Stripes and Related Phenomena (Selected Topics in Superconductivity Volume 8)

Stripes and Related Phenomena SELECTED TOPICS IN SUPERCONDUCTIVITY Series Editor: Stuart Wolf Naval Research Laborato...

Author: Antonio Bianconi | Naurang L. Saini

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Stripes and Related Phenomena

SELECTED TOPICS IN SUPERCONDUCTIVITY Series Editor: Stuart Wolf Naval Research Laboratory Washington, D.C. CASE STUDIES IN SUPERCONDUCTING MAGNETS Design and Operational Issues Yukikasu Iwasa ELECTROMAGNETIC ABSORPTION IN THE COPPER OXIDE SUPERCONDUCTORS Frank J. Owens and Charles P. Poole, Jr. INTRODUCTION TO HIGH-TEMPERATURE SUPERCONDUCTIVITY Thomas Sheahen THE NEW SUPERCONDUCTORS Frank J. Owens and Charles P. Poole, Jr. NONEQUILIBRIUM ELECTRONS AND PHONONS IN SUPERCONDUCTIVITY Armen M. Gulian and Gely F Zharkov QUANTUM STATISTICAL THEORY OF SUPERCONDUCTIVITY Shigeji Fujita and Salvador Godoy STABILITY OF SUPERCONDUCTORS Lawrence Dresner STRIPES AND RELATED PHENOMENA Edited by Antonio Bianconi and Naurang L. Saini

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

Stripes and Related Phenomena Edited by

Antonio Bianconi Università di Roma “La Sapienza” Rome, Italy

and

Naurang L. Saini Istituto Nazionale di Fisica della Materia Rome, Italy

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-47100-0 0-306-46419-5

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2000 Kluwer Academic/Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

Preface The problem of superconductors has been a central issue in Solid State Physics since 1987. After the discovery of superconductivity (HTSC) in doped perovskites,

it was realized that the HTSC appears in an unknown complex electronic phase of condensed matter. In the early years, all theories of HTSC were focused on the physics of a homogeneous 2D metal with large electron–electron correlations or on a 2D polaron gas. Only after 1990, a novel paradigm started to grow where this 2D metallic phase is described as an inhomogeneous metal. This was the outcome of several experimental evidences of phase separation at low doping. Since 1992, a series of conferences on phase separation

were organized to allow scientists to get together to discuss the phase separation and related issues. Following the discovery by the Rome group in 1992 that “the charges move freely mainly in one direction like the water running in the grooves in the corrugated iron foil,” a new scenario to understand superconductivity in the superconductors was open. Because the charges move like rivers, the physics of these materials shifts toward the physics of novel mesoscopic heterostructures and complex electronic solids. Therefore, understanding the striped phases in the perovskites not only provides an opportunity to understand the anomalous metallic state of cuprate superconductors, but also suggests a

way to design new materials of technological importance. Indeed, the stripes are becoming a field of general scientific interest.

This book is a collection of papers in the field of stripes and related phenomena. The most relevant theoretical and experimental contributions, presented at the second international conference on Stripes and Superconductivity from experts in the field of stripes and related phenomena are selected for the publication. Apart from the relevant contribution on stripes in the cuprates, the book includes contributions on other stripe phases observed in manganites, nikelates, spin ladders, and heterostructures. Because a large stream of research is converging toward the stripe scenario with a growing community, this book serves as an important reference in the field of striped phases and superconductivity. We would like to thank Anna De Grossi for her secretarial help, and Kevin Sequeira, Diana Osborne, and Robert Maged at Kluwer Academic/Plenum Publishers for their continuous support.

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Contents

INTRODUCTORY OVERVIEW From Phase Separation to Stripes K. A. Miiller

1

Lattice-Charge Stripes in the Superconductors A. Bianconi, S. Agrestini, G. Bianconi, D. Di Castro, and N. L. Saini

9

STRIPES, CDW, AND SDW INSTABILITIES IN CUPRATES: THEORETICAL ASPECTS Stripes, Electron-Like and Polaron-Like Carriers, and J. Ashkenazi Charge Ordering and Stripe Formation in

in the Cuprates

Cuprates

27 39

A. Bussmann-Holder

The Stripe-Phase Quantum-Critical-Point Scenario for Superconductors S. Caprara, C. Castellani, C. Di Castro, M. Grilli, A. Perali, and M. Sulpizi Phase and Amplitude Fluctuation in Superconductors: Formation of Gap Stripes Due to Lack of Electron-Hole Symmetry in Cuprate Oxides B. K. Chakraverty and K. P. Jain

45

55

Stripe on a Lattice: Superconducting Kink/Soliton Condensate Yu. A . Dimashko and C. Morais Smith

63

Microscopic Theory of High-Temperature Superconductivity V. J. Emery and S. A. Kivelson

69

Two Reasons of Instability in Layered Cuprates I. Eremin, M. Eremin, and S. Varlamov

77

Influence of Disorder and Lattice Potentials on the Striped Phase N. Hasselmann, A. H. Castro Neto, and C. Morais Smith

83

Stripe Liquid, Crystal, and Glass Phases of Doped Antiferromagnets S. A. Kivelson and V. J. Emery

91

Dynamical Mean-Field Theory of Stripe Ordering A . I. Lichtenstein, M. Fleck, A. M. Oles, and L. Hedin

101

Tunneling and Photoemission in an SO(6) Superconductor R. S. Markiewicz, C. Kusko, and M. T. Vaughn

111

vii

viii Spin, Charge, and Orbital Ordering in 3d Transition-Metal Oxides Studied by Model Hartree–Fock Calculation T. Mizokawa, and A. Fujimori

Contents 121

Sliding Stripes in 2D Antiferromagnets C. Morais Smith, Yu. A. Dimashko, N. Hasselmann, and A. O. Caldeira

129

Quantum Interference Mechanism of the Stripe-Phase Ordering S. I. Mukhin

135

Spontaneous Orientation of a Quantum Lattice String O. Y. Osman, W. van Saarloos, and J. Zaanen

143

Domain Wall Structures in the Two-Dimensional Hubbard Model with Long-Range Coulomb Interaction G. Seibold, C. Castellani, C. Di Castro, and M. Grilli

151

POLARONS, TWO COMPONENTS, AND LATTICE INSTABILITIES IN CUPRATES Boson–Fermion Mixtures, d-Wave Condensate, and Tunneling in Cuprates A. S. Alexandrov

159

The Small Polaron Crossover: Role of Dimensionality M. Capone, S. Ciuchi, and C. Grimaldi

169

CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas V. Cataudella, G. De Filippis, and G. Iadonisi

175

The Charge-Ordered State from Weak to Strong Coupling S. Ciuchi and F. de Pasquale

183

Low-Temperature Phonon Anomalies in Cuprates T. Egami, R. J. McQueeney, Y. Petrov, G. Shirane, and Y. Endoh

191

Enhanced Thermoelectric Power and Stripes in Cuprate Superconductors J. B. Goodenough and J. -S. Zhou

199

A Refined Picture of the YBa 2 Cu 3 O x Structure: Sequence of Dimpling-Chain Superstructures, 1D-Modulation of the Planes, Phase Separation Phenomena 211 E. Kaldis, E. Liarokapis, N. Poulakis, D. Palles, and K. Conder Evolution of the Gap Structure from Underdoped to Optimally Doped from Femtosecond Optical Spectroscopy D. Mihailovic, J. Demsar, and B. Podobnik Local Lattice Distortions in YBa2Cu3Oy : Doping Dependence H. Oyanagi, J. Zegenhagen, and T. Haage

219 227

STRIPE EFFECTS ON THE ELECTRONIC STRUCTURE Fermi Surface of Superconductor by Angle-Scanning Photoemission M. C. Asensio, J. Avila, N. L. Saini, A. Lanzara, A. Bianconi, S. Tajima, G. D. Gu, and N. Koshizuka

237

Local Lattice Fluctuations and the Incoherent ARPES Background J. Ranninger and A. Romano

245

Evidence for Strongly Interacting Electrons with Collective Modes at and in the Normal Phase of Superconductors N. L. Saini, A. Lanzara, A. Bianconi, J. Avila, M. C. Asensio, S. Tajima, G. D. Gu, and N. Koshizuka

253

Contents

ix

Angle-Resolved Photoemission Study of 1D Chain and Two-Leg Ladder T. Sato, T. Yokoya, H. Fujisawa, T. Takahashi, M. Uehara, T. Nagata, J. Akimitsu, S. Miyasaka, M. Kibune, and H. Takagi

263

Optical Study of Spin/Charge Stripe Order Phase in S. Tajima, N. L. Wang, M. Takaba, N. Ichikawa, H. Eisaki, S. Uchida, H. Kitano, and A. Maeda

271

SPIN AND LATTICE DYNAMICS AND SPIN STRIPES: NMR/NQR AND NEUTRON SCATTERING

Vibrational Pseudo-Diffusive Motion of the Oxygen Octahedra in from Anelastic and 139 La Quadrupolar Relaxation F. Cordero, C. R. Grandini, R. Cantelli, M. Corti, A. Campana, and A. Rigamonti

279

Charge and Spin Dynamics of Cu-O Chains in An NMR/NQR Study B. Grévin, Y. Berthier, G. Collin, and P. Mendels

287

Cuprates

Mobile Antiphase Domains in Lightly Doped Lanthanum Cuprate P. C. Hammel, B. J. Suh, J. L. Sarrao, and Z. Fisk

295

On the Structure of the Cu B Site in La1.85Sr0.15CuO4 J. Haase, R. Stern, D. G. Hinks, and C. P. Slichter

303

On the Estimate of the Spin-Gap in Quasi-1D Heisenberg Antiferromagnets from Nuclear Spin-Lattice Relaxation R. Melzi and P. Carretta Magnetic and Charge Fluctuations in Superconductors H. A. Mook, F. Dogan, and B. C. Chakoumakos

309

315

Neutron Scattering Study of the Incommensurate Magnetic Fluctuation in YBa2Cu3O6+x T. Nishijima, M. Arai, Y. Endoh, S. M. Bennington, R. S. Eccleston, and S. Tajima

323

Rare Earth Spin Dynamics in the Nd-Doped Superconductor La2–x Srx CuO4 M. Roepke, E. Holland-Moritz, B. Büchner, R. Borowski, R. Kahn, R. E. Lechner, S. Longeville, and J. Fitter

329

Static Incommensurate Magnetic Order in the Superconducting State of La2–x Srx CuO4+y K. Yamada, R. J. Birgeneau, Y. Endoh, M. Fujita, K. Hirota, H. Kimura, C. H. Lee, S. H. Lee, H. Matsushita, G. Shirane, S. Ueki, and S. Wakimoto

335

THEORETICAL ASPECTS: GENERAL Marginal Stability of d-Wave Superconductor: Spontaneous P and T Violation in the Presence of Magnetic Impurities A. V. Balatsky and R. Movshovich

343

Skyrmions in 2D Quantum Heisenberg Antiferromagnet Static Magnetic Susceptibility S. I. Belov and B. I. Kochelaev

349

Spin Peierls Order and d-Wave Superconductivity

355

Partha Bhattacharyya

On Localization Effects in Underdoped Cuprates C. Castellani, P. Schwab, and M. Grilli Interpolative Self-Energy Calculation for the Doped Emery Model in the Antiferromagnetic and in the Paramagnetic State J. Fritzenkötter and K. Dichtel

361

369

x

Contents

The Quasi-Particle Density of States of Optimally Doped Bi 2212: Break-Junction vs. Vacuum-Tunneling Measurements

377

R. S. Gonnelli, G. A. Ummarino, and V. A. Stepanov Long-Range Terms in the Dynamically Screened Potential of R. Grassme and P. Seidel

385

Chemical Analysis of the Superconducting Cuprates by Means of Theory

391

Itai Panas Superconductivity with Antiferromagnetic Background in a S. Saito, S. Kurihara, and Y. Y. Suzuki

Hubbard Model

399

d-Wave Solution of Eliashberg Equations and Tunneling Density of States in Optimally Doped Superconductors G. A. Ummarino and R. S. Gonnelli

407

Enhancement of Electron–Phonon Coupling in Exotic Superconductors near a Ferroelectric Transition M. Weger and M. Peter

413

STRIPE EFFECTS IN MANGANITES, LADDERS AND RELATED PEROVSKITES Features of the

Structural Phase Transitions in

421

M. Arao, S. Miyazaki, Y. Inoue, and Y. Koyama Infrared Signatures of Charge Density Waves in Manganites P. Calvani, P. Dore, G. De Marzi, S. Lupi, I. Fedorov, P. Maselli, and S.-W. Cheong

427

Recent Results in the Context of Models for Ladders Elbio Dagotto, George Martins, Claudio Gazza, and André Malvezzi

437

Charge-Ordered States in Doped AFMs: Long-Range “Casimir” Attraction and Instability

447

Daniel W. Hone, Steven Kivelson, and Leonid P. Pryadko Features of the Modulated Structure in the Layered Perovskite Manganate Y. Horibe, N. Komine, Y. Koyama, and Y. Inoue

455

Numerical Studies of Models for Manganites

459

Adriana Moreo and Seiji Yunoki

Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds with Ba, Nd S. Pachot, P. Bordet, C. Chaillout, C. Darie, R.J. Cava, M. Hanfland, M. Marezio and H. Takagi

X-Ray Scattering Studies of Charge Stripes in Manganites and Nickelates Y. Su, C.-H. Du, B.K. Tanner, P. D. Hatton, S. P. Collins, S. Brown,

465

473

D. F. Paul, and S.-W. Cheong

Colossal Negative Magnetoresistivity of

Films in Field up to 50 T

481

P. Wagner, I. Gordon, L. Trappeniers, V. V. Moshchalkov, and Y. Bruynseraede MATERIALS CHARACTERIZATION: FUNDAMENTAL PROPERTIES Synthesis and Characteristics of the Indium-Doped Tl-1212 Phase R. Awad, N. Gomaa, and M. T. Korayem

487

High-Frequency Optical Excitations in YBCO Measured from Differential Optical Reflectivity I. M. Fishman, W. R. Studenmund, and G. S. Kino

495

Contents Low-Temperature Structural Phase Transitions and in Zn-Substituted Y. Inoue, Y. Horibe, and Y. Koyama

xi Suppression 501

Josephson Nanostructures and the Universal Transport and Magnetic Properties of YBCO J. Jung, H. Yan, H. Darhmaoui, and W. K. Kwok

507

Evidence of Chemical Potential Jump at Optimal Doping in Z. G. Li and P. H. Hor

515

Studies of the Insulator to Metal Transition in the Deoxygenated System P. Starowicz, and

521

Differential Optical Reflectivity Measurements of W. R. Studenmund, I. M. Fishman, G. S. Kino, and J. Giapintzakis

529

On Some Common Features in High- and Superconducting Perovskites I. Vobornik, D. Ariosa, H. Berger, L. Forró, R. Gatt, M. Grioni, G. Margaritondo, M. Onellion, T. Schmauder, and D. Pavuna

535

MATERIALS CHARACTERIZATION: APPLICATION ASPECTS Pinning Mechanisms in a-Axis-Oriented and Multilayers E. M. González, J. M. González, Ivan K. Schuller, and J. L. Vicent Angular Dependence of the Irreversibility Line in Irradiated a-Axis-Oriented Films J. I. Martín, W.-K. Kwok, and J. L. Vicent Defect-Modulated Long Josephson Junctions as Source of Strong Pinning in Superconducting Films

539

545

551

E. Mezzetti, E. Crescio, R. Gerbaldo, G. Ghigo, L. Gozzelino, and B. Minetti Bulk Confinement of Fluxons by Means of Surface Patterning of Columnar Defects in BSCCO Tapes E. Mezzetti, R. Gerbaldo, G. Ghigo, L. Gozzelino, B. Minetti, P. Caracino, L. Gherardi, L. Martini, G. Cuttone, A. Rovelli, and R. Cherubini

559

OTHER MATERIALS A Finite-Size Cluster Study of

567

M. Cuoco, C. Noce, and A. Romano

New Copper-Free Layered Perovskite Superconductors: and Related Compounds Yoshihiko Takano, Yoshihide Kimishima, Hiroyuki Taketomi, Shinji Ogawa, Shigeru Takayanagi, and Nobuo Môri Author Index

Subject Index

573

579 581


From Phase Separation to Stripes K. A. Müller1

An overview of the evolution of the concept of phase separation in the superconductors that has been developed since the mid-1990s is presented with

describing a historical perspective on what has happened before this conference and how we arrived at the stripes scenario starting from phase separation.

The basic concept of Jahn-Teller polarons [ 1 ] was behind the search for superconductivity in doped cuprate perovskites [2]; however, after 1986, the majority of theoretical works had been focused on the physics of a doped homogeneous antiferromagnetic 2D lattice. During my visits to Stuttgart University, Sigmund and Hyzhnyakov, back in 1988,

proposed from their theoretical calculations some kind of phase separation of the doped holes driven by magnetic interactions. Their work showed the formation of small ferromagnetic clusters (magnetic polarons in the antiferromagnetic background which they called ferrons [3]) with a characteristic size (a is the lattice constant), and which was quasi-static on the short time scale but dynamic on a longer time scale. These clusters were reported to have only low mobility, whereas the holes inside the clusters could move freely. The metal-to-insulator transition in cuprate superconductors by increasing the doping observed was assigned to the percolation of charge carriers in the clusters in this inhomogeneous system. I paid only mild attention to that, as I was, and still am, more interested in the role of electron-lattice interactions of the doped holes forming Jahn–Teller

polarons. Later, in 1990, I became aware of the work of Di Castro’s group while visiting the University “La Sapienza” in Rome. Their theoretical calculations, using mean field theory, revealed a tendency toward clustering of the doped holes, formed by spin zero singlets, in an antiferromagnetic background. This resulted in a phase separation of the doped holes in macroscopic metallic domains separated by the undoped antiferromagnetic domains in the normal phase [4]. This phase separation was found to be derogative for inducing a good homogeneous metallic phase, but good for superconductivity because Di Castro et al. found 1

Physikalisches Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.

Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.

1

2

Müller

that the superconducting phase was competing with the phase separation below a critical temperature. That visit at Rome University made me more attentive to the topic of electronic inhomogneities.

A year later, I gave a talk at the Brookhaven National Laboratory on something quite different and came to know the work of Victor Emery and collaborators [5] on dynamic

metallