposal and thecounting statistics could be considerably enhanced. Figure 3 shows constantenergy scans across the incommensurate signal along the [100] direction. Thesescans clearly indicate that the shape of thenesting signal is not symmetric butexhibits a shoulder at (qx,0.3,0) with 0.2 < qx < 0.3. In contrast we do notfind strong susceptibility in the ridges proposed in [20]. We have also performedan analysis of the LDA-band structure within RPA using the parameters givenin [27,28] and can thereby conclude that the shoulders presented in Fig. 3 stemfrom a contribution of the γ-sheet. The dashed lines in Fig. 3 represent the resultsof these calculations: not only the position but also the intensity of the secondcontribution to the incommensurate peak fits reasonably well to the experiment,giving further support to the RPA analysis of our data.

Servant et al. [29] have documented the two-dimensional nature of the in-commensurate scattering, which does not depend on the ql-component. Thisroughly reflects the pronounced two-dimensional character of Sr2RuO4 whichmay already be seen in the extremely anisotropic transport properties [30]. Con-sidering in more detail the Ql-dependence of the incommensurate scattering,we can deduce some anisotropy of the susceptibility, see Eq. (3). The out-of-plane component might be slightly enhanced, but such anisotropy cannot bestrong [24].

Magnetic Excitations in 214-Ruthenates 21

Fig. 3. Intensity distribution in energy scans across the incommensurate positionalong the [100]-direction. Solid circle denote the data, solid lines fits with Gaussiansand dashed lines the calculated imaginary part of the generalized susceptibility (at4.1meV; shifted in y-axis), from [24]

0

200

400

600

800

1000

1200

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500

17 O NMR INS

Σ q χ

"(q

,ω)

/ ω

ω→

0 (

µ B

2.

eV-

2 )

17 ( 1/T

1 T ) (sec

-1. K

-1 )

Temperature (K)

0

20

40

60

80

100

χ "(Q

0,6

.2 m

eV)

( µ B

2.e

V-1 )

0.05

0.1

0.15

0.2

0.25

0 50 100 150 200

∆q

(F

WH

M)

( Å

-1 )

Temperature (K)

( a )

( b )

( c )

0 50 100 150 200 250temperature (K)

Fig. 4. Temperature dependence of the imaginary part of the susceptibility at 6.2meV,from [23]

As already seen in Fig. 1 the incommensurate scattering rapidly disappearsupon heating. This may be understood due to the common weakening of Fermi-surface nesting at high temperature. Figure 4 shows the temperature dependenceof the imaginary part of the generalized susceptibility at 6.2meV. This resultmay be compared to those obtained inNMR [16]. 1

T1T in NMR also measures theimaginary part of the susceptibility but divided by the frequency, which is lowcompared to the accessible range in INS, and averaged over the Brillouin-zonewith a weight described by the hyper-fine fields. The incommensurate fluctua-tions with a propagation-vector (0.3,0.3,0) can contribute to O- and to Ru- 1

T1T ,

22 M. Braden et al.

Fig. 5. (a) Observed imaginary part of the generalized susceptibility as function of en-ergy and temperature; lines are fits with a single relaxor. (b) Temperature dependenceof the averaged full width at half maximum (FWHM) corrected for the experimentalresolution and its square. (c) Temperature dependence of the amplitude and its inverse(c) and of the characteristic energy (d) in the relaxor behavior fitted to part (a). Linesin (b-d) are guides to the eye

and it can qualitatively explain the NMR-data. The quantitative analysis [23,24]may explain most of the temperature dependence by the incommensurate scat-tering leaving only little space for mainly temperature independent fluctuationswith a more ferromagnetic character.

In the recent experiments we have studied the combined temperature and en-ergy dependence of the incommensurate scattering with better statistics. In thatpurpose, scans were performed across qi in the direction transversal to the totalscattering vector as function of temperature and energy. The incommensuratescattering was deduced from these scans by fitting with Gaussian distributions.The results are given in Fig. 5. Due to phonon contaminations [31] and dueto the increase of the background with temperature we could not extend thesemeasurements to energies higher than 12meV and to temperatures above 160 K.

The spectral functions are well described by the behavior of a single re-laxor [32]:

χ”(qi, ω) = χ′(qi, 0)Γω

ω2 + Γ 2 , (6)

where Γ is the characteristic damping energy. Equation 6 relates the spectralfunction observed by INS to the real part of the susceptibility. By comparisonwith phonon scattering we obtain an absolute estimate of χ′(qi, 0)=180µB

2/eV[23] which is very large compared to the macroscopic susceptibility in Sr2RuO4of only 30µB

2/eV . Using the bare-susceptibility reported by Mazin and Singhwe then estimate the enhancement factor at qi to 1

1−I(qi)χ0(qi)∼30 which again

Magnetic Excitations in 214-Ruthenates 23

is stronger than that of the macroscopic susceptibility. Already this analysisindicates that Sr2RuO4 is rather close to the related SDW magnetic ordering.The temperature dependence of the spectral function shown in Fig. 5 supportsthis conclusion.

At higher temperature the characteristic energy shifts towards larger val-ues and out of the analyzed energy range. The relaxor formula (6) was fittedto the spectral functions and the results are given in Fig. 5c) and d). For ametallic system close to antiferromagnetic or SDW ordering the self consistentrenormalization theory [32] predicts several entities to be determined by a sin-gle parameter: theStoner-like enhancement δ = 1 − I(qi)χ0(qi). The square ofthe width of the magnetic scattering, κ2, the characteristic energy Γ and theinverse of the real part of the static susceptibility at qi, 1

χ′(qi,0), should all be

proportional to δ and vanish when the system approaches the ordering. Thetransition is characterized by a sharpening of the magnetic response in energyand in Q-space. As it is shown in the right part of Fig. 5, the three quantitiesindeed decrease upon cooling but remain finite. This analysis clearly shows thatSr2RuO4 is approaching the SDW transition associated to the nesting but is notreaching it. In spite of careful search we did not find any elastic scattering atthe nesting position down to the lowest temperatures.

Since Sr2RuO4 is very close to the SDW transition one may ask whetherquantum criticality plays a role [33,34]. The pure compound is certainly not atthe quantum critical point since the transition is sufficiently suppressed. How-ever, for such a non-critical composition one would expect a cross-over betweenFermi-liquid behavior at low temperatures and a regime of quantum fluctuationswhere scaling concepts should be applicable. Such scaling concepts were used toanalyze the magnetic scattering in the high temperature superconductors [35,36]and in CeCu5.9Au0.1 [37]. Indeed our data on Sr2RuO4 presented in Fig. 5 and4 exhibit scaling behavior, χ”(qi, ω, T ) ∝ T−αg( ω

T ), but only at temperaturesabove 30 K [24]. This roughly corresponds to the temperature range where thetransport properties are not Fermi-liquid like too [30].

3.2 Additional Magnetic Scattering

Since the discovery of the incommensurate fluctuations several groups have elab-orated theories to explain spin-triplet superconductivity as being mediated bythese fluctuations [38,39,40]. However, there is no consensus till now on this topic.The more straightforward model is based on ferromagnetic fluctuations [14] andit appears therefore essential to characterize such contributions as detailed aspossible. In our first experiment we did not find any evidence for ferromagneticexcitations [23], and also the recent experiments with much better statisticsclearly demonstrate that at low temperature and energies below 12 meV thenesting signal is by far the strongest magnetic excitation [24]. We have mappedout a part of the two-dimensional Brillouin-zone in order to get a complete pic-ture at the energy of 4.1 meV. These scans are shown in Fig. 6. The nestingsignal clearly dominates, but there is some evidence that additional scatteringoccurs near (100) but not peaking at this 2D-zone-center.

24 M. Braden et al.

Fig. 6. Mapping of the INS intensity by constant energy scans with fixed Qy. Thescattering angle dependent background was subtracted and scans are vertically andhorizontally shifted (by 100 counts for a Qy step of 0.1); in scans crossing the nestingpeaks (grey) the highest intensities had to be suppressed for clarity

In the meanwhile, there have been several groups performing the calcula-tions to analyze the magnetic excitations in Sr2RuO4 [41,42,43,44]. There isgeneral agreement about the dominating nesting peak, but the results differconcerning additional weaker contributions. In none of these calculations onesees evidence for pure ferromagnetic excitations but some of them present en-hanced susceptibility close to the zone-center. Also our own calculation gives arelatively large susceptibility for q=(0.15,0.15,0), which however is still about afactor six smaller than the nesting peak at 4 meV. These features in general re-flect the original instability in the γ-sheet towards ferromagnetism, which resultsfrom the van-Hove singularity close to the Fermi-level in the dxy-band [27]. Thequalitative agreement between the calculations and the weak additional scatter-ing in Fig. 6 strongly suggests to interpret it as of magnetic origin. Also thetemperature and energy dependence of the additional scattering roughly agreeswith the RPA-predictions [24].

Sr2RuO4 seems to exhibit different types of magnetic excitations, the strongand dominating nesting signal, and some weak scattering which is still incom-mensurate but situated closer to the zone-center.

4 SDW-Ordering in Sr2Ru1−xTixO4

Upon doping Sr2RuO4 with non-magnetic titanium, Minakata et al. [22] observedthe appearance of a ferromagnetic moment by measurements of the magnetic sus-

Magnetic Excitations in 214-Ruthenates 25

Fig. 7. Scans across the incommensurate magnetic Bragg peaks in Sr2Ru0.91Ti0.09O4

at different temperatures obtained on the 4F-spectrometer with ki=1.48A−1. The y-axis scales of the scans at 20 and 60 K are shifted for clarity by 40 and 80 counts,respectively. The horizontal bar indicates the spectrometer resolution, from [8]

ceptibility. However, the small value of the ferromagnetically ordered momentalready suggests that this does not represent the primary order parameter of theunderlying phase transition. Attempts to observe the ferromagnetic ordering byelastic neutron scattering were unsuccessful as it was expected in view of theweak moment. Also we could not observe an elastic signal at the commensu-rate antiferromagnetic ordering. However, elastic scans across the q-position ofthe nesting signal in Sr2RuO4 revealed a clear peak at low temperature whichdisappears upon heating above 25 K, hence, above the temperature where thestatic ferromagnetic moment in the susceptibility vanishes [8]. Some results aredisplayed in Fig. 7. Even at the lowest temperature analyzed, the peak due to theincommensurate ordering remains of finite width. One may, however, emphasizethat the elastic correlation length of the ordering in Sr2Ru0.91Ti0.09O4 is muchlarger than the correlation length of the excitations in Sr2RuO4, which amountsonly to a few lattice spacings. The ordering in Sr2Ru0.91Ti0.09O4 exhibits onlylittle correlation between the layers which reflects the two-dimensional charac-ter of the nesting signal in Sr2RuO4. The finite in-plane-correlation length inSr2Ru0.91Ti0.09O4 can be attributed to the fact that Ti plays a twofold role inthis transition. First, Ti induces the transition through some modification ofthe band structure and second it acts as an impurity for the SDW since the

26 M. Braden et al.

Fig. 8. Constant energy scans across the incommensurate inelastic signal at Qi=(0.6930.307 0), all scans were performed transverse to Qi. For the two scans at 4.1 and 6.2 meVperformed at 1.5 K the background was shifted by 150 counts. For the scan at 2.1 meVand 1.5 K the background was shifted by -150 counts. Lines correspond to Gaussiansand a sloped background [8]

Ti itself does not carry a magnetic moment. The vacancies in the SDW thencan be considered as free moments and align under an applied field which mayexplain the susceptibility results. A rather small amount of Ti is needed to in-duce the magnetic ordering. For concentrations higher than 3 %, Minakata etal. [22] observe the susceptibility signal and for a concentration of 2.5% elas-tic neutron diffraction already finds the beginning of short range ordering be-low ∼15 K. This confirms our conclusion that pure Sr2RuO4 is very close tothe incommensurate magnetic ordering. Another important issue of the order-ing in Sr2Ru0.91Ti0.09O4 concerns the orientation of the spins along the c-axis,which indicates some anisotropy of generalized susceptibility along this direc-tion. Inelastic experiments on the Sr2Ru0.91Ti0.09O4 crystal reveal qualitativelythe same excitations as observed in Sr2RuO4, results are shown in Fig. 8. Inparticular, at high energies there is no indication for any difference. However, atthe lowest studied energy, 2 meV, we find some more complicated temperaturedependency which may be explained by the opening of the gap at the nestingvector. At low energies in the SDW ordered phase, one can expect the occurrenceof spin-waves, which have to merge into the continuum of excitations at higher

Magnetic Excitations in 214-Ruthenates 27

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ca2 - x

SrxRuO

4

TS

TN

TP

Tem

pera

ture

(K

)

Sr content x

I41/acd

Paramagnetic Metal

I4/mmm"Tilted"-P

hase

Superconductivity

L-P

bcaS-P

bca

0.2<x<0.5 1.5<x<2.0

x=0.0

metallic

insulatinglong RuO1 bond

short RuO1 bond

0.5<x<1.5a

b

0<x<0.2

Fig. 9. Phase diagram of Ca2−xSrxRuO4 including the different structural and mag-netic phases and the occurrence of the maxima in the magnetic susceptibility, TP [3].TS denotes the critical temperature of structural phase transitions and TN that of an-tiferromagnetic ordering. In the right part, we schematically show the tilt and rotationdistortion of the octahedra (only the basal square consisting of the Ru (small points)and the O(1) (larger points) is drawn) together with the elongation of the basal planes.Note that all phases are metallic besides S-Pbca, from [6]

energies. Qualitatively, the observed features correspond to such a picture, butmore work on larger crystals is needed to analyze this problem in detail.

The phase diagram of Sr2Ru1−xTixO4 shows that the Fermi-surface nest-ing instability in Sr2RuO4 can condensate into magnetic ordering and that thequantum critical composition lies near a Ti concentration of 2.5 %, hence veryclose to the spin-triplet superconductor.

5 Magnetic Scattering in Ca2−xSrxRuO4

The rich variety of the phase diagram of Ca2−xSrxRuO4 might astonish in view ofthe fact that Ca and Sr differ only in their ionic radius. The smaller ionic radiusfirst implies a series of structural phase transitions characterized by rotations ofthe RuO6-octahedra which are typical in layered perovskites [6,19].

Starting with pure Sr2RuO4, first a rotation around the c-axis develops fol-lowed by a tilt of the octahedron around an axis lying in the plane. The com-bination of different rotational and tilt schemes gives then rise to the complexstructural phase diagram. Sr2RuO4 itself seems to be on the border of struc-tural instability with a rather low corresponding phonon frequency. Close tothe Ca end-member a structural first order transition occurs [19,6] which is di-rectly coupled to the electronic metal-insulator transition [3,7]. In Fig. 9 weshow the phase diagram of Ca2−xSrxRuO4 together with the schematic pictureof the shape and the rotation or tilts of the octahedra. In Sr2RuO4 octahedra areslightly elongated along the c-direction and this peculiarity is kept till the first

28 M. Braden et al.

Fig. 10. Magnetic scattering in Ca2−xSrxRuO4 for x=0.62. Above left: schematicdrawing of the reciprocal space indicating the position of the nesting induced fluctua-tions and the scattering near (.2,0,0) most likely related to the γ-sheet. Lower left: com-parison of the scans across the zone center for pure Sr2RuO4 and for Ca1.38Sr0.62RuO4.Right: constant energy scans across (1,0,0) in transverse direction for different energiesin Ca1.38Sr0.62RuO4

order transition into the insulating phase. In the latter phase RuO6-octahedraare even flattened in addition the mean Ru-O-bond distance has increased; botheffects clearly favor electronic localization. For the comparison of the magneticexcitations one, therefore, should not include these non-metallic compounds. Onemay note that the octahedra in the Sr3Ru2O7-compound exhibit shapes similarto that of Sr2RuO4 [45]. The distinct structural phases in Ca2−xSrxRuO4 mostinterestingly seem to determine the magnetic and electronic properties. The lowtemperature susceptibility increases with the increasing rotation angle in the Sr-concentration range 0.5< x ≤ 1.6 [2,3]. For Sr-concentrations smaller than 0.5the tilt-distortion sets in and concomitantly the low-temperature susceptibilitystarts to decrease. The critical concentration for the tilt transition seems to ex-hibit the largest low temperature magnetic susceptibility. More recently, Nakat-suji et al. found ferromagnetic ordering for this concentration below 1 K [7]. Thesearch for the ferromagnetic excitations which remained almost success-less inSr2RuO4 is hence more promising in this material.

Scans across the incommensurate position in Ca1.38Sr0.62RuO4 immediatelyrevealed that the incommensurate fluctuations have less intensity than inSr2RuO4. In contrast scans across (100), which may be considered as a zone-center of the two-dimensional lattice, yielded strong scattering which, however,appears to be quite broadened. Some results are shown in Fig. 10. The broadenedpeaks observed in Fig. 10 are still incommensurate, they appear near q=(0.2,0,0).Furthermore, these features appear not to be single peaks, but to form deformedrings around the 2D-zone-center. The magnetic origin of these effects was ascer-

Magnetic Excitations in 214-Ruthenates 29

tained through their Q-dependence. A complete description of these experimentswill be given elsewhere [46].

These newly discovered features are much closer to the zone-center and,hence, should be related to the initially invoked ferromagnetic instability. Ac-tually the broad peaks have significant extension to q=(0,0,0). We think thatthis signal and the rather weak scattering in pure Sr2RuO4 have a common ori-gin, which is found in the contribution of the γ-sheet to the Lindhard-function.Fang and Terakura [47] have discussed the phase diagram of Ca2−xSrxRuO4on the base of LDA-calculations They argue that the rotation of the octahedrayields a downshift in energy of the γ-band, which pushes the van-Hove singu-larity closer to the Fermi-level. In consequence its influence on the magneticsusceptibility gets enhanced. Fang and Terakura argue that the tendency to-wards ferromagnetism gets thereby enhanced in Ca1.5Sr0.5RuO4 [47]. We wantto transfer this interpretation to the magnetic excitations. In general the γ-sheet causes contributions to the magnetic susceptibility near the zone-centerin all 214-ruthenates. Due to the enhanced influence of the γ-band for composi-tions near Ca1.5Sr0.5RuO4, these contributions get strong and the correspondingq-vectors exhibit the strongest generalized susceptibility.

6 Conclusions

The INS studies on the magnetic excitations in pure Sr2RuO4 reveal incommen-surate fluctuations which unambiguously dominate the spectrum. These excita-tions arise from the nesting between the quasi-one-dimensional α- and β-bands.The position and the precise shape of these excitations agree well to RPA calcu-lations basing on the LDA band structure. There are clear indications that thespin-triplet superconductor approaches the related SDW magnetic transitionupon cooling, although it remains paramagnetic down to the lowest tempera-tures. Upon temperature decrease, the spectrum of the incommensurate fluctu-ations exhibits sharpening in q-space and in energy, in addition its amplitudebecomes stronger. One may describe the incommensurate fluctuation spectrumwithin ω/T -scaling concepts, but only for temperatures higher than about 30K. All these observations suggest that the incommensurate fluctuations are notonly dominating the magnetic response, but are also relevant for the electronicproperties of Sr2RuO4.

There are some weak indications that Sr2RuO4 still exhibits additional mag-netic fluctuations which are situated much closer to the zone-center in reciprocalspace, and which are, hence, more ferromagnetic in nature. The weakness of thisscattering in Sr2RuO4, however, prevents a quantitative study.

Magnetic instabilities frequently manifest themselves upon some doping; there-fore, the analysis of substituted samples may help to identify the underlyingmagnetic interaction. In Sr2RuO4 replacements on the Ru- (Ti) and on the Sr-sites (Ca) may be studied. Only a small amount of Ti drives the system into aSDW-ordered phase with a magnetic excitation spectrum which remains similarto that in Sr2RuO4. This elastic magnetic ordering for a few percent of Ti fur-

30 M. Braden et al.

ther supports the conclusion that the incommensurate fluctuations in Sr2RuO4are the dominant ones and that Sr2RuO4 itself is very close to such order.

In the Ca-doped samples one finds the ferromagnetic instability, which hasbeen looked for since a long time [3,7,14]. Also, the excitation spectrum in sucha sample is no longer dominated by the nesting excitations; instead, we find astrong scattering situated near q=(0.2,0,0), which most likely arises from theγ-sheet. The two-dimensional dxy-band exhibits a van-Hove singularity near theFermi-level which induces a general instability towards ferromagnetism. How-ever, the ferromagnetic instability is found to become relevant only for ratherhigh Ca-concentrations, near Ca1.5Sr0.5RuO4, where the crystal structure is al-ready quite different to that in Sr2RuO4 [6]. In particular, the strong rotation ofthe RuO6-octahedra in this compound seems to induce quite dramatic changesin the electronic band-structure [47]. We think that the excitations observed forcompositions near Ca1.5Sr0.5RuO4 are characteristic for 214-ruthenates in gen-eral and play also some role in Sr2RuO4. But, due to the different band-structuretheir intensity seems to be efficiently reduced in the superconducting compound.

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Unconventional Superconductivitywith Either Multi-component or Multi-band,or with Chirality

K. Machida, M. Ichioka, M. Takigawa, and N. Nakai

Department of Physics, Okayama University, Okayama 700-8530, Japan

Abstract. In this review article, we discuss unconventional superconductivity wherethe unconventionality comes either from the multi-component order parameter, or fromthe multi-band order parameter, or from the chirality of the pairing function. Thesethree typical form of unconventional superconductivity are analyzed in connection withmaterials such as UPt3 and UGe2 as the first example. MgB2 is mentioned as thesecond case where two distinctive energy gaps exist on different Fermi surface sheets.Sr2RuO4 is taken up as an example for chiral superconductivity. Here the vortex statein the mixed state in each superconductor is focused on from the view point of theirunconventionality.

1 Introduction

Superconductors could have multiple order parameters, and also could breakchirality if their pairing symmetry is the triplet one. Actual superconductors, ingeneral, could have multiple bands with the Fermi surface consisting of multi-sheets. This does not affect the parity of the pairing function. These featurescharacterize unconventionality of a given superconductor. Since the third char-acteristic of the multi-band effect is quite general, there are abundant examplesof this kind of superconductors. However, if the interband coupling is strongenough, the overall superconducting properties can be effectively regarded as asingle band with an anisotropic gap function because each band mingles thor-oughly. The multiplicity of the bands reduces to an effective one-band case asin usual anisotropic s-wave superconductors. The new high-Tc superconductorMgB2 is interesting in this respect: there are two distinct Fermi surfaces, onecoming from the σ band formed by the px and py orbitals of B atom and the otherfrom the π band mainly formed by the pz orbital of B. It is known now that thegap value on the quasi-two dimensional cylindrical Fermi surface in the formerband is different from that on the three-dimensional Fermi surface in the latterband. This results in several physical consequences which make MgB2 differentfrom ordinary s-wave superconductors. Here we discuss the T -linear coefficientγ(H) of the specific heat under a magnetic field H which is proportional to Hα

with α being extremely small. This anomalously small index α is attributed tothe multi-band effect.

A certain class of superconductors belong to a triplet pairing scheme wherethe order parameter is conveniently described in terms of the d-vector (theWerthamer notation):

C. Noce et al. (Eds.): LNP 603, pp. 32–45, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Unconventional Superconductivity 33

∆(k) = i(d(k) · σ)σy . (1)

Here d(k) is a three-dimensional complex vector and a function of the relativemomentum k for a Cooper pair. The dimensionality of the d-vector characterizesthe number of components of the order parameter. Thus, superconductivity withmulti-components has a multi-dimensional d-vector when the multiplicity of theorder parameter is due to its spin part. The multiplicity of the order parametercould also come from the orbital part when it belongs to a multi-dimensionalrepresentation, such as the two-dimensional or the three-dimensional one.

The k-dependence of the vector d(k) represents the topology of the energygap on the Fermi surface, yielding a line or point node in some cases. If the d-vector components are complex numbers, we call it the non-unitary state wherea spontaneous moment ∝ d(k) × d∗(k) is induced and breaks the time rever-sal symmetry. Even when the d-vector component is unity which belongs tothe unitary state, the time reversal symmetry can also be broken by makingthat component a complex number. This corresponds to a multi-dimensionalrepresentation as mentioned above. Thus, there are two possibilities of chiralsuperconductivity which breaks the time reversal symmetry, either coming fromthe spin part of a Cooper pair (non-unitary case), or from the orbital part of aCooper pair (unitary case). These correspond to the above classification for theorigin of the multiplicity of the order parameter. A typical example of the orbitalcase is thought to be Sr2RuO4 where the existing experiments so far point tothis direction. These include the µSR experiment claims [1] that the spontaneousmoment appears below Tc. It is believed that this is direct evidence for chiral su-perconductivity in Sr2RuO4. Historically, however, the µSR experiment reportedin [2], which was taken as evidence for the time reversal symmetry breaking inUPt3, is now inconsistent with other abundant experiments and is never repro-duced by other µSR experiments. Thus, it is considered now that in UPt3 thetime reversal symmetry-unbroken state is realized although the order parameterapparently is a multi-component one. Indeed, as shown later, in order to explainthe three different phases A, B, and C in the H–T plane a multi-componentorder parameter is required [3].

In order to theoretically identify the pairing function for a given supercon-ductor, we first classify the possible pairing functions allowed under a givencrystalline symmetry based on group theory [4,5]. We must choose one of theclassification schemes belonging either to strong spin-orbit coupling [4] or toweak coupling [5]. The classified pair functions are widely different for the twoschemes. Then, it is not obvious or trivial to consider under the strong spin-orbit scheme superconductors containing heavy elements such as U or Ce asin heavy-fermion superconductors. Indeed as for the typical multi-componenttriplet superconductor UPt3, we have demonstrated that it must belong to theweak spin-orbit coupling scheme as shown later.

In the analysis of the unconventional superconductors, of particular impor-tance is the study of the vortex structure because the physical properties inthe mixed state distinctively depend on the realized pairing functions or on theunconventionality. Thus the magnetic field is a suitable probe to narrow down

34 K. Machida et al.

possible pairing functions. There are two theoretical methods to analyze theabove mentioned properties. One is to use the Bogoliubov–de Gennes (BdG)equation and the other is to rely on the quasi-classical Eilenberger equation [6].These two methods are complimentary in the following sense: BdG can fully takeinto account the quantized core, which becomes important for small kFξ0 (kF isthe Fermi momentum and ξ0 is the BCS coherence length). This quantum lim-iting behavior [7] is thought to be realized in high-Tc cuprate superconductorsbecause ξ0 is short and comparable to the lattice constant. However, the over-whelming majority of the superconductors, including the exotic heavy fermionsuperconductors, are categorized into the other class for which kFξ0 1. Thisparameter region is well describable by the quasi-classical theory except for theextremely low-temperature regime (T/Tc 1/kFξ0) in which the discretizedquantum levels with the level spacing O(1/kFξ0) around a core become visible.In the following we introduce these two methods to understand the multi-bandeffects in MgB2 in terms of the BdG equation and chirality effects in Sr2RuO4in terms of the quasi-classical equation.

2 Multi-component Superconductivity

There are two classes of heavy fermion materials which definitely exhibit multi-component superconductivity: One is UPt3 (see for review [8,9,10,11]) and theother class is the ferromagnetic superconductors UGe2 [12,13] and URhGe [14](possibly ZrZn2 [15]). Here we analyze the possible pairing functions for theseclasses of materials.

2.1 UPt3

UPt3 was discovered in 1984 [16] and later found to have two successive super-conducting transition temperatures [17] Tc1 = 0.58 K and Tc2 = 0.53 K. Namely,the hypothetical transition temperature at which the order parameter with themulti-component allowed under the hexagonal symmetry is degenerate, is splitinto two Tc1 and Tc2 by some symmetry-breaking field. After long discussionsconcerning the origin of this symmetry-breaking field [18,19,20], we are now leftwith the last possibility where the actual crystalline symmetry is lowered to thetrigonal symmetry from the hexagonal one. The assumed hexagonal symmetry isalready weakly broken at a higher temperature from the outset [21]. Initially, itsorigin was attributed to the weak antiferromagnetism which apparently breaksthe hexagonal symmetry, if it exists. However, it turns out eventually that thisantiferromagnetism observed by elastic neutron scattering [22] is not a trulystatic order, but fluctuates quicker than 10−6 sec and slower than 10−11 sec,enabling us to use the origin of the symmetry breaking field as a result.

Since there are at least three phases identified in UPt3, the A (higher Tand low H), B (low T and low H) and C (high T and high H) [23,24] phases,we must consider at least a two-component order parameter to describe this

Unconventional Superconductivity 35

multiple phase diagram. The degeneracy can either come from the spin part orfrom the orbital part of the triplet pairing function, or from the combined form.

According to Knight shift experiments performed on high-quality UPt3 singlecrystals [25], the rich details of the spin structure of the pairing function are nowrevealed: (i) the spin susceptibilities χa and χb for the a- and b-axis directionsdecrease below T = Tc2; (ii) the parity of the pairing function is odd and the spintriplet state is realized; (iii) the B phase is characterized by a two-component d-vector. By increasing H (‖ c), one of the d-vector components rotates to lower theZeeman energy around 2 kG [25], implying that the effective spin-orbit couplingwhich pins the d-vector to the crystal lattice is weak.

Among the various possible two-dimensional representations allowed underthe hexagonal symmetry, which is still the crystal symmetry to start with, onlytwo states are left as candidates to explain the existing experiments and, in par-ticular, the above mentioned Knight shift measurement [3], namely, the planarstate

d(k) = λx(k)τx + λy(k)τy (2)

and the bipolar state

d(k) = λx(k)τx + iλy(k)τy (3)

with τ = iσσy. The former (latter) is a unitary (non-unitary) state, where

λx(k) = kz(k2x − k2

y) (4)

and

λy(k) = kz2kxky (5)

are a two-dimensional representation in the hexagonal symmetry. The nodalstructure of the planar state is characterized by two quadratic point nodes atthe north and south poles and a linear line node on the equator for the Fermisphere. On the other hand, the bipolar state has four line nodes parallel to thekz-axis and one line node on the equator. Thus, the thermodynamic behavior isdifferent for the two states. In particular, direction dependent transport prop-erties, such as the thermal conductivity and the ultrasound attenuation, couldhelp to discriminate between the two states. In fact, a detailed analysis indicatesthat the planar state describes these experimental data better than the bipo-lar state [3]. We note that the nodal structure of the planar state is identicalto that of the so-called E2u state proposed by Sauls [20] in which the degen-eracy comes exclusively from the orbital part of the pairing function and thed-vector is one-component. Thus it is impossible to explain the facts that thesusceptibility decreases for two crystalline axes, as mentioned above, and alsothat upon increasing H (‖ c) the d-vector rotates, because in the E2u state thed-vector is rocked to the c-axis by the strong spin-orbit coupling. The seeminglymost complicated and mutually contradicting experimental facts on UPt3 arenow consistently understood in terms of the planar state (2), which is a two-component order parameter belonging to the odd-parity triplet pairing function.

36 K. Machida et al.

2.2 UGe2

Ferromagnetism and superconductivity are thought to be mutually exclusive.Ginzburg [26] points out a possibility of their coexistence under the conditionthat the magnetization is less than the thermodynamic critical field. The recentdiscoveries on UGe2 [12], URhGe [14] and ZrZn2 [15] where these two long-range orders coexist either under applied pressure or at ambient pressure areconsidered as a surprise. w UGe2 is an itinerant metallic ferromagnet whose Curietemperature is TFM=52 K and the spontaneous moment is ∼ 1.4 µB/U-atomunder ambient pressure. Upon increasing P , both TFM and the moment decreasegradually and for P ∼ 1.0 GPa superconductivity appears. After reaching amaximum Tc at around ∼ 1.3 GPa both orders disappear simultaneously uponfurther increasing P at ∼ 1.7 GPa.

There are two crucial theoretical and experimental features: (i) the exchangesplitting due to ferromagnetism is of the order of meV according to band cal-culation [27], and (ii) the Mossbauer spectroscopy on uranium [28] shows anhyperfine field about equal to 240 T at the U-site. This huge exchange field ap-parently excludes not only any singlet pairing, but also certain forms of tripletpairing, namely the unitary triplet states. The only possible state survivingunder such a huge exchange field is the non-unitary triplet pairing which is,as mentioned above, characterized by a multi-component order parameter, i.e.∆(k) = i(d(k) · σ)σy is a 2×2 matrix. If d(k) is a complex number, the product

∆(k)∆†(k) = |d(k)|2σ0 + i [d(k) × d∗(k)] · σ (6)

is not a multiple of the unit matrix and ∆(k) becomes non-unitary. Since thenon-unitary state breaks the time reversal symmetry, a spontaneous momentm(k) at the k-point of reciprocal space

m(k) ∝ i d(k) × d∗(k) (7)

is necessarily induced. In the bipolar (non-bipolar) state the average moment〈m(k)〉 over the Fermi surface vanishes (appears). Thus an additional Zeemanenergy is gained for the non-bipolar state. It is plausible for that reason that thenon-bipolar state is likely to be realized in UGe2, as well as in URhGe, which isanalogous to UGe2 in various respects.

The quasi-particle excitation spectrum for a non-bipolar type of state of theform d(k) = ηφ(k), where the spin direction η and the orbital part φ(k) areseparable, has two branches:

Ek,σ(k) =√

ε2σ(k) + ∆2

σ(k) (σ = ±) . (8)

The gap functions are given by

∆±(k) = |ηx ± ηy|φ(k) = |η±|φ(k) . (9)

Thus, one of the two branches of the energy gap vanishes identically, meaningthat on the spin-down or spin-up Fermi surface there is no superconducting gap

Unconventional Superconductivity 37

formed, leaving it in the normal phase. We naively expect a residual linear spe-cific heat coefficient γ, corresponding to the minority Fermi surface. This resid-ual minority Fermi surface shows up in other various thermodynamic quantities,such as the low temperature thermal conductivity, etc.

3 Multi-band Superconductivity

In order to investigate multi-band effects on superconductivity, let us considerthe simplest and minimum extension from an isotropic single-band s-wave su-perconductor. We examine the following tight-binding model [29] for a two-bandsuperconductor under a magnetic field:

H = H0 + Hpair (10)

H0 =∑

i,j,σ,γ

(−tijγ − µγδi,j)a†iσγajσγ (11)

Hpair =12

∑i,σ,γ,γ′

gγγ′(aiσγai−σγ)†aiσγ′ai−σγ′ . (12)

Here the nearest-neighbor hopping integral is

tijγ = tγ exp[iπ

φ0

∫ rj

ri

A(r) · dr

], (13)

where A(r) is the vector potential and φ0 = hc/2e is the unit flux. A two-dimensional square lattice with lattice constant equal to one is assumed. Thesubscript γ denotes the two bands L and S. Assuming the singlet pairing, wecan derive the BdG equations for γ = L and S as

∑i

(Kjiγ δi,j∆γ(ri)

δi,j∆†γ(ri) −K∗

jiγ

) (uγε(ri)vγε(ri)

)= Eγε

(uγε(rj)vγε(rj)

),

where

Kijγ = −tijγ − µγδi,j . (14)

This model Hamiltonian describes a situation where on two Fermi surfaces twodistinctive energy gaps appear and two kinds of pairing mix via a pair tunnelingprocess whose coupling parameter gSL is defined below. The gap equation isgiven by

∆L(ri) = gLLdL(ri) + gLSdS(ri) (15)∆S(ri) = gSSdS(ri) + gLSdL(ri) (16)

with the order parameter

dγ(ri) = 〈ai↓γai↑γ〉= −

∑ε

v∗γε(ri)uγε(ri) tanh

Eγε

2T. (17)

38 K. Machida et al.

It should be noted for a uniform system without H that when all the interactionchannels are characterized by the same negative sign, namely the attractive one,the two gap functions ∆L and ∆S have the same phase. When gLL ≤ 0, gSS ≤ 0and gLS ≥ 0, the relative phase is out of phase by π. The signs of the twogap functions are opposite. While this out-of-phase case gives rise to the samebulk thermodynamics as in the in-phase case, the boundary effects such as thetunneling characteristics may differ. Since in two-band superconductors the signof gLS can in general be positive or negative with equal probability, the studyof the out-of-phase case is worth being further investigated.

The local density of states (LDOS) at site i for the γ band is calculated as

Nγ(ri, E) =∑

ε

|uγε(ri)|2δ(E − Eγε) + |vγε(ri)|2δ(E + Eγε)

. (18)

As mentioned above, we assume an isotropic s-wave pairing for both bands Land S, characterized by the order parameters (the energy gaps) dL (∆L) anddS (∆S). In the following, the attractive interactions are chosen as gLL = 0,gLS = gSL = 0 and gSS = 0, namely the gap ∆S on the S-band is induced bythe Cooper pair tunneling via gLS . As for the normal-state band parameters, wetake tL = tS = t (≡ 1) and µL = −1 and µS = +1, so that the Fermi surface forγ = L (S) is close (open) around the Γ -point. The density of states (DOS) forboth bands is the same at the Fermi level. As two vortices are accommodatedin a unit cell of Na × Na atomic sites, the applied magnetic field is given byHNa×Na ≡ 2φ0/N

2a . By introducing the quasi-momentum of the magnetic Bloch

state, we obtain the wave function with periodic boundary condition for a largenumber of unit cells.

The spatial profiles of the LDOS are shown in Fig. 1, where NL(r, E ∼ 0)and NS(r, E ∼ 0) have a peak at the vortex center and the ridges connecting thevortex cores are clearly seen. While the density of states is mostly concentratedat the vortex core in NL(r, E ∼ 0), it rather spreads out in NS(r, E ∼ 0). Thisis because the vortex bound states are highly confined in the L-band vortexcorresponding to the narrow core radius, while in the S-band vortex the corestates are loosely bounded. The spatial profiles for NL(r, E ∼ 0) and NS(r, E ∼0) resemble those of the low-field case and the high-field case in the single-band superconductor. In NS(r, E ∼ 0), the low energy states extending fromthe vortex cores overlap with each other, and the LDOS is suppressed alongthe line connecting the nearest-neighbor or next-nearest-neighbor vortices. Withincreasing H, the effect of the overlap becomes dominant, and the LDOS isreduced to the flat profile NS(r, E ∼ 0)/N(EF) ∼ 0.5 in the S-band (N(EF) isthe total DOS in the normal state at the Fermi level).

The spatial average of NL(r, E ∼ 0) and NS(r, E ∼ 0) gives rise to the totalDOS under a given field, leading to γ(H) which is defined by

γ(H) = γL(H) + γS(H) (19)

with

γL,S(H) = 〈NL,S(r, E ∼ 0)〉r∈unit cell . (20)

Unconventional Superconductivity 39

1 0.5

0

1

2

0.5 0.4

0

1

0.5 0.4

0

1

1

0

1

2

0.5

(a)

(b)

Fig. 1. Zero energy local density of states NL(r, E ∼ 0) for the L-band andNS(r, E ∼ 0) for the S-band. They are normalized by N(EF), the normal state DOSat the Fermi level. (a) At lower field H30×30. (b) At higher field H18×18

In Fig. 2 it is seen that γ(H) is described by the power law γ(H) ∝ Hα withsmall α. If only the low-field points are fitted, we obtain α = 0.38 (thick line). Thefitting by γ(H) ∼ γN (H/Hc2)α under the condition that γ(H) is reduced to thenormal state value γN gives α = 0.33 (thin line). The small exponents α, or thesharp rise of γ(H) in small fields, can be attributed to the S-band contributionγS(H) ∝ H0.20, while γL(H) ∝ H1.00 in the L-band. That is, the small α is dueto the overlap of the low energy states outside the vortex cores at the S-band.Physically, this happens because the energy gap for the S-band is suppressedby a weak field, while the total superconductivity is maintained by the largerenergy gap up to Hc2. This intuitively appealing picture is actually confirmed bythe present microscopic calculation. This is, however, different from the case oftwo independent gaps with different transition temperatures and different Hc2.In such a case, we would have double transition and γ(H) would be a simpleaddition of two independent curves, which have a kink structure at the lowerHc2. This is not the case for MgB2.

4 Chiral Superconductivity

In this section we discuss the possible pairing function realized in Sr2RuO4,where the time reversal symmetry is likely to be broken due to the orbital partof the triplet superconductivity. Several physical consequences associated withthis chiral superconductivity is now investigated in connection with the behavior

40 K. Machida et al.

0

0.5

1

0

Fig. 2. Field dependence of γ(H) for ∆S/∆L = 0.27. Points of γ(H) (circles ), γS(H)(triangles ) and γL(H) (squares ) are numerical data. The thick line is fitting for lowerfield data of γ(H). The thin line is fitting by γ(H) ∼ γN (H/Hc2)α. We also show fittinglines in low field for γS(H) (thick dotted line ) and γL(H) (thin dotted line)

under an applied field. In particular, the vortex core excitations of the low-lyingquasi-particles around a vortex manifest themselves in γ(H).

4.1 Possible Pairing Function

The original proposal by Rice and Sigrist [30] for the pairing function of Sr2RuO4is

d(k) = ∆0z(kx + iky) (21)

which is analogous to the Anderson-Brinkman-Morel (ABM) state realized inthe A phase of superfluid 3He. This proposed state is able to explain variousexperimental facts, including notably the µSR and the Knight shift data. Bothare consistent with the idea of chiral triplet superconductivity with the limitationthat the susceptibility does not decrease below Tc only for the basal plane ofthe tetragonal crystal. Thus, no information for the perpendicular susceptibilityalong the c-axis is obtained.

On the other hand, the Rice and Sigrist state [30] in which the energy gapopens all over the Fermi surface contradicts a variety of low-temperature ther-modynamic measurements, rather pointing to a line node. Hasegawa et al. [31]classify the possible triplet pairing states allowed under the tetragonal symmetryD4h by going to angular momentum states higher than the p-wave pairing con-sidered by Rice and Sigrist [30]. It turns out that there are two kinds of f -wavepairing states. One class has a horizontal line node

d(k) = ∆0z(kx + iky)(1 + β cos kz) (22)

Unconventional Superconductivity 41

with |β| ≥ 1. The other class has vertical line nodes running along the c-axissuch as

d(k) = ∆0 z (k2x − k2

y)(kx + iky) (23)

or

d(k) = ∆0 z kxky(kx + iky) . (24)

According to the recent direction-dependent thermal conductivity experi-ments in a magnetic field by Izawa et al. [32], there is no in-plane anisotropy,implying the absence of the vertical line node. This immediately leads to theidentification that the state with the horizontal node associated with d(k) givenin (23) is most consistent with the existing data.

4.2 Quasi-classical Theory for Chiral Superconductivity

In order to focus on the influence of chirality, we examine the quasi-classicalEilenberger equation for a simplified pairing function described by the followingorder parameter under an applied field perpendicular to the two-dimensionalFermi surface, namely along the c-axis. In the p+-chiral state

∆(kF, r) = ∆1(r)φ+(kF) + ∆2(r)φ−(kF) , (25)

the dominant component comes from

φ+(kF) = kx + iky (26)

and the sub-dominant component from

φ−(kF) = kx − iky . (27)

The coefficients ∆1,2(r) describe the spatial dependence of the center of massfor a Cooper pair. In the p−-chiral state the roles of φ+(kF) and φ−(kF) areinterchanged.

The Eilenberger equations readωn +

i2vF ·

(∇i

+2π

φ0A

)f(iωn,kF, r) = ∆(kF, r)g(iωn,kF, r) (28)

ωn − i

2vF ·

(∇i

− 2π

φ0A

)f†(iωn,kF, r) = ∆∗(kF, r)g(iωn,kF, r) (29)

withg(iωn,kF, r) =

√1 − f(iωn,kF, r)f†(iωn,kF, r) . (30)

Here g(iωn,kF, r), f(iωn,kF, r) and f†(iωn,kF, r) (Re g(iωn,kF, r) ≥ 0) arethe quasi-classical Green’s functions, iωn is the imaginary fermionic Matsubarafrequency and kF is the Fermi momentum vector. The gap equation is given by

∆(kF, r) = N0 2πT∑

ωn>0

⟨V (k′

F,kF)f(iωn,k′F, r)

⟩k′

F(31)

42 K. Machida et al.

0 10

1

H

N(E

=0)

pp

s

+ −

0 10

1

H1/2

N(E

=0)

pp

s

+−

Fig. 3. Field dependence of the zero-energy density of states N(E = 0)/N0 in the p−state (•), the p+ state (), and the s-wave pairing case (∗). The square vortex latticeand the triangular vortex lattice give the same results. Lines are fitting for numericaldata in the p− state by N(0) ∝ H0.74 (dotted line ) or N(0) ∝ √

H (solid line ). Inset:N(0)/N0 is plotted as a function of

√H. The solid line shows the relation N(0) ∝ √

H

and the associated Maxwell equation is

∇ × ∇ × a(r) = − πφ0

κ2∆0ξ302πT

∑ωn>0

⟨vF

ig(iωn,kF, r)

⟩kF

(32)

with the vector potential A(r) = 12H × r + a(r) (H is the applied magnetic

field). In the above equations the symbol 〈· · · 〉kF denotes the average over theFermi surface, ξ0 = πvF/∆0 is the BCS coherence length (∆0 is the uniform gapat T = H = 0), N0 is the density of states at the Fermi level and κ is given byκ =

√7ζ(3)/72 ∆0 κBCS/Tc, with κBCS being the Ginzburg-Landau parameter.

The local density of states at r-position and energy E is given by

N(E, r) = N0 〈Re g(iωn → E + iη,kF, r)〉kF. (33)

The spatial average of N(E, r) gives the density of states N(E). In the followingwe fix T = 0.5 Tc and κBCS = 2.7, appropriate for H ‖ c for Sr2RuO4. Themagnetic field is scaled by φ0/ξ2

0 .It is found that the p− state has a higher Hc2 than the p+ state, meaning

that the former is more stable than the latter under a magnetic field. As shownin Fig. 3, the density of states at E = 0, normalized by its normal-state valueN(E = 0) proportional to γ(H), behaves differently, depending on the chirality.In the higher field region near Hc2 the γ(H) curve for the p− state exhibitsa

√H-like behavior, which is observed in H ‖ c for Sr2RuO4. This provides

an explanation for the observed γ(H) ∝ √H in a high field. Note that this

is not the so-called Volovik effect [33] γ(H) ∝ √H in lower field for d-wave

Unconventional Superconductivity 43

superconductors, as it should be limited to low fields [34] near Hc1. We alsoremark that the p+-chiral state does not show such a

√H-like behavior. Its

H-dependence is similar to that for isotropic s-wave superconductors. Thereforeγ(H) is found to be a good physical quantity for narrowing down possible pairingfunctions, as concerns not only the nodal structure but also the chirality of thetriplet superconductivity.

5 Conclusion and Prospect

In this article we review the present status of unconventional superconductorswhich are characterized by an order parameter other than a simple isotropics-wave pairing. Namely we discuss the pairing functions with multi-component,multi-band and chirality. These are exemplified by UPt3 and UGe2 as the firstcategory, MgB2 as the second category and Sr2RuO4 as the third category.

UPt3 is one of the most studied heavy-fermion superconductors, yet there ismuch room to finally settle the pairing function. Experimentally, (i) the NMRmeasurement of the Knight shift must be repeated for other samples and ex-tended to further unexplored H and T regions. Some hints to show a new highfield phase other than the A,B and C phases come from the c-axis NMR ex-periment. (ii) The determination of the precise crystalline symmetry must beestablished under pressure. If the symmetry breaking field is due to the trigonaldistortion, the crystal symmetry must be hexagonal beyond P ≥ 1 GPa wherethe double transition ceases to exist.

Theoretically, (i) the microscopic origin of why the spin-orbit coupling isso weak must be clarified, and (ii) the pairing mechanism which stabilizes theidentified complex, yet rich pairing function of the planar state is still unknown.

As for the ferromagnetic superconductors UGe2, URhGe and ZrZn2, the spe-cific heat measurements at low T are crucial by using high-quality samples.So far, only a broad specific heat jump at Tc is observed for the pressurizedUGe2 [13] and URhGe [14]. Naively, a substantial electronic residual γ coeffi-cient is expected to be observed.

MgB2 is interesting because multi-band superconductors are omnipresentand here MgB2 is an extreme case. Thus, if we carefully examine “conventional”superconductors in this respect, there are some chances to find unconventionalbehaviors even there. For example, the power law index α in γ(H) ∝ Hα isoften rather small for usual materials where we expect α ∼ 0.67 for the isotropicone-band s-wave superconductors and α ∼ 0.41 for the d-wave superconductors.Any deviation from this law, 0.41 ≤ α ≤ 0.67, may imply the multi-band effectas one possibility.

Sr2RuO4 is focused recently. Yet there are several loopholes which mightturn down the identified pairing function. The γ(H) behavior for H ⊥ c isunusual and does not follow the simple power law mentioned above. Near Hc2the superconducting transition to the normal state is also not simple; either twosuccessive transitions or first-order transition with latent heat. Thus there areseveral outstanding unsolved issues in the high-field region in Sr2RuO4.

44 K. Machida et al.

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15. C. Pfleiderer, M. Uhlarz, S.M. Hayden, R. Vollmer, H.v. Lohneysen, N.R. Bern-hoeft, G.G. Lonzarich: Nature 412, 58 (2001)

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21. D.A. Walko and J.-I. Hong, T.V. Chandrasekhar Rao, Z. Wawrzak, D.N. Seidman,W.P. Halperin, M.J. Bedzyk: Phys. Rev. B 63, 054522 (2001)

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Unconventional Superconductivity 45

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A. Hasegawa, K. Machida: Phys. Rev. B 59, 8902 (1999)

The Behaviour of a Triplet Superconductorin a Spin Only Magnetic Field

B.J. Powell, J.F. Annett, and B.L. Gyorffy

H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, BS8 1TL, UK

Abstract. We investigate the order parameter of Sr2RuO4 in an exchange-only mag-netic field. A Ginzburg-Landau symmetry analysis implies three possibilities: a pure3He A phase, a 3He A1 or a 3He A2 phase. We explore the exchange field dependence ofthe order parameter and energy gap in a one-band model of Sr2RuO4. The numericalsolutions show no A1 phase and that the A2 phase is lower in free energy than the Aphase. We explore heat capacity as a function of temperature and field strength andfind quantitatively different behaviours for the A and A2 phases.

1 Introduction

A spin triplet superconductor should show a number of interesting magnetic-fieldeffects which are direct consequences of the magnetic moment of the Cooperpairs. In particular, for spin-triplet superconductors the Zeeman coupling be-tween the quasiparticle spins and the external magnetic field need not lead toPauli limiting, unlike the case of spin-singlet superconductors. In the extremehigh field limit with completely exchange-split bands we could expect singlespin pairing of the majority spin Fermi surface. We may also expect possiblephase transitions or symmetry changes of the order parameter in a magneticfield, which are analogous to the transitions seen in superfluid 3He [1,2,3]. The3He B-phase is destroyed in a magnetic field in a qualitatively similar mannerto a singlet superconductor. On the other hand, if the zero field ground stateis one of equal-spin-pairing (ESP), then the gap function can deform continu-ously as a function of a Zeeman field the 3He A-phase evolves first smoothlyinto A2 phase and then, via a phase transition A1 phase, as it progresses fromequal-spin-pairing to single spin pairing with increasing field [4].

The superconductor Sr2RuO4 should be an ideal candidate to examine theseeffects. There is strong evidence for spin-triplet pairing [5] from direct measure-ments of spin-susceptibility in the superconducting state [6,7]. It has a simpleand well understood Fermi surface [8], and is in the clean limit. The detailedgap function is still somewhat controversial, but is generally believed [9] to beof tetragonal Eu symmetry [10], and more specifically to be a two-dimensionalanalogue of the 3He A-phase, with d(k) ∼ (sin kx + i sin ky)(0, 0, 1). This orderparameter would agree with the spin-susceptibility measurements in the super-conducting state [6,7], and also would lead to time-reversal symmetry breakingbelow Tc [11]. More recently, specific heat [12], penetration depth [13] and ther-mal conductivity [14] experiments have shown that the gap must have line-nodes

C. Noce et al. (Eds.): LNP 603, pp. 46–59, 2002.c© Springer-Verlag Berlin Heidelberg 2002

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 47

on the Fermi surface. However, for the cylindrical Fermi surface geometry [8]of Sr2RuO4 a complete group theoretic analysis of symmetry distinct pairingstates does not show any which both break time-reversal symmetry and haveline-nodes [10]. A possible resolution to this dilemma has been developed in theorbital dependent pairing model of Zhitomirsky and Rice [15] and in a relatedmodel by Litak et al. [16]. For different reasons both groups proposed that thegap function is of the form

d(k) ∼ (sin kx + i sin ky)(0, 0, 1) (1)

on the dominant γ-Fermi surface sheet, and of the form

d(k) ∼(

sin(

kx

2

)cos(

ky

2

)+ i sin

(ky

2

)cos(

kx

2

))cos(

ckz

2

)(0, 0, 1) (2)

on the α and β sheets. Both of these functions possess the same Eu symmetry,but correspond to intra-plane and inter-plane pairing interactions respectively.This gap function has horizontal line nodes at kz = ±π/c on the α and βsheets, and was shown to be in good agreement with experimental temperaturedependences for specific heat, penetration depth and thermal conductivity [16].

In a magnetic field Sr2RuO4 shows a number of unusual features. Firstlythe vortex lattice is square [17,18] which agrees well with the predictions of atwo-component Eu symmetry Ginzburg-Landau theory [19,20]. Secondly thereis an anomalous second feature close to Hc2, which only occurs when the fieldis aligned within 1o of the a-b plane [12]. At the present time the origin ofthis feature is uncertain. It may be a vortex lattice phase transition, or it maycorrespond to a change in pairing symmetry with field, perhaps analogous to thedouble superconducting transition in UPt3 [21].

In this paper we will focus specifically on the unique effects of the Cooperpair spin in a triplet superconductor. Therefore we neglect the effects of thevector potential on the quasiparticles, and instead focus solely on the Zeemancoupling of the quasiparticle spin to the magnetic field. We can justify this modelby appealing to the strong Stoner enhancement in Sr2RuO4 [22,23], and so theexchange field will be large. Alternatively, our model may be appropriate for theferromagnetic superconductor ZrZn2 [24].

This paper is organised as follows. Firstly we write a simple single-bandmodel Hamiltonian for p-wave pairing in the γ-band of Sr2RuO4 . Next weexamine how a spin-only magnetic field enters the corresponding Eu symmetryGinzburg-Landau theory. In Sect. 4, we present detailed numerical results forthe field dependent energy gap, and specific heat for the two relevant cases ofthe exchange field either parallel or perpendicular to the d(k) order parameter.We show that the lower free energy state in analogous to the 3He A2 phase.Finally, in Sect. 5 we present our conclusions.

48 B.J. Powell, J.F. Annett, and B.L. Gyorffy

2 A Microscopic Model for a Triplet Superconductorin a Spin Only Magnetic Field

We consider the effect of a spin-only magnetic field, H, on an attractive, nearestneighbour, Hubbard model. We use a one-band model, appropriate for the γsheet of the Sr2RuO4 Fermi surface. The set of interaction constants, Uσσ′

ij ,describe attractions between electrons on sites i and j with spins σ and σ′. TheHamiltonian for this model is:

H =∑ijσ

((ε − µ)δij − tij)c†iσ cjσ − 1

2

∑ijσσ′

Uσσ′ij niσnjσ + µB

∑iσσ′

c†iσ(σσσ′ · H)ciσ ,

(3)where tij is the hopping integral, ε is the site energy, c

(†)iσ are the usual an-

nihilation (creation) operators and niσ is the number operator. σσσ′ are thecomponents of the vector of Pauli matrices:

σ = (σ1, σ2, σ3) . (4)

By making the Hartree-Fock-Gorkov approximation and taking a lattice Fouriertransform the following spin-generalised Bogoliubov-de Gennes (BdG) equationcan be derived from the above Hamiltonian.

εk + µBH3 µB(H1 − iH2) ∆↑↑(k) ∆↑↓(k)µB(H1 + iH2) εk − µBH3 ∆↓↑(k) ∆↓↓(k)

−∆†↑↑(−k) −∆†

↑↓(−k) −ε−k − µBH3 µB(−H1 − iH2)−∆†

↓↑(−k) −∆†↓↓(−k) µB(−H1 + iH2) −ε−k + µBH3

u↑σ(k)u↓σ(k)v↑σ(k)v↓σ(k)

= Eσ(k)

u↑σ(k)u↓σ(k)v↑σ(k)v↓σ(k)

, (5)

where εk is the (Fourier transformed) normal, spin independent part of theHamiltonian. The order parameters ∆σσ′(k) are determined self consistently by

∆σσ′(k) = −12

∑qσ′′

Uσσ′(q)(uσσ′′(−q)v∗

σ′σ′′(−q) − v∗σσ′′(q)uσ′σ′′(q))(1 − 2fqσ′′) ,

(6)where Uσσ′

(q) is the lattice Fourier transform of Uσσ′ij and fqσ is shorthand for

the Fermi function f(Eqσ). It is natural to separate the spin-generalised BdGequation into triplet and singlet parts:

∆(k) ≡(

∆↑↑(k) ∆↑↓(k)∆↓↑(k) ∆↓↓(k)

)= (d0(k) + σ · d(k))iσ2 . (7)

d0(k) is the (scalar) singlet order parameter and d(k) is the (vector) tripletorder parameter. The singlet order parameter is symmetric under spatial inver-sion while the triplet order parameter is anti-symmetric under spatial inversion.

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 49

Hence, the BdG equation can be rewritten as

εk + µBH3 µB(H1 − iH2) −d1(k) + id2(k) d0(k) + d3(k)µB(H1 + iH2) εk − µBH3 −d0(k) + d3(k) d1(k) + id2(k)−d∗

1(k) − d∗2(k) −d∗

0(k) + d∗3(k) −εk − µBH3 µB(−H1 − iH2)

d∗0(k) + d∗

3(k) d∗1(k) − d∗

2(k) µB(−H1 + iH2) −εk + µBH3

u↑σ(k)u↓σ(k)v↑σ(k)v↓σ(k)

= Eσ(k)

u↑σ(k)u↓σ(k)v↑σ(k)v↓σ(k)

. (8)

If there is no superconductivity in the triplet channel we regain the standardresult [3] for the spectrum of a singlet superconductor in a spin only magneticfield:

E(k) = ±√

ε2k + |d0(k)|2 ± µB |H| . (9)

By setting the singlet order parameter to zero we find that the equivalent resultfor a triplet superconductor is

E(k) = ±√

ε2k + µ2

B |H|2 + |d(k)|2 ±√

Λ(k) , (10)

where

Λ(k) = |d(k) × d(k)∗|2 + 4ε2kµ2

B |H|2 + 4µ2B |H · d(k)|2

+4iεkµBH · d(k) × d(k)∗ . (11)

It should be noted that this does not assume a unitary order parameter1.It is a relatively straightforward process to calculate thermodynamic prop-

erties for a triplet superconductor. For example the specific heat is given by

CV = T∂S

∂T(12)

= −kBT∂

∂T

∑kσ

(fkσln(fkσ) + (1 − fkσ)ln(1 − fkσ)) (13)

=∑kσ

fkσ(1 − fkσ)(

Eσ(k)2

kBT 2 +1

kBTEσ(k)

d

dTEσ(k)

)(14)

=∑kσ

fkσ(1 − fkσ)kBT 2

(Eσ(k)2 − T

2d

dT|d(k)|2

). (15)

1 A unitary state is any state for which d(k) × d∗(k) = 0.

50 B.J. Powell, J.F. Annett, and B.L. Gyorffy

3 Ginzburg–Landau Theory of a Quasi–two DimensionalTriplet Superconductor in a Magnetic Field

Before considering numerical solutions of the self consistent Bogoliubov–de Gennesequations, we will examine the possible results by deriving a Ginzburg–Landautheory from our microscopic theory.

Consider a quasi–two dimensional system with two orbital degrees of freedom(which we label x and y) and three spin degrees of freedom (labelled 1, 2 and3). Hence, instead of the familiar 3 by 3 order parameter of 3He this system isdescribed by the complex 2 by 3 matrix A, which is related to the microscopicorder parameter, d(k), by

d1(k)d2(k)d3(k)

=

A1x sin kx + A1y sin ky

A2x sin kx + A2y sin ky

A3x sin kx + A3y sin ky

=

(Ax Ay

)(sin kx

sin ky

). (16)

In zero field the free energy for a tetragonal crystal is given by [10]

F = α(T − Tc)(|Ax|2 + |Ay|2) + β1(|Ax|2 + |Ay|2)2 + β2|Ax · Ax + Ay · Ay|2+β3((Ax · A∗

y)2 + (A∗x · Ay)2 + (A∗

x · Ax)2 + (A∗y · Ay)2)

+β4(2|Ax · A∗y|2 + |Ax|4 + |Ay|4)

+β5(2|Ax · Ay|2 + |Ax · Ax|2 + |Ay · Ay|2)+β6(|Ax · Ax|2 + |Ay · Ay|2) + β7(|Ax|4 + |Ay|4) . (17)

Only the first five quartic terms (β1 −β5) are required to describe 3He [25]. Theadditional two terms here (β6 and β7) appear because the rotational symmetryof the crystal is discrete, where as rotational symmetry is continuous in the fluid.Gradient terms can also be calculated [10,19,20], but we will not make use ofthese here.

To second order in A the free energy in a finite magnetic field, FH , is

FH =1β

∑iωn

∫dk tr

(G

0(k, iωn)∆(k)G∗

0(−k, iωn)∆†(−k)

), (18)

where,G

0(k, iωn) = (iωn + εk − µ + µBσ · H)−1

, (19)

and ωn are the Matsubara frequencies.Thus to all orders in H

FH = − 1β

∑iωn

tr∫

dk(iωn − εk + µ + µBσ · H)[(iωn − εk + µ)2 − |H|2]

×(σ · Ax sin kx + σ · Ay sin ky)σ2(iωn + εk − µ − µBσ∗ · H)[(iωn + εk − µ)2 − |H|2]

×(σ∗ · A∗x sin kx + σ∗ · A∗

y sin ky)σ2 . (20)

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 51

Hence,FH = Axχ

xxA∗

x + Axχxy

A∗y + Ayχ

yxA∗

y + Ayχyy

A∗y , (21)

where,

χαβij = − 1

β

∑iωn

∫dk sin ki sin kj (22)

×tr

( (iωn − εk + µ + µBσ · H)σασ2(iωn + εk − µ − µBσ∗ · H)σ∗βσ2

[(iωn − εk + µ)2 − |H|2] [(iωn + εk − µ)2 − |H|2])

.

By x–y symmetry χxy = χyx = 0. Some algebra then leads to

FH = (α0 + α2|H|2)(|Ax|2 + |Ay|2) + iα1H · (Ax × A∗x + Ay × A∗

y)

−2α2(|H · Ax|2 + |H · Ay|2) , (23)

where,

α0 =2β

∑iωn

∫dk

sin2 kx

((εk − µ)2 + ω2

n

)[(iωn − εk + µ)2 − |H|2] [(iωn + εk − µ)2 − |H|2] , (24)

α1 = −4µB

β

∑iωn

∫dk

sin2 kx(εk − µ)[(iωn − εk + µ)2 − |H|2] [(iωn + εk − µ)2 − |H|2] ,(25)

and

α2 = −2µ2B

β

∑iωn

∫dk

sin2 kx

[(iωn − εk + µ)2 − |H|2] [(iωn + εk − µ)2 − |H|2] .(26)

Clearly α0 reduces to α (17) in zero magnetic field, but the α1 and α2 terms donot have an analogue in the zero field Ginzburg–Landau expansion. It is inter-esting to note the similarity of these extra terms to the change in the Hartree–Fock–Gorkov quasiparticle spectrum caused by the magnetic field (See 11). Thecross product of any complex vector with its complex conjugate is purely imag-inary2 so the square root of minus one before the α1 term in the expression forthe free energy is to be expected.

As we have expanded in A but not in H the above expression for the freeenergy is valid for small gaps at all field strengths. It is therefore valid close toHc. But, note that, since we assumed an exchange-only magnetic field we do notconsider the vortex lattice here. Agterberg and Heeb [19,20] have discussed thevortex lattice using Ginzburg–Landau theory, but did not include the Zeemanterms of (23).

In the Ginzburg–Landau formalism the superconducting phase transition oc-curs when the quadratic terms go to zero. In a zero field this condition is simply

α(T − TC) = 0 . (27)2 This can easily be confirmed. Consider the cross product of the most general complex

vector, v = (a+ ib, c+ id, e+ if). It is trivial to show that v ×v∗ = −2i(cf −de, be−af, ad − bc).

52 B.J. Powell, J.F. Annett, and B.L. Gyorffy

In a finite spin only magnetic field the equivalent condition is that the matrix

α = αijAiA∗j (28)

has (at least) one zero eigenvalue, but no negative eigenvalues, where the indicesi and j run over both orbital and spin degrees of freedom. In this case

α =

(β 00 β

). (29)

where,

β =

α0 + α2|H|2 − 2α2H

21 iα1H3 −iα1H2

−iα1H3 α0 + α2|H|2 − 2α2H22 iα1H1

iα1H2 −iα1H1 α0 + α2|H|2 − 2α2H23

.

(30)The condition for there being a zero eigenvalue of α is(

α0 + α2|H|2 − 2α2H21) (

α0 + α2|H|2 − 2α2H21) (

α0 + α2|H|2 − 2α2H21)

− (α0 + α2|H|2 − 2α2H21)α2

1H21(α0 + α2|H|2 − 2α2H

22)α2

1H22

− (α0 + α2|H|2 − 2α2H23)α2

1H22 = 0 . (31)

This expression can be greatly simplified by choosing our coordinate system sothat H lies parallel to one of the axes. With, for example, H = (0, 0, H) we find

β =

α0 + α2H

2 iα1H 0−iα1H α0 + α2H

2 00 0 α0 − α2H

2

. (32)

Which has at least one zero eigenvalue when

(α0 − α2H2)((α0 + α2H

2)2 − α21H

2) = 0 . (33)

The eigenvectors of α are

A1x

A2x

A3x

A1y

A2y

A3y

=

001000

,

000001

,

1iκ0000

,

0001iκ0

,

−iκ10000

and

000

−iκ10

. (34)

Where κ is real. To second order in A, κ is given by

κ = −α0 + α2H2

α1H. (35)

Much recent work (see introduction) has suggested that Sr2RuO4 is likelyto be in an state analogous to the A-phase of 3He. If the pairing interaction

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 53

favours the A-phase in zero magnetic field there are three possible solutions ina magnetic field:

Ax = −iAy = (0, 0, 1) , (36)Ax = −iAy = (1, iκ, 0) , (37)Ax = −iAy = (−iκ, 1, 0) . (38)

Equation 36 is the A-phase with d(k) parallel to H. Equations 37 and 38both give the A2-phase for 0 < |κ| < 1 and the A1-phase for |κ| = 1. Analogymay be drawn to the description of elliptically polarised light in optics [26].One can think of the three A-like phases as being described by an ellipse ofeccentricity

√1 − κ2. The A-phase is the special case of linear polarisation when

the ellipse reduces to a line parallel to d(k). The A1 phase is the special case ofcircularly polarised light a circle which lies in the 1,2-plane. The A2 correspondsto any ellipse between these two extremes. In the A1 and A2-phases by takingthe appropriate superposition of (37) and (38) the major axis of the ellipse canbe made to point in any direction in the plane perpendicular to H.

4 Numerical Results

To progress further we must resort to solving the self consistent Bogoliubov–deGennes equations numerically. To do this we fit the hopping integral and siteenergy to the experimentally determined Fermi surface of the γ-sheet of Sr2RuO4[8]. The interaction potential is restricted to include nearest neighbour terms onlyand chosen to give the experimentally observed critical temperature (1.5 K).

4.1 d(k) Parallel to H

We begin by studying the first solution of the Ginzburg–Landau theory (36), inwhich d(k) is parallel to H. In zero field we find that the ground state of themodel is a triplet state analogous to the A-phase of 3He, specifically the state is

d = ∆0(sin kx + i sin ky)e . (39)

Here we have defined the vector order parameter to point in the e direction.In zero field, all directions in spin space are degenerate if spin–orbit couplingis neglected. When an external field is applied the ground state has d(k) per-pendicular to the field, as we will show below. However, in Sr2RuO4 the orderparameter is thought to be aligned with the c-axis [5], by spin-orbit coupling.Therefore despite the low critical field along the c-axis, Sr2RuO4 presents uswith the possibility of studying a triplet superconductor with a magnetic fieldparallel to the order parameter. It is therefore interesting to predict what wouldbe observed in such experiments. To do this we simply discard any A-phaselike solutions with d(k) not parallel to H. We then consider the remaining selfconsistent solution of the BdG equation with the lowest free energy.

54 B.J. Powell, J.F. Annett, and B.L. Gyorffy

(a) (b)

(c) (d)

Fig. 1. (a)The Fermi surface and the gap at T/TC = 0.5 with (b) µBH/kBTC = 0, (c)µBH/kBTC = 0.5, (d) µBH/kBTC = 0.9

A field applied parallel to d(k) does not cause a change in the symmetry ofthe gap. It follows that at zero temperature the gap is independent of magneticfield strength (see appendix). At finite temperature, a field applied parallel tothe order parameter causes a change in the magnitude of the gap (see Fig. 1). Itshould be noted that the gap is nodeless but has minima at kx = 0 and ky = 0.

We calculate the heat capacity, magnetisation and magnetic susceptibility asfunctions of temperature and field strength. For an isotropic nodeless gap in zerofield it is well known [3] that these properties behave as

Cv, M, χ ∼ exp(− ∆

kBT) . (40)

We find that for an anisotropic, nodeless, p-wave gap the thermodynamicshave the same form, even in the presence of a magnetic field (see inset Fig. 2) Wetherefore define the effective gap, ∆eff ‘seen’ by the thermodynamic functionsas

Cv, M, χ ∼ exp(−∆eff

kBT) . (41)

We find that ∆eff is the mean gap at the Fermi surface, |d(kF )| in zero fieldand that ∆eff is a linear function of magnetic field strength (see Fig. 2). Thatis to say that

∆eff = |d(kF )| − µB |H| . (42)

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 55

0 0.5 1 1.5µBH/kBT

0

0.5

1

∆eff

0 5000 10000 15000 200001/T

−60

−50

−40

−30

−20

−10

ln(C

V)

Fig. 2. ∆eff (normalised to |d(kF )| at T = H = 0) as a function of magnetic fieldparallel to d(k) extrapolated from heat capacity (circles), magnetisation (squares) andmagnetic susceptibility (diamonds). The line is |d(kF )| − µBH. Inset - Logarithmicplot of heat capacity with inverse temperature at various fields. From the bottom up:H=0 T, 0.28 T, 0.42 T, 0.71 T, 0.85 T, 1.13 T, 1.41 T, 1.76 T, 2.12 T, 2.47 T and 2.82 T

4.2 d(k) Perpendicular to H

Recall that the ground state of the model in zero field is

d = ∆0(sin kx + i sin ky)(1, 0, 0) . (43)

(See Fig. 3a). We will now examine the numerical solutions of the full BdGequations corresponding to the second solution of the Ginzburg–Landau theory(37) and(38). In a magnetic field the ground state is when the vector orderparameter points perpendicular to the field. There is also a change in the pairingstate to a phase analogous to the A2-phase of 3He, where

d = ∆0(sin kx + i sin ky)(1, iκ, 0) , (44)

where κ is a real function of temperature and field strength (Fig. 3b,c). Physicallythis corresponds to the majority of the spin 1 Cooper pairs aligning themselvesantiparallel to the magnetic field.

In 3He as the field and temperature increase κ increases until κ = 1. This isthe A1 phase which is the ground state of 3He near to TC in finite fields. TheA1-phase has order parameter

d = ∆0(sin kx + i sin ky)(1, i, 0) (45)

and corresponds to single spin pairing with all of the Cooper pairs aligningthemselves with the magnetic field (Fig. 3d). However, even near TC and in

56 B.J. Powell, J.F. Annett, and B.L. Gyorffy

(a) (b)

(c) (d)

Fig. 3. The Fermi surface (circles), spin up gap (crosses) and the spin down gap(squares). In the (a) A-phase (T = H = 0), (b) the A2-phase (T=0, H=1.4 T), (c)the A2-phase with a larger κ (T = 1.8 K, H = 1.4 T ) and (d) the A1phase - not ob-served

large fields we do not find that the A1-phase is the ground state of our model.If such a transition does occur then it is certainly well above the experimentallyobserved upper critical field. This is in agreement with experiment as no A1-phase has been observed to date.

Due to the nodeless gap in the A2 phase the specific heat has an exponentialtemperature dependence. Hence we can calculate the effective gap for this fieldorientation (Fig. 4). We find a linear field dependence in low fields but its de-pendence is much weaker than for d(k) parallel to H and there is an upturn inlarge fields. There is known to be a qualitative change in heat capacity in thisfield orientation [12]. It remains to be seen if these are related.

5 Conclusions

We investigated the order parameter of Sr2RuO4 in an exchange-only magneticfield. A Ginzburg–Landau symmetry analysis implied three possibilities: eithera 3He A1 or A2 phase with d(k) perpendicular to the magnetic field or a pure3He A phase with d(k) parallel to the magnetic field. We explored the exchangefield dependence of the order parameter and energy gap in a one-band model of

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 57

0 0.5 1 1.5µBH/kBTC

0

0.2

0.4

0.6

0.8

1

∆ eff

Fig. 4. ∆eff (normalised to |d(kF )|(T = H = 0)) as a function of H perpendicular tod(k) (solid line) extrapolated from heat capacity. For comparison we plot ∆eff for Hparallel to d(k) (dashed line)

Sr2RuO4. The numerical solutions showed no A1 phase for physically reasonablefield strengths and that of the two remaining phases the A2 phase is lower in freeenergy. We did not include the effect of spin-orbit coupling which could changethe ground state for particular orientations of the magnetic field (particularlywith H parallel to the c-axis of the crystal). We investigated the behaviour ofthe heat capacity as a function of both field and temperature for both of thesesolutions. We have shown that the variation of the exponential cutoff below TC asa function of H is quantitatively and qualitatively different for these two phases.This makes heat capacity an excellent experimental probe of the symmetry statein a magnetic field.

Acknowledgments

We acknowledge support of BJP from an EPSRC studentship.

58 B.J. Powell, J.F. Annett, and B.L. Gyorffy

Appendix

For an A-phase triplet superconductor with H parallel to d(k) and z the BdGequations are

εk + µBH 0 0 d3(k)0 εk − µBH d3(k) 00 d∗

3(k) −ε−k − µBH 0d∗3(k) 0 0 −ε−k + µBH

u↑σ(k)u↓σ(k)v↑σ(k)v↓σ(k)

= Eσ(k)

u↑σ(k)u↓σ(k)v↑σ(k)v↓σ(k)

.(46)

Hence, the eigenvalues are

Eσ = E0(k) + σµBH , (47)

whereE0 =

√εk + |d3(k)|2 (48)

is the spectrum in zero field. The eigenvectors are

uσσ(k) =d3(k)√

(E0(k) − εk)2 + |d3(k)|2 , (49)

and

vσ−σ(k) =E0(k) − εk√

(E0(k) − εk)2 + |d3(k)|2 . (50)

Substituting these into the self-consistency condition (6) we find that the gapequation is

d3(k) =12

∑kσ

Uσ−σ(k)d3(k)E0(k)

tanh(E0(k) + σµBH

2kBT) . (51)

At T = 0 this becomes

d3(k) =12

∑kσ

Uσ−σ(k)d3(k)E0(k)

. (52)

which is independent of H.

References

1. D.M. Lee, R.C. Richardson: ‘Superfluid 3He’. In: The Physics of Liquid and SolidHelium Part II ed. by K.H Bennemann, J.B Ketterson (Wiley, New York 1978)

2. D. Vollhardt, P. Wolfe Superfluid Phases of Helium 3 (Taylor and Francis, London1999)

The Behaviour of a Triplet Superconductor in a Spin Only Magnetic Field 59

3. V.P. Mineev, K.V. Samokhin: Introduction to Unconventional Superconductivity(Gordan and Breach, Amsterdam 1999)

4. P. Brussaard, H.W. Capel: Physica A 265, 370 (1999)5. Y. Maeno, T.M. Rice, M. Sigrist: Physics Today 54, 42 (2001)6. K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori, Y. Maeno:

Nature 396, 658 (1998)7. J.A. Duffy, S.M. Hayden, Y. Maeno, Z. Mao, J. Kulda, G.J. McIntyre: Phys. Rev.

Lett. 85, 5412 (2000)8. C. Bergemann, S.R. Julian, A.P. Mackenzie, S. NishiZaki, Y. Maeno: Phys. Rev.

Lett. 84, 2662 (2000)9. D.F. Agterberg, T.M. Rice, M. Sigrist: Phys. Rev. Lett. 73, 3374 (1997)

10. J.F. Annett: Adv. in Phys. 39, 83 (1990)11. G.M. Luke, Y. Fudamoto, K.M. Kojima, M.I. Larkin, J. Merrin, B. Nachumi,

Y.J. Uemura, Y. Maeno, Z.Q. Mao, Y. Mori, H. Nakamura, M. Sigrist: Nature394, 558 (1998)

12. S. NishiZaki, Y. Maeno, Z. Mao: J. Phys. Soc. Japan 69, 572 (2000)13. I. Bonalde, B.D. Yanoff, D.J. Van Harlingen, M.B. Salamon, Y. Maeno: Phys. Rev.

Lett. 85, 4775 (2000)14. K. Izawa, H. Takahashi, H. Yamaguchi, Y. Matsuda, M. Suzuki, T. Sasaki,

T. Fukase, Y. Yoshida, R. Settai, Y. Onuki: Phys. Rev. Lett. 86, 2653 (2001)15. M.E. Zhitomirsky, T.M. Rice: Phys. Rev. Lett. 8705, 7001 (2001)16. G. Litak, J.F. Annett, B.L. Gyoffy, K.I. Wysokinski: cond-mat/010537617. T.M. Riseman, P.G. Kealey, E.M. Forgan, A.P. Mackenzie, L.M. Garvin,

A.W. Tyler, S.L. Lee, C. Ager, D.McK. Paul, C.M. Aegerter, R. Cubitt, Z.Q. Mao,T. Akima, Y. Maeno: Nature 396, 242 (1998)

18. P.G. Kealey, T.M. Riseman, E.M. Forgan, L.M. Galvin, A.P. Mackenzie, S.L. Lee,D.McK. Paul, R. Cubitt, D.F. Agterberg, R. Heeb, Z.Q. Mao, Y. Maeno: Phys.Rev. Lett. 84, 6094 (2000)

19. D.F. Agterberg: Phys. Rev. Lett. 80, 5184 (1998)20. R. Heeb, D.F. Agterberg: Phys. Rev. B 59, 7076 (1999)21. J.A. Sauls: Adv. Phys. 43, 114 (1994)22. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz,

F. Lichtenberg: Nature 372, 532 (1994)23. I. Mazin, D.J. Singh: Phys. Rev. Lett. 79, 733 (1997)24. C. Pfleiderer, M. Uhlarz, S.M. Hayden, R. Vollmer, H.v. Lohneysen, N.R. Bern-

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Coexistence of Spin-Triplet Superconductivityand FerromagnetismInduced by the Hund’s Rule Exchange

J. Spalek1, P. Wrobel1, and W. Wojcik2

1 Marian Smoluchowski Institute of Physics, Jagellonian University,ulica Reymonta 4, 30-059 Krakow, Poland

2 Institute of Physics, Tadeusz Kosciuszko Technical University,ulica Podchorazych 1, 30-084 Krakow, Poland

Abstract. We discuss general implications of the local spin-triplet pairing amongcorrelated fermions that is induced by the Hund’s rule coupling in orbitally degeneratesystems. The quasiparticle energies, the magnetic moment, and the superconductinggap are determined for the principal superconducting phases, in the situation withthe exchange field induced by both the local Coulomb and the Hund’s rule exchangeinteractions. The phase diagram, as well as the evolution in an applied magnetic fieldof the spin-triplet paired states near the Stoner threshold, is provided for a modeltwo-band system. The appearance of the spin-polarized superconducting phase makesthe Stoner threshold a hidden critical point, since the pairing creates a small butdetectable uniform magnetization. The stability of the superconducting state againstthe ferromagnetism with an alternant orbital ordering appearing in the strong-couplinglimit is also discussed.

1 Introduction

The discovery of superconductivity in Sr2RuO4 [1], and particularly of its co-existence with ferromagnetism in UGe2 [2], ZrZn2 [3], and URhGe [4] showedclearly that the long awaited spin-triplet superconducting state is realized innature. The above three systems have a weak and an itinerant nature of theelectrons involved in both ferromagnetism and the pairing. Therefore, the mod-els of correlated electrons generalized to orbitally degenerate systems (such asthe degenerate Hubbard model) should be a starting point in theoretical con-siderations of the phases involved, since they are certainly applicable to the de-scription of the itinerant magnetism. Furthermore, they should also be regardedas a providing proper pairing mechanism for those systems, since the supercon-ductivity disappears at high applied pressure together with ferromagnetism andhence, it is unlikely that it is caused by a nonmagnetic mechanism (e.g. by theelectron-phonon coupling), which should not be influenced by the presence (orabsence) of ferromagnetism to such an extent. Moreover, the superconductivityappears together with or inside the ferromagnetic phase only when magnetismis rather weak (magnetic moment is small), i.e. when the system is susceptibleto a local exchange-field enhancement (by formation of a pair bound state inspin-triplet state). In other words, the spin-triplet pairing should be enhanced

C. Noce et al. (Eds.): LNP 603, pp. 60–75, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 61

in the vicinity of the Stoner critical point provided that the quantum spin fluc-tuations do not introduce too strong scattering of the individual carriers. In thepresent article, we extend our original treatment [5,6] of spin-triplet supercon-ductivity and discuss its coexistence with the itinerant ferromagnetism within asingle mechanism responsible for the appearance of both of them - the Hund’srule ferromagnetic exchange among correlated and orbitally degenerate d statesof narrow-band electrons. The structure of the paper is as follows. In the nexttwo Sections we build a formal structure of the theoretical approach. Namely, weintroduce the concept of the real-space spin-triplet pairing, as well as generalizethe Nambu-Bogoliubov-de Gennes formalism to the situation with spin-tripletpairing. In Sect. 4 we discuss the spin-triplet pairing below the Stoner threshold.The most important message there is that the Stoner critical point is actuallya hidden critical point in the spin-triplet paired state. In Sect. 5 we presentanalytic estimates of the superconducting gaps and critical temperatures in theweakly ferromagnetic states. Finally, in Sect. 6 we provide the phase diagramin the strong correlation limit and, in particular, discuss the orbital ordering aswell.

2 Real-Space PairingInduced by the Local Ferromagnetic Exchange

We start from an extended Hubbard model of correlated and orbitally degeneratenarrow-band electrons represented by the parametrized Hamiltonian

H =∑

ijll′σ

′′ tll′

ij a†ilσajl′σ + U

∑il

nil↑nil↓ +12

U ′ ∑ill′σσ′

′ nilσnil′σ′

− J∑ill′

′(Sil · Sil′ +

34

nilnil′

)+ J

∑ill′

′ a†il↑a

†il↓ail′↓ail′↑ . (1)

In this Hamiltonian the first term describes the electron hopping between theatomic sites i and j and between the orbitals l and l′; the double primed sum-mation means that both l = l′ and i = j. The next two terms describe the directCoulomb interactions, intra- and inter-orbital terms, respectively. The last twoterms represent the Hund’s rule ferromagnetic exchange and the pair hopping,respectively. In what follows we are interested in the spin-triplet correlations andpairing, so the first task is to construct an effective Hamiltonian with pairingrenormalized by the Coulomb interactions U (the third and the last terms playonly a minor role, at least in the weak-coupling regime). This procedure has beencarried out earlier [5] for the two-band case within the auxiliary (slave) bosonformalism in the saddle-point approximation. However, to introduce the startingeffective Hamiltonian in an explicit form in a weak-coupling limit we introducereal-space spin-triplet pairing operators via the following relations

A†

i1ll′ = a†il↑a

†il′↑ for Sz

l + Szl′ ≡ m = 1

A†i0ll′ = 1√

2

(a†

il↑a†il′↓ + a†

il↓a†il′↑)

for m = 0

A†i−1ll′ = a†

il↓a†il′↓ for m = −1

(2)

62 J. Spalek, P. Wrobel, and W. Wojcik

and neglect as irrelevant the third and the fifth term in (1). In effect, we obtain

H =∑ill′σ

′ tll′

ij a†ilσail′σ + U

∑il

nil↑nil↓ − J∑imll′

A†imll′Aimll′ . (3)

The first term provides the hybridized band states, the second the repulsion be-tween electrons on the same orbital but with opposite spins (the Hubbard term),and the third introduces local spin-triplet correlations for electrons located onthe orbitals l and l′ (l = l′).

The simplest solution of the model is to make the Hartree-Fock approxima-tion. We will study the solution for which the ferromagnetic moment 〈Sz

l 〉 =〈nil↑ − nil↓〉/2, and the anomalous superconducting averages ∆ll′m ≡ 2J(d −1)〈A†

imll′〉 are nonzero and represent a stable solution. We assume additionallythat the bands are equivalent, i.e. put 〈Sz

l 〉 = Sz and ∆ll′m = ∆m. Such aprocedure leads to the Hartree-Fock Hamiltonian of the form

H =∑

ijll′σ

′′ tll′

ij a†ilσajl′σ − 4J(d − 1)Sz

∑il

Szil + Jd(d − 1)NSz2

−∑

iml =l′

(∆mA†

imll′ + ∆∗mAimll′

)+ Jd(d − 1)|∆m|2N

− 2USz∑il

Szil + UdNSz2

, (4)

where d is the orbital degeneracy and N is the number of atomic sites. Equiva-lently, we can write

H =∑

ijll′σ

′′ tll′

ij a†ilσajl′σ − 2[U + 2J(d − 1)]Sz

∑il

Szil (5)

−∑

iml =l′

(∆mA†

imll′ + ∆∗mAimll′

)+ [U + J(d − 1)]Sz2 +

|∆m|22J(d − 1)

Nd .

We see that the quantity I = 2[U + 2J(d− 1)] is the magnetic coupling constantand the coupling constant for spin-triplet pairing is J(d − 1). We have a ferro-magnetism coexisting with a spin-triplet paired phase if both Sz and at leastone of the gap parameters ∆m (m = +1, 0, −1) are nonzero simultaneously forthe energetically stable solution. In what follows we provide the solution of theHamiltonian (6) in a model two-band situation, i.e. neglect the hybridization ofthe bands (put t12 = 0). As long as the d-fold degenerate bands are regarded asalmost equivalent, such two-band model should catch the essential qualitativefeatures of the solutions.

3 Spin-Triplet Superconducting State

In the absence of spin-triplet superconductivity a two-band system is param-agnetic below the Stoner threshold, i.e. when (εF )I < 1, where (εF ) is the

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 63

density of states at the Fermi energy εF . If we have a degenerate two-band sys-tem with flat density of states and of bandwidth W each, then the conditiontakes the form 2I/W < 1.

To solve the system of self-consistent equations for ∆m, Sz and the chemicalpotential µ we have generalized [5,6,7] the Nambu-Bogoliubov-de Gennes nota-tion and have constructed a 4 × 4 matrix representation of the Hamiltonian bydefining composite creation operators of the form f†

k = (f†k1↑, f

†k1↓, f−k2↑, f−k2↓),

and annihilation operators as fk = (f†k)†. We have

H =∑

k

f†k Afk +

∑k

Ek2 + I(Sz)2N +|∆m|2

J(d − 1)N , (6)

with

A =

Ek1 − ISz, 0, ∆1, ∆00, Ek1 + ISz, ∆0, ∆−1

∆1, ∆0, −Ek2 + ISz, 0∆0, ∆−1, 0, −Ek2 − ISz

. (7)

The quantities Ek1 ≡ Ek1 − µ and Ek2 ≡ Ek2 − µ are the band energies (withthe chemical potential as a reference point). This matrix can be brought to adiagonal form analytically for the case of interest for us here with ∆0 = 0 (thephase with ∆0 = 0 is almost always energetically unstable). In such a situationwe obtain the following four eigenvalues

λkσ1,2 =12

(Ek1 − Ek2) ∓[

14(Ek1 + Ek2 − σISz

)2 + |∆σ|2]1/2

, (8)

where the sign (∓) corresponds to the label (1, 2) of the eigenvalues λkσ1,2.The quasiparticle spectrum separates into a pair of spin subbands with thespin splitting δ ≡ λk↓i − λk↑i, determined mainly by the exchange field, sinceI is substantially larger than J . The spectrum is fully gapped if both ∆1 ≡∆↑↑ and ∆−1 ≡ ∆↓↓ are nonzero; this phase is called, in analogy to superfluidhelium, as the anisotropic A phase (in general, ∆↑↑ = ∆↓↓ in the ferromagneticphase). However, if only one component (∆↑↑) of the gap is nonzero, then λk↓1 =−Ek2 + ISz and λk↓2 = Ek1 + ISz. This means that the minority spin spectrumis ungapped and will thus produce a nonzero linear specific heat γ↓T , withγ↓ ∼ ↓(εF ) if the bands are symmetric with respect to their middle point (i.e.respect the electron-hole symmetry). The phase with ∆↑↑ = 0, ∆↓↓ = 0 will becalled the A1 phase (we take here a convention that in the ferromagnetic phasethe magnetic moment Sz > 0 and ∆↑↑ = 0; a physically equivalent but distinctstate is that with (−Sz) and ∆↓↓ = 0). Note also that the appearance of the A1phase does not necessarily mean that we are in the ferromagnetically saturatedphase, i.e. with 〈Sz〉 = n/4, where n is the band filling, defining here as thenumber of electrons per site.

One should also mention that in an applied magnetic field B = 0, all theabove results are valid except that we have to make the replacement ISz →ISz + µBB, where µB is the Bohr magneton.

64 J. Spalek, P. Wrobel, and W. Wojcik

The Bogoliubov quasiparticle operators can also be calculated [6] when di-agonalizing (6). In this paper we provide the explicit results only for the caseEk1 = Ek2 = Ek, since they have a simple interpretation. Namely, under thiscondition the quasiparticle energies take the usual form

λkσ1,2 = ± (E2

kσ + |∆σ|2)1/2 ≡ ±λkσ , (9)

with Ekσ = Ek − σ(µBB + ISz). Also, the quasiparticle operators α and βcorresponding respectively to the eigenvalues +λk and −λk are given by

(αkσ

β†−kσ

)=

1√2

(u

(σ)k , v

(σ)k

−v(σ)k , u

(σ)k

)(fk1σ + f†

−k2σ

fk1σ − f†−k2σ

), (10)

with the coherence factors(u

(σ)k

v(σ)k

)=

1√2

(1 + ∆σ

λkσ

1 − ∆σ

λkσ

). (11)

We see that the equations for the quasiparticles in the spin subband σ have inthis limit the same form as in the BCS case. In other words, we have two gapsin an anisotropic A phase in the system and they are induced by the presence ofthe molecular field (when B = 0). Hence in the paramagnetic state (Sz = 0) weshould have an isotropic A phase (∆↑↑ = ∆↓↓ = ∆ = 0, ∆0 = 0) as the stablephase. We shall see that this is not always the case, i.e. the superconductingpairing may produce a nonzero spin polarization even below the Stoner threshold,as we discuss next.

4 Spin-Triplet Paired State Below the Stoner Threshold:Phase Diagram and a Hidden Critical Point

One should note that for d-electron systems J is of the order of 0.1 − 0.3 U .In the numerical calculations we therefore take J = 0.25 I. If we take also thedensity of states of the single band as (1/W ) we have that the Stoner thresholdfor the onset of ferromagnetism is reached when J/W = 0.125. In Fig. 1 wehave plotted the ground-state energy (in units of W ) for the A, A1 and normal(∆m ≡ 0) states as a function of the applied magnetic field B (all the energiesand parameters are expressed in units of W ). The energy difference between theA and A1 phases is small and for an applied field of the order of µBB = 5.10−4 Wthe A → A1 transition takes place (for W = 1 eV this critical field is ∼ 50 T). Forthis applied field magnitude the gap anisotropy is ∆↑↑/∆↓↓ ∼ 3, as displayed inthe lower panel in Fig. 2 (note that the ∆↑↑(B) dependence is almost the samein both A and A1 states).

In Fig. 2 we display also the value of the magnetic moment per orbital 〈Szl 〉

and (in the inset) the field dependence of the chemical potential in both A and A1paired states. Again, the magnetic moment in the A1 state (dashed line) follows

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 65

0 0.0005 0.001 0.0015 0.002

APPLIED FILED, µBB / W

-0.127

-0.1265

-0.126

-0.1255

-0.125

EN

ER

GY

, E

/ W

A phaseA1 phasenormal state

n = 1J = 0.25 IJ/W = 0.12

Fig. 1. Phase diagram involving the spin-triplet superconducting A and A1 states in anapplied magnetic field B for a two-band model at quarter filling and with a constantdensity of states. The dot marks the transition from the anisotropic A phase (with∆1 ≡ ∆↑↑ > ∆−1 ≡ ∆↓↓) to the A1 phase (∆↓↓ = 0)

0 0.002 0.004 0.006 0.008 0.01

APPLIED FIELD, µBB/W

0

0.005

0.01

0.015

GA

PS, ∆

/W

0

0.05

0.1

0.15

0.2

0.25

MA

GN

ET

IC M

OM

EN

T, <

Slz >

0 0.005 0.01

-0.2501

-0.25005

-0.25

A1A

A1

A

A1

A

∆↓↓

∆↑↑

J/W = 0.12

n = 1

µ(B)

SATURATIONJ = 0.25 I

Fig. 2. Upper panel: Magnetic moment 〈Szl 〉 ≡ Sz in the field for the same situation

as in Fig. 1. Lower panel: Field dependence of the superconducting gaps as marked.Inset: Field dependence of the chemical potential in the A and A1 phases

66 J. Spalek, P. Wrobel, and W. Wojcik

0.1 0.11 0.12 0.13

J/W

0

0.02

0.04

0.06

0.08

MA

GN

ET

IC M

OM

EN

T, <

Slz >

A phaseA1 phase

0 0.05 0.1 0.15 0.2

0

0.1

0.2

n = 1

Stoner Criterion

J = 0.25 IB = 0

PM

saturated

FM

normal state

Fig. 3. Magnetic moment 〈Szl 〉 induced by the spin-triplet pairing below the Stoner

threshold (marked as Stoner criterion, J/W ≈ 0.125). The effect is spectacular whenthe system approaches the critical point. Inset: magnetic moment vs. J/W if the spin-triplet pairing were absent (the paramagnetic (PM) to ferromagnetic (FM) transition isdiscontinuous at the Stoner point for the constant density of states selected to illustratethe hiding of the Stoner critical point in the paired state)

essentially the same straight-line dependence Sz(B) for both paired states. Inthis sense, magnetic properties are not influenced much by the pairing. In viewof this last feature of the solution, it is not strange that the A1 phase is stableeven though the system is not yet magnetically saturated.

One very interesting feature of our mean-field approach should be mentioned.Namely, the dependence of Sz on B in the paired state does not approach ex-actly the value Sz = 0 for B = 0, even though the system is below the Stonerthreshold. The effect is small to become visible in the lower left-hand corner ofthe upper panel, but it is certainly well above the numerical accuracy of theresults. To test our conjecture that the pairing itself may introduce a uniformferromagnetic polarization, we have calculated this residual value of the spinmagnetic moment in the field B = 0 when approaching the Stoner critical point.The result is displayed in Fig. 3. We observe a beautiful critical dependence ofthe moment as we approach the Stoner point. So, indeed, the pairing washes outthe critical Stoner point, i.e. makes it a hidden point. It is interesting to ask towhat extent the quantum critical fluctuations can change this mean-field result.The result also means that the superconducting coherence length becomes infi-nite at the Stoner point. It remains to be seen whether it is unbound wheneverthe A1 phase sets in.

The results displayed in Fig. 3 contain also one additional feature exhibited inthe inset. Namely, the inset shows that if no pairing were present, then the mean-field para-ferro-magnetic transition would be discontinuous (for the assumedconstant density of states) and directly to the saturated state. The pairing smears

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 67

0.05 0.075 0.1 0.125

J / W

0

0.01

0.02

0.03

0.04

MA

GN

ET

IC F

IEL

D

SATURATION

A PHASE

A1 PHASE

STONER CRITERION

n = 1

J = 0.25 I

Fig. 4. Critical magnetic field for the A→A1 transition (solid line) and the field satu-rating the moment (i.e. making Sz = 1/4). The Stoner threshold is also marked. Notethat the threshold for the appearance of the polarized paired (A1) state does not coin-cide with the onset of the saturated ferromagnetic (SF) state. The A phase disappearsat the Stoner threshold

out this discontinuity and, therefore, we have an extended critical regime forJ/W → 0.125. Additionally, because of the absence of the critical point for theJ-dependence of Sz, it is difficult to say where ferromagnetism disappears as afunction of e.g. pressure. This is exactly what is actually observed for the newlydiscovered superconducting ferromagnets [2,3].

The fact that the spin-triplet pairing can induce a weak ferromagnetic or-dering must mean that the coherence length ξ of the paired states is larger thanthe classical distance (V/N)1/3 between the electrons in this system of volume Vcontaining N electrons. The overlap between the Cooper pairs effectively inducesa spin-spin interaction, which can be understood in the following manner. Thesuperconducting gap creates an effective magnetic field Hjm = χji ∆mi, whichin turn induces a magnetic moment Mi ∼ χji∆mi (χji is the superconduct-ing susceptibility) and, in turn, a negative contribution to the magnetic energy∼ (Sz)2.

Of particular interest is the stability of the paired states when approachingthe Stoner threshold from the paramagnetic side. The border line between the Aand A1 phases is drawn as a solid line in Fig. 4. The A phase disappears exactlyat the Stoner point, but the A1 survives. The reason why only the A1 phase cansurvive at the critical point is very simple. Namely, the magnetic susceptibilityis infinite at this point, so even a weak field induces total polarization. However,strictly speaking, the A1 phase should not be the stable state since in the mag-netically saturated state there is no way we can increase the polarization locally

68 J. Spalek, P. Wrobel, and W. Wojcik

to form Cooper pairs as proper bound states. This is the reason to assume thatthe superconducting coherence length becomes infinite at the Stoner point.

5 Spin-Triplet State in a Weak Ferromagnetic State:Analytic Estimates

We discuss now the situation for a weak itinerant (Stoner-Wohlfarth) ferromag-net, i.e. the system for which (εF )I is above but close to unity. In the mean-fieldapproximation, the equation for the magnetic moment m = 2Sz is determinedfrom a Landau-type expansion, which is obtained from the low-temperature ex-pansion [8]

m = I(εF )m +I3

24

[3′(εF )2

(εF )− ′′(εF )

]m3 = 0 , (12)

where ′ and ′′ are the derivatives of (ε) taken at ε = εF . The nonzero solutionis thus of the form

Sz =12

[I(εF ) − 1

B

]1/2

, (13)

with

B = I3 (εF )8

[′(εF )2

(εF )2− ′′(εF )

(εF )

]. (14)

In a similar fashion, one obtain the following expression for the Curie tempera-ture

TC =√

6π

[′(εF )2

(εF )2− ′′(εF )

(εF )

]−1/2 [I(εF ) − 1

I(εF )

]1/2

. (15)

Obviously, TC express the critical temperature for ferro- to para-magnetic phasetransition. The magnetization diminishes with temperature in the low-temperatureregime according proportionally to T 2 as observed in URhGe [4]. This provesdirectly that these systems are weak itinerant ferromagnets (the standard con-tribution due to the spin wave excitations is ∼ T (2n+1)/2, with n = 1, 2, ...).

In order to estimate the value of the superconducting gap in the A phase, weuse the corresponding BCS equation, which for the simplest case with Ek1 = Ek2reads

1 =J

N

∑k

12λkσ

tanh(

λkσ

2kBT

). (16)

To estimate the gap ∆σ at T = 0 and the temperature TS of the transition tothe superconducting state, we assume that the dispersion relation in the spinsubbands σ is linear, i.e. Ekσ − µ vσk, where vσ is the Fermi velocity inthat subband. Then, making the usual BCS-type approximation we obtain theequation for ∆σ at T = 0

1 = Jσ

∫ kmσ

−kmσ

d3(vσk)√(vσk)2 + ∆2

σ

, (17)

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 69

where the density of states at the Fermi level is σ = 12 ε2Fσ/W , εFσ being the

Fermi energy for the quasiparticles in the σ-subband. We perform the integrationonly within the spin-split region of the bands, which extends from −ISz to +ISz.Therefore, the border wave vector is determined from the relation ±vσkm = ISz

(“ + ” for electrons, “ − ” for holes).As a result, the zero-temperature gap ∆σ takes the form

∆σ = ISz exp(

− 1σJ

). (18)

This gap vanishes identically if we reach the Stoner point, at which Sz = 0.Likewise, the estimate of TS is determined from the equation

1 = J

∫ kmin

−kmin

d3k1

vF k − ISztanh

(vF k − 2ISz

2kBTS

), (19)

where we have taken vσ vF , with vF being the Fermi velocity in the param-agnetic phase, and kmin is determined from the equation vF kmin = ISz. As aresult, we obtain

TS 2.26 I Sz exp(

− 1J

), (20)

where (↑ + ↓)/2. From these expressions we can estimate the gap ratio

∆↑∆↓

∼ exp[(

2I

3J

)′(εF )(εF )2

Sz

]. (21)

A flat density of states favors isotropic A-phase solution for B = 0 (cf. Sec. 4),whereas for the steep density of states near εF we observe a strong anisotropy.Additionally, the ratio increases exponentially with increasing the magnetic mo-ment. Also, formulae (15) and (20) allow for a determination of the TS/TC ratioexplicitly.

One can notice immediately that this ratio must be small for weak itinerantmagnet, for which I ≈ 1, and then J ∼ 0.2. So, the two critical temperaturescan differ by two orders of magnitude easily (the actual ratio in the newly dis-covered superconducting ferromagnets [2,4] is above 30 in the applied-pressureregime placing the systems not too close to the Stoner threshold).

To summarize, we have three energy scales in the system: (i) the eV energyscale of U and J , (ii) the scale of the exchange splitting (2ISz) and, associatedwith it, the value of TC , and (iii) the magnitude of the superconducting gaps(∆↑↑, ∆↓↓) and, associated with it, the value of TS . Both the weak itinerant-electron ferromagnetism and the spin-triplet superconductivity are induced bya single mechanism - the local ferromagnetic exchange interaction. The pairingis induced by the exchange interaction itself. In this respect, our mechanismbelongs to the same class of models as the t–J model [9], for which the spin-singlet pairing in that case is induced by the kinetic exchange (superexchange).

The appearance of the exchange-induced paired state is here as natural asthe presence of ferromagnetism above the Stoner threshold; the phase of super-conducting ferromagnet is energetically more favorable than either the purely

70 J. Spalek, P. Wrobel, and W. Wojcik

ferromagnetic or the spin-triplet superconducting states. The effect of the ex-change field is stronger than that coming from the spin fluctuations [10]. In away, our approach represents the simplest approach to the spin-triplet supercon-ductivity, in a complete parity with the Stoner theory of ferromagnetism.

6 Spin-Triplet Pairing for Strongly Correlated Electrons:Role of Ferromagnetic Superexchangeand Orbital Ordering

The approach in the preceding Sections is based on the notion that electrons ina narrow band are weakly correlated (i.e. placed physically close to the Stonerboundary for ferromagnetism). The question arises what would happen if theparticles were strongly correlated? This question is important because the answerin the affirmative would provide a strong indication that the mechanism is quiteuniversal and hence, is relevant in the most interesting (and difficult) regime ofintermediate correlations (U ∼ W ).

The simplest situation arises for the quarter-filled doubly degenerate band,for which we have a ferromagnetic insulator with orbital ordering [7]. We haveapplied the same type of formalism to the case with filling around the quarterfilling and have obtained the following Hamiltonian, with the help of which onecan study the spin-triplet pairing

H =∑ijlσ

′ tijb†ilσbjlσ − 2

K − J

∑ijk

1∑m=−1

tijtjkB†ijmBjkm . (22)

Here b†ilσ and bjlσ represent the so-called projected creation and annihilation

operators, e.g.

b†i1σ = a†

i1σ(1 − ni1σ)(1 − ni2σ)(1 − ni2σ), etc. (23)

The pairing operators are

B†ij1 = b†

i1↑b†j2↑ for m = 1

B†ij0 = 1√

2

(b†i1↑b

†j2↓ + b†

i1↓b†j2↑)

for m = 0

B†ij−1 = b†

i1↓b†j2↓ for m = −1 .

(24)

In other words, the local triplet-pair creation operators are composed of pro-jected creation operators, one taken for the site i, the other for the neighbor-ing site j. Thus, the Hamiltonian (22) has a form similar to that for the t–Jmodel [11] except that we have here the triplet and interband pairing. This ef-fective pairing Hamiltonian should be compared with the standard form of thetwo-band Hamiltonian expressed through the spin and pseudospin operators,

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 71

given respectively by

Si =12

∑lσσ′

b†ilσ τσσ′ bilσ′ (25)

Ti =12

∑σll′

b†ilσ τ ll′ bil′σ (26)

(ταξξ′ are the elements of the Pauli matrix τα, α = 1, 2, 3). The effective Hamil-

tonian has then the form

H =∑ijlσ

′ tijb†ilσbjlσ − 2

K − J

∑〈ij〉

t2ij

(Si · Sj +

34ninj

)(14ninj − Ti · Tj

)+ ... ,

(27)where the antiferromagnetic kinetic exchange part has not been included. Thisform is equivalent to the form (22). In effect, the multiband model in the strong-correlation limit may exhibit spin-triplet pairing (with 〈B†

ijm〉 = 0), spin ordering(with 〈Sz

il〉 = 0), and orbital ordering (with 〈T zl 〉 = 0), or two of the orderings

simultaneously.One can solve the ferromagnetic t-J model (22). This has been performed [12]

within the slave-boson formalism [13] in which the projected fermion operatorsare decomposed into the pseudofermion operator f and the boson operator b,namely b†

ilσ = f†ilσbi. In effect, the effective Hamiltonian takes the form

H =∑ijlσ

′ tijf†ilσfjlσb†

jbi − 2t2

K − J

∑〈ij〉〈jk〉

F †ijmFjkm

+∑

i

λi

(b†i bi +

∑lσ

f†ilσfilσ − 1

)− µ

(∑ilσ

f†ilσfilσ − Ne

), (28)

where λi is the constant expressing the constraint

b†i bi +

∑lσ

f†ilσfilσ = 1 (29)

and the F †ijm operators have the same form as B†

ijm with b†ilσ being replaced by

f†ilσ.

When constructing the phase diagram for the system described by the effec-tive Hamiltonian (28), we have to consider also the possibility of the appearanceof ferromagnetism which coexists with the alternant (antiferromagnetic) order-ing (AFO). In Fig. 5 we have plotted a simpler version of the phase diagram,on which we have marked only the saturated ferromagnetic phase (SF), the su-perconducting phase (S), the paramagnetic metallic (PM) and the paramagneticsuperconducting (PS) phases, as well as the coexisting phases AFO-SF and S-SF.This phase diagram is for a quarter-filled, doubly degenerate band (n = 1, d = 2),with a constant density of states. The superconductivity is stable for low val-ues of U and a rather strong Hund’s rule coupling J . All the lines determining

72 J. Spalek, P. Wrobel, and W. Wojcik

Fig. 5. Magnetic phase diagram on the plane U − J for a quarter-filled doubly degen-erate band. The symbols label the corresponding phases: PM - paramagnetic metallic,AFO-SF - antiferromagnetic orbital ordering of saturated ferromagnet, PS - paramag-netic spin-triplet superconductors, S-SF - A1 superconducting phase coexisting withsaturated ferromagnetism. All the lines mark discontinuous phase transitions at tem-perature T = 0

Fig. 6. The evolution of the AFO-SF and SF states with increasing Hund’s rule cou-pling J (the phase labelling is the same as in Fig. 5), plotted as a function of the bandfilling n =

∑lσ〈nilσ〉, for a doubly-degenerate band with a constant density of states

phase border lines mark first-order phase transition lines. The AFO-SF state isa Mott insulating state, whereas the remaining phases are metallic. Hence thetransition from the AFO-SF to the S-SF phase is also an insulator to metaltransition. The system is ferromagnetic on both sides of the transition and this

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 73

Fig. 7. Upper panel: Phase diagram for n → 1 for a quasi-two-dimensional modelinvolving spin-triplet pairing with a q-dependent gap obtained in the strong-correlationregime. TD represents the temperature of the Bose condensation of auxiliary bosons,TRVB the formation of a BCS-like state (the RVB state) for pseudofermions. Lowerpanel: The condensation temperatures in a wide range of the band filling

transition is possible only in orbitally degenerate systems with non-half-filledband configuration. This type of transition complements the standard canonicalMott transition, which takes place from the antiferromagnetic insulator to eitherantiferromagnetic or paramagnetic metal.

In order to see the regimes of stability of the AFO and the SF states (andtheir coexistence) with respect to the spin-triplet superconducting state as afunction of the band filling, we have plotted in Fig. 6 the phase diagram in-volving the magnetic and orbitally ordered phases. The dashed lines representa continuous phase transformation. The antiferromagnetic orbital ordering isstable only within 15% filling difference from the quarter filling. The nature ofthe transition evolves with increasing the ratio J/U . The slave boson approachpresented in [5] and [7] has been used to obtain results valid for arbitrary U andJ . One should mention that the metallic AFO state is not in conflict with thespin-triplet superconducting state. This question requires a separate discussion.

In Fig. 7 we have shown (upper panel) the regimes of the existence of vari-ous superconducting states (explained below) and the temperature TRVB belowwhich the gap parameters 〈f†

i1σfj2σ′〉 are nonzero, as well the temperature TD

below which the slave bosons condense. Since the physical superconducting gapis ∼ 〈B†

ijm〉 ∼ 〈F †ijm〉〈bibj〉, a nonzero critical temperature for superconductivity

is realized for 0.1 n < 1. Note that in this Figure the exchange integral is

74 J. Spalek, P. Wrobel, and W. Wojcik

defined as J ≡ 2t2/(K − JH), where JH is the Hund’s rule coupling (taken as Jin all preceding discussion). So, a pure superconducting phase should appear inthe regime, in which both AFO and SF states are absent and TRVB = 0, i.e. for0.60 n 0.85.

The superconducting phases specified in the upper panel of Fig. 7 are definedthrough the wave-vector (q) dependent two-dimensional representation of thesuperconducting gap ∆q as follows

∆q = 2∆(cos qx + eiΘ cos qy

)(30)

with

∆ ≡ i (d · τ) σy =(−dx + idy dz

dz dx + idy

)(31)

being the standard form of the spin-triplet gap [14]. For Θ = 0 we have extendeds-wave pairing, whereas for Θ = π we have a d-wave pairing. We have selecteda two-dimensional case for the numerical analysis since Sr2RuO4 is regarded asa quasi-two-dimensional system. The qualitative features of the phase diagramdo not depend much on the J/t ratio if only J is substantially smaller thant ≡ |t〈ij〉|.

Summarizing, in this Section we have discussed the coexistence of the satu-rated ferromagnetic and the A1 superconducting state for the quarter-filled bandin the strongly correlated regime, as well as its competition with the AFO-SFstate. The transformation for n = 1 of the AFO-SF insulating state into the S-SF metallic state for J ≥ U/3 is accompanied by a closure of the Mott-Hubbardgap [7]. Hence, not only the appearance of the spin-triplet superconductivity,but also the metallic character are in this case both induced by the Hund’s rulecoupling J . As we have obtained a stable spin-triplet superconductivity (withan isotropic gap (∆q = ∆) in the weak-coupling (Hartree-Fock BCS limit) aswell as in the strong-correlation limit (this time with q-dependent gap), we cansay that this type of superconductivity is a generic phenomenon of our modelof correlated electrons with orbital degeneracy, at least in the mean-field ap-proximation. It would be desirable to discuss the stability of the present resultsagainst the quantum Gaussian fluctuations.

7 Concluding Remarks

In this article we have briefly reviewed the mean-field approach to the spin-tripletsuperconductivity in orbitally degenerate narrow band systems that is inducedby the local ferromagnetic (Hund’s rule) exchange. Both weak (Hartree-Fock)-and strong-correlation regimes were discussed and concrete quantitative resultshave been presented for the case of a doubly degenerate band. The hybridiza-tion of the bands has been neglected and therefore the microscopic parameters(W , εF , U , and J) represent effective values. Nonetheless, explicit calculationsfor a hybridized model should be performed and this should allow for an anal-ysis of concrete materials. Hybridization will introduce a q-dependence for thesuperconducting gap even in the weak-coupling regime.

Coexistence of Spin-Triplet Superconductivity and Ferromagnetism 75

The Hund’s rule coupling allows to treat a weak itinerant-electron ferro-magnetism and real-space spin-triplet pairing on equal footing within a singlemechanism. However, the spin fluctuation contribution should be included to seetheir relative role in scattering the carriers (particularly near TC) and providingthe pairing in the temperature regime T TS TC .

Acknowledgements

The authors are grateful to Leszek Spalek for his technical help and a constantencouragement. The discussions with Karol Wysokinski, Mark Jarrell, Ben Pow-ell, Canio Noce, and Mario Cuoco are appreciated. The work was supported bythe State Committee for Scientific Research (KBN) of Poland.

References

1. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, S. Fujita, J.G. Bednorz: Nature372, 532 (1994)

2. 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) A. Hux-ley, I. Sheikin, E. Ressouche, N. Kernavanois, D. Braithwaite, R. Calemczuk,J. Flouquet : Phys. Rev. B 63, 144519 (2001); N. Tateiwa, T.C. Kobayashi,K. Hanazono, K. Amaya, Y. Haga, R. Settai, Y. Onuki: J. Phys.: Cond. Matter13, L17 (2001)

3. C. Pfleiderer, M. Uhlarz, S.M. Hayden, R. Vollmer, H.v. Lohneysen, N.R. Bern-hoeft, G.G. Lonzarich: Nature 412, 58 (2001)

4. D. Aoki, A. Huxley, E. Ressouche, D. Brithwaite, J. Floquet, J-P. Brison, E. Lhotel,C. Paulsen: Nature 413, 613 (2001)

5. A. Klejnberg, J. Spalek: J. Phys.: Cond. Matter 11, 6553 (1999)6. J. Spalek: Phys. Rev. B 63, 104513 (2001)7. A. Klejnberg, J. Spalek: Phys. Rev. B 61, 15542 (2000)8. See e.g. E.P. Wohlfarth: J. Appl. Phys. 39, 1061 (1968). A more refined theory

invokes the spin fluctuations, cf. T. Moriya: Spin Fluctuations in Itinerant-ElectronMagnetism (Springer-Verlag, Berlin, 1985)

9. P.W. Anderson: in Frontiers and Borderlines in Many-Particle Physics, edited byR.A. Broglia and J.R. Schrieffer (North-Holland, Amsterdam, 1988) pp.1-40

10. The role of spin fluctuations in the pairing for a single-band case has been treatedin: D. Fay and J. Appel: Phys. Rev. B 16, 2325 (1977) A. Layzer, D. Fay: Int. J.Magn. 1, 135 (1971) Solid State Commun. 15, 599 (1974). The role of the Hund’srule in purely qualitative terms has been discussed in: G. Baskaran: Physica B222-224, 498 (1996)

11. J. Spalek: Phys. Rev. B 37, 533 (1988)12. A. Klejnberg, J. Spalek: unpublished13. This is the so-called one-boson approach. For a brief review see J. Spalek,

W. Wojcik: in Spectroscopy of Mott Insulators and Correlated Metals (Springer Se-ries in Solid State Sciences, vol.119, Berlin, 1995) pp.41-65, and references therein

14. See e.g. M. Sigrist, K. Ueda: Rev. Mod. Phys. 63, 239 (1991)

Quasi-particle Spectra of Sr2RuO4

A. Lichtenstein1 and A. Liebsch2

1 University of Nijmegen, 6525 ED Nijmegen, The Netherlands2 Institut fur Festkorperforschung, Forschungszentrum, 52425 Julich, Germany

Abstract. Recent theoretical and experimental results for the electronic structureand correlation effects in Sr2RuO4 are discussed. Multi-band quasi-particle calculationsbased on perturbation theory and dynamical mean field methods show that the creationof a photoemission hole state in Sr2RuO4 is associated with a highly anisotropic self-energy resulting from the coexistence of Ru-derived narrow and wide t2g subbands. Thenarrow, nearly one-dimensional dxz,yz bands are more strongly distorted by Coulombcorrelations than the wide, approximately two-dimensional dxy band. Thus, charge ispartially transferred from dxz,yz to dxy. This correlation-induced inter-subband chargetransfer gives rise to a substantial redshift of the dxy van Hove singularity towardsthe Fermi level by about 50 meV and to a slight deformation of the shape of theFermi surface. The theoretical results are in good agreement with recent angle-resolvedphotoemission spectra.

1 Introduction

The single-layered ruthenate Sr2RuO4 has recently attracted considerable inter-est due to the discovery of unconventional superconductivity at low tempera-ture [1]. This system shares the crystal structure with the archetypal cuprateparent compound La2CuO4, but has entirely different physical properties. Whilethe triplet p-wave symmetry and the mechanism of the superconducting state arestill under discussion [2], the electronic structure of this layer-perovskite materialwhich was investigated using several experimental and theoretical techniques hasnow converged to a clear physical picture. Nevertheless, the complicated Fermisurface consisting of three different sheets and showing large mass renormal-ization makes this system a very interesting object for a modern fermiology:On the one hand, for symmetry reasons single-particle hybridization betweenthe partially filled t2g subbands is absent. On the other hand, these bands arestrongly mixed via local Coulomb interactions. As a result, because of the si-multaneous presence of narrow and wide subbands, correlation effects in thishighly anisotropic multi-band system distort the Fermi surface compared to theone obtained using mean-field band-structure calculations within the Local Den-sity Approximation (LDA) [3]. Normally, the LDA gives rather accurate Fermisurfaces even for highly correlated copper oxide (cuprate) superconductors [4]since according to the Luttinger theorem the total volume of the Fermi surfacein many-body and single-particle schemes is preserved. How this volume is dis-tributed between various Fermi sheets in a multi-band case, however, dependson local Coulomb interactions. Thus, the possibility of deforming Fermi-surface

C. Noce et al. (Eds.): LNP 603, pp. 76–90, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Quasi-particle Spectra of Sr2RuO4 77

sheets due to anisotropic renormalization factors for various bands is an inter-esting issue and can be directly investigated.

Correlation effects play an important role in the ruthenate series and smallchanges of the electronic hopping parameters due to crystal structure distor-tion in the isoelectronic alloys Sr2−xCaxRuO4 can change the Fermi liquid state(x = 0) to an antiferromagnetic insulating state (x = 2) [5,6]. Moreover, thecorresponding perovskite compounds SrRuO3 and CaRuO3 are ferromagneticand paramagnetic metals, respectively. Thus, ruthenate compounds can serveas an ideal tool for the investigation of ferro- and antiferro-magnetic fluctua-tions in itinerant multi-band systems. It is clear that the strength of Coulombcorrelations in the Ru 4d-shell is weaker than in the Cu 3d-shell. Many mag-netic properties of rhuthenates can therefore be understood within the densityfunctional theory using the local density or generalized gradient approximation(GGA). It should be noted, however, that the latter approach gives an artificialferromagnetic ground state in Sr2RuO4 [7,8,9].

Angle-resolved photoemission spectroscopy (ARPES) is one of the key tech-niques providing detailed information on the quasi-particle spectra and Fermisurface topology of high-temperature superconductors. In the layered copper ox-ide compounds high-resolution photoemission spectra can be obtained below andabove the superconducting transition temperature and for different hole dopingregimes [4]. Although de Haas–van Alphen (dHvA) experiments in principle yieldmore reliable bulk Fermi surface data, they are less useful for the investigationof cuprates since they require extremely pure samples. Thus, it has so far notbeen possible for any of the high-Tc cuprates to obtain consistent Fermi surfacedata from both photoemission and dHvA measurements.

The detection of superconductivity in the pure crystal Sr2RuO4 [1] is of greatimportance since this system is the only layered perovskite compound known sofar that is superconducting in the absence of copper and without requiring dop-ing. Thus, a critical comparison of photoemission Fermi surface data with thosederived from dHvA measurements is feasible. Surprisingly, independent studiesof the dHvA effect [10] and earlier angle-resolved photoemission [11,12] yieldedcontradictory Fermi surface topologies. This discrepancy raised serious questionsconcerning the interpretation of photoemission data also in cuprate supercon-ductors. Recently this discrepancy was resolved by accounting for the surfacecontribution to photoemission spectra [13,14,15,16,17]. According independentsurface-sensitive experiments [18] the first Sr2RuO4 layer exhibits a

√2 × √

2-reconstruction driven by the freezing of a zone boundary soft phonon. As shownby surface electronic structure calculations, [8,9,13] this reconstruction leads to aslight redistribution of Ru valence charge so that the shape of the Fermi surfacein the first layer differs indeed from that in the bulk.

In the following section we discuss the electronic structure of Sr2RuO4 withinthe single-particle picture and the modifications arising from local Coulomb cor-relations. Both perturbation theory and dynamical mean field theory based onthe Quantum Monte Carlo method are used to evaluate quasi-particle spectra. In

78 A. Lichtenstein and A. Liebsch

Sect. 3 we compare these calculated spectra with angle-resolved photoemissiondata. A summary is given in Sect. 4.

2 Theory

Because of the layered structure of Sr2RuO4 (see Fig. 1), the electronic bandsclose to the Fermi level may be qualitatively understood in terms of a sim-ple tight-binding picture. These bands are derived mainly from Ru t2g states.The wide xy band exhibits two-dimensional character, while the narrow xz andyz bands are nearly one-dimensional. All three bands are roughly 2/3 occu-pied, giving about 4 Ru d electrons per formula unit. Density functional cal-culations based on the local density approximation [8,19] place the (π, 0), (0, π)saddle point van Hove singularity (vHs) of the xy band about 60 meV above theFermi energy. Taking into account gradient corrections lowers this singularityonly slightly to about 50 meV above EF [8]. As we demonstrate below, localCoulomb correlations push the xy van Hove singularity approximately to within10 meV above EF because of charge transfer from the xz, yz bands to the xyband.

XY XZ

Ru-O

Ru-O

Ru-O

Sr-O

Sr-O

Fig. 1. Crystal structure of Sr2RuO4 with xy and xz orbitals

Quasi-particle Spectra of Sr2RuO4 79

Figure 2 provides a qualitative picture of the t2g bands and of the Fermisurface exhibiting one hole sheet (α) and two electron sheets (β, γ). Whereas thedHvA data [10] are consistent with these results, earlier photoemission spectrarevealed a fundamentally different topology [11,12]: the xy van Hove singularitynear M appears below the Fermi level, converting the γ sheet from electron-liketo hole-like. Nevertheless, both experiments were reported to be in accord withLuttinger’s theorem. Various effects were proposed to explain this discrepancybetween dHvA and photoemission data. For example, surface ferromagnetismwas suggested for Sr2RuO4 by LDA band structure calculations [7]. On theother hand, since photoemission is surface sensitive, it is clear that the spectrareveal contributions due to the surface and bulk electronic structure. As willbe discussed in detail below, this superposition is now believed to explain thedifference between dHvA and photoemission data.

Let us discuss first the quasi-particle electronic structure resulting from localCoulomb interactions . According to the reduced dimensionality of Sr2RuO4,creation of a photohole should be associated with highly anisotropic screeningprocesses which reflect the nature of the different electronic states involved. Asshown in Fig. 2, the relevant bands near EF comprise a roughly 3.5 eV wide bandformed by in-plane hopping between Ru dxy and O 2p orbitals, and 1.4 eV narrowdxz, dyz bands. Assuming an on-site Ru dd Coulomb interaction U ≈ 1.5 eV, wehave the peculiar situation: Wxz,yz < U < Wxy, where Wi is the width of theith t2g band. A value U ≈ 1.5 eV had been deduced from the observation of avalence band satellite in resonant photoemission from Sr2RuO4 [20]. Accordingto this picture, intra-atomic correlations have a much larger effect on the xz, yzbands than on the xy band, giving rise to a strongly anisotropic self-energy. Aswe discuss below, because of the ∼ 2/3 filling of the xz, yz bands, their narrowingleads to a charge flow from the xz, yz bands to the xy band [3]. For reasonablevalues of U this charge transfer is large enough to push the xy vHs close to oreven below the Fermi level.

Because of the relatively simple form of the partly filled single-particle bandsclose to EF , it is possible to obtain an accurate tight-binding fit to the knownLDA band structure. In the case of t2g orbitals in the two-dimensional perovskitelattice the effective 3 × 3 Hamiltonian is diagonal in k-space:

H(k) =

εxy(k) 0 0

0 εxz(k) 00 0 εyz(k)

, (1)

where the energy dispersions within the next-nearest-neighbor hopping approx-imation are given by:

ε(k) = −ε0 − 2tx cos akx − 2ty cos aky + 4t′ cos akx cos aky (2)

with (ε0, tx, ty, t′) = (0.50, 0.44, 0.44, −0.14), (0.24, 0.31, 0.045, 0.01), (0.24,0.045, 0.31, 0.01) eV for xy, xz, yz, respectively (see Fig. 2). These parametersensure that the xy band has edges at −2.8 and 0.7 eV, with a van Hove singularityat 0.05 eV, while the xz, yz bands have edges at −0.9 and 0.5 eV, with van Hovesingularities at −0.80 and 0.26 eV, in agreement with the LDA band structure [8].

80 A. Lichtenstein and A. Liebsch

-3

-2

-1

0

1

Γ M X

(a)

xy

xz

yz

Γ

En

erg

y (

eV

)

γ

α

β

(b)

Fig. 2. (a) Dispersion of t2g bands of Sr2RuO4 in simplified two-dimensional BrillouinZone (EF = 0). The xy band contributes to the γ sheet of the Fermi surface, whilethe xz, yz bands form the α and β sheets. (b) Solid lines: schematic Fermi surfaceconsistent with LDA band structure and dHvA measurements, with hole sheet α andelectron sheets β, γ (after accounting for hybridization). Dashed line: approximate xyFermi surface derived from early photoemission data, indicating that γ is hole-like

Quasi-particle Spectra of Sr2RuO4 81

For the evaluation of the quasi-particle spectra we need to specify the on-siteCoulomb and exchange integrals. In the present case involving only t2g states,there are three independent elements (i = j) [21]: U = 〈ii||ii〉, U ′ = 〈ij||ij〉,and J = 〈ij||ji〉 = 〈ii||jj〉 = (U − U ′)/2, where i = 1 . . . 3 denotes xy, xz, yz.Thus, the Hartree-Fock energies are

ΣHF1 = n1U + 2n2(2U ′ − J) (3)

ΣHF2,3 = n1(2U ′ − J) + n2(U + 2U ′ − J) . (4)

Since the band occupations ni are rather similar, it is convenient to define anaverage occupation n = 2/3, so that n1 = n − 2δ, n2,3 = n + δ, with δ ≈ 0.01,and

ΣHF1 = 5n(U − 2J) + 2δ(U − 5J) (5)

ΣHF2,3 = 5n(U − 2J) − δ(U − 5J) . (6)

To achieve a qualitative understanding of the effect of Coulomb correlations onthe quasi-particle spectra it is useful to consider the second-order contributionto the local self-energy. The important feature, namely, the large difference be-tween the quasi-particle shifts of the xy and xz, yz bands, can be convenientlyillustrated using this approximation. Because the t2g bands do not hybridize,the self-energy has no off-diagonal elements. Thus, the imaginary parts of thediagonal second-order Coulomb and exchange terms are given by

Im Σi(ω) = π∑jkl

Rjkl(ω) 〈ij||kl〉 [2〈kl||ij〉 − 〈kl||ji〉] , (7)

where

Rjkl(ω) =( ∫ ∞

0

∫ 0

−∞

∫ 0

−∞+

∫ 0

−∞

∫ ∞

0

∫ ∞

0

)dω1dω2dω3

× ρj(ω1)ρk(ω2)ρl(ω3) δ(ω + ω1 − ω2 − ω3) . (8)

Here, ρj(ω) denotes the single-particle density of t2g states. Exploiting the sym-metry properties of the Coulomb matrix elements, (7) reduces to

Im Σ1(ω) = U2 R111(ω) + 2J2 R122(ω)+ 4(U ′2 + J2 − U ′J) R212(ω) (9)

Im Σ2,3(ω) = (U2 + 2U ′2 + 3J2 − 2U ′J) R222(ω)+ J2 R211(ω) + 2(U ′2 + J2 − U ′J) R112(ω) . (10)

The above expressions demonstrate that even for J = 0 the self-energy of agiven band depends on scattering processes involving all three t2g bands. Nev-ertheless, Σxy is dominated by interactions within the wide xy band or betweenxy and xz, yz. On the other hand, Σxz,yz primarily depends on interactionswithin the narrow xz, yz bands or between xz, yz and xy. These differences are

82 A. Lichtenstein and A. Liebsch

a consequence of the layered structure of Sr2RuO4 and give rise to anisotropicrelaxation shifts.

For a more accurate description of charge transfer among quasi-particlebands, we include self-consistency in the spirit of dynamical mean-field the-ory (DMFT) [22,23,24]. The DMFT maps the many-body system onto a multi-orbital quantum impurity, i.e., a set of local degrees of freedom in a bath de-scribed by the Weiss field function G [24]. The impurity action (here niσ = c+

iσciσ

and c(τ) = [ciσ(τ)] is a vector of Grassman variables) is given by:

Seff =∫ β

0dτ

∫ β

0dτ ′ Tr[c+(τ)G−1(τ, τ ′)c(τ ′)]

+12

∑i,j,σ

∫ β

0dτ [Uijni,σnj,−σ + (Uij − Jij)ni,σnj,σ] . (11)

It describes the spin, orbital, energy and temperature dependent interactions ofa particular magnetic 3d-atom with the rest of the crystal and is used to computethe local Green’s function matrix:

Gi(τ − τ ′) = − 1Z

∫D[c, c+]e−Seff c(τ)c+(τ ′) , (12)

where Z is the partition function. The impurity self energy is given by

Σi(ωn) = G−1i (ωn) − G−1

i (ωn) . (13)

The Weiss field function is required to obey the self-consistency condition,which can be specified for a given case as

Gi(ωn) =∑k

[(iωn + µ)1 − HLDA(k) − Σdcσ (ωn)]−1

ii . (14)

The local matrix Σdcσ is the sum of two terms, the impurity self-energy and a so-

called “double counting” correction, Edc, which is meant to subtract the averageelectron-electron interactions already included in the LDA Hamiltonian. In thecase of a simple tight-binding model the constant Edc can be absorbed in theshift of chemical potential µ.

Within the DMFT frame work, the local self-energy Σi(ωn) is a functionalof the effective bath Green’s function G−1

i = G−1i + Σi, where the local Green

function Gi in the tight-binding model for t2g states with a diagonal matrixHLDA(k) is given by

Gi(ω) =∫ ∞

−∞dω′ ρi(ω′)

ω + µ − Σi(ω) − ω′ . (15)

We use a two-dimensional linear-tetrahedron (“triangular”) method for the k-space integration of the bare single-particle density of states:

ρi(ω) =∑k

δ(ω − εi(k)) . (16)

Quasi-particle Spectra of Sr2RuO4 83

-3 -2 -1 0 1

-2

-1

0

1

Re Σxy

Re Σxz

Im Σxy

Im Σxz

EF

Se

lf-e

ne

rgy

(eV

)

Energy (eV)

Fig. 3. Real and imaginary parts of self-consistent second-order self-energy for U =1.2 eV, J = 0.2 eV. Solid curves: xy, dashed curves: xz

A typical frequency variation of the self-consistent second-order self-energycomponents Σi(ω) is shown in Fig. 3. Near EF , the imaginary parts vary quadrat-ically with frequency and the real parts satisfy Σxz,yz Σxy, i.e., the energyshift of the narrow xz, yz bands is much larger than for the wide xy band.Moreover, the difference Σxz,yz − Σxy at EF is much larger than the differencebetween the Hartree-Fock energies ΣHF

xz,yz − ΣHFxy .

Qualitatively similar results are derived from more refined treatments ofon-site Coulomb correlations using multi-band self-consistent Quantum MonteCarlo (QMC) methods for solving the impurity problem in DMFT scheme [24,25].The temperature of the simulation was 15 meV (200 K) with 128 imaginarytime slices and ∼ 300 000 Monte Carlo sweeps. Figure 4 shows the quasi-particledensity of states Ni(ω) = − 1

π Im Gi(ω), obtained via maximum entropy recon-struction [26], together with the bare density of states ρi(ω). The van Hovesingularities near the edges of the xz, yz bands are shifted towards EF , causinga band narrowing of about a factor of two. Because of the ∼ 2/3 filling of thesebands, this effect is not symmetric, giving a stronger relaxation shift of the oc-cupied bands than for the unoccupied bands. There is also some band narrowingof the xy bands, but since U < Wxy this effect is much smaller than for thexz, yz bands.

An important point is now that in order to satisfy the Luttinger theoremthe more pronounced band narrowing of the xz, yz bands requires a transferof spectral weight to the xy bands. Thus, the xy vHs is pushed towards theFermi level. In the example shown in Fig. 4, it lies only about 10 meV above

84 A. Lichtenstein and A. Liebsch

-3 -2 -1 0 10

1 (b)

xz

xy

TB

De

nsity o

f sta

tes

(eV

-1)

Energy (eV)

0

1(a)

EF

QMC

Fig. 4. (a) Quasi-particle density of states Ni(ω) derived from self-consistent QMCscheme for U = 1.2 eV, J = 0.2 eV. (b) Single-particle density of states ρi(ω) derivedfrom tight binding bands fitted to LDA band structure. Solid curves: xy, dashed curves:xz

EF , compared to 60 meV in the single-particle spectrum. We emphasize thatthis shift is much larger than the 10 meV shift obtained within the GGA (seeabove). Also, this result is a genuine multi-band correlation effect where thefilling of a relatively wide quasi-particle band is modified by correlations withinother narrow bands of a different symmetry. Of course, since the values of U andJ are not very well known, and considering the approximate nature of our single-particle bands and self-energy calculations, it is not possible at present to predictthe exact position of the xy singularity. Nevertheless, qualitatively it is evidentthat correlations lead to an appreciable redshift of this critical point very close toEF . As a consequence, the shape of the Fermi surface in the interacting-electronpicture differs slightly from the one obtained in LDA calculations.

3 Comparison with Photoemission Spectra

As indicated in Fig. 2, the topology of the Fermi surface of Sr2RuO4 dependscritically on the position of the xy van Hove singularity with respect to EF .It is evident therefore that the charge transfer from xz, yz to xy due to thecreation of the photohole must be taken into account when using angle-resolvedphotoemission to determine the shape of the Fermi surface. To compare ourresults with photoemission spectra, we show in Fig. 5(a) the dispersion of the

Quasi-particle Spectra of Sr2RuO4 85

M X

-1

0

1

Energ

y (

eV

)

(a)

Γ

(b)

M

M MX

ΓΓ

Fig. 5. (a) Quasi-particle bands along ΓM and MX derived from self-consistentsecond-order self-energy. Symbols: LDA bands. (b) Quasi-particle Fermi surface afteraccounting for energy broadening and resolution (see text)

86 A. Lichtenstein and A. Liebsch

t2g quasi-particle bands along ΓM and MX derived from the spectral function

Ai(k, ω) = − 1π

Im[ω + µ − εi(k) − Σi(ω)

]−1. (17)

The xy vHs at M lies 10 meV above EF , so that considerable spectral weightappears below EF in the immediate vicinity of M . To account for the finiteenergy resolution, and following the experimental procedure for determining thespectral weight near EF , [15] we show in Fig. 5(b) the Fermi surface obtainedfrom the partially integrated spectral function

Ai(k) =∫ ∆

−∆

dω Ai(k, ω + i∆) (18)

with ∆ = 25 meV. Considering in addition the finite aperture of the detector(typically ±1o, corresponding to ±5% of k‖ near M for 25 eV photon energy),it is unavoidable to pick up spectral weight from occupied regions near M ,even when the detector is nominally set at M . Thus, the near-degeneracy ofthe xy singularity with EF makes it extremely difficult using angle-resolvedphotoemission to determine the k-point at which the xy band crosses the Fermienergy. Obviously, only photoemission data taken with high energy and angleresolution can provide a conclusive answer.

Earlier photoemission data [11,12] indicated that the xy van Hove singularitynear M is occupied. This finding makes the γ Fermi surface sheet hole-like. Incontrast, dHvA data [10] cleary show this sheet to be electron-like, suggesting thexy critical point at M to be unoccupied. This discrepancy was recently resolvedby accounting for the surface sensitivity of photoemission. Thus, by associatingthe feature near M with emission from the deformed xy band within the recon-structed first layer of Sr2RuO4, an interpretation of the observed photoemissionspectra that is consistent with all available experimental and theoretical resultscan be achieved [16,17].

According to recent surface electronic structure calculations [8,9,13], scanningtunneling microscopy and low energy electron diffraction data [18] the followingpicture emerges for the properties of the first layer: The reconstruction is drivenby the freezing of a zone boundary soft phonon mode giving a slight rotationof the RuO6 octahedra around the surface normal. The rotation reduces theeffective d-d hopping within the first Ru plane and causes band narrowing. Inthe non-magnetic case (which seems to be observed in the photoemission work)the band narrowing yields a shift of the xy van Hove singularity below EF , givingrise to strong emission near M .

The photoemission spectra in this picture consist of a superposition of twosets of t2g bands originating in the surface layer and the deeper layers. Bothsets are energetically very similar and difficult to resolve, with the exceptionof the xy vHs near M : in the bulk it is above EF while at the surface it isbelow. Accordingly, the γ sheet of the Fermi surface is electron-like in the bulk,but hole-like at the surface. Whereas the bulk xy band crosses the Fermi levelalong ΓM , the reconstruction-induced xy band remains below EF near M and

Quasi-particle Spectra of Sr2RuO4 87

M

M X

Fig. 6. ARPES Fermi surface map of Sr2RuO4 [13] compared with the LDA calcula-tion [8]

crosses along MX. Both bands cross the Fermi level along ΓX. In addition, thereconstruction generates weak shadow bands [14].

Figure 6 shows the Fermi surface intensity map of Sr2RuO4 as obtained fromrecent ARPES measurements [13]. The sample was cleaved at 180 K in orderto suppress the

√2 × √

2 surface reconstruction and the emission signal wasintegrated within EF ± 5 meV. The Fermi surface derived within the LDA isalso indicated [8]. As pointed out above, the Fermi surface predicted by LDAcalculations is usually quite accurate since due to the Luttinger theorem the totalvolume is preserved. Nevertheless, the α and γ sheets of Fermi surface in theARPES experiments are slightly more separated than in the LDA calculations(see arrows in Fig. 6). Qualitatively, this trend agrees with the results of ourquasi-particle calculations: Due to the slight charge transfer from the xy to thexz, yz bands, the γ sheet is forced to move away from the origin Γ of the two-dimensional Brillouin Zone, while the β sheet is attracted towards it.

Figs. 4 and 5 also show that due to local Coulomb interactions the narrowxz, yz bands are only about half as wide as predicted by the LDA. Accordingly,the lower edge of these weakly dispersive bands is shifted from −0.9 eV to about−0.5 eV, in excellent agreement with photoemission data [11,12,15]. Moreover,for k‖ between M and X, this band is observed to cross EF at about (π, 0.6π),in good accord with our calculations. In addition, the calculations indicate theexistence of a satellite below the xz, yz bands which might be related to thespectral feature observed near 2.5 eV binding energy using resonant photoemis-sion [20]. The precise location of this satellite is however difficult to determine

88 A. Lichtenstein and A. Liebsch

Fig. 7. ARPES spectral density map of Sr2RuO4 (data from [13])

because of the uncertainty of U and the approximate nature of our self-energycalculations.

We point out here that because of the proximity of the quasi-particle xy vanHove critical point to the Fermi level, the imaginary part of the self-consistentself-energy exhibits a small linear contribution near EF , indicating that thesystem may partially behave like a marginal Fermi liquid. In fact, in (9), it isonly the first term ∼ R111(ω) that gives rise to a linear term if the singularitycoincides with EF . As a result of multi-band effects, however, this contributionis rapidly dominated by stronger quadratic terms involving the narrow xz, yzbands. Thus, we find the marginality to be rather weak.

We finally discuss the mass renormalization derived from our quasi-particlebands. For Coulomb and exchange matrix elements in the range U = 1.2−1.5 eV,J = 0.2 − 0.4 eV we find m∗/m ≈ 2.1 − 2.6, in agreement with photoemissionestimates m∗/m ≈ 2.5−3.2 [15,13], as well as with the dHvA measurements [10]and specific heat data [27], which suggest a factor of 3 − 4.

4 Summary

We have performed multi-band quasi-particle calculations for Sr2RuO4 whichdemonstrate that the simultaneous existence of nearly one- and two-dimensionalt2g bands near EF leads to a highly anisotropic self-energy of the photoemissionhole state. Because of Luttinger’s theorem, this anisotropy gives rise to a chargeflow from the narrow xz, yz bands to the wide xy band, thereby shifting the xyvan Hove singularity very close to EF . These correlation effects cause a slightdeformation of the shape of the Fermi surface compared to the one predicted by

Quasi-particle Spectra of Sr2RuO4 89

LDA band structure calculations. Recent high-resolution angle-resolved photoe-mission data support this conclusion. Also, the narrow xz, yz bands which aremore strongly affected by local Coulomb interactions than the wider xy band,are only about half as wide as calculated within the LDA. This band narrowingis also observed in photoemission spectra. Finally, the quasi-particle calculationsreveal a considerable renormalization, in qualitative agreement with photoemis-sion estimates, dHvA measurements and specific heat data.

The previous discrepancy between photoemission and dHvA data was shownto be associated with the surface sensitivity of photoemission spectra which canbe well interpreted in terms of a superposition of bulk t2g states consistent withdHvA measurements and deformed t2g states existing in the reconstructed firstlayer of Sr2RuO4. The main deformation arises near M where the vHs of the xyband is below EF as a result of a slight narrowing of the t2g bands, while thebulk vHs is unoccupied.

Acknowledgments

We are grateful to Andrea Damascelli for the help with manuscript and usefuldiscussions. The work has been supported by the Netherlands Organization forScientific Research a grant of supercomputing time at NIC Julich.

References

1. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz,F. Lichtenberg: Nature 372, 532 (1994)

2. T. M. Rice, M. Sigrist: J. Phys.: Condens. Matter 84, L643 (1995); Y. Maeno,T. M. Rice, M. Sigrist: Phys. Today 54, 42 (2001)

3. A. Liebsch, A. Lichtenstein: Phys. Rev. Lett. 84, 1599 (2000)4. A. Damascelli, D.H. Lu, Z.-X. Shen: J. Electron Spectr. Relat. Phenom. 117-118,

165 (2001)5. S. Nakatsuji Y. Maeno: Phys. Rev. Lett. 84, 2666 (2000)6. V.I. Anisimov, I.A. Nekrasov, D.E. Kondakov, T.M. Rice, M. Sigrist: Eur. Phys.

J. B 25, 191 (2002)7. P.K. de Boer, R. A. de Groot: Phys. Rev. B 59, 9894 (1999)8. I.I. Mazin, D.J. Singh: Phys. Rev. Lett. 82, 4324 (1999)9. A. Fang, K. Terakura: Phys. Rev. B 64, R020509 (2001)

10. A.P. Mackenzie, S.R. Julian, A.J. Diver, G.J. MacMullan, M.P. Ray, G.G. Lon-zarich, Y. Maeno, S. Nishizaki, T. Fujita: Phys. Rev. Lett. 76, 3786 (1996);A.P. Mackenzie, S.R. Julian, G.G. Lonzarich, Y. Maeno, T. Fujita: Phys. Rev.Lett. 78, 2271 (1997); C. Bergemann, S. R. Julian, A.P. Mackenzie, S. NishiZaki,Y. Maeno: Phys. Rev. Lett. 84, 2662 (2000)

11. T. Yokoya, A. Chainani, T. Takahashi, H. Katayama-Yoshida, M. Kasai, Y. Tokura:Phys. Rev. Lett. 76, 3009 (1996); Phys. Rev. Lett. 78, 2272 (1997); T. Yokoya,A. Chainani, T. Takahashi, H. Ding, J.C. Campuzano, H. Katayama-Yoshida, MKasai, Y. Tokura: Phys. Rev. B 54, 13311 (1996)

12. D.H. Lu, M. Schmidt, T.R. Cummins, S. Schuppler, F. Lichtenberg, J.G. Bednorz:Phys. Rev. Lett. 76, 4845 (1996)

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13. K.M. Shen, A. Damascelli, D.H. Lu, N.P. Armitage, F. Ronning, D.L. Feng, C. Kim,Z.-X. Shen, D.J. Singh, I.I. Mazin, S. Nakatsuji, Z.Q. Mao, Y. Maeno, T. Kimura,Y. Tokura: Phys. Rev. B 64, R180502 (2001)

14. A. Damascelli, D.H. Lu, K.M. Shen, N.P. Armitage, F. Ronning, D. L. Feng,C. Kim, Z.-X. Shen, T. Kimura, Y. Tokura, Z. Q. Mao, Y. Maeno: Phys. Rev.Lett. 85, 5194 (2000)

15. A.V. Puchkov, Z.X. Shen, T. Kimura, Y. Tokura: Phys.Rev. B 58, R13322 (1998)16. A. Liebsch, Phys. Rev. Lett. 87, 239701 (2001)17. A. Damascelli, K.M. Shen, D.H. Lu, Z.-X. Shen,: Phys. Rev. Lett. 87, 239702

(2001)18. R. Matzdorf, Z. Fang, Ismail, J.D. Zhang, T. Kimura, Y. Tokura, K. Terakura,

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N. Shanthi, D.D. Sarma: Phys. Rev. B 53, 8151 (1996)21. J. Kanamori: Progr. Theoret. Phys. 30, 275 (1963); J. Igarashi, P. Unger, K. Hirai,

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(1996)

Normal State Properties of Sr2RuO4

M. Cuoco and C. Noce

I.N.F.M. -Unita di Salerno, Dipartimento di Fisica “E.R. Caianiello”,Universita di Salerno, I-84081 Baronissi (Salerno), Italy

Abstract. There is large evidence that Sr2RuO4 is a spin-triplet superconductor, andthe simplicity of the Fermi surface combined with detailed knowledge of the quasipar-ticle properties means that Sr2RuO4 may be a good candidate for testing mechanismsof unconventional superconductivity. The understanding of the normal state propertiesis the key issue that we address in this paper by investigating the intriguing featuresof the electronic structure, the role of multi-orbital and magnetic scattering in themagneto-transport properties, and the origin of the orbital dependent magnetic fluc-tuations.

1 Introduction

Superconductivity of the conventional phonon-mediated type is one of the fewmany-body problems that is well understood. The unconventional superconduc-tivity of the cuprates and heavy fermion systems on the other hand is provingvery difficult to understand. Some years ago, Maeno and collaborators [1] suc-ceeded in preparing high-quality of the strontium ruthenate compound Sr2RuO4and found a metallic state with hallmarks of a Landau-Fermi liquid at lowtemperatures, and a superconducting transition at Tc ∼1K. Sr2RuO4 sharesthe same crystal structure as high-Tc superconductors of copper oxides withthe quasi-two-dimensional network of CuO2 planes replaced by the RuO2 ones.Thus, Sr2RuO4 offers an opportunity to study normal and superconducting stateproperties in a system isostructural to cuprate superconductors, but with a dra-matically lower critical temperature.

Preliminary investigations led to speculate [2] that the electrons in the Cooperpairs would have aligned spins indicating a triplet superconductivity, and anodd orbital wavefunction as for example p-wave. Recent experiments supportthis picture. Ishida et al. [3] showed that the Knight shift of the NMR peaksof the oxygen remains unaffected by the transition to superconductivity. Thisclearly indicates a signature of triplet pairing, because singlet pairing necessar-ily has a vanishing spin susceptibility when the temperature goes to zero. Theunconventional nature of the superconductivity is exemplified by the extremesensitivity of the transition temperature Tc to disorder introduced by nonmag-netic impurities [4]. Luke et al. [5] performed muon spin resonance experimentsand observed induced magnetic fields in the superconducting state. This obser-vation shows that the angular momentum of the Cooper pairs is not quenchedby the crystal environment, and that the superconducting state has a brokentime-reversal symmetry.

C. Noce et al. (Eds.): LNP 603, pp. 91–107, 2002.c© Springer-Verlag Berlin Heidelberg 2002

92 M. Cuoco and C. Noce

The proof of the sharp Fermi surface in the normal state, which is centralto the perturbative Landau theory, is the observation of periodic oscillationsin the magnetization when the applied magnetic field is varied. Mackenzie etal. [6] observed these oscillations and determined the shape of the Fermi sur-face, which agrees with band structure calculations [7]. In particular, they foundthree bands of predominantly 4d character with dε symmetry, all of which con-tribute to give three Fermi surface sheets with corrugated cylindrical topology.In Sect. 2, we introduce a method which combines the extended Huckel theoryand the tight-binding approximation [8] to get a description of the molecularorbitals and the related relevant bands at the Fermi level. We will show that themain features observed experimentally and derived within LDA (Local densityapproximation), can be captured and reproduced with the advantage to havean analytic description of the low energy single particle excitations close to theFermi level.

Concerning the magneto-transport properties, the experimental scenario isquite complex. The Hall coefficient exhibits a complicated temperature behav-ior [9]: it is negative at very low temperatures, it changes sign at temperaturesapproximately above 30 K and shows a return to negative values above approxi-mately 130 K. As far as resistivity is concerned [10], in contrast with the metallicin-plane resistivity ab, the out-of-plane resistivity ρc takes a maximum at about130 K and changes to non-metallic temperature dependence at higher tempera-tures. Below about 25 K both ρab and ρc exhibit a T 2 power law behavior quiteaccurately and thus showing a Fermi-liquid like behavior up to TFL ∼25 K.The in-plane magnetoresistance is positive and large at low temperatures, thendecreases as T is raised up to 80 K without showing any particular tempera-ture crossover. All these quantities show that though this material is consideredan anisotropic Fermi liquid at low temperatures, its properties at temperatureof approximately 30 K and above are more anomalous, raising the question ofwhat should set such a low crossover scale in a material with a relatively highFermi temperature of the order of 1000 K. In Sect. 3 and 4, we will address thisquestion by determining the T dependence of all the relevant magneto-transportquantities. The main purpose is to show that the anomalous transport behavioris due to the presence of large critical magnetic scattering at small momenta, andthat this anomaly does manifest differently in each magneto-transport quantity.

In Sect. 5 and 6 we will discuss in more detail the origin of the crossover scale,the role of the orbital fluctuations in the setting of the magnetic correlations andtheir influences in the class of ruthenates belonging to the Ruddlesden-Popper se-ries. In particular, the origin and the consequences of the closeness to a quantumcritical point will be carefully investigated. A theoretical analysis, performed onthe basis of the band structure calculation presented in Sect. 2, will be developedwithin a self consistent spin fluctuation approach, to clarify the intriguing ob-served magnetic properties of Sr2RuO4. 17O NMR in 17O isotope-enriched [11]probed spin correlations in Ru dxy and dxz,yz orbitals separately. These mea-surements would provide the evidence that only the spin susceptibility χxy fromthe electrons in the dxy band shows significant temperature dependence. More-

Normal State Properties of Sr2RuO4 93

over, χxy increases monotonically with decreasing temperature down to about40 K, turns over, and then tends to level below TFL in a Pauli paramagnetic liketrend, indicating a clear connection to the crossover in the charge transport mea-surements. On the other hand, the temperature dependence of χxz,yz is muchweaker. These results indicate that the Hund’s coupling is not strong enough toalign all four electron spins, and the spin degrees of freedom in the dxy and dxz,yz

orbitals behave more or less independently at least down to TFL. Therefore, thespin correlations seem to be predominantly ferromagnetic in origin, and orbitaldependent. We consider these issues within a simplified model which takes com-pletely into account the on site intra- and inter-orbital correlations together withthe Hund coupling, with the aim to understand the mechanisms responsible forthe orbital dependent magnetic correlations and the effect induced by the changeof the local orbital symmetry on the magnetic configurations.

2 Huckel-Tight-Binding Model

In this Section we describe the method employed to calculate the energy bandsand the Fermi surface of Sr2RuO4. The calculation is performed in two steps.First, we apply the extended Huckel theory (EHT) to Sr2RuO4 determining theenergy of molecular orbitals and their composition, and then we introduce atight-binding model for this compound, to get the dispersion of the main bandsclose to the Fermi level.

Let us summarize how EHT works.EHT allows for the calculation of the electronic structure of relevant phys-

ical systems and it is very useful in understanding how physical properties ofmolecules, solids and surfaces are related to their geometrical structures. In EHT,the electronic structure of the system under study are represented as linear com-binations of valence atomic orbitals and the exact form of the Hamiltonian isnot specified, but the matrix representation of the Hamiltonian in atomic orbitalbasis is constructed semi-empirically.

Unlike first principles calculations, EHT calculations are not designed topredict the optimum structure of a molecule or a crystal. For systems of knowngeometry, EHT has been useful in uncovering the structure-property relation-ship for discrete molecules and solid state materials. These studies reveal thatapproximate electronic structures obtained by EHT calculations are mostly ade-quate for structure-property correlation analysis, although they may not providequantitative predictions. Indeed, the relevance of EHT lies in this role of facil-itating the search for structure-property correlations. Summarizing, EHT takesfull account of the symmetry of the network of atoms through which the bondsextend, and leads to expressions for the relative energies of the different molecu-lar orbitals in terms of a single parameter. The molecular orbitals of the systemare expressed as linear combinations of atomic orbitals. The appropriate coef-ficients can be obtained from the Schrodinger equation making the followingsimplifications: 1) the diagonal elements of the Hamiltonian, corresponding tothe Coulomb integrals, are given fixed values which have been assigned from

94 M. Cuoco and C. Noce

Fig. 1. Schematic representation of the symmetry allowed hybridization processes fromthe ruthenium ion to the neighbour oxygens

the spectroscopic data; 2) the off-diagonal elements of the Hamiltonian are cal-culated in terms of the Coulomb integrals and the overlap integrals from theHoffmann formula [12].

Using the crystal parameters reported on literature, we have applied theEHT to Sr2RuO4. The numerical results show that all the orbitals have predom-inantly Ru 4d character, with an orbital composition of 84% Ru(4d) and 16%O(2p). This result agrees quite well with the near-edge X-ray-absorption andvalence band photoemission spectroscopy experimental data [13] that estimatethe orbital contribution at EF as 80% Ru(4d) and 20% O(2p). Besides, one ofthe MO is planar because it is composed by orbitals that lie in the a − b planeand the orbitals entering its composition decouple completely from Ru(4d) andO(2p) orbitals that give rise to the other MO. Finally we notice that from EHTwe find that the MO containing the dxy orbital has the lowest energy comparedto other two MO that are degenerate.

Now, let us introduce a tight-binding model for Sr2RuO4.Taking advantage of EHT results we assume that one of the bands contains

only Ru dxy and O1 2py and O2 2px orbitals along a and b axes, respectively.If we assume that only the non-zero hopping integrals between neighboring Ruand O orbitals and those between two neighboring O are important (see Fig. 1),it is straightforward to show that the low energy excitation of electrons with dxy

character is described by the following tight-binding Hamiltonian:

Hxy =∑

k

D†

εxy it1x(k) it1y(k)−it1x(k) εp t2(k)−it1y(k) t2(k) εp

D , (1)

where D† = (d†xy, p†

1y, p†2x) are the electron creation operators in the k-space

corresponding to the above three orbitals, specifically, p†1y (p†

2x) refers to the

Normal State Properties of Sr2RuO4 95

creation of electrons on 2py (2px) orbital located at O1 (O2 ) and d†xy creates

electrons on dxy orbital; εxy and εp are the energy of dxy and p states, respec-tively. The off diagonal elements contain the dispersion relation for the differenthybridization processes between dxy Ru and O(2p) orbitals.

On the contrary, the orbitals contributing to the “z”-bands are the dxz, dyz

and 2pz belonging to the Ru and O atoms in the plane, respectively, and the2px,y of the O atoms out of plane (see Fig. 1). As for xy-band, we assume thatare non-vanishing only the hopping integrals between Ru and O orbitals andthose between two neighboring O. Under these assumptions the tight-bindingHamiltonian is

Hz =∑

k

C†

εd 0 −iek 0 −igk 00 εd 0 −ifk 0 −igk

iek 0 εp ak −ibk 00 ifk ak εp 0 −ick

igk 0 ibk 0 εp 00 igk 0 ick 0 εp

C , (2)

where C† = (d†xz, d

†yz, p

†1z, p

†2z, p

†3x, p†

3y) are the creation operators in the k-spacecorresponding to the above six orbitals. In particular, p†

1z (p†2z) denotes the

creation operator of electrons on 2pz orbital along a (b) axis and p†3x (p†

3y) refersto the creation operator of electrons on apical 2px ( 2py) orbital at O3 site. εd

and εp are the energies of dxz, dyz and px,y,z states, respectively. The off diagonalelements contain the dispersion relation for the different hybridization processesbetween Ru(dxz,dyz) and O(2p) orbitals.

To determine the parameters in Hxy and Hz, we have required that thevolumes enclosed by the three bands coincide with those measured in quantumoscillations and the composition of the molecular orbitals is the same as deducedfrom EHT, i.e. the relative values of the hopping amplitude depend on the overlapbetween molecular orbitals. Besides, LDA one gets εd − εp = 1.5eV [7], to setthe overall energy scale.

The energy dispersion of the effective bands at the Fermi level can be ob-tained by projecting out all low lying O(2p) degrees of freedom using the Lowdindown folding formula [14]. The spectrum for the three 4d bands along a few high-symmetry lines is shown in the left panel of Fig. 2. The electronic structure nearand below EF is strongly anisotropic and all the bands show a very little disper-sion along the kz axis, as is evident from the small bandwidth along the shortΓ −Z direction. It is worth mentioning that the “xy”-band is kz− dispersionlessby construction. Furthermore, this simple approach reproduces the main fea-tures of LDA band structure, i.e. two bands are degenerate at Γ point, the bandmaxima locate at X point for all three bands and the “xy”-band is more dis-persive. The agreement between LDA computations and our calculations givesconfidence that our simplified model captures the essential single particle physicsof Sr2RuO4 at least near EF .

As one can see in the right panel of Fig. 2, the Fermi surface consists ofthree almost cylindrical sheets: two of such sheets are large electron-like cylinders

96 M. Cuoco and C. Noce

−2.50

−2.00

−1.50

−1.00

−0.50

0.00

0.50

Ene

rgy(

eV)

Γ Z X Γ Z

Fig. 2. Left panel: electronic structure as obtained from the method presented inSect. 1. There are three bands at the Fermi level which have dxy character (red line),and mixed dxz-dyz symmetry (blue and green line), respectively

centered at Γ point and the last surface is hole-like cylinder centered at X point.In particular the surface associated with the Ru dxy orbital forms a cylindricalsheet showing no dispersion along kz−direction while the other two bands giverise to corrugated cylinders. The calculated Fermi surfaces in the Brillouin zoneare shown in the right part of Fig. 2. The largest electron like Fermi surfacecorresponds to the hybridized dxy-O(px,py) band. The center of this Brillouinzone is the Γ point.

3 Magneto-Transport Properties

The explicit knowledge of the energy spectrum renders also possible the calcu-lation of several magneto-transport properties. Indeed, one has to determine theconductivity tensor up to second order and this in turn implies the knowledgeof the electronic structure close to EF . A second aspect related to the trans-port analysis concerns the study of the relaxation rate processes which a prioricould come from different sources of scattering. In the present case, the lossof coherence of the conduction electrons is simulated by assuming a tempera-ture dependence for the relaxation rates of the electrons belonging to differentbands, which would include all the main scattering processes. Indeed, we con-sider both scattering due to electron-electron interactions and those which mayarise from the scattering due to magnetic fluctuations. An important assump-tion to reproduce the experimental observations will be that the temperaturebehavior of the carrier scattering rate depends on the presence of an appliedmagnetic field. We argue that the relaxation rate produced by spin fluctuationsis suppressed in presence of an external magnetic field implying that the main

Normal State Properties of Sr2RuO4 97

contribution comes only from the electron-electron scattering, yielding a ∼ T 2

term at low temperature. On the other hand, in the absence of magnetic field,according to the Matthiessen rule, the induced spin-fluctuation scattering rateadds to the previous relaxation mechanisms in determining the total resistivityof the system.

The relevance of magnetic scattering rate in the transport is supported by theexperimental evidence that the Sr2RuO4 is close to magnetic instabilities [11,15]which are the source of enhanced spin fluctuations at different q points of theBrillouin zone, thus giving rise to large scattering amplitudes between the chargecarriers and the spin fluctuations themselves.

The derivation of the physical quantities will be done in the framework ofthe Boltzmann theory generalized to a multi-band case.

To calculate the normal resistivity, the Hall coefficient and the magneto-resistance for a multi-band case, we write the total current as the sum of thecontributions coming from the three bands and then we invert the matrix con-necting J and E. Neglecting powers above B2, we have [16]:

ρ0 =1

σtot0

, (3)

ρH = −ρ20σ

totH , (4)

ρMR = −ρ20[σtot

MR + ρ0(σtotH )2

], (5)

where σtot0 , σtot

H and σtotMR denote the total conductivity, the total Hall conduc-

tivity and the total conductivity derived at the second order of the external field,respectively.

The explicit computation of the magneto-transport quantities requires theknowledge of the band spectra as well as the relaxation times for describing thecollision of the electrons in the bands produced by the dxy, dxz and dyz Ruorbitals. For the energy spectra, we use the electronic energy band structureof Sr2RuO4 above derived, while for the relaxation times, assuming that theydepend on the magnetic field as above discussed, in the case of the Hall- andmagneto-resistance the following expressions have been considered, which con-tain only the scattering due to intra and inter-orbital particle-hole correlations:

(τi)−1 = ηi + αiT2 ,

where i=(xz, yz, xy) indicates the band and η = (2.75, 2.75, 3.25), and α =(0.035, 0.04, 0.06). The values of ηi have been chosen in a way to get the exper-imental observed resistivity at T=4 K of ∼ 0.7µΩcm−1. The constraint on thevalues of αi is given by the complicated temperature dependence of RH togetherwith the behaviour of the transverse in-plane magnetoresistance. We notice thatthe behaviour of the RH can be reproduced only if αxz ∼ αyz but smaller thanαxy. These values are consistent with the expected effective mass dependence

98 M. Cuoco and C. Noce

0 50 100 150T(K)

0

10

20

30

40

50

ρ ab(µ

Ωcm

−1)

Fig. 3. Temperature dependence of the in-plane zero field resistivity. The solid lineindicates the outcome of the calculation performed within the Boltzmann theory, whilethe circles stand for the experimental data [9]

of the coefficient in the T 2 like scattering rate. Indeed, the effective mass forthe electrons with dxy character is larger than that one of the carriers derivedfrom the hybridized almost one dimensional dxz and dyz bands. The fit to theexperimental data is reported in the left panel of Fig. 3, for the Hall coefficientand in right panel of the same figure for the magnetoresistance. The experimen-tal data are taken from [9] for RH and from [17] for the magnetoresistance. Inboth cases, we find a good agreement between the experimental results and thetheoretical prediction indicating that the main contribution to scattering ratefollows a T 2 power law. It is interesting to notice that the change of sign in RH

at a temperature of about TFL is not related to the anomalous crossover in thein plane resistivity from a T 2 to a T like behavior. Moreover, paramagnetic andferromagnetic materials can have a large contribution to the Hall effect mainlydue to skew scattering. Indeed, in this case moving charge carriers experience aforce due to the magnetic field produced by localized magnetic moment and arescattered asymmetrically. However, there is no sign of the saturation of the Hallresistivity commonly observed when is present a large magnetic contribution tothe Hall effect. Therefore, we argue that the experimental data are dominatedby the standard orbital Hall resistivity, ruling out some magnetic contributionaccording with our assumption on the relaxation rates [9].

Let us now consider the zero-field in plane resistivity.As mentioned in the introduction, ρab exhibits a Fermi-liquid behavior up

to TFL; above this temperature rises monotonically with a curvature which isless stronger than T2. We assume that this effect may be probably induced by aspin scattering mechanism which does not contribute to the determination of theHall coefficient and the magneto-resistance. Therefore, we add to the scattering

Normal State Properties of Sr2RuO4 99

0 50 100 150T(K)

−1.5

−1.0

−0.5

0.0

0.5

RH(1

0−10 m

3 /C)

0 5000 10000T

2 (K

2)

0

200

400

600

1/

∆ρ ab

/ρab

Fig. 4. Left (right) panel: hall coefficient (magneto-resistance) vs temperature. Thesolid line stands for the theoretical results within the Boltzmann approximation, thetriangles are the experimental data [17]

rate for the γ band electrons above used (we will see later the origin of thisassumption) a term linear in temperature, while we keep unchanged the othertwo scattering rates. Hence, the temperature dependence of the scattering ratein xy band contains also a contribution:

(τ sfxy )−1 = βT ,

where β = 0.6 as obtained from the comparison with the experimental results.The fit is reported in Fig. 4 where the experimental data are taken from [17].The quite good agreement between the experimental observations and the the-oretical prediction gives confidence that spin fluctuations affects only the zerofield relaxation time for the xy band.

Finally, we notice that we have assumed for the relaxation rates an isotropick-independent form. We have also evaluated the above mentioned quantities as-suming for τi the suitable form in the case of a tetragonal environment. Theresults are only slightly modified, so that we have confined ourselves to temper-ature dependent but k-independent τi.

4 Correlation Effects and Quantum Criticality

The first question on the nature of the magnetic fluctuations appears in the inter-play between the charge fluctuations and the formation of local moment on theruthenium ion. When in the Sr2RuO4 we substitute Ru with Ir ions, the systemundergoes a transition from a metallic paramagnet with small net momentumon the Ru ion, to an insulating paramagnet, with almost saturated moment onthe ruthenium ion, which corresponds to a spin one in the t2g configuration [18].This transition is induced by a so called bandwith control parameter as the sub-stitution of Ir in the compound under study, can be considered isoelectronic,with the main effect to induce a change in the dispersion width by means of

100 M. Cuoco and C. Noce

local distortions. Moreover, it is enough to substitute few percent of Ru to getthe change in the local Ru spin configuration. Hence, our first purpose, in theanalysis of the change of the magnetic correlation, has been devoted to under-stand the subtle interplay between the charge fluctuations controlled by the onsite Coulomb repulsion and the Hund coupling.

We have found, by means of exact diagonalization study on a cluster whichincludes two RuO4 cells [19], that a finite value of the Hund coupling JH is neededin order to have a spin alignment of the electrons in different d orbitals belongingto the same atom. This critical value turns out to be strongly dependent onthe on site Coulomb repulsion U , thus indicating that the existence of a localtriplet configuration for the d electrons is a combined effect of U , that tends toforbid double occupation on the same d orbital, and the Hund coupling, thatinduces parallel alignment. The fact that a critical value of JH must be exceededin order to have local triplet states, is also a consequence of the high degreeof hybridization existing in Sr2RuO4 between ruthenium and oxygen orbitals.There is evidence from photoelectron spectroscopy experiments performed byInoue [20] that the intra-orbital Coulomb repulsion is about 2.4 eV, whereascalculations performed by Baskaran [21] give an estimate for the Hund couplingof the order of 0.5 eV. When we consider the above mentioned parameters forU and JH , the values of the inter-orbital local spin correlations indicate thatthere is small magnetic moment on the Ru ion, and that the non-local spincorrelations are ferromagnetic. Moreover, this state in the phase diagram is closeto an instability, induced by a change in the hopping amplitude between Ru(4d)and O(2p) orbitals, toward a configuration with on site spin one and antiferro-likecorrelations on neighbors Ru atoms. We argue that this scenario may address theproblem of the easy tuning instability induced by the Ir substitution as discussedbefore.

The previous example shows that the magnetic phase diagram of the Sr2RuO4is fairly sensible to any change on a microscopic scale of the parameters whichcontrol the electron dynamics in the lattice.

Different arguments tend to indicate that ferromagnetic fluctuations are rel-evant in this compound.

The Ruddlesen-Popper series Srn+1RunO3n+1 contains multilayer compoundswith n as the number of RuO2-planes per unit cell. In particular, there is a seriesof ferromagnetic compound related to Sr2RuO4: the infinite-layer (3D) SrRuO3is ferromagnetic with Tc 165K [22]; for n = 3 one finds Tc 148K [23] and forn = 2 the system is an itinerant metamagnet [24]. These considerations seem toindicate the tendency that with decreasing the layer number n, Tc is reduced andfinally vanishes. Thus, n may be thought as the parameter controlling a quan-tum phase transition between a ferromagnetic and a magnetically disorderedphase [25]. In this picture the single-layer compound Sr2RuO4 would lie verynear to a quantum critical point from a ferromagnetic to a paramagnetic stateso that the ferromagnetic spin fluctuations would play an important role. Us-ing this argument and considering the experimental data in [11], we propose anexplanation of the normal state magnetic properties of Sr2RuO4 assuming that

Normal State Properties of Sr2RuO4 101

0 200 400 600T(K)

0.2

0.3

0.4

0.5

0.6

0.7

χ xy−1

(10−4

em

u/m

ol)−1

TFL

0 100 200 300 400 500 T(K)

0.15

1.15

1/T

1T (

sec−1

K−1

)

TFL

Fig. 5. Left panel: inverse of the xy spin susceptibility vs T . The blue solid line isthe theoretical result, the red line stands for the crossover temperature TFL, while thesquares indicate the experimental data [11]. Right panel: nuclear lattice relaxation ratevs T . The solid line represents the theoretical behaviour as got within the self consistentspin fluctuation approach, while the squares stand for the experimental data [11]

this compound is close to a ferromagnetic quantum critical point, and withinthe same assumption we also argue about the possible origin of a p-wave type ofsuperconductivity. To calculate various physical properties of the Sr2RuO4, weconsider a ferromagnetic transition in two dimensions in the limit of vanishingtransition temperature, assuming that the main contribution manifests in the γband (we will discuss later this issue). The microscopic calculation can be donewithin the self-consistent spin fluctuation theory developed by Ramakrishnan etal. [26] and many others [27]. It is found that there are two different regimes:when T/TF < α(0) one has an enhanced Pauli susceptibility behavior, whileat higher temperature, i.e. for α(0) < T/TF < 1, the susceptibility follows theclassical Curie Weiss behavior. Here, α(T ) is the inverse of spin susceptibilityexpressed in terms of the Pauli susceptibility, TF being the Fermi tempera-ture. Therefore, the quantity α(0)TF plays the role of the low energy scale andnaturally comes out from the fluctuation theory. The comparison between theexperimental data from [11] and the xy spin susceptibility calculated within theabove assumptions, is reported in Fig. 5. As we can see, there is a good agree-ment between the experimental results and the theoretical prediction. It is worthstressing that using the experimental data for the spin magnetic susceptibilityand the Fermi temperature, calculated using the Fermi wavevector and the ef-fective mass reported in [6], we obtain for α(0)TF the value ∼70K which givesthe right order of magnitude for the temperature TFL of the crossover into aFermi liquid state.

Finally, we have performed the same analysis to determine the critical Tdependence of the nuclear spin-lattice relaxation rate above TFL. As due to thelow momentum spin fluctuations, one gets a ∼ T−3/2 temperature dependencefor (T1T )−1 above TFL while a usual Korringa law for the other two bands isderived by the band structure calculations. Considering these results with respect

102 M. Cuoco and C. Noce

to the observed T dependence of the xy static susceptibility, we deduce that thesame ferromagnetic spin fluctuations are also responsible of the T behavior inthe nuclear lattice relaxation rate. The final outcome is reported in Fig. 5 wherewe have compared the theoretical result with the experimental one of [11] aboveand below TFL.

The quite good agreement between the experimental results and the the-oretical predictions based on the self-consistent spin fluctuation theory givesconfidence that the normal state properties of Sr2RuO4 are strictly related tothe proximity to a quantum critical point and that the ferromagnetic spin fluc-tuations which manifest in the dxy band control the T dependence of the staticsusceptibility and of the nuclear spin relaxation rate, respectively. Finally, wepoint out that these spin fluctuations are also responsible of the linear T scat-tering rate which manifests in the resistivity above TFL.

5 Orbital Dependent Magnetic Correlations:Dynamic Double Exchange vs Superexchange

Let us discuss why in the electrons of the xy band the effect of the ferromagneticfluctuations is more pronounced [28]. Our starting point is to consider the dy-namics of two holes (four electrons) in the t2g bands, projecting out the oxygendegrees of freedom and taking fully into account the on site atomic correlations.

The first target is to show that the removal of the cubic symmetry has astrong influence on the magnetic character of the ground state and that in thiscontext we derive a scenario where orbital dependent spin correlations (ferro andantiferro) develops as due to a competition between a dynamic double exchangeand a superexchange coupling (see Fig. 6).

The Hamiltonian under study is the following:

H =∑i,α

εi,αni,α + (U + 2JH)∑iα

ni,α↑ni,α↓ + (U − 12JH)

∑i,α<β

ni,αni,β

− 2JH

∑i,α<β

Si,αSi,β + JH

∑i,αβ

d†iα↑d

†iα↓diβ↓diβ↑ + t

∑〈i,j〉σ

(d†α,iσdα,jσ + H .c.) ,

xz−yz

xy

Ru Ru

∆<0 : superexchange

xz−yz

xy

Ru Ru

∆>0 : double exchange

Fig. 6. Schematic representation of the magnetic exchange processes activated in thecase of negative(left panel) and positive (right panel) splitting of the degenerate γzand xy orbital energy

Normal State Properties of Sr2RuO4 103

where H is the complete Hamiltonian containing intrasite correlation terms fort2g orbitals [29], and the symmetry allowed hopping between electrons in differentRu atoms. The operator dα,iσ destroys an electron with spin σ in the orbitalα on site i; ni,ασ is the electron density with spin σ on the orbital α, andSi,α is the spin of the electron in the α orbital on site i, respectively. Here,εi,α denotes the on-site energy of the α orbital on the site i, U and JH standfor the Coulomb and Hund’s exchange interaction, respectively, and t denotesthe hopping amplitude. The effective hopping is diagonal, i.e. between orbitalswith the same symmetry (we have neglected the small hybridization process viain plane O(2pz) orbitals between the dxz and dyz configuration). We controlthe cubic-tetragonal symmetry crossover by means of the zero energy splitting∆ ≡ (ε(dγz) − ε(dxy)) between the xy and z orbitals. It is clear that the removalof the degeneracy involves also a renormalization of the effective hopping viathe oxygens between the t2g orbitals implying a self consistent procedure, whichwill not be considered in this scheme of computations. We calculate the phasediagram for the case of four effective ruthenium atoms, by means of the Lanczosalgorithm to determine the ground state of H. The results of the numericalsimulation are reported in Fig. 7 where the possible magnetic configurations areshown as a function of the ratio JH/U and the energy level splitting ∆.

For the cubic case, i.e. when ∆=0, the ground state is ferrimagnetic (totalspin is equal to 2), that is an intermediate value of the spin between the maxi-mum allowed by the filling and zero, in a very small region of JH/U above that itbecomes ferromagnetic passing through a spin zero configuration with non-localantiferro like correlations. When the value of the exchange interaction JH/U isenough large, crystal field effects, associated with the elongation of the RuO6octahedra (∆ >0), tend to stabilize ferromagnetic spin configurations. Indeed,for positive ∆, the dxy orbital is lower in energy than the others two degeneratedxz,yz. Then, if the Coulomb repulsion is larger than ∆ and the other parametersinvolved, one would occupy the first lower energy orbital and put the other holein one of the two degenerate dxz,yz. In this case there is a gain in the kineticenergy if the holes on the neighbor site have the same spin of the hole moving.We have a kind of dynamical double-exchange mechanism (see the right panelof Fig. 6), since the spin of the neighbor site is fluctuating in amplitude andphase, instead of having only phase fluctuation as happens in manganese oxides.In other words, the exchange mechanism does not occur between localized spins.It is worth noticing that if one looks at the orbital dependent spin correlations,it turns out that in the ferromagnetic region, the xy spin correlations are com-pletely ferromagnetic while for the z orbitals they are not fully ferromagneticdue to the orbital fluctuations induced by the degeneracy. This aspect confirmsthe hypothesis that the ferromagnetic fluctuations are more pronounced for theelectrons in the xy band, while the orbital degeneracy for the dγz gives rise toa kind of frustration in the spin sector reducing the strength of the ferromag-netic correlations. On the other side of the phase diagram, the gain in kineticenergy due to the large d-p hybridization, inverting the order of the molecularorbitals respect to the result of crystal field [8], and/or the compression of the

104 M. Cuoco and C. Noce

-0.50 -0.25 0.00 0.25 0.50

0.1

0.2

0.3

0.4

0.5

FERRI

FERRO

PARAJ H/U

∆/t

Fig. 7. Phase diagram for the case of two holes in the t2g sector. AF, Ferro, and Ferristand for antiferromagnetic, ferromagnetic and ferrimagnetic phase. In particular theantiferromagnetic phase corresponds to a ground state configuration with zero totalspin and antiferro like correlations in neighbors Ru atoms

RuO6 octahedra, would produce negative value for ∆ and stabilize a low spinconfiguration. Indeed, for negative ∆, the lowest energy configuration is realizedby occupying with two holes the dxz,yz orbitals (see the left panel of Fig. 6). Inthis case for intermediate Coulomb repulsion we would have and effective anti-ferromagnetic Hamiltonian with a net superexchange J ∼ 4t2/(U + JH) whichyields the lowest spin configuration for the ground state.

These two limiting cases come in competition when |∆| is smaller than thekinetic energy scale t. In this case the gain in the kinetic energy allow to haveconfigurations with non integer occupation number on each t2g orbital. This isthe relevant situation which occurs in the ruthenium oxides either for the cubicand the tetragonal compound.

We claim that the case of the Sr2RuO4 would be for negative values of ∆close to zero. In this regime the system is globally paramagnetic (total spin zero)with strong competition between ferro and antiferro like correlations.

Some comments on the limits of this toy model calculations are worthwhile.Within this two-sites problem one cannot include any band effect which, for ex-ample, is relevant to understanding the dynamical nesting at incommensuratewave vector in the Sr2RuO4. Nevertheless, the present calculation yields use-

Normal State Properties of Sr2RuO4 105

ful insights on the interplay between the magnetic correlations and the orbitaldegrees of freedom in the complete series of the SrRuO compounds.

We argue that ∆ may represent the parameter which control the quantumphase transition from the ferromagnetic to paramagnetic state, that would simu-late the effect of reducing the number of layers in the Ruddlesden-Popper series.Indeed, as we have obtained in our calculation, the magnetic phase diagram forthe t2g system is highly asymmetric depending on the sign of ∆ and by oppor-tune tuning of the local environment one may easily drive the system toward achange in the magnetic configuration.

6 Conclusions

In conclusion, in this paper we described the normal state properties of Sr2RuO4from microscopic principles. The electronic energy band structure of Sr2RuO4has been calculated by using a simple combined extended Huckel theory/tight-binding method. We have derived an appropriate tight-binding Hamiltonian,as well as analytical expressions for energy bands and for the constant-energycontour near the Fermi surface. Our calculations reproduce with good accuracyLDA results, showing almost dispersionless Fermi surfaces along kz−directionand thus supporting the quasi two dimensional nature of Sr2RuO4. We pointout that the analytical formulas for energy bands of electrons around the Fermilevel have been used to analyze physical quantities in which the topology of theFermi surface is important. Using the calculated electron energy band struc-ture, we have computed the temperature dependence of the Hall coefficient,the magneto-resistance and the in-plane resistivity by solving the Boltzmannequation for a multi-orbital system. The reasonably good fit of these physicalquantities suggests that the assumption of two contributions in the relaxationrate for the xy-electrons used to quantify the galvanomagnetic transport is es-sentially correct. Then, we have introduced in our model the strongly correlatedinteractions and we have performed an exact diagonalization study on smallplanar Ru-O cluster, based on the use of the Lanczos algorithm. We have dis-cussed that the minimum value of the Hund coupling JH needed to have on eachruthenium atom a local triplet spin state is a rapidly decreasing function of theintra-orbital d-electron Coulomb repulsion U , and that this result would explainthe subtle interplay between charge and magnetic moment fluctuations in thesubstituted SrIrRuO system. Finally, we have shown that, assuming that thiscompound lies very near to a quantum critical point, the temperature variationof various physical quantities is governed by ferromagnetic spin fluctuations,whose effect is observed over a wide temperature range at low temperatures. Aconsistent explanation of the experimental results is obtained within this pic-ture and assuming that only the electrons belonging to dxy band are involvedin this mechanism. Moreover, we have discussed the effect of local symmetrychange on RuO6 octahedra with respect to the magnetic phase diagram. Themain finding is that the parameter which control the splitting between the xyand z orbitals which depends both on local distortions and on the covalency

106 M. Cuoco and C. Noce

between the Ru(4d) and O(2p) atoms, may drive a quantum phase transitionfrom a paramagnetic to ferromagnetic state for a system with partially filled t2g

orbitals.

References

1. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz,F. Lichtenberg: Nature 372, 532 (1994)

2. T.M. Rice, M. Sigrist: J. Phys.: Cond. Matter 7, 643 (1995)3. K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori, Y. Maeno:

Nature 396, 658 (1998)4. A.P. Mackenzie, R.K. Haselwimmer, A.W. Tyler, G.G. Lonzarich, Y. Mori,

S. Nishizaki, Y. Maeno: Phys. Rev. Lett. 80, 161 (1998)5. G.M. Luke, Y. Fudamoto, K.M. Kojima, M.I. Larkin, J. Merrin, B. Nachumi,

Y.J. Uemura, Y. Maeno, Z.Q. Mao, Y. Mori, H. Nakamura, M. Sigrist: Nature394, 558 (1998)

6. A.P. Mackenzie, S.R. Julian, A.J. Diver, G.J. McMullan, M.P. Ray, G.G. Lonzarich,Y. Maeno, S. Nishizaki, T. Fujita: Phys. Rev. Lett. 76, 3786 (1996)

7. T. Oguchi: Phys. Rev. B 51, 1385 (1995); D.J. Singh: Phys. Rev. B 52, 1358 (1995)8. C. Noce, M. Cuoco: Phys. Rev. B 57, 11989 (1999)9. A.P. Mackenzie, N.E. Hussey, A.J. Diver, S.R. Julian, Y. Maeno, S. Nishizaki,

T. Fujita: Phys. Rev. B 54, 7425 (1996)10. A.W. Tyler, A.P. Mackenzie, S. NishiZaki, Y. Maeno: Phys. Rev. B 58, 10107

(1998)11. T. Imai, A.W. Hunt, K.R. Thurber, F.C. Chou: Phys. Rev. Lett. 81, 3006 (1998)12. R. Hoffmann: J. Chem. Phys. 39, 1937 (1963)13. M. Schmidt, T.R. Cummins, M. Burk, D.H. Lu, N. Nucker, S. Schuppler, F. Licht-

enberg: Phys. Rev. B 53, 14761 (1996)14. O.K. Andersen, A.I. Liechtenstein, O. Jepsen, F. Paulsen: J. Phys. Chem. Sol. 56,

1573 (1995)15. Y. Sidis, M. Braden, P. Brouges, B. Hennion, S. NishiZaki, Y. Maeno, Y. Mori:

Phys. Rev. Lett. 83, 3320 (1999)16. C. Noce, M. Cuoco: Phys. Rev. B 62, 9884 (2000)17. N.E. Hussey, A.P. Mackenzie, J.R. Cooper, Y. Maeno, S. Nishizaki, T. Fujita: Phys.

Rev. B 57, 5505 (1998)18. R.J. Cava, B. Batlogg, K. Kiyono, H. Takagi, J.J. Krajewski, W.F. Peck, Jr.,

L.W. Rupp, Jr., C.H. Chen: Phys. Rev. B 49, 11890 (1994)19. M. Cuoco, C. Noce, A. Romano: Phys. Rev. B 57, 11989 (1998)20. quoted in: H. Yoshino, K. Murata, N. Shirakawa, Y. Nishihara, Y. Maeno, T. Fujita:

J. Phys. Soc. Japan 65, 1548 (1996)21. G. Baskaran: Physica B 223-224, 490 (1996)22. T.C. Gibb: J. Solid State Chem. 11, 17 (1974)23. G. Cao, S.K. McCall, J.E. Crow, R.P. Guertin: Phys. Rev. B 56, 5740 (1997)24. R.S. Perry, L.M. Galvin, S.A. Grigera, L. Capogna, A.J. Schofield, A.P. Mackenzie,

M. Chiao, S.R. Julian, S.I. Ikeda, S. Nakatsuji, Y. Maeno: Phys. Rev. Lett. 86,2661 (2001)

25. M. Sigrist, D. Agterberg, A. Furasaki, C. Honerkamp, K.K. Ng, T.M. Rice,M.E. Zhitomirsky: Physica C 317-318, 134 (1999)

Normal State Properties of Sr2RuO4 107

26. T.V. Ramakrishnan: Solid State Commun. 14, 449 (1974); T.V. Ramakrishnan:Phys. Rev. B 10, 4014 (1974)

27. See for example: T. Moriya: Spin Fluctuations in Itinerant Electron Magnetism(Springer Verlag, Heidelberg, 1985)

28. C. Noce, G. Busiello, M. Cuoco: Europhys. Lett. 51, 195 (2000)29. J.S. Griffith: The Theory of Transition Metal Ions, (Cambridge University Press,

Cambridge) 1971; A.M. Oles: Phys. Rev. 28, 327 (1983)

The Nature of the Superconducting Statein Rutheno-Cuprates

C.W. Chu1,2,3, Y.Y. Xue1, B. Lorenz1, and R.L. Meng1

1 Department of Physics and TCSUH, 202 Houston Science Center,University of Houston, Houston TX 77204-5002, USA

2 Lawrence Berkeley National Laboratory,1 Cyclotron Road, Berkeley CA 94720 USA

3 Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong

Abstract. The magnetism and superconductivity (SC) of rutheno-cuprates RuSr2L-Cu2O8−δ (Ru1212) and RuSr2(L,Ce)2Cu2O10+δ (Ru1222) were investigated, where L= Gd or Eu. The normal state carrier concentration p, the Tc, the intragrain pen-etration depth λ, and the diamagnetic field-cooled magnetization were measured invarious annealed rutheno-cuprate samples. The p varies with annealing only slightly(from 0.09 to 0.12 holes/CuO2), but the intragrain Tc by a factor of 2.3 (from 17 to40 K). The 1/λ2, on the other hand, was enhanced more than tenfold (from 0.3 to6 µm−2). The data are in disagreement with both the universal Tc vs. p and Tc vs.1/λ2 proposed for bulk cuprates. These, together with the unusually large dTc/dH ≈100 K/T observed in both Ru1212 and Ru1222, suggest that even the intragrain SC ofrutheno-cuprates is granular. A Josephson-junction-array model was then proposed tointerpret the data. The memory effect observed in Ru1222 far above the main magnetictransition temperature further suggests that the root of the granularity may be a phaseseparation, resulting in the mesoscopic ferromagnetic and antiferromagnetic species inthese rutheno-cuprates.

The reported coexistence of antiferromagnetic (AFM), ferromagnetic (FM),and supercondutive long-range orders in rutheno-cuprates RuSr2LCu2O8−δ (Ru-1212L) and RuSr2(L,Ce)2Cu2O8−δ (Ru1222L) have attracted intense interestrecently, where L = Gd, Eu, or Y [1,2,3,4]. The situation for their magnetism,unfortunately, is still unclear after intensive investigations of magnetization, neu-tron powder diffraction (NPD), and nuclear magnetic resonance (NMR) [3,5,6,7].The significant spontaneous magnetization (≈ 0.28 µB/Ru in Ru1212Gd and0.8 µB/Ru in Ru1222Eu after a geometric correction for grain orientations) re-ported at low temperature and the internal field of ≈ 720 Oe detected by the zero-field µSR were initialy taken as evidence for an FM spin order in Ru1212Gd [2,3].Later NPD data, however, suggested a G-type AFM long-range order instead [5].The suggested upper limit, 0.1 µB/Ru, of any possible FM component at zeroexternal field by NPD is significantly smaller than that suggested by the spon-taneous magnetization of the same sample at 5 K. The situation becomes moreconfused when two independent NMR studies reported that the Ru-spins areFM aligned as a mixture of 60% Ru5+ and 40% Ru4+ with the aligned momentsof 2 and 0.9 µB/Ru, respectively [7,8]. While the dc magnetization, NPD, and

C. Noce et al. (Eds.): LNP 603, pp. 108–117, 2002.c© Springer-Verlag Berlin Heidelberg 2002

The Nature of the Superconducting State in Rutheno-Cuprates 109

NMR data have all been verified by independent measurements, the almost ten-fold difference in the FM aligned moments observed by different probes presentsa serious challenge [9,10,11]. It will be impossible to reconcile the differences, inour opinion, if the compounds are microscopically homogeneous.

Our understanding of their SC is no better. While the resistivity, , dropsrapidly below T ′

c ≈ 20–30 K to zero, consistent with the large diamagnetic shiftin the zero-field-cooled χ, no Meissner signal was detected in the field-cooled χfor early Ru1212Gd samples. Later, a specific heat Cp anomaly was detected atTb ≈ 45 K [12]. This was taken as evidence for a bulk superconducting transitionat Tb. Also cited as support for such a suggestion is the diamagnetic shift belowT ′

c in the field-cooled χ, which is extremely field-sensitive and disappears in afield 20 Oe or higher, whereas T ′

c is unchanged in such low fields [13]. However,the χ of powder samples demonstrates that the T ′

c so quoted is actually an inter-grain phase-lock temperature and the actual intragrain transition temperature,Tc ≈ 40 K, is somewhere between T ′

c and Tb, with a very small diamagneticshift between T ′

c and Tc [14]. This very small shielded volume-fraction abovethe intergrain transition temperature suggests an unusually large penetrationdepth, λ, which is in disagreement with the proposed Uemura line of bulk super-conductors [15]. The diamagnetic drop ∆MFC in the field-cooled magnetizationMFC , which has been considered to be associated with a bulk Meissner effect,becomes comparable to the 1/4π expected for bulk SC only below a few Oe,and again suggests an unusually small supercarrier density 1/λ2. It should bepointed out that the argument that a spontaneous-vortex-state above a few Oemay be responsible for the diminishing of ∆ MFC is in conflict with the Ginzburg-Landau (G-L) model. The ∆ MFC ≈ φo lnκ/(16π2λ2) ln(H/Hc2) expected fora type II superconductor in a mixed state should only weakly depend on Haround Hc1 [16], i.e. when a superconductor evolves from the Meissner stateto the spontaneous-vortex-state. The absence of bulk Meissner effect or a dia-magnetic MFC above a few Oe in Ru1212, therefore, should be an indication ofJosephson-junction-array-like (JJA) superconductivity in rutheno-cuprates, aswill be discussed below.

To explore the topic, single-phase ceramic Ru1212Gd, Ru1212Eu, Ru1222Gd,and Ru1222Eu samples were investigated. Our data suggest that the SC detectedin Ru1212 and 1222 can be described well by a Josephson-junction-array model,and that phase separation resulting in the FM and AFM species may be theroot of the JJA-like granularity.

Ceramic Ru1212 and Ru1222 samples were synthesized following the stan-dard solid-state-reaction procedure. Precursors were first prepared by calcinedcommercial oxides at 400–900 C under flowing O2. Mixed powder with a propercation ratio was then pressed into pellets and sintered at 800–900 C in air for24 hr. The final heat treatment of the ceramics was done at 1060 and 1090 C for60 hr and 120 hr, for Ru1212 and Ru1222, respectively, after repeatedly sinteringat 1000 C and regrinding. Powder samples with different particle sizes were pre-pared according to the procedures previously reported [6]. The structures of thesamples were determined by powder X-ray diffraction (XRD), using a Rigaku

110 C.W. Chu et al.

2θ

20 30 40 50 60

counts

100

1000

counts

100

1000

10000

Ru1222Gd

Ru1212Gd

Fig. 1. XRD of a Ru1222Gd sample (top); and a Ru1212Gd sample (bottom). Data(dots); refinements (solid lines)

DMAX-IIIB diffractometer. Refinement was done using the program of Rietan-2000 [17]. There are no noticeable impurity lines in the X-ray diffraction patternwithin our resolution of a few percent (Fig. 1). The grain sizes (≈ 2–20 µm) ofthe ceramic samples, as well as the particle sizes of the powder samples, weremeasured using a JEOL JSM 6400 scanning electron microscope (SEM). Themagnetizations were measured using a Quantum Design SQUID magnetometerwith an ac attachment.

The multistage transition in Ru1212 has been suggested previously, but basedonly on indirect evidence [3,12]. To clarify the situation, several powder sampleswith different particle sizes were prepared from the same ceramic. Particles withdifferent sizes were sorted using either sieves or the descending speed of the parti-cles in acetone. The details have been reported before [6]. An obvious correlationwas observed between the particle size and the ac susceptibility χ(T) (Fig. 2).The 27 K transition in the bulk ceramic is severely suppressed in the powder

The Nature of the Superconducting State in Rutheno-Cuprates 111

T (K)

10 20 30 40 50

m'/H (10

-3 emu/g)

1.0

1.5

bulk

40 micron

20 micron

8 micron

3 micron

0.8 micron

Fig. 2. The ac χ of powder with different particle sizes

sample with particle size decrease, e.g. to 10 µm. It, therefore, is interpretedas the intergrain phase-lock temperature, T ′

c, consistent with the average grainsize 2–10 µm observed in the ceramic [6,14]. The 40 K transition, on the otherhand, is essentially not affected by the size down to 8 µm, but is systematicallysuppressed by the further decrease in the size. It is attributed to the intragraintransition, Tc, associated with grains which further consist of mesoscopic do-mains and are thus not homogenous [6,14].

A data analysis procedure was developed based on the calculated shielding ofχ = −3/(8π)[1 − 6(λ/d) coth(d/2λ) + 12(λ/d)2] of an isotropic superconductivesphere with a diameter of d. By assuming that the χ of magnetic domains isd-independent, a randomly oriented powder sample j (= 1...n), which containsparticles i = 1...m with sizes of dj,i, has:

χj = χm +∑

i

∫[1 − 6(λab/dj,i) coth(dj,i/2λab) + 12(λab/dj,i)2]

d3j,icos

2θ sin θ + [1 − 6(λc/dj,i) coth(dj,i/2λc) + 12(λc/dj,i)2]

d3j,i sin2 θ sin θdθ/

∑i

d3j,i , (1)

where λc and λab and θ are the penetration depths along c and ab and the polarangle, respectively. When λc λab (highly anisotropic approximation), one has:

112 C.W. Chu et al.

T (K)

0 10 20 30 40 50

λ (µm)

0

2

4

6

8

Fig. 3. The deduced λ for the Ru1212Gd sample

χj = χm +∑

i

[1 − 6(λab/dj,i) coth(dj,i/2λab) + 12(λab/dj,i)2]d3j,i/3

∑i

d3j,i .

(2)A regression was used to calculate λ = λab and χm. The χj of the powder

with the smallest average particle size was used as the initial value of χm. Theinitial value of λ was deduced from the χ of the powder sample with the largestparticle size. The new χm was then regressively calculated through a least-squarefit using the approximate λ value and the χj , dj,i data observed. The regressionof λ followed. Typically, the result will be convergent to within 1% after threeregression cycles. The λ(5 K) so obtained for this Ru1212Gd sample with Tc ≈40 K is ≈ 3 µm, unusually long for typical cuprates but reasonable for a JJAsuperconductor (Fig. 3).

To further explore the nature of these rutheno-cuprates, several Ru1222Gdsamples, which do not display an intergrain transition due to poor grain coupling,were systematically annealed to change the intergrain transition temperatureTc. Nine samples in three different batches were measured. A0, B0, and C0 areas-synthesized samples with Tc’s varying between 26 and 30 K. All others arepieces from the respective as-synthesized ceramic after a 2 hr/600 C annealingin different gas atmospheres. The gases used in annealing were 300 atm. O2for samples A1, B1, and C1; 20 atm. O2 for sample A2; Ar+0.01 atm. O2 forsample C2; and high purity Ar (99.99%) for sample C3. The Tc observed in

The Nature of the Superconducting State in Rutheno-Cuprates 113

1/λ2 (µm

-2)

0 5 10 15

TC

(K)

0

20

40

1/λ2 (µm

-2)

0 5

∆MFC

/H at 5 Oe (0.01 emu/cm

3)

0

5

Fig. 4. Tc vs. 1/λ2 for Ru1222Gd. •: samples C0–C3; : samples B0–B1; : samplesA0–A2. The proposed Uemura line (solid line). Inset: The Meissner effect of the nineannealed samples

the nine samples increases with oxidation from 17 to 40 K, but the normalstate carrier concentration varies only from 0.09 to 0.12 holes/CuO2 based onboth the 295 K thermal power and the oxygen stoichiometry measured. Thedoping level, therefore, will not be the dominant factor in the Tc enhancement.However, the intragrain penetration depths deduced change more than tenfold,and a systematic correlation between Tc and 1/λ2 exists (Fig. 4).

In typical cuprates with low pair-breaker concentrations (e.g. Zn), Nachumiet al. have observed that Tc is a universal linear function of 1/λ2 [15]. Althoughdisputes remain as to whether the interpretation is valid at high Zn-level, allreported Tc data approach zero with the suppression of 1/λ2 by pair-breaking.In particular, both the strong-coupling d-wave model in the unitarity limit andthe “swiss cheese” model predicts Tc 10 K when 1/λ2 ≈ 0.3 µm−2 (i.e. for therelaxation time σ ≈ 0.02 µs−1 in a µSR measurement) [15,18]. It is, therefore,interesting to note that all of our data points fall on the far left of this line. Inparticular, the Tc of sample C2 with 1/λ2 ≈ 0.3 µm−2 is still 15 K or higher. Thetrend is similar to that of the data of Ru1212, where samples with λ(5 K) as largeas 2–3 µm still have Tc > 20 K [6]. A simple pair-breaking mechanism, therefore,may have difficulty to accommodate the data. This view is supported by the factthat there is no systematic correlation between the Tc (or 1/λ2) and the MFC

(or the remanent moment) above Tc, i.e. the FM aligned spins. In fact, the MFC

114 C.W. Chu et al.

at 5 Oe and 50 K differs less than 5% between samples C3 and C1 with Tc of 17and 37 K, respectively. Intragrain inhomogeneity, therefore, seems to be a morereasonable interpretation, suggesting the existence of microdomains inside thesegrains. It was proposed that the Tc of a JJA is 2.25J , with J being the couplingenergy of a junction [19]. The λ, in such a case, may depend on the length ofthe junctions involved, but the Tc will not. A non-zero phase-lock temperature,therefore, may coexist with a unusually long λ if the junction length is large.

The ∆ MFC across Tc, the diamagnetic shift, also changes with 1/λ2 (inset,Fig. 4). The large value of ∆ MFC in a Ru1212 sample below 1 Oe and itsdisappearance above 10 Oe have previously been taken as evidence for a bulksuperconducting to a spontaneous vortex state (SVS) transition induced by amagnetic field. Within the framework of the Ginzburg-Landau(G-L) model, thereversible part of ∆ MFC of a type II superconductor should have a maximum ofHc1/4π at Hc1; and a few times smaller (≈ φo ln(Hc2/H)

32π2λ2 = Hc1 ln(Hc2/H)8π ln κ ) in mixed

states far above Hc1, where Hc1, Hc2, and κ are the lower- and upper-criticalfields and the G-L parameter, respectively [16]. This is simply the result of acompetition between the magnetic energy M· H and the carrier kinetic-energy,and should hold even in the existence of spontaneous vortex and an internalmagnetic field BM (= 4π M in homogeneous superconducting ferromagnets).In particular, the ∆ MFC should be ≈ φo lnκ/(16π2λ2) (> 1 emu/cm3 withλ < 0.5 µm) over a broad H-range regardless of the value of BM if the pinningis weak. The maximum ∆ MFC , we would argue, is a far better parameter forferromagnetic superconductors than the widely used χ, whose interpretation maybe ambiguous due to the uncertainties in the BM and the possible SVS.

The ∆ MFC of sample A1, for example, is ≈ 0.3 emu/cm3 at 20 Oe, andseems to increase continuously with H, although the large magnetic backgroundmakes a quantitative estimation difficult at larger fields. This value is not toofar from that expected based on the deduced λ ≈ 0.4 µm, considering the cor-rections needed for the random grain-orientations and vortex pinning. The muchsmaller ∆ MFC(< 0.04 emu/cm3 over the whole H range) reported by Bernhardet al., on the other hand, may imply a unusually long λ, i.e. severe inhomogene-ity, if the pinning is not too strong. This inhomogeneity is also obvious fromthe H-dependency of Tc in a Ru1212Gd sample (Fig. 5) [14]. The differentialsusceptibility, i.e. the ac χ with a dc bias of H, shifts parallel with H (inset,Fig. 5). The decrease rate dTc/dH ≈ 100 K/T (Fig. 5), however, is a hundredtimes higher than that of typical cuprates. It can be understood only if theSC of Ru1212/Ru1222 is actually associated with the JJA. The nature of theimhomogeneity has also been explored in Ru1222Gd by an unusual memory ef-fect (Fig. 5) [20]. Three distinct FM-like transitions have been identified at 80,120, and 160 K in Ru1222Gd, respectively (inset, Fig. 5). The weak FM do-mains of the 80 K phase, therefore, should be formed under a combinationalfield of the demagnetized field of the 120/160 K phase and the external fieldH, which causes a memory effect. It was demonstrated that the memory effectexists in individual grains, and excludes the possibility of chemical impurities.It has been further demonstrated that the strength of the field requires a special

The Nature of the Superconducting State in Rutheno-Cuprates 115

H (Oe)

0 2000 4000 6000 8000 10000

TC

(K)

10

15

20

25

30

35

T (K)

8 12 16 20 24 28 32 36 40

χ' (10

-3 emu/cm

3)

1.2

1.6

Fig. 5. The intergrain Tc vs. H. : Tc deduced from ; •: Tc deduced from the differ-ential susceptibility. Inset: The differential susceptibility, i.e. ac χ with a dc bias. :H = 0; ©: 0.2 T; : 0.4 T. Linear fits of χ above (below) Tc (solid lines). The Tc wastaken as the intersection of the two lines

space correlation between the species of the 80 K and the 120/160 K species.Ru1222Gd, therefore, seems to be made of an AFM matrix and three types ofFM-like species. Such a possible phase mixture caused by phase separations mayoffer a self-consistent explaination for the disagreement between NPD and NMR:while the NPD data are dominated by the AFM matrix due to both the volumefraction and the mesoscopic length scale of the FM species, the NMR may onlybe sensitive to the FM species due to the so-called H1-enhancement [7]. We pro-pose that a phase separation resulting in the AFM and FM species causes theJJA-like superconductivity in the rutheno-cuprates.

In summary, the data suggest that the Tc, the suppression of Tc by thefield, and the extremely field-sensitive diamagnetic effect of rutheno-cupratesare dominated by their unusually large intragrain penetration depth. A JJAmodel is proposed to interpret the data. The unusually large memory effect inRu1222 further suggests that the mixture of FM and AFM species resulting fromphase separation that is common in cuprate superconductors is the root of themicroscopic inhomogeneity.

116 C.W. Chu et al.

T (K)

60 80 100 120

M (

emu/

cm3 )

0.0

0.2

0.4

B

E D

C

T or TS (K)

0 50 100 150

MF

C a

t 5

Oe

(em

u/cm

3 )

0.01

0.1

1

A

∆M

FC (

emu/

cm3 )

0.0

0.2

TC

Tm1

Tm2

Tm3

Fig. 6. M(T) determined in the following thermo-magneto sequence: A (field-cooledunder a fixed field of 10 Oe from 200 K to a temperature TS) – B (switching field from10 Oe to a lower field of HS at TS) – C (field-cooled under HS from TS to 60 K) – D(raising the temperature under HS to 200 K) – E (field-cooled under HS from 200 Kto 60 K). From top to bottom: HS = 1.4, 1.02, 0.63, -0.04, -0.22, and -0.6 Oe. Data insteps B–C (solid symbols); Data in steps D–E (open symbols). Inset: © – MFC under5 Oe; the four vertical arrows show Tc, Tm1, Tm2, and Tm3, respectively; – ∆ MFC

at HS = 0.05 Oe but with different TS

Acknowledgments

The work in Houston is supported in part by NSF Grant No. DMR-9804325,the T. L. L. Temple Foundation, the John J. and Rebecca Moores Endowment,and the State of Texas through the Texas Center for Superconductivity at theUniversity of Houston; and at Lawrence Berkeley Laboratory by the Director,Office of Science, Office of Basic Energy Sciences, Division of Materials Sciencesand Engineering of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

References

1. L. Bauernfeind, W. Widder, H.F. Braun: Physica C 254, 151 (1995)2. I. Felner, U. Asaf, Y. Levi, O. Millo: Phys. Rev. B 55, 3374 (1997); I. Felner,

U. Asaf, S. Reich. Y. Tsabba: Physica C 311, 163 (1999)

The Nature of the Superconducting State in Rutheno-Cuprates 117

3. C. Bernhard, L.T. Tallon, C. Niedermayer, T. Blasius, A. Golnik, E. Brucher,R.K. Kremer, D.R. Noakes, C.E. Stronach, E.J. Ansaldo: Phys. Rev. B 59, 14009(1999)

4. C.W. Chu, Y.Y. Xue, S. Tsui, J. Cmaidalka, A.K. Heilman, B. Lorenz, R.L. Meng:Physica C 335, 231 (2000)

5. J.W. Lynn, B. Keimer, C. Ulrich, C. Bernhard, J.L. Tallon: Phys Rev. B 61, 14964(2000)

6. Y.Y. Xue, B. Lorenz, R.L. Meng, A. Baikalov, C.W. Chu: Physica C 364-365, 231(2001)

7. Y. Tokunaga, H. Kotegawa, K. Ishida, Y. Kitaoka, H. Takagiwa, J. Akimitsu: Phys.Rev. Lett. 86, 5767 (2001)

8. K. Kumagai, S. Takada, Y. Furukawa: Phys. Rev. B 63, 180509 (2001)9. A. Butera, A. Fainstein, E. Winkler, J. Tallon: Phys. Rev. B 63, 05442 (2001)

10. O. Chmaissem, J.D. Jorgensen, H. Shaked, P. Dollar, J.L. Tallon: Phys. Rev. B61, 6401 (2000)

11. H. Takagiwa, J. Akimitsu, H. Kawano-Furukawa, H. Yoshizawa: J. Phys. Soc. Jpn.70, 333 (2001)

12. J.L. Tallon, J.W. Loram, G.V.M. Williams, C. Bernhard: Phys. Rev. B 61, 6471(2000)

13. C. Bernhard, J.L. Tallon, E. Brucher, R.K. Kremer: Phys. Rev. B 61, 14960 (2000)14. B. Lorenz, Y.Y. Xue, R.L. Meng, C.W. Chu: Phys. Rev. B 65, 174503 (2002)15. B. Nachumi, A. Keren, K. Kojima, M. Larkin, G.M. Luke, J. Merrin, O. Tch-

ernyshov, Y.J. Uemura, N. Ichikawa, M. Goto, S. Uchida: Phys. Rev. Lett. 77,5421 (1996)

16. Z.D. Hao, J.R. Clem: Phys. Rev. Lett. 67, 2371 (1991)17. F. Izumi: J. Rigaku 6, 10 (1989)18. C. Bernhard, J.L. Tallon, C. Bucci, R. De Renzi, G. Guidi, G.V.M. Williams,

Ch. Niedermayer: Phys. Rev. Lett. 77, 2304 (1996); Phys. Rev. Lett. 80, 205(1998)

19. Y.H. Li, S. Teitel: Phys. Rev. B 40, 9122 (1989)20. Y.Y. Xue, D.H. Cao, B. Lorenz, C.W. Chu: Phys. Rev. B 65, 020511 (2002)

Coexistence of Superconductivityand Weak-Ferromagnetismin Eu2−xCexRuSr2Cu2O10−δ

I. Felner

Institute of Physics, The Hebrew University, Jerusalem, Israel 91904

Abstract. Eu2−xCexRuSr2Cu2O10−δ(Ru-2122) is the first Cu-O based system in whichsuperconductivity (SC) in the CuO2 planes and weak-ferromagnetism (W-FM) in theRu sublattice coexists. The hole doping in the CuO2 planes, is controlled by appropriatevariation of the Ce concentration and/or increasing the oxygen concentration. SC oc-curs for Ce contents of 0.4-0.8, with the highest TC=35 K for Ce=0.6. The as-preparednon-SC EuCeRuSr2Cu2O10 sample exhibits magnetic irreversibility below Tirr=125 Kand orders anti-ferromagnetically (AFM) at TM=165 K. The saturation moment at5 K is Msat=0.89 µB /Ru. Annealing under oxygen pressures, does not affect theseparameters, whereas depletion of oxygen shifts both Tirr and TM up to 169 and 215 Krespectively. TM , Tirr and Msat decrease with x, and the Ce dependent magnetic-SCphase diagram is presented. A simple model for the SC and the W-FM states is pro-posed. We argue that: (i) the system becomes AFM ordered at TM ; (b) at Tirr < TM ,W-FM is induced by the canting of the Ru moments, and (c), at lower temperaturesthe appropriate samples become SC at TC . The magnetic features are not affected bythe SC state, and the two states coexist.

1 Introduction

The general antagonism nature between (SC) and long range magnetic orderis one of the fundamental problems of condensed matter physics and has beenstudied experimentally and theoretically for almost four decades. In conventionalsuperconductors, local magnetic moments break up the spin singlet Cooper pairsand hence strongly suppress SC, an effect known as pair-breaking. Therefore, alevel of magnetic impurity of only 1%, can result in a complete loss of SC.In a limited class of intermetallic systems, SC occurs even though magneticions with a local moment occupy all of one specific crystallographic site, whichis well isolated and de-coupled from the conduction path. The study of thisclass of magnetic-superconductors was initiated by the discovery of RRh4B4and RMo6S8 compounds (R=rare-earth), and has been recently revitalized bythe discovery of the RNi2B2C system. In all three systems, both SC and anti-ferromagnetic (AFM) order states coexist. The onset of SC takes place at TC ∼2-15 K, while AFM order appears at lower temperatures (except for DyNi2B2C),thus, the ratio TN/TC is ∼ 0.1-0.5. Many of the high TC superconducting ma-terials (HTSC) contain magnetic R and are AFM ordered at low temperatures,e.g. in GdBa2Cu3O7 (TC=92 K), and TN (Gd)=2.2 K. The R sublattice is elec-tronically isolated from the Cu-O planes, and has no adverse effect upon the

C. Noce et al. (Eds.): LNP 603, pp. 118–141, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Coexistence of Superconductivity and Weak-Ferromagnetism 119

Fig. 1. The crystal structure of YBa2Cu3O7 and Eu1.5Ce0.5RuSr2Cu2O10−δ

superconducting state. Much attention has been focused on a phase resemblingthe RBa2Cu3O7 materials, having the composition R1.5Ce0.5MSr2Cu2O10 (M-2122, M= Nb, Ru or Ta) [1]. The tetragonal M-2122 structure (space groupI4/mmm) evolves from the RBa2Cu3O7 structure by inserting a fluorite typeR1.5Ce0.5O2 layer instead of the R layer in RBa2Cu3O7, thus shifting alternateperovskite blocks by (a+b)/2 (Fig. 1). The M ions reside in the Cu (1) siteand the hole doping of the Cu-O planes, which results in metallic behavior andSC, can be optimized with appropriate variation of the R/Ce ratio [2]. In Ru-2122, SC occurs for Ce contents of 0.4-0.8, and the highest TC was obtainedfor Ce=0.6. The Nb-2122 and Ta-2122 materials are SC with TC ∼ 28-30 K.Coexisting of weak-ferromagnetism (W-FM) and SC was discovered in 1997 inR1.5Ce0.5RuSr2Cu2O10 (R=Eu and Gd) materials [3,4,5,6], and more recently [7]in GdSr2RuCu2O8 (Ru-1212). In both systems, the magnetic order coexists withSC and remains unchanged when SC sets in at TC .

The Ru-2122 materials (for R=Eu) display a magnetic transition at TM=125-180 K and bulk SC below TC = 32-50 K (TM >TC) depending on oxygenconcentration and sample preparation [6]. Here TM/TC ∼ 4, a trend which iscontrary to that observed in the intermetallic systems. The SC charge carriersoriginate from the CuO2 planes and the W-FM state is confined to the Ru lay-ers. Specific heat studies show a sizeable typical jump at TC and the magnitudeof the ∆C/T (0.08 mJ/gK2) indicates clearly the presence of bulk SC [8]. The

120 I. Felner

specific heat anomaly is independent of the applied magnetic field. SC survivesbecause the Ru moments probably align in the basal planes, which are practicallyde-coupled from the CuO2 planes, so that there is no pair breaking. Scanningtunneling spectroscopy (STM) [3,6], muon-spin rotation [9] and magneto-opticexperiments [10] have demonstrated that all materials are microscopically uni-form with no evidence for spatial phase separation of SC and magnetic regions.That is, both states coexist intrinsically on the microscopic scale. In the Ru-2122, the W- FM state, as well as irreversibility phenomena, arise as a result ofan antisymmetric exchange coupling of the Dzyaloshinsky-Moriya (DM) type [3]between neighboring Ru moments, induced by a local distortion that breaks thetetragonal symmetry of the RuO6 octahedra. Due to this DM interaction, thefield causes the adjacent spins to cant slightly out of their original direction andto align a component of the moments with the direction of applied field. Belowthe irreversible temperature (Tirr), which is defined as the merging temperatureof the zero-field (ZFC) and field-cooled (FC) curves, the Ru-Ru interactions be-gin to dominate, leading to reorientation of the Ru moments, which leads toa peak in the magnetization curves. The most remarkable magnetic propertiesof the Ru-2122 materials are: (a) a negative magnetic moments in the ZFCbranches measured at low applied fields (H) is observed, (b) the ferromagnetic-like hysteresis loops and strong enhancement of coercive field which appear onlyin the SC state at T < TC , (c) the so-called spontaneous vortex phase (SVP)model, which permits magnetic vortices to be present in equilibrium without anexternal field [4]. The vortices in the SC planes, are caused by the internal fieldBint= 4πM (higher than Hc1) of a few hundreds of G of the FM Ru sublat-tice [9]. (d) No diamagnetic signal, in the FC branch (the Meissner state (MS)-the conventional signature of a bulk SC), has been observed. The absence of theMS, may be a result of the SVP, and/or the high Ru moment induced by theexternal field at TM , which masks this SC signature. On the other hand, whenRu is partially replaced by Nb, the small positive contribution of the W-FM Rusublattice decreases the internal field and the MS is readily observed.

2 Experimental Details

Samples of Eu2−xCexRuSr2Cu2O10 and Eu1.5Ce0.5Ru0.6Nb0.4Sr2Cu2O10(Ru,Nb) – 2122) were prepared by a solid state reaction technique as describedelsewhere [3,6]. Eu3+ is used in order to diminish the paramagnetic contribu-tion of Gd3+. Parts of the as prepared samples (ASP) were re-heated for 24h at 800oC under various pure oxygen pressures up to 150 atm and will beidentified according to applied pressure. Determination of the absolute oxygencontent in the ASP material and in the samples annealed under oxygen pres-sures, is difficult because CeO2 is not completely reducible to a stoichiometricoxide when heated to high temperatures. Thermo – gravimetric measurements ofEu1.5Ce0.5RuSr2Cu2O10 show that the materials are stable up to 600oC and nooxygen weight loss is detected. Above this temperature a small weight decreasebegins and our analysis indicates that the sample annealed at 150 oxygen atm

Coexistence of Superconductivity and Weak-Ferromagnetism 121

(150 atm) contains ∼4 at % more oxygen than the ASP sample. The hydrogencharging procedure is described in [6]. ZFC and FC dc magnetic measurementsin the range of 5-300 K were performed in a commercial (Quantum Design)super-conducting quantum interference device (SQUID) magnetometer. The re-sistance was measured by a standard four contact probe and the ac susceptibilitywas measured by a home-made probe with excitation frequency and amplitudeof 733 Hz and 30 mOe respectively, both inserted in the SQUID magnetometer.

3 X-ray Diffraction Studies

Powder X-ray diffraction (XRD) measurements of Eu1.5Ce0.5RuSr2Cu2O10−δ

confirmed the purity of the compounds (∼ 97%) and indicate within the instru-mental accuracy, that all samples have the same lattice parameters a=3.846(1)A and c=28.72(1) A. Due to the similarity of the ionic radii of Eu3+ (0.94 A)and Ce4+(0.87 A) and within the instrumental accuracy, the lattice parame-ters of Ru-2122 materials are independent of Ce content. The diffraction scansof Eu1.5Ce0.5RuSr2Cu2O10−δ at various temperatures indicate clearly that allpeaks and their relative intensities remain unaltered down to 10 K [11]. Thediffraction scans, do not show any anomalies around TC and TM . The a andc lattice parameters are linear down to 100 K, and the typical flattening ofthe thermal expansion behavior is observed at low temperatures. The detailedcrystal structure and the atomic positions Ru-2122 were studied by synchrotronX-rays diffraction [12,13] and neutron diffraction [14] experiments, which showthat the RuO6 octahedra are rotated ∼ 14o around the c-axis and that this ro-tation is essentially the same for x=1 and x=0.6 as well as for Ru-1212. Thereis no evidence for super-cell peaks in the Ru-2122 samples.

4 The Effect of Ce on TC and TM

in Eu2−xCexRuSr2Cu2O10−δ

4.1 Superconductivity in Eu2−xCexRuSr2Cu2O10−δ:Results and Model

The temperature dependence of the normalized real ac susceptibility curves (atHdc=0) of Eu2−xCexRuSr2Cu2O10−δ are presented in Fig. 2. It is readily ob-served that the x=1 and 0.9 are not SC and that SC occurs for Ce contents ofx=0.8-0.4. Figure 3 exhibits the onset of SC deduced from these ac plots whichexhibit a bell shape behavior with a peak at 35 K for Ce=0.6. Similar values areobtained by our resistivity measurements (not shown). A similar bell shape be-havior was observed in the Eu2−xCexNbSr2Cu2O10−δ (x=0.4-1) system, whichserve as our reference materials. It is apparent in Fig. 2 that the SC transitions,are much broader than those observed in many other HTSC materials. Thistransition width is comparable to data published in [12,13]. Such a broadeningis typical of under-doped Cu based high TC materials where inhomogeneity inthe oxygen concentration causes a distribution in the TC values [6]. In addition,the SC transition width may be due to the SPV [4].

122 I. Felner

Fig. 2. Normalized ac susceptibility, (Hdc =0) of ASP Eu2−xCexRuSr2Cu2O10 sam-ples. Note, that the absence of SC in the two x=1 and x=0.9 materials

Fig. 3. The bell shape SC onset temperature as a function of Ce content

Coexistence of Superconductivity and Weak-Ferromagnetism 123

Fig. 4. XAS spectra at the K edge of Ru of Ru-2122 and reference compounds

4.2 Superconductivity in the M-2122 System

Given the variety of crystal structures and the chemical methods used to intro-duce holes into the CuO2 layers, it is well accepted that a ‘generic’ electronicphase diagram can be sketched for all compounds. Hole (or carrier) density(p) in the CuO2 planes, or deviation of the formal Cu valence from Cu2+,is a primary parameter which affects TC in most of the HTSC compounds.In the well-established phase diagram La2−xSrxCuO4, the insulating parentLa2CuO4 is AFM ordered. The magnetic interactions are well described by asimple Heisenberg model, with a large exchange interaction (J= 1500 K) value.In La2−xSrxCuO4 the charge carrier concentration, can be varied by replacingSr2+ for La3+. The variation of TC as a function of hole doping exhibits a bellshape behavior, with a peak for the optimally doped material (x=0.15). The va-lence of Eu, Ce and Ru ions in Eu1.5Ce0.5RuSr2Cu2O10−δ have been studied byMossbauer spectroscopy (MS) and X-ray-absorption spectroscopy (XAS) tech-niques [15,16]. MS performed on 151Eu show a single narrow line with an isomershift =0.69(2) and a quadrupole splitting of 1.84 mm/s, indicating that the Euions are trivalent with a nonmagnetic J=0 ground state. This is in agreementwith XAS taken at LIII edges of Eu and Ce that shows that Eu is trivalent andCe is tetravalent.

The local electronic structure in several Ru based compounds was studiedby XAS at the K edge of Ru, and the results obtained at room temperatureare shown in Fig. 4. Since the valence of Gd3+, Sr2+ and O2 are conclusive,

124 I. Felner

a straightforward valence counting for GdSr2RuO6 and SrRuO3 yields Ru, asRu+5 and Ru+4 ions respectively. The similarity between the XAS spectra ofRu-2122 and GdSr2RuO6 indicates clearly, that in Ru-2212 the Ru ions are ina pentavalent state. It is apparent that SC in the M-2122 system exists only forpentavalent M ions such as Nb, Ta and Ru. We argue that the RCeMSr2Cu2O10(R/Ce=1) samples serve as the parent stoichiometric insulator compounds (sim-ilar to La2CuO4). Since the valence of R3+, Ce4+, M5+ Sr2+ Cu2+ and O2 areconclusive, a straightforward valence counting yields a fixed oxygen concentra-tion of 10. Hole doping of the Cu-O planes, which results in metallic behaviorand SC, can be optimized with appropriate variation of the R3+/ Ce4+ ratio(trivalent R3+ ions are replaced for Ce4+). SC occurs for Ce contents of 0.4-0.8(Figs. 2-3), and the optimally doped sample is obtained for Ce=0.6. Unlike theLa2−xSrxCuO4 system, this substitution does not appear to significantly alterthe hole carriers on the Cu-O planes. This is apparent in Fig. 3 which shows thatthe change of x from 0.8 to 0.6, results in a small increase in TC . Indeed, if all thecarriers were induced into the Cu-O planes, then for the under-doped (x=0.8) tothe optimally doped (x=0.6) samples, p should vary by 0.2 and result in a largeshift in TC , as observed in La2−xSrxCuO4 and in other HTSC materials. It isthus possible, that in all M-2122 compounds, the extra holes introduced by re-ducing the Ce content, are partially compensated for by depletion of oxygen [12]and in R2−xCexMSr2Cu2O10−δ the oxygen deficiency (d) increases with R3+.

4.3 The Magnetic Properties of EuCeRuSr2Cu2O10

Figure 2 shows that EuCeRuSr2Cu2O10 is not SC. ZFC and FC dc magneticmeasurements for the ASP sample, were performed over a broad range of appliedmagnetic fields and typical M/H curves measured at 50 Oe, are shown in Fig. 5.Note the ferromagnetic-like shape of the FC branches. The two curves merge atTirr=125 K. TM (Ru) is not at Tirr. The M/H(T ) curves do not lend themselvesto an easy determination of TM (Ru), and TM (Ru)=165 K, was obtained directlyfrom the temperature dependence of the saturation moment (Msat), discussedbelow. Tirr is field dependent, and shifted to lower temperatures with the appliedfield. Tirr =91, 64, 39, 28 and 14 K for H= 250, 500, 1000, 2000 and 3000 Oerespectively. For higher external fields the irreversibility is washed out, and bothZFC and FC curves collapse to a single ferromagnetic-like behavior [3]. It appearsthat the magnetic properties of Ru in EuCeRuSr2Cu2O10 are all enhanced, ascompared to the ASP Eu2−xCexRuSr2Cu2O10−δ samples with x<1 as shownbelow.

Similar measurements were performed to EuCeRuSr2Cu2O10 annealed (at800 o C) under oxygen at 60 atm (hop), and Fig. 5 shows that both Tirr andTM (Ru) remain unchanged. Since the oxygen concentration in EuCeRuSr2Cu2O10is fixed, it does not change during the annealing process. As shown below, this isin contrast to Eu1.5Ce0.5RuSr2Cu2O10−δ where annealing under the same oxy-gen pressure affects Tirr and TM (Ru) and shift them to higher temperatures (seeFigs. 13-15). On the other hand, when the the ASP material is quenched from1050oC to ambient temperature, a small amount of oxygen is depleted and both

Coexistence of Superconductivity and Weak-Ferromagnetism 125

Fig. 5. ZFC and FC susceptibility curves for EuCeRuSr2Cu2O10 samples, ASP, an-nealed under 60 oxygen atmosphere and quenched materials measured at 50 Oe

Tirr and TM (Ru) are shifted to 169 and 215 K respectively. It is reminiscentof the magnetic phase-diagram of YBa2Cu3Oz (z<6.5), where the depletion ofoxygen increases the magnetic transition of the CuO2 planes.

M(H) measurements at various temperatures for the ASP, hop and quenchedsamples have been carried out, and the results obtained for the ASP sample areexhibited in Figs. 6-8. All M(H) curves below TM , are strongly dependent onthe field (up to 2-4 kOe), until a common slope is reached (Fig. 6 inset). M(H)can be described as: M(H) = Msat+χ H, where Msat corresponds to the W-FMcontribution of the Ru sub lattice, and χ is the linear paramagnetic contributionof Eu and Cu). The saturation moment obtained at 5 K is Msat = 0.89(1)µB.Similar M(H) curves have been measured at various temperatures and Fig. 7(inset) shows the decrease of Msat with temperature. Msat becomes zero atTM (Ru)=165(2). Similar Msat and TM (Ru) values were obtained for the hopmaterial. However, for the quenched material, Msat at 5 K remains unchanged,but TM is shifted to 215 (2) K. Thus, only the magnetic transitions are sensitiveto the oxygen concentration. Msat=0.89 µB, is somewhat smaller than the fullysaturated moment 1µB expected for the low-spin state of Ru5+, i.e. gµBS forg=2 and S=0.5. This means that a small canting on adjacent Ru spins occurs,

126 I. Felner

Fig. 6. The temperature dependence of 5 K remanent moment of EuCeRuSr2Cu2O10.The inset shows the high field magnetization and the saturation moment at 5 K

and the saturation moments are not the full moments of the Ru5+ ions. Theexact nature of the local structural distortions causing DM exchange couplingin Ru-2122 (see above) is not presently known.

At low applied fields, the M(H) curve exhibits a typical ferromagnetic-likehysteresis loop (Fig. 8). The positive virgin curve at low fields, indicates clearlythat SC is totally suppressed. Two other characteristic parameters of the hys-teresis loops are marked, namely, the remanent moment, (Mrem= 0.41µB/Ru)and the coercive field(HC =-190 Oe at 5 K). This Mrem is much larger thanMrem=0.035 µB obtained for EuSr2RuCu2O8 [12]. The same Mrem and HC

values (at 5 K) were obtained for the hop and quenched materials. Figure 6shows the temperature dependence of Mrem (5 K) which disappears at Tirr.The large Mrem/Msat ratio (at 5 K) is consistent with ferromagnetic-like orderin EuCeRuSr2Cu2O10. M(H) curves measured at various temperatures yieldthe Mrem(T) and HC(T ) values which are plotted in Fig. 7 (inset). For boththe ASP and quenched samples, Mrem(T ) also disappear at Tirr, and HC(T )becomes zero around 80 K and 130 K respectively (Fig. 8).

Coexistence of Superconductivity and Weak-Ferromagnetism 127

Fig. 7. The coercive field HC as function of temperature for ASP and quenchedEuCeRuSr2Cu2O10 samples. The inset shows the temperature dependence of the sat-uration and the remanent moments

In the paramagnetic range (above 165 K), the χ(T ) curve measured at 10kOe for ASP EuCeRuSr2Cu2O10, has the typical paramagnetic shape and hasa behaviour close to the Curie-Weiss (CW) law: χ = χ0 + C/(T − θ), where χ0is the temperature independent part of χ, C is the Curie constant, and θ is theCW temperature. The net paramagnetic Ru contribution to χ(T ), was obtainedby subtracting χ(T ) of EuCeNbSr2Cu2O10 (the reference material) from themeasured data. This procedure yields: χ0=0.0063 and C=0.57(1) emu/mol Oeand θ= 146(1) K, which corresponds to an effective moment Peff =2.13 µB.

The positive θ obtained is in fair agreement with TM and indicates ferromag-netic fluctuations. But Peff is greater than the expected value of the low-spinstate of Ru5+ and S=0.5 (Peff =1.73 µB).

4.4 The Effect of Ce on the Magnetic Propertiesof Eu2−xCexRuSr2Cu2O10−δ

All the Eu2−xCexRuSr2Cu2O10−δ compounds reported here have been preparedsimultaneously under the same conditions and our extensive magnetic studyshows, that the magnetic behavior of all materials is quite similar to these de-scribed in Figs. 5-8. For the sake of brevity, we display in Fig. 9 only the data

128 I. Felner

Fig. 8. The hysteresis low field loop at 5 K

obtained for x=0.1 and x=0.5. It is readily observed, that for x=1, the magneticproperties due to the Ru are all enhanced, as compared to the x=0.5.

The latter sample is SC and its ZFC branch starts from negative values.At 50 Oe the diamagnetic signal due to high shielding fraction (SF) of the SCstate dominates, and the net moment at low temperatures is negative. The weakferromagnetic component of Ru and the high paramagnetic effective moment ofEu3+ do not permit a quantitative determination of the SF from these curves.The inflection in the FC branch agrees well with TC determined from the ac curve(Figs. 2-3). The absence of a complete Meissner effect in the Ru-2122 system isdiscussed below. The variation of Tirr and TM as a function of Ce concentrationin Eu2−xCexRuSr2Cu2O10−δ, is summarized in Fig. 10. The enhancement ofthe magnetic properties is manifested by the monotonic rise of Tirr and TM asx increases. Msat values at 5 K, increase gradually with x, Msat =0.43, 0.46,0.60 0.67 and 0.86 µB for x=0.4, 0.6, 0.7, 0.8 and 0.9 respectively. It is apparentthat this trend is not affected by the SC state which is induced for x=0.8.Due to the presence of a tiny amount of SrRuO3 in the x=0.4 sample (notdetectable by XRD), its TM was not determined. Above TM , the paramagneticparameters extracted using the CW law, have been obtained by subtracting theparamagnetic moment of the relevant Nb-2122 compound, as described above.

Coexistence of Superconductivity and Weak-Ferromagnetism 129

Fig. 9. ZFC and FC susceptibility curves for ASP x=1 and x=0.5 samples

It appears that the C and θ parameters for all materials, are very close to thoseof the x=1 sample i.e. for x=0.5, C=0.58 emu/mol Oe, (Peff=2.15 µB) and θ=134(1) K, indicating similar net paramagnetic Ru contribution in all Ru-2122compounds.

4.5 Mossbauer Effect of 57Fe Doped in Gd1.4Ce0.6RuSr2Cu2O10−δ

Mossbauer effect studies (ME) on 57Fe doped samples has been proved to be apowerful tool in the determination of the magnetic nature of the Fe site location.When the Ru ions become magnetically ordered, they produce an exchange fieldon the Fe ions residing in this site. The Fe nuclei experience a magnetic hyperfinefield leading to a sextet in the observed ME spectra. As the temperature is raised,the magnetic splitting decreases and disappears at TM . Figure 11 shows thereal and imaginary ac susceptibility of Gd1.4Ce0.6RuSr2Cu2O10−δ, from whichTirr=92 K and TM (Ru)=175 K are deduced and Fig. 12 shows the ME studieson the 57Fe doped material. The main effect to be seen in Fig. 12 is that the 4.2and 180 K spectra, consist of one site only.

A least square fit to the spectrum at 180 K yields an isomer shift (IS) of0.30(1) mm/s (relative to Fe metal) with a line width of 0.35 mm/s, and a

130 I. Felner

Fig. 10. TM and Tirr as function of Ce in Eu2−xCexRuSr2Cu2O10

quadrupole splitting (∆Q = 1/2eqQ) of 1.00(1) mm/s3. This doublet is at-tributed to paramagnetic Fe ions in the Ru site [3]. At low temperatures, all spec-tra display magnetic hyperfine splitting, which is a clear evidence for long-rangemagnetic ordering. The fitting parameters of the single sextet obtained at 4.1 Kare: IS= 0.40(1)mm/s, Heff (0)=467(3) kOe and an effective quadrupole split-ting value of ∆eff = 1/2eQqeff = -0.33(21) mm/s. Using the relation: ∆eff=∆Q/2(3cos 2θ-1), we obtained for the Ru site a hyperfine field orientation, θ=72o, relative to the tetragonal symmetry c axis. As the temperature is raised,Heff decreases and disappears completely at TM (Ru)= 175(5) K. Heff valuesobtained at 110, 130, 150 K and 160 K are: 399(3), 358(5), 312(2) and 279(5)kOe. Figure 12 also shows that ME spectrum at 90 K (below Tirr) is much differ-ent than that of 110 K (above Tirr). In between these temperatures the cantingspin reorientation occurs, well felt by the Fe probe.

5 The Magnetic Structure of Eu2−xCexRuSr2Cu2O10−δ

Our general picture is that in Eu2−xCexRuSr2Cu2O10−δ all compounds havea similar magnetic structure. We suggest two scenarios that could lead to the

Coexistence of Superconductivity and Weak-Ferromagnetism 131

Fig. 11. Real and imaginary ac susceptibility of Gd1.4Ce0.6RuSr2Cu2O10−δ

shift to higher values, of the magnetic parameters (such as TM , Tirr and Msat)with increasing Ce (see Fig. 10 a). The small difference between Eu3+ and andCe4+ ionic radii discussed above, decreases the mean Ru-Ru distance, and asa result the magnetic exchange interactions become stronger with Ce, and (b)this enhancement arises from a change of the anti-symmetric exchange couplingof the DM type between the adjacent Ru moments, which causes the spins tocant out of their original direction to a larger (or smaller) angle, and as a re-sult, a different component of the Ru moments forms the W-FM state. Whilethe data do not include any determination of the magnetic structure of the Rusublattice in Ru-2122, our results are compatible with a simple model whichmay be used for understanding the qualitative features at low applied fields.The magnetic behavior is basically divided into 4 regions. (i) At elevated tem-peratures, the paramagnetic net Ru moment is well described by the CW law,and the extracted Peff =2.15 µB and θ=134-146 K values, practically do notalter with Ce. (ii) At TM (depends on Ce content Fig. 10), the Ru sub latticebecomes AFM ordered. (iii) At Tirr < TM , which is also Ce dependent and varieswith the external field, a weak ferromagnetism is induced, which originates fromcanting of the Ru moments. Tirr is defined as the merging point of the lowfield ZFC and FC branches, or alternatively, as the temperature in which theremanent moment disappears. This canting arises from the DM antisymmetric

132 I. Felner

Fig. 12. Mossbauer spectra of Gd1.4Ce0.6RuSr2Cu2O10−δ

Coexistence of Superconductivity and Weak-Ferromagnetism 133

superexchange interaction, which by symmetry, follows from the fact that theRuO6 octahedra tilt away from the crystallographic c axis. At high magneticfield (H >3000 Oe) the irreversibility is washed out and the M(T ) curves ex-hibit a ferromagnetic-like behavior.(iv) At lower temperatures SC is induced.Figure 3 shows that TC depends strongly on the R3+/Ce4+ (as hole carriers)and on oxygen concentrations [6]. Below TC , both SC and weak-ferromagneticstates coexist and the two states are practically decoupled. This model is sup-ported by our Mossbauer studies (Fig. 12) and also from unpublished non-linearac susceptibility measurements, which show non-linear signals up to TM . Thisinterpretation, differs completely from the phase separation of AFM and FMnano-domain particle scenario, suggested in [17]. Neutron diffraction measure-ments are required to precisely determine the nature of the magnetic order inthe Ru-2122 system.

6 The Effect of Oxygen on the SC and Magnetic Behaviorof Eu1.5Ce0.5RuSr2Cu2O10−δ

The temperature dependence of the normalized resistance R(T) for ASPEu1.5 Ce0.5 Ru Sr 2 Cu 2 O 10−δ and 22 atm samples (measured at H=0) isshown in Fig. 13. The onset of the SC transition for the ASP (TC =32 K) isshifted to 38 K. At high temperatures, a metallic behavior is observed, and forthe ASP sample, an applied field of 5 T smears the onset of SC and shifts it to28 K. The SC transition for the ASP sample is more easily seen in the derivativedR/dT plotted in the inset. At TC =32 K the derivative rises rapidly and doesnot fall to zero until the percolation temperature around 19 K is obtained. Thisbehavior is typical for under-doped HTSC materials, where inhomogeneity inoxygen concentration causes a distribution in the TC values (see also Fig. 3).This distribution is also reflected in the broad range of gap values observed inour STS data, as shown below. The dependence of TC on the applied oxygenpressure obtained from resistivity measurements, is presented in Fig. 14, exhibit-ing a monotonic increase from 32 to 49 K. R(T ) curves of the 75 atm samplemeasured at various applied fields are shown in Fig. 15. In contrast to the ASPsample, an applied field of 5 T only smears the onset of SC at 46 K, but doesnot shift it to lower temperatures. The temperature dependence of dR/dT atH=0 T, shows two peaks (Fig. 15 inset). This provides clear evidence for thetwo major SC phases having TC at 32 and 46 K, where the latter is below thepercolation threshold. This is consistent with the STM data shown in Fig. 16and Fig. 13. Normalized resistivity measured at H=0 of the ASP Ru-2122 andthe sample annealed under 22 atm. The inset shows the derivative of the R(T )for the ASP sample.

The spatial distribution of the SC gap (∆) on the surface of the ASP and 75atm samples are exhibited by the histograms in Fig. 16(a) and (b). The gaps wereextracted by fitting the Dynes’ function [18] to tunneling I-V curves acquired atvarious lateral tip positions. In the ASP sample, the I-V curves show an ohmicgap-less structure, and the values of ∆ range mainly between 3 and 5.5 meV.

134 I. Felner

Fig. 13. Normalized resistivity measured at H=0 of the ASP Ru-2122 and the sampleannealed under 22 atm. The inset shows the derivative of the R(T ) for the ASP sample

This broad distribution probably results from spatial variations in hole-doping,and is consistent with the broad SC transition exhibited in Figs. 2 and 13.

In Fig. 16 b, two peaks are clearly observed in the distribution, showing theexistence of two SC phases, (a trace of the higher TC phase is already present inthe ASP sample). The ratio between the large and small gap values is 1.45, inagreement with the ratio between the two peaks extracted from Fig. 15 (inset).Note that the small-gap phase (lower TC) is dominant, consistent with the factthat the higher TC phase should be below the percolation threshold. The ZFCand FC values of the 75 atm sample are much higher, and both Tirr and TM (Ru)are shifted to 137(2) and 168(2) K respectively [6]. The estimated SF (takinginto account contributions from Ru and Eu3+) are ∼30 % and 65%, for the ASPand 75 atm samples respectively. For the 150 atm sample we obtain TC= 49 K(Fig. 14), Tirr=178 K, and TM (Ru)=225 K [6]. The STM results are similar tothe 75 atm sample (Fig. 16 b), but exhibit an increase of the relative abundanceof the large gaps.

Oxygen pressure, enhance TM and changes other W-FM characteristic fea-tures of the ASP material. This effect, which was also observed in several rare-earth based intermetallic hydrides, is probably an electronic effect. As describedabove, in addition to the change in the hole density of the Cu-O planes, thereis a transfer of electrons from oxygen to the Ru 4d sub-bands, resulting in an

Coexistence of Superconductivity and Weak-Ferromagnetism 135

Fig. 14. The effect of the annealing oxygen pressure on TC

increase of the exchange interactions between the Ru sublattice and hence toan increase in TM of the materials. An alternative way is to assume that thisenhancement arises from an alternation of the anti-symmetric exchange couplingof the DM type between the adjacent Ru moments, which causes the spins tocant out of their original direction with a smaller (or larger) angle and as aresult, a different component of the Ru moments forms the W-FM state. Theexact nature of the local structure distortions causing the W-FM behavior inthis system, as well as the extra oxygen location, are not presently known.

7 The Effect of Hydrogenon the SC and Magnetic Behavior of Ru-2122

Figure 17 shows that in Ru-2122H0.07, the effect of hydrogen is to suppress SCand to enhance the W-FM properties of the Ru sublattice (TM increased to 225K). Two scenarios that could lead to this phenomenon are: (a) in addition to thechange of p in the CuO2 planes, there is a transfer of electrons from hydrogen tothe Ru 4d sub-bands, resulting in an increase in the Ru moments, and hence toenhance the magnetic parameters. Indeed, XAS data indicate, that in contrast toASP Ru-2122 (Fig. 4), the hydrogen loaded samples exhibit a dominant Ru4+

valence [16];(b) the enhancement arises from a change of the anti-symmetric

136 I. Felner

Fig. 15. Normalized resistivity measured at various magnetic applied field of sampleannealed under 75 oxygen atmosphere. The inset shows the derivative of the resistivitycurve at zero applied field

exchange coupling of the DM type between the adjacent Ru moments, whichcauses the spins to cant out of their original direction to a larger angle. Thepicture emerges from the STM measurements [6] is that hydrogen doping leadsto phase separation. Even at very low doping (Ru-2122H0.03), insulating regionsstart to form. As doping is increased, the density and size of the insulatingregions increase, until they coalesce and the sample becomes globally insulating.Figure 18 presents the ZFC curves obtained for several hydrogen loaded samples.For the ASP and the Ru-2122H0.03 samples, the peak is around 80 K, and forthe samples with H>0.14 atm the peaks are shifted to about 160 K. For theintermediate hydrogen concentration H=0.07 a superposition of both peaks isobserved which leads to a somewhat flat curve. The hydrogen atoms reside ininterstitial sites, and their effect is reversible. Depletion of hydrogen leads tothe original charge density and SC is restored. This is reflected in both themacroscopic magnetization studies and in the SC gap distribution extractedfrom STM result. TM drops back to 122 K and the peak in ZFC curve is shiftedback to 80K [6]. Data for the regenerated sample are not presented here. Sincehydrogen loading affects both (SC and W-FM) phenomena, we tend to believethat H atoms occupy interstitial sites, presumably inside the Sr-O planes.

Coexistence of Superconductivity and Weak-Ferromagnetism 137

Fig. 16. Histograms showing the spatial distribution of the SC energy gaps for theASP sample (a) and the sample annealed at 75 atm oxygen (b). Inset: Two tunnelingdI/dV vs. V curves obtained on the annealed sample, one taken on a region of smallgaps (dotted), the other on a large-gap region (solid)

8 The Mixed (Ru,Nb)-2122 System

ZFC and FC curves for the mixed Eu1.5Ce0.5Ru0.6Nb0.4Sr2Cu2O10−δ sample(annealed under 50 atm of oxygen), are shown in Fig. 19. TC is shifted to 41K (confirmed also by four point resistivity measurements), and the ZFC andFC branches merge at Tirr=142 K. All M (H) curves below TM , are stronglydependent on the field until a common slope is reached (see Fig. 6). At 5 K,Msat = 0.15 µB a value which is much smaller than 0.72(1) µB obtained forASP Eu1.5Ce0.5RuSr2Cu2O10−δ sample, Msat decreases with T, and becomes

138 I. Felner

Fig. 17. ZFC and FC susceptibility curves for ASP and Ru-2122H0.07

Fig. 18. ZFC susceptibility curves for various hydrogen loaded samples

Coexistence of Superconductivity and Weak-Ferromagnetism 139

Fig. 19. ZFC and FC susceptibility curves for oxygen annealed Ru,Nb -2122. Note theflux expulsion at TC

zero at TM (Ru)=160(2) K. Thus, reduction of the Ru content does not changethe typical coexistence of SC and W-FM found in the Ru-2122 system. (Theenhancement of TC and TM is consistent with Fig. 14). The clear peak at TC

in both the ZFC and FC branches, and the negative signal in the ZFC curvebelow TC are quite evident. On the other hand, in contrast to Fig. 5, the typicalexpulsion of magnetic flux lines at TC (the Meissner state) in the FC curve,as well as the flatness at low temperatures, are readily observed. The smallpositive contribution of the W-FM Ru sublattice to the total magnetic momentdecreases the Ru moment, and as a result the internal field is weaker. Here, theSC properties are not masked by the reduced W-FM features and the typicalMsat is clearly observed, although the M/H values in the FC branch are positive.This proves that the absence of the MS in Ru-2122 is caused either (or both)by the significantly high Ru moment contribution to the overall moment, andby the enhanced internal fields which lead to the SVP. The hysteresis loop at 5K, at low applied fields is shown in Fig. 20. Note: (a) the increase of negativemoments up to 200 Oe in the virgin curve, and (b) the particular hysteresis loopobtained at low applied fields. In contrast to the FM hysteresis loop shown inFig. 8 this loop is a superposition of SC and W-FM properties of the sample.

140 I. Felner

Fig. 20. A superposition of SC and W-FM properties in the hysteresis loop at 5 K for(Ru,Nb) -2122

9 Conclusions

We have shown that both SC and weak-ferromagnetism coexist in Ru-2122and are an intrinsic property of this system. In contrast to other intermetallicmagnetic-SC systems, the present materials exhibit magnetic order well abovethe SC transition (TM/TC ∼4). We attribute the magnetic order to the Ru sub-lattice, whereas SC is confined to the CuO2 planes. Both sites are practicallydecoupled from each other. Hole doping of the Cu-O planes, which results in SC,can be optimized with either (i) appropriate variation of the Eu3+/ Ce4+ ratioand the optimally doped material is obtained for Ce=0.6, and (ii) by annealingunder high oxygen pressures which leads to increase the oxygen concentration.The magnetic insulator parent EuCeRuSr2Cu2O10 (x=1), is used to describe themagnetic behavior of the Ru-2122 system. For x=1 (d=0), annealing under highoxygen pressure does not affect the magnetic properties, whereas TM is enhancedby oxygen depletion. The magnetic structure of all Eu2−xCexRuSr2Cu2O10−δ

materials studied is practically the same, but the magnetic parameters, such asTM and Msat, increase with increasing oxygen (as well as hydrogen) content,but decrease with decreasing Ce content. In Ru-2122, the ASP compound, isunder-doped, and annealing under high oxygen pressure shifts TC to higher tem-peratures. On the other hand, the influence of hydrogen on Ru-2122 is reversibleand not destructive, which means that hydrogen changes the hole density of theCuO2 planes, either by increasing or decreasing the ideal effective charge of the

Coexistence of Superconductivity and Weak-Ferromagnetism 141

planes. Two steps in the magnetic behavior are presented. At TM ranging from122 K (for x=0.5) up to 225 K (for hydrogen loaded samples), all materials be-come AFM ordered. At Tirr (depends on various parameters) a W-FM state isinduced, originating from canting of Ru moments. This canting arises from theDM anti-symmetric super-exchange interaction and follows from the fact thatthe RuO6 octahedra tilt away from the crystallographic c axis. A direct magneticstructure determination by neutron diffraction or 99Ru Mossbauer spectroscopystudies are warranted to confirm our assumptions.

Acknowledgments

This research was supported by the Israel Academy of Science and Technologyand by the Klachky Foundation for Superconductivity.

References

1. R.J. Cava, J.I. Krajewski, H. Takagi, H. Zandbergen, H.W. Van Dover, R.B. PeckJr, B. Hessen: Physica C 191, 237 (1992)

2. L. Bauernfeind, W. Widder, H.F. Braun: Physica C 254, 151 (1995)3. I. Felner, U. Asaf, Y. Levi, O. Millo: Phys. Rev. B 55, R3374 (1997)4. E.B. Sonin, I. Felner: Phys. Rev. B. 57, R14000 (1998)5. I. Felner, I. Asaf U. Goren, S.D. Goren, C. Korn.: Phys. Rev. B 57, 550 (1998)6. I. Felner, U. Asaf, Y. Levi, O. Millo: Physica C 334, 141 (2000)7. D.J. Pringle, J.L. Tallon, J.L. Walker, H.J. Trodahl: Phys. Rev. B 59, R11679

(1999)8. X.H. Chen, Z. Sun, K.Q. Wang, S.Y. Li, Y.M. Xiong, M. Yu, L.Z. Cao: Phys. Rev.

B 63, 064506 (2001)9. A. Shengelaya et al. (to be published)

10. S.Y. Chen et al.: Phys. Rev. B (submitted); cond-mat/010551011. I. Felner, U. Asaf, F. Ritter, P.W. Klamut, B. Dabrowski: Physica C 364-365, 368

(2001)12. G.V.M. Willaams, M. Ryan: Phys. Rev. B 64, 094515 (2001)13. G.V.M. Willaams et al.: Phys. Rev. B (2001) submitted14. C.S. Knee, B.D. Rainford, M.T. Weller: J. Mater. Chem. 10, 2445 (2000)15. I. Felner, U. Asaf, C. Godart, E. Alleno: Physica B 259-261, 703 (1999)16. Y. Hirai, I. Zivkovic, B.H. Frazer, A. Reginelli, L. Perfetti, D. Ariosa, G. Margari-

tondo, M. Prester, D. Drobac, D.T. Jiang, Y. Hu, T.K. Sham, I. Felner, M. Ped-erson, M. Onellion: Phys. Rev. B 65, 054417 (2002)

17. Y.Y. Xue, D.H. Cao, B. Lorenz, C.W. Chu: Phys. Rev. B 65, 020511 (2002) (2001)18. R.C. Dynes, V. Narayanamutri, J.P. Garno: Phys. Rev. Lett. 41, 1509 (1978)

A Phase Diagram Approachto Superconductivity and Magnetismin Rutheno-Cuprates

H.F. Braun, L. Bauernfeind, O. Korf, and T.P. Papageorgiou

Universitat Bayreuth, Physikalisches Institut, D-95440 Bayreuth, Germany

Abstract. The layered rutheno-cuprates RuSr2LnCu2O8 (Ru-1212 structure) andRuSr2(Ln1+xCe1−x)Cu2O10 (Ru-1222 structure; Ln = lanthanide or Y for both struc-tures) consist of pairs of CuO2 planes alternating with perovskite-like sheets of vertexsharing RuO6 octahedra. Samples of Ru-1212 and Ru-1222 materials were known toshow both superconducting and magnetic transitions. However, these perovskite-likesheets are also the characteristic structural feature of some strontium ruthenates. Thepresence of such impurity phases and an apparent dependence of superconducting andmagnetic properties on sample preparation conditions made it difficult to attribute theordering phenomena to the Ru-1212 and Ru-1222 phases. Here we report on investiga-tions of the phase equilibria in the system Sr–Gd–Ru–Cu–O and on a precursor routeto the synthesis of RuSr2GdCu2O8 involving Sr2GdRuO6 and CuO.

1 Introduction

The structures of high-Tc cuprate superconductors may be described as a stack-ing of blocks containing conducting CuO2 layers alternating with insulatingcharge reservoir blocks. The immediate consequence are strongly anisotropicelectronic properties. In LnBa2Cu3O7 (Ln = Y and most of the trivalent lan-thanides), the intrinsic electronic properties are much closer to three-dimensionalbehavior than in other high-Tc cuprates, since the charge reservoir containsmetallic chains formed by CuO4 squares that couple the CuO2 layers along thec-axis.

The layered rutheno-cuprates RuSr2(Gd,Ce)2Cu2O10 (Ru-1222) and RuSr2-GdCu2O8 (Ru-1212) were discovered in an attempt to introduce different con-ducting metal-oxide layers in the charge reservoir block [1]. First reported in1995 [1,2], they appear today to provide the first example of superconductiv-ity [3] developing at temperatures far below the transition into a weakly ferro-magnetic state, with coexistence of both ordering phenomena [4,5,6,7]. The weakferromagnetism is associated with the ordered antiferromagnetic state of the Rumoments which appears at about 138 K in Ru-1212, while the Gd moments orderantiferromagnetically below 3 K [8,9,10].

While superconductivity was readily observed in Ru-1222, Ru-1212 has beena more problematic case, with strong effects of the details of sample preparationon its superconducting properties and reports on the absence of superconduc-tivity. We are convinced that answers to the apparent contradictions must besought in the nature of the phase diagram of the these systems consisting of five

C. Noce et al. (Eds.): LNP 603, pp. 142–159, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Phase Diagram of Rutheno-Cuprates 143

(Ru-1212) or six (Ru-1222) elemental constituents. In this article, we present abrief review of the properties of the rutheno-cuprates and then concentrate onRu-1212, presenting our study of the quinary system Sr–Gd–Ru–Cu–O.

The naming scheme of the chemical formulae and their shorthand nota-tion used throughout this paper follows that introduced with the discovery ofbismuth-, thallium-, or mercury-containing cuprates. It has the advantage toreflect the different crystal-chemical nature of the cationic sites in the CuO2conducting layers and the interspersed “charge reservoirs”, see e.g. [11,12,13].However, we keep the popular notation for YBa2Cu3O7 (Y-123) which shouldbe replaced by CuBa2YCu2O7 (Cu-1212) in the above scheme.

2 Structure and Properties of Rutheno-Cuprates

The structure of the rutheno-cuprates shown in Fig. 1 is closely related withthat of the “123-cuprates” of the YBa2Cu3O7 type . In this Y-123-structure, thecharge reservoir blocks contain chains of CuO4 squares which in Ru-1212 arereplaced by corner-sharing RuO6 octahedra forming planar sheets.

This “T -1212” structure has originally been observed with the transitionmetals T = Nb, Ta [14,15], later in solid solutions (Nb,Ru) [16], with a varietyof rare earth and alkaline earth elements replacing the Y and Ba of Y-123. Ithas been described in spacegroup P4/mmm [9,14,16,17] (with a 0.384 nm,c 1.155 nm for RuSr2GdCu2O8 [1,18]) leading to straight T -O-T bonds inthe octahedron layer. This description might, however, correspond to an aver-age structure. The mismatch between Cu-O and Ru-O distances [8,17] inducesordered rotations of the RuO6 octahedra. The resulting structure has been de-scribed in spacegroups P4/mbm [8] or I4/mcm [19,20]. Ru-1212 is found to be

Fig. 1. The structures of Ru-1212 (left) and Ru-1222 (right, part of unit cell shown)

144 H.F. Braun et al.

cation and oxygen stoichiometric, but disorder arises from c/a being close tothree and the RuO6 rotations [17]. Solid solutions with (at least partial) substi-tution of Ru by Sn [21], V, Ti [22], and Cu [23] have also been reported.

In RuSr2(Ln1+xCe1−x)Cu2O10, a fluorite-like (Ln,Ce)2O2-layer is insertedbetween the Cu-O-planes (Fig. 1) which introduces a body-centering operation(I4/mmm [24]; a 0.384 nm, c 2.855 nm for Ru-1222 [1,2]). The “T -1222”structure was found with the transition metals Nb, Ta [24], Ru [1,2,5,25,26],(Ru,Fe) solid solutions [4,27], and Ti [28].

Ru-1222 is readily made superconducting through doping of the CuO2-planesby the mutual replacement of tetravalent Ce and the trivalent lanthanide Ln =Sm, Eu, Gd, and by a variable oxygen content in the fluorite block [1,18]. Onostudied superconducting solid solutions (Nb,Ru)Sr2(Sm,Ce)2Cu2Oz, but did notestablish superconductivity in the pure Ru-1222 samples [25]. The simultane-ous occurrence of superconductivity and unusual magnetism in RuSr2(Gd,Ce)2-Cu2O10 was first shown by Bauernfeind et al. [2]. Coexistence of magnetic orderand superconductivity in Ru-1222 on a microscopic scale was reported by Felneret al. [4].

The potential of T -1212-type compounds for high-Tc superconductivity hasbeen pointed out by Mattheis [29]. In the first study of Ru-1212 [1], as-sinteredsamples were semiconducting. Resistive transitions to superconductivity werereported in samples that were slowly cooled or annealed under oxygen pres-sure (Fig. 2. of [1], but see also Sect. 4). However, since no shielding signal wasdetected, the superconductivity of this phase remained unclear. No supercon-ductivity or only trace amounts [20] were detected for T = Nb, Ta. Absenceof superconductivity in RuSr2GdCu2O8 was also reported [30,31,32], while themagnetic transition observed at 138 K could have been due to trace amountsof SrRuO3, a known ferromagnet.

With improved sample preparation, involving a treatment step in inert at-mosphere which results in the intermediate formation of Sr2GdRuO6, or usingthis compound and CuO as precursors, and final sintering in flowing oxygen,Ru-1212 became metallic and superconducting [3]. This preparation route, dis-cussed in Sect. 4, has been essential to avoid the appearance of perovskite im-purities [3,18,33], and has subsequently been adopted by others [6,7,17,34,35].While these and other authors, e.g. [23,36,37,38,39,40], obtain superconductingRu-1212, including evidence for a bulk Meissner effect [41] and a jump of the heatcapacity at Tc [42], Chu et al. suggest non-uniform, non-bulk superconductiv-ity [43,44,45]. Given the strong effect of details of the sample preparation [36] andthe granular nature of sintered RuSr2GdCu2O8 [17,18], these findings might notcontradict the above evidence for a bulk homogeneous superconducting phase.

3 Phase Equilibria in the Sr–Gd–Ru–Cu–O System

RuSr2GdCu2O8 (Ru-1212) was the first known quinary phase in the system Sr–Gd–Ru–Cu–O. With respect to synthesis and, eventually, single crystal growthof the Ru-1212 phase, knowledge of the corresponding phase diagram is of par-

Phase Diagram of Rutheno-Cuprates 145

ticular interest at temperatures in the range of 1000 C and above and at oxygenpartial pressures of the order of 105 Pa = 1 bar, for example in air under ambientconditions pO2 ≈ 0.21 bar.

Under these conditions, the metal oxides often have a fixed cation/oxygenratio and in such a case it is possible to treat the system consisting of N metallicelements and oxygen as a system consisting of N constituents, the binary metallicoxides. Likewise, it is tempting to treat this system of five elemental componentsas a quaternary system made up of the four binary metal oxides: SrO–GdO1.5–RuO2–CuO. This is perfectly possible for strontium and gadolinium which havefixed valences and are strictly divalent (Sr) and trivalent (Gd), respectively.However, both copper and ruthenium show variable valences and thus the oxygenconcentration is not fixed by cationic composition (we use valence and oxidationstate as synonyms). The preferred formal valence of copper in the solid phasesof the system is two or somewhat higher as in (Sr14−xGdx)Cu24O41−z, whereit reaches 2.25+. In liquid phases, copper tends to assume a valence below two,but for the subsolidus phase diagrams we may suppose copper to be divalent,since in air, the decomposition of 4 CuO(s) → 2 Cu2O(s)+O2(g) proceeds abovethe solidus temperature of the systems under consideration.

In contrast, solid phases with tetravalent and pentavalent ruthenium areknown. Among the former are RuO2, Gd2Ru2O7, and the strontium basedruthenates Srn+1RunO3n+1(n = 1, 2,∞). Phases with pentavalent rutheniumare, e.g., Sr2GdRuO6 and Gd3RuO7. In Ru-1212, the (average) Ru valence isbetween four and five [46,47]. We show below that there exist solid solutionsSr(Ru1−xCux)O3 in which the average valence of ruthenium varies from four tofive in a continuous fashion. Higher valences however, as do occur in RuO3 andRuO4 [48,49], were not observed for solid phases at high temperatures.

Because of the variable cation/oxygen ratio and the possible loss of volatileRuOx at elevated temperature, we are working in an open quinary system, andour pseudoquaternary representation SrO–GdO1.5–RuO2+δ–CuO (0 ≤ δ ≤ 0.5)is to be considered an approximation which we have chosen for the convenienceof visualization.

In the constitutional tetrahedron depicted in Fig. 2 only phases of immediateinterest are indicated by a black sphere (that is, neglecting any homogeneityranges), while phases without relevance for our considerations have been omitted.Among the latter is, for example, SrCu3Ru4O12 [50], which decomposes above950 C. For convenience, we consider only subsolidus equilibria. Since the solidustemperature is determined by “low melting point” copper-rich phases, we shouldkeep in mind that some considerations presented below also apply at highertemperatures in the Ru-rich part of the system.

In the next Sects. we discuss the pseudoternary boundary systems and asection of the pseudoquaternary system where (XSr + XGd)/(XCu + XRu) = 1.Here, the XM are cationic mole fractions defined according to XM = nM/(nSr +nGd + nRu + nCu), and nM is the number of moles present of element M. Thissection is represented by the square outlined in Fig. 2 and contains both theRu-1212 phase and the solid solution (Sr1−xGdx)(Ru1−yCuy)O3.

146 H.F. Braun et al.

SrCuO2

Sr2GdRuO6

1212

Gd2Ru2O7

SrRuO3

Gd2CuO4

GdO1.5

RuO2+δ

SrO

CuO

Fig. 2. Schematic overview of the system SrO–GdO1.5–CuO–RuO2+δ at 1000 C inair. For clarity, all homogeneity ranges and all pseudoternaries except Sr2GdRuO6

have been omitted. Grey square: section parallel to the SrO–GdO1.5 and RuO2+δ–CuOedges, see Sect. 3.2. Solid line connecting CuO and Sr2GdRuO6: see Sect. 4

Finally, we describe the precursor route [3] for the synthesis of RuSr2Gd-Cu2O8 from a mixture of CuO and Sr2GdRuO6 (connected by a straight line inFig. 2) at a molar ratio 2 : 1.

3.1 Pseudoternary Subsolidus Phase Diagrams

We studied phase equilibria in the pseudoternary subsystems of the pseudo-quaternary phase diagram in air, and, for selected cases, in streaming oxygen.Samples were prepared from the binary oxides or carbonate in the case of stron-tium, by calcination and sintering at appropriate temperatures, with interme-diate grindings. Details of sample preparation and characterization have beenpublished elsewhere [1,3,18]. Sample properties were monitored by resistivity,ac susceptibility, and dc magnetization measurements. The interior of the pseu-doquaternary system has not been examined in detail, however, we have reason-able evidence that besides solid solutions extending from the border systems,Ru-1212 is the only stable phase inside the constitutional tetrahedron under theconditions chosen.

SrO–GdO1.5–CuO – Gadolinium is intermediate in ionic size between La andY and therefore one might expect the SrO–GdO1.5–CuO phase diagram to re-flect properties of the well-studied SrO–LaO1.5–CuO and SrO–YO1.5–CuO sys-tems [51,52]. Although the sole binary oxide is Ln2CuO4 for both Ln = Gd andLa (with different crystal structures, however) while it is Ln2Cu2O5 for Ln = Y,gadolinium in many oxides behaves more like the slightly smaller yttrium thanlike the larger lanthanum.

The pseudoternary subsolidus phase diagram of the system SrO–GdO1.5–CuO is depicted in Fig. 3, where, for the sake of simplicity, we have assumed the

Phase Diagram of Rutheno-Cuprates 147

I

II

IIISr1.8Gd1.2Cu2O5.6

SrGd2O4Gd2CuO4

SrO

GdO1.5

CuO Sr2CuO3(Sr,Gd)14Cu24O41SrCuO2

Fig. 3. Subsolidus phase equilibria in the system SrO–GdO1.5–CuO in air. I, II, andIII denote three-phase regions touched along the dotted line: see text

phase Sr2−xGd1+xCu2O5.5±δ = (Sr,Gd)3Cu2Oz to have the unique compositionSr1.8Gd1.2Cu2O5.6. In fact, a homogeneity range of this phase 0.1 ≤ x ≤ 0.3has been reported [53]. However, our neglect of this homogeneity range and theresulting omission of various narrow two-phase regions in this representation ofthe phase diagram has no severe consequences for its gross features.

The phase with the lowest melting point is Sr14Cu24O41. Its peritectic decom-position temperature varies with oxygen partial pressure and has been reportedbetween 955 C [54] and 982 C [55] in air and at 1030 C at pO2 = 1 bar [55].These temperatures are only slightly higher than the eutectic temperatures ofthe system Sr-Cu-O, reported at 955 C [54] to 973 C [55] in air and 1011 C atpO2 = 1 bar [55].

The dotted line connecting SrCuO2 and the point of equimolar ratio be-tween CuO and Gd2CuO4 is of particular interest. Along this line we moveacross the three-phase regions (I) SrCuO2–(Sr,Gd)14Cu24O41–(Sr,Gd)3Cu2Oz,(II) Gd2CuO4–(Sr,Gd)14Cu24O41–(Sr,Gd)3Cu2Oz and (III) CuO–Gd2CuO4–(Sr,Gd)14Cu24O41, and the corresponding two-phase regions. These regions markthe left boundary of the section of the pseudoquaternary system with (XSr +XGd)/(XCu + XRu) = 1, depicted in Figs. 2 and 7 and discussed in Sect. 3.2.

SrO–GdO1.5–RuO2+δ – The only pseudoternary phase in this system isSr2GdRuO6. None of the phases appear to have non-negligible homogeneityranges. Therefore, all phases are considered point compounds, yielding the rathersimple subsolidus phase diagram depicted in Fig. 4. The solidus temperatureswere not determined, but are believed to be much higher than the melting tem-perature of Ru-1212, which is about 1125 C at pO2 = 1 bar. In the open system,however, where the investigations have been carried out, Gd2Ru2O7 tends tobe unstable with respect to decomposition into Gd3RuO7(s) and RuOx(g) al-ready above 1050 C. No loss of ruthenium was observed from Sr-Ru-O phases,

148 H.F. Braun et al.

Sr2RuO4 SrRuO3

Sr3Ru2O7

Sr2GdRuO6

Gd2Ru2O7

Gd3RuO7SrGd2O4

SrO RuO2+δ

GdO1.5

Fig. 4. Phase equilibria in the system SrO–GdO1.5–RuO2+δ at 1000 C in air

BA

Gd2CuO4

Gd3RuO7

Gd2Ru2O7

CuORuO2+δ

GdO1.5

Fig. 5. Phase equilibria in the system GdO1.5–RuO2+δ–CuO at 1000 C in air. A andB denote three-phase regions touched along the dotted line: see text

from Gd3RuO7 and Sr2GdRuO6 below 1200 C. The latter phase is stable after12 h at 1500 C in flowing oxygen, while SrRuO3 decomposes into Sr3Ru2O7,Sr2RuO4, and RuOx(g) under these conditions.

GdO1.5–RuO2+δ–CuO – This system (Fig. 5) is even simpler than the pre-vious one since it contains neither ternary phases nor extended homogeneityranges nor a pseudobinary phase in the system RuO2+δ–CuO. The dotted linerepresents the horizontal upper boundary of the square section shown in Figs. 2and 7 and discussed in Sect. 3.2. Along this line, one moves from the three-phase region (A) Gd2Ru2O7–Gd3RuO7–CuO into the three-phase region (B)Gd2CuO4–Gd3RuO7–CuO.

Phase Diagram of Rutheno-Cuprates 149

Sr3(Ru1-xCux)2O7

Sr14Cu24O41

Sr(Ru1-xCux)O3SrCuO2

Sr2CuO3Sr2(Ru1-xCux)O4

RuO2+δ CuO

SrO

Fig. 6. Phase equilibria in the system SrO–RuO2+δ–CuO at 1000 C in air

SrO–RuO2+δ–CuO – Much more interesting is this system, presented inFig. 6. Here we found that copper substitutes for ruthenium in all strontiumbased ruthenates with general formula Srn+1RunO3n+1 (n = 1, 2,∞; compoundswith other values of n could not be synthesized). The solubility limit is the samein all three phases: copper replaces up to one third of ruthenium, which is enor-mous for elements of different valence. However, this striking behavior can beexplained quite naturally if one assumes that for each divalent copper ion sub-stituted, two ruthenium ions transform from tetra- to pentavalent, whereby thestructure is left intact and the oxygen content does not change: Sr(Ru1−xCux)O3= Sr(Ru4+

1−3xRu5+2x Cu2+

x )O3, and similar for the other phases. At x = 1/3, onlyRu5+ is present and both further oxidation of ruthenium and depletion of oxygendo not occur, instead, impurity phases form for x > 1/3.

Although SrRuO3 is orthorhombic, the deviation from cubic metric is onlyslight and the X-ray powder diagram corresponds qualitatively to a perovskitecell with ap 0.3925(4) nm. Substitution of copper for ruthenium leaves thisvalue almost unchanged for x < 1/6. This unexpected behavior can be explainedby the fact that the mean ionic radius [56] of the elements at the octahedral siteis left unchanged r(Ru4+) =

(r(Cu2+) + 2 r(Ru5+)

)/3 upon this substitution.

At higher concentrations of copper (x ≥ xJT 1/6), the Jahn–Teller char-acter of the divalent copper ion apparently begins to break through and thelattice undergoes an abrupt orthorhombic to tetragonal structural transitionwith tetragonal lattice parameters a = 0.3895(5) nm and c = 0.400(1) nm. Themagnetic transition temperature TM was found to decrease linearly with increas-ing copper content: TM ≈ (1 − 4 x) 160 K. For x > 0.19, no magnetic transitionwas observed. This value is in close vicinity of xJT, but the correlation is notclear, since three samples with 0.17 ≤ x ≤ 0.19 that were tetragonal at roomtemperature did show magnetic transitions at low temperatures; possibly, xJTincreases with decreasing temperature.

150 H.F. Braun et al.

0,0 0,2 0,4 0,6 0,8 1,0

0,0

0,2

0,4

0,6

0,8

1,0B A

III

II

I

perovskite

(Sr,Gd)14Cu24O41+ (Sr,Gd)3Cu2O5.6

1/2 Gd2CuO4

+1/2 CuO

(Sr,Gd)(Cu,Ru)O3

1212

1/2 (Gd2Ru2O7)

SrRuO3SrCuO2

[Gd]

[Ru]

Fig. 7. Section of the pseudoquaternary system at (XSr + XGd)/(XCu + XRu) = 1,containing Ru-1212 and the solid phase which forms at its peritectic decomposition

3.2 The Section (XSr + XGd)/(XCu + XRu) = 1

Three corners of the square of Fig. 7 are formed by the pseudobinary phasesSrCuO2, SrRuO3, and Gd2Ru2O7. In the fourth corner, CuO and Gd2CuO4coexist in equimolar ratio. Along the lower and right hand edges of the square,SrCuO2–SrRuO3 and SrRuO3–Gd2Ru2O7, respectively, form pseudobinary equi-libria. Along the left and upper edges we cut across two- and three-phase regionsof the border systems; the latter are denoted I, II, III, and A, B as defined above(compare Figs. 3 and 5). In this section we find the Ru-1212 phase and theSrRuO3-derived solid solution (Sr,Gd)(Ru,Cu)O3±δ. Other than in the solid so-lution Sr(Ru,Cu)O3, it is likely that there occur deviations from the nominaloxygen content in this Gd-substituted phase.

Thermogravimetric measurements at pO2 = 1 bar show (Fig. 8) that theRu-1212 phase decomposes at about 1132 C. The weight loss, which corre-sponds to 1/2 oxygen per formula unit, is reversible upon cooling, and theliquid solidifies at about 1030 C (in Al2O3 crucibles). In the solidified sam-ples, X-ray and microprobe investigations reveal the presence of the disorderedperovskite (Sr,Gd)(Ru,Cu)O3±δ, of (Sr14−xGdx)Cu24O41 with x 5, and, to amuch smaller extent (sometimes only trace amounts), of CuO, Gd2CuO4, and(Sr,Gd)3Cu2Oz. This implies that we observe a peritectic decomposition into aliquid which is rich in copper and poor in – possibly almost free from – ruthe-nium and a solid perovskite of the solid solution (Sr,Gd)(Ru,Cu)O3±δ. Twosuch possible decompositions of Ru-1212 are indicated by arrows in Fig. 7. TheRu-1212 phase (here represented as a point compound) is in equilibrium withthe disordered perovskite phase (Sr1−xGdx)(Ru1−yCuy)O3±δ with x 0.25(10),y 0.35(10), and δ 0. In contrast to pure SrRuO3, no magnetic ordering wasobserved in the solid solution at this composition and in fact everywhere near

Phase Diagram of Rutheno-Cuprates 151

700 800 900 1000 1100 1200 1300

-1,0

-0,5

0,0

0,5

1,0

1139.4 oC1022.5 oC1032.8 oC

1131.6 oCendothermic

DT

A s

igna

l / (

µV/m

g)

T / oC

700 800 900 1000 1100 1200 1300

99,0

99,5

100,0

100,5

101,0

1.2%

TG

/ %

Fig. 8. Differential thermal analysis (DTA) and thermo-gravimetric (TG) study of theperitectic decomposition of Ru-1212 at pO2 = 1 bar

the Cu-rich boundary of its homogeneity range depicted in Fig. 7. Rutheniummost probably is nearly pentavalent along this boundary.

With these values of x and y, we estimate the composition of the liquid to bebetween XSr : XGd : XCu = 0.25 : 0.25 : 0.5 and 0.35 : 0.15 : 0.5. The region ofprimary crystallization of Ru-1212 may be expected in the tetrahedron spannedby these two compositions, Ru-1212, and CuO [18]. Lin et al. obtained their bestsingle crystals with a starting composition somewhat more rich in ruthenium [57].This higher Ru concentration is probably the reason that complete melting didnot occur at 1300 C in their experiments. Our knowledge of the open quinarysystem is not advanced enough to fully understand which reactions are involvedin this crystal growth process.

4 The Precursor Route to Superconducting Ru-1212

One serious problem of Ru-1212 synthesis is the formation of impurity phaseswhich proved to be very stable under the synthesis conditions generally used,that is oxidizing atmosphere and temperatures in the range of 1000 C. Thepresence of impurity phases in samples synthesized from nominally stoichiomet-ric mixtures of binary oxides and SrCO3 could indicate that (a) some volatilecomponent is lost in the processing, (b) RuSr2GdCu2O8 simply does not existwith ideal composition or (c) RuSr2GdCu2O8 is a stoichiometric compound, butphase formation is kinetically hindered due to the stability of impurity phasesforming prior to Ru-1212.

152 H.F. Braun et al.

In cases (a) and (b), single phase synthesis could only succeed with off-stoichiometric starting compositions, for (a) possibly leading to stoichiometricRu-1212. In case (b), vacancy formation, or, more likely, mutual cationic sub-stitution should occur. Since the pairs Sr/Gd and Ru/Cu have comparable sizeand chemical preferences (e.g. electronegativity), the most plausible composi-tions appeared to be those with mutual substitutions of these cation pairs. Re-lated substitutions have been suggested for Nb-based phases, where, for instance,stoichiometric NbBa2LaCu2O8 is reported to not exist but single-phase samplesare obtained at the off-stoichiometric composition Nb0.9Ba1.9La1.1Cu2.1O8 [15].

We synthesized a series of samples with nominal compositions Ru1−ySr2−x-Gd1+xCu2+yOz at 1030 C, pO2 = 0.21 bar and a sintering time of 72 h. Theexistence of a homogeneity range of the Ru-1212 phase is indicated by a varia-tion of its unit cell c/a ratio in these samples and of other properties, like Tc.Although a slight reduction of the impurity content was achieved for the sampleswith moderate nominal excess of Cu and Gd (x, y 0.1), this cannot be takenas proof that the stoichiometric compound does not exist.

Rather, case (c) appears to hold. When the Ru-1212 phase was synthesizedfrom the binary oxides in air or oxygen, we found appreciable amounts of SrRuO3at temperatures as low as 600 C, while non-negligible formation of Ru-1212 wasobserved only at temperatures approaching 1000 C. In previously publishedstudies [3] we have shown that it was possible to eliminate phases like SrRuO3and (Sr,Gd)(Ru,Cu)O3 by an intermediate synthesis step in inert atmosphere attemperatures around 1000 C, which resulted in the decomposition of Ru-1212to yield Sr2GdRuO6 and Cu2O. Subsequent sintering of this two-phase mixturein oxidizing atmosphere led to highly improved phase purity of the Ru-1212samples. Similar results were obtained starting from a stoichiometric mixture ofSr2GdRuO6 and CuO [3] as precursors. It would appear that the presence of pen-tavalent ruthenium in Sr2GdRuO6 effectively inhibits the formation of SrRuO3(with tetravalent ruthenium) under oxidizing conditions. More important mightbe that the reaction path most likely follows the straight line of Fig. 2 involvingpseudobinary equilibria and thus giving no room for the appearance of otherphases.

In order to find the optimum synthesis parameters, thoroughly ground pow-ders of Sr2GdRuO6 and CuO were pressed into pellets and sintered in flowingoxygen (pO2 1 bar; a sintering time of 12 h was used for convenience) at tem-peratures between 1000 and 1150 C. As can be seen in Fig. 9, the best phasepurity was obtained for sintering temperatures around 1050–1060 C. At tem-peratures below 1050 C, the reaction was not complete after 12 h, while above1060 C the formation of impurity phases was observed (Fig. 9).

The resistivity of these samples reveals a correlation between phase purityand metallic behavior. Samples sintered below 1050 C show semimetallic behav-ior at room temperature (with (T ) only slightly varying) and a semiconductingincrease in resistivity below 100 K (Fig. 10). Samples sintered between 1050–1110 C at pO2 = 1 bar are metallic with positive d/dT down to about 100 K,where a less pronounced increase in resistivity occurs. Samples sintered above

Phase Diagram of Rutheno-Cuprates 153

30 31 32 33 34 35 36

(Sr,Gd)(Ru,Cu)O3

Sr2GdRuO

6

1139°C

1198°C

1122°C

1101°C

Inte

nsity

/ a.

u.

2θ / degrees

1090°C

1080°C

1070°C

1060°C

1050°C

1040°C

1030°C

Fig. 9. Powder X-ray diffraction (CuKα) of samples prepared via the precursor route.The main peaks of Ru-1212, Sr2GdRuO6, and (Sr,Gd)(Ru,Cu)O3 are visible, see theupper two traces included for comparison

0 50 100 150 200 250 3000,0

0,4

0,8

1,2

0,0

0,4

0,8

1,2

R /

R30

0K

T / K

1030°C 1040°C 1050°C 1060°C

Fig. 10. Resistivity of Ru-1212 prepared via the precursor route in flowing oxygen atdifferent sintering temperatures

1110 C are semiconducting over the whole temperature range, all others show abroad transition into the superconducting state, with an onset below 30 K whichis almost independent of sintering temperature.

Should this improvement be due to an increase of the oxygen content of Ru-1212, then the absolute change is very small and below the limit of resolution ofour thermo-gravimetry. We could not detect any weight change during the reac-tion Sr2GdRuO6 + 2 CuO → RuSr2GdRuCu2O8±δ and thus assume the oxygencontent to be quite close to eight: 0 δ < 0.05 [18]. This is compatible withthe observation by Henn et al. who for non-superconducting Ru-1212 find only

154 H.F. Braun et al.

0 50 100 150 200 250 3000,0

0,5

1,0

1,5 1060°C, P(O2) = 1 bar

12 h 120 h 720 h

T / Kρ

/ m

Ω c

m

Fig. 11. Resistivity of RuSr2GdCu2O8 prepared via the precursor route after prolongedsintering at 1060 C under flowing oxygen (pO2 = 1 bar)

0 10 20 30 40 50

-1,2

-0,8

-0,4

0,0

1/16 G1/8 G

1/4 G1/2 G

2 G1 G

4 G

8 G

16 G

32 G

χ'm

T / K

Fig. 12. Diamagnetic screening as a function of ac field amplitude of a spherical sampleof RuSr2GdCu2O8 prepared via the precursor route (720 h, 1060 C, pO2 = 1 bar)

a negligible change in oxygen content, independent of heat and Ar/O gas flowtreatment [32].

The resistivity evidently is governed by trace amounts of impurity phasessegregated at the grain boundaries of the sintered samples, while the supercon-ductivity appears to be a property of the Ru-1212 phase. Since Sr2GdRuO6 isnot susceptible to loss of volatile RuOx under these preparation conditions, thepresence of impurity phases could affect the chemical composition of the Ru-1212phase.

Prolonged sintering at the optimum temperature of 1060 C improves themetallic behavior and enhances the onset of the resistive transition to supercon-ductivity to about 48 K, with zero resistance achieved at 35 K [18,58] (Fig. 11).

Samples prepared under optimum conditions show sharp transitions to fulldiamagnetic shielding (97 ± 3% after correction for geometric demagnetizationof the spherical sample) at 33 K [18,58] as shown in Fig. 12. The apparentdepression of Tc and decrease of the shielding fraction with ac field amplitude inthe susceptibility measurements is indicative of granular superconductivity witha low inter-granular critical (Josephson) current density [18]. Superconductivityis not destroyed at 32 G, since in a dc magnetic field of 5 T superimposed ona small ac field (1/4 G), the onset is at 8 K and the shielded volume fractionhigher than one third at 5 K. Granularity effects are also seen in the shape of

Phase Diagram of Rutheno-Cuprates 155

the resistive transition in an applied dc magnetic field, while the midpoint of theresistive transition shifts down from Tc 38 K to Tc 10 K at 9 T [18].

Scanning electron microscope investigations of our samples indicate that theimproved transport properties are (at least partly) due to an increase of grain sizeand improved contacts between grains [18]. The apparent changes of resistivitywith different oxygen treatment observed in the first investigation of Ru-1212 [1]must thus be attributed to a variation (due to the oxygen treatment) of the typeand amount of foreign phases present in those samples.

Other effects of the long-term heat treatment could be the loss of ruthenium,cation ordering, and long range ordering of RuO6-octahedra rotations. This inturn could affect the average Ru valence, magnetic, electronic, and transportproperties of the Ru-1212 phase. With pentavalent ruthenium, we expect insu-lating or semiconducting behavior in our simple valence-counting scheme. Theloss of ruthenium at fixed oxygen content would contribute to doping accordingto Ru5+ + Cu2+

2 → Ru5+1−δ + Cu(2+5δ/2)+

2 . As already suggested by Bauern-feind [18], a change of the average Ru valence gives rise to a doping of the CuO2

layers according to Ru5+ + Cu2+2 → Ru(5−ε)+ + Cu(2+ε/2)+

2 . Such deviationswith a mixture of Ru4+ and Ru5+ have indeed been observed [46,47]. Dopingcan also arise from off-stoichiometric compositions of single-phase Ru-1212. Un-der high oxygen pressure, a large fraction of the Ru can be substituted by Cu,increasing Tc to 72 K [23]. Electron-doping by the substitution of Ce4+ for Gd3+

decreases Tc [37].Sr2GdRuO6 shows magnetic ordering below about 35 K and a magnetiza-

tion peak around 20 K [59,60]. In dc magnetization measurements of Ru-1212,a peak in this temperature range has been observed occasionally [36,60], whichcould have been due to traces of precursor material present in the samples. Aquantitative study reveals, however, that the observed magnetization anoma-lies of Ru-1212 are associated with its superconductivity [60] and not due toSr2GdRuO6 impurities.

5 Discussion and Conclusions

In non-single phase samples, properties of the impurities could erroneously beattributed to the majority phase. We have identified the phases that are inequilibrium with Ru-1212 and those that survive the preparation process if thestandard oxide/carbonate route is used. Among those phases of the system thatmight coexist with the Ru-1212 phase in equilibrium, only semiconductors andinsulators were found. In view of the results on the pseudoternary phase dia-grams where at temperatures above 1000 C equilibration of samples with non-negligible content of copper was found to occur within some hours, it is highlyimprobable that metallic compounds used as starting material (RuO2) or whichmight have formed during preparation did persist through the prolonged syn-thesis procedures, at least at starting compositions at or near the ideal Ru-1212composition and with the exception of SrRuO3. Some of the neighboring phasesshow magnetic transitions, but we can separate their properties from those of

156 H.F. Braun et al.

the Ru-1212 phase. The impurities, if present, mostly affect transport propertiesof the granular sintered samples. Metallic behavior and a large resistivity ratio(see Fig. 11) will be an indication for the absence of such impurity phases.

We have described a precursor route to the formation of essentially phase pureRu-1212, by which in particular the formation of SrRuO3 and (Sr,Gd)(Ru,Cu)O3can be suppressed. The availability of sufficiently pure Ru-1212 is prerequisiteto any detailed characterization of this phase, by neutron diffraction or localprobes.

Leaving aside impurities, let us consider properties of a stoichiometric com-pound, i.e. a phase that exists only with ideal chemical composition. Even here,complications may arise by mutual cationic substitution Sr/Gd and Ru/Cu, orthe disproportionation into tetra- and pentavalent ruthenium. These effects maybe present either in a homogeneous way or give rise to domain formation. Siteordering usually is a thermally activated process, leading to properties that de-pend on the thermal history of the sample under consideration. In this context,it is interesting to note that a preparation procedure resembling our precursorroute, involving annealing under moderately reducing conditions followed by aheat treatment in oxygen (in that case under high pressure, though) is requiredin order to obtain Fe/Cu ordering in Fe-1212 or (Fe,Cu)-1212, which are reportedto become superconducting [61,62].

The situation becomes even more complicated in a phase with a homogene-ity range. The superconducting and magnetic properties of such a phase canvary with its chemical composition as has been observed in other ternary andquaternary systems, e.g. for ternary Tm2Fe3Si5 [63] or the quaternary boro-carbides [64]. It is possible that the Ru-1212 phase, which is known to have ahomogeneity range, is metallic and superconducting only for a limited range ofchemical composition. In this case, the apparently contradicting findings on theRu-1212 phase being superconducting or not can both be true for the respectivesamples with their specific composition. In this case the apparent contradictionscould be resolved, were the exact chemical compositions known.

For these considerations, we have tacitly assumed the samples to be homo-geneous. There is evidence (see the references in Sects. 1 and 2) that the mag-netically ordered and superconducting states develop in a homogeneous way andcoexist on a microscopic scale in the rutheno-cuprates. However, the existenceof a homogeneity range could lead to single-phase, non-homogeneous Ru-1212samples. Considering the small and loosely packed grains in sintered material,once formed most of these Ru-1212 crystallites might be effectively removed fromsolid state diffusion, thereby rendering thermal equilibration almost unachiev-able, and inhomogeneities might be frozen in. Intermediate grinding should re-duce this problem. Residual strains could affect the physical properties. Thus,it appears easy to prepare single-phase but non-homogeneous materials. It willnot be surprising if granular properties possibly with hierarchies of length scalesare observed in such materials.

Even single crystals could be subject to these difficulties. In this peritecticsystem, crystals will inevitably be grown from an off-stoichiometric melt. The

Phase Diagram of Rutheno-Cuprates 157

crystal composition will be restricted by the requirement to have liquid and solidphase in equilibrium during crystal growth and thus not all chemical composi-tions of the homogeneity range will be accessible in single crystals. If the solidusconcentration depends on temperature, which is likely for a phase with a ho-mogeneity range, the as-grown crystal may not be homogeneous, but annealingmay be required. Domain formation and all the temperature-dependent effectsmentioned above could then also be present in single crystals. In our opinion, themain advantage of single crystals will be the possibility to study the electronicand magnetic anisotropies of the rutheno-cuprates.

The challenge is to prepare homogeneous material. Does homogeneous Ru-1212 exist or will there be intrinsic formation of domains? Local probes shouldanswer this problem, if those methods have sufficient sensitivity and resolution.We have to be aware that even single-phase samples or single crystals are notan ideal material. The knowledge of the phase equilibria in the multicomponentsystems should help in two ways, to improve materials preparation and to aidin the interpretation of “unexpected” observed properties.

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The Synthesis, Structure and PhysicalProperties of the Layered RuthenocupratesRuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl

A.C. Mclaughlin and J.P. Attfield

Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB21EW and Interdisciplinary Research Centre in Superconductivity, Department ofPhysics, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK.

Abstract. Studies of the structure and physical properties of the layered rutheno-cuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl are reviewed. RuSr2GdCu2O8 is aweak ferromagnetic superconductor and doping studies have shown that it is possibleto tune the magnetic and superconducting transitions simultaneously. The averagecrystal structure of RuSr2GdCu2O8 is tetragonal at both 10 and 295 K (space groupP4/mmm), but a

√2a x

√2a x c superstructure resulting from coherent rotations of the

RuO6 octahedra within subdomains of 50-200 A is observed by selected area electrondiffraction (SAED). The same tilts and rotations of the RuO6 octahedra are observedin semiconducting Pb2Sr2Cu2RuO8Cl, which has strikingly similar magnetic propertiesto RuSr2GdCu2O8. Antiferromagnetic order is observed in the 10 K neutron diffractionpattern with a Ru moment of 1.1(1) µB, but a spin-flop transition is observed above afield of 0.5 T.

1 Introduction

The (ferro)magnetic superconductor RuSr2GdCu2O8 [1-21] (Fig. 1) is an ex-tremely interesting material, with a maximum Tc = 37 K and TM = 136 K,where Tc is the superconducting temperature and TM is the Curie tempera-ture. The superconductivity occurs in the CuO2 layers and the ferromagnetismarises in the RuO2 layers. µSR studies have demonstrated that the materialis microscopically uniform with no evidence of spatial phase separation of thesuperconducting and magnetic regions [4]. Initial SQUID magnetometry resultsshowed that the magnetic order in the ruthenate planes was predominantly fer-romagnetic and this persists through the onset of superconductivity at 37 K tothe lowest temperatures investigated (1.9 K) [1]. Variable field measurements ofRuSr2GdCu2O8 showed hysteresis with a remanent moment of 0.12 µB indicativeof a ferromagnetic component in zero field. However G-type antiferromagneticorder within the RuO2 planes was subsequently observed from neutron scatter-ing [10] experiments. In this model the Ru spins are aligned antiparallel to theirneighbours in the ab plane and along c resulting in a doubling of the unit cellin all three directions. The direction of the spins in this model is parallel to thetetragonal c axis. An upper limit of 0.1 µB was obtained for the ferromagneticcomponent, which appears to contradict results from SQUID magnetometry andelectronic paramagnetic resonance experiments [18].

C. Noce et al. (Eds.): LNP 603, pp. 160–175, 2002.c© Springer-Verlag Berlin Heidelberg 2002

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 161

Sr Gd Cu Ru O

Fig. 1. The average crystal structure of RuSr2GdCu2O8 showing the disordered rota-tions and tilts of the RuO6 octahedra

Variable field neutron diffraction studies of RuSr2GdCu2O8 showed that theRu spins cant into a ferromagnetic arrangement upon the application of a mag-netic field and at 7 T the Ru spins are fully ferromagnetically ordered. Gd3+

is paramagnetic down to 2.5 K and orders with the G-type antiferromagneticstructure below this temperature [10]. It is difficult to perform neutron diffractionexperiments on RuSr2GdCu2O8 because Gd has an extremely high absorptioncross section for thermal neutrons and isotopic enrichment with 160Gd is neces-sary. Only two antiferromagnetic diffraction peaks have been observed in studiesto date [10,19] and it is therefore difficult to know whether the Ru spins arecanted in the ground state.

RuSr2YCu2O8 has recently been synthesised under a pressure of 5.5 GPa [22].It is superconducting at Tc ≥ 25 K and appears to be ferromagnetic, TM = 149K, from SQUID magnetometry experiments. Neutron diffraction on this mate-rial has also evidenced antiferromagnetic order with a G-type structure and anincreased intensity on the [001] peak giving an estimate for the ferromagneticmoment of 0.3 µB. Only the [12

12

12 ] magnetic diffraction peak was observed from

neutron diffraction on RuSr2YCu2O8 due to the low signal to noise ratio andso it is still impossible to know whether the spins are canted in the groundstate. However the observation of a ferromagnetic component on the [001] peakat ∼145 K which is equivalent to 0.3 µB at 10 K gives increasing evidence ofthis. Upon application of a magnetic field the intensity of the antiferromagnetic[ 12

12

12 ] peak decreases and the magnetic structure becomes increasingly ferro-

magnetic as for RuSr2GdCu2O8 [10]. Hence it appears that coexisting super-conductivity and weak ferromagnetism occurs in these ruthenocuprates with a

162 A.C. Mclaughlin and J.P. Attfield

transition to full ferromagnetism at high fields. Coexisting ferromagnetism andsuperconductivity have also been observed in the 1222-type ruthenocupratesRuSr2RE2−xCexCu2O10 [1,2] RE = Eu, Gd. So far no evidence of antiferro-magnetism has been observed from neutron diffraction experiments on thesematerials [23] and SQUID magnetometry experiments on RuSr2EuCeCu2O10have indicated a sizeable ferromagnetic component.

In this paper our studies of the structural and physical properties of RuSr2Gd-Cu2O8 will be reviewed. A new phase Pb2Sr2Cu2RuO8Cl which has strikinglysimilar magnetic properties to RuSr2GdCu2O8 has recently been synthesised [24]and its basic structural and physical properties are discussed within.

2 The Structure and Microstructure of RuSr2GdCu2O8

The physical properties of RuSr2GdCu2O8 are strongly dependent on the prepa-ration conditions [1]. An “as prepared” sample [5] synthesised by heating inflowing oxygen for 10 hours at 1050 C and 1055 C has Tc ≥ 21 K. Howeverannealing the same sample for 7 days in flowing oxygen at 1050 C increases Tc

to 37 K and halves the 300 K resistivity. We have shown that this occurs due tofeatures of the structure and microstructure of RuSr2GdCu2O8. RuSr2GdCu2O8has a tetragonal unit cell at all temperatures [5,6] (space group P4/mmm, a =3.83955(1), c = 11.57239(7) A) and is cation and oxygen stoichiometric (Fig. 1).Thermogravimetric analysis performed on the annealed sample gave an oxygenstoichiometry of 7.99 ± 0.03. Disorder of the oxygen atoms in the ruthenateplanes and the apical atoms linking the CuO5 units and the RuO6 octahedrahave been observed from synchrotron X-ray diffraction measurements. The in-plane Ru-O bond length (1.969(2) A) is larger than the in-plane Cu-O bondlength (1.9268(4) A) at 295 K and it is this bond mismatch which results inrotations of the RuO6 octahedra around the c axis, by 13 at 295 K, with anet stabilisation of the structure. There is also a slight tilting of the octahedra,which reduces the Cu-O-Ru angle to ∼173. The thermal contraction of the in-plane Cu-O bond is greater than that of the Ru-O bond distances and thereforeresults in an increase of the bond mismatch between the two bond lengths. Thishas the effect of rotating the RuO6 octahedra around c with a slightly increas-ing angle as the temperature decreases. Hence the displacement of the oxygenatoms actually increases with decreasing temperature proving that they are dueto static disorder within the average structure.

The rotations of the RuO6 octahedra around c that are observed from syn-chrotron X-ray diffraction on RuSr2GdCu2O8 would give rise to a

√2a x√

2a x c superstructure if long-range ordered. This superstructure was observedin SAED patterns for the “as prepared” RuSr2GdCu2O8 sample viewed downthe [001] axis (Fig. 2) [5]. All of the main diffraction spots in the SAED pat-tern could be indexed by the basic tetragonal unit cell and the additional weakspots indicated the formation of a

√2a x

√2a x c supercell. However there

was no evidence for this supercell from synchrotron X-ray diffraction measure-ments [5,6] which have an extremely high sensitivity to weak diffraction peaks.

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 163

Fig. 2. The SAED pattern from the as prepared RuSr2GdCu2O8 sample viewed downthe [001] direction. The additional weak spots arrowed evidence the

√2a x

√2a x c

superstructure

Fig. 3. HRTEM image of the annealed RuSr2GdCu2O8 sample viewed down the [001]zone axis showing rectangular anti-phase boundaries

A High Resolution Transmission Electron Microscopy (HRTEM) image for theannealed RuSr2GdCu2O8 sample shows dark rectangular boundaries which di-vide the structure into sub-domains of 50-200 A (Fig. 3). It was concluded thatthere is no long range order of the RuO6 octahedra and the ordering of theRuO6 octahedra over many unit cells is shown in Fig. 4. Anti-phase boundariesof width a occur every 50-200 A as seen in the HRTEM image. At an anti-phaseboundary the sense of rotation of the RuO6 octahedra is reversed, but the re-mainder of the structure is unaffected. This therefore explains why the electrondiffraction but not the X-ray diffraction sees the superstructure, because 50-200A is too short to give rise to superstructure peaks in the X-ray data.

The c/a axis ratio for RuSr2GdCu2O8 at 3.015 (295 K) is very close to idealfor a triple perovskite, for comparison c/a varies between 3.032 and 3.066 as δvaries from 0 to 1 in YBa2Cu3O7−δ [25]. The near coincidence of a and b withc/3 in RuSr2GdCu2O8 results in the formation of many small domains with cin one of the three equivalent directions as shown in the HRTEM image for the“as prepared” RuSr2GdCu2O8 sample (Fig. 5). The areas labelled A, B, and Care individual domains. The 90 angle between the CuO2 planes meeting at the

164 A.C. Mclaughlin and J.P. Attfield

Fig. 4. A model for the rotations of the RuO6 octahedra around c in theRuSr2GdCu2O8 structure, resulting in the

√2a x

√2a x c superstructure shown in

the xy plane by broken lines. An anti-phase boundary is also shown

Fig. 5. HRTEM image of a regions of the “as prepared” RuSr2GdCu2O8 sample show-ing the multi-domain nature of the microstructure. The areas labelled A, B, and C areindividual domains. A and C are both orientated in the [100] direction, but rotatedby 90o from one another whilst domain B ([001] orientation) is perfectly intergrownbetween A and C without any amorphous boundaries. Area D shows an intergrowthof A and B with an interface on the [100] plane for both domains. The intergrowth ofA and B in the area E is between the [010] plane of A and the [001] plane of B

boundaries strongly reduces the supercurrent transport leading to high granular-ity, a low transport Tc and a high residual resistivity in the “as prepared” sample.The microstructure of RuSr2GdCu2O8 therefore depends on the synthesis andannealing time; when annealing the sample in oxygen at high temperature fora long time, the improvement in the superconducting properties is due to anincrease in domain size, rather than a change in cation composition or oxygencontent. This was later confirmed by heat capacity studies which showed thatthe thermodynamic critical temperature occurs at 46 K in both samples [9].

No large changes in any of the bond lengths or angles have been observed bypowder X-ray diffraction when the Ru moments order at 132 K [6]. The atomicdisplacement U-factor for the in-plane O(2) changes from 0.009 to 0.011 A2 andis the only observed structural anomaly accompanying the magnetic orderingtransition. However a variable temperature powder neutron diffraction study of

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 165

a 160Gd-substituted RuSr2GdCu2O8 sample observed abrupt changes in the Cu-O(2)-Cu buckling angle and the Cu-Cu interlayer distance at TM [11] and hencethe anomaly in the U-factor of O(2) suggests that similar structural changesoccur over a shorter range.

The observation of this anomaly at TM in the CuO2 planes rather than inthe RuO6 octahedra is surprising but correlates with magnetoresistance mea-surements performed on RuSr2GdCu2O8 [26] which show a strong exchangeinteraction (J = 35 meV) between the spins and the carriers comparable to thesuperconducting energy gap. It was concluded that the carriers are associatedwith both CuO2 and RuO2 bands i.e. itinerant electron ferromagnetism becausesuppression of superconductivity due to exchange is expected if the carriers areonly on the CuO2 planes. Therefore if the Ru moments are itinerant then thiscould cause anomalies due to magnetic ordering to manifest in the CuO2 planesrather than in the RuO6 octahedra as observed in the synchrotron X-ray [5]and neutron diffraction [11] data. However this contradicts the recent observa-tion of antiferromagnetic order in the ruthenate planes from a different neutrondiffraction study [10].

RuSr2GdCu2O8 is thought to be a canted antiferromagnet (a weak ferromag-net). The slight canting of the Ru moments can arise due to a Dzyaloshinsky-Moriya [27,28] interaction (antisymmetric exchange interaction between neigh-bouring Ru moments) which is non-zero due to the tilts and rotations of theRuO6 octahedra. Previous transport measurements on RuSr2GdCu2O8 indi-cated a hole concentration on the CuO2 planes of p = 0.07 [1]. Bond valencesummations [29] on RuSr2GdCu2O8 resulted in a large estimation of the holeconcentration in the cuprate planes; p = 0.44. The canting of the Ru momentsdue to the Dzyaloshinsky-Moriya interaction could lead to magnetic trappingor scattering of the holes and hence transport properties typical of underdopedcuprates, whilst the apparent hole concentration measured crystallographicallyis much higher. However Brown et al. [30] have shown that bond valence sumcalculations do not give accurate results if the bond lengths are strained. TheCu-O bonds are strained in RuSr2GdCu2O8 due to the bond mismatch betweenthe ruthenate and cuprate layers which could therefore explain the discrepancyin the two estimations of p from transport measurements and bond valence sums.

3 Doping Studies of RuSr2GdCu2O8

The substitution of Ru by the non-magnetic, fixed valent cations Nb5+ and Sn4+

has helped in the understanding of the charge distribution and magnetism ofRuSr2GdCu2O8. The hole doping of the copper oxide planes necessary to inducesuperconductivity arises from the overlap of the minority spin Ru: t2g and theCu: 3dx2−y2 bands and the formula is written as Ru5−2p0Sr2Gd(Cu2+p0)2O8 toshow the average Ru and Cu oxidation states [31,32].

The Ru1−xMxSr2GdCu2O8 solid solutions have been studied by powder X-ray diffraction [31,32] and the observation of overall increases in the lattice pa-rameters, cell volume, Sr-O and Ru/M-O bond lengths are in accordance with

166 A.C. Mclaughlin and J.P. Attfield

Temperature (K)

0 50 100 150 200 250 300 350

0.0 % Sn2.5 % Sn5.0 % Sn7.5 % Sn

-1M

/H (

emu.

mol

)

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 % Nb5 % Nb

10 % Nb15 % Nb20 % Nb

0 50 100 150 200 250 300

a b

Fig. 6. Variation of molar susceptibility with x in (a) the Ru1−xNbxSr2GdCu2O8 solidsolutions and (b) Ru1−xSnxSr2GdCu2O8 solid solutions

the substitution of the slightly larger Sn4+ and Nb5+ for Ru4+/5+. The RuO6octahedra were found to rotate around c with a greater angle as x increases inthe Ru1−xMxSr2GdCu2O8 solid solutions due to the increased bond mismatchbetween the in plane Ru-O and Cu-O bonds. An increase in the average apicalCu-O bond was also observed with Nb substitution due to the change in chargetransfer from the minority spin Ru: t2g to the Cu: 3dx2−y2 bands [31,32]. Similarresults were found upon reduction of YBa2Cu3O7 [25] in which the apical bondincreases from 2.328(3) A to 2.380(4) A and the Tc decreases from 69 to 56 K.

The magnetic properties of RuSr2GdCu2O8 were found to change dramati-cally with Sn/Nb substitution for Ru. Magnetic hysteresis loops recorded at 10 Kconfirmed the ferromagnetic order in all samples. A reduction of the Curie tem-perature (TM ) and a broadening of the magnetic transition were clearly observedwith both Sn and Nb substitution (Fig. 6). The Ru moment, remanent momentand coercive field measured for all samples were also found to decrease smoothlywith x in the Ru1−xMxSr2GdCu2O8 solid solutions. The fixed valent diamag-netic cations do not contribute electronic states close to the Fermi level andtherefore the substitution of diamagnetic Sn4+ and Nb5+ dilutes the ferromag-netism in the RuO2 layers leading to a rapid decrease in the Curie temperaturefrom 136 K to 103 K in Ru0.8Nb0.2Sr2GdCu2O8. The estimated moment per Ruatom decreases with x due to the disorder of the spins incurred by substitutionof the non-magnetic cations.

Superconducting transitions were observed for the Ru1−xNbxSr2GdCu2O8solid solutions with x = 0 - 0.15 (Fig. 7(a)) but not for x = 0.2 down to 7 K. Allthe Ru1−xSnxSr2GdCu2O8 samples were superconducting (Fig. 7(b)) and theonset Tc was found to increase from 38 K for RuSr2GdCu2O8 to 50 K in the 7.5% Sn sample. All samples were observed to be metallic (d/dT > 0) althougha semiconducting upturn was observed close to Tc. Therefore substitution ofRu4.84+ by Nb5+ leads to the removal of holes from the CuO2 planes so that thematerials become more underdoped. This is supported by the increase in the 290K Seebeck coefficient with x in the Ru1−xNbxSr2GdCu2O8 solid solutions [31,32]and the decrease in Tc to 19 K in the 15 % Nb sample. The opposite effect oc-

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 167

Temperature/K

0

1

2

3

4

5

6

0 50 100 150 200 250 300

0 % Sn2.5 % Sn5 % Sn7.5 % Sn

Temperature/K0 50 100 150 200 250 300

0

2

4

6

8

10

Res

istiv

ity/m

.c

m

Res

istiv

ity/m

.c

m

Ω Ω

0 % Nb5 % Nb

10 % Nb15 % Nb20 % Nb

a b

Fig. 7. Variation of the resistivity with x for (a) the Ru1−xNbxSr2GdCu2O8 solidsolutions and (b) the Ru1−xSnxSr2GdCu2O8 solid solutions

64

72

80

88

96

104

112

120

0 5 10 15 20 25

S (2

90 K

)/

V.K

% Nb

µ-1

S (2

90 K

)/

V.K

µ

-1

65

70

75

80

85

90

95

0 2 4 6 8% Sn

a b

Fig. 8. Variation of the room temperature Seebeck coefficient with x for (a) theRu1−xNbxSr2GdCu2O8 solid solutions and (b) the Ru1−xSnxSr2GdCu2O8 solid so-lutions

curs upon Sn substitution; the hole concentration increases with a subsequentincrease in Tc to 50 K and a decrease in the Seebeck coefficient (Fig. 8). Thermo-gravimetric analysis on the Ru1−xMxSr2GdCu2O8 solid solutions showed thatthere is no change in oxygen content with increasing Nb5+/Sn4+ substitution.The charge distribution in the doped ruthenocuprates was therefore written as(Ru5−2p0)1−xMq

xSr2Gd(Cu2+p0+∆p)2O8 where the extrinsic doping introducedby the substituents M of charge q is ∆p = (5 - q - 2p0)x/2, assuming the initialdoping level p0 remains constant. The fitted value of p0 is 0.08.

The maximum Tc for RuSr2GdCu2O8 has been estimated at 65(10) K byfitting the values of the Tc onset measured for the Ru1−xMxSr2GdCu2O8 solidsolutions to the quadratic equation Tc = Tmax

c [1 - 82.6(p - 0.16)2] [33] (Fig. 9).This is in agreement with the Tmax

c = 72 K recently recorded for Ru1−xSr2Gd-Cu2+xO8−y at optimal doping [34]. This is much lower than the highest Tc of105 K obtained for the 1212 cuprate (Tl0.5Pb0.5)Sr2(Ca, Y) Cu2O7 [35]. It wasspeculated that this suppression could reflect a pairbreaking interaction withthe ferromagnetic moments in the RuO2 plane. However this effect would be ex-pected to be greater in the undoped compound and it was concluded that the low

168 A.C. Mclaughlin and J.P. Attfield

0

20

40

60

80

100

120

140

0.06 0.07 0.08 0.09 0.1 0.11 0.12

Tem

pera

ture

/K

P

7.52.5 5.00.010.020.0

% Sn% Nb

T

T

M

c

Fig. 9. The variation of the superconducting critical temperature (Tc) and the Curietemperature (TM ) with doping level p (lower scale) and % Nb or Sn (upper scale) inthe Ru1−xMxSr2GdCu2O8 solid solutions. The Tc values are fitted by the quadraticexpression supplied in the text

estimate of Tmaxc is due to the lattice strain from the bond mismatch between

the cuprate and ruthenate layers which increases upon substitution of both Snand Nb. The unusually short apical Cu-O bond of 2.16 A in RuSr2GdCu2O8provides evidence that the geometry of this tetragonal 1212 structure is not op-timal for superconductivity. As a consequence bond valence sums give p ∼0.4for RuSr2GdCu2O8 whereas results from transport measurements led to an esti-mate of p ∼0.1. It was originally thought that this discrepancy was due to a largenumber of holes trapped by defects or by the ferromagnetic order in the sample.However upon dilution of the weak ferromagnetism there was no evidence of adisproportionately large increase in p from the room temperature Seebeck coef-ficient or the transport properties of the Ru1−xMxSr2GdCu2O8 samples. Earlierresults on YBa2Cu3O7−δ [30] have shown that bond valence sum calculationsdo not work well on structures under strain. The structure of RuSr2GdCu2O8is strained due to the bond mismatch between the ruthenate and cuprate layersand it is therefore concluded that the actual hole concentration in the cuprateplanes is p ∼0.08 and that there are no additional, magnetically trapped holes.

4 The Structure and Magnetic Propertiesof Pb2Sr2Cu2RuO8Cl

A new material Pb2Sr2Cu2RuO8Cl (Fig. 10) has recently been synthesised [24]enabling further study of the electronic and magnetic properties of the rutheno-cuprates. This material is of similar structure to RuSr2GdCu2O8 but caesiumchloride type Pb2Cl layers replace Gd making the material more convenient tostudy by neutron diffraction. Such a CsCl-type Pb2Cl layer also exists in the

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 169

Sr

Pb

CuRuOCl

Fig. 10. The average crystal structure of Pb2Sr2Cu2RuO8Cl showing the tilts androtations of the RuO6 octahedra

mineral ferrate Pb4Fe3O8Cl known as hematophanite [36,37] and the isostruc-tural insulating materials Pb2Sr2Cu2MO8Cl (M = Ta, Nb, Sb) [38 - 40].

Pb2Sr2Cu2RuO8Cl has been studied by TOF neutron diffraction between295 and 10 K [24]. This phase is difficult to prepare free of secondary phases andthe sample contained 73 % Pb2Sr2Cu2RuO8Cl by mass with 8 % CuO and 19% “SrRuO3”. An excellent Rietveld fit was obtained at all temperatures witha tetragonal P4/mmm symmetry structural model for the principal phase (a= 3.86681(9) A, c = 15.3688(7) A at 295 K). The oxygen atoms within theRuO6 planes and the oxygen atoms linking the CuO5 units and RuO6 octahedra(Fig. 10) were found to be disordered as observed for RuSr2GdCu2O8 [5,6];the RuO6 octahedra are rotated by 13.4o around the z axis and are tiltedaway from this axis by 7.1o at room temperature. A recent BVS calculationon Pb2Sr2Cu2RuO8Cl has shown that the hole transfer to the CuO2 planes is∼0.1 less than in RuSr2GdCu2O8; the apical Cu-O distance (2.24 A) is longerthan that in RuSr2GdCu2O8 (2.16 A) at room temperature. Hence since p wasestimated at 0.08 in RuSr2GdCu2O8 it was concluded that the CuO2 planes inPb2Sr2Cu2RuO8Cl are essentially undoped. Pb2Sr2Cu2RuO8Cl is semiconduct-ing with a room temperature resistivity of 160 Ω.cm and there is no evidencefor a superconducting transition at low temperatures, consistent with the copperoxide planes being too underdoped to superconduct.

A variable temperature neutron study has shown that Pb2Sr2Cu2RuO8Clappears to be antiferromagnetic below 120 K as the [12

12

12 ] magnetic diffraction

peak is observed below this temperature (Figs. 11 and 12). The low temperatureneutron diffraction patterns were fitted with the same G-type antiferromag-

170 A.C. Mclaughlin and J.P. Attfield

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250 300C

ount

s (a

.u)

Temperature (K)

Fig. 11. Temperature dependence of the intensity of the magnetic [ 1212

12 ] neutron

diffraction peak

Fig. 12. Part of the neutron diffraction pattern of Pb2Sr2Cu2RuO8Cl showing theweak nuclear [003] peak along with the [ 12

12

12 ] antiferromagnetic Bragg peak at 10 K.

The same portion at 290 K is shown below for reference

netic model as for RuSr2GdCu2O8 [10]. The spins were assumed to lie parallelto the c axis giving a refined Ru moment of 1.1(1) µB. This value is withinerror of the value µRu = 1.18(6) µB in RuSr2GdCu2O8 [10]. There was no ob-servation of a change in the nuclear Bragg intensity of any of the [00l ] peakswhere the ferromagnetic contribution would be expected between 295 K and 10K. A variable field neutron diffraction study has recently been performed onPb2Sr2Cu2RuO8Cl [41]. These measurements have shown that at fields higherthan 0.5 T the intensity of the [12

12

12 ] magnetic peak decreases whilst an increase

of the [003] peak intensity is observed corresponding to induced ferromagnetismin the xy plane (Fig. 13). No significant [12

12

12 ] intensity was observed above 1.1

T and the effect of the magnetic field was found to be reversible; returning tozero field, the [12

12

12 ] peak recovers its original intensity. Evidence of such a spin-

flop transition has previously been reported in RuSr2GdCu2O8 at 0.4 T [8] andRuSr2YCu2O8 [22]. It was concluded that the field dependent magnetic order iscommon to the ruthenocuprate structures but further neutron diffraction exper-

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 171

(003)(1/2 1/2 1/2)

Pea

k In

tens

ity

(a.u

.)

Field (T)

0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 13. Field dependence of the intensity of the magnetic [ 1212

12 ] and nuclear [003]

neutron diffraction peaks

0.0

0.5

1.0

1.5

2.0

0 50 100 150 200 250 300 350

M/H

(em

u. m

ol )

Temperature (K)

-1

FC

ZFC

Fig. 14. Variable temperature molar susceptibility for Pb2Sr2Cu2RuO8Cl

iments on a phase pure Pb2Sr2Cu2RuO8Cl sample will be necessary in order toconfirm this.

A ferromagnetic transition at TM = 117(1) K was evidenced from SQUIDmagnetometry measurements on Pb2Sr2Cu2RuO8Cl (Fig. 14) despite the ob-servation of antiferromagnetism in the variable temperature neutron diffractionstudy. Similar SQUID magnetometry results were observed for RuSr2GdCu2O8citej1. No separate ferromagnetic transition was observed for the “SrRuO3” sec-ondary phase showing that substitutions by Pb or Cu have suppressed the fer-romagnetism found in pure SrRuO3 (TM = 165 K); a suppression of the fer-romagnetic state in the SrRu1−xPbxO3 system [42] has been reported previ-ously. Furthermore there was no observable ferromagnetic contribution from the“SrRuO3” phase in the 10 K neutron pattern. Magnetic hysteresis loops recordedat 10 K (Fig. 15) yielded a moment of 0.8(1) µB per Ru atom. The presence of19% SrRuO3 which has a µRu = 1.4 µB at 5 T [42] could only contribute ∼0.3µB to the sample magnetisation per Ru, even if it were stoichiometric. Henceit was concluded that the saturated Ru moment in Pb2Sr2Cu2RuO8Cl is 0.5 -0.8 µB which is comparable to the value of 1.09 µB in RuSr2GdCu2O8 [31,32].

172 A.C. Mclaughlin and J.P. Attfield

-1.0

-0.5

0.0

0.5

1.0

-60 -40 -20 0 20 40 60

µ (µ

)

B

H (kOe)

Fig. 15. Variable field magnetisation data for the Pb2Sr2Cu2RuO8Cl sample

A narrowing of the hysteresis loop for Pb2Sr2Cu2RuO8Cl was observed at lowfields, indicative of a spin flop transition from weak ferromagnetism to full fer-romagnetism above 0.5 T consistent with the variable field neutron diffractionresults. It was therefore concluded that Pb2Sr2Cu2RuO8Cl is a canted antifer-romagnet with an ordered moment of 1.1 µB per Ru in zero field below TM =117 K. The Ru spins cant into a fully ferromagnetic arrangement above H = 0.5T giving a saturated Ru moment of 0.5 - 0.8 µB.

5 Conclusions

The structures of the (ferro)magnetic superconductor RuSr2GdCu2O8 and thenew layered ruthenocuprate Pb2Sr2Cu2RuO8Cl have been studied. There aremarked similarities in both the structure and basic magnetic properties of thesematerials. Both materials contain CuO5 units separated by RuO2 planes (Figs. 1and 10). The RuO6 octahedra are rotated by approximately 13o in both materi-als due to a bond mismatch between the in-plane Ru-O and Cu-O bond lengths.Tilts of the RuO6 octahedra which reduce the Cu-O-Cu angle to ∼173o arealso observed for both RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl. RuSr2GdCu2O8and Pb2Sr2Cu2RuO8Cl are believed to be canted antiferromagnets in zero fieldbelow TM = 132 K and 117 K respectively. The canting of the Ru momentsarises due to a Dzyaloshinsky-Moriya interaction between neighbouring Ru mo-ments which is non-zero due to the tilts and rotations of the RuO6 octahedra.Upon the application of magnetic field the Ru moments cant into a ferromag-netic arrangement above H = 0.5 T giving a saturated Ru moment of 0.5 - 0.8µB and 1.09 µB in Pb2Sr2Cu2RuO8Cl and RuSr2GdCu2O8 respectively. Themaximum TM is plotted against the Ru-Ru interplanar distance for the threeruthenocuprate structure types RuSr2GdCu2O8, RuSr2Eu1.4Ce0.6Cu2O10 [43]and Pb2Sr2Cu2RuO8Cl in Fig. 16. The inverse correlation shows that TM islimited by the interplanar superexchange coupling between RuO2 planes, as ex-pected for these layered magnetic systems.

The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 173

115

120

125

130

135

140

11 12 13 14 15 16

T

(K)

interplanar distance (A)

M

o

Fig. 16. The variation of TM with Ru-Ru interplanar distance in ruthenocuprates

Doping studies of RuSr2GdCu2O8 have shown that the overlap of the Cu3dx2−y2 and the Ru t2g bands leads to the hole doping of the CuO2 planes re-quired for superconductivity. A tuning of both the ferromagnetic and supercon-ducting transitions is observed with increasing x in the Ru1−xMxSr2GdCu2O8solid solutions (M = Nb, Sn). Despite the structural and magnetic similarities ofRuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl, the carrier distribution in Pb2Sr2Cu2-RuO8Cl appears to be different. RuSr2GdCu2O8 is superconducting below Tc =37 K whereas Pb2Sr2Cu2RuO8Cl is semiconducting and the longer apical Cu-O distance suggests that the CuO2 planes are essentially undoped. The RuO2planes of Pb2Sr2Cu2RuO8Cl thus contain Ru5+ with little or no electron dopingto the Ru4+ state. This confirms that the weak ferromagnetism in the layeredruthenocuprates arises from the local symmetry breaking distortions rather thana mixed Ru4+/Ru5+ state.

Acknowledgements

We thank EPSRC for the provision of research grant GR/M59976, synchrotronand neutron beam time, and a studentship for ACM. We thank I. Pape andA.N. Fitch (ESRF), P. Radaelli (RAL), J. Tallon and colleagues (DSIR, NewZealand) and our coworkers W. Zhou, J.A. McAllister, V. Janowitz and L.D. Stout.

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(1997)8. I. Felner, U. Asaf, S. Reich, Y. Tsabba: Physica C 163, 311 (1999)9. J.L. Tallon, J.W. Loram, G.V.M. Williams, C. Bernhard: Phys. Rev. B 61, 6471

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The Layered Ruthenocuprates RuSr2GdCu2O8 and Pb2Sr2Cu2RuO8Cl 175

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Magnetism and Superconductivityin Ru1−xSr2RECu2+xO8−d (RE=Gd, Eu)and RuSr2Gd1−yCeyCu2O8 Compounds

P.W. Klamut1,5, B. Dabrowski1, S.M. Mini1, S. Kolesnik1, M. Maxwell1,J. Mais1, A. Shengelaya2, R. Khazanov2,3, I. Savic2,4, H. Keller2,C. Sulkowski5, D. Wlosewicz5, M. Matusiak5, A. Wisniewski6, R. Puzniak6,and I. Fita6

1 Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA2 Physik-Institut der Universitat Zurich, CH-8057 Zurich, Switzerland3 Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut,

CH-5232 Villigen PSI, Switzerland4 Faculty of Physics, University of Belgrade, 11001 Belgrade, Yugoslavia5 Institute of Low Temperature and Structure Research

of Polish Academy of Sciences, 50-950 Wroclaw, Poland6 Institute of Physics of Polish Academy of Sciences, 02-668 Warszawa, Poland

Abstract. We discuss the properties of new superconducting compositions of rutheno-cuprates Ru1−xSr2RECu2+xO8−d (RE=Gd, Eu) that were synthesized at 600 atm. ofoxygen at 1080oC. By changing ratio between the Ru and Cu, the temperature ofsuperconducting transition (TC) raises up to TmaxC = 72 K for x=0.3, 0.4. The holedoping achieved along the series increases with Cu→Ru substitution. For x = 0, TCcan be subsequently tuned between TmaxC and 0 K by changing oxygen content in thecompounds. The magnetic characteristics of the RE=Gd and Eu based compounds areinterpreted as indicative of constrained dimensionality of the superconducting phase.Muon spin rotation experiments reveal the presence of the magnetic transitions at lowtemperatures (Tm=14-2 K for x=0.1-0.4) that can originate in the response of Ru/Cusublattice. RuSr2Gd1−yCe1−yCu2O8 (0 ≤ y ≤ 0.1) compounds show the simultane-ous increase of TN and decrease of TC with y. The effect should be explained by theelectron doping that occurs with Ce→Gd substitution. Properties of these two seriesallow us to propose phase diagram for 1212-type ruthenocuprates that links their prop-erties to the hole doping achieved in the systems. Non-superconducting single-phaseRuSr2GdCu2O8 and RuSr2EuCu2O8 are reported and discussed in the context of theproperties of substituted compounds.

1 Introduction

Recent reports of the apparent coexistence of superconductivity (SC) and ferro-magnetism (FM) in ruthenocuprates [1,2] have triggered intense interest in theproperties of these materials. The compounds that exhibit this unusual behaviorare RuSr2RECu2O8 (Ru-1212) [3] and RuSr2(RE2−xCex)Cu2O10−y (Ru-1222)(RE=Gd, Eu) [2] and they belong to the family of high temperature supercon-ductors (HTSC). Structurally similar to the well-known GdBa2Cu3O7 (Gd123)

C. Noce et al. (Eds.): LNP 603, pp. 176–193, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Magnetism and Superconductivity in Rutheno-Cuprates 177

superconductor, the ruthenocuprate RuSr2GdCu2O8 is a layered perovskite con-taining both CuO2 and RuO2 planes in its crystal structure. The correspondencebetween the two structures can be described by replacing the so called chain-Cu atoms in the Gd123 by Ru ions coordinated with full octhaedra of oxygensthus forming RuO2 planes. The positions of Sr atoms in Ru-1212 correspond toBa positions in Gd123, and the structural block containing CuO2 double planesremains similar for both compounds. In RuSr2GdCu2O8, the magnetically or-dered state manifests itself at temperatures TN=130-136 K, much higher thanthe superconducting transition reported at 45 K for the highest TC samples. Themagnetic order persists in the superconducting state [2]. What makes these com-pounds unique in the family of HTSC is that the magnetic ordering originatesin the sublattice of the d-electron Ru ions. Recent muon spin rotation and mag-netization results provided evidence for the coexistence of the magnetic orderingof Ru moments with superconductivity at low temperatures [2]. Although theferromagnetic ordering was initially proposed for the Ru sublattice below TN [2],recent neutron diffraction experiments show that the dominant magnetic inter-actions are of the G-type antiferromagnetic (AFM) structure [4,5]. Based onthese results, the observed ferromagnetism should originate from the cantingof the AFM lattice that gives a net moment perpendicular to the c-axis [5].This scenario resembles the description of the properties of Gd2CuO4, a non-superconducting weak ferromagnet, where the distortions present in the CuO2plane permit the presence of antisymmetric superexchange interactions in thesystem of Cu magnetic moments [6,7]. In the ruthenocuprates, however, the weakferromagnetism originates in the RuO2 planes and should result in the effectivemagnetic field being parallel to the CuO2 planes [2,5,8].

2 Ru1−xSr2GdCu2+xO8−d (0 ≤ x ≤ 0.75)

In order to address how the properties of RuSr2RECu2O8 can be affected by thedilution of the magnetic sublattice of Ru, we attempted to partially substituteRu with Cu ions. With this substitution the nominal formula of the resultingcompound should change toward the hypothetic GdSr2Cu3O7, a Sr containinganalogue of the GdBa2Cu3O7 TC ≈ 92 K superconductor. We have found thatfor Ru1−xSr2GdCu2+xO8−d series, the layered Ru-1212 type structure becomesstable only during synthesis at high pressure oxygen conditions. Polycrystallinesamples of Ru1−xSr2GdCu2+xO8−d (x=0, 0.1, 0.2, 0.3, 0.4, 0.75) were preparedby solid-state reaction of stoichiometric RuO2, SrCO3, Gd2O3 and CuO. Aftercalcination in air at 920oC the samples were ground, pressed into pellets, andannealed at 970oC in flowing oxygen. Then the samples were sintered at 1060oCfor 10 hours in a high pressure oxygen atmosphere (600 bar). Repeated annealingat high pressure oxygen conditions improved the phase purity of the materialwhile not changing their superconducting and magnetic properties. Figure 1presents changes of the lattice parameters with x. The insets to Fig. 1 show theX-ray diffraction patterns for the x=0.4 and 0.75 compositions.

178 P.W. Klamut et al.

Fig. 1. Lattice constants for Ru1−xSr2GdCu2+xO8−d. Insets show XRD patterns forx=0.4 and 0.75 samples

Both a and c dimensions decrease with the substitution of Cu for Ru inthe Ru-O planes. The observed change can indicate increased hole doping withx for the series, which will be confirmed in further reported X-ray AbsorptionNear Edge Spectroscopy (XANES), thermopower, and Hall effect experiments.Figure 2 presents the set of temperature dependencies of resistivity and ac sus-ceptibility measured for this series. They are also compared to the same de-pendencies obtained for the parent x = 0 compound, which was synthesized inflowing oxygen at 1060oC followed by slow cooling. The onset temperatures forthe resistive superconducting transition increase from T onC = 45 K for x=0 to72 K for x=0.3 and 0.4, and then decrease to 62 K for the 0.75 sample. The acsusceptibility results indicate that the upturn at TN=132 K, associated withthe onset of the magnetic transition for the x=0 compound, is absent for allx = 0 samples. Figure 3 presents the magnetic field dependencies of dc magneti-zation for the series measured at 4.5 K. The comparison of the M(H) dependencefor RuSr2GdCu2O8 with the behavior of non-superconducting Gd3+Ba2Cu3O6.2(open circles in Fig. 3, for the response of paramagnetic sublattice of Gd3+

ions) reveals additional contribution to the magnetization observed in the par-ent ruthenocuprate. This may represent substantial ferromagnetic alignment ofthe Ru moments (approximately 1 µB per formula unit at Hdc=6 T) for the

Magnetism and Superconductivity in Rutheno-Cuprates 179

Fig. 2. The temperature dependencies of ac susceptibility (Hac=1 Oe, f=200 Hz) andresistivity for Ru1−xSr2GdCu2+xO8−d

values of the magnetic field that are still well below the critical field for thesuperconducting phase [9]. This leads to the important conclusion that the weakferromagnetism observed for superconducting Ru-1212 is significantly enhancedin the presence of the magnetic field. The x = 0 samples follow practically thesame dependence originating in the response of paramagnetic ions in the Gd3+

sublattice. This shows that the type of magnetic order characteristic of the par-ent ruthenocuprate is absent in the Cu rich samples. It should be noted that,although the low temperature magnetization did not reveal any extra magneticcomponent present, we observed a slight irreversibility in field cooled (FC) andzero field cooled (ZFC) magnetization that opens below ∼120 K and 100 K, forx=0.1 and 0.2, respectively [9]. This behavior could be attributed to the re-sponse of the Cu diluted Ru sublattice. Muon spin rotation (µSR) experimentsperformed for the x=0.1 sample indicated that the increase of the relaxationrate observed below ∼120 K should not be attributed to the bulk response ofthe material. It can be tentatively assumed that this ZFC-FC irreversibility, ifnot reflecting the magnetic response of diluted RuO2 planes, arises from com-

180 P.W. Klamut et al.

Fig. 3. M(H) dependencies at 4.5 K for Ru1−xSr2GdCu2+xO8−d and GdBa2Cu3O6.2

Fig. 4. dc magnetization for x=0.4 and x=0.75 powder samples. Dashed line: non-superconducting GdBa2Cu3O6.2; Hdc= 500 Oe

positional inhomogeneity (for example, the formation of Ru rich clusters in theRu/Cu-O planes). For the x=0.3, 0.4 and 0.75 compositions we did not observeany irreversibility of the magnetization in the normal state.

The temperature dependencies of the ac susceptibility (Fig. 2) always showtwo characteristic temperatures for the onset of the superconducting transi-tions (see inset to this Figure). The onset of the intrinsic transition at Tc1(marked with squares) and the Tc2 (marked with crosses) at which the suscepti-bility changes slope in reflecting the establishment of the bulk superconductingscreening currents. It is worth noting that the diamagnetic contribution to thesignal measured between Tc1 and Tc2 substantially increases with x along theseries. Figure 4 depicts the low temperature reentrant behavior of the magne-

Magnetism and Superconductivity in Rutheno-Cuprates 181

tization at Hdc=500 Oe found for every composition in the series (x=0.4 and0.75 shown) at sufficient values of the magnetic field [9]. The results presentedin this Figure show the response of the samples, which were powdered to mini-mize the intergrain diamagnetic screening at low temperatures. The dashed linein this Figure shows the behavior of Gd3+Ba2Cu3O6.2 that again delineatesthe low temperature paramagnetic contribution of the Gd3+ ions. Regardlessof this reentrant behavior of magnetization, the low temperature zero resistiv-ity state is preserved at much higher magnetic fields (H=6.5 T was the highestfield used). In [9] we propose that the paramagnetic response in the presenceof superconductivity in these samples can be qualitatively understood assumingquasi-two-dimensional character of superconducting layers that are separatedby non-superconducting regions. For polycrystalline samples with randomly ori-ented crystallites, the paramagnetic response would arise from the crystallitesfor which superconducting layers are oriented parallel to the external field thatcan penetrate the space between them. Similar effect was recently proposed toexplain the anisotropy of the magnetic susceptibility (for H ⊥ ab and H ‖ ab)observed in highly oxygen deficient (i.e. strongly underdoped) superconductingGdBa2Cu3O7−d single crystals [10].

Figure 5 presents the M(H) dependencies measured for the x= 0, 0.3 and0.75 samples at 4.5, 20, and 50 K and small magnetic fields from -500 Oe to 500Oe. The hysteresis loops can be interpreted as the superposition of the magneticand superconducting components. The first penetration fields at T=4.5 K, seethe local minima at corresponding virgin parts of M(H) loops, are 5, 8 and14 Oe for x=0, 0.3 and 0.75, respectively. Larger positive contributions to themagnetization are observed for smaller x (see also Fig. 4). This can suggestthe more constrained dimensionality of the superconducting phase for sampleswith smaller x, in particular for the parent RuSr2GdCu2O8. For this compound,the additional magnetic component originating from the weak-ferromagnetismof the Ru sublattice also contributes to the magnetization above TC and belowTN ≈ 130 K. With regard to the data presented in Fig. 5, we should notethat for RuSr2GdCu2O8 the ferromagnetic coercive field at 4.5 K is approx. 400Oe when measured after the magnetic field was cycled up to the fully reversiblerange of magnetization. Contrary to the behavior of this compound, for all x = 0samples the remnant magnetization was observed only at the temperatures belowthe superconducting transition and thus can be attributed exclusively to theirreversible movement of the vortices in the material.

In order to further investigate the low temperature magnetic behavior ofRu1−xSr2GdCu2+xO8−d we performed a series of temperature dependent muonspin rotation (µSR) experiments. This method is especially suitable to revealingthe presence of even weak magnetic correlations persisting in the superconduct-ing state in the samples [11]. Figure 6 presents the results of zero-field muon spinrotation measurements for x=0.1, 0.3 and 0.4 compositions. The initial asymme-try parameters measured in this experiment allow us to conclude that we observebulk (90-100% of sample volume) magnetic transitions resulting in the presenceof an internal magnetic field below Tm=13, 6 and 2 K, for x=0.1, 0.3 and 0.4

182 P.W. Klamut et al.

Fig. 5. The magnetic field dependencies of dc magnetization measured at 4.5, 20 and50 K for x=0, 0.3 and 0.75 samples. The field cycled between -500 and 500 Oe

respectively. Values of the internal magnetic field, as well as the relaxation ratespresented in this Figure, have been calculated from the time dependent spectrafit with an equation:

A =23B cos(γµBmt+ φ) (1)

for the time evolution of the muon spin polarization described by:

P (t) = A exp(−12

(γµ∆Bµt)2) +13B exp(−λt) , (2)

where B is the initial asymmetry measured in the external field in the normalstate, γµ=851.4 MHz/T, and Bm is the average internal field at the muon site.Since the sublattice of paramagnetic Gd3+ moments orders antiferromagneti-cally at 2.8 K [4,12], the Ru/Cu-diluted sublattice or Cu moments in the CuO2planes seem only to be candidate systems that can be responsible for observedbehavior. Thus, assuming the bulk nature of the superconducting phase, theµSR data should be interpreted as indicative of a coexistence of the AF andSC order parameter as has been observed in other underdoped systems: see [11]for the AF correlations in underdoped La2−xSrxCuO4 and Y1−xCaxBa2Cu3O6for which the microscopic inhomogeneity of the charge distribution in the CuO2

Magnetism and Superconductivity in Rutheno-Cuprates 183

Fig. 6. Temperature dependencies of the internal magnetic field forRu1−xSr2GdCu2+xO8−d measured in zero-field muon spin rotation experiment.Samples: x=0.1 (circles), 0.3 (squares), and 0.4 (triangles). Inset shows the corre-sponding temperature dependencies of the relaxation rate σ

Fig. 7. Thermopower for Ru1−xSr2GdCu2+xO8−d superconductors. Inset shows itstemperature dependence for x=0.4 sample. Open circle represents the value for non-superconducting RuSr2GdCu2O8. T=293 K

planes and associated stripes formation have been proposed [13]. Further exper-iments aimed at understanding the microscopic nature of the low temperaturemagnetism observed for these compounds are in progress.

The charge carrier density at 293 K for the Ru1−xSr2GdCu2+xO8−d series, asestimated from Hall effect measurements (nH=1/[RHec]) ranges from 1.7×1027(x=0) to 3.9 × 1027 1/m3 (x=0.4) and saturates for far x. Figure 7 shows thevalues of room temperature thermopower for the x=0, 0.1, 0.3, 0.4 samples inthis series. The inset from this Figure presents the temperature dependence of

184 P.W. Klamut et al.

Fig. 8. XANES Cu-K edge energy vs. Cu content for Ru1−xSr2GdCu2+xO8−d series.T=293 K

thermopower measured for the x=0.4 sample. Positive values in the normal statesuggest the underdoped nature of this superconductor. Figure 8 presents the val-ues of characteristic Cu-K edge energy obtained in XANES measurements forthe same set of samples. All three experiments reveal that the Cu doping intothe Ru positions result in enhanced effective hole doping with its probable sat-uration occurring for x ≈ 0.4. This composition has the highest TC in the series(onset at 72 K vs. 45 K for x=0 and 62 K for x=0.75 [9]). We should note, thatsince all samples were prepared at the same oxidizing conditions, one can expectthe oxygen content per formula unit could decrease with x and become closer to7 for compositions with larger x. This effect, through its probable influence onthe effective charge doping achieved, could contribute to the underdoped char-acteristic of the x = 0.4 sample. Samples with x = 0 support variable oxygenstoichiometry that effect TC , resembling the properties of Gd123. For x = 0.4composition, the post annealing at 800oC in flowing air and in 1% of oxygen de-creases T onc from 72 to 55 and 43 K respectively, whereas the annealing in argonat 800oC leads to the non-superconducting material. Measurements of the tem-perature dependencies of dc magnetization for high-pressure oxygen synthesizedseries, performed at external pressures up to 1.2 GPa reveal that TC increases ata rate of approx. 5.5 K/GPa for all investigated samples. This also suggests theunderdoped character of the compounds. Figure 9 presents the specific heat jumpat the temperature of superconducting transition for Ru0.6Sr2GdCu2.4O8−d. Themagnitude of ∆Cp/T is approximately 12 mJ/mol·K2.

Following the same high-pressure oxygen synthesis route as were applied forRu1−xSr2GdCu2+xO8−d materials, we have synthesized the isostructural super-conducting samples of Ru1−xSr2EuCu2+xO8−d for x=0.4 and 0.6 (TC=70 and52 K respectively, being the onsets of resistive transitions). Since Eu3+ ions donot carry the net magnetic moment in their ground state, these samples pre-sented as with an opportunity to investigate their low temperature properties

Magnetism and Superconductivity in Rutheno-Cuprates 185

Fig. 9. Cp/T vs. temperature for Ru0.6Sr2GdCu2.4O8−d in the vicinity of supercon-ducting transition

with no extra paramagnetic contribution present as with Gd based compounds.Figure 10 presents comparison of M(H) dependencies for Ru0.6Sr2GdCu2.4O8−dand Ru0.6Sr2EuCu2.4O8−d samples measured at 4.5 K. The comparison showsagain that the positive values of magnetization measured forRu0.6Sr2GdCu2.4O8−d in the superconducting state (at magnetic field well be-low Hc2) originate from the paramagnetic response of Gd ions. A similar ef-fect, although considerably smaller because of the negligible contribution of theEu moments, is also observed for Ru0.6Sr2EuCu2.4O8−d and suggests the con-strained dimensionality of the superconducting phase in this compound. TheZF-µSR experiment revealed the presence of low temperature magnetism for su-perconducting Ru0.6Sr2EuCu2.4O8−d, which is similar to the behavior found forRu0.6Sr2GdCu2.4O8−d [14].

The asymmetry of the muon decay spectrum registered in a zero externalfield experiment at 1.8 K for Ru0.6Sr2EuCu2.4O8−d (not shown) is indicative ofthe presence of fast damped oscillations due to the precession of muons’ spinin an inhomogeneous internal magnetic field. The initial value of the observedasymmetry also indicated the bulk nature of the phenomena. The temperaturedependence of the muon spin relaxation rate (σ measures the rate of muon spindepolarization due to the field distribution in the specimen) measured in a trans-verse field experiment (TF-µSR) is presented in Fig. 11. In this experiment thesample was cooled down in an external field of 2000 Oe with an orientationperpendicular to the spins of the incoming muons. The initial increase of σ at60 K, due to the distribution of the magnetic field penetrating the supercon-ducting volume in the form of vortices, is followed by a steep increase observedat low temperatures that can be attributed to internal magnetism in the sam-ple. The T=0 value of σ inferred for the superconducting phase by cutting offthe magnetic contribution, equals 1.6 µs−1 and scales well with TC according tothe universal Uemura relation for underdoped HTSC compounds. The inset from

186 P.W. Klamut et al.

Fig. 10. M(H) dependencies for Ru0.6Sr2GdCu2.4O8−d and Ru0.6Sr2EuCu2.4O8−d at4.5 K

Fig. 10 presents the temperature dependence of the internal magnetic field as cal-culated from the frequency of asymmetry oscillations at different temperatures.The field decreases at the superconducting transition (Meissner effect) and thenincreases at low temperatures in the magnetically ordered state that sets in belowTm ≈ 5K. The low temperature magnetism detected for Ru0.6Sr2EuCu2.4O8−dshould also be considered for its possible positive contribution to the field de-pendent low temperature magnetization of this specimen (see Fig. 10).

3 RuSr2Gd1−yCeyCu2O8 (0 ≤ y ≤ 0.1)

In a separate approach to address the question of which electronic and struc-tural parameters control TC and TN in RuSr2GdCu2O8, we partially substitutedtrivalent Gd by Ce4+ ions. This substitution by intervening into the layer lo-cated between the two CuO2 planes, resembles substitutions frequently studiedin other cuprates where the hole doping, and thus TC , can be effectively con-trolled by varying the amount of substituted ion. By leaving the Ru and Cu sitesintact, we attempted to achieve the modification of magnetic and superconduct-ing properties of RuO2 and CuO2 layers primarily as a function of charge doping.

Magnetism and Superconductivity in Rutheno-Cuprates 187

Fig. 11. Temperature dependence of the muon spin relaxation rate σ calculated fromtransverse field (HTF=2000 Oe) µSR spectra for Ru0.4Sr2EuCu2.6O8−d sample. Insetshows temperature dependence of the internal magnetic field

Polycrystalline samples of RuSr2Gd1−xCexCu2O8 (x = 0, 0.02, 0.05, 0.1) weresynthesized by solid state reaction of stoichiometric oxides of RuO2, CeO−2,Gd2O3, CuO and SrCO3. After calcination in air at 900oC, the material wasground, pressed into pellets and annealed in flowing Ar at 1010oC. This mini-mizes the preformation of the SrRuO3 impurity phase. Subsequently, the sam-ples were annealed in flowing oxygen at increasing temperatures from 1030oC to1060oC with frequent intermediate grinding and pelletizing. The resulting ma-terial was slowly cooled to 500oC and then quenched to room temperature. Thestructure of all samples was indexed in the tetragonal 4/mmm symmetry simi-lar to the Ru1−xSr2GdCu2+xO8−d compounds. We found the lattice constantsincrease with y [15], which suggests effective electron doping into the antibond-ing part of Cu-O orbitals. This resembles the effect observed in superconductingNd2−xCexCuO4±d a-axis dimensions.

Figure 12 presents the temperature dependencies of the ac susceptibility andresistivity for this series. For y=0.02, T onc decreases to 36 K and an upturn ofthe resistivity above this transition is indicative of increased charge localization.The y=0.05 and 0.1 samples are not superconducting. The slight decrease ofthe resistivity around 130 K for x=0 corresponds to the temperature of mag-netic ordering of the Ru sublattice. This agrees with a similar feature reportedpreviously [2]. The decrease of resistivity at TN becomes more pronounced withincreasing Ce substitution. This can be interpreted as increased contributionfrom RuO2 layers, in their magnetically ordered state, to the conductivity of thesystem; or alternatively as the indication of charge redistribution between RuO2and CuO2 layers which takes place at TN .

The onset of the superconducting transition as seen in the real part of acsusceptibility (χ

′) coincides with the temperatures at which the material attains

188 P.W. Klamut et al.

Fig. 12. The temperature dependencies ac susceptibility and resistivity forRuSr2Gd1−yCeyCu2O8

zero resistivity (see Fig. 12). The maxima of χ′

near and above 130 K distin-guish the temperatures of the magnetic ordering of the Ru sublattice (TN ). TheTN , as also reflected on the resistivity dependencies, increases with Ce doping.Figure 13 (triangles) presents the temperature dependence of the internal fieldfor RuSr2Gd0.9Ce0.1Cu2O8, calculated from the results of zero-field µSR mea-surement. The temperature of magnetic ordering of the Ru sublattice can beestimated to TN ≈ 195 K; which is in quite good agreement with magnetic andtransport data. Figure 13 also presents temperature dependencies of the internalfield measured for RuSr2GdCu2O8 and RuSr2EuCu2O8; these will be addressedfurther in a later part of the article.

Because the observed decrease of TC , as well as changes of the lattice param-eters with y [15] remains consistent with the expected effect of electron dopingachieved by heterovalent Ce substitution, we can attempt to combine the charac-teristics of RuSr2Gd1−yCeyCu2O8 and Ru1−xSr2GdCu2+xO8−d compounds toconstruct a qualitative phase diagram that presents properties of both series vs.

Magnetism and Superconductivity in Rutheno-Cuprates 189

Fig. 13. Temperature dependencies of the internal magnetic field forRuSr2Gd0.9Ce0.1Cu2O8 (triangles) and non-superconducting samples ofRuSr2GdCu2O8 (closed circles) and RuSr2EuCu2O8 (open circles) measured in zero-field µSR experiment. Closed squares show data for superconducting RuSr2GdCu2O8

after [2]

Fig. 14. Properties of RuSr2Gd1−yCeyCu2O8 and Ru1−xSr2GdCu2+xO8−d vs.Ce→Gd and Cu→Ru substitutions, which scales with hole doping in the series. Opentriangles: temperatures of magnetic phase transitions (TN , Tm), as determined fromtemperature dependencies of the internal field measured in zero-field µSR experiment.Closed triangles and circles: temperatures of the magnetic (TN ) and superconducting(TC) phase transitions, as from the results of magnetic and transport measurements

changing hole doping. Figure 14 illustrates this approach, where the horizontalaxis reflects the Ce→Gd and Cu→Ru doping, for left and right parts of the dia-gram respectively. The important general conclusion is how the superconductingand magnetic properties depend on the charge doping in the Ru-1212 system,and, quite unexpectedly, that the low temperature magnetic correlations seen

190 P.W. Klamut et al.

in µSR experiments seem to be well preserved into the superconducting state.That of course raises the questions regarding the microscopic nature of this mag-netic ordering, and description for its’ coexistence with superconductivity. Bothquestions expose the challenge that currently defines our research effort.

4 Superconducting and Non-superconducting Samplesof RuSr2RECu2O8 (RE=Gd, Eu)

The usual route for the synthesis of 1212-type ruthenocuprates (see [2,3] or syn-thesis of RuSr2Gd1−yCeyCu2O8 as presented above) results in superconductingmaterial with maximum TC=45 K for RE=Gd, and a somewhat lower valueof TC reported for Eu. Recently, we have found [14] that modified synthesisconditions, with a final annealing at 930oC in 1% of oxygen in argon, leadto non-superconducting single-phase samples for both Gd and Eu based Ru-1212. Figure 15 presents the temperature dependence of dc magnetization fornon-superconducting RuSr2GdCu2O8. The inset (1) shows the low temperaturemagnetization, which, as one could expect, forms a maximum at 2.8 K - thetemperature of the antiferromagnetic phase transition of the Gd sublattice.

Interestingly, our thermogravimetric measurements did not show any notice-able changes in the oxygen content between superconducting and non-super-conducting materials. Thus, we should consider that the structural differences

Fig. 15. Temperature dependence of dc susceptibility for non-superconductingRuSr2GdCu2O8. Inset (1): Low temperature behavior on the expanded scale. Inset (2):ac susceptibility (Hac=1 Oe, f=200 Hz) of RuSr2GdCu2O8 at the temperature rangeof magnetic phase transition in Ru sublattice; open circles: for non-superconductingmaterial synthesized at 930o C in 1% of oxygen in argon, open squares: for the samesample after additional 24 hour annealing at 1060o C in oxygen (T onC ≈ 5 K), closedcircles: for the same sample after additional 180 hour annealing at 1060o C in oxygenthat led to an inducement of superconductivity with T onC ≈ 45 K

Magnetism and Superconductivity in Rutheno-Cuprates 191

between both compounds occur in the form of slight, but still decisive, changesof cation site defects/substitutions or in the difference in the structural dis-tortions. Comparison of the magnetic properties of superconducting and non-superconducting samples of RuSr2GdCu2O8, utilizing the dc magnetization, acsusceptibility [14] and µSR data (see Fig. 13), reveal that the weak ferromagneticcomponent is always enhanced for the superconducting compounds, whereas thetemperature of the magnetic ordering is slightly lowered (130 vs. 136 K). Thehole doping, as indicated by the values of thermopower presented in Fig. 7, isenhanced for the superconducting compound (compare open and closed circlesfor x=0). Superconducting RuSr2EuCu2O8 can still be obtained by repeatedoxygen annealing at 1060oC, however, our experiments indicate that this occursonly for materials containing a small fraction of secondary phases (predomi-nantly SrRuO3) [14]. That, in turn, can create a favorable situation for theformation of structural defects. The superconductivity in these samples shouldbe considered within the scope of our finding that partial Cu→Ru substitutionin Ru1−xSr2RECu2+xO8−d (RE=Gd, Eu) leads to a significant increase of thesuperconducting TC . The superconducting RuSr2EuCu2O8 for which neutrondiffraction results are reported in [16] has an even smaller magnetic transitiontemperature (TN ≈ 120K) than its Gd based superconducting counterpart. ZF-µSR data reveal that for the non-superconducting single phased RuSr2EuCu2O8the magnetic order exists below approximately 150 K (see Fig. 13). Since the acsusceptibility of RuSr2EuCu2O8 below TN remains significantly smaller than forRuSr2GdCu2O8, and the paramagnetic contribution of the Gd3+ sublattice istoo small to fully account for this difference (see our results presented in [14]),one can conclude that the weak-ferromagnetic component in RuSr2GdCu2O8 isalways enhanced compared to its Eu-based analogue compound. Detailed neu-tron diffraction experiments on Gd160 enriched Ru-1212, to further elucidate onthe role of structural disorder in determining the properties of this material arecurrently in progress.

5 Conclusions

We report two heterovalent substitutions in RuSr2RECu2O8 that expand thisparent Ru-1212 ruthenocuprate to two new series of compounds: hole dopedRu1−xSr2GdCu2+xO8−d and electron doped RuSr2Gd1−yCeyCu2O8. The char-acteristics of these materials allow us to propose the qualitative phase diagramthat links their properties to different hole doping realized in the series. Themagnetic properties of the series of Ru1−xSr2GdCu2+xO8−y superconductors(maximum TC=72 K for x=0.4) reveal the dominant contribution of the param-agnetic response of the Gd3+ sublattice at low temperatures. The results are in-terpreted as indicative of the constrained dimensionality of the superconductingphase that apparently evolve along the series toward the quasi-two dimensionalbehavior characteristic for the x=0 parent compound.

The non-superconducting samples of RuSr2RECu2O8 (RE=Gd, Eu) weresynthesized by modified synthesis at 930oC in 1% of oxygen in argon. Comparison

192 P.W. Klamut et al.

of the magnetic properties between superconducting and non-superconductingcompounds of the same nominal composition reveal that the weak-ferromagneticcomponent is always enhanced for the superconducting material, whereas thetemperature of the magnetic ordering is always slightly lowered. The effect canbe attributed to different levels of cation site defects/substitutions or to thedifference in details of structural distortions present in the crystal structure.This, in turn, would influence the weak ferromagnetic response observed for theantiferromagneticaly ordered and canted sublattice of the Ru moments. We alsoshould consider that even minute effective Cu→Ru substitution can provide thehole doping mechanism that stabilizes the superconducting phase in the system.

Acknowledgments

Research was supported by the National Science Foundation in the U.S. (DMR-0105398), and by the State of Illinois under HECA. P.W.K., A.S., R.K., I.S. andB.D. would like to thank Dr. D. Herlach and Dr. A. Amato of PSI, Villigen fortheir valuable assistance with the µSR experiments. S.M.M. acknowledges sup-port by the NSF (CHE-9871246 and CHE-9522232) and the use of the AdvancedPhoton Source was supported by the U.S. Department of Energy, Basic EnergySciences, Office of Science, under Contract No. W-31-109-Eng-38. A.W., R.P.,and I.F acknowledge the Polish State Committee KBN contract No. 5 P03B12421.

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Comm. 97, 957 (1996)11. Ch. Niedermayer, C. Bernhard, T. Blasius, A. Golnik, A. Moodenbaugh, J. I. Bud-

nick: Phys. Rev. Lett. 80, 3843 (1998)12. R. Kruk, R. Kmiec, P.W. Klamut, B. Dabrowski, D.E. Brown, M. Maxwell,

C.W. Kimball: Physica C, in press

Magnetism and Superconductivity in Rutheno-Cuprates 193

13. J.M. Tranquada: Phys. Rev. Lett. 78, 338 (1997)14. P.W. Klamut, B. Dabrowski, S. Mini, M. Maxwell, S. Kolesnik, M. Mais, A. Shen-

gelaya, R. Khasanov, I. Savic, H. Keller, T. Graber, J. Gebhardt, P.J. Viccaro,Y. Xiao: Physica C 364-365, 313 (2001)

15. P.W. Klamut, B. Dabrowski, J. Mais, M. Maxwell: Physica C 350, 24 (2001)16. J.D. Jorgensen, O. Chmaissem, H. Shaked, S. Short, P.W. Klamut, B. Dabrowski,

J.L. Tallon: Phys. Rev. B 63, 054440 (2001)

ESR Studies of the Magnetismin Ru-1212 and Ru-2212

O. Sigalov1 and A. Shames1, S.D. Goren1, H. Shaked1, C. Korn1, I. Felner2,and A. Vecchione3

1 Department of Physics, Ben Gurion University, Beer Sheva, Israel2 Racah Institute of Physics, The Hebrew University, Jerusalem, Israel3 I.N.F.M. -Unita di Salerno, Dipartimento di Fisica “E. R. Caianiello”

Universita di Salerno, Salerno, Italy

Abstract. We have studied by electron spin resonance (ESR) the nonsuperconductingsystem Ru-1212 and the superconducting and nonsuperconducting Ru-2212 over thetemperature range 100–300 K, in the normal and magnetic phases. In the nonsuper-conducting Ru-1212 we found that the ferromagnetic line appears at a much highertemperature than the magnetic ordering as determined by neutron diffraction. We alsoshow a correlation between the temperature dependence of the intensity and widthof the ESR resonance lines, the intensity of magnetic absorption lines and the DCmagnetization measurements.

1 Introduction

The rutheno-cuprates are known to exhibit coexistence of ferromagnetism andsuperconductivity [1,2]. GdSr2RuCu2O8 (Ru-1212) is a weak ferromagnet (WF)below 133 K [2] and is both superconducting and WF below 40 K [2]. Neutrondiffraction [3,4] studies show that at 133 K the Ru sublattice becomes anti-ferromagnetic (AFM), hence, the WF results from AFM canting and is con-fined to the RuO2 plane. Superconductivity is assumed to occur in the CuO2planes as in other high-temperature superconductor cuprates. The compoundRE2−xCexSr2RuCu2O10, (Ru-2212) where RE = Gd or Eu, shows a more com-plicated behavior. For certain values of x the material is only magnetically or-dered while for larger values of x coexistence of magnetism and superconduc-tivity has been found [5]. The related crystal structure of these compounds isshown in Fig. 1. The structure of 1212 is similar to YBa2Cu3O7 where theCuO chains have been replaced by RuO2 planes. This structure is a buildingblock for the 2212 compound where layers of oxygen and rare earths have beeninserted between units of Ru-1212, and neighboring layers of 1212 are shiftedby (1/2,1/2,0). Contrary to the Ru-1212 case, no detailed magnetic structuredetermination is available for the Ru-2212 compounds. Based on magnetic andMossbauer measurements, the following phase diagram (for Ru-2212) has beenproposed [5] as the temperature is decreased

NORMAL =⇒ AFM =⇒ WF =⇒ (SUPERCONDUCTORS + WF)

where the temperatures of the various transitions depend on the detailed com-position of the material.

C. Noce et al. (Eds.): LNP 603, pp. 194–204, 2002.c© Springer-Verlag Berlin Heidelberg 2002

ESR Studies of the Magnetism in Ru-1212 and Ru-2212 195

Fig. 1. The structure and relation between Ru-1212 and Ru-2212

The fact that these samples exhibit magnetic phases makes it possible to useferromagnetic resonance (FMR) [6] to study the magnetic phase diagrams. Weperformed our resonance measurements at 9.4 GHz. The appearance of the FMRlines is in addition to the usual electron spin resonance (ESR) lines that appearwhenever one or more of the constituents of the sample has unpaired electrons,which in our case is the Gd+3 ion. A previous study of the resonance spectra inthe microwave region [7] did not consider the intensity of the resonance lines.

2 Experimental Results

2.1 Ru-1212

A sample of 1212 was prepared according the procedure reported in [8]. Powderx-rays diffraction (XRD) measurements confirmed the absence of impurities,

196 O. Sigalov et al.

Fig. 2. The magnetic moment of nonsuperconducting GdSr2RuCu2O8 as a functionof temperature. Note that the magnetic moment deviates from zero at a temperaturemuch higher than the Curie temperature, which is 133 K

0 200 400 600 800 1000

0

10

20

30

40

50

GdSr2RuCu2O8T = 145 K

T = 150 K

T = 155 K

T = 160 K

T = 165 K

T = 170 K

ESR

Sign

alIn

tens

ity(a

.u.)

Magnatic Field (mT)

Fig. 3. Typical ESR spectra of nonsuperconducting GdSr2RuCu2O8 at various tem-peratures. The ESR line of the Gd+3 ion is observed at all temperatures, whereas theFMR line is observed at low temperatures

without any lines that belong to traces of SrRuO3 or GdSr2RuO6 which maycomplicate the FMR or ESR spectra. The Zero-Field-Cooled (ZFC) and Field-Cooled (FC) curves vs. temperature are shown in Fig. 2 These magnetic curvesshow (a) a magnetic transition at TN = 133 K [2] and (b) the sample is notsuperconducting. It can be seen also that the magnetic moment starts growingat a much higher temperature, around 160 K. This point will be discussed later inconjunction with the FMR results. Figure 3 shows typical spectra of the Ru-1212sample at selected temperatures. Above 160 K only one-resonance line appears

ESR Studies of the Magnetism in Ru-1212 and Ru-2212 197

-10

0

10

20

30

0

50

100

T = 110 K

T = 115 K

T = 120 K

T = 105 K

T = 100 K

T = 125 K

Narrow

W ide

T = 130 KF

MR

Line

Inte

nsity

(a.u

.)

0 200 400 600 800 1000

0

50

100T = 135 K

T = 140 K

T = 145 K

T = 150 K

T = 155 K

T = 160 K

Magnetic F ield (m T)

Fig. 4. The FMR spectra over a series of temperatures with the effect of the Gdresonance removed. Note that the width of the line changes drastically at TN (133 K)(Spectra for 155 K and 160 K were recorded with a gain of 10)

at ∼ 350 mT which is attributed to the ESR of the Gd3+ ions. Below 160 K, wellabove the magnetic transition (133 K), a second line at low magnetic fields isobserved, indicating the emergence of a ferromagnetic phase. A closer look intothe FMR lines (Fig. 4) reveals that they are much broader at temperatures abovethe transition than below the transition. At the transition (TN ≈ 130 K) bothlines are observed. The intensity of the FMR absorption line can be obtainedfrom the double integration (DIN) of the resonance spectra shown in Fig. 4,giving the temperature dependence of the intensity which is shown in Fig. 5.This will be discussed more fully later.

The temperature dependence of the Gd resonance spectrum is illustratedin Fig. 6. The internal magnetic field acting on the Gd ion, obtained from theposition of the Gd ESR line, is plotted as a function of the temperature in Fig. 7.The value obtained, equal to 20 mT for T below ∼ 100 K, can be compared to30 mT reported by Fainstein et al. [7]. The value of the internal magnetic fieldas measured by the Gadolinium line is zero for temperatures above TN (130 K),contrary to the results of the intensity of the FMR lines, which emerges athigher temperature (160 K) (Fig. 5). The plot of the inverse intensity of theGd line vs. temperature given in Fig. 8 shows that the Curie law is obeyed.The extrapolation of the plot intersects the temperature axis at a very smallpositive value indicating that the Gd ions order antiferromagnetically at a verylow temperature. It was indeed shown that Gd orders at 2.6 K [3].

198 O. Sigalov et al.

90 100 110 120 130 140 150 160 170

0

250

500

750GdSr2RuCu2O8

DIN

ofFM

Sign

al(a

.u.)

Temperature (K)

Fig. 5. The intensity of the FMR lines in GdSr2RuCu2O8 as a function of temperature.The FMR lines appear 30 K above TN

2.2 Ru-2212

The system RE2−xCexSr2RuCu2O10−δ, where RE=Gd, Eu, orders magneti-cally below certain temperatures [1]. Three samples, Eu2−xCexSr2RuCu2O10−δ

(x = 1, 0.5) and Gd1.5Ce0.5Sr2RuCu2O10−δ were studied. The Eu material withx = 0.55 as well as Gd1.5Ce0.5Sr2RuCu2O10 order magnetically and are super-conductors. Figure 9 shows the magnetization vs. temperature of the two Eubased samples. For x = 0.5, we observe that the magnetic moment starts deviat-ing from zero at Tm ≈ 122 K. The ZFC and FC Gd1.5Ce0.5Sr2RuCu2O10 curvesseparate at the irreversibility temperature Tirr = 92 K, and at Tc = 35 K thesample starts to superconduct. For the non-superconducting x = 1 sample we ob-tain Tm = 165 K and Tirr = 125 K. The Gd sample (not shown) has Tm = 180 K,Tirr = 80 K and Tc = 42 K [1]. Figures 10 and 11 show the experimental ESRspectra for all three materials.

The appearance of the FMR lines at temperatures below 160 K (Eu, x=1) and190 K (Gd) clearly correspond to the Tm of the two compounds. Figures 12, 13and 14 exhibit the temperature dependence of the intensity of the FMR lines ofEu2−xCexSr2RuCu2O10−δ (x = 1, 0.5) and Gd1.5Ce0.5Sr2RuCu2O10−δ.

3 Discussion

There is a difference between the ESR line and the FMR lines, although both sig-nals originate from the precession of the electron spin around the magnetic field.In the ferromagnetic phase, the precession of the electrons is coherent, due tothe strong exchange interaction between the electron spins, causing the magneticmoment of the magnetic domain to precess around the magnetic field. In a ferro-magnetic sample there are internal and demagnetization magnetic fields, whichmust be taken into account, in addition to the applied (external) field. These

ESR Studies of the Magnetism in Ru-1212 and Ru-2212 199

0 200 400 600 800 1000

-6

-4

-2

0

2

4

6

GdSr2RuCu2O8

T = 150 K

T = 100 K

ESR

Sign

al(a

.u.)

Magnetic Field (mT)

250 300 350 400

-6

-4

-2

0

2

4

6

T = 150 K

T = 100 K

Magnetic Field (mT)

Fig. 6. Resonance spectra at various temperatures of the Gd ESR lines. The insetshows only the central part of the ESR lines, revealing the shift in the line positionwith temperature

added fields can be anisotropic, and the demagnetization field can be shape de-pendent. All these effects can shift significantly the position of the resonance line.The appearance of FMR line indicates the existence of a ferromagnetic phase.A similar situation occurs for long-range antiferromagnetic order. However, theAFM resonance lines usually appear at a much higher frequency [9] than theone used in our measurements. In these cases where the AFM resonance linesappear at low frequency, the Neel temperatures are much lower than Tm foundhere [9].

200 O. Sigalov et al.

100 120 140 160

0

5

10

15

20

Hin

t(m

T)

Temperature (K)

Fig. 7. A plot of the internal magnetic field at the Gd site in GdSr2RuCu2O8, as afunction of the temperature

0 50 100 150 200 250 3000.0

0.2

0.4

0.6

0.8

1.0

1.2

GdSr2RuCu2O8

1/D

IN(a

.u.)

Temperature (K)

Fig. 8. The inverse ESR line intensity of the Gd ion in GdSr2RuCu2O8 vs. temperature

Fig. 9. Magnetization of EuxCe1−xSr2RuCu2O10−δ as a function of temperature

ESR Studies of the Magnetism in Ru-1212 and Ru-2212 201

0 100 200 300 400 500 600 700 800

0

50

100

150

200

250T = 100 K

T = 110 K

T = 120 K

T = 130 K

T = 140 K

T = 150 K

T = 160 K

T = 170 K

EuCeSr2RuCu2O10

FMR

Lin

eIn

tens

ity(a

.u.)

Magnetic Field (mT)

Fig. 10. The experimental spectra of EuCeSr2RuCu2O10 at selective temperaturesshowing the emergence of the FMR lines upon cooling

Fig. 11. The experimental spectra of Gd1.5Ce0.5Sr2RuCu2O10 at selective tempera-tures showing the emergence of the FMR lines upon cooling

202 O. Sigalov et al.

100 150 200 250 300 350 400-10

0

10

20

30

40

50

EuCeSr2RuCu2O10-δ

DIN

,FM

RSi

gnal

Inte

nsity

(a.u

.)Temperature, K

Fig. 12. The temperature dependence of the intensity of the FMR line ofEuCeSr2RuCu2O10

100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0Eu1.5Ce0.5SrRuCu2O10

DIN

,FM

RL

ine

Inte

nsity

(a.u

.)

Temperature (K)

Fig. 13. The temperature dependence of the intensity of the FMR line ofEu1.5Ce0.5Sr2RuCu2O10−δ

50 100 150 200 250 300 350 400 450

0

100

200

300

400 Gd1.5Ce0.5Sr2Cu2RuO10-δ

DIN

,FM

RL

ine

Inte

nsity

(a.u

.)

Temperature, K

50 100 150 200 250 300 350 400

1.0

10.0

100.0

FM

Rin

tens

ity

Temperature (K)

Fig. 14. The temperature dependence of the intensity of the FMR line ofGd1.5Ce0.5Sr2RuCu2O10. Inset shows the intensity on a log scale

ESR Studies of the Magnetism in Ru-1212 and Ru-2212 203

3.1 Ru-1212

The intensity of the FMR lines, as well as the magnetic moment (Fig. 2), raiseabove zero approximately 30 degrees above the temperature TN = 133 K ob-tained from neutron diffraction studies. We attribute this difference to the in-homogeneity of the sample where magnetic islands, too small to be detectedby neutron diffraction, begin to form. Such inhomogeneity may result from theoxygen nonstoichiometry. The intensity of the FMR lines is a measure of thenumber of correlated spins that give rise to FMR lines in these islands. Sincethe internal magnetic field that acts on the Gd ion emerges at TN , one mightexpect it to reflect the inhomogenuity of the sample. However, close to TN thevalue of this field is quite small, less than 10 mT (Fig. 7), and only a small partof the sample is magnetically ordered. Thus at the most we can expect somebroadening of the Gd ESR line. Since the Gd line width is large, of the order of30 mT, such broadening is too small to be observed with our equipment.

3.2 Ru-2212

The ESR measurements show the appearance of FMR lines at different values ofTm for the different Ru-2212 samples. One of the samples, EuCeSr2RuCu2O10,also shows a maximum in the value of the intensity of the FMR lines at Tirr

(Figs. 9 and 12). The temperature range of our spectrometer did not permitmeasurements below 100 K. In Ru-1212, it was established that the magneticorder consists of canted antiferromagnetism of the Ru sublattice [3,4]. It is thecanting of the Ru moments which gives rise to the weak ferromagnetism [2].The similarity of the temperature behavior of the intensity of the FMR lines ofRu-1212 and Ru-2122, leads to the conclusion that in Ru-2212 a weak ferromag-netism is extended into the temperature range Tirr-Tm, and is not replaced inthis range by antiferromagnetism, as was suggested in [5]. This is supported bythe additional fact that at the low frequency of 9.4 GHz, one does not observeAFM resonance lines in AFM samples unless they have a very low Neel tem-perature. However, only the determination of the magnetic structure by neutrondiffraction can supply a definite answer.

References

1. I. Felner, U. Asaf, Y. Levy, O. Millo: Phys. Rev. B 55, R3374 (1997)2. C. Bernhard, J.L. Tallon, Ch. Niedermayer, Th. Blasius, A. Golnik, E. Brucher,

R.K. Kremer, D.R. Noakes, C. Estronach, E.J. Ansaldo: Phys. Rev. B 59, 14099(1999)

3. J.W. Lynn, B. Keimer, C. Ulrich, C. Bernhard, J.L. Tallon: Phys. Rev. B 61, R14964(2000)

4. J.D. Jorgensen, O. Chmaissem, H. Shaked, S. Short, P.W. Klamut, B. Dabrowski,J.L. Tallon: Phys. Rev. B 63, 054440 (2001)

5. I. Felner, U. Asaf, E. Galstyan: Phys. Rev. B (submitted)6. A.H. Morrish: The Physical Principal of Magnetism (Wiley, 1965)

204 O. Sigalov et al.

7. A. Fainstein, E. Winkler, A. Butera, J. Tallon: Phys. Rev. B 60, R12597 (1999)8. I. Felner, U. Asaf, S. Reich, Y. Tsabba: Physica C 311, 163 (1999)9. See 6, page 620

Structure and Morphologyof NdSr2RuCu2Oy and GdSr2RuCu2Oz

L. Marchese1, A. Vecchione2, M. Gombos2, C. Tedesco3, A. Frache1,H.O. Pastore4, S. Pace2, and C. Noce2

1 Dipartimento di Scienze e Tecnologie Avanzate, Universita del Piemonte Orientale“A. Avogadro”, C.so Borsalino, I-15100 Alessandria, Italia

2 Unita I.N.F.M. di Salerno and Dipartimento di Fisica “E.R. Caianiello”,Universita di Salerno, Via S. Allende, I-84081 Baronissi (SA), Italia

3 Dipartimento di Chimica, Universita di Salerno, Via S. Allende,I-84081 Baronissi (SA), Italia

4 Instituto de Quımica, Universitade Estadual de Campinas,CP6154, CEP 13083-970, Campinas, SP, Brazil

Abstract. The structural and morphological properties of the NdSr2RuCu2Oy (Nd1212)compound, which has been recently synthesized as a pure perovskite phase, are reportedand compared with the properties of the magnetic superconducting triple perovskiteGdSr2RuCu2O8 (Gd1212). The preparation of the RESr2RuCu2O8 (RE1212, RE =Rare Earth) phases is also critically reviewed and the nature of the magnetic orderingof these materials has been investigated and correlated to the structural properties. Thecrystal structure of RE1212 compounds have been determined by a combined studyof high-resolution transmission electron microscopy (HRTEM), selected area electrondiffraction (SAED) and high-resolution X-ray powder diffraction (HRXRPD) usingsynchrotron radiation. These measurements show that Nd1212 has a symmetric cubicstructure with a high degree of disorder between cation sites in contrast with the wellordered tetragonal structure of Gd1212. The cation intermixing avoids the formationof extended CuO2 and RuO2 planes, responsible for the superconducting and magneticproperties. Thus, Nd1212 does not show any superconductivity and/or magnetic or-dering, while Gd1212 shows the coexistence of superconductivity and ferromagnetism.To have a better insight into the formation of RE1212 phases, temperature XRPDmeasurements have been performed. The morphology of the sample has been mon-itored by a scanning electron microscope (SEM) equipped with an energy dispersivespectrometer (EDS) attachment by which the microanalysis of the crystallites has beenalso performed.

1 Introduction: Historical Review

1.1 From RE123 to RE1212 (RE = Nd, Gd)

The superconductors REBa2Cu3O7 (RE = Y, or Rare Earth), usually shortenedas RE123, are among high Tc cuprates the most studied compounds [1]. Indeed,REBaCuO systems are very promising ceramic superconductors for the develop-ment of large-scale applications based on high-temperature superconductivity.Therefore, a deepened knowledge of their properties is fundamental in the search

C. Noce et al. (Eds.): LNP 603, pp. 205–221, 2002.c© Springer-Verlag Berlin Heidelberg 2002

206 L. Marchese et al.

of methodologies to produce devices of practical interest. As far as power appli-cation are concerned, the attention has been focused mainly on materials withRE = Y, Nd, Sm which present the best superconducting properties. In particu-lar, NdBaCuO has been extensively studied because it has shown higher criticaltemperatures and critical current densities than any other compound of the RE-BaCuO family. RE123 compounds have a triple perovskite crystal structure andpresent a structural transition from orthorhombic to tetragonal cell (associatedwith the oxygen content) that is generally correlated to the transition from ametallic-superconducting behavior to an insulating one. The superconductivityis based on the presence of an ordered CuO chain structure constituted by O−2

oxygen ions linking Cu+3 copper ions. This structure disappears when oxygencontent decreases. Thus RE123 compounds must undergo several oxygenationtreatments, after the syntheses, to achieve a superconducting behaviour.

The crystal structure of fully oxygenated superconducting Nd123 and Gd123is, in both cases, orthorhombic, belonging to Pmmm space group, with verysimilar crystallographic axes sizes (Fig. 1). The crystallographic axes for Nd123are: a=0.3856 nm, b=0.3895 nm, and c=1.1712 nm, while for Gd123 they are:a=0.3847 nm, b=0.3909 nm, and c=1.1682 nm [2]; the b/a ratio is equal to1,0101 for Nd123 and to 1,016 for Gd123. Gd123 and Nd123 present both anantiferromagnetic ordering below 2.2 K and 1 K, respectively [3], while theirsuperconducting transition temperatures are respectively TcNd

= 96 K [4] andTcGd

= 92.9 K [5].Another interesting feature of these compounds is that they present the

phenomenon of RE-Ba substitution [6], resulting in the formation of RE123(x)(=RE1+xBa2−xCu3O7+x/2−δ) solid solutions. Although from statistical model-ing this process appears to be entropy driven [7], the tendency toward cationicsubstitution is not the same for all rare earth ions. The possibility of an in-terchange between Ba2+ and RE3+ ions depends on their relative ionic radii.Moreover, rare earth ions are quite indistinguishable under a chemical point ofview, they have different atomic radii. The more similar are Ba2+ and RE3+

radii, the easier would be the interchange between them, and the greater wouldbe the stability region of RE123(x). Being Ba2+ greater than any RE3+, themost easily substituted ion would be Nd3+, which has the largest ionic radius. Arelevant tendency of Nd3+ ions to substitute for Ba2+ ions was in fact observed,leading to a Nd123(x) stability region of 0< x <0.6 [6]. On the contrary, thetendency of Gd3+ to substitute for Ba2+ ions is by far more moderate, and thestability region of Gd123(x) is 0< x <0.2 (Fig. 2). The main effect of Nd-Ba sub-stitutions on Nd123(x) is a severe weakening of superconducting properties [8,9].Indeed, as x increases, a decrease of the transition temperature is observed, untila total disappearance of superconductivity occurs at x ∼ 0.4. On the other side,the substituted phases seem to play a role in magnetic flux pinning [10], deter-mining, for high magnetic fields, higher critical current densities than the Y123ones [11]. This phenomenon is correlated to a phase transition from orthorhombicto tetragonal structure for x ∼ 0.2 [9].

Structure and Morphology of Nd1212 and Gd1212 207

Fig. 1. Crystal structure of GdBa2Cu3O7−δ and NdBa2Cu3O7−δ RE-123 systems

Fig. 2. Gd1+xBa2−xCu3O7+x/2−δ (left) and Nd1+xBa2−xCu3O7+x/2−δ (right) simpli-fied ternary diagrams in air at fixed temperature showing their respective solid solutionregions

1.2 The Oxygen Problemand the Synthesis of the First 1212 Phases

The occurrence of cation substitutions is favoured by thermal treatments underhigh oxygen pressure. Substitutions are correlated to an increase in the oxygencontent of the compound due to charge balance, as can be seen from the formulaNd1+xBa2−xCu3O7+x/2−δ. Moreover, it is generally observed that high temper-ature treatments, except those in inert atmosphere, increase the tendency of Ndions to substitute for Ba3+. Thus, fabrication processes of Nd123 samples areusually performed under reduced oxygen pressure (PO2 ≤ 0.01 atm) [11], fol-lowed by low temperature oxygenation treatments [9,12] in order to minimizecationic substitution. The substitution of Cu3+ ions in the chains with differentM4+ and M5+ ions has been attempted in order to stabilize the oxygen content inthe solid [13] and to avoid problems due to structural variations in RE123 com-pounds. In these compounds CuO chains are substituted by MO2 planes, leadingto a stoichiometry of the type RE1212 (REA2MCu2O8 with A=Ba2+, Sr2+), as

208 L. Marchese et al.

Fig. 3. Comparison between RE-123 and RE-1212 crystal structure

shown in Fig. 3. The attention of researchers toward RE1212 barium-based com-pounds was firstly attracted by samples containing Ta and Nb ions [14]. AmongNd- and Gd-based compounds, only the tetragonal NdBa2NbCu2O8−δ phasewith insulating properties has been successfully synthesised. NdBa2TaCu2O8−δ,GdBa2NbCu2O8−δ, and GdBa2TaCu2O8−δ were not formed. Attempts havebeen also made to synthesize Sr-based RE123 (namely RESr2Cu3O7) compounds,and very recently GdSr2Cu3O7 have been successfully prepared by the use of highpressure techniques [15]. Sr-based RE1212 tetragonal phases such asNdSr2NbCu2O8−δ, NdSr2TaCu2O8−δ, GdSr2NbCu2O8−δ, andGdSr2TaCu2O8−δ have been successfully synthesized, and all these materials areinsulators [14]. These results are summarized schematically in Table 1.

1.3 The RE1212 Rutheno-Cuprates

Interestingly results have been obtained by substituting Ru5+ ions in Cu(1)sites: the synthesis of GdSr2RuCu2Oz has been successfully achieved in 1995 byBauernfeind et al. [16], while in 1996 the superconductivity has been observedin this compound [17]. We notice that the attention was firstly focused on thesubstitution of the Cu ions by different metal ions, and the most frequently usedformula to represent this compound was RuSr2GdCu2O8 shortened as Ru1212.Despite its quite widespread diffusion this formula is not very practical for studiesfocused on the variation of properties due to substitutions of the rare earth ions.

Structure and Morphology of Nd1212 and Gd1212 209

Table 1. RE-123 and RE-1212 compounds: phase formation and superconductingproperties.

Compound Nd-based Phase Gd-based Phasetype compound formed compound formed

123NdBa2Cu3O7 Y, Tc=96 K GdBa2Cu3O7 Y, Tc=92 K

NdSr2Cu3O7 N GdSr2Cu3O7 N

1212

NdBa2NbCu2O8 Y, insulator GdBa2NbCu2O8 N

NdBa2TaCu2O8 N GdBa2TaCu2O8 N

NdSr2NbCu2O8 Y, insulator GdSr2NbCu2O8 Y, insulator

NdSr2TaCu2O8 Y, insulator GdSr2TaCu2O8 Y, insulator

Moreover, the historical derivation of these compounds from the RE123 cupratefamily of superconducting materials suggests to use a different arrangement ofthe elements, such as GdSr2RuCu2O8 (Gd1212) and NdSr2RuCu2O8 (Nd1212).

1.4 Coexistence of Superconductivity and Magnetic Orderingin Gd1212

The first report on the existence of magnetic ordering in Gd1212, with a tran-sition temperature TM=185 K, was based on measurements performed on non-superconducting samples [18]. Coexistence of superconductivity and ferromag-netism in Gd1212 was observed for the first time in 1999 by Bernhard, Tallon etal. [19,20]. This observation caused a huge increase of interest in RE1212 mate-rials [21,22,23,24,25,26,27]. These two cooperative phenomena were consideredmutually exclusive, and the observation of their coexistence poses non-trivial the-oretical problems, and, at the same time, very promising applications. The natureof magnetic the magnetic ordering of Gd1212 is not yet completely understoodbecause there are controversial reports on this subject [28,29,30,31,32,33,34].Due to its peculiar properties, several structural studies have been performed onGd1212. Rietveld refinement of synchrotron data showed that it has a tetragonalcell with a = 3.83841(2) A and c = 11.5731(1) A, (Rp=0.059, Rwp=0.074) [21].The interest raised by the unusual behavior of Gd1212, has stimulated attepmptsto synthesize several RE1212 materials. RE1212 based on Eu, Sm [16] and Pr [35]have a tetragonal crystal structure similar to the Gd1212 one. As far as theelectronic properties are concerned, Eu1212 and Sm1212 are superconductors,whereas Pr1212 is insulator. The synthesis of NdSr2RuCu2Oy has been also at-tempted by several groups, but detailed results have not been reported. In par-ticular no X-ray powder diffraction (XRPD) pattern was reported until recently.Bauernfeind et al. [16] showed that using Nd a cubic (or pseudocubic) perovskiteof unknown stoichiometry is produced, presumably with mixed A and B cationsites, e.g. (Sr,Nd)(Cu,Ru)O3. Tang et al. [36] reported that RE1212 sampleswith RE = Eu, Sm, and Nd contained a large amount of SrRuO3 phase which

210 L. Marchese et al.

increases with the increase of the RE3+ ionic radius. A dependence on rare earthionic radius has been then hypothesized to explain both the different behavior ofRE1212 compounds during synthesis and their different physical properties [16].However, no systematic studies on Nd1212 have been published until now. Inthe following, we will focus on the synthesis and characterization of Nd1212 incomparison with the analogous Gd1212 compound. This comparative study willclarify the role of the Rare Earth ions on the chemical and physical propertiesof rutheno-cuprates.

2 NdSr2RuCu2Oy and GdSr2RuCu2Oz

2.1 Synthesis Procedure

Two main routes have been identified for the formation of the RE1212 phasesfrom the precursor powders. The most diffused one [16,19] starts from a stoichio-metric mixture of oxides and/or carbonate compounds and consists in a heatingstep followed by several annealing cycles under oxygen or inert gases. An alter-native route [17] consists of heating under air a mixture of oxides and carbonatecompounds with the exclusion of CuO and a RE1210 phase (RESr2RuO6) isobtained in this case. Only after this step the CuO is added and the processcontinues with annealing cycles.

The syntheses of RE1212 (RE = Gd, Nd) powders have been performed bystarting from Gd2O3 or Nd2O3 and SrCO3, CuO, and RuO2. Stoichiometricamounts of these compounds have been mixed together by ball-milling and thenprocessed. The reaction procedures consist in several high temperature treat-ments in various atmospheres: oxidizing atmospheres (either air or pure oxygenflow) and inert atmospheres (either pure argon or pure nitrogen flow). A firstreaction cycle is performed by calcination in air in a muffle furnace. The pow-ders, disposed in alumina crucibles, are heated at 5 C/min up to 960 C andheld at this temperature during 10h, subsequently they are cooled down to roomtemperature. Although this step is crucial to produce the desired phase, the as-prepared powders obtained after this step are only partially reacted and presentspurious phases, such as SrRuO3, that require a long series of high temperatureannealing treatments to be eliminated [19,37]. The use of Ar atmosphere hasbeen tried besides the more widely used N2 with the aim of preventing possiblephenomenon of Gd-Sr substitution in analogy with Nd123 and Gd123 prepa-ration procedures. Ar treatments are known to reduce the formation of highlyoxygenated substituted phases. The occurrence of substitutions may result in asevere worsening, or even the loss of superconductivity and magnetic ordering ofthe sample. The possibility of Gd-Sr and Nd-Sr substitutions was suggested bythe structural analogy of RE1212 and RE123 and by the high similarity of Ba2+

and Sr2+ ionic radii [38]. However, no evidence of any substitution phenomenonhas been observed in Gd1212. Moreover, Ar annealing treatments are not effec-tive in reducing substitution effects in Nd1212, although it appears to be slightlymore efficient than N2 annealing in preventing the formation of spurious pha-ses. Although the procedures followed in the preparation of Nd1212 and Gd1212

Structure and Morphology of Nd1212 and Gd1212 211

compounds are similar, slight variations of the processing parameters are morecrucial for obtaining a Nd1212 pure phase in comparison with Gd1212. A possi-ble explanation of this difference has been searched monitoring step-by-step theformation of the perovskite RE1212 compounds by variable temperature XRPDanalysis performed on the starting mixtures.

2.2 Phase and Morphological Characterization

Thermo-Gravimetric analysis (TGA) is a useful tool to investigate the RE1212phase formation from the reaction mixture during the calcination cycle. Twomain abrupt weight losses at 500-700 C and 800-900 C occur in the case ofGd1212 during the cycle (Fig. 4). The first of these losses can be correlated tothe formation of a perovskite-like phase, presumably SrRuO3 as can be seen byvariable temperature XRPD analysis (Fig. 5). The formation of Gd1212 phase(main reflections at 2θ = 32.9, 47.2 and 58.8) occurs only after several hoursof thermal treatment in air at 960C. In particular, as results from Fig. 6, after10 h the Gd1212 phase is not yet formed. At this stage, the main reflections canbe attributed probably to SrRuO3 and to an unidentified phase containing Gdand Cu. Only after 17 h the Gd1212 phase begins to form. Several annealingcycles in oxygen at different temperatures are needed to obtain a pure Gd1212phase (top diffractogram in Fig. 5).

TGA profiles of the starting mixtures of powders, precursors of the RE1212phases, are profoundly different and this suggests that different reaction routestake place during the formation of the Nd1212 and Gd1212 compounds (Fig. 4).The first significant weight loss for Nd1212 is in fact at much lower temperature(around 350 C) and involves a transformation of Nd2O3, as may be seen in theXRPD patterns of Fig. 7. This fact allows more opportunities for the formation of

Fig. 4. TGA of the starting mixture leading to Gd1212 and Nd1212 phases. The tem-perature profile (heating rate) is also reported as dotted line

212 L. Marchese et al.

Fig. 5. Variable temperature XRPD patterns obtained during the calcination thermaltreatment of the reacting mixture for Gd1212

Fig. 6. XRPD patterns at different preparation stages for Gd1212 compound

several intermediate phases from the reaction mixture at temperatures between300 and 900 C. This is well visible both in the TGA profile, where severalsteps of weight losses are present, and by variable temperature XRPD analysis(Fig. 7). The largest weight loss for Nd1212 is observed between 900 and 960 Cand after 3 h at 960 C the weight loss is negligible. This is very different fromthe TGA profile of Gd-based mixture, which continuously decreases during thetreatment at 960 C and the process does not appear complete even after 3 hat this temperature (Fig. 5). As-prepared Nd1212 samples show low resolvedXRPD patterns (Fig. 8), due to the partial overlap of Nd1212 reflections withthose of secondary phases. Extra peaks are observed around the (110) reflectionat 2θ = 32.4 of the main phase. Scanning electron micrograph of as-preparedNd1212 powder samples shows irregular aggregates of crystallites of about 10

Structure and Morphology of Nd1212 and Gd1212 213

Fig. 7. Variable temperature XRPD results for Nd1212 calcination cycle

Fig. 8. XRPD patterns for Nd1212 at different preparation stages

µm size (Fig. 9). Higher magnification images show that these crystallites havesubmicrometre particles of average sizes down to 0.5 µm.

SEM analysis have been also performed on Nd1212 after a short annealing inoxygen (Fig. 10), they show rounded shape aggregates with average size largerthan in the as-prepared sample, reaching several tens of micrometres.

Energy Dispersive Spectrometry (EDS) analysis shows a great variability ofcomposition on the surface of aggregates that is probably due to the presence ofsmall amounts of binary and ternary oxides on the surface of Nd1212 particles(Fig. 11 and Tab. 2). The round shape of the aggregates is probably due to thefact that these oxides form initially as a liquid-like phase, which solidifies duringthe cooling step of the oxygenation processes. The amount of these oxides shouldbe very small since almost no trace of their presence is visible in the XRPDpatterns, even as amorphous phases. These liquid oxides, on their solidification,

214 L. Marchese et al.

Fig. 9. SEM image of as-prepared Nd1212 sample, from [39]

Fig. 10. SEM images of Nd1212 after the first oxygenation. On the right side is shownan higher magnification of the region marked with a rectangular frame on the left partof the figure

Fig. 11. On the left side is shown a SEM image for a Nd1212 sample obtained after thefirst oxygenating cycle and analyzed by EDS; on the right side the corresponding EDSdata collected from the regions labelled in the micrography are reported as histogram

may act as a kind of glue, cementing together the Nd1212 particles; the almostround shape of the crystallites would then be a trace of the incoherent isotropicnature of this process.

Structure and Morphology of Nd1212 and Gd1212 215

Table 2. EDS data for Nd1212 particle of Fig. 11

% Atomic

A B C D

Cu 23.02 34.78 44.59 16.11

Sr 15.11 11.29 9.18 12.99

Ru 7.71 3.73 3.80 9.14

Nd 8.98 6.76 11.52 9.23

Fig. 12. SEM image for long oxygenated Nd1212 sample

The average composition of the particle shown in Fig. 11, determined byEDS, gives a mean formula Nd1.07Sr2.42Ru1Cu2.54O7.26 that is compatible withNd1212 stoichiometry. Nd1212 powders that underwent long annealing treat-ments in N2 and O2, with the same thermal schedules of Gd1212 have beenalso analyzed. SEM micrographs show a visible improvement in the quality andhomogeneity of the crystallites of less than 1 µm size (Fig. 12). XRPD patternshows a good resolution of peaks and the complete absence of secondary phases(Fig. 8). With respect to the short time annealed particles, these particles appearwell detached and this is probably due to the disappearance of the cementingoxides, recombined to form the Nd1212 phase. Transmission electron microscopy(TEM) measurements have been performed to reveal details invisible at SEMmagnification level. Moreover, high-resolution transmission electron microscopy(HRTEM) and selected area electron diffraction (SAED) have allowed to monitorthe microstructure of the Nd1212 phase (Fig. 13).

TEM micrograph shows a large aggregate of Nd1212 (Fig. 13, left side),that appears to be composed by several single microcrystals as revealed by their

216 L. Marchese et al.

Fig. 13. TEM (left) and HRTEM (right) images of Nd1212 sample. The upper leftinset shows the SAED pattern for A region of TEM image. The lower right inset showsa magnified image of B region of HRTEM image, from [39]

SAED patterns. The top left inset in Fig. 13 shows the SAED pattern of region Aand corresponds to a cubic structure oriented along the [010] direction. HRTEMimage of region B (Fig. 13, right side) shows single crystal structure allowingto detect fringes separated by 3.65 A, that correspond to (101) planes of theNd1212 phase. Many particles have a similar regular fringe structure in largedomains (200-300 A2) and this suggests that large areas of the microcrystalshave regular bulk structure. High-resolution XRPD analyses were performed inorder to determine the crystalline structure of Nd1212 and its cell parameterswith high accuracy [39]. HR-XRPD patterns of Nd1212 samples corresponds toa cubic perovskite structure as results from the absence of low angle peaks withrespect to the XRPD pattern of Gd1212 (Fig. 14).

Rietveld refinement (Fig. 15) performed in space group Pm3m gives a crys-tallographic axis a = 3.90727(3) A (final disagreement indexes: Rp = 0.142, Rwp

= 0.191, RF 2 = 0.075). The high degree of symmetry of this structure is theeffect of disorder between cation sites in the unit cell. A-type sites in the cubicperovskite appear to be occupied indifferently by Nd3+ or Sr2+ ions in a 1:2ratio, and B-type sites occupied by Ru5+ or Cu3+ [39]. These two substitutionsappear to be strictly correlated. Indeed, to properly fit the experimental data,it was supposed that Nd-Sr segregation triggers the analogous phenomenon forRu-Cu. Nd1212 behaviour is coherent to what was already observed in RE123compounds, where Nd3+ ions showed a tendency to substitute Ba2+ ions by fargreater than for Gd3+ ones. A comparison between the structures of Gd1212and Nd1212 is shown in Fig. 16.

Structure and Morphology of Nd1212 and Gd1212 217

Fig. 14. XRPD patterns for Nd1212 and Gd1212

Fig. 15. Rietveld refinement for Nd1212 performed on HR-XRPD pattern, from [39]

2.3 Magnetisation and Susceptibility Measurements

Now, let us analyze the transport properties of fully oxygenated Gd1212 sampleand the magnetic properties of Nd1212 compound. The experimental results forGd1212 compound are reported in Fig. 17 where the temperature dependence ofthe resistance is shown together with the temperature behavior of the magneticsusceptibility presented in the inset of the same Figure. It can be observed asuperconducting transition with an onset at about 39 K and a zero-resistancetemperature at Tc=9 K. This result is also supported by magnetic susceptibilitycurve shown in the inset where it can be clearly identified the superconductingtransition with an onset between 20 K and 30 K as well as the ferromagnetictransition at TM about 130 K. These experimental results agree well with the

218 L. Marchese et al.

Fig. 16. Comparison of tetragonal structure of Gd1212 and cubic structure of Nd1212

0 20 40 60 80 100 120 1400

20

40

60

0 50 100 150 200 2500.0

0.2

0.4

0.6

R (

mΩ

)

T (K)

χ (

emu

103 g

-1 G

-1)

T (K)

Fig. 17. Resistivity and susceptibility as function of the temperature in Gd1212

data reported in literature that find a magnetic ordering temperature at 133 Kand a superconducting temperature transition up to 44 K [19,20].

Referring to the magnetic properties of Nd1212 compound, in Fig. 18 wereport the temperature dependence of the dc susceptibility. The data presentedin this figure indicate that there is no signature of both the magnetic and su-perconducting transition. These remarkable differences between the conductingand magnetic properties of Gd1212 and Nd1212 compounds could be ascribedto two main reasons. The high resolution X-ray powder diffraction data suggesta strong Cu/Ru site mixing as well as between Nd and Sr ions; this intermixingof cations producing a disordered network of CuO2 and RuO2 planes stronglyaffects the formation of an ordered magnetic structure and inhibits the forma-tion of Cooper pairs in the CuO2 channel. Secondly, the magnetic propertiesof Nd1212 compound are dominated by the Nd3+ ions that produce a purelyparamagnetic behavior and this fact prevents the formation of a ferromagneticas well as a superconducting ordered phase.

Structure and Morphology of Nd1212 and Gd1212 219

0 50 100 150 200 2500,0

0,1

0,2

0,3

0,4

0,5

0,6

χ (e

mu

10-3

g-1 G

-1)

T (K)

Fig. 18. Susceptibility as function of temperature on Nd1212

3 Conclusions

Gd1212 and Nd1212 phases have been successfully synthesized. The formation ofNd1212 phase follows a pathway different from that of Gd1212 one. In particular,Nd2O3 transforms at 350 C, while Gd2O3 reacts at 900C. This leaves enoughroom for several intermediate phases to form in the temperature range 350-960 C during the production of the Nd1212 phase and critical fine tuning of thepreparation conditions is therefore required. However, after only 2 h treatmentat 960 C, in air, there are not significant amounts of precursors, and Nd1212is the most abundant phase present although the composition of the aggregatesis not homogeneous. They result to be composed by sub-micrometric Nd1212crystals cemented together by a small amount of oxides. However, the amountof these oxides is so low that they are hardly detected in the XRPD. Very longannealing cycles at 1050-1080 C both in inert and oxidizing atmospheres areneeded for obtaining a homogeneous Nd1212 phase. SEM observation of longannealed samples shows that the Nd1212 particles, whose mean dimensions donot change very much, are well separated from each other.

SrRuO3 is formed as intermediate phase during the preparation of Gd1212phase, and requires very long annealing processes under inert and oxidizing atmo-spheres to be transformed completely to a pure Gd1212 solid. Moreover, crys-tal structure investigation has shown that Nd1212 presents a cubic structure,whereas Gd1212 has a tetragonal triple perovskite structure. The cubic Nd1212phase shows a high disorder between cation sites, analogously to what observedin Nd123.

The centre of Nd1212 unit cell is occupied by Nd:Sr cations in a 1:2 ratio andthe vertices by Ru:Cu in a 1:2 ratio too. The difference in the crystallographicstructure of the two RE1212 phases has a profound effect on their physical prop-

220 L. Marchese et al.

erties. Gd1212 shows a coexistence of superconductivity and ferromagnetism,while the absence of any superconducting and magnetic order is observed inNd1212. The cation intermixing avoids the formation of extended CuO2 andRuO2 planes, responsible for the superconducting and magnetic properties.

Acknowledgements

We acknowledge Dr. A. Nigro, Dr. B. Savo, Dr. D. Zola and Dr. G. Carapella(Dipartimento di Fisica “E.R. Caianiello”, Universita di Salerno and INFM UdRSalerno, Italia) for the transport and magnetic measurements; Ms. D. Sisti (Di-partimento di Chimica, Universita di Salerno, Italia) for assistance during thepreparation of powders; Prof. P. W. Stephens (Department of Physics and As-tronomy, State University of New York, Stony Brook, NY, USA) for HR-XRPDmeasurements; Dr. G. Cerrato (Dipartimento di Chimica IFM, Universita diTorino, Italia) for TEM and HRTEM/SAED analysis, and Mrs. R. E. Miller(Instituto de Quımica, Universitade Estadual de Campinas, Brazil) for assis-tance during variable temperature XRPD measurements.

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Synthesis Effects on the Magneticand Superconducting Propertiesof RuSr2GdCu2O8

R. Masini1, C. Artini2, M.R. Cimberle3, G.A. Costa2, M. Carnasciali2,and M. Ferretti2

1 CNR – IENI, Sezione di Milano, Via Cozzi 53, 20125, Milano, Italy2 INFM and DCCI, University of Genoa, Via Dodecaneso 31, 16146 Genova, Italy3 CNR – IMEM, Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy

Abstract. A systematic study on the synthesis of the Ru-1212 compound by preparinga series of samples that were annealed at increasing temperatures and then quenchedhas been performed. It results that the optimal temperature for the annealing liesaround 1060-1065 C; a further temperature increase worsens the phase formation.Structural order is very important and the subsequent grinding and annealing improvesit. Even if from the structural point of view the samples appear substantially similar,the physical characterizations highlight great differences both in electrical and magneticproperties related to intrinsic properties of the phase as well as to the connectionbetween the grains as inferred from the resistive and the Curie-Weiss behaviour at hightemperature as well as in the visibility of field cooled and zero field cooled magneticsignals.

1 Introduction

There has been a number of reports on the coexistence of magnetic order andsuperconductivity in the rutheno-cuprate RuSr2GdCu2O8, synthesized for thefirst time in 1995 [1]. Its peculiarity lies in the fact that, unlike previous com-pounds, magnetic order occurs at a temperature much higher than the supercon-ducting transition temperature. This compound is characterised by a triple per-ovskitic cell similar to the high-temperature superconducting cuprate (HTSC)YBa2Cu3Ox, in that it contains two CuO2 layers while the CuO chains are re-placed by a RuO2 layer. However, various experimental reports came to differentconclusions. It has been suggested on the basis of transport measurements thatits electronic behavior is similar to an underdoped HTSC [2] while, on the con-trary, NMR measurements resulted comparable to those of an optimally dopedHTSC [3]. Some other reports concluded that the magnetic order is ferromag-netic in the RuO2 layers [2,4,5,6] in which case there should be competitionbetween the superconducting and magnetic order parameters resulting even-tually in a spontaneous vortex phase formation or spatial modulation of therespective order parameters. However, powder neutron diffraction showed thatthe low-field magnetic order is predominantly antiferromagnetic [7], with a smallferromagnetic component presumably produced by spin canting. The spectrumof published data includes also non-superconducting samples showing similar

C. Noce et al. (Eds.): LNP 603, pp. 222–238, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 223

macroscopic magnetic behaviour [8], samples showing zero resistance but nodiamagnetic signal and finally samples with evidence of a resistive and magnetictransition. Since the physical properties of this rutheno-cuprate material arestrongly dependent on the details of the preparation procedure, and can be verydifferent even in samples that turn out to be formally identical to a standardstructural and chemical-physical characterization, we have conducted a system-atic study on the effects of sample preparation conditions on the properties ofsuch hybrid compound.

2 Experimental

The crystal structure was determined by powder X-ray diffraction (XRD) usingCu Kα radiation. dc resistivity and magnetic measurements were performed bythe standard four-probe technique with 1 mA current in a closed-cycle heliumcryostat in the temperature range 15–300 K and by a Quantum Design SQUIDmagnetometer, respectively. Measurements were performed on similar size bar-shaped sintered polycrystalline specimens allowing comparison of the results.

2.1 Sample Preparation

Polycrystalline samples with nominal composition RuSr2GdCu2O8 (hereafterreferred as Ru-1212) are commonly prepared by solid-state reaction techniquefrom a mixture of high purity RuO2 (99.95 %), Gd2O3 (99.99 %), CuO (99.9 %)and SrCO3 (99.99 %) [1,4,5,9,10,11]. The raw materials are:i - first reacted in air at about 960 C to decompose SrCO3,ii - heated in flowing N2 at 1010 C,iii - annealed in flowing O2 at temperatures ranging from 1050 to 1060 C andiv - finally, a prolonged anneal in flowing O2 at 1060 C is performed, duringwhich the material densifies, granularity is substantially reduced [12] and order-ing within the crystal structure develops [13].Because the superconducting and magnetic properties are affected by the detailsof the preparation process, which in turn affect the microscopic structure, a sys-tematic work on the synthesis of Ru-1212 and the effects of sample preparationon the magnetic and superconducting properties was developed [14]. Basically,a procedure as described commonly in literature and sketched in Fig. 1 has beenadopted with the aim of giving insight on the formation and stability of thevarious phases involved in the synthesis of this complex system. Each reactionstep was carried out on a MgO single crystal substrate to prevent reaction withthe alumina crucible. Between each step the products were thoroughly groundand pressed into pellets.i - The stoichiometric oxides were first calcined in air at different temperatures,TA, for 12 h. XRD spectra performed on these calcined samples are shown inFig. 2 (a), (b), (c). The spectra show the peaks of the Ru-1212 phase whoseamount increases with the temperature of the thermal treatment. There are

224 R. Masini et al.

calcination stepair (12 h)900 - 960oC

calcination stepN2 flow (12h)

1010oC

annealing stepO2 flow (15 h)1030 - 1085oC

long time sintering step

O2 flow (7 days)

full characterizationgrinding

Tf = Tprev + 7oC

quench

Fig. 1. Sample synthesis and sintering of RuSr2GdCu2O8

however reflections of second phases identified as SrRuO3 and Gd2CuO4, withhigher amounts in samples calcined at the lower temperature, diminishing withincreasing the calcination temperature.ii - The pellets were then annealed in flowing nitrogen at 1010 C for 12 h.The sintering in N2 gas is required to suppress the SrRuO3 phase [10]. Thisstep resulted in fact in the formation of a mixture of Sr2GdRuO6 and Cu2Oindependently of the starting calcined mixture from step (i) (hereafter namedL-serie). Typical XRD pattern is shown in Fig. 2 (d) obtained from the samplecalcined at 900 C which contained the highest amount of SrRuO3. No detectabletraces, within the resolution of the technique, of such very stable in oxidizingenvironment [10] impurity phase, were observed. In the light of these results,the synthesis of Ru-1212 by using SrO2 as starting reagent in place of SrCO3was investigated (sample I). Raw materials were then heated directly in N2flow at 1010 C avoiding thus the first calcination step (i) in air. No significantdifferences were obtained in the composition of the products as inferred fromXRD analysis with respect to previous results shown in Fig. 2 (d).iii – The L- serie mixture was then subjected to eight successive sintering stepsin flowing O2, each one lasting 15 h, at successively increasing temperatures inthe range 1030–1085 C. Each successive thermal treatment was performed at atemperature about 7 C higher than the previous one. In order to investigate theeffects of the thermal treatments, the product was quenched to room temperatureat the end of every step, fully characterized, reground, pressed into pellets andsubjected to the successive thermal treatment.

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 225

a

b

c

20 30 40 50 60

2

Inte

nsity

[a.u

.]

d

++

+

+

++ +

++

Fig. 2. X-ray spectra for samples calcined at (a) TA = 900, (b) 940 and (c) 960 C inair for 12h. © Ru–1212 , SrRuO3 and + Gd2CuO4; (d) after annealing in N2 for15h Sr2GdRuO6, Cu2O

Powder XRD patterns of all our samples show Ru-1212 as the major phase,with zero to some amount of SrRuO3 as minor impurity depending on the samplepreparation condition. Traces of second phase SrRuO3 (2% vol. for sample L1)with decreasing amount up to sample L3 were detected. Single-phase materialswere obtained afterwards. All peaks can be indexed assuming a tetragonal latticeand Table 1 lists the lattice parameters calculated for these Ru-1212 samples. InFig. 3 the X-ray powder diffraction pattern of sample L5, synthesized after fivesintering steps up to 1067 C for a total time t= (15 x 5) = 75 h, is reported.

226 R. Masini et al.

20 30 40 50 60

2

Inte

nsity

[a.u

]

101

110

104,

005

113

114,

105 20

0,00

6

007

213,

116

107

214

103

Fig. 3. XRD pattern of sample L5

Table 1. Synthesis, structural and electrical data of L–serie Ru–1212 samples

sample Tann(C) a (A) c (A) 290(mΩcm) TR=0(K) Tmax(K)

L1 1031 3.826(2) 11.516(7) 154.8 –a 37

L2 1037 3.821(3) 11.528(8) 141.7 –a 44

L3 1044 3.831(1) 11.545(7) 140.9 –a 46

L4 1053 3.831(1) 11.547(6) 70.6 17 48

L5 1061 3.828(1) 11.552(6) 30.6 25 49

L6 1067 3.844(1) 11.585(3) 14.0 21 44

L7 1073 3.845(1) 11.610(5) 23.7 21 47

L8 1084 3.835(1) 11.580(5) 22.0 –a 45

a

No information available below 15 K. See text for a complete discussion.

The same XRD spectra have been obtained for sample I, subjected to asubsequent thermal treatment at 1050 C for 24 h and successively to a prolongedanneal at 1060 C for a week in O2 flow.

Parallel checks have been performed allowing us to conclude that reachingthe “optimal” temperature directly in one step for a time which is the sum of thecorresponding partial times of each single step covered up to the same tempera-ture does not produce the same results of the longer procedure described above.Single-phase formation seems to be kinetically hindered by the slow decompo-sition rate of the impurities which already formed upon calcination. Repeatedhomogenizations, related to the sequence of grinding and annealing, improve thephase purity of the material and control the superconducting behaviour.

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 227

Fig. 4. SEM picture of sample L2

Fig. 5. SEM picture of sample L6

Morphologically, all the L-series samples show a high grain homogeneity withclean grain boundaries as probed by SEM and microprobe analysis. Fig. 4 showsthe typical granular morphology detectable at the beginning of the thermal treat-ment cycles (sample L2) with an average grain size of about 2 µm.

A progressive grain growth and a corresponding increase in grain connectivitydue to the different thermal treatments can be observed (Fig. 5, sample L6, itcan be noticed how some grains begin to coalesce into big aggregates dispersed inan almost unchanged granular matrix), without reaching, by the way, completesintering at the highest temperature.

228 R. Masini et al.

Such behaviour is related to the difference between the decomposition tem-perature of the 1212 phase and the maximum temperature of the thermal pro-cesses considered in this work.

A geometric density variation of about 5 % between the first and the finalbulk sample has been measured (d ≈ 4.2 g/cm3 corresponding to about 63 % ofthe theoretical crystallographic density).

3 Electrical Properties

In general, resistivity measurements of Ru-1212 show a superconducting transi-tion at 45 K with a very slight upturn in the vicinity of Tc reaching zero resistivityat a lower temperature between 20–30 K. The resistivity transitions at H = 0are much broader than those observed in many of the other HTSC. A metal-lic behaviour, with a linear T dependence at high temperatures above 100 K,is usually observed. A magnetic transition at TM = 132 K manifests itself as asmall yet noticeable kink/minimum in the resistivity related to the onset of themagnetic ordering of the Ru lattice.

Since the oxygen stoichiometry is practically unchanged in Ru-1212, the an-nealing turns out to influence mainly the granularity and ordering within thecrystal structure. Previous studies have shown that the semiconductor-like up-turn and the zero resistivity temperature are critically dependent on the sampleprocessing [10,15]. In particular, according to [6] the slight upturn in the vicin-ity of Tc is related to grain boundary effects. High-resolution TEM study onRu-1212 has shown that prolonged thermal treatment at 1060 C in O2 removesmost of a multidomain structure, consisting predominantly of 90 rotations, aswell as significantly reduces the semiconductor-like upturn [15]. Part of the su-perconducting transition width may be due to structural disorder. However itmust be underlined that a broad superconducting transition is also expectedwithin thespontaneous vortex phase model [16].

Curves of (T ) of selected samples (L-serie) considered significative, for thesake of clarity, of the overall process of synthesis, are shown in Fig. 6. All samplesexhibit weakly pronounced or local minima in the dc resistivity near the magnetictransition temperature, of the order of about 132 K. This feature is more clearlyvisible in the inset of the figure where the derivative of the resistivity (sampleL4) is plotted.

At low temperatures the dc resistance shows a semiconductor-like upturnfollowed by a sudden decrease in resistivity starting at Tmax and achieving zeroresistivity state for temperatures below 30 K as reported in Table 1. There isa small increase in the zero resistivity temperature for our best sample (L6)and only a small reduction in the semiconductor-like upturn. Summarizing thegeneral trend, it can be stated that the resistivity is progressively decreased anda crossover from a semiconducting to metallic normal state resistivity behaviouris observed on going from L1 to L8 sample. We underline that zero resistivityhas not been reached for samples from L1, L2, L3 and L8, even if, consideringtheir strong resistivity drop detected below 45 K, a R = 0 value is expected

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 229

0

100

200

300

0

0 50 100 150 200 250 3000

10

20

30

T (K)

(mcm

)

L1

L4

L5

L6

L8

100 150 200 250

Fig. 6. Resistivity vs T curves of some selected L-serie samples synthesized underdifferent conditions. Inset: d/dT temperature dependence for sample L4 around TM

0 50 100 150 200 250 300150

200

250

300

350

T (K)

(mcm

)

Fig. 7. Resistivity temperature dependence of sample I

at a temperature lower than 13 K for samples L2, L3 and L8. A comparisonbetween the resistivity behaviors, independently of their granular nature, hasbeen possible because different values are not related to the sample densityvariations, which, as already noted, is almost unchanged for all the samples.

A semiconducting-like transport with no indications of transition to super-conductivity at low temperatures is observed for sample E in all the temperaturerange considered, as shown in Fig. 7. It is noteworthy that a kink in resistivity

230 R. Masini et al.

10 20 30 40 50 60-1

0

1

2

3

4

T (K)

d/d

TL5L6L8

0

10

20

30

40L3L4

Fig. 8. d/dT temperature dependence

is observed in the vicinity of TM (arrow in Fig. 7). This anomaly is due to areduction of spin scattering. Such a behaviour is also observed in SrRuO3 singlecrystals [17] at around its ferromagnetic transition temperature.

The derivative of resistivity, shown in Fig. 8 for L3 – L8 samples, clearly showstwo overlapping maxima, indicating that the resistive transition proceeds in twosteps: a high-temperature contribution, associated with the thermodynamic su-perconducting transition temperature, and another one, at a lower temperature,which critically depends on sample processing conditions [10,14,15] as well asthe zero resistance temperature value.

Since from the analysis of X-ray patterns our superconducting and non-superconducting samples are indistinguishable, more insight about the physicalnature of the superconducting and magnetic states is expected from magneticmeasurements.

4 Magnetic Properties

The magnetic characterization of the rutheno-cuprate materials is a crucial andnot trivial point. Magnetic measurements are obviously a key tool to observeboth the superconducting and magnetic behaviour of these samples, but manyyears after their successful synthesis [1] and in spite of a great experimental effortdevoted to this problem, many doubts still survive about the magnetic orderingpresent in these type of samples [7, 11, 18, 19, 20, 21, 22].

First of all, we recall that measurements are usually performed on granularsamples. Therefore, all the problems related to the granular behaviour of HTSC

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 231

and, in general, to the distinction between intrinsic and extrinsic properties(intra-granular and inter-granular) must be born in mind.

A first problem encountered in the observation of the superconducting be-haviour is the fact that the standard diamagnetic signals, both in the FieldCooled (FC) and Zero Field Cooled (ZFC) mode, are not always seen in all thesamples of such compounds [13, 14, 18, 23]. What is more often observed is theshielding signal, rarely the diamagnetism related to the FC procedure. Both sig-nals are quickly removed by the application of even a small external magneticfield (few tens of Gauss). In contrast, even when in the magnetic measurementthere is no trace of superconducting behaviour, it may be observed resistivelyand the application of even a high external magnetic field (up to Tesla) doesnot destroy it [24]. The reason for such a contradictory phenomenology may beunderstood bearing in mind the simultaneous presence of magnetic and super-conducting ordering in these samples. This fact has consequences both on thesample physical behaviour and on the measurement technique used to monitorit. We recall that µSR measurements [5] indicate the homogeneous presence ofan internal field that, at low temperature, may reach hundreds of Gauss andmay give rise to a spontaneous vortex phase (SVP) in the temperature rangewhere it exceeds the first critical field Hc1(T ) [25]. In a type II superconductorat H > Hc1 the Meissner effect is practically never observed for the presenceinside the materials of “pinning centers” that are able to block the flux linesand prevent their expulsion. This is the reason why the FC diamagnetic signalmay be very small and its difference from the ZFC signal is an indication ofthe critical current density that a sample can carry. A vast literature relatedto high-Tc superconductors illustrates unambiguously this item [26, 27]. More-over, as noted in [28], the magnetization of rutheno-cuprate materials containsmagnetic signals arising from different contributions: the Gd paramagnetic spinlattice, the Ru spin lattice and, finally, the diamagnetic signal related to thesuperconducting behaviour. Both for Gd and Ru spin lattice the antiferromag-netic ordering is coupled with a ferromagnetic component that, in the case ofRu, is attributed to a canting of the lattice and in the case of Gd is simplyrelated to the presence of the net ferromagnetic moment of the Ru lattice [20].The simultaneous presence of such opposite magnetic signals makes the mag-netic measurement unsuitable for the observation of the superconductivity: infact, such measurement cannot separate the magnetic signal related to super-conductivity from that related to the magnetic ordering. Moreover, it is clearthat the application of an external magnetic field enhances the magnetic signalsand depresses the superconducting one, destroying very quickly the visibility ofthe superconductivity. In the light of these considerations we can understandthe fact that the superconducting behaviour is often observed resistively but notmagnetically: it depends on the competition between two opposite magnetic sig-nals, one related to the magnetic ordering, the other to the superconducting one.“More superconductivity” is obviously related to many factors: the amount ofsuperconducting phase inside the sample, the quality of the connection betweenthe grain that makes the shielded volume and therefore the related magnetic

232 R. Masini et al.

signal smaller or larger, and the intrinsic properties of the Ru-1212 phase that,as we will see, may change in connection with the degree of order of the mate-rial. Now, dealing with the experimental problems, we point out the following.In order to enhance the superconducting behaviour it is suitable to apply mag-netic fields as small as possible. This fact, due to the peculiar features of theinstrumentation commonly used, must be considered in detail. The first problemis the exact knowledge of the field that is effectively seen by the sample, andthe second is strictly related to the complexity of the magnetic signal presentin these samples. A small residual magnetic field in the superconducting coil ofthe experimental set-up is often present. It may be eliminated by a procedurethat, starting from a value of some Tesla, applies coercive fields of decreasingintensity. In such a way the field becomes vanishing but for a few Gauss thatmay be reduced to zero in the central point of the magnet by applying a smallcounterfield. Anyway, a very small residual field survives and turns out to be ofthe order of fractions of Gauss. In the light of what has been said, a real ZFCmeasurement cannot be made and, since the FC magnetic moment is about oneorder of magnitude greater than the ZFC, also a residual field of fractions ofGauss may give a considerable magnetic signal whose polarity depends on thefield polarity. In addition, the basic condition of a homogeneous magnetic mo-ment required by the SQUID magnetometer is not fulfilled, in particular at lowtemperatures, where, as a consequence of the applied field, magnetic momentsof opposite polarity will be present in the sample. Finally, we recall that duringthe measurements the sample is moved for a length that is usually of few cen-timeters, so that it travels in a non uniform magnetic field that makes it followa minor hysteresis loop. If the value of the moment is not constant during thescan, an asymmetric scan wave form will be observed and the quality of themeasurement will drastically degrade [29].

All we have said is illustrated in Fig. 9 where magnetization measurements arereported for both ZFC and FC conditions. For the sake of clarity, we report datafor some representative samples only. The cuspid at T 30 K marks the mag-netic ordering. A small variation is found in this temperature, which is smallerin the sample with higher superconducting temperature, in agreement with theliterature data [8]. It is remarkable to observe the different behaviour exhibitedby the various samples: L3 gives no hint of superconductivity, L5 exhibits a veryclear shielding corresponding to about 75 % of the maximum diamagnetic signalat µHext = 0.5 G while at µHext = 5 G its transition is strongly worsened, L6shows a diamagnetic shift after an ascent of the magnetization (probably dueto the instrumental effects we outlined before, for the presence of two oppositemagnetic signals of similar magnitude), and L8 shows a behaviour very similarto L3. At the lowest temperatures, a large contribution from the Gd sublattice,which orders antiferromagnetically at 2.5 K, is clearly visible in FC magnetiza-tion curves for the magnetically non-superconducting samples L3 and L8. If thesuperconductivity is marked by the visibility of a diamagnetic shift of the ZFCor FC signals, such behaviour is surely absent in L1, L2, L3, L4 and L8 while,to a different extent, it is observed in L5, L6 and L7. In the resistivity mea-

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 233

0

0.5

1

1.5(1

0-3em

u/g)

L3L6L8

ZFC

-10

-8

-6

-4

-2

0

2

(10-3

emu/

g)0.5 G5.0 G

L5 sample

ZFC

0 50 100 1500

2

4

6

8

10 L3L6L8

FC

0 50 100 150 2000

5

10

15

20

T (K)

0.2 G5.0 G

L5 sampleFC

Fig. 9. ZFC and FC susceptibility vs temperature curves for some samples of theL–serie: left L3 (5.5 G), L6 (5.5 G) and L8 (3.0 G); right L5 sample

surements, on the contrary, all the samples show a large drop of resistivity, butat the temperature of T = 13 K (the minimum value reached in the resistivitymeasurements, while in magnetic measurements we reached T = 5 K) zero isreached for L4, L5, L6 and L7 samples.

In the lower panels of Fig. 9 the FC data for samples L3, L5, L6 and L8 areshown. A sudden onset of a spontaneous magnetic moment appears, related toa ferromagnetic component arising from Ru spin ordering in RuO2 planes. Sucha spontaneous magnetization develops at temperatures in the range 130–135 K,then rising almost linearly below 110 K as the temperature decreases down toabout 50 K. We remark the very similar behaviour of L3 and L8, already observedin the upper part of Fig. 9. A clear diamagnetic behaviour is seen only in L5: atthe minimum applied field of 0.2 G, and to a minimum extent even at 1.2 G, adiamagnetic behaviour that quickly reenters is seen in the FC curve. At 5.5 G thediamagnetic effect is only seen as a constant value hindering the Gd magneticordering. Such behaviour has been already observed [30].

In Fig. 10 we present the inverse of the ruthenium susceptibility as a functionof the temperature for all the samples in the series L1-L8. In the calculation ofthe ruthenium susceptibility we have followed the procedure suggested by Buteraet al. [22]. Such a procedure calculates the Ru susceptibility by subtractingthree magnetic contributions to the experimental value: 1) the paramagnetic

234 R. Masini et al.

220 240 260 280 30050

100

150

200

T (K)

-1(m

ole/

emu)

L1L2L3L4L5L6L7L8

Fig. 10. χ−1Ru vs temperature for all the samples. Fitting results are shown as solid lines

1 2 3 4 5 6 7 81.5

2

2.5

3

3.5

4

annealing order

eff(

B)

80

100

120

140

160

180

(K)

Fig. 11. µRueff and θ calculated from a best fit of χ−1

Ru vs temperature data

contribution from Gd ions, 2) the core diamagnetism for the 1212 compounds asdeduced from the Landolt-Bornstein tables, and 3) a temperature-independentPauli-like contribution coming from the conduction electrons. The rutheniumsusceptibility correspondingly obtained is fitted by the Curie-Weiss relationshipχRu = CRu/(T −θ) and allows to calculate both the Curie temperature θ and theeffective magnetic moment µeff for Ru atom. Although a maximum content ofabout 2 vol. % of the SrRuO3 impurity phase was detected from X-ray analysis insample L1, when this phase is decreased to zero in sample L4 a similar negligibleerror on the absolute values of µRu

eff and θ has been calculated with no significanteffect on their general behaviour.

The obtained results are reported in Fig. 11 and in Table 2 as a function ofthe number of annealing steps that give rise to the sequence L1-L8.

Starting from L1 the values of θ increase, reach a maximum (around L5-L6 ofabout 160 K), and then decrease going up to L8. The µeff values have a specular

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 235

Table 2. µRueff and θ values as a function of annnealing steps and “superconductivity”

status of all L–serie samples

sample µRueff (µB) θ(K) resa ma

ZFC maFC

L1 3.77 85.3 – b no no

L2 2.63 149.0 – b no no

L3 2.36 155.6 – b no no

L4 2.21 159.7 yes no no

L5 2.18 161.8 yes yes yes

L6 1.95 168.3 yes yes yes

L7 2.74 148.4 yes yes no

L8 2.88 142.7 – b no noa measurement technique utilized to detect superconductivity: resistivity, ZFC and FCmagnetization.b No information available below 15 K. See the text for a complete discussion.

trend, decreasing from the value 3µB for L1 down to a minimum value of about2 µB at L6, and then slightly increasing once again. Since the superconductivityis better observed in the samples L5, L6, L7 both by resistivity and magneticmeasurements, these data suggest that an improved superconducting behaviourmay be related to small intrinsic variations in the structure of the sample thatproduce smaller effective magnetic moments for Ru atom and higher Curie tem-peratures. We give here only some suggestions to be explored. The values of µeffderived by the best fit imply that Ru is in a mixed valence state between Ru4+

and Ru5+. Such a result has been firstly proposed by Liu et al. [21] throughXANES spectroscopy and successively confirmed by Butera et al. [22] throughmagnetic measurements by means of the procedure we have outlined. These re-sults definitely contradict the hypothesis that Ru exhibits an effective momentµeff 1µB/Ru atom, as proposed in [4]. On passing from L1 up to L8 the pro-portion of Ru4+ and Ru5+ changes. Possible consequences of this fact are: slightvariations in the carriers number and, as a consequence, in the critical temper-ature (see the resistivity data in Fig. 6 and Table 1), different coupling betweenthe superconducting and the magnetic planes both in term of total magnetic mo-ment seen by the conduction electron (with increased or decreased pair-breakingeffect) and in term of coupling between orbitals of superconducting and magneticelectrons [31]. The origin of these variations may be found in a different degreeof lattice disorder following the various annealing steps performed at differenttemperatures that, as we have widely observed, produce significant variationsin all the physical properties. Moreover, the lattice disorder can imply a certainamount of Cu→Ru substitutions that are a possible candidate for the observedvariations of the effective magnetic moment. Also the variation of θ may be the

236 R. Masini et al.

0 10 20 30 40 50 600

2

4

6

8

0Hext (10+3 Gauss)

m(

B) L2

L3L4L5L6L7L8

T = 5K

Fig. 12. Magnetic moment versus magnetic field at 5K for L–series samples

consequence of the different coupling between Ru atoms following the differentvalence state.

The same trend we have seen in µeff is observed in the saturation momentas seen by measuring magnetization up to the maximum field of 5.5 Tesla atT = 5 K. Results are shown in Fig. 12. The values change from the minimumvalue of 6.5µB for L5 and L6 samples up to a maximum value of 7µB in the L2and L8 samples. We remark that in the experimental conditions we have used,the saturation is not completely reached, but the hierarchy of the saturatedmagnetic moments is surely correct.

5 Conclusions

The magnetic and superconducting properties of Ru–1212 have been studiedand compared for a series of samples synthesized under different conditions withthe aim of finding out the fundamental parameters ruling out the phase forma-tion and its related structural and physical properties. From our experimentalwork it results that the optimal annealing temperature lies in a narrow rangearound 1060–1065 C. A further temperature increase worsens the phase forma-tion. Subsequent grinding and annealing steps up to this temperature improvethe phase homogeneity. A wide variety of physical properties has been obtainedon quenched samples coming from the same batch and differing only in the syn-thesis procedure parameters. No other substantial differences were detected forthese samples, all showing similar compositional and structural characteristics.It emerges that the preparation method plays an important role when dealingwith the magnetic and superconducting properties of this hybrid compound.So far, published data on the Ru–1212 phase show the same general trend asregards the measured physical properties. Because most of the samples are chem-ically and structurally comparable, great care must be taken in the details of the

Synthesis Effects on Magnetic and Superconducting Properties of Ru-1212 237

preparation process, such as the final sintering temperature and the number ofhomogenization steps (if any) performed up to that temperature. Only sampleswith the same thermal history/parameters can be compared.

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Comparison of Electronic Structure,Magnetic Mechanism, and Symmetry of Pairingin Ruthenates and Cuprates

S.G. Ovchinnikov

L.V. Kirensky Institute of Physics, Siberian Branch of RAS,and UNESCO Chair of New Materials and Technologyof the Krasnoyarsk State Technical University, Krasnoyarsk, 660036, Russia

Abstract. A generalized tight-binding (GTB) method is developed to calculate quasi-particle band structure with explicit account for strong electronic correlations. TheGTB method combines exact diagonalization of the multi-orbital p–d model Hamil-tonian within the unit cell and the perturbation theory in the Hubbard X-operatorsform for the inter-cell hopping and interactions. For undoped cuprates we obtain thevalence band structure in excellent agreement with ARPES data, evolution of theband structure with doping with an in-gap state at small doping, impurity-like bandat higher doping to optimally doped metal. The effective low-energy Hamiltonian hasthe form of singlet-triplet t–J model. For ruthenates the t–J–I model is proposedwith both antiferromagnetic J and ferromagnetic I couplings. In the strong correla-tion limit, the mean-field theory of superconductivity results in d-wave singlet pairingmediated by J (cuprates) and p-wave triplet one mediated by I (ruthenates) with ra-tio T

(d)c /T

(p)c ∼ 100 due to the different symmetry of the gap. The competition of

ferromagnetic and antiferromagnetic order in rutheno-cuprates is also discussed.

1 Introduction

More than a decade of intensive research of the cuprate superconductors andrelated systems has raised fundamental challenges to our understanding of themechanism of high-temperature superconductivity (SC). One of the most im-portant questions is what is so specific in copper oxides, i.e. is it the uniquechemistry of the planar Cu–O bond that determines the high value of Tc? Thediscovery of SC in Sr2RuO4 with Tc ∼ 1 K [1] is of a particular interest be-cause it has similar crystal structure to the parent compound La2CuO4, of oneof the best studied families of the cuprate superconductors, La2−xSrxCuO4, buthas four valence electrons (for Ru4+) instead of one hole per unit formula. It isgenerally believed that comparison of normal and SC properties of the cupratesand the ruthenates will give deeper understanding of the nature of high-Tc SC.While the normal state of doped cuprates looks like that of an almost antifer-romagnetic Fermi-liquid [2], the normal state of Sr2RuO4 is characterised bystrong ferromagnetic fluctuations [3]. The properties of the SC state are alsodifferent: the singlet pairing with major contribution of the dx2−y2 symmetrywas suggested for the cuprates [4], while the triplet pairing with p-type sym-metry similar to the 3He A1 phase is proposed for Sr2RuO4 [5]. The triplet SC

C. Noce et al. (Eds.): LNP 603, pp. 239–255, 2002.c© Springer-Verlag Berlin Heidelberg 2002

240 S.G. Ovchinnikov

in Sr2RuO4 is induced by the ferromagnetic spin fluctuations [6]. Experimentalevidence for ferromagnetic spin fluctuation mechanism of pairing in Sr2RuO4results from NMR measurements [7] and ferromagnetism (FM) in the closelyrelated SrRuO3.

However, more recent evidence has suggested that this simple picture forSr2RuO4 may be incomplete. Antiferromagnetism (AFM) in Ca2RuO4, the ob-servation in Sr2RuO4 of an incommensurate peak at Q = (0.6π, 0.6π, 0) by neu-tron scattering [8] predicted by band theory calculations [9] due to nesting atQ = (2/3π, 2/3π, 0), and strong competition of FM and AFM in Ca2−xSrxRuO4[10,11] system all seem to imply that AFM correlations should not be ne-glected. Further experimental evidence for AFM correlations in RuO2 layerscomes from rutheno-cuprate RuSr2GdCu2O8 study where a coexistence of mag-netism and superconductivity was reported [12]. SC takes place in CuO2 layersbelow Tc ∼ 40 K while RuO2 initially was considered as FM below the Curietemperature Tm = 133 K. Later neutron diffraction experiments revealed AFMRu ordering [13]. Canted AFM is a possible way to reconcile different measure-ments concerning the magnetic properties of the RuO2 layer. Nevertheless, a netmoment is found of less than 0.1µB [13].

In the LDA approach, Sr2RuO4 is an itinerant system with a good agreementof the calculated Fermi surface [14] with the experimental results found in mea-surements of the Haas-van Alphen effect [15]. A competition of FM and AFM inthe band theory is governed by structural distortions (rotations and flatteningof RuO6 octahedra) as was shown by LDA calculations for Ca2−xSrxRuO4 [16]and RuSr2GdCu2O8 [17]. The insulating AFM ground state of Ca2RuO4 in theitinerant electron picture can be obtained only as a spin density wave (SDW)state that exists below TN with an insulator-metal transition at T = TN . Con-trary to this picture, Ca2RuO4 is an insulator both below and above TN , anda Mott-Hubbard ground state looks more appropriate. Thus, effects of strongelectronic correlations (SEC) have to be considered in ruthenates.

There are several experimental indications for SEC in Sr2RuO4:a) high effective mass of carriers in the dHvA oscillations [15] (mα = 3.4me,

mβ = 6.6me, mγ ≈ 12me);b) suppression of the single-electron density of states (DOS) in the region of

energy 0.5–2.5 eV below EF , as measured by valence-band photoemission spec-tra [18,19]. The experimental value of the DOS at the Fermi level N(EF ) is 3times less than the one calculated by LDA [19];

c) the calculation of the self-energy effects in the framework of the multi-band Hubbard model [20] allows to decrease N(EF ) to the experimental value.According to [20], Sr2RuO4 is a moderately correlated system with U ∼ W ;

d) ultraviolet photoemission spectroscopy measurements [21] reveal statesreminiscent of the lower Hubbard band at a binding energy of 1.5 eV. This allowsto estimate the on-site Coulomb repulsion U to be equal to 2.4 eV. Being theone-electron bandwidth W equal to 1.4 eV, one gets a ratio U/W = 1.7 whichsupports SEC. However, it was argued in [19] that this broad peak at 1.5 eV isrelated to a contaminated sample surface.

Comparison of Electronic Structure 241

In this paper we analyze the electronic structure of different ruthenates inthe framework of the multiband p–d model in the SEC limit and compare itwith a similar treatment of cuprates [22]. The different nature of the chemicalbonding in cuprates and ruthenates results in different magnetic properties too.In order to compare the electronic structure of cuprates and Sr2RuO4, we haveproposed a t–J–I model with both AFM coupling J and FM one I that allowsto treat SC in both oxides on the same footing [23]. Similar ideas have beenused recently to discuss the phase diagram of Ca2−xSrxRuO4 [24].

2 Generalized Tight-Binding Methodfor Quasiparticle Band Structurein Strongly Correlated Electron Systems

The necessary ingredient for discussing possible mechanisms of high-Tc super-conductivity is the band structure of the fermion-like quasiparticles. However,ab initio calculations are made difficult by the presence of strong electroniccorrelations. For this reason, here we use a model approach which is based onthe possibility of writing the exact full Hamiltonian of the system in terms oflarger-scale variables. Briefly described, the essence of the approach is as follows.Suppose we have a set of Wannier functions belonging to a chosen elementarycell, which is generated by some single-electron problem for the crystal symme-try of interest. Then the full electron field operator can be written in the formof an expansion:

ψσ(r) =∑jL

χjL(r) ajL .

Here the index j ≡ Rj gives the coordinate of the chosen elementary cell ofthe lattice, L is a composite index, i.e. L = λ, σ, where λ labels bands and σstands for the spin projection. Due to the orthogonality of the Wannier functions,we have the anti-commutation relations ajL, a

+j′L′ = δjL,j′L′ . A full non-

relativistic Hamiltonian can be written in the form of a sum of intra- (Hc) andinter-cell (Hcc) terms:

H = Hc +Hcc =∑

i

Hci +

∑i =j

Hccij , (1)

Hci =

∑L L′

hLL′i a+

iLaiL′ +12

∑L L′

νiL, iL′, iL′′, iL′′′a+iLa

+iL′aiL′′aiL′′′ ,

Hccij =

∑L L′

hLL′ij a+

iLajL′ +12

∑L L′

νiL, jL′, jL′′, iL′′′a+iLa

+jL′ajL′′aiL′′′ −

−12

∑L L′

νiL, jL′, iL′′, jL′′′a+iLa

+jL′aiL′′ajL′′′ .

Suppose now that we are able to find the many-electron states for the intra-cellHamiltonian exactly, so that strong intra-atomic and intra-cluster interactions

242 S.G. Ovchinnikov

are fully taken into account. Then

Hci |i, Γn〉 = E

(0)nΓ |i, Γn〉 (2)

and the full Hamiltonian can be rewritten in terms of the Hubbard’s operators

Xmi ≡ X

Γn−1,Γn

i ≡ |i, Γn−1〉 〈i, Γn| , Zξi ≡ Z

Γn,Γ ′n

i ≡ |i, Γn〉 〈i, Γ ′n| . (3)

The Hubbard model is often used to study the electronic structure of stronglycorrelated electron systems (SCES). To take into account the chemistry of metaloxides, it is generalized to the p–d model. A simple version of such a model hasbeen proposed by Emery [25] and Varma et al. [26]. In this three-band p–d modelonly dx2−y2 Cu and pσ O orbitals are considered. There are many indications ofthe importance of the dz2 Cu orbital (see the review in [27]). The multiband p–dmodel with both dx2−y2 and dz2 states has been proposed by Loktev et al. [28]. Tocalculate the electronic structure in SCES, the generalized tight-binding (GTB)method combining the exact diagonalization of the Hamiltonian for small clus-ters (intra-cell part of the Hamiltonian) with cluster perturbation theory for theinter-cell part of the Hamiltonian has been proposed by Ovchinnikov and San-dalov [29]. Recently, the electronic structure of the undoped antiferromagnetic(AF) insulator Sr2CuO2Cl2 and its evolution with doping have been studied byGavrichkov et al. [22]. The dispersion equation within the GTB method for thetwo-sublattice AF state is given by

∥∥∥∥∥(E −ΩG

m

)FG

σ (m)δmn − 2

∑λλ′

γ∗λσ (m)TPG

λλ (k) γλ′σ (n)

∥∥∥∥∥ = 0 , (4)

where the local excitation energy Ω(0)i,m = E

(0)n+1,Γ −E

(0)n,Γ is renormalized by the

Coulomb interactions within the analogue of the Hubbard-I approximation as

δmm1Ωi,m = δmm1Ω(0)i,m +

∑j ( =i)

(BΓ1Γ2,Γ3Γ4

ij εm,[Γ1,Γ2]m1

⟨ZΓ3,Γ4

j

⟩−

− BΓ1Γ2,Γ3Γ4ji εm,[Γ3,Γ4]

m1

⟨ZΓ1,Γ2

j

⟩)(5)

with the matrix elements BΓ1Γ2,Γ3Γ4ij given by

BΓ1Γ2,Γ3Γ4ij = νiL,jL′,jL′′,iL′′′

[a+

iLaiL′′′]Γ1Γ2

[a+

jL′ajL′′]Γ3Γ4

. (6)

Here the transitions [Γ1, Γ2], [Γ3, Γ4] are Bose-like ones, and the splitting infermion-like transitions originates, obviously, from the crystal-field splitting ofthe states Γ by the Coulomb interactions with the neighbouring clusters. BeingΩ

(0)i,m of the order of the Hubbard U of the cluster, these effects are small. Below

we consider the antiferromagnetic state and the lattice index i runs over thesublattices P,G.

Comparison of Electronic Structure 243

Here m ↔ (p, q) enumerate a quasiparticle described by the Hubbard opera-tor Xp q, and γλσ(m) is a parameter of X-operator representation for the single-electron annihilation operator with orbital λ and spin σ, af λσ =

∑mγλσ(m)Xm

f .

Eq. (4) has the same structure as the usual tight-binding equation of the single-electron approach but differs in the following: i) the local energies Ωm are givenby the multielectron resonance Ωm = En+1(p)−En(q) , between n– and (n+1)–electron terms of the cell, ii) the filling factors F (m) = 〈Xpp〉 + 〈Xqq〉 lead to aconcentration and a temperature dependence of the band structure.

3 Evolution of the Electronic Structure with Dopingin Cuprates

For cuprates we calculate the electronic structure of the CuO2 layer with apicaloxygen ions, because all other electron states are far from the Fermi level. Theessential atomic orbitals are dx2−y2 and dz2 of Cu, px and py of in-plane Oand pz of the apical ion (O or Cl). In the hole representation the multielectronconfigurations involved in the formation of the single-electron band structureare:

i) the vacuum state |0〉, corresponding to the d10p6 configuration, with num-ber of holes per unit cell nh = 0;

ii) single-hole states (nh = 1) corresponding to the usual molecular orbitalsof the CuO6 (CuO4Cl2) cluster with b1g and a1g symmetry (d9p6 + d10p5);

iii) two-hole states (nh = 2) corresponding to spin singlet and triplet states(d9p5 + d10p4 + d8p6 + d10p5p5). The well known Zhang-Rice singlet (d9p5) isonly one of the components of the lowest two-hole level 1A1g. The usual beliefthat its triplet partner is more than 2 eV higher in energy and is not importantfor a low-energy physics is valid only in the three-band model. Two main factorsdecrease the energy of the 3B1g triplet up to 0.5 ÷ 0.7 eV above the 1A1g term:the Hund exchange coupling in the dx2−y2dz2 configuration and the effect of theapical anion [22].

The exact diagonalization of the intra-cell part of the Hamiltonian Hc givescluster eigenstates |p〉 with nh = 0, 1, 2 and energies Ep that have been used toconstruct the Hubbard X-operators and treat the CuO2 lattice as a generalizedHubbard model in GTB (see details in [22]).

The electronic structure for the undoped case with hole concentration nh = 1is shown in Fig. 1. It was calculated for the following set of the model parameters(in eV):

tpd = 1, ε(dx2−y2) = 0, ε(dz2) = 2, ε(px) = 1.6, ε(pz) = 0.5,tpp = 0.46, t′pp = 0.42, Ud = 9, Up = 4, Vpd = 1.5, Jd = 1 .

The wide charge-transfer gap Ect ≈ 2 eV separates the empty conductivityband, having mainly dx2−y2 character, from the filled valence band formed by acomplex mixture of different copper and oxygen orbitals.

244 S.G. Ovchinnikov

Fig. 1. Band structure of the undoped CuO2 layer

Fig. 2. The dispersion of the top part of the valence band in comparison with ARPESdata on Sr2CuO2Cl2

The dispersion of the valence band is shown in Fig. 2 together with theARPES data [30]. It should be noted that the valence band is formed by hole ex-citations from the initial single-hole state |1, σ〉 to different final two-hole states,i.e. the singlet |S〉 and the triplet |T,M〉 (M = −1, 0, +1), which are bothessential. One more unusual result of GTB calculations [22] is the dispersionlessvirtual level shown by the dotted line in Fig. 2. It has zero weight in the un-

Comparison of Electronic Structure 245

Fig. 3. Concentration dependence of the band structure in the antiferromagnetic(x=0.01, solid line; x=0.1, dot-dashed line) and paramagnetic (x=0.2, dashed line)phases

doped case but acquires a spectral weight ∼ x in the hole-doped system withnh = 1 + x, giving rise to the formation of an in-gap state at the top of the va-lence band. The dispersion of the in-gap state and its concentration evolution isshown in the Fig. 3. In the underdoped region with strong AFM correlations thein-gap state evolves in the impurity-like band separated from the main valenceband by a pseudogap at x = (π, 0). A maximum of the band lies at (π, π). Inthe optimally doped paramagnetic state (x = 0.2 in the Fig. 3) the dispersion issimilar to that obtained by LDA with a much smaller bandwidth: a maximumis at (π, π) and a saddle point at (π, 0) provides the van Hove singularity in theDOS. A detailed discussion of ARPES results for Sr2CuO2Cl2 and Ca2CuO2Cl2,as well as the analysis of the Fermi surface and of the polarization dependenceof ARPES is given within the GTB method in [31].

246 S.G. Ovchinnikov

4 Comparison of Superconductivity in Cupratesand Ruthenates in the t–J–I Model

To compare the SC in Sr2RuO4 and in the cuprates, we have proposed a t–J–I model containing both an antiferromagnetic coupling J and a ferromagneticcoupling I between neighboring cations [32]. An important difference from thecuprates is that the relevant orbitals for the states near the Fermi energy are Rudε(dxy, dyz, dxz) and O pπ, instead of Cu dx2−y2 and O pσ states. Due to theσ-bonding in the cuprates, a strong p–d hybridization takes place resulting in thestrong antiferromagnetic coupling J , while a direct dx2−y2 Cu-Cu overlappingis negligible. In Sr2RuO4 with π-bonding the Ru-O-Ru 180 antiferromagneticsuperexchange coupling is weak [33], while a direct dxy Ru-Ru overlapping is notsmall. This is the reason why we add the Heisenberg-type ferromagnetic Ru-Ruexchange interaction in the Hamiltonian of the t–J model.

The Hamiltonian of the t–J–I model is written in the form

H =∑fσ

(ε− µ)Xσσf − t

∑fδσ

Xσ0f X0σ

f+δ + J∑fσ

K(−)f,f+δ − I

∑fσ

K(+)f,f+δ (7)

withK

(±)fg = Sf · Sg ± 1

4nfng X↑↑

f +X↓↓f +X00

f = 1 .

Here the Hubbard X-operators Xpqf = |p〉〈q| are determined in the reduced

Hilbert space containing empty states |0〉 and singly-occupied states |σ〉 (σ =↑,↓).The X-operators algebra takes into account the constraint condition, which isone of the important effects of the strong electronic correlations. The operatorsSf and nf in (7) are the usual spin and particle number operators at the site f ,and δ is the vector between nearest neighbour sites. For the cuprates J I, butfor Sr2RuO4 I ∼ J . To get SC, the copper oxides require doping while Sr2RuO4is self-doped.

The t–J–I model results from the electronic structure of Sr2RuO4 in thestrongly correlated limit. A discussion of this model is given below in Chapter5. The strong electronic correlations split the γ-band into a filled lower Hubbardband (LHB) with ne = 1 and a partially filled upper Hubbard band (UHB)with electron concentration ne = n0. We use the hole representation where theelectron UHB transforms into the hole LHB with hole concentration nh = 1−n0.The other bands (α and β) are treated here as an electron reservoir. Observationof a square flux-line lattice in Sr2RuO4 allows to suggest that SC mainly developsin the γ-band [34]. For the cuprates the quasiparticle is a hole in the electronLHB with electron concentration ne = 1 − n0, for La2−xSrxCuO4 n0 = x.

Among the many possible ways of getting the mean-field solutions for thesuperconducting state, we have used the irreducible Green function method [35]projecting the higher-order Green functions onto the subspace of the normal⟨⟨X0σ

k

∣∣Xσ0k

⟩⟩and the anomalous

⟨⟨X−σ0

−k

∣∣Xσ0k

⟩⟩Green functions, coupled via

the Gorkov system of equations. Three different solutions have been studied:singlet s- and d- and triplet p-types. The gap equation has for the s-state the

Comparison of Electronic Structure 247

form

1 =1N

∑p

2ωp + (2g − Λ)ω2p

2Ep0tanh

(Ep0

2τ

)(8)

and for the p and d-states the form

1λl

=1N

∑p

ψ2l

2Epltanh

(Epl

2τ

). (9)

Here for s, p, d-states

Epl =√c2(n0)(ωp −m)2 + |∆pl|2

m =1

c(n0)

[µ− ε

zt+ (g + Λ)

1 − n0

2

],

where ωp = −γp = −(1/z)∑

δ exp(ipδ), m is a dimensionless chemical potential,c(n0) = (1 + n0)/2, g = J/t, Λ = I/t and τ = kBT/zt is a dimensionlesstemperature. The superconducting gap is equal to

∆k0 = [2 + (2g − Λ)ωk]∆0 (10)

for s-type (l = 0) pairing, and to

∆kl = λl ψl(k)1N

∑p

ωlpBp

c(n0), (11)

where l = p, d distinguishes among p- and d-type states. Here we have defined

∆0 =1N

∑p

ωpBp

c(n0)

withBp =

⟨X0↓

−pX0↑p

⟩.

The coupling constants and the gap anisotropy in the l–th channel are givenby

λp = Λ ψp(k) =12(sin kx + i sin ky) (12a)

λd = (2g − Λ) ψd(k) =12(cos kx − cos ky) . (12b)

Here we have considered only a two-dimensional square lattice with lattice pa-rameter a = 1. The equation for the chemical potential has the form

1 − n0 =1N

∑kσ

⟨Xσ0

k X0σk

⟩. (13)

248 S.G. Ovchinnikov

Fig. 4. Concentration dependence of Tc, λp = I/t, λd = J/t. Here t = 0.1 eV

An important effect of the strong electronic correlations is the constraint condi-tion excluding doubly-occupied states

1N

∑k

Bk =1N

∑k

⟨X0↓

−kX0↑k

⟩= 0 . (14)

The first term in the r.h.s. of (8) (∝ ωp) for the singlet s-type pairing isproportional to 2tz and appears due to the kinematic mechanism of pairing [36].The s-type solution does not satisfy the constraint condition (14) [37] while forp- and d-type it is fulfilled. The equation for Tc in p- and d-states is given by

2c(n0)λl

=1N

∑p

ψ2l (p)

|ωp −m| tanh(c(n0)|ωp −m|

2τ (l)c

). (15)

The same equation for the dx2−y2 -pairing has been derived by a diagram tech-nique applied to the t–J model [38].

In the numerical solution of (15), more than 106 points of the Brillouin zonehave been taken into account. Results for Tc(n0) are shown in Fig. 4 for sev-eral values of the coupling constants λl. These results have revealed remarkabledifferences in the Tc values: T (p)

c T(d)c when λp = λd. Moderate values of λl

(≈ 0.4–0.5) and zt (≈ 0.5 eV) give rise to T (p)c ∼ 1 K and T (d)

c ∼ 100 K.The reason why Tc in the cuprates is much higher than in Sr2RuO4 lies in

the different gap anisotropy. For the p-state the k-dependence of the gap resultsin the cancellation of the van-Hove singularity, while for the d-state the gapanisotropy permits a large van-Hove singularity contribution in the equationfor Tc.

The coexistence of magnetism and superconductivity in RuSr2CdCu2O8 hasbeen discussed also in the framework of the t–J–I model assuming that this

Comparison of Electronic Structure 249

model can be applied to both CuO2 and RuO2 layers with JCu ICu andJRu < IRu [39]. In the CuO2 layer the dx2−y2-pairing mediated by AFM spinfluctuations occurs with the Tc value (46 K) determined by the hole concentra-tion in the underdoped regime. In the RuO2 layer with both AFM and FM spincoupling, a competition of singlet dx2−y2 and triplet p superconducting phasesas well as long-range ordered AFM and FM phases takes place. The couplingparameter for the dx2−y2 SC is given by λd = (2J − I)/t (see Eq. (7). In thestrongly correlated limit of the electronic structure of rutheno-cuprates discussedin the next chapter, J ∼ 4t2α,β/U and I ∼ txy. For tzg orbitals with O–π bond-ing, tαβ ≈ txy/2 and thus we can estimate λd ∼ (2txy/U − 1). Close the theMott-Hubbard transition U ∼ ztxy, and λd ∼ (2/z − 1) < 0. This estimationshows that the singlet d-type SC cannot appear in RuO2 layer, while p-triplet SCis competing in energy with FM and AFM order. In a three-dimensional phasediagram of the model in (n, I, J) coordinates, there are regions of stability forboth FM and AFM phases and the triplet p-wave SC. A two-dimensional cross-section of this phase diagram by the plane J = 0 has been considered in [39]where the stabilization of the FM phase vs. SC at decreasing carrier concen-trations has been obtained. Due to the neutron scattering data [13] that revealAFM order in the RuO2 layer, a competition of FM and AFM order is to bestudied. A qualitative analysis of the electronic structure and FM/AFM ordercompetition in rutheno-cuprate is given in the next chapter.

5 Electronic Structure of Ruthenatesin the Multiband p–d Model

From a general point of view, electronic correlations in 4d Ru orbitals are weakerthan in 3d Cu orbitals. This conclusion is also supported by experimental datadiscussed above in the Introduction. This is why we assume that U ∼ Uc ∼ Wfor ruthenates, contrary to U W in cuprates. In this region, the Hubbard-Iapproximation is not very reliable, and here we shall qualitatively analyze theeffect of correlations on the electronic structure.

Due to the π-bonding of Ru t2g and O p electrons, the p–d hybridization inruthenates is weaker than in cuprates with σ-bonding of Cu eg and O p electrons.This is why the oxygen contribution to the bands near the Fermi level is small.For example, oxygen p contribution is 16 % while Ru 4d weight is 84 % [40]. LDAcalculations [3,14] have revealed three bands α, β and γ crossing the Fermi level.Analytical expressions for each band Eλ(k) (λ = α, β, γ) are obtained in thetight-binding method [40]. Here the γ-band forms from the dxy orbital, whilethe (α, β)-bands come from the mixture of the dyz and dzx orbitals. The β andγ bands form an electron Fermi surface centered at the Γ point and the α-bandforms a hole pocket around the x = (π/a, π/a) point [15]. The degeneracy ofdyz and dzx bands is removed by a small hybridization of these bands witht = 0.1 eV [41]. An inter-orbital Coulomb interaction between α- and β-statesresults in the further increase of the splitting of these bands due to the different

250 S.G. Ovchinnikov

Fig. 5. Three-band model of the electronic structure of ruthenates: a) Sr2RuO4; b)Ca1,5Sr0,5RuO4 and RuO2 layer in RuSr2GdCu2O8; c) Ca2RuO4

filling of electron and hole bands. Indeed, in the mean field approximation

Vαβ nαnβ = Vαβ〈nα〉nβ + Vαβnα〈nβ〉 , (16)

one obtains a renormalization of the band energies

εα → εα = εα + Vαβ〈nβ〉 (17a)εβ → εβ = εβ + Vαβ〈nα〉 (17b)

so that the α–β band splitting is equal to

εβ − εα = εβ − εα + Vαβ (〈nα〉 − 〈nβ〉) ∼ t⊥ +23Vαβ .

For the typical value Vαβ ∼ 1 eV this splitting is 0.7 ÷ 0.8 eV, which is compa-rable with the bandwidth Wα = Wβ ≈ 1.5 eV. As for the Hubbard intra-atomicCoulomb parameter U , all estimations in the literature give U ≈ 2÷3 eV, so thatwe assume that Sr2RuO4 is near the Mott-Hubbard transition. In order to modelthe effect of strong correlations, we consider the results obtained within the dy-namical mean-field theory (DMFT) [42], which are similar to those presentedin [24]. Typically, the effects of strong electronic correlations on the DOS are thefollowing: at U > Uc a band is splitted into upper and low Hubbard bands (UHBand LHB) with a gap between them. At U < Uc UHB and LHB are overlapping,this resulting in a metallic state. Moreover, near Uc a very narrow and sharpKondo resonance develops near εF [42].

Taking into account α- and β-bands splitting due to correlation effects, weassume for Sr2RuO4 the three-band model depicted in Fig. 5a. In the limit U → 0and Vαβ → 0 this model DOS is similar to the model considered in [11]. Due

Comparison of Electronic Structure 251

to different bandwidth of γ- and (α, β)-bands, Wγ ≈ 2Wα,β , the γ-band in ourmodel has overlapping γ-UHB and γ-LHB while for α, β-bands U > Wα andUHB and LHB are separated by a gap. The hole α-band is almost occupied witha small concentration of holes nh

α ≈ 0.3 [14]. For this reason, the spectral weightof the α-LHB is small (∼ nh

α/2), and the Mott-Hubbard splitting for the almostfilled α-band is not important. The metallic state of Sr2RuO4 is provided bythe crossing of the γ-band, the α-UHB, and the β-UHB by the Fermi level, sothat one has one hole-like Fermi surface (α-UHB) and two electron-like Fermisurfaces (γ- and β-UHB). We do not discuss here the dispersion and the Fermisurface in our model, but it is known from the study of the Hubbard and the t–Jmodels that the Fermi surface at U = ∞ and ne = 2/3 in a square lattice is thesame as for U = 0 and ne = 1. Moreover, our calculations of the band structureof La2CuO4 (Ch. 3 above) lead for optimal doping nh ≈ 0.18 to a dispersionwhich in shape is similar to that given by LDA, but with a bandwidth 3÷4times smaller than LDA. So, given the DOS of Fig. 5a, it looks possible toget a self-consistent solution for the chemical potential equation resulting ina Fermi surface in agreement with that given by LDA calculations and dHvAmeasurements.

Recently, the electronic structure of Sr2RuO4 has been calculated in theframework of the LDA+U method which accounts for correlations and self-energy effects [48].

Ca substitution in Ca2−xSrxRuO4 is known to result in the rotation, tilting,and flattening of the RuO6 octahedron [43]. In the region between Sr2RuO4,having symmetry I4/mmm, and Ca1.5Sr0.5RuO4, having symmetry I41/acd, thereduction of x leads to a rotation of RuO6 around the c axis, with the rotationangle at x = 0.5 being equal to ϕ = 12.78. As shown by LDA calculations [16],this rotation couples with the dxy orbital but not with the dyz, dzx orbitals,because while the pdπ type of hybridisation between the oxygen 2p and the dxy

states is significantly reduced by the RuO6 rotation, that between O-2p and dyz,dzx states is not affected so much. Thus the bandwidth Wγ reduces and thesystem moves closer to the Mott-Hubbard transition, with the γ-band almostsplitted (Fig. 5b). The concentration of carriers (electrons) in the γ-UHB issmaller than in Sr2RuO4.

Exactly the same type of RuO6 rotation takes place in RuSr2GdCu2O8 withthe angle ϕ = 14 as determined by neutron diffraction [44]. We assume thatthe DOS of the RuO2 layer in rutheno-cuprates may be described by the samemodel as in Fig. 5b.

The further reduction of x in Ca2−xSrxRuO4 results in RuO6 tilting andsymmetry is further reduced to Pbca until x = 0 (Ca2RuO4), where ϕ = 11.93

and the tilting angle θ 12 at low temperatures [43]. The tilting reduces allt2g bandwidths, providing the Mott-Hubbard gap between γ-UHB and γ-LHB aswell as between β-UHB and β-LHB. In this model (Fig. 5c), the α-band is totallyoccupied and Ca2RuO4 is a two-band Mott-Hubbard insulator with one hole inthe γ-UHB and one hole in β-UHB, i.e. nh

γ : nhβ = 1 : 1. It should be stressed that

this model describes the insulator phase of Ca2RuO4 both below and above the

252 S.G. Ovchinnikov

Neel temperature, in agreement with the experimental phase diagram [10,11]and disagreement with LDA+U [24] and Hartree-Fock [45] calculations thatascribe the insulator gap to spin-up/spin-down band splitting due to AFM order.Another difference between our model and the LDA+U results [24] for Ca2RuO4is in the different filling of the bands. The γ-band is fully occupied in [24] sothat 2 holes are in the (α, β)-band, and thus the ratio nh

γ : nhα,β = 0 : 2 is in

disagreement with the O 1s X-ray absorption spectra (XAS) of Ca2RuO4 [45]where nh

γ : nhα,β = 1 : 1 at T = 300 K and 1/2 : 3/2 at T = 90 K. Our model of

Ca2RuO4 in Fig. 5c is in agreement with O 1s XAS at T = 300 K.The temperature dependence of XAS results from the flattening of the RuO6

octahedron with cooling [45]. Up to now we have considered elongated or regularRuO6 octahedron. The ratio r of apical Ru-O to in-plane Ru-O distances is equalto 1.07 for Sr2RuO4 and it is not affected by rotation around the c axis. In therutheno-cuprate RuSr2GdCu2O8, r ≈ 1 as well as in Ca2RuO4 at T = 300 K.Decreasing temperature in Ca2RuO4 below 300 K, RuO6 octahedron flatteningoccurs with r ≈ 0.98 at T = 100 K [46]. In the free-electron approach, the yz/zxorbitals become higher in energy compared to the xy orbital in the compressedoctahedron. Nevertheless, the Hartree-Fock calculations, with account for theintra-atomic Coulomb and exchange interactions, reveal that two holes at T =90 K are in the yz and (xy + izx)/

√2 orbitals [zx and (yz + ixy)/

√2] [45],

providing a ratio nhγ : nh

α,β = 1/2 : 3/2 in agreement with 1s–XAS. To obtainthe insulating state, the magnetic gap due to spin-up/spin-down splitting isinvolved again [45], while in our approach the strong electronic correlations splittwo half-filled orbitals into a filled LHB and an empty UHB. So, the total DOSpicture looks like the one in Fig. 5c, but with some redistribution of the partialDOS with temperature.

6 Competition of Ferromagnetismand Antiferromagnetism in Ruthenates

The competition of FM and AFM is an essential property of SEC systems.Indeed, in the one-band Hubbard model on a bipartite lattice where the Fermisurface at ne = 1 has nesting, the AFM ground state is stable at all valuesof U/W . Nevertheless, in the U = ∞ limit only one extra carrier is enoughto establish the Nagaoka FM state. It is known from the Hubbard model thatvirtual excitations between LHB and UHB result in an AFM exchange couplingJ = 4t2/U while in the SEC limit U W the FM coupling is of intrabandkinematic origin, I ∼ t. Let x be a concentration of carriers near half-filling, i.e.ne = 1 + x at ne > 1 (electron doping) and ne = 1 − x at ne < 1 (hole doping).Then, qualitatively, the competition of FM and AFM may be estimated in thefollowing way. The energy gain per bond is in the FM phase EFM ∼ Ix, and inthe AFM phase EAFM ∼ J(1 − x2) ∼ 4t/gz(1 − x2), where z is the number ofnearest-neighbour sites, W = zt, g = U/W , and the factor (1−x2) = ne(2−ne)takes into account the absence of AFM for empty and totally occupied band.Comparing EFM and EAFM , we see that below the critical concentration xc =

Comparison of Electronic Structure 253√

(gz/8)2 + 1 − gz/8 the AFM state is stable, while above xc the FM statestabilizes. In the limit U → ∞ xc = 4/gz → 0 in agreement with the Nagaokatheorem. This simple estimation is confirmed by a generalized RPA treatmentof the t–J model [47] that at small x yields

TN =12zJ − 1

4xzt . (18)

Here the second term in the right side represents the kinematic suppression ofAFM by carriers.

Keeping all this in mind, we come back to the band models in Fig. 5 todiscuss the competition of FM and AFM in various ruthenates. For Sr2RuO4the γ-band is at the metallic side of the Mott transition and a band approachwith a significant Stoner factor IN(εF ) ≤ 1 provides the FM contribution tothe effective spin coupling, while both nesting in (α, β)-bands (weak correlation)and interband virtual excitations (strong correlation) provide the AFM coupling.According to LDA calculations [9], J ∼ I in Sr2RuO4. In the SEC limit U W the FM coupling is I ∼ tγ and the AFM one J = 4t2α,β/U ≈ t2γ/U withdominating FM coupling. Near the Mott transition U ∼ ztγ and then J/I ∼tγ/U ∼ 1/z = 1/4 for 2d RuO2 lattice. Both LDA and SEC limits are not veryaccurate close to the Mott transition. Nevertheless, both give that J and I areof the same order of magnitude. In this situation a compensation of FM andAFM couplings results in the non-magnetic ground state of Sr2RuO4.

For the band structure of Fig. 5b, the AFM coupling via (α, β)-bands shouldbe the same as for Sr2RuO4 while the FM coupling will decrease due to thesmaller concentration of electrons at the bottom of the γ-UHB, with simulta-neous increasing of the AFM coupling through the γ-UHB ↔ γ-LHB interbandvirtual excitations. Both these effects may stabilize the AFM state, that fromneutron measurements [13] is known to develop in RuSr2GdCu2O8. Previously,a similar conclusion has been obtained by LDA calculations [17].

In Ca2RuO4 the FM coupling is suppressed by the absence of carriers inthe Mott-Hubbard insulator state while the AFM coupling due to the interbandexcitations is produced by both γ and β bands.

7 Conclusions

The explicit account for SEC in a generalized tight-binding method allows tocalculate the electronic band structure of the Mott-Hubbard insulators. For un-doped cuprates it gives a quantitative agreement with the ARPES spectra. Thesmooth evolution of the band structure with hole doping in cuprates is obtained,with the formation of a pseudogap at the X = (π, 0) point in the underdopedregion and the development of the typical LDA band dispersion in the optimallydoped region.

Assuming SEC to be valid for ruthenates, we can compare the magneticmechanism of pairing and the symmetry of the order parameter of cuprates andruthenates in the framework of the t–J–I model. For ruthenates a three-band

254 S.G. Ovchinnikov

model accounting for SEC is proposed that allows to explain qualitatively theconnection between lattice distortion, electronic phase diagram, and FM/AFMcompetition both in Ca2−xSrxRuO4 and in rutheno-cuprate materials.

Acknowledgements

The author is thankful to C. Noce and M. Cuoco for their hospitality duringthe visit to Salerno, and to E.V. Kuz’min, I.O. Baklanov, V.A. Gavrichkov, andA.A. Borisov for the fruitful collaboration, and to N. Lishneva and M. Korshunovfor technical assistance. This work has been supported by the Russian FederalProgramm “Integratsia” , grant A0019, by RFFI grant 00-02-16110, and by theKrasnoyarsk Regional Science Foundation, grant 10F003.

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Magnetism, Spin Fluctuationsand Superconductivity in Perovskite Ruthenates

D.J. Singh and I.I. Mazin

Naval Research Laboratory, Washington, DC 20375 U.S.A.

Abstract. The triplet superconducting state of Sr2RuO4 and the rich variety of mag-netic ground states observed in perovskite ruthenates are discussed using results fromdensity functional calculations. We emphasize the importance of itinerant effects, whichlead to a high sensitivity of the magnetic state to structural degrees of freedom as wellas connections between the Fermiology and properties.

1 Introduction

Perovskite derived ruthenates have been the focus of considerable recent activ-ity both because of the interesting magnetic properties of these phases and thetriplet superconductivity of Sr2RuO4 [1,2]. The phases, which may be genericallywritten, (Sr,Ca)n+1RunO3n+1, are all based on Ru4+, octahedrally coordinatedby O, with electronically inactive A-site counter-ions. Nonetheless, they show arich variety of magnetic and electronic states, often with experimental signaturesof strong coupling to the lattice. These include robust ferromagnetism, antifer-romagnetism, paramagnetism, with both metallic and insulating ground states.Besides the interest arising from this magnetic variety, it is now widely sus-pected that the pairing mechanism in Sr2RuO4 is at least partially magnetic inorigin [3,4,5,6,7,8,9]. This further motivates investigation of the magnetic prop-erties of ruthenates. Here we discuss the magnetic properties of ruthenates fromthe point of view of density functional (DF) calculations and relate these resultsto the superconductivity of Sr2RuO4. We begin with the n = ∞ end-members,CaRuO3 and SrRuO3.

2 SrRuO3 and CaRuO3

Both 3D perovskite end-members CaRuO3 and SrRuO3 occur in the GdFeO3,Pnma structure. This is a distortion of the cubic perovskite structure consistingof a combination of rotations of the octahedra. The structural difference betweenCaRuO3 and SrRuO3 is an approximately twice larger rotation angle in the Cacompound [10,11]. DF calculations [12,13] show no contribution to the electronicstructure near the Fermi level EF from the A-site, and furthermore results forCaRuO3 forced to assume the crystal structure of SrRuO3 yield magnetic andelectronic properties practically identical to SrRuO3. This underscores the ex-pected electronic inactivity of the A-site cations in these materials. Nonetheless,

C. Noce et al. (Eds.): LNP 603, pp. 256–270, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Magnetism, Spin Fluctuations and Superconductivity 257

the magnetic properties are very different. CaRuO3 is a highly enhanced param-agnetic metal, while SrRuO3 is a robust ferromagnet (m ≈1.6µB/Ru), with thehighest Curie temperature (TC ≈165K) in the series [14,15,16,17]. Several DFcalculations have been reported using various methods [12,13,18,19,20]. Thosewith full potential methods agree in the ground state properties and electronicstructure. In particular, using the experimental crystal structures, SrRuO3 isfound in the local spin density approximation (LSDA) to have a spin moment of1.59 µB/Ru, while CaRuO3 is found to be a paramagnetic metal on the vergeof an itinerant ferromagnetic instability, i.e. at the critical point. The calculatedmoment for SrRuO3 is in excellent agreement with experiment. However, whileCaRuO3 is correctly predicted to be a highly renormalized, it would seem thatthe LSDA puts the critical point too close to this composition. Yoshimura and co-workers [21], place the ferromagnetic – paramagnetic transition in Sr1−xCaxO3near x=0.7 based on NMR investigations of a series of samples. One may specu-late that this reflects the neglect of critical fluctuations, which may be expectedto reduce the tendency to ordering in the vicinity of the critical point. On theother hand, Mukuda and co-workers [22], using Ru NMR measurements, esti-mate that CaRuO3 has a Stoner factor of 0.98, very close to the calculated valueof 1, and assert that the properties are dominated by soft, long wavelength spinfluctuations. Further experimental investigation of the position of the criticalpoint in the phase diagram would be desirable.

In an ionic model, the 4d states of the nominal Ru4+ ions are split by theoctahedral crystal field into t2g and eg manifolds, with the t2g below the eg.In the low spin state, this would result in three majority and one minorityt2g electrons – seemingly a good starting point for describing the properties ofSrRuO3. However, the DF band structure reveals rather an electronic structuredominated by strongly hybridized Ru d – O p states.

Fig. 1 shows the calculated LSDA band structure for SrRuO3 in the idealundistorted cubic perovskite structure. The ground state for this structure isferromagnetic as with the actual structure, though the magnetization is reducedto 1.17µB . Interestingly, simple antiferromagnetic configurations cannot be sta-bilized in the LSDA, implying itinerant as opposed to local moment magnetism.In any case, the band structure shows a single continuous manifold of hybridizedRu 4d – O 2p bands extending from ≈-8 eV to ≈ 6 eV relative to EF withsubstantial exchange splittings through this range. The bands near the bottom(≈ -8 eV – -5 eV have mainly O 2p – eg bonding character, those from ≈ -5 eV– -2 eV, are mainly O pπ like with an add-mixture of t2g character, those from≈ -2 eV to 1 eV are t2g – O p hybrids, and those above are antibonding eg – Opσ in character. However, it should be emphasized that the bands have mixedcharacter. Near EF the bands have approximately 2/3 Ru t2g character, withthe remaining 1/3 being O p in origin. Formally, these are t2g – pπ antibondingbands.

The bonding topology of the perovskite structure, which strongly disfavorsdirect d – d hopping, often results in bands that are non-dispersive along someCartesian directions. This often results in highly nested cubic or sheetlike Fermi

258 D.J. Singh and I.I. Mazin

-10

-8

-6

-4

-2

0

2

4

6

Γ X M R Γ M

E(e

V)

Fig. 1. LSDA band structure of SrRuO3 with the ideal cubic perovskite structure, afterRef. [12]. Majority (minority) bands are shown as solid (dashed) lines. The calculatedspin moment for this structure is 1.17µB / Ru

surfaces as opposed to rounded shapes. In the idealized cubic structure, SrRuO3has flat t2g derived bands at EF , with a Fermi surface that may be described asthe intersection of three equivalent 2-D sections, one from each t2g orbital, i.e.three intersecting circular cylinders along kx, ky and kz. In any case, there is alarge peak almost at EF in the non-spin-polarized density of states (DOS); thisleads to a Stoner instability and is responsible for the ferromagnetic ground state.Furthermore, in the ferromagnetic state, the bands are almost rigidly exchangesplit. This reflects the similar strongly hybridized characters of the bands and thefact that there is substantial Hund’s coupling on both the Ru and O sites. Sincethe DOS near EF contains partial O character, the magnetization, resultingfrom rigidly exchange splitting this DOS, resides partly on O. This amounts toapproximately 1/3 of the total.

The actual Pnma structure of SrRuO3 has a lower symmetry and four for-mula units per cell, which complicates the band structure and broadens the DOSnear EF . Nonetheless, the basic picture is unchanged. Detailed calculations [12]show hybridized d – p bands around EF and a DOS peak of the same origin,though broader. This leads to a Stoner instability and a ferromagnetic groundstate with a moment of 1.59 µB / Ru. The bands around EF are rigidly exchangesplit by 0.65 eV, and since they are roughly 1/3 O p derived, approximately 1/3of the total magnetization is distributed among the O sites. In CaRuO3, thestructural distortion is larger. This results in a further broadening of the DOSpeak near EF . The result is that even though the overall t2g band width issmaller in CaRuO3 than in SrRuO3, the peak at EF is not as high. Thus the

Magnetism, Spin Fluctuations and Superconductivity 259

Stoner criterion is not exceeded and the ground state as predicted by LSDAcalculations is paramagnetic [13].

The ferromagnetic ground state of SrRuO3 can be analyzed using extendedStoner theory. Extended Stoner analysis helps shed some additional light on thegeneral features of the magnetic instabilities in ruthenates. The key parameterin this theory is N(EF )I, where the Stoner I is a normally atomic-like quan-tity giving the local exchange enhancement. Generally, I is determined by thedensity distribution on an ion, and is larger for more compact orbitals, as in 3dions relative to 4d ions. In compounds, I is replaced by a material dependentaverage I. The appropriate averaging for calculating the energetics is with thedecomposed DOS, I = IAn2

A + IBn2B , for two components, A and B, where IA

and IB are the Stoner I for atoms A and B, and nA and nB are the fractionalweights of A and B in N(EF ) (normalized to nA + nB = 1). The O2− ion ishighly polarizable (it does not exist outside crystals) and because of this thevalue of IO may be expected to be material dependent. Nonetheless, O2− is asmall ion and so IO may also be large. Mazin and Singh got IRu=0.7 eV and IO

= 1.6 eV for SrRuO3, yielding I = 0.38 eV including O and I = 0.31 eV withoutthe O contribution. The O contribution to I is generic to perovskite derivedruthenates, as it simply reflects the hybridization of the t2g orbitals of nominallytetravalent octahedrally coordinated Ru with O. This provides a ferromagneticinteraction between Ru ions connected by a common O [13]. The interactioncomes about because for a ferromagnetic arrangement the O polarizes, and thiscontributes to the energy, while for a strictly antiferromagnetic arrangement, Odoes not polarize by symmetry, and so in this case there is no O contribution tothe magnetic energy. This is local physics and so this contribution to the para-magnetic susceptibility, while peaked at the zone center, is smooth in reciprocalspace.

3 Band Structure of Sr2RuO4

The triplet superconductor Sr2RuO4 is perhaps unique among the perovskitederived ruthenates in that it occurs in an ideal undistorted structure, in thiscase the bct P4/mmm layered perovskite structure. As a result, substantiallynested Fermi surfaces may be anticipated, both because of the 2-D nature ofthe compound and because of the bonding topology of the layered perovskitestructure. This in fact is confirmed by calculations [23,24,25,26,27,28,29,30].

The LSDA band structure of Sr2RuO4 is shown in Fig. 2 and the corre-sponding basal plane Fermi surface in Fig. 3. As expected, these are highly 2Din character. The band structure, like that of SrRuO3, shows strong Ru 4d - O2p hybridization, resulting in the same pattern, i.e, bonding eg − pσ states atthe bottom of the valence bands, the corresponding anti-bonding states aboveEF , and hybridized t2g derived bands around EF . There are three bands, cross-ing EF . These correspond to the three 4d t2g orbitals, dxy, dxz and dyz. In theI4/mmm structure, the dxy orbital may have a crystal field splitting from thedxz and dyz orbitals, and may also disperse differently; (the dxz and dyz hy-

260 D.J. Singh and I.I. Mazin

-8

-4

0

4

8

Γ Z X Γ Z

E(e

V)

Fig. 2. LSDA band structure of Sr2RuO4 after [24]. The long Γ-Z direction is in thebct basal plane; the short Γ-Z direction is the c-axis direction. EF is at 0

bridize with one in-plane O and the apical O 2p states, which are further awayand do not have a direct hopping channel to a neighboring Ru, whereas thedxy hybridizes with only in-plane O 2p states). In this simple picture [3,4], thedxy band would have twice the width of the dxz and dyz bands, because it hashopping via both neighboring in-plane O as opposed to only the one along x ory, respectively. This is close to what is found. It may also be noted that becauseof the tetragonal crystal field, the dxy is centered lower than the dxz and dyz

bands, so the band maxima at X are quite close in energy.The Fermi surface, which as mentioned is highly 2D, consists of three sections.

These are a nearly circular cylindrical section centered around Γ (denoted γ)and two nearly square cylindrical sections, α and β centered around Γ and X,respectively. The γ section is the 2D surface that comes from the dxy orbital.The dxz and dyz bands yield 1D sections – flat sheets perpendicular to x and y,respectively. As may be seen, the α and β are quite close to this, allowing forsome second neighbor hopping and minor reconnection at the intersections.

Sr2RuO4 is found to be paramagnetic within the LSDA, but ferromagneticin the generalized gradient approximation [26] perhaps reflecting an overly mag-netic tendency for 4d and 5d transition metals in the GGA [31]. In any case,the Fermi surface has been measured by quantum oscillations [32,33], and angleresolved photoemission [34,35,36]. These measurements confirm the LDA pre-dictions in detail, although they find a mass enhancement of 3–5 and that thec-axis dispersion is smaller than the already very small LDA value, consistentwith transport data [37]. This is a very high mass enhancement for a conven-

Magnetism, Spin Fluctuations and Superconductivity 261

Z Γ

Γ ZFig. 3. LSDA basal plane Fermi surface of Sr2RuO4 after [30]. The solid lines are fromdirect calculations [24], while the dashed lines are a tight binding fit. The corners ofthe plot are Γ and Z points; X is at the center. Note the slight deviation from 4-foldsymmetry in this plot which reflects kz dispersion

tional superconductor, no doubt reflecting the importance of spin-fluctuationsand maybe other many body effects.

4 Spin Fluctuations in Sr2RuO4

Two contributions to χ(q) may be identified based on the above [3,4]. The firstis a weakly ferromagnetic tendency due to the O contribution to the Stoner I,as discussed above for (Sr,Ca)RuO3. This factor is important because of thehigh Stoner renormalization, 1/(1 − IN) = 9, somewhat larger than the valuefrom the experimental susceptibility [1], χ/χband = 7.3. The difference in thedenominator makes χ/χband very sensitive to the precise value of I. An estimateof the q dependence was made using the Stoner model with LSDA calculationsof the partial Ru and O contributions to N(EF ) and the atomic-like IRu and IO

from the calculations [3]. Specifically, a smooth interpolation was made betweenthe full value for Sr2RuO4, i.e. I(q)=0.43 eV at Γ and the 14 % smaller Ru onlyvalue at the zone corner, (π, π). Using only these ferromagnetic fluctuations, amass renormalization due to spin fluctuations, (1+λs) ≈ 3 results. This is large,but still significantly smaller than the experimental values of 3.4 to 5 dependingon the Fermi surface sheet.

262 D.J. Singh and I.I. Mazin

Fig. 4. Bare band structure contribution to χ in Sr2RuO4 after [4]

The second contribution is due to nesting. As mentioned, the α and β sheetsare close to one-dimensional, and therefore are strongly nested. Calculation ofthe band structure contribution to χ involves matrix elements and I(q).

In the absence of a proper linear response calculation, an estimate of thiscontribution to χ may be made by setting the matrix element to its full valuebetween bands of the same character and zero otherwise. With the Stoner renor-malization discussed above, one obtains

χ(q) =χ0(q)

1 − I(q)χ0(q)≈ χ0(q)

1 − I(q)N(0) − I(q)χn(q), (1)

where χn is the nesting dependent contribution [4]. The bare susceptibility χ0(q)is shown in Fig. 4. It shows strong ridges corresponding to the nesting of thedxz and dyz Fermi surfaces. The peaks at the intersection of these ridges, occur-ring at q ≈ (2π/3a, 2π/3a), are particularly pronounced. These nesting relatedfeatures in χ(q) have been clearly seen in spin-polarized neutron experimentswith a temperature dependence consistent with an itinerant band structure ori-gin [38,39,40]. The clear observation of the nesting related structure in χ(q) andthe quantum oscillations clearly show the Fermi liquid character of Sr2RuO4.At present, experimental evidence for the ferromagnetic spin fluctuations is lessclear. 17O and Ru NMR measurements [22,40,41] do show exchange enhancedspin fluctuations, which may suggest nearness to ferromagnetism. However, themeasurements can be interpreted in terms of a nearly q-independent suscepti-bility as well. In any case, the predicted smooth ferromagnetic background inχ(q) was found to be strongly pairing in the triplet p-wave channel, while theband structure part (nesting) is pairing in the singlet dx2−y2 channel (and more

Magnetism, Spin Fluctuations and Superconductivity 263

weakly so in the p-wave channel) [4]. So in our simple spin-fluctuation pairingscenario, triplet superconductivity is associated mainly with the spin fluctuationscorresponding to the ferromagnetic background.

5 Magnetism in (Sr,Ca)2RuO4 and Sr3Ru2O7

Braden and co-workers [42] investigated the phonon dispersions of Sr2RuO4using neutron scattering. They found that the Σ3 phonon branch exhibits asharp drop near the zone boundary at (π, π, 0). This branch corresponds to arotation of the RuO6 octahedra around the c-axis. The steepness of this dropnear the zone boundary is indicative of the rigidity of the RuO6 octahedra,while the zone boundary frequency indicates the proximity to an instability.The branch has practically no kz dispersion, indicating that there is very littlecoupling between rotations in different layers.

In perovskites, the stability of the zone boundary rotational modes is as-sociated with the A-site – O interaction; substitutions of smaller A-site ionsdestabilize these modes. This happens in Ca2−xSrxRuO4 upon Ca substitu-tion [43,44,45,46]. The ideal bct I4/mmm structure is retained from x = 2 tox = 1.5, at which point the Σ3 phonon finally becomes unstable. Condensationof this mode leads to a doubling of the unit cell into spacegroup I41/acd in theregion from x = 1.5 to x = 0.5. The octahedral rotation reaches 12.8 at x = 0.5.

Ca2−xSrxRuO4 is a paramagnetic metal, with no magnetic ordering at lowtemperature in this range, x = 1.5 to x = 0.5. Nakatsuju and Maeno [46] mea-sured the susceptibility χ(0) at 2K for in this composition range finding thatχ(0) increases strongly as x is lowered from x = 1.5 to x = 0.5 reaching a valuemore than 100 times χ(0) for Sr2RuO4, with a high Wilson ratio and a shapethat suggests a critical point at x = 0.5. However, at x = 0.5 a structural transi-tion occurs to a so-called “T” phase. The crystal structure of this phase has notbeen fully refined. However, it is known that it has both octahedral rotationsabout the c-axis and octahedral tilting.

The onset of the tilts at x = 0.5 coincides with a drop in the susceptibilityand the onset of low temperature magnetism as evidenced by a temperature de-pendent susceptibility peak. However, a long range ordered magnetic structure,corresponding to this peak, has not yet been found by neutron diffraction. Re-lated to this apparent lack of long range order, it should be noted that the peakin χ occurs at slightly different temperatures depending on whether the in-planeor out-plane component is measured [46]. This is not the expected behavior fora true magnetic ordering transition and suggests instead that dynamical fluctu-ations may be important in the phase below the susceptibility peak.

For Sr concentrations x ≤ 0.2 the tilts are ordered into space group Pbca,with long range ordered antiferromagnetism at low temperature. Two distinctstructural phases, “S” and “L” occur between x = 0 and x = 0.2. These differin the c/a ratio reflecting the geometry of the RuO6 octahedra. The octahedrain the “L” phase are like those in the “T” and I41/acd regions, while in the “S”phase the apical oxygen is ≈ 0.07 A closer to the Ru ion and the in plane O’s are

264 D.J. Singh and I.I. Mazin

≈ 0.05 A further. The “S” phase is an antiferromagnetic insulator, TN ∼ 110K,and is characterized as a Mott insulator, since it is insulating both above andbelow the magnetic ordering temperature. For x > 0 there is a sharp temperaturedependent metal-insulator transition between the “S” and “L” phases with aseveral orders of magnitude resistivity change [46].

To understand this phase diagram one needs to take into account several mag-netic interactions present in the system: first, the Hund’s rule coupling on oxygen,which is ferromagnetic and relatively weakly q-dependent. Second, the itiner-ant antiferromagnetism due to the nesting; this interaction is sharply peakedat q= (2π/3a, 2π/3a). Third, there is (as always) a superexchange interactionbetween the Ru d orbitals, which is weakly q-dependent with a maximum atq= (2π/a, 2π/a). All three are probably important, but their role is different inthe different regions of the phase diagram of Ca2−xSrxRuO4.

Since the LSDA describes the pure compound x = 2 reasonably well, onemay believe that the effect of the lattice distortions in the Sr-rich part of thediagram can be described in the LSDA as well. Both tilting and rotation ofthe octahedra narrow the bands, which is favorable both for ferromagnetismand nesting related antiferromagnetism. Narrower bands are also more localizedand thus more prone to correlation effects; this does not, however, imply strongsuperexchange. Recently, Fang and Terakura performed a very detailed LDAstudy of SrxCa2−xRuO4 in distorted crystal structures [47]. As expected, theyfound competing ferromagnetic and antiferromagnetic tendencies with substan-tial coupling to the lattice. In particular, they found that rotation of the RuO6octahedra favors ferromagnetism, while antiferromagnetism is stabilized by oc-tahedral tilts. They trace this down to the different effects of rotations and tiltson the dxy band, which tends to favor ferromagnetism, and the nested dxz anddyz bands, which favor antiferromagnetism. For example, rotation narrows thedxy in the first order, but the dxz and dyz only in the second order, and indeedFang and Terakura observed this effect numerically.

We find particularly interesting the fact that Fang and Terakura’s calcula-tions yield a ferromagnetic ground state already for an octahedral rotation angleclose to that at the presumed critical point at x=0.5, with an energy 25 meVbelow the non-spin-polarized state and a sizeable magnetization of 0.74 µB/Ru(the GGA has an even greater tendency to ferromagnetism, giving, for exam-ple, a magnetic ground state for Sr2RuO4 itself [26]). This poses something ofa conundrum as DF is a ground state theory, and provided that nothing goesseriously wrong in the particular approximations used, it should yield the propermagnetic ground state. The LSDA often does not properly predict the magneticground states of Mott insulators and related strongly correlated systems (e.g.cuprates) reflecting the inadequate treatment of on-site Coulomb correlations.However, incorrect LSDA predictions of magnetic ground states for materialsthat are not magnetic are very rare, although not unknown. It appears that thisresult of Fang and Terakura is not even too sensitive to the fine details of thecrystal structure: we confirmed it by LSDA calculations at x=0.5 in the full lowtemperature crystal structure refined by Friedt and co-workers [44] (our calcula-

Magnetism, Spin Fluctuations and Superconductivity 265

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

E (

meV

/Ru)

m ( /Ru)µB

Fig. 5. Fixed spin moment energy of Ca1.5Sr0.5RuO4

tions were for Ca2RuO4 in the x=0.5 crystal structure). For these, we used thelinearized augmented planewave (LAPW) method [48] with local orbitals andwell converged basis sets.

The calculated fixed spin moment energy vs. magnetization curve is shownin Fig. 5. We see that, in the full crystal structure, the ferromagnetic stateis still the ground state in the LSDA, consistent with [47]. We obtain for theferromagnetic state an energy 40 meV/Ru below the non-spin-polarized state,with a magnetization of 1.0 µB/Ru. Of this 0.64 µB/Ru is contained in the RuLAPW spheres (radius 1.99 a0) with the remainder distributed over the inter-stitial and the in-plane O sites. Interestingly, while approximately 1/3 of thetotal magnetization is on the O sites, as in other magnetic ruthenates, in thiscase all of it is on the in-plane O, with negligable polarization of the apical O.We also find a metastable in-plane c(2×2) antiferromagnetic solution with anenergy 18 meV/Ru below non-spin-polarized state, but with a low moment of0.3 µB within each Ru LAPW sphere, indicating that the magnetism has morelocalized character that in SrRuO3. There is some similarity with NaCo2O4,which is one of the very few other layered oxides that is paramagnetic experi-mentally, but predicted to be an itinerant ferromagnet in the LSDA [49]. Onepossible explanation is that strong spin-fluctuations associated with the criti-cal point (see [50]) suppress the magnetic ordering, shifting the critical point inthese magnetically soft itinerant systems. Such quantum fluctuation effects arenot included in the LSDA or GGA, as these approximations are parameterizedby the uniform electron gas, which is very far from any magnetic critical pointfor the densities relevant to solids. Although such effects are not accounted forin the LDA, they are totally different from on-site Coulomb (Hubbard) correla-

266 D.J. Singh and I.I. Mazin

tions. An attempt to explain the increasing tendency to magnetism from x = 2to x = 0.5 by including Hubbard correlation [51] implies orbital ordering, notobserved experimentally, and it appears to be rather difficult to obtain the effec-tive magnetic moment of 1 µB , deduced from magnetic susceptibility [45], whileLDA calculations explain this rather naturally.

On the other side of the critical point, x < 0.5, the octahedra are not onlyrotated, but tilted. This leads to narrowing of the zx and yz bands [47], andthus increases nesting and favors antiferromagnetism. Magnetic fluctuations inthis region acquire a substantial antiferromagnetic component, as observed ex-perimentally [45]. Again, there is no reason to believe that Hubbard correlationsplay a decisive role in this region, 0.5 > x > 0.2. It is worth noting that in theoriginal paper [45], Nakatsui and Maeno mentioned the possibility that the maineffect of the rotation and tilting is a change in crystal field splitting between thedxy and the dxz and dyz states. Numerical calculations [47] rather suggest thatthe main effect is modification of hopping parameters and thus bandwidths.

The Ca-rich phase, x < 0.2, seems to be described much better by theLDA+U approximation, which includes Coulomb correlations in a static way [51],than by the LSDA [47]. Although GGA renders Ca2RuO4 a band insulator [47],this does not seem to be the correct physical picture, as there is now accumulat-ing experimental evidence of a Mott-Hubbard state. Anisimov et al. successfullydescribed Ca2RuO4 in the LDA+U approximation, but they did not address thechange in the shape of the RuO6 octahedra (which remain rigid for the wholerange 2 > x > 0.2), and of the strong first-order character of the metal-insulatortransition. A natural explanation of this effect belongs to Khomskii [52], whoobserved that inside the insulating phase a substantial energy can gained bymaking the pd bands narrower by means of increasing the in-plane Ru-O bondlength, while in the metallic phase the kinetic energy favors wider bands andshorter Ru-O in-plane bonds. Note that while this interpretation formally sug-gests full occupation for the xy band and half-occupancy for the zx and yzbands, in agreement with both [51] and [45], the driving force in the formeris narrowing of the bands (sudden in the Khomskii picture and gradual in theAnisimov et al. picture), while in the latter it is band shifting with respect toeach other.

The bilayer Sr compound, Sr3Ru2O7 is arguably the most similar compoundto Sr2RuO4. Besides the obvious structural relationship, it is highly two di-mensional, metallic and paramagnetic [53] with an extremely strongly enhancedsusceptibility and high Wilson ratio (i.e. on the verge of ferromagnetism) [54,55]like Ca1.5Sr0.5RuO4. Shaked and co-workers [56] have refined the crystal struc-ture into spacegroup Bbcb with a rotation of the RuO6 octahedra of approxi-mately 7 with respect to the ideal bct structure. DF calculations of the electronicband structure have been reported both for the ideal bct crystal structure [57]and for the actual structure [58]. Qualitatively, the electronic structure is verymuch what might be expected based on results for Sr2RuO4. Each RuO2 layercontributes three t2g bands, which cross EF , yielding six in total. These (espe-cially the dxz and dyz combinations) are split into symmetric and anti-symmetric

Magnetism, Spin Fluctuations and Superconductivity 267

combinations by the inter-layer interaction. Significantly, however, there is con-siderably less nesting than in Sr2RuO4, especially when the actual Bbcb crystalstructure is used.

This reduced nesting suggests that the incommensurate antiferromagneticspin-fluctuations seen in neutron scattering experiments [38] on Sr2RuO4 maybe less prominent in Sr3Ru2O7. On the other hand, tetragonal Sr3Ru2O7 ismuch closer to a ferromagnetic instability. This is not surprising considering thehigh value of the calculated density of states (DOS) at the Fermi energy (EF ),N(EF ) = 4.5 states/eV Ru, compared with 4.1 states/eV Ru for Sr2RuO4 [24].According to LSDA calculations, tetragonal Sr3Ru2O7 is close to a Stoner insta-bility, but is paramagnetic. However, orthorhombic Sr3Ru2O7 has a still higherN(EF ) = 5.0 eV−1, leading to a magnetic ground state of itinerant characterwithin the LSDA [58]. The ferromagnetic (FM) solution was found to have aspin magnetization of 0.80 µB/Ru and an energy of −23 meV/Ru relative to thenon-spin-polarized case. Calculations were also done for antiferromagnetic con-figurations with a c(2×2) in-plane ordering, and having adjacent Ru ions in thetwo planes of the bilayer polarized parallel. However, no self-consistent magneticconfiguration was found. Thus it may be concluded that the LSDA magneticcharacter within each plane is itinerant. However, calculations in which the Ruions in a layer were ferromagnetically aligned, but the layers were stacked an-tiferromagnetically (so each bi-layer had one spin up and one spin down RuO2layer) did yield a stable magnetic solution (denoted AF-A), with an energy of−20 meV/Ru and a Ru moment (as measured by the moment in a Ru sphere)only 14% smaller than the ferromagnetic solution. In an earlier work [58], wespeculated that these results could be reconciled with the experimental obser-vation of a susceptibility peak without long range ordering via the Kosterlitz-Thouless mechanism [59]. In this scenario, the highly 2D magnetic interactionsand the in-plane magnetic anisotropy of layered ruthenates, would result in alogarithmic suppression of TC , with a separation between the long range order-ing temperature and the susceptibility peak [59]. This mechanism, by itself,cannot, however, lower the ordering temperature to 0. On the other hand, ithas now been shown, in extremely clean samples, that Sr3Ru2O7, although nearferromagnetism, remains a paramagnetic Fermi liquid down to extremely lowT [50]. So it should be concluded that our speculation was incorrect, and thatthe suppression of ferromagnetic ordering in Sr3Ru2O7 is due to some othercause, perhaps again fluctuations associated with the quantum critical point, orperhaps quantum critical fluctuations combined with some softening associatedwith the 2D magnetic character.

6 Conclusions

Detailed study of perovskite derived ruthenates has revealed an unexpectedlyrich physics including itinerant 4d magnetism, triplet superconductivity, andquantum critical phenomena. Although much can be learned about these ma-terials from DF calculations, there is a lot of interesting physics beyond this

268 D.J. Singh and I.I. Mazin

level. For example, there is strong evidence for important k-dependent electroncorrelation effects (e.g. strong scattering by various sorts of spin-fluctuations),and, at least in Ca2RuO4, Hubbard-type correlations.

Acknowledgments

We are grateful for helpful discussions with G. Cao, J.E. Crow, Z. Fang, R.P. Gue-rtin, S. Ikeda, S.R. Julian, D. Khomskii, A.P. Mackenzie, Y. Maeno, A.J. Millis,S.E. Nagler, S. Nakatsuji, D.A. Papaconstantopoulos, W.E. Pickett and K. Ter-akura. We especially thank A.J. Schofield for an illuminating discussion on theferromagnetic critical point. Work at the Naval Research Laboratory is supportedby the office of Naval Research.

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Metamagnetic Quantum Criticality in Sr3Ru2O7

A.J. Schofield1, A.J. Millis2, S.A. Grigera3, and G.G. Lonzarich4

1 School of Physics and Astronomy, University of Birmingham,Edgbaston, Birmingham, B15 2AD, United Kingdom

2 Department of Physics, Columbia University,538 W 120th St, New York, NY 10027 USA

3 School of Physics and Astronomy, University of St. Andrews,North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom

4 Cavendish Laboratory,Madingley Road, Cambridge, CB3 0HE, United Kingdom

Abstract. We consider the metamagnetic transition in the bilayer ruthenate, Sr3Ru2O7,and use this to motivate a renormalization group treatment of a zero-temperaturequantum-critical end-point. We summarize the results of mean field theory and give apedagogical derivation of the renormalization-group equations. These are then solvedto yield numerical results for the susceptibility, the specific heat and the resistivity ex-ponent which can be compared with measured data on Sr3Ru2O7 to provide a powerfultest for the standard framework of metallic quantum criticality. The observed approachto the critical point is well-described by our theory explaining a number of unusualfeatures of experimental data. The puzzling behaviour very near to the critical pointitself, though, is not accounted for by this, or any other theory with a Fermi surface.

1 Introduction

The presently interesting ruthenium-oxide metals provide a useful test-bed forbasic theoretical ideas in the strongly correlated electron problem. The single-layer ruthenate Sr2RuO4, being isostructural and isoelectronic with the parentcompound of a cuprate superconductor, La2CuO4, was original studied as anexample of a conventional quasi-2D Fermi-liquid metal [1] against which to com-pare the more exotic cuprates. The discovery [2] of what now appears to betriplet superconductivity [3] shows that this compound is fascinating in its ownright as well as providing a new perspective on cuprate physics. In this paperwe discuss the bilayer version of this compound, Sr3Ru2O7, and argue that themetamagnetism in this material [4,5] provides an important test for the conven-tional framework [6,7] of quantum-critical points in metals.

A metamagnetic transition is empirically defined as a rapid increase in mag-netization at a particular value of applied magnetic field. Because there is nobroken symmetry involved, one expects a first-order transition from a low-magnetization to a high-magnetization state as an applied magnetic field, H,is swept through a (temperature-dependent) critical value, Hmm(T ). Near a firstorder transition, kinetics may be complicated but thermodynamic fluctuationcorrections to observables should not be divergent or particularly large. How-ever in general, the curve of first-order transitions, Hmm(T ), terminates in a

C. Noce et al. (Eds.): LNP 603, pp. 271–289, 2002.c© Springer-Verlag Berlin Heidelberg 2002

272 A.J. Schofield et al.

Fig. 1. (a) Schematic phase diagram in the H, T plane showing a line of first-ordermetamagnetic transitions terminating in a critical end-point. (b) This critical end-pointcan by suppressed to lower temperatures using a suitable tuning parameter (such aspressure). At a critical pressure, p0, we have pushed the end-point to T = 0 giving thequantum critical end-point discussed in this paper

second-order critical point (H∗, T ∗) which is characterized by divergent fluctua-tions [see Fig. 1(a)]. Furthermore, by appropriately tuning material parameters(by applying pressure for example) it should be possible to reduce T ∗ to zerotemperature, yielding a quantum-critical end-point. This situation is illustratedin Fig. 1 (b) which shows a possible variation of the temperature of the criticalend-point with pressure. At a critical end-point there are important fluctua-tion effects and near a quantum critical end-point the quantal nature of thesefluctuations also needs to be taken into account.

The bilayer ruthenate is not unique in showing metamagnetism. A number of‘strongly correlated metals’, including UPt3 [8], CeRu2Si2 [9,10,11], CeRh2Si2 [12]and other heavy fermion compounds [13], as well as d-electron systems such asMnSi [14] exhibit metamagnetic transitions with properties suggestive of prox-imity to a quantum critical point. What makes Sr3Ru2O7 [4] stand out is thatat ambient pressure and moderate applied field it seems that this material istuned almost exactly to such a quantum-critical end-point [5]. By a direct com-parison of our theory with experiments we show that this system is indeed closeto a quantum-critical end-point. In addition, we are able to explain a number ofheretofore puzzling features observed in this system: the finite temperature peakin the weak-field susceptibility [15] and the paramagnetic ground state in a metalthat should, according to band-structure calculations, be ferromagnetic [16].

Quantum critical transitions in itinerant electron systems have themselvesalso attracted widespread interest. Generally they are reached by tuning theordering temperature of a second order phase transition to absolute zero using acontrol parameter, such as pressure [17] or chemical composition [18]. The criticalfluctuations of the slow mode associated with the ordering often influences a wideregion of the temperature/tuning parameter phase diagram. The effect of thesefluctuations is the appearance of non-Fermi liquid temperature dependences inquantities such as the resistivity, specific heat and the Pauli susceptibility [7].Indeed it has been suggested that the proximity to a quantum critical point maybe the root cause of much of the non-Fermi liquid seen in nature even when

Metamagnetic Quantum Criticality in Sr3Ru2O7 273

there is no obvious symmetry breaking phase transition in the phase diagram.One possibility is that the order is present but hidden [19]. Alternatively, thequantum critical point is argued to be close to the physical system in the sensethat one could tune to it but only through the application of some unphysicaltuning parameter (such as negative pressure), though the physical system isaffected by the “almost critical” modes of this transition [20]. In this paper weinvestigate a third possibility: a quantum critical end-point.

An appealing feature of quantum criticality from a theoretical perspectiveis that an established and tractable theory has been developed [6,7]. Howevera number of the experimental realizations of quantum-critical systems show de-viations from the predicted behaviour [21] which have caused some to ques-tion its validity [22]. Factors omitted in the original theory include other, non-critical, slow modes [23], the possible influence of disorder [24] and the Kondoeffect [22]. We shall show that these factors should not be present at a metam-agnetic quantum-critical end-point and so the theory we present will provide aclean test of the standard [6,7] framework when compared with experiments.

There have been a number of mean-field treatments of metamagnetism inmetals recently [25,26] and some discussion was given in the context of a treat-ment of weak ferromagnets via the ‘SCR’ method [27,28]. Our work gives, webelieve, the first analysis of the critical phenomena at a metamagnetic point. Inthis paper we give a pedagogical account of our renormalization group theory ofmetamagnetic quantum criticality—specializing here to the case of a two dimen-sional system as is appropriate for the layered ruthenates. A brief account of ourresults for Sr3Ru2O7 and the more general case has been recently published [29].

2 Deriving the Action

The starting point for the treatment is the standard functional integral ap-proach [30] to the interacting-electron problem where we have assumed that theimportant interaction is a spin-density interaction. This is then decoupled byintroducing a Hubbard-Stratonovich field

Z =∫

D(ψ, ψ)e− ∫β0 dτ

∑q[ψq(τ)(∂τ+εq−µ)ψq(τ)−J(q)Sq(τ)·S−q(τ)] , (1)

=∫

D(ψ, ψ,φ) exp

[−

∫β

0dτ

∑q,α

14J(q)λ2 φq,αφ−q,α

+∫dDrψα(r, τ) δα,β (∂τ + ε(−i∇) − µ) + iλφ(r) · σαβψβ(r, τ)

].

(2)

The key assumption lying behind the standard approach developed by Hertz [6]and refined by Millis [7] is that the electronic degrees of freedom may be inte-grated out leaving a model of an over-damped bosonic modes which may then beanalysed by renormalization-group methods. It is this procedure that we follow

274 A.J. Schofield et al.

here. Doing the integration over the fermion Grassmann fields, ψ, yields

Z = Z0

∫Dφe

[− ∫

β0 dτ

∑q,α

14J(q)λ2 φq,αφ−q,α+

∫d2x ln det∂τ+ε(−i∇)−µ+iλφ·σ

].

(3)This “ln det” term is then expanded in powers of the field to give the effectiveaction. The leading order term contains the important dynamics

SG[φ] =12

∑ωn,q

[1

4J(q)λ2 +λ2

2Π(q, ω)

]φ−ωn,−q · φωn,q . (4)

Here Π(q, ωn) is the Lindhard function, which comes from the fermion bubbleobtained by expanding the determinant. Redefining the fields (using λ) to makethem dimensionless and expanding the Lindhard function we obtain the Gaussianaction

SG[φ] =12

∑ωn,q

E

[r + ξ2q2 +

|ωn|vq

]φ−ωn,−q · φωn,q . (5)

Here ξ ∼ 1/kF , v ∼ vF , E is an electronic energy scale and r = 1 − J(0)/E is ameasure of the deviation of the system from the Stoner instability [31]: when thereduction in interaction energy from mean-field ordering more than compensatesfor the consequential increase in kinetic energy. We have used a conventional gra-dient expansion for the static part of the action and assumed that the coefficientsare simple numbers and that the parameters vary weakly with temperature (asT 2, as usual in Fermi-liquid theory). For an O(3) ferromagnetic quantum crit-ical point this may not be the case [32,33], but in the presence of a symmetrybreaking field, the T 2 and gradient expansions are believed [23] to apply.

The frequency dependence above is a consequence of the conservation-lawsof the order parameter, φ. Since φ is essentially the difference in position of the“spin-up” and “spin-down” Fermi surfaces, then fluctuations at nonzero q corre-spond to locally increasing the number of “spin-up” electrons and decreasing thenumber of “spin-down” electrons. If “spin” is conserved such a fluctuation canrelax only via propagation or diffusion of electrons within each spin manifold.This must vanish as q → 0. Quotes are placed about “spin up” and “spin down”because in many metamagnetic materials spin-orbit coupling is large and spinis not a good quantum number. However, for most purposes one may adopt a‘pseudo-spin’ notation [34] labelling the two Kramers-degenerate states in zero-field. The Kramer’s degeneracy is broken by an applied field, leading to two Fermisurfaces and the theory carries through as in the non-spin-orbit case: since ina clean spin-orbit-coupled system, pseudo-spin is conserved (at least for fieldsaligned along a crystal symmetry axis). There is one exception to this—disorderscattering in the presence of strong spin-orbit coupling can allow pseudo-spinrelaxation even as q → 0. Then the vq term would be replaced by a momentumindependent constant leading to a dynamical exponent of z = 2 (see later).

Further expansion in the fields φ adds interaction terms to the action givingthe final form of the action

S[φ] = SG +∫d2xdτ

14uabφ

2aφ

2b +

16vabcφ

2aφ

2bφ

2c − gµBH · φ(x, τ) + · · · . (6)

Metamagnetic Quantum Criticality in Sr3Ru2O7 275

In writing the action above we have in mind the case of Sr3Ru2O7 so arespecializing to the case of two dimensions. The subscripts on the fields are com-ponents of the vectors and are summed over. Experimentally it is known thatat zero temperature the weak-field susceptibility is isotropic which implies nodirectional dependence of r or g. It becomes anisotropic at finite magnetic fieldand/or temperature where, as we will see, higher order terms in the expansionbecome important—hence in general u and v are anisotropic tensors. Obtainingthe anisotropy theoretically requires inclusion of spin-orbit effects not explicitlywritten in (3) above. We choose units such that the field φ is measured relative tothe saturation magnetization: 2µB/Ru. (The important electrons are in d shellswith four in t2g which leaves two holes and the g factor should be close to 2.)

3 Mean-Field Theory

To develop an understanding of this model we consider it first in mean-field the-ory ignoring the effect of fluctuations, both quantal and spatial, by suppressingthe τ , ω and x, q dependence and treating the action as a free energy:

F [φ] =12rφ2 +

14uφ4 +

16vφ6 + gµBHφ . (7)

We have suppressed the vector nature of the field φ for simplicity here. Toobtain a metamagnetic transition we require that r > 0 so that the groundstate is paramagnetic, but that the fourth order term u is negative (v > 0controls the expansion). The equilibrium magnetization, φ, is then found byminimizing the action. The solutions are when the tangent to the free energy,∂F/∂φ, is equal to gµBH. Such solutions are shown graphically in Fig. 2(a)–(c).The form of the phase diagram is shown in Fig.2 where it can be seen that theratio rv/u2 determines the transitions. For 9/20 < rv/u2 < 3/16 we have afirst order metamagnetic transition signalled by a jump in the magnetization atfinite magnetic field [see Fig. 2(e)]. The critical end-point of the metamagnetictransitions occurs when F [φ] first develops an inflection point: at rv/u2 = 9/20.The magnetization then has infinite slope at a critical field, H∗. At the criticalend-point and for fields in the c-direction (restoring the tensorial nature of theterms)

m∗c =

√−3ucc10vccc

, geffµBH∗c =

√−3ucc10vccc

6u2cc

25vccc. (8)

Comparison with experimental data on Sr3Ru2O7 allows us to fix parame-ters for the theory. Fig. 1 of [4] shows that at low T and low applied field thesusceptibility is about 0.025µB/T implying r ≈ 160µB − T ≈ 100K. This smallvalue of r implies a very large enhancement of the susceptibility, χ ≈ 1/r, overthe band value—the material is near a paramagnetic-ferromagnetic transition.For fields directed along the c axis the observed metamagnetic transition occursat a magnetization of about 0.25 − 0.3µB/Ru implying ucc = 3000 − 4300K andvccc = 40, 000 − 80, 000K with the larger values corresponding to the smaller m.The consistency of these estimates may be verified by substitution into (8); use

276 A.J. Schofield et al.

Fig. 2. Mean field treatment of the magnetization from (7). The phase diagram de-pends on rv/u2 and the equilibrium magnetization is found when the tangent to thefree energy curves equals the magnetic field (here measured in units of gµB). This isillustrated in (a) to (c). The general phase diagram is shown in (d) where the linesrepresent the metamagnetic transition for 9/20 < rv/u2 < 3/16. For rv/u2 > 9/20 wehave crossover behaviour and no transition. For rv/u2 < 3/16 the system is unstableto a first order ferromagnetism even in the absence of an applied field. The criticalend-point of the metamagnetism occurs at rv/u2 = 9/20 where the magnetization hasinfinite slope at the critical field as show in (e)

of geff = 2 yields an estimate of 5 − 6T for the metamagnetic field, in the rangefound experimentally. The scale E is of order 6000K-8000K. The dimensionlesscritical field gµBH

∗/u ∼ 0.001. If we are concerned with the behaviour in thevicinity of the critical end-point then we should look at small deviations withrespect to this already small value. At present rather less information aboutthe spin fluctuation frequencies is available [35]. Therefore when comparing theresults of our renormalization group analysis with experiment, we will normal-ize our results to the temperature T0 at which the differential susceptibility atthe critical field is equal to the zero-field zero temperature susceptibility, i.e∂m∂h (δ = 0, T = T0) = χ(H = 0, T = 0) = χ0.

Metamagnetic Quantum Criticality in Sr3Ru2O7 277

4 Tree-Level Scaling

Having considered the model at mean field, we now proceed to treat the fullquantum action of (6) via the renormalization group. The method we are usingis that of [7] and for completeness we outline the steps involved. The first stepis to perform “tree-level scaling” of the parameters in the action, which followsfrom a dimensional analysis. This only involves power-counting so we write theaction to emphasize this

S = H·φ+∫ Λ

d2kdω

[r + ξ2k2 +

|ωn|vk

]φ2+u

∫ Λ

(d2kdω)3φ4+v∫ Λ

(d2kdω)5φ6+· · · .(9)

Now the idea is to let the momentum bandwidth be rescaled: Λ(λ) = Λe−λ.Then all the coupling constants and fields will be rescaled to preserve the samephysics and so become functions of λ. Writing b = eλ we have Λ → Λ/b. Torestore the momentum cutoff we must have k = bk. Now this forces the gradientterm (proportional to k2) to have an extra factor of 1/b2 so to maintain thesame form of the Gaussian part of the action we need to be able to pull out ab−2 factor from every term. This dictates the scaling of r and ω. So, r = r/b2

and hence we have

r(λ) = re2λ ⇔ ∂r

∂λ= 2r . (10)

Also ω/k → ω/(b2k) so this means that ω = ω/b3 → [ω] = 3. This is the dynam-ical exponent and is often denoted z. We then absorb all of the b factors into arescaling of the field φ. There are two sources, the Jacobians from the change ofvariables which will generate b−5, and the 1/b2 factor from the coefficient of theGaussian term. Thus we have an overall factor of b−7. To absorb this, the fieldφ must scale as φ = b−7/2φ. This, in turn, dictates the scaling of the magneticfield: h = b+7/2h. It also then fixes the form of the u and v part of the action.In the φ4 term we have 3 powers of phase space integrals and 4 powers of thefield giving b−1 which must be absorbed by a renormalization of u : u = b−1u.In the φ6 term we have 5 powers of phase space integrals and 6 powers of thefield giving b−4 which must be absorbed by a renormalization of v : v = b−4v.Thus writing, λ = ln b, we have the following scaling equations at tree level:

∂r

∂λ= 2r , (11)

∂h

∂λ=

72h , (12)

∂u

∂λ= −u , (13)

∂v

∂λ= −4v . (14)

The remaining scaling equation is for the temperature. This is not so obviousperhaps from the above analysis. However, if we consider any physical quantity

278 A.J. Schofield et al.

calculated in the theory then the scaling of temperature becomes apparent. Wewill use the free energy computed from the Gaussian part of the action. Thefree energy is found from the partition function, βF = − lnZ. Now working toGaussian order only we find

Z = Z0

∫Dφ exp [−SG(φ)]

= Z0

∏n,k

1r + ξ2k2 + |ωn|/Γk , (15)

So F = F0 +V

β

∫d2k

(2π)2∑n

ln[r + ξ2k2 + |ωn|/Γk

]. (16)

where Γk = vk is the Landau damping rate.Handling Matsubara sums involving |ωn| will recur in this paper so we give

some of the details here of how these are treated. The Matsubara sum is overωn = 2πn/β and is formally divergent. We can regularize it by considering∮

dz2πinB(z)f(z), where f(z) is the function we wish to sum over Matsubara

frequencies (here the logarithm) and nB is the Bose distribution. The contour ofintegration is taken as a large radius circle centred on the origin. This contourintegral is divergent just as the sum is, however its divergence is temperatureindependent as the contour is taken to infinity and so can be subtracted. Thuswe essentially do the Matsubara sum using the usual contour integral techniqueas if it did converge and the contour integral was zero.

Now nB(z) has poles of residue 1/β at the Matsubara frequencies so the sumover its residues will give the required Matsubara sum. The analytic structureof the logarithm requires knowing, in term, the analytic structure of |ωn|. Thiscomes from its origin in the Lindhard function where we find that

z

πΓln

(z + Γ

z − Γ

)→ |ω|

Γfor z → iω . (17)

Thus f(z) has a branch cut along the real axis between −Γ and Γ . Finally theMatsubara sum plus the integral around the branch-cut must combine to give thevalue of the large radius contour integral (which we take to be zero as detailedabove). This allows us to evaluate the sum. (There is a pole on the branch-cutbut this will be taken care of by the principle part of the line integral.) So wehave, for each mode k (and using η = r + ξ2k2),

F =1β

∑n

ln (η + |ωn|/Γ ) , (18)

= −∫ Γ

−Γ

dω

2πinB(ω) ln(η − iω/Γ ) +

∫ Γ

−Γ

dω

2πinB(ω) ln(η + iω/Γ ) , (19)

= −∫ Γ

−Γ

dω

2πnB(ω)2 tan−1

(ω

ηΓ

). (20)

Metamagnetic Quantum Criticality in Sr3Ru2O7 279

This can be viewed as the definition of the free energy of an overdamped simpleharmonic oscillator—so confirming our regularization procedure. Now note thatwe can write the Bose factor in terms of an odd and an even function

nB(ω) =12

(coth

[ω

2kBT

]− 1

). (21)

Since the other term in the free energy is odd then we only need the odd partof Bose factor and we can halve the integration range, thereby giving for eachmode

F = −∫ Γ

0

dω

πcoth

(ω

2kBT

)tan−1

(ω

ηΓ

). (22)

Thus our full expression for the free energy (per unit volume) is

F = −∫ Λ d2k

(2π)2

∫ Γk

0

dω

πcoth

(ω

2kBT

)tan−1

(ω/Γk

r + ξ2k2

). (23)

Now we imagine reducing the momentum cutoff as before: Λ → Λ/b whereb ∼ 1 + ε, and then perform the rescaling to restore, this time, the form of theequation for the free energy to the original one ( 23). Just as before, we musthave k = kb. Then in order for the argument of tan−1 to be preserved, we musthave r = b2r and ω = b3ω. This now forces the scaling of T from the argumentof the coth such that T = b3T . So we have the RG equation for T

∂T

∂λ= 3T . (24)

The rescaling of ω implies that to restore the form of the equation for the freeenergy, we need to change the frequency cutoff to Γk. So as well as removing highmomentum modes the rescaling means that we are losing some high frequencymodes at all momenta. The effect of rescaling on the cutoffs is illustrated inFig. 3, where the shaded region shows the modes that are being removed.

5 One-Loop Corrections

We now consider the one-loop corrections to the scaling equations to lowestorder. The modes above the cutoff (as shaded in Fig. 3) are denoted > while theremaining modes are superscripted as <. We perturbatively include the effectsof the modes above the cutoff by renormalizing terms in the effective action ofthe remaining < sector:

Z =∫

D[φ<]D[φ>]e−SG[φ<]−SG[φ>]−SI[φ<,φ>] , (25)

=∫

D[φ<]D[φ>]e−SG[φ<]−SG[φ>] (1 − SI[φ<, φ>] + · · · ) , (26)

=∫

D[φ<]e−SG[φ<] (1 − 〈SI[φ<, φ>]〉>[φ<] + · · · ) , (27)

=∫

D[φ<]e−SG[φ<]−〈SI[φ<,φ>]〉>[φ<]+··· . (28)

280 A.J. Schofield et al.

Fig. 3. When rescaling the k cutoff from Λ to Λ = Λ/b preserving the form at aGaussian level requires that the frequency cutoff is also rescaled: Γk becomes Γk =Γk/b2. The rescaling is independent of dynamical exponent z. The modes which arelost by this process are shaded

Fig. 4. The 1 loop contribution to the RG equations which renormalize r and u. Theloop is summed over the high momentum and frequency modes > illustrated in Fig. 3

Thus we obtain, perturbatively, a low energy effective action where the averageis taken by integrating out the regions above the cut-off.

At lowest order in u we would have a correction coming from one loop whichrenormalizes r as shown in Fig. 4. It produces an identical term to that renormal-izing u from v except for a combinatorial factor for an n component field whichcomes from the number of ways of closing the loop. Doing the combinatoricsgives 2n+ 4 for the u-loop and 3n+ 12 for the v-loop.

Now in evaluating the diagrams of Fig. 4 we must integrate over the elimi-nated modes within the Gaussian approximation. (Strictly this is the Gaussianapproximation for fluctuations of the field about the equilibrium value as dis-cussed later ( 51) so we should replace the zero-field inverse-susceptibility, r,by the differential value at the field where scaling is being done, δ. Ultimately

Metamagnetic Quantum Criticality in Sr3Ru2O7 281

though the modes we are integrating over are at the cut-off scale so we will beable to set this mass to zero and the distinction is not important.) The diagramas a whole (integrating over all modes) would in general have the following form:

fall =∫ Λ

0

d2k

(2π)21β

∑n

1r + ξ2k2 + |ωn|/Γk . (29)

We do the frequency sum in exactly the same way as we calculated the freeenergy giving

fall =∫ Λ

0

d2k

(2π)2

∫ Γk

0

dω

πcoth

(ω

2kBT

)ω/Γk

(r + ξ2k2)2 + (ω/Γk)2, (30)

= A

∫ Λ

0kdk

∫ Γk

0dω coth

(ω

2kBT

)ω/Γk

(r + ξ2k2)2 + (ω/Γk)2. (31)

where A = 12π2 . However, for the RG equations we only need this function

summed over the missing modes. This can be done in two parts: the momentabetween Λ/b and Λ and the frequencies between Γk/b2 and Γk.

Considering the high k modes first. The integral over k then becomes trivialsince the limits are coalescing. Thus we can replace k by Λ and multiply by thewidth of the integration region, Λ−Λ/b. Using the fact that for small deviationsof b from 1, we may write 1 − b−a ∼ a ln b we can write

fk = AΛ2(1 − b−1)∫ ΓΛ

0dω coth

(ω

2kBT

)ω/ΓΛ

(r + ξ2Λ2)2 + (ω/ΓΛ)2, (32)

= AΛ2 ln b∫ vΛ

0dω coth

(ω

2kBT

)ω/vΛ

(r + ξ2Λ2)2 + (ω/vΛ)2. (33)

Now putting x = w/vΛ we may rewrite the integral as

fk = A ln b vΛ3∫ 1

0dx coth

[vΛx

2kBT

]x

(r + ξ2Λ2)2 + x2 . (34)

Now we do the same for the high ω modes. For each k I have a trivial ωintegral to do with the limits coalescing: Γk/b2 < ω < Γk. Thus we have

fω = A

∫ Λ

0k dkΓk(1 − b−2) coth

(Γk

2kBT

)1

(r + ξ2k2)2 + (1)2, (35)

= 2A ln b∫ Λ

0dk k2v coth

[vk

2kBT

]1

(r + ξ2k2)2 + 1. (36)

Now putting y = k/Λ we may rewrite this terms as

fω = 2A ln b vΛ3∫ 1

0dy coth

[vΛy

2kBT

]y2

(r + ξ2Λ2y2)2 + 1. (37)

282 A.J. Schofield et al.

Now we can identify the energy scales in the problem. One recurring scale is

vFΛ = ΓΛ = ωsf , (38)

which is the damping rate of spin-fluctuations at the cut-off.Finally, as noted above, we can set r = 0 since the >-sector modes are far

from the criticality, so defining the following function (where t = kBT/ωsf andsetting ξΛ = 1)

f(t) = A

∫ 1

0du coth

( u2t

) [u

u2 + 1+ 2

u2

u4 + 1

]. (39)

Thus at 1-loop the RG equations become modified (λ = ln b) so that

dr

dλ= 2r +

n+ 22

u(λ) f(t) , (40)

du

dλ= −u+

n+ 42

v(λ) f(t) . (41)

6 Integrating the RG Equations

To use these scaling equations (12,14,24, 40 and 41), together with the defini-tion (39), we need to solve them. However, the RG equations are simply a set ofcoupled linear differential equations which may be integrated directly.

Solving the equations for v, T and h is trivial

v(λ) = v0e−4λ , t(λ) = t0e

3λ , h(λ) = h0e7λ/2 . (42)

We can therefore substitute these results into the equation for u:

du

dλ− u(λ) =

n+ 42

v0e−4λf(t0e3λ) , (43)

to be solved subject to the initial condition that u(0) = u0. This again is straight-forward

u(λ) = e−λ[u0 +

n+ 42

v0

∫ λ

0e−3xf(t0e3x)dx

]. (44)

Now again we can substitute this into the equation for r. Solving this yields

r(λ)=e2λ[r0 +

n+ 22

∫ λ

0dye−2yf(t0e3y)u(y)

],

=e2λr0 +

n+ 22

∫ λ

0dy e−3yf(t0e3y)

[u0 +

n+ 42

v0

∫ y

0dx e−3xf(t0e3x)

].

(45)

Now we can rewrite the solution of the RG equations into parts which ex-plicitly are temperature dependent, and the remaining T = 0 values which are

Metamagnetic Quantum Criticality in Sr3Ru2O7 283

renormalized from their initial values by quantal fluctuations. Consider the def-inition of the function f(t) in (39). We may identify coth(y/2t) = 1 + 2nB(y/t)where, for t = 0, the Bose factor is zero for all positive y. Thus we may write

f(t) = f0 + g(t) , (46)

where

f0 = A

∫ 1

0du

[u

u2 + 1+ 2

u2

u4 + 1

]∼ 0.0422 . (47)

g(t) = A

∫ 1

0dunB

(ut

) [u

u2 + 1+ 2

u2

u4 + 1

]. (48)

The dependence of r and u on f0 reflects how quantal fluctuations renormalizethe bare parameters at T = 0 and this can be an appreciable effect (determinedby the size of u and v). The sign of these effects is such that they move thesystem away from the magnetically ordered phase. This provides an explana-tion for the band-structure predictions that Sr3Ru207 should be ferromagneticat zero magnetic field [16]: the quantal effects are pushing the metal into theparamagnetic phase.

It is more transparent to write the solution to the scaling equations explicitlyin terms of the T = 0 renormalized parameters

r(λ) = e2λ

[reff +

n+22

∫ λ

0dye−3y

ueffg(t0e3y) +

n+42

v0

∫ y

0e−3xg(t0e3x)dx

].

(49)Finally, it is also more convenient for the numerical evaluation to use t as therunning variable rather than λ. This then amounts to a change of variables inthe equation for r: t = t0e

3w. Thus dw = dt/(3t). Doing this change of variablesin each of the integrals we will have

r(λ) = e2λ

[reff +

n+ 22

∫ t0e3λ

t0

dt

3t

(t0t

) ueff(3−1 ln t/t0)g(t)

+n+ 4

2v0

∫ t

t0

(t0t′

)g(t′)

dt′

3t′

]. (50)

This is the expression we will use for subsequent numerical evaluation.

7 Numerical Procedure

The process by which we turn the solution of the scaling equations into physicalquantities is described as follows (though performed numerically). For a fixedset of initial parameters reff , ueff , v0 and t0, integrate the RG equations untilr = 1. This effectively sets the H = 0 and T = 0 susceptibility to 1 (which is

284 A.J. Schofield et al.

the normalization for our results). We then use the rescaled parameters at thevalue of λ which gives r = 1.

To analyse the experimental consequences of proximity to metamagnetismwe consider Gaussian fluctuations about the equilibrium magnetization as deter-mined from the action now with the renormalized parameters. We expand thefield about its equilibrium point: ϕ = (|φ| − φ)/φ0 normalized to the saturationmagnetization (φ0). At the metamagnetic point itself we find φ2 = −3u/10v. Wemeasure the field relative to the critical field h = (H −H∗)/H∗. The resultingaction at quadratic order looks exactly the same as the Gaussian action for thetotal magnetization (5) since its form is determined by symmetry

Smeta =∫d2q

∑n

[δ + ξ2q2 +

|ωn|vq

]ϕq,ωn

ϕ−q,−ωn. (51)

However here r has been replaced by δ, the “mass” of the fluctuations, whichmeasures the distance from the quantum critical end-point. Experimentally it isthe inverse of the differential susceptibility: δ = r+3uφ2 +5vφ4. The differentialsusceptibility, δ−1, diverges at the critical end-point. At the metamagnetic criti-cal end-point (rv/u2 = 9/20) but not quite at the critical magnetic field (h = 0)then we find, from an expansion of (7), that

δ =3r(2φ2r−1h)2/3

φ2. (52)

Analysis of this action alone can be used to determine the qualitative behaviournear the metamagnetic point. The results are reminiscent of a ferromagneticquantum critical point except that the “mass” is also related to the field h.Results are summarized in [29]. Note that the detailed temperature dependenceof quantities requires integration of the RG equations.

We compute three key quantities: the differential susceptibility, the resistivityexponent and γ = C/T . The differential susceptibility (δ−1) is obtained directlyfrom the second derivative of the free energy about the equilibrium value (withthe rescaled parameters). The linear term in the heat capacity is determinedfrom the second-temperature derivative of the free energy of (23) but evaluatedusing the parameters of the Gaussian model of (51). The resistivity is moreinvolved since we now need to reconsider the electrons which were integratedout. We essentially “undo” the Hubbard Stratonovich of (2) only now using theGaussian action for φ ( (51) determined using the renormalized parameters). Theresistivity then comes from considering the lowest order term in the self-energythat comes from interactions of the electrons with the mode ϕ. This self-energymust be corrected for the forward scattering physics: the dominant scattering isat long wavelength and so is ineffective at degrading a current.

At lowest order the electron self-energy is

Σ(p, iΩn′) = J2∫

d2q

(2π)21β

∑n

G(p − q, iΩn′ − iωn)D(q, iωn) , (53)

Metamagnetic Quantum Criticality in Sr3Ru2O7 285

where G is the non-interacting electron Green’s function and D(q, iωn) = (δ +ξ2q2 + |ωn|/Γq)−1 is the Gaussian propagator of the spin fluctuations. Doing thissummation and integral is straightforward particularly since we are interestedin the quasiparticle scattering rate at the Fermi surface. For this we need theimaginary part of the self-energy calculated on shell

1τpF

= −2ImΣ(pF , iΩn′ → εpF+ i0+) , (54)

and corrected for the 1 − cos θ ∼ (q/kF )2 forward scattering factor (θ is thescattering angle). After some algebra we find

1τ2DpF

∼∫ ∞

0

qdq

π2

(q

kF

)2 ∫ 0

−vF

dω′

vF q

1sinh(ω′β)

2ω′/Γq(δ + ξ2q2)2 + (ω′/Γq)2

. (55)

8 Results

We have started with an isotropic n = 3 vector theory for the magnetizationvariable (just as would be the case for a ferromagnetic quantum critical point).References [32,36,33] have argued that the spatial gradient structure in this caseis dramatically affected by interaction corrections. However in the presence of amagnetic field, the ‘mass’ (coefficient of the quadratic part of the fluctuations)becomes anisotropic, with the component corresponding to fluctuations alongthe field becoming reff = r+3uφ

2+5vφ

4. A Heisenberg-XY or Heisenberg-Ising

crossover occurs when then larger of reff or r passes through unity and scalingstops when the smaller of the two becomes of order unity. Thus in the vicin-ity of the metamagnetic transition we are only dealing with a one-component,Ising, field as described in (51) which does not suffer from the deviations men-tioned above. Moreover the bilayer ruthenate material is stoichiometric so dis-order should not be so important and, being a d metal system, Kondo physicsis not relevant. So it seems that none of the possible mechanisms for discrepan-cies from the conventional theory should apply to the metamagnetic quantumend-point in Sr3Ru2O7. Indeed comparison of the theory developed here withexperiment provides an excellent test of the theoretical framework.

We now present the results of a numerical solution of the scaling equations.Fig. 5 shows the h dependence of the differential susceptibility for several val-ues of T , obtained in the two dimensional case using parameters reasonable forSr3Ru2O7. The inset shows the temperature dependence of the differential sus-ceptibility for different h. Note the non-monotonic temperature dependence forfields different from h = 0 if the control parameter is tuned to criticality. Thusour theory shows that the similar peak seen in the experiments [15] is not anindication of some hidden magnetic order but rather is a natural signature ofproximity to a metamagnetic quantum critical end-point.

Fig. 6 shows the specific heat coefficient γ = C/T ; in this quantity thecrossover is much less sharp, in part because a 2d nearly critical Fermi liquidhas a specific heat coefficient γ ∼ A + BT with both A and B divergent as

286 A.J. Schofield et al.

0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

12D

iffe

renti

alS

usc

epti

bil

ity

0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

12

14

Dif

fere

nti

alS

usc

epti

bil

ity

Fig. 5. Differential susceptibility, χ−10 (∂m/∂h), as a function of applied field H at

temperatures T/T0 = 0.05, 0.1, 0.2, for a two dimensional metamagnetic critical point.Inset: Dependence of χ−1

0 (∂m/∂h) on temperature T at h = .01, .02, .04, .08. (Normal-izations discussed in text)

the critical point is approached. This is an example of the corrections to scalingfrom the one-loop corrections of the irrelevant operators. The inset shows theresistivity exponent α = −∂ ln ρ/∂ lnT plotted against temperature for h = 0and h = 0.1. The high-T resistivity exponent at a ferromagnetic critical pointwould naively expected to be 4/3. However the 1-loop corrections modify thisvalue as shown in the inset. The crossover to the expected low-T T 2 behaviouris very sharp. The coefficient of this T 2 resistivity diverges h−2/3 as the criticalpoint is approached since it is determined δ (see (52)). This power-law divergenceis seen in Sr3Ru2O7 as the critical field is approached.

There are, however, two features of the experimental resistivity that thistheory cannot explain. The first is that the residual (elastic) part of the resistivityshows a sharp maximum at the critical field. This is presumably a consequenceof the interplay between static disorder and the critical fluctuations—which isbeyond the scope of this treatment [37]. The second is that at the critical pointitself a new temperature dependent resistivity is seen: ρ ∼ ρ0 + AT∼3 [5]. Thistemperature dependence is not found in this theory. Indeed it is hard to seehow such behaviour could arise in any theory involving a Fermi surface: anyresidual electron-electron scattering at the Fermi surface should at least give aT 2 resistivity which would dominate.

Metamagnetic Quantum Criticality in Sr3Ru2O7 287

0.1 0.2

40

60

80

0.2 0.4 0.6 0.8 1.

1.2

1.4

1.6

1.8

2.

Res

isti

vit

yE

xp

on

ent

Fig. 6. Dependence of specific heat coefficient C/T on temperature T for h =0.01, .0.1, 0.2, 0.4 calculated for a two dimensional metamagnetic critical point. Inset:Dependence of resistivity exponent ∂ ln ρ/∂ ln T on T for h/H∗ = 0 (lower curve) and0.1 (upper curve). (T0 defined in text)

The unusual resistivity at the critical point itself is very mysterious. It couldperhaps be a consequence of a first order metamagnetic transition occurring atthe lowest temperatures. The process whereby the higher magnetization phasenucleates in the lower magnetization one and the result of this on the resistivityis a possible avenue to be explored. A more speculative idea is that a novel metal-lic state has been found. Most metals at quantum critical points are unstableto other ordered phases. Either the transition goes first order (as is the case forMnSi) or superconductivity is induced. At the metamagnetic critical end-pointneither of these states are possible: the magnetic field is too strong for super-conductivity and we have deliberately tuned away the first order transition.Without the usual “escape routes” perhaps we are forcing the metamagneticquantum critical metal to do something entirely different and new.

9 Conclusions

In conclusion, we have presented a theory of metamagnetic quantum criticalityas it applies to the bilayer ruthenate Sr3Ru2O7. We have obtained numericalsolutions to the RG equations and used them to compute the differential suscep-tibility, the specific heat and the resistivity. Our theory accounts for the unusual

288 A.J. Schofield et al.

temperature dependence of χ(T,H = 0), the paramagnetic ground state and themain features of behaviour seen in Sr3Ru2O7 as the metamagnetic critical pointis approached. This lends weight to the assumptions underlying much of the re-cent work on quantum phase transitions and indeed on non-Fermi liquids. How-ever, the puzzling behaviour of Sr3Ru2O7 at the critical point itself [5] remainsoutside this framework. Subsequent papers [29] will present the most generalcase, and in particular apply the theory in three dimensional materials such asMnSi. We have shown already that the assumptions implicit in the conventionalapproach, which underlie much recent work on quantum phase transitions inmetals and indeed on non-Fermi-liquid physics, are apparently consistent withmuch of the observed behaviour of Sr3Ru2O7.

Acknowledgements

AJS acknowledges the support of the Royal Society and Leverhulme Trust. AJMwas supported by NSF-DMR-0081075 and the EPSRC and thanks the CavendishLaboratory, the Theoretical Physics group at the University of Birmingham andthe Aspen Center for Physics for hospitality while parts of this work were un-dertaken. We thank A. P. Mackenzie for advice, encouragement and commentson the manuscript and D. J. Singh for helpful comments.

References

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M. Chiao, S. R. Julian, S.I. Ikeda, S. Nakatsuji, Y. Maeno, C. Pfleiderer: Phys.Rev. Lett. 86, 2661 (2001)

5. S.A. Grigera, R.S. Perry, A.J. Schofield, M. Chiao, S.R. Julian, G.G. Lonzarich,S.I. Ikeda, Y. Maeno, A.J. Millis, A. P. Mackenzie: Science 294, 329 (2001)

6. J. A. Hertz: Phys. Rev. B 14, 1165 (1976)7. A. J. Millis: Phys. Rev. B 48, 7183 (1993)8. M.B. Suvasini, G.Y. Guo, W.M. Temmerman, G.A. Gehring: Phys. Rev. Lett. 71,

2983 (1993)9. P. Haen, J. Flouquet, F. Lapierre, P. Lejay, G. Remenyi: J. Low Temp. Phys. 67,

391 (1987)10. H. Aoki, S. Uji, A. K. Albessard, Y. Onuki: Phys. Rev. Lett. 71, 2110 (1993)11. J. Flouquet, S. Kambe, L. P. Regnault, P. Haen, J.P. Brison, F. Lapierre, P. Lajey:

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Y. Onuki: Physica B 281-282, 64 (2000)13. See, e.g. articles in Proceedings of the 6th International Conference on Research

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14. C. Thessieu, C. Pfleiderer, A.N. Stepanov, J. Flouquet: J. Phys.: Cond. Matter9,6677 (1997)

15. S. Ikeda, Y. Maeno, M. Kosaka, Y. Uwatoko: Phys. Rev. B 62, R6089 (2000)16. D.J. Singh, I.I. Mazin: Phys. Rev. B, 63, 165101 (2001)17. S.R. Julian, C. Pfleiderer, F.M. Grosche, N.D. Mathur, G.J. McMullan, A.J. Diver,

I.R. Walker, G.G. Lonzarich: J. Phys.: Cond. Matter 8, 9675 (1996)18. Hilbert von Lohneysen: J. Phys.: Cond. Matter 8, 9689 (1996)19. S. Chakravarty, R.B. Laughlin, D.K. Morr, C. Nayak: Phys. Rev. B 63, 094503

(2001)20. P. Gegenwart, F. Kromer, M. Lang, G. Sparn, C. Geibel, F. Steglich: Phys. Rev.

Lett. 82, 1293 (1999)21. A. Schroder, G. Aeppli, R. Coldea, M. Adams, O. Stockert, H. v. Lohneysen,

E. Bucher, R. Ramazashvili, P. Coleman: Nature 407, 351 (2000)22. P. Coleman, C. Pepin, Q. Si, R. Ramazashvili: J. Phys.: Cond. Matter 13, R723

(2001)23. D. Belitz, T.R. Kirkpatrick, Thomas Vojta: Phys. Rev. B 65, 165112 (2002)24. A. Rosch: Phys. Rev. Lett. 82, 4280 (1999)25. D. Meyer, W. Nolting: Phys. Rev. B 64, 052404 (2001)26. H. Satoh, F.J. Ohkawa: Phys. Rev. B 63, 184401 (2001)27. H. Yamada: Phys. Rev. B 47, 11211 (1993)28. H. Yamada, K. Terao: Phys. Rev. B 59, 9342 (1999)29. A.J. Millis, A.J. Schofield, G.G. Lonzarich, S.A. Grigera: Phys. Rev. Lett. 88,

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Metamagnetic Transition and Low-Energy SpinDensity Fluctuations in Sr3Ru2O7

L. Capogna1,2, E.M. Forgan2, S.M. Hayden3, G.J. McIntyre4, A. Wildes4,A.P. Mackenzie2,5, J.A. Duffy3,6, R.S. Perry2,7, S. Ikeda7, and Y. Maeno7,8

1 Max Planck Institute for Solid State Research,Stuttgart, D-70569, Germany

2 School of Physics and Astronomy, University of Birmingham,Birmingham B15 2TT, U.K.

3 H.H. Wills Physics Laboratory, University of Bristol,Bristol BS8 1TL, U.K.

4 Institut Laue-Langevin,6 Rue Jules Horowitz, F38042 Grenoble Cedex, France

5 School of Physics and Astronomy, University of St. Andrews,St. Andrews KY16 9SS, U.K.

6 Department of Physics, University of Warwick,Coventry CV4 7AL, U.K.

7 Department of Physics, Kyoto University,Kyoto 606-8502, Japan

8 CREST, Japan Science and Technology Corporation,Kawaguki, Saitama 332-0012, Japan

Abstract. The magnetic properties of the bilayer ruthenate Sr3Ru2O7 as investigatedby neutron scattering are reviewed. No long range magnetic order is observed down to40 mK, but instead a metamagnetic transition which is induced by a moderate magneticfield at low temperatures. In zero field, the spectrum of the spin fluctuations revealsthat at low temperature the magnetic correlations in this material are antiferromag-netic in nature. The wavevectors at which the signal is strongest are incommensurateand appear to be determined by Fermi surface nesting. The low energy scale of thefluctuations suggests that they are intimately related to the metamagnetic transition.

1 Introduction

The strontium ruthenate Srn+1RunO3n+1 series is acknowledged to be wellsuited to the study of correlated electron phenomena in itinerant systems. Theposition of the Fermi level in bands resulting from the hybridisation of oxygen2p and ruthenium 4d levels leads to a variety of ground state behaviours whichis almost unprecedented in any other transition metal oxide series. Furthermore,in contrast to 3d oxides such as cuprates and manganites, no explicit chemicaldoping is required to produce metallic conduction. The main source of disorderin stoichiometric ruthenates is limited to either chemical impurities (in crystalsgrown from fluxes in hot crucibles) or structural defects (in thin films or crystalsgrown using crucible free methods). Major advances in the floating zone method(FZ) have proven crucial to unveiling a fascinating range of exotic properties in

C. Noce et al. (Eds.): LNP 603, pp. 290–302, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Metamagnetic Transition and Low-Energy Spin Density Fluctuations 291

the ruthenates. In the single layer compound Sr2RuO4 superconductivity [1] hasbeen discovered because free paths longer than 1000 A have been achieved. Thestrong dependence of the superconducting critical temperature of Sr2RuO4 uponnon-magnetic impurities demonstrates that the wave function of the Cooper pairsin this material can not have s-wave parity [2]. Furthermore, even the metallicstate of the ruthenates is highly sensitive to disorder. The Fermi liquid state inthe itinerant ferromagnet SrRuO3 and in Sr3Ru2O7 appears to be remarkablysensitive to both thermal and impurity scattering. Anomalous behaviour hasbeen observed in zero applied field in these compounds [3] but it does not ap-pear to extend to the region of parameter space where Fermi liquid theory makesits most robust predictions: low temperatures and high purity samples [4]. Ap-plication of a moderate magnetic field [5] reveals however that Sr3Ru2O7 is infact characterised by an intrinsic anomaly.

Sr3Ru2O7 has been reported as an antiferromagnet [6] , a paramagnet [7]and as a weak itinerant ferromagnet [8] , depending on the technique employedto grow the samples. In general, single crystals grown with the floating zonetechnique are significantly purer than those grown in a platinum crucible. Thevery low level of impurities is reflected in their residual resistivity, which in smallcrystals is of the order of a few µΩ cm [5,9]. FZ crystals do not show any longrange ferromagnetic ordering. The susceptibility χ has a maximum at approxi-mately 17 K; below 5 K, χ becomes isotropic and temperature independent [9].The peak in χ is reminiscent of that seen at the Neel point in an antiferromag-net, but it is too broad to correspond to a phase transition and shows no signof a magnetic easy axis. On the other hand, applying pressure [9] or introducingimpurities induce ferromagnetism. Combined with a T 2 dependence of the re-sistivity below 10 K, this experimental evidence strongly suggests that at lowapplied magnetic field Sr3Ru2O7 is an exchange-enhanced Fermi liquid on theverge of a ferromagnetic instability. The fragility of the Fermi liquid state in thiscompound is well exemplified by the anomalous dependence of the resistivity on temperature in a moderate magnetic field [5]. At T=2 K and in a field of5.5 T applied in the ab plane, the resistivity of very high-purity single crystalsof Sr3Ru2O7 displays a striking linear dependence upon temperature, violatingthe T 2 law valid for a Fermi liquid at low temperatures. This is suggestive of theexistence of a critical point near T=0 which is induced by the field.

The field-induced anomalous behaviour in Sr3Ru2O7 is associated with meta-magnetism, i.e. a rapid and reversible increase in magnetisation at a particularvalue of applied magnetic field at low temperatures [5,10,11]. In contrast, nometamagnetism has been observed in the sister compound Sr2RuO4 in fieldsup to 33 T . The close structural similarity between the bilayer Sr3Ru2O7 andthe single layer Sr2RuO4 is not reflected in their physical properties. WhileSr3Ru2O7 does not show any superconductivity even in high purity single crys-tals, in paramagnetic Sr2RuO4 there is mounting evidence for triplet super-conductivity. However, no evidence has been found for spin fluctuations at lowq [12,13] which are commonly thought to provide the binding mechanism for p-wave superconductivity. The underlying reasons for the pronounced differences

292 L. Capogna et al.

in the two materials may ultimately lie in the orthorombicity of Sr3Ru2O7 [14]which could differentiate the two Fermi surfaces by much more than a bilayersplit. In this context, a knowledge of the spin fluctuation spectrum of Sr3Ru2O7can provide valuable though indirect information on the structure of the Fermisurface and its possible modifications induced by an applied field.

In this paper we review details of the metamagnetic transition and the spec-trum of the spin fluctuations, which provide a detailed and unambigous micro-scopic picture of the low energy excitations. First we describe the experimentalprocedures and then the models employed to interpret the experimental results.Our co-workers in the individual neutron scattering experiments are acknowl-edged in the specific references given under Sects. 3 and 4.

2 Experimental Techniques

In this Section we summarize details of the neutron scattering techniques em-ployed to probe the metamagnetic transition and the spectrum of magnetic fluc-tuations in Sr3Ru2O7.

2.1 Neutron Diffraction

A search for long-range magnetic ordering and the measurement of the inducedmoments at the metamagnetic transition was carried out on the four-circle crys-tal diffractometer D10 at the Institute Laue-Langevin in Grenoble. In the two-circle mode, D10 can be equipped with a helium-flow cryomagnet. In the dilutionmode, this allows temperatures in the range between 0.04 K and 10 K to be set,while a vertical magnetic field up to 6 T is applied. A medium-sized crystal ofSr3Ru2O7 of mass ∼ 0.2 grams was used to minimise extinction problems whilekeeping an acceptable intensity in the diffracted beam. The measurements wereperformed with a λ = 2.358 A wavelength and a pyrolitic graphite filter insertedbefore the sample to remove higher order contamination.

2.2 Inelastic Neutron Scattering

The energy and momentum resolved neutron scattering cross section of the spinfluctuations was observed using the three-axis spectrometer IN14 also at theInstitute Laue-Langevin. The three-axis technique is to date the most versatiletool to probe excitations in solids in single-crystalline form. In addition IN14 isparticularly suited to the detection of low-energy excitations in systems with asmall magnetic moment and a small mass. Due to its position relatively nearthe source on a straight guide tube, IN14 exploits a particularly wide range ofincident wavelengths (∼ 2 to 6 A) with a concomitant wide range of resolution(∼ 1 to 0.02 meV). The instrument offers a high neutron flux at small energytransfer in a high resolution configuration.

Metamagnetic Transition and Low-Energy Spin Density Fluctuations 293

(1 1 0)

(1 1 0)

Q

i

(0 0 0)

(4 0 0)(2 0 0)

(2 0 0)

(0 2 0)

f

φ Ψ

k k

Fig. 1. The reciprocal space of Sr3Ru2O7 at kz = 0 approximated to the tetragonalcrystal structure. The energy and momentum conservation laws require that the tri-angle of scattering defined by the initial and the final neutron momentum ki, kf andthe scattering vector Q be closed

In a three-axis experiment the neutron initial Ei and final Ef energies areselected independently by Bragg diffraction tuning the angle θM and θA at themonocromator and at the analyser crystal respectively:

Ei/f =

2

2mk2

i/f =

2

2mπ2

d2M/A sin2θM/A

. (1)

The momentum Q exchanged between the neutrons and the sample is de-termined by the scattering angle φ defined by the direction of the outcomingneutrons relative to the incoming beam.

A measurement of the scattered intensity in a given instrument configura-tion yields the intensity of the excitation at a particular point (Q, ω) of themomentum-energy four-dimensional space. The two relations stating the conser-vation of energy and momentum must hold [15]:

2k2

i

2m−

2k2f

2m= ω , (2)

k2i + k2

f − 2kikfcosφ = Q2 . (3)

The energy transfer ω and the momentum components Qx and Qy in thescattering plane can be appropriately selected (Fig. 1) by tuning the experimen-tal parameters: ki, kf , φ and the sample orientation ψ relative to the incomingneutron beam. Since the number of available instrument parameters is higherthan the number of physical variables, there are several ways of carrying out ameasurement.

In the so-called Q-scan, the energy transfer from the neutron to the sampleis kept constant and the intensity of the scattered beam is studied as a functionof the wavevector Q. In this case θM and θA are kept fixed and Q is scanned

294 L. Capogna et al.

Analyser

Be filter

Detector

Monitor

Sample

Monocromator

Neutron beam

A4

A3

A1

A5

A2 A6

Fig. 2. In the so-called “Long Chair” configuration the detector is placed at a negativeangle (A6) with respect to kf , whilst the analyser and the sample are at positiveangles A4 and A2 with respect to ki and the neutron beam, respectively. A berylliumfilter with a cut off at 1.57 A−1 is inserted between the sample and the analyzer. Theintensity of the incoming beam is monitored before the sample by means of a lowefficiency detector

by varying the sample rotation angle ψ and the scattering angle φ in correlatedsteps. A large portion of the in−plane reciprocal space can be explored compat-ibly with the dynamical range of the instrument which is set by the requirementthat (2) and (3) be satisfied. All measurements were taken in the “Long Chair”configuration (Fig. 2). In the so-called energy scans, the momentum transfer iskept constant and the energy transfer is scanned by varying three of the fouravailable parameters: ψ, φ and either ki or kf . In Sr3Ru2O7 the final energy Ef

was kept constant to the value of 4.97 meV to measure the excitations in theab plane. In this case a beryllium filter was inserted between the sample andthe analyser to remove contamination from higher order Bragg reflections frommonochromator and analyser. For measurements in the ac plane, Ei was insteadfixed to the value of 14.67 meV in order to increase the observed scattered inten-sity: this would be otherwise too small due to the size of the resolution ellipsoidin the ac plane (see Appendix) and the small mass of the sample (∼ 0.9 grams).

At Ei constant, a pyrolitic graphite higher order filter was used in the incidentbeam.

Finally, to establish the nature and properties of the excitations, a cryostatwith background shielding and a neutron window was used; in combination witha heater this allows the whole temperature range between 1.5 K and 320 K to beinvestigated. A temperature study combined with a wavevector Q dependenceallows to distinguish effectively between magnetic and phonon scattering. Theintensity of phonon scattering increases with temperature and has a quadraticdependence on the wavevector Q, while the magnetic intensity decreases with Q

Metamagnetic Transition and Low-Energy Spin Density Fluctuations 295

because of the magnetic form factor. At high temperature the phonon absorptioncross section was measured to provide a signal against which absolute calibrationof the magnetic cross section was performed (see Appendix).

The use of single crystals of Sr3Ru2O7 free of ferromagnetic phases (i.e.SrRuO3) has precluded contamination of the spectra from magnetic inhomo-geneities. Details on the crystal growth and characterisation may be foundin [7,9].

3 The Metamagnetic Transition

Metamagnetism in insulating antiferromagnetic systems refers to the phase tran-sition produced by an applied magnetic field when the strength of the field equalsor exceeds the exchange coupling between the magnetic moments. In this case themagnetic moments are realigned through a spin-flip or spin-flop process depend-ing on the relative orientation of the field to the spin direction. In an itinerantparamagnetic system the term metamagnetism describes the transition from aless polarised state to a more polarised state through either a phase transition ora crossover. The extra moment appearing at the metamagnetic field is generatedby a change in polarization of the spin-up and spin-down Fermi surfaces and thefluctuations can be thought of as fluctuations of the Fermi surface itself. Thispicture holds at q = 0 but the polarised Fermi surface and unpolarised Fermisurfaces are likely to have different nesting properties in a nearly two-dimensionalmaterial, so a coupling with fluctuations at higher q is to be expected. Itinerantelectron magnetism has previously been observed in MnSi and CeRu2Si2 butSr3Ru2O7 is the first example of a compound in which the magnetic field isdriving the system closer to criticality [5]. In Sr3Ru2O7 a moderate magneticfield (B∼ 5.3 T in the ab plane) re-aligns the spins in a long range ferromagneticfashion.

The experimental evidence is the significant and reversible increase of thenuclear Bragg peak intensities Ihkl (Figs. 3 and 4) upon application of a magneticfield B:

I(B)hkl = I(0)hkl +∆I × 12

(1 + tanh

B −Bm

Bw

), (4)

with

I(0)hkl =λ3V

V 2c

1ωL×A× T × E |F(h)|2 . (5)

Here λ is the neutron wavelength, V the crystal volume, Vc the unit cellvolume, ω the angular scanning velocity, F (h) is the structure factor for thereflection hkl with scattering vector h. L is geometrical correction, the polarisa-tion factor. In addition, in a real experiment one needs to correct for absorptionA, extinction E, which is the reduction of the incident beam by scattering or thediffracted beam by rescattering back parallel to the incident beam, and finallyfor thermal diffuse scattering T . The intensity jump ∆I induced by the field isdescribed by means of a mathematically convenient function (Eq. 4).

296 L. Capogna et al.

Fig. 3. The intensity of the (0 0 6) Bragg reflection as a function of the magnetic fieldB applied in the ab plane. The jump observed at the metamagnetic field is fitted bya phenomenological function, (4). The fitted parameters were Bm = 5.23(2); Bw =0.188(3); ∆I/I = 0.054

Fig. 4. The difference scan between a rocking curve of the (0 0 6) reflection at 5.5 T,just above the metamagnetic field, and that at a “background” field of 4.4 T, both ata temperature T = 40 mK

The intensities of a set of 21 nuclear reflections taken at 4.4 T were used todetermine extinction corrections and refine the crystal structure using the modelin [14] for the crystal structure of Sr3Ru2O7 and assuming negligible effects of

Metamagnetic Transition and Low-Energy Spin Density Fluctuations 297

Fig. 5. The Ru+4 form factor measured from the increase in the intensity of the nuclearBragg reflections at the metamagnetic transition (experimental points), compared withthe theoretical value for Ru+ given in [16] (solid line)

magnetisation. The tetragonal lattice parameters are a = b = 5.476 A andc = 20.830 A and do not appear to vary significantly through the metamagnetictransition (Fig. 4). The observed metamagnetic intensity changes can then beadded to the calculated low-temperature structural intensities to obtain a set ofobserved total intensities against which ferromagnetic models are refined [10].Assuming that the moments are only induced on the Ru ions and that they areparallel to the applied magnetic field along the (0 1 0) direction, we obtain forthe induced moment the value m = 0.25(3) µB per Ru. Figure 5 shows the Ru4+

magnetic form factor derived from our data as a function of scattering vector.Our curve is in reasonable agreement with the theoretical dependence of theRu+ magnetic form factor [16].

An important question to address in Ru-based perovskites is the level ofhybridisation between the Ru 4d orbitals and the O 2p orbitals. In particular,in Sr3Ru2O7 a significant contribution from the O ions to the density of statesat the Fermi level has been predicted [10]. On the basis of the present data it isnot possible to establish the extent to which the metamagnetic moments appearalso on the oxygens. If a moment is allowed to appear on the O ions in the RuO2planes, we obtain an O moment antiparallel to the Ru one. However, the totalmagnetic moment is still m = mRu + 2mO = 0.33 − 0.08(12) = 0.25 µB, themagnitude of the oxygen moment is not well defined and the overall fit is nobetter.

At the sample temperature of 40 mK, we found no evidence for antiferro-magnetic ordering [10] in FZ single crystals. A similar result was obtained on

298 L. Capogna et al.

polycrystalline samples [17] by powder diffraction down to 1.5 K and disagreeswith the suggestion from Liu et al. [18] of canted antiferromagnetic ordering.

4 The Magnetic Fluctuation Spectrumand the Dynamical Susceptibility

4.1 Energy Dependence

The statistical behaviour of spontaneous fluctuations in the local spin magneti-sation µ(r, t) can be characterised by the spectrum of the Fourier transformµ(Q, ω) =

∫d3r

∫µ(r, t) exp[−i(Q · r − ωt)] dt in a large volume V and long

time interval. The dynamical susceptibility χ(Q, ω) generalises then the staticsusceptibility and describes the response of the system to a magnetic field thatvaries in space and time.

µα(Q, ω) =∑

β

χαβ(Q, ω)Hβ(Q, ω) . (6)

In the limit Q and ω → 0, the dynamical susceptibility is proportional tothe static magnetic susceptibility. In general, χαβ has complex values whichmeans that the the magnetisation and the external field are out of phase. Viathe fluctuation-dissipation theorem, the magnetic fluctuations are related to theimaginary part χ′′ of the magnetic susceptibility. In the case of low Q and ω thedominant contribution arises from fluctuations of the local magnetic field due tothe fluctuations in the spin magnetisation. The inelastic part of the differentialcross section for the scattering of unpolarised neutrons can then be derived fromthe interaction potential between the neutron magnetic moment and the mag-netic field produced by the electrons in the system or, equivalently, between thesample magnetisation and the magnetic field locally generated by the neutrons.If energy is transferred from the neutrons to the system, the differential crosssection may be expressed in the form:

d2σ

dΩdEf= σ0

ki

kf[n(ω) + 1)] χ′′(Q, ω) , (7)

where n(ω) is the Bose factor (exp(ω/kBT ) − 1)−1. A suitable model at lowtemperatures and low frequencies is often a Lorentzian one. In this case, theimaginary part of susceptibility reduces to [19]

χ′′(Q, ω) = χ′(Q)ωΓ (Q)

Γ 2(Q) + ω2 . (8)

In Sr3Ru2O7 we find that the characteristic energy Γ = 2.3 ± 0.3 meV [20],which is much less than 9 meV reported in the single layer compound Sr2RuO4.Using the calculated scattering cross section [21] and the integrated intensity ofacoustic phonons, the magnetic scattering can be converted into the equivalentof effective magnetic moments to allow a comparison with static susceptibility.

Metamagnetic Transition and Low-Energy Spin Density Fluctuations 299

In Sr3Ru2O7 the wavevector dependent susceptibility is large, translating to amaximum value of χ′ of 1.6 × 10−2 emu/mol Ru. This indicates that Sr3Ru2O7is much closer to magnetic order than its sister compound.

4.2 Momentum Dependence

Figure 6 shows a summary of the location of the spin fluctuations in recipro-cal space: the excitation intensity peaks at two incommensurate wavevectorsof the form qδ ≈ 0.25, 0, 0 and qε ≈ 0.09, 0, 0, distributed symmetricallyabout (1, 0, 0). Measurements in a moderate magnetic field indicate that the in-commensurate spanning vectors are affected by the field and change on passingthrough the metamagnetic field.

Unlike Sr2RuO4, where the excitations have been observed along the [h, h,0] direction (π − π) [12,13], in Sr3Ru2O7 the fluctuations are present along the[h, 0, 0] direction (Γ − Z). In the single layer compound Sr2RuO4 the antifer-romagnetic fluctuations are due to nesting features in the Fermi surface withspanning vectors corresponding to maxima in the wavevector-dependent sus-ceptibility. The susceptibility of Sr3Ru2O7 has not yet been calculated. Thetheoretical Fermi surface [23,22] shows that the coupling between the two halvesof a bilayer splits each of the three sheets observed in Sr2RuO4. This, and therotation of the octahedra cause hybridisation between the bands.

It appears from the calculations [23], that compared with Sr2RuO4, much ofthe nesting at the Fermi level is removed, except between parts of the α sheets(Ru dxz and dyz orbitals). The calculated sheets have nesting vectors along the(tetragonal) 1, 0, 0 directions with spanning vectors close to our experimentalfindings. Differences of our results from those on single-layer Sr2RuO4 may arisefrom the effects of bilayers and octahedral rotation on the Fermi surface in oursystem.

Like Sr2RuO4, Sr3Ru2O7 is structurally and electronically a highly 2D sys-tem. Of great interest is the extent of magnetic correlations along the c-axis.

b*

(1, 0, 0)

a*δε

Fig. 6. A schematic indication of where in reciprocal space peaks are observed in theinelastic neutron scattering from Sr3Ru2O7 at 1.5 K at an energy transfer of 2 meV [23]

300 L. Capogna et al.

If the magnetic fluctuations are also of 2D character, scattering around qδ =(0.25, 0, 0) should extend as a rod along q = (0.25, 0, l). Any variation of the in-tensity at small l due to the ruthenium form factor is expected to be negligible,but phase relationships should have a big effect. Because of the bilayer structure,correlations between layers can have an intra- or inter-bilayer character. Let usconsider first the case of correlation between fluctuations on one bilayer and thenext bilayer at the body centre of the tetragonal cell. If these two bilayers fluc-tuate in antiphase, the intensity should vary as I(q, 0, ) ∝ cos2(π/2) and if inphase as sin2(π/2). Neither of these predictions is in agreement with our exper-imental observation; hence the scattering must be an incoherent superpositionof fluctuations on different bilayers. Let us consider now the phase relationshipswithin a bilayer. If the two Ru layers were fluctuating in phase, then the in-tensity of the signal would vary as I(q, 0, ) ∝ cos2(2πz) where 2z = 0.194c isthe distance between the RuO2 planes in a bilayer [14], and if in antiphase assin2(2πz). Measurements along c of the intensity of the signal at qδ [20] arewell-represented by I ∝ f(Q)2 cos2(2πlz/c), where f(Q) is the Ru form fac-tor. The fundamental fluctuating unit in Sr3Ru2O7 in the low energy range istherefore given by the bilayer with no correlations between bilayers so that thefluctuations are effectively two-dimensional.

Finally, the temperature dependence of the fluctuations suggests a crossoverin the nature of the low-energy magnetic correlations in this material. As thetemperature is increased above ∼ 17 K, the intensity of the incommensuratepeaks falls off, and is replaced by a broad peak of similar intensity around the2D reciprocal lattice point (1, 0, 0) [20]. This position is not a Bragg peak ofeither the tetragonal or the orthorhombic cell, so does not give rise to a lowenergy acoustic phonon. Hence the peak at (1, 0, 0) is most likely of magneticorigin.

The picture of a crossover is largely supported by measurements of thestatic suceptibility and longitudinal magnetoresistance as a function of tem-perature [18,24]. A strong temperature dependance of the electronic propertiesand magnetic excitations is also observed in High Tc superconductors [25] andheavy fermion systems [26]. Thus the behaviour of Sr3Ru2O7 may ultimately berelated to its proximity to a quantum critical point [27].

5 Conclusions

In summary, we have reviewed details of the metamagnetic transition observed inthe itinerant ruthenate Sr3Ru2O7 by neutron diffraction. The induced momenton the Ru ions at low temperature amounts to m = 0.25(3) µB per Ru. Thespectrum of the spin fluctuations investigated by inelastic neutron scatteringreveals that the excitations in this system are localised at spots in the reciprocalspace with incommensurate wavevectors corresponding to nesting vectors of thecalculated Fermi surface. The small characteristic energy of these fluctuationsand the fact that they are affected by the application of a magnetic field suggests

Metamagnetic Transition and Low-Energy Spin Density Fluctuations 301

that they are implicated in and related to the metamagnetic transition observedat low temperatures.

Acknowledgments

The neutron scattering experiments have been carried out at the Institute Laue-Langevin in Grenoble and supported by the Science and Engineering ResearchCouncil of the United Kingdom. We are grateful to Jane Brown for her calcula-tion of the oxygen magnetic form factor and to the ILL techniqual staff for theirvaluable support.

Appendix

The observed intensity associated with the magnetic signal is given by the mag-netic scattering cross section convolved with the resolution function:

Imagn = α

∫R(dQ, dω)

ki

kf

d2σ

dΩdEdQdE . (9)

The resolution function is related to the transmission of the monochromator andanalyser crystals and needs to be calculated for each instrument configuration.The absolute calibration [28] of the dynamical susceptibility can be performedusing the cross section for acoustic phonon scattering. Within a damped har-monic oscillator model the scattering function for a single phonon mode in asolid is given by

Sph(Q, ω) =ki

kf

d2σ

dΩdE∝ 4π

(n(ω) + 1)Z(Q) ωγ(ω2 − ω2

0)2 + 4ω2γ2 , (10)

where ω0 is the harmonic phonon frequency, γ is the damping frequency andZ(Q) is the one-phonon structure factor. The instrument factor α which isneeded in Eq. 9 to yield χ′′(Q, ω) in absolute units is therefore extracted fromthe experimental integrated intensity of the phonon scattering:

Iph = α

∫R(dQ, dω)Sph(Q, ω) dQdE . (11)

References

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(1984)

Decrease of Ferromagnetic TransitionTemperature in NonstoichiometricSrRu1−vO3 Perovskites

B. Dabrowski1, P.W. Klamut1, O. Chmaissem1, S. Kolesnik1, M. Maxwell1,J. Mais1, C.W. Kimball1, J.D. Jorgensen2, and S. Short2

1 Physics Department, Northern Illinois University,DeKalb, IL 60115, U.S.A.

2 Materials Science Division, Argonne National Laboratory,Argonne, IL 60439, U.S.A.

Abstract. We have discovered that synthesis at high-pressure oxygen of 600 atmat 1100 C produces nonstoichiometric SrRu1−vO3 perovskites with randomly dis-tributed vacancies on the Ru-sites. The increased amount of Ru vacancies rapidlysuppresses the ferromagnetic TC from 162 to 86 K for ∆v ∼ 0.06. We associate thedecrease of TC with an increased formal valence of Ru and the induced structuraldisorder.

1 Introduction

The relation of physical and structural properties to cation and oxygen non-stoichiometry in complex transition-metal oxides has been a subject of intenseresearch for several decades, heightened recently by the discovery of high-tem-perature superconductivity in cuprates and colossal magnetoresistivity in man-ganites [1,2]. Complex effects of non-stoichiometry were studied most extensivelyfor 3d transition metals that acquire less extended electronic orbitals than thecorresponding 4d or 5d ions. Among the 4d metals, ruthenates with the ABO3perovskite and related crystal structures have been found to display a fasci-nating mixture of magnetic (Sr1−xCaxRuO3, Sr3Ru2O7) and superconducting(Sr2RuO4, RuSr2GdCu2O8) properties that for several of them depend sensi-tively on the synthesis conditions and compositions [3,4,5,6].

SrRuO3 is known as a highly correlated, narrow-band metallic ferromagnetwith a robust TC ≈ 162 K [7]. The ferromagnetism of SrRuO3 arises from aparallel alignment of the low-spin electronic configuration of t4 electrons. Thiselectronic configuration of Ru4+ is different from the high-spin t3e1 configurationof Mn3+ due to a larger magnitude of the crystal-field splitting of the energylevels observed for 4d transition metals. The low-temperature ordered magneticmoment of SrRuO3, reported to be between 0.8µB/Ru and 1.6µB/Ru, showsa lack of saturation to the full S = 1 moment, 2 µB/Ru atom, in dc fields to30 T. This lack of saturation of the magnetic moment is consistent with itin-erant ferromagnetism [8]. To explain the varying measured magnetic moments,a difficulty of correctly measuring the saturation magnetization was invokedbased on the observed abnormally high magneto-crystalline and magneto-optic

C. Noce et al. (Eds.): LNP 603, pp. 303–311, 2002.c© Springer-Verlag Berlin Heidelberg 2002

304 B. Dabrowski et al.

anisotropy for this pseudocubic material [9]. Greater overlap and hybridizationbetween Ru 4d orbitals and O 2p orbitals are also expected to lead to greateritinerancy and more interplay between structural degrees of freedom and mag-netic and electronic properties [10]. In addition to the intriguing fundamentalproperties arising from proximity to magnetic, superconducting, and quantumcritical transitions observed for ruthenates [11,12], the SrRuO3 compound hasbeen intensively studied for possible applications as an electrode material inmicroelectronic circuits [13].

Moderately reduced Curie temperatures observed for thin films of SrRuO3deposited on substrates with mismatched lattice parameters have been explainedin terms of strain effects consistent with the observations of a decrease of TC

with applied hydrostatic pressure in bulk samples [14]. On the other hand, thefree standing thin films showed a TC comparable to bulk samples confirmingthat by removal of the strain imposed on thin film by the substrate the intrinsicTC is recovered. Reduced Curie temperatures were also observed for some of thesingle-crystals that were grown in aluminum crucibles [15] but these suppressedvalues of TC are most probably caused by unintentional impurity substitutions.Recently [15] it has been reported that the Curie temperature of SrRuO3 maydepend on the synthesis temperature in oxygen. They reported the existenceof two different phases, a phase with TC = 141 K formed below 1100 C and anormal TC = 160 K phase obtained for synthesis at higher temperatures. Bothphases had orthorhombic crystal symmetry with a marginal difference of theirlattice parameters.

Several isovalent and heterovalent substitutions have been studied for SrRuO3.Ca substitution for Sr while retaining 4+ formal oxidation state of Ru suppressesTC to 0 K [16]. An increased orthorhombic distortion and a larger deviation ofthe Ru–O–Ru bond angle from 180 were invoked to explain this behavior [17].Heterovalent substitution of 12 % of Na for Sr, that increases Ru formal valenceto 4.12+, was found to rapidly suppress TC to 70 K. More complex substitutionson the Ru site have also been found to rapidly decrease TC by several inter-connected effects like valence change, magnetic impurity ion scattering, chargelocalization, and increase of structural distortions and disorder [18,19]. The re-sults of these substitution studies can be summarized as follow: The dominantfactor controlling the decrease of TC is a change of the formal valence of Ru.For substitutions retaining the formal Ru4+ oxidation state, the TC decreasesas a function of the decreasing Ru–O–Ru bond angle. In this paper we reporton the decrease of TC in nonstoichiometric SrRu1−vO3 perovskites caused byan effective increase of the Ru formal valence. The nonstoichiometric sampleshave been synthesized for the first time at high-pressure oxygen conditions thatfavor formation of the Ru5+ ions. Detailed neutron powder diffraction measure-ments indicate a random distribution of vacancies on the Ru crystallographicsite, similar to the LaMn1−vO3 perovskites.

Decrease of Ferromagnetic Transition Temperature 305

2 Sample Preparation and Experimental Details

Polycrystalline SrRuO3 samples were synthesized from mixtures of SrCO3 andstoichiometric RuO2. Samples were processed using the solid state reactionmethod and fired in air several times at various temperatures up to 1100 Cfollowed by slow cooling to 400 C. Using these conditions, three single-phase,∼ 2 gram-size batches of SrRuO3 (n. 1–3) were obtained. The batches were di-vided into smaller parts and processed under various oxygen pressure conditions:flowing oxygen, air, and argon at temperatures 950 - 1150 C, and high-pressureoxygen of 600 atm (a total pressure of 3,000 atm of 20 % O2 in Ar). The high-pressure oxygen samples were annealed at 1050 C (Batch n. 1), 1080 (Batchn. 2), and 1120 C (Batch n. 3). The effective oxygen contents of samples an-nealed at various conditions were studied by thermogravimetric measurementson a Cahn TG171 thermobalance in argon with slow heating (1 - 2 deg/min.)rates. The relative changes of mass for the high-pressure oxygen and air syn-thesized samples were found to be quite large, implying changes of the oxygencontents of 0.09 and 0.03, respectively. These changes of oxygen contents indi-cate a large changes of the Ru formal valence of ∼ 0.24 between high-pressureoxygen and argon annealing for the sample from batch n. 3. Because thermo-gravimetric analysis (TGA) could not provide conclusive information about theexact nature of the cation/oxygen non-stoichiometry, we have performed a de-tailed neutron diffraction study for all three high-pressure oxygen synthesizedsamples and for selected samples obtained in argon, air, and oxygen. Magneti-zation and resistivity measurements were performed using a Quantum DesignPhysical Properties Measurement System-Model 6000 at temperatures between10 and 350 K. Time-of-flight neutron powder diffraction data were collected onthe Special Environment Powder Diffractometer (SEPD) at the Intense PulsedNeutron Source (IPNS). Diffraction data were acquired at room temperature us-ing the ten-position sample changer. High-resolution backscattering data, from0.5 to 4 A, were analyzed using the Rietveld method and the General StructureAnalysis System code (GSAS).

3 Magnetic and Resistive Properties

The temperature dependence of the ac susceptibility in zero-magnetic field ispresented in Fig. 1 for SrRuO3 samples annealed in argon and high-pressureoxygen (batch n. 3). For the argon annealed sample we observe a sharp increasein ac susceptibility on decreasing temperature at 162 K followed by a slow de-crease at lower temperatures. This behavior, in agreement with dc magnetiza-tion data (shown as inset in Fig. 1), indicates a uniform ferromagnetic transitionthat is characteristic of high-quality homogenous SrRuO3 samples. For the high-pressure oxygen annealed sample, the initial increase in ac susceptibility is lesssharp and appears at significantly lower temperatures. The transition is slightlybroader, again in agreement with dc magnetization data, indicating a wider dis-tribution of TC ’s. We define the Curie temperature TC as the temperature of

306 B. Dabrowski et al.

Fig. 1. ac susceptibility (main frame) and “Field-cooled” and “Zero-Field-Cooled” dcmagnetization (inset) vs. temperature for SrRuO3 synthesized in argon and at 600 atmO2. Both measurements were done at low magnetic fields

the maximum in the ac susceptibility (or the maximum slope in the dc magneti-zation) versus temperature curves. Both ac susceptibility and dc magnetizationmeasurements give the same values of TC = 86, 91, and 135 K for batches n. 3,2, and 1, respectively.

The temperature dependence of the resistivity in several dc fields is pre-sented in Fig. 2 for argon and high-pressure (batch n. 3) oxygen annealed sam-ples. Both samples show metallic behavior. However, higher resistivity for ourpolycrystalline samples than the single-crystal samples indicates a considerablecontribution of grain boundary resistance to the measured values. Low synthesistemperatures, used to prevent the loss of Ru due to volatilization, may be respon-sible for a weak connectivity between small-size grains. Despite this significantgrain boundary contribution to the resistivity, the ferromagnetic transitions canbe clearly observed at 162 and 87 K for the argon and high-pressure oxygenannealed samples, respectively. Thus, both the resistive and magnetic measure-ments give the same values of TC . In addition, we can observe a small negativemagnetoresistance present for both samples near TC and extending to lower tem-peratures. Existence of magnetoresistance over an extended temperature rangeindicates a magnetic field dependent scattering at the grain boundaries and ahigh spin polarization of the electron density at the Fermi energy.

4 Neutron Powder Diffraction

Neutron diffraction measurements were used to search for the structural andchemical roots of the reduced Curie temperatures in the high-pressure oxygen

Decrease of Ferromagnetic Transition Temperature 307

Fig. 2. Resistivity as a function of temperature for SrRuO3 synthesized in argon andat 600 atm O2 using several magnetic fields

synthesized samples. All measured samples were phase pure according to X-ray and neutron powder diffraction. The room temperature crystal structure ofSrRuO3 was refined in the orthorhombic space group Pbnm consistent with pre-vious reports [20]. Figure 3 shows neutron diffraction patterns for three samplesprepared in argon, air, and at high-pressure oxygen. The structural refinementsusing Pbnm structural models show good fits. Neither superstructure diffractionpeaks nor unusual peak broadening or atom displacements, that could indicatenovel cation vacancy or excess oxygen orderings, were detected for the high-pressure annealed samples.

The refined lattice parameters at room temperature are a = 5.5715 (1) and5.5739 (2), b = 5.5348 (1) and 5.5419 (1), and c = 7.8488 (2) and 7.857 (3) A forargon and high-pressure (batch n. 3) annealed samples, respectively. Lattice pa-rameters display only a slight dependence on synthesis conditions. The values oflattice parameters for samples synthesized in air and argon are virtually identi-cal to those reported previously by Jones et al. [20]. The high-pressure oxygensynthesized samples have slightly increased lattice parameters.

308 B. Dabrowski et al.

Fig. 3. Best-fit Rietveld refinement using neutron powder diffraction data at roomtemperature for SrRuO3 synthesized in argon, air, and at 600 atm O2

To differentiate among various possible origins of the suppressed TC ’s, thecation and oxygen site occupancies were compared for the high-pressure and at-mospheric pressure synthesized samples. Figure 4 shows occupancies of the Rusites (a) and O sites (b) versus TC for various samples. During the refinementthe Sr site occupancies were kept constant at 1. The data clearly reveal emer-gence of numerous vacancies on the Ru sites for samples with suppressed TC ’swhile the O sites remain fully occupied. The synthesis at high-pressure oxygenseems, thus, to introduce vacancies on the Ru sites while the A-site (Sr) and oxy-gen site occupancies remain unchanged. The maximum amount of Ru vacanciesintroduced by high pressure annealing can be estimated at ∆v ∼ 0.06.

As a result of the decreased amount of Ru in the compound, the Ru formal va-lence should increase. The magnitude of the introduced defects can be estimatedby considering a difference of the Ru site occupancy between samples annealedat atmospheric pressure and at high-pressure oxygen. Using the obtained magni-tudes of the vacant sites, the change of the Ru formal valence can be estimatedfrom the formula ∆(FV) (0.98 - 0.91) x 4 = 0.28, (0.96 - 0.91) x 4 = 0.20, and(0.975 - 0.96) x 4 = 0.06 for samples n. 3, n. 2, and n. 1, respectively. These largechanges of the Ru formal valence are consistent with the large changes of theoxygen content found from the TGA experiments. Using a model of the superex-change interaction it was shown that in rare earth orthoferrites, for which theFe oxidation state remains constant at 3+ (i.e., the electronic configuration isthe high-spin t3e2), the change in TN relates primarily to departures from 180

of the B-O-B interaction angle, and that TN and cos2 θ (where θ is the B-O-Bbond angle) are linearly related (B and O stand for the transition metal andthe oxygen atoms, respectively). Similar dependence of TN on the B-O-B inter-action angle was seen for RCrO3 (the t3 electronic configuration) [21]. We have

Decrease of Ferromagnetic Transition Temperature 309

Fig. 4. Site occupancies for Ru (a) and oxygen (b) versus ferromagnetic transitiontemperature for several for SrRuO3 synthesized in argon, air, and at 600 atm O2

observed a similar bond angle effect on TN in Sr1−x−yCaxBayMnO3 perovskiteswith 0 ≤ x ≤ 1 and y ≤ 0.2 for which the Mn formal valence remains constant at4+ (the t3 electronic configuration) [22]. Analogous dependence between the fer-romagnetic transition temperature TC and 〈cos2 θ〉 can be seen for orthorhombicruthenates for which the Ru formal valence remains constant at 4+ [23]. Figure 5shows this TC dependence on the measured 〈cos2 θ〉 for ARuO3 samples (A =Sr, Ca). In Fig. 5 is also shown the TC dependence on the measured 〈cos2 θ〉for our samples synthesized at high-pressure oxygen. The Ru–O–Ru bond angleincreases only slightly for these samples while TC decreases quite rapidly. Thisrelationship is clearly opposite to what is observed for the Ru, Fe, Cr, and Mnsamples for which the formal valences remain constant. Clearly, the decreaseof TC is not caused by the weakening of the superexchange interactions arisingfrom a departure of the B-O-B interaction angle from 180. The data presented

310 B. Dabrowski et al.

Fig. 5. Ferromagnetic transition temperatures, TC , vs. the measured 〈cos2 θ〉 forCaRuO3 and SrRuO3 samples synthesized in air [23] and for SrRuO3 synthesized at600 atm O2

here indicates that the main causes of the decreased TC are formation of vacantsites on Ru network, that increase Ru formal valence, and, possibly, enhanceddisorder that is induced by these defects [24].

5 Conclusions

The high oxygen partial pressure conditions used during synthesis at ∼ 1100 Cappear to induce vacancies on the Ru sites that considerably increase the Ruformal valence state in SrRu1−vO3. Samples with the largest amount of vacanciesand the highest Ru formal valence have the lowest observed TC ’s. At the high-pressure oxygen synthesis conditions the easiest way to increase the Ru formalvalence would be to increase the oxygen content of the material. However, sincethe oxygen network is already fully occupied this process is not possible for theperovskite structure, and the other way to increase the Ru formal valence is togenerate vacancies on the cation sites. This process is similar to an increase of theMn formal valence by formation of randomly distributed Mn vacancies for therare earth manganites . The systematic decrease of TC with increased amount ofthe Ru vacancies arises at increased synthesis temperatures during high-pressureoxygen annealing. The extraction of the Ru atoms from the normal sites shouldlead to the formation of the Ru-containing second phases. We have not seensuch phases in the high-resolution neutron diffraction experiments indicatingthat their domain size is very small or that they may be distributed at the verythin grain boundary region. Annealing experiments of the high-pressure oxygensynthesized material in argon have shown that the 160 K ferromagnetic phaseis recovered, i.e., the Ru atoms remain in the sample and can fill up the vacant

Decrease of Ferromagnetic Transition Temperature 311

sites. Electron microscopy experiments are currently underway to unequivocallyfind the Ru-containing second phases and to confirm an increase of the Ru formalvalence for the high-pressure oxygen synthesized samples.

Acknowledgements

Work at NIU was supported by the NSF grant n. 0105398 and by the State ofIllinois under HECA. At ANL work was supported by the U.S. Department ofEnergy - Office of Science under contract No. W-31-109-ENG-38.

References

1. J. Orenstein, A.J. Millis: Science 288, 468 (2000)2. Y. Tokura, N. Nagaosa: Science 288, 462 (2000)3. G. Cao, S. McCall, M. Shepard, J.E. Crow, R.P. Guertin: Phys. Rev. B 56, 321

(1997)4. R.S. Perry, L.M. Galvin, S.A. Grigera, L. Capogna, A.J. Schofield, A.P. Mackenzie,

M. Chiao, S.R. Julian, S.I. Ikeda, S. Nakatsuji, Y. Maeno: Phys. Rev. Lett. 86,2661 (2001)

5. C. Bernhard, J.L. Tallon, Ch. Niedermayer, Th. Blasius, A. Golnik, E. Brucher,R.K. Kremer, D.R. Noakes, C.E. Stronach, E.J. Ansaldo: Phys. Rev. B 59, 14099(1999)

6. A.P. Mackenzie, R.K. Haselwimmer, A.W. Tyler, G.G. Lonzarich, Y. Mori,S. Nishizaki, Y. Maeno: Phys. Rev. Lett. 80, 161 (1998)

7. I.I. Mazin, D.J. Singh: Phys. Rev. B 56, 2556 (1997)8. G. Cao, S. McCall, M. Shepard, J.E. Crow, R.P. Guertin: Phys. Rev. B 56, 321

(1997)9. A. Kanbayashi: J. Phys. Soc. Jap. 44, 108 (1978)

10. I.I. Mazin, D.J. Singh: Phys. Rev. B 56, 2556 (1997)11. A.P. Mackenzie, Y. Maeno: Physica (Amsterdam) B 280, 148 (2000)12. I.I. Mazin, D.J. Singh: Phys. Rev. B 56, 2556 (1997)13. C.B. Eom, R.J. Cava, R.M. Fleming, J.M. Philips, R.B. van Dover, J.H. Marshall,

J.W.P. Hsu, J.J. Krajewski, W.F. Peck Jr.: Science 258, 1766 (1992)14. J.J. Neumeier, A.L. Cornelius, J.S. Schilling: Physica B 198, 324 (1994)15. P.A. Joy, S.K. Date, P.S.A. Kumar: Phys. Rev. B 56, 2324 (1997)16. G. Cao, S. McCall, M. Shepard, J.E. Crow, R.P. Guertin: Phys. Rev. B 56, 321

(1997)17. I.I. Mazin, D.J. Singh: Phys. Rev. B 56, 2556 (1997)18. G. Cao, F. Freibert, J.E. Crow: J. Appl. Phys. 81, 3884 (1997)19. T. He, R.J. Cava: Phys. Rev. B 63, 172403 (2001)20. C.W. Jones, P.D. Battle, P. Lightfoot, W.T.A. Harrison: Acta Cryst. C 45, 365

(1989)21. J.B. Goodenough, J.M. Longo: Landolt-Bornstein vol. 4a, (Springer Verlag, Berlin

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B. Pyles, J.D. Jorgensen: Phys. Rev. B 64, 134412 (2001)23. H. Kobayashi, M. Nagata, R. Kanno, Y. Kawamoto: Mater. Res. Bull. 29, 1271

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huri, A.J. Schofield, C.S. Alexander, G. Cao, S.R. Julian, Y. Maeno: Phys. Rev.Lett. 88, 076602 (2002)

Strain Effects in SrRuO3 Thin Filmsand Heterostructures

G. Balestrino, P.G. Medaglia, P. Orgiani, and A. Tebano

I.N.F.M. - Dipartimento di Scienze e Tecnologie Fisiche ed Energetiche,Universita di Roma “Tor Vergata”, Via di Tor Vergata 110, 00133 Roma, Italy

Abstract. In the present work we report structural and transport properties of epi-taxial SrRuO3 films grown by Laser Molecular Beam Epitaxy (Laser MBE) equippedwith in-situ Reflection High Energy Electron Diffraction (RHEED) analysis at verylow oxygen pressure (less than 10−4 mbar). Several works have given first evidence ofa strong correlation between transport and magnetic properties and structural prop-erties of SrRuO3 (such as the influence of the strain, substrate–film interface, etc.).We worked on the determination of the conductive and magnetic properties of SrRuO3

films as a function of the strain. Different strains were induced varying the SrRuO3

films thickness from ultrathin (few unit cells) to thick (thousands of Angstrom) struc-tures. Superlattices based on SrRuO3 and SrTiO3 films were also grown. In such astructure, the largest value of strain was achieved and the transport properties confirmour simple SrRuO3 conducting model.

1 Introduction

In the last few years several research groups have been able to use ReflectionHigh Energy Electron Diffraction (RHEED) diagnostic technique [1], in-situ, incombination with Pulsed Laser Deposition (PLD), i.e. the Laser MBE technique,to obtain the two-dimensional (2D) growth of artificial materials otherwise dif-ficult or impossible to synthesize. The advantage of the Laser MBE techniqueis due to the possibility to monitor the RHEED intensity oscillations duringthe growth process, enabling very precise control of the layer by layer epitaxialgrowth (phase-locked growth), thus reducing as much as possible occasional fluc-tuations in the thickness of each constituent layer. Presently, the Pulsed LaserDeposition technique is used to grow a wide variety of thin oxide films, most ofthem with a complex structure: high-Tc superconductors, ferroelectrics, piezo-electrics, colossal magnetoresistance oxides, electro-optics materials. With thepossibilities offered by the utilization, in-situ, of the RHEED diagnostic, it ispossible to predict the realization of functional materials of very good quality,with very interesting applications in several fields.

The family of perovskite oxides displays a broad range of technologicallyimportant phenomena such as high-temperature superconductivity, ferroelec-tricity, ferromagnetism, colossal magnetoresistance and metallic conductivity.The structural and chemical similarities of these materials make possible epi-taxial heterostructures opening new perspectives in the field of electronic, mag-netic and optical devices. Because of its electrical and magnetic properties [2,3]

C. Noce et al. (Eds.): LNP 603, pp. 312–324, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Strain Effects in SrRuO3 313

and its structural compatibility with high-temperature superconductors such asYBa2Cu3O7−δ [4], the conductive magnetic oxide SrRuO3 represents one of themost interesting materials not only for its fundamental physical property of be-ing a ferromagnetic material [5] but also for applications. Able to furnish spinpolarized electrons, SrRuO3 epitaxial films have been found useful for junctions,electrodes and capacitors in the field of integrated “spintronics” [6].

We report our results on the hetero-epitaxial growth by laser MBE and onthe RHEED intensity oscillations of the SrRuO3 thin films and SrRuO3 /SrTiO3artificial structures. The structure of this superlattice is composed by stacking,along the c axis, the SrRuO3 compound and the SrTiO3 compound. In this re-spect, the laser MBE technique could allow a better control of the interface dis-order. We report structural and transport properties of epitaxial SrRuO3 filmsgrown by Pulsed Laser Deposition (PLD) technique at high oxygen pressure(about 1 mbar) and by Laser Molecular Beam Epitaxy (Laser MBE) equippedwith in-situ Reflection High Energy Electron Diffraction (RHEED) analysis atvery low oxygen pressure (less than 10−4 mbar). Several works have given firstevidence of a strong correlation between transport and magnetic properties andstructural properties of SrRuO3 (such as the influence of the strain, substrate–film interface, etc.). Nevertheless, the precise determination of the relationsamong structural properties and transport properties is strongly dependent onthe capability to control the sample growth. The crystallographic quality of thesematerials is a crucial aspect both for understanding physical properties and forpractical applications. Laser MBE is able to ensure the required control on thefilms growth. Here we present our results about the magnetic and transportproperties of SrRuO3 epitaxial films. The properties of thick films (thicker than1000 A) are similar to those reported in literature. A ferromagnetic transition ata Curie temperature of about 150 K and magneto-resistance hysteresis loop atlow temperature are reported. Moreover we worked on the determination of theconductive and magnetic properties of SrRuO3 films as a function of the strain.Different strains were induced varying the SrRuO3 films thickness from ultrathin(few unit cells) to thick (thousands of Angstrom) structures. Finally superlat-tices based on SrRuO3 and SrTiO3 films were also grown by Laser MBE. In sucha structure, the largest values of the strain were achieved (as shown later).

2 Experimental

The SrRuO3 crystal structure was found to be orthorhombic with lattice con-stants ao = 5.56 A, bo = 5.53 A and co = 7.84 A. It has a subunit cell that isa pseudocubic perovskite with lattice constants ap ∼= bp ∼= cp = 3.93 A for bulkmaterial [2] and slightly larger, i.e. 3.94 A, for epitaxial films [7]. For the sakeof simplicity in this paper, the Miller indices of SrRuO3 will be referred to theperovskite cell (indicated by the superscript p). SrRuO3 shows a ferromagnetictransition at a Curie temperature of ∼ 150 ÷ 160 K [2]. Even if the origin of theferromagnetic ordering is not completely cleared up [5,8,9], several works havegiven preliminary evidence of a strong correlation between transport, magnetic

314 G. Balestrino et al.

0 50 100 150

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Fig. 1. RHEED intensity oscillations of the specular spot during the SrTiO3 homoepi-taxial film growth as a function of laser pulses

and structural properties of SrRuO3 [10,11,12,13,14,15]. The depositions wereperformed using an excimer laser charged with KrF, generating 248 nm wave-length pulses of 25 ns width with 1 Hz repetition rate. The laser beam, withan energy of 60 mJ per pulse, was focused in a high vacuum chamber onto thecomputer controlled multi-target rotating system. Substrates used for depositionwere nominally zero miscut (100) SrTiO3 single crystal (with lattice parametersa = 3.904 A), placed at a distance of about 70 mm from the target on a heatedholder. Before growth, substrates were chemically etched in a buffered solution ofNH4F-HF (pH = 4.6) for 8 minutes [16]. This etching treatment leaves a termi-nating substrate layer of TiO2 and decreases the surface roughness. Few mono-layers of SrTiO3 were deposited at the beginning, to have an optimal 2D startingdeposition surface. The incident RHEED electron beam was parallel to the [100]substrate direction. The homoepitaxial SrTiO3 deposition was performed undera molecular oxygen pressure of 10−5 mbar and with the substrate temperatureat 530 C. Usually four monolayers of SrTiO3 were deposited monitoring fourRHEED intensity oscillations of the specular spot (Fig. 1). The SrTiO3 deposi-tion was stopped when the specular spot intensity reached the fourth maximum,after about 200 laser pulses. SrRuO3 films and SrRuO3/SrTiO3 heterostruc-tures were grown starting from a stoichiometric target, prepared by solid-statereactions. For all samples, structural characterization was carried out by X-raydiffraction (XRD) using Cu Kα radiation. Resistance measurements were per-formed by a standard four-probe DC technique.

2.1 SrRuO3 Thin Films

For SrRuO3 thin films, the growth temperature ranged from 450 C to 570 C.Recent powder XRD experiments show a transition from an orthorhombic phase,at room temperature, to a tetragonal phase at higher temperature [17,18]. In or-

Strain Effects in SrRuO3 315

0 50 100 150 200 250 300

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Fig. 2. Left panel: RHEED intensity oscillations of the specular spot during theSrRuO3 film growth as a function of laser pulses. Right panel: Typical 2D final RHEEDpattern of the SrRuO3 surface

der to analyze the effect of such a structural transition on the film strain, severalsamples were grown varying the deposition temperature. For our growth con-dition, we do not detect any sizeable effect of growth temperatures on the filmstrain. SrRuO3 films were grown at different oxygen pressures (from 10−4 to100 mbar). The magnetoresistance and electrical resistivity properties resultedto be independent from oxygen pressure. This feature made possible the SrRuO3films growth by Laser MBE (namely at an oxygen pressure 10−4 mbar) usingthe RHEED control. The deposition process was monitored by in-situ RHEEDanalysis. The direct calibration of the deposition rate through the intensity os-cillation of the specular diffraction spot in the RHEED pattern guaranteed theprecise control of the film thickness. RHEED oscillations of a typical SrRuO3film, detected at the first deposition stage, are shown in Fig. 2a. In accordancewith the well accepted interpretation, a complete oscillation in the intensity ofthe RHEED pattern corresponds to the growth of one unit cell. There is a size-able difference between the first oscillation width and the following ones. Asshown in a recent work, this difference is probably due to a change in the mobil-ity of the adatoms, switching the surface termination layer from the substrate tothe film after the first SrRuO3 layer [19]. Nevertheless, starting from the secondoscillation in the RHEED pattern, the number of the laser shots to completeone oscillation resulted to be constant. The possibility to monitor the depositionprocess by RHEED analysis, combined with the very low deposition rate (about35÷40 laser shots per unit cell starting from the second oscillation), allows aprecise control of the film thickness. Moreover, the RHEED pattern maintainsa 2D nature for the whole deposition process. In Fig. 2b the final 2D pattern isshown for a SrRuO3 film about 1000 A thick. SrRuO3 epitaxial films were alsogrown on (100) LaAlO3 and (110) NdGaO3 substrates (with in-plane lattice pa-rameter of 3.79 A and 3.85 A, respectively). Due to the larger mismatch with the

316 G. Balestrino et al.

SrRuO3 lattice parameter, the epitaxial strain is already relaxed in the first fewlayers. As a consequence the growth on such substrates was not pure 2D and nooscillations in RHEED pattern were detected.

2.2 SrRuO3 / SrTiO3 Heterostructures

To grow the SrRuO3/SrTiO3 heterostructures, the deposition temperature waskept at about 500 C and the molecular oxygen pressure at 10−4 mbar. All thedeposition were started with the SrRuO3 layer. In these conditions, the growthrates of SrRuO3 and SrTiO3 were calibrated, monitoring the RHEED intensityoscillations of the specular spot. The layer by layer deposition of SrRuO3/SrTiO3superlattices was carried out by stacking in sequence m SrRuO3 unit layers and nSrTiO3 unit layers (the so called m×n superstructure, [(SrRuO3)m/(SrTiO3)n]S ,where S represents the total number of deposition cycles). The bidimension-ality of the growth process is proved by the RHEED pattern which, at theend of the growth, shows typical 2D features. Monitoring the RHEED inten-sity oscillations, the laser pulses on the two targets were adjusted to obtain the(SrRuO3)4/(SrTiO3)4 superlattice. As reported in Fig. 3, the RHEED inten-sity monitoring makes possible the precise control of the supercell during thegrowth. A typical optimization process is reported as an example: in the firstpattern the number of laser shots results to be smaller than those required tocomplete four SrRuO3 cells (a in Fig. 3), while in the second one it exceeds sucha value (b in Fig. 3). Acting on the number of laser shots during the growth,the desired superstructure was obtained (as shown in c in Fig. 3). In Fig. 4 theRHEED intensity oscillations of the specular streak are shown during one of theoverall 20 cycles of the (SrRuO3)2/(SrTiO3)2 and (SrRuO3)4/(SrTiO3)4 super-lattice deposition, respectively. The time evolution of the RHEED intensity ina single cycle remains unchanged throughout the superlattice growth. Its majorfeature is the sizeable variation of intensity when the growth is switched fromthe SrRuO3 oxide to the SrTiO3 oxide. This effect can possibly be ascribed tothe electron scattering factors of atoms in the two layers, which differ in theirabsolute magnitude, in the phase shift upon scattering and phase shift due tothe height difference. Such RHEED intensity oscillations were observed duringall of the cycles. From the period of the RHEED intensity oscillations, a growthrate of 50 laser pulses per unit SrRuO3 layer and 50 laser pulses per unit SrTiO3layer was estimated.

3 Structural Characterization

Standard XRD characterization was performed for all samples. θ − 2θ XRDdiffraction patterns show only the (00l)p peaks, indicating that films are [001]p

oriented. A typical XRD spectrum of a SrRuO3 thick film (1000 A) is reported inFig. 5a. To check the crystalline quality of the films, rocking curve measurementswere also performed (inset of Fig. 5a). For thick films typical full width at halfmaximum (FWHM) of the rocking curve for the (002)p reflection is about 0.1 .

Strain Effects in SrRuO3 317

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a

Fig. 3. Using of RHEED diagnostic analysis during the (SrTiO3)4 /(SrRuO3)4 super-lattice: number of laser shots smaller (a) and larger (b) than those required to realizethe 4x4 superlattice; c) optimized number of laser shots to realize the desired structure

The cp lattice parameter of films having thickness larger than 1000 A results tobe 3.96 A (as reported in literature, the cp lattice parameter of epitaxial filmsresults to be slightly larger than the 3.93 A bulk value). On the contrary, thinnerfilms are heavily strained. As a consequence of the in-plane compressive epitaxialstrain, a sizeable increase of the cp lattice parameter occurs. θ−2θ XRD spectrafor three SrRuO3 samples with different thickness (1000 A, 400 A and 80 A re-spectively) are reported in Fig. 5. Thinner films are more and more strained and,as a consequence, a shift of the (00l)p SrRuO3 diffraction peaks toward lowerangles is detected. The cp lattice parameter values, calculated for 1000 A, 400 Aand 80 A thick films, result to be 3.96 A, 4.01 A and 4.06 A (±0.01), respectively.The in-plane ap lattice parameter was also measured. Using asymmetric (303)p

reflection, ap was found to be 3.93 A and 3.91 A (±0.02) for 1000 A and 400 A

318 G. Balestrino et al.

0 20 40 60 80 100 120 140

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.)

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Fig. 4. Typical RHEED pattern during one of the sequence to realize a(SrTiO3)2/(SrRuO3)2 and a (SrTiO3)4/(SrRuO3)4 superlattices, respectively

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**

*

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01)

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Fig. 5. θ−2θ XRD spectra of three SrRuO3 films grown on (001) SrTiO3 substrate(*);the film thickness are 1000A (top), 400A (center) and 80A (bottom) respectively. Inthe inset of Fig. 2a typical XRD ω-scan of the (002)p SrRuO3 reflection

thick films, respectively. Moreover, φ-scan measurements (Fig. 6) show that theSrRuO3 films grow with the in-plane ap axis perfectly aligned with the SrTiO3substrate ones. The above values confirm that the relaxed film lattice parametersare slightly larger than the bulk ones.

Strain Effects in SrRuO3 319

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Fig. 6. Asymmetric (303)p reflection φ-scan measurement of 400A thick SrRuO3 films

19 20 21 22 23 24 25 26

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Fig. 7. θ − 2θ XRD spectrum of (SrTiO3)4/(SrRuO3)4 films grown on (001) SrTiO3

substrate(*)

In Fig. 7 the θ − 2θ X-ray diffraction spectrum of the (SrRuO3)4/(SrTiO3)4superlattice is shown. From the angular distance between the first order satellitepeaks (SL−1 and SL+1) it is possible to obtain the period Λ of the superlattice,Λ = λ/(sin θ+1 − sin θ−1), where θ+1 and θ−1 represent the angular positions ofthe first-order satellite peaks and λ represents the X-ray wavelength. The aver-age lattice parameter c can be estimated from the angular positions of the zerothorder SL0 (00l) peak, θ0, c= lλ/2 sin θ0. The total number of layers composingthe superlattice, N = m+n, can then be calculated, N = Λ/ c . From this simpleanalysis of the spectrum, it was calculated that Λ = 31.35 A and N = 8, con-firming the deduction by RHEED pattern analysis. Moreover, from the formula

320 G. Balestrino et al.

Fig. 8. Resistance vs. temperature for 1000A (a), 400A (b) and 80A (c) thick SrRuO3

films

Λ = n1λ1 + n2λ2 (where n and λ are the number of unit cells and the latticeparameter, respectively) it is possible to calculate the SrRuO3 lattice parame-ter. Assuming the SrTiO3 lattice parameter λ1 = 3.904 A, the SrRuO3 latticeparameter was found to be λ1 = 4.06 ± 0.01 A. This simple calculation confirmsthe hypothesis of a very large strain in (SrRuO3)m/(SrTiO3)n superlattices.

4 Electrical Measurements

Electrical transport measurements were carried out by standard four-probe dctechnique, with a bias pulsed and reversed current density of about 100 A/cm2.The thickest sample (a in Fig. 8) shows electrical resistance and magnetore-sistance properties very similar to those reported in literature for thick films.Namely, for films thicker than 1000 A, the resistivity behaviour is metallic for thewhole temperature range with a ferromagnetic transition at about 150 K. Valuesof the Residual Resistance Ratio (RRR, defined as the ratio between the resis-tance measured at 300 K and the value calculated from a linear extrapolation,made at high temperatures, to 0 K) are similar to those reported in literature.However, there is a striking difference between the RR-value (defined as the ratiobetween the measured values of the resistance at 300 K and at 0 K) in our bestfilms and higher values reported elsewhere in literature [10]. Preliminary mea-surements on SrRuO3 films grown on (0001) Al2O3 and (111) Si substrates showhigher RR-values compared to those grown on (100) SrTiO3 substrates (unpub-lished). These data seem to confirm that the origin of such RR-values is not re-lated to crystallographic quality of the samples. Furthermore, derivative analysisconfirms the T 2 behaviour at low temperature (Fig. 9a) [2]. Magnetoresistancemeasurements were performed with the applied magnetic field parallel to theelectric current, namely in the Longitudinal Magneto Resistance configuration

Strain Effects in SrRuO3 321

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T(O

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)

T (K) 80 100 120 140 160 180

-3

-2

-1

0

R[H

=1.

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-R

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OT]

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Fig. 9. Left panel: 1000A thick SrRuO3 film resistance derivative vs temperature. Rightpanel: difference between zero-field resistance and in-field resistance measurements asa function of temperature

(LMR). A zero-field cooled resistance measurement and a magneto-resistancemeasurement in a field of 1.2 T were carried out as a function of the tempera-ture. Measurements in a magnetic field were performed cooling the sample to20 K in zero-field and then applying the external magnetic field. The differencebetween zero-field resistance and in-field resistance measurements in the vicinityof the Curie temperature is reported in Fig. 9b. As expected, in the paramagneticstate (T > TC) no difference between the two values is detected. As the externalmagnetic field increases, the resistivity decreases. Decreasing the thickness, theferromagnetic transition takes place at lower and lower temperatures. The fer-romagnetic ordering disappears for a thickness smaller than few hundreds of A.In particular, the resistance of a 400 A thick SrRuO3 film is reported in Fig. 8(curve b). Even if the behaviour is metallic in the whole temperature range, noferromagnetic transition occurs. A further decrease in the sample thickness re-sults in a metallic behaviour at relatively high temperatures and a semiconduct-ing behaviour at low temperatures appears. In Fig. 8 (curve c), the behaviourof resistance versus temperature is reported for a 80 A thick SrRuO3 film. Alocalization transition seems to take place at about T ∼ 30 K. Transport mea-surement on (SrRuO3)m/(SrTiO3)n superlattices were performed too. In such astructure the largest strains can be achieved. The periodical insertion of SrTiO3blocks in the structure does not permit the progressive relaxation of the SrRuO3cell. According to the SrRuO3 conducting model, the (SrRuO3)m/(SrTiO3)n su-perlattices (for instance, we report 4 × 4 superlattice resistance measurementsin Fig. 10) shows a strong semiconducting behaviour in the whole temperaturerange.

5 Discussion

In the distorted SrRuO3 perovskite, the RuO2 planes are corrugated, in the sensethat the O atoms are slightly displaced out of the planes of the Ru atoms. A mi-crostructural parameter characterizing such distortion is the angle between the

322 G. Balestrino et al.

0 50 100 150 200 250 300240

280

320

360

400

440

480

Res

ista

nce

(Ohm

)

T (K)

Fig. 10. Resistance vs. temperature for (SrTiO3)4/(SrRuO3)4 superlattice

Ru-O bond and the Ru plane (buckling angle). Studies on ruthenate compoundssuggest that the Ru-O-Ru structural arrangement is crucial for the magneticproperties of this material [20,21,22,23]. A mechanism influencing the Ru-O-Ruarrangement in SrRuO3 films is the epitaxial strain induced by the substrates. Inthe case of the SrTiO3 substrate, due to the relatively small mismatch betweenfilm and substrate lattice parameters (less than 1%), it is possible to obtain co-herently strained thin SrRuO3 films. However, above a certain critical thickness,the epitaxial strain is progressively relaxed until, for thick films, the SrRuO3 lat-tice parameter approaches the bulk value. The buckling angle for SrRuO3 relaxedfilms was found to be about 6.6 [24]. The in-plane strain (εab) is defined as thedifference between the bulk and the strained film lattice parameters, normalizedto the bulk value. Because of the Poisson effect, the compression of the SrRuO3cell in the Ru plane (εab > 0) produces an increase in the lattice parameter alongthe cp axis direction. Reasonably, both the interatomic distance between the al-kaline earths and the oxygen ions in the Ru planes and the distance betweenRu and the octahedrally coordinated oxygens remain almost constant. This fea-ture results in an increase of the Ru-O buckling angle. All the data presentedin this paper support the existing theories about the magnetic and conductingproperties of SrRuO3. As reported elsewhere, the feature that SrRuO3 is both agood metal and a ferromagnetic material suggests that ferromagnetic orderingis due to Ru4d band electrons. The width of the Ru4d conduction band stronglydepends on the superimposition of the oxygen O2pσ and O2pπ orbitals and Ru4d

orbitals. In a relaxed structure (where the buckling angle has a minimum value)this superposition is maximized. As a consequence, the conduction band widthalso has a maximum value and seems to be connected with the onset of theferromagnetic ordering. Decreasing the film thickness, the strain simultaneouslyinduces an increase along the cp axis direction and a compression in the apbp

plane. This feature results in an increase of the Ru-O buckling angle which, inturn, reduces the superposition of the ruthenium and the oxygen orbitals, de-

Strain Effects in SrRuO3 323

creasing the conduction band width. A semiconducting behaviour arises whenthe conduction band width becomes smaller than the localization energy.

6 Conclusions

In conclusion, the transport properties of SrRuO3 and (SrRuO3)m/(SrTiO3)n

superlattices were studied. Epitaxial strain in the SrRuO3 films was varied chang-ing sample thickness from a few unit cells to hundreds of unit cells (∼1000 A).Due to its small mismatch with the SrTiO3 lattice parameter, SrRuO3 epitaxialfilms were grown. All depositions processes were monitored by in-situ RHEEDpattern oscillations analysis. All samples exhibit very high crystallographic qual-ity showing a pure 2D growth mechanism. Moreover, the RHEED pattern oscil-lations guaranteed a very precise control on the samples thickness. As expected,the conducting and magnetic properties of SrRuO3 films are strongly dependenton the epitaxial strain.

Relaxed films show metallic behaviour in the whole temperature range anda ferromagnetic ordering at about 150 K. As the thickness decreases (and con-sequently the strain increases), magnetic ordering does not occur even for filmswhich are still metallic in the whole temperature range. Thin SrRuO3 films(only a few unit cells thick) loose their metallic behaviour at low temperatureshowing a semiconducting behaviour below 30 K. Transport measurements on(SrRuO3)m/(SrTiO3)n superlattices were performed too. Assuming the super-lattice as the larger strained structure, the semiconducting behaviour confirmsour simple SrRuO3 conducting model.

The strong relationship between transport properties and magnetic proper-ties confirms the itinerant nature of ferromagnetism in SrRuO3 films.

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(1994)

Subject Index

µSR– experiments, 7– Ru1−xSr2GdCu2+xO8−d, 179

A phase, 53– BCS gap equation, 68– transition temperature, 69A1 phase, 55, 63A2 phase, 46

Band structure– Sr2RuO4, 249, 259– SrRuO3, 257Bogoliubov-de Gennes equation, 37, 48– spin singlet, 48– spin triplet, 48, 63Boltzmann theory, 97Buckling angle, 322– SrRuO3, 322

Canted antiferromagnet, 172Canting– Ru moments, 165Charge distribution– ruthenocuprates, 167Charge fluctuations, 99Chiral superconductivity– f -wave pairing, 40– p-wave pairing, 40– quasi-classical theory, 41Coercive field, 120, 126Coexisting phases, 71– ferromagnetism and superconductivity,

36Coherence factors, 64Conductivity– in plane– – Sr2RuO4, 96Crystal field, 103

– SrRuO, 257Cubic-tetragonal distortions, 103Curie law, 197Curie temperature, 68, 234– CaRuO3, 309– SrRuO3, 305, 309

Diamagnetic shielding, 154Diamagnetism– shielding fraction, 128Differential thermal analysis, 151Disordered perovskite, 150Doping– hole, 167Dynamical mean-field theory, 82Dzyaloshinsky-Moriya interaction, 165

Eilenberger equation, 41Electron diffraction, 215Electron spin resonance (ESR)– Eu2−xCexSr2RuCu2O10−δ , 198Electron spin resonance (ESR)– Gd1.5Ce0.5Sr2RuCu2O10−δ, 198– GdSr2RuCu2O8, 196Electronic structure– cuprates, 243– ruthenates, 249Epitaxial strain, 315Extended Huckel theory, 93

Fabrication– GdSr2RuCu2Oz, 210– NdSr2RuCu2Oy, 210– REBa2Cu3O7, 205Fermi surface– nesting, 19, 21, 262– polarised, 295– Sr2RuO4, 87, 251, 260– unpolarised, 295

326 Subject Index

Ferromagnetic resonance (FMR)– Eu2−xCexSr2RuCu2O10−δ, 198– Gd1.5Ce0.5Sr2RuCu2O10−δ, 198– GdSr2RuCu2O8, 197Ferromagnetic superconductors, 3Ferromagnetic transition, 66, 321– SrRuO3, 305Ferromagnetism– fluctuations, 23– instability, 29– Stoner theory, 259– transition temperature, 100Fluctuation-dissipation theorem, 17Free energy, 57– spin triplet superconductor, 50

G-type antiferromagnetism, 161Gap function, 53, 55– spin triplet, 74Ginzburg-Landau theory– anisotropic effective mass approxima-

tion, 12

H-T phase diagram, 9, 12– UPt3, 34Hall coefficient– Sr2RuO4, 97Hartree-Fock approximation, 62Hartree-Fock-Gorkov approximation, 48Heterostructures– SrRuO3/SrTiO3, 316Hidden critical point, 66High-resolution transmission electron

microscopy (HRTEM), 165, 215Hubbard model– extended, 61Hubbard operators, 242Hubbard-Stratanovich transformation,

284Hund coupling, 61, 100, 243Hybridization, 95Hysteresis loop, 120

Incommensurate spin fluctuations, 21,267

Incommensurate wavevectors, 299Inelastic neutron scattering, 17– Sr3Ru2O7, 292Integrals– exchange, 81

– hopping, 95– on-site Coulomb, 81Interband excitations, 253Internal magnetic field, 197Intragrain inhomogeneity, 114Irreducible Green function method, 246Itinerant ferromagnetism– SrRuO3, 323

Jahn-Teller effect, 149Josephson-junction-array, 108, 109

Korringa law, 102Kramers degeneracy, 274

Lanczos algorithm, 105Landau expansion, 68Laser MBE, 315Lattice structure, 143– Ca2−xSrxRuO4, 263– CaRuO3, 256– Eu2−xCexSr2RuCu2O10, 194– Gd2−xCexSr2RuCu2O10, 194– GdSr2RuCu2O8, 194– Sr2RuO4, 259– Sr3Ru2O7, 266– SrRuO3, 256, 307, 313Lindhard function, 274Local density approximation (LDA), 76– Sr2RuO4, 20Local moment, 99Local spin density approximation (LSDA)– SrRuO3, 257Luttinger theorem, 83

M-H– Gd3+Ba2Cu3O6.2, 178– RuSr2GdCu2O8, 178Magnetic excitations, 18Magnetic fluctuations, 97Magnetic form factor– Sr3Ru2O7, 295Magnetic moment, 62, 64– Eu2−xCexSr2RuCu2O10−δ

– – field cooling, 198– – zero field cooling, 198– GdSr2RuCu2O8

– – field cooling, 196– – zero field cooling, 196Magnetic scattering, 23

Subject Index 327

– Ca2−xSrxRuO4, 27– cross section, 301– Sr3Ru2O7, 290Magnetic susceptibility– Cooper pairs contribution, 5Magnetism– competition ferro-antiferro, 252Magnetization– dc– – SrRuO3, 305– dc Ru1−xSr2GdCu2+xO8−d, 178– dc RuSr2GdCu2O8, 190Magneto-transport– Sr2RuO4, 97Magnetoresistance– Sr2RuO4, 97– SrRuO3, 306– SrRuO3, 320Matthiessen rule, 97Metal-insulator transition, 72Metamagnetic critical field, 299Metamagnetic transition, 292– Sr3Ru2O7, 297Metamagnetism, 100, 271– critical end point– – differential susceptibility, 284– – resistivity, 284– – specific heat, 285– critical exponents, 286– effective action, 274– mean-field approximation, 275– one loop corrections, 280– renormalization group, 273– RG equations, 281– spin-orbit coupling, 274– tree level scaling, 277Mixed valence, 124Mott-Hubbard transition, 250Multiband p–d model, 249

Nagaoka theorem, 253Nesting signal, 20Neutron diffraction, 161, 292– SrRuO3, 306Nuclear Bragg peak, 295Nuclear magnetic resonance (NMR), 21– Knight shift, 7Nuclear spin-lattice relaxation rate, 102

Orbital degeneracy, 62

Orbital ordering– antiferromagnetic, 73Order paramater– hexagonal symmetry, 35

Peritectic decomposition, 147, 150Perovskite, 144Perovskite cell, 149Phase diagram– Ca2−xSrxRuO4, 264Phase equilibria– GdO1.5-RuO2+δ-CuO, 148– Sr-Gd-Ru-Cu-O, 144– SrO-GdO1.5-CuO, 146– SrO-GdO1.5-RuO2+δ, 147– SrO-RuO2+δ-CuO, 149Photoemission spectra, 86Polarisation factor– Sr3Ru2O7, 295Pseudobinary equilibria, 150Pseudoquaternary phase diagram, 146Pseudospin operators, 70Pseudoternary system, 146

Quantum critical end-point, 284Quantum criticality, 23Quantum Monte Carlo, 83Quantum phase transition, 105Quasi-particle structure– Sr2RuO4, 79

Reflection High Energy ElectronDiffraction (RHEED)

– SrRuO3, 314Relaxation rate, 96Resistance– (SrRuO3)m/(SrTiO3)n superlattices,

321– SrRuO3, 320Resistivity, 152– Ru1−xSr2GdCu2+xO8−d, 178– RuSr2Gd1−yCeyCu2O8, 187– RuSr2GdCu2O8, 228– Sr2RuO4, 97– SrRuO3, 306Rietveld refinement, 216RPA approximation, 18Ru effective magnetic moment, 234RuO6 octahedra– flattening of, 252

328 Subject Index

– rotation of, 27, 251, 264

Sample preparation– Ru1−xSr2GdCu2+xO8−d, 177– Ru-1212, 151– RuSr2EuCu2O8, 190– RuSr2Gd1−yCeyCu2O8, 186– RuSr2GdCu2O8, 190, 223– SrRuO3, 305SDW ordering, 23– Sr2Ru1−xTixO4, 24Seebeck coefficient– Ru1−xNbxSr2GdCu2O8, 166Self-energy, 82Skew scattering, 98Slave bosons, 71Sn/Nb susbstitution, 166Specific heat– jump, 184– linear term coefficient γ, 37, 38, 42– Sr2RuO4, 7Spectral function, 86Spectroscopy– angle-resolved photoemission, 77– Mossbauer, 123– X-ray absorption (XAS), 123Spin fluctuations, 261, 291– spectrum, 292Spin magnetization– Sr3Ru2O7, 298Spin moment energy, 265Spin susceptibility, 18Spin triplet operators, 61, 70Spontaneous vortex phase, 120, 228SQUID magnetometry, 161Stoner– critical point, 66– enhancement, 23– interaction, 19– threshold, 62Stoner-Wohlfarth ferromagnet, 68Structural properties– RuSr2GdCu2O8, 162Structure– Sr2RuO4, 1Structure factor– Sr3Ru2O7, 295Superconducting coherence length, 67Superconducting order parameter– spin triplet, 32

Superconductivity– d-wave, 246– p-wave, 246– s-wave, 246– Ginzburg-Landau theory, 9, 114– heavy fermions, 2– multi-band model, 37– multi-component order parameter– – bipolar state, 35– – non-bipolar state, 36– – planar state, 35– – UGe2, URhGe, ZrZn2, 36– – UPt3, 34– multiple phases, 8– orbital dependent– – Sr2RuO4, 7– specific heat, 10– – field dependence, 12– spin singlet, 4, 49– spin triplet, 3, 23, 49, 62– – gap function, 47– – Ginzburg-Landau theory, 50– – magnetic susceptibility, 54– – magnetization, 54– – specific heat, 49, 54– thermal conductivity– – field dependence, 12– tight-binding model, 37– two-band model, 37– – density of states, 38– – Fermi surface, 38– Zeeman field, 48Superstructure, 162Susceptibility– ac, 110– – Ru1−xSr2GdCu2+xO8−d, 178– – RuSr2Gd1−yCeyCu2O8, 187– – RuSr2GdCu2O8, 190– – SrRuO3, 305– differential, 114– dynamical magnetic, 17, 298– magnetic, 91– non-interacting, 18– Pauli like, 101– RuSr2GdCu2O8, 232Synthesis– precursor route, 152– RuSr2(L,Ce)2Cu2O10+δ, 109– RuSr2L-Cu2O8−δ, 109

Subject Index 329

t-J model– ferromagnetism, 71t-J-I model, 246Ternary diagrams, 206Thermo-Gravimetric, 150, 211Thermopower, 183Thin films– SrRuO3, 314Three-band model– Sr2RuO4, 250Tight-binding, 79, 241– Sr2RuO4, 94Time reversal symmetry, 33Transition– intergrain temperature, 111– spin flip, 295– spin flop, 172, 295Transmission electron microscopy

(TEM), 215– RuSr2GdCu2O8, 228

Transverse field µSR, 185Two-band model, 62

van Hove singularity, 29, 78, 245

Wannier functions, 241Weak ferromagnetic superconductor, 160Weak ferromagnetism, 179Werthamer-Helfand-Hohenber formula,

12

X-ray diffraction, 316– powder– – NdSr2RuCu2Oy, 205– refinement, 109– RuSr2GdCu2O8, 223– RuSr2GdCu2O8, 164XANES measurements, 184

Zhang-Rice singlet, 243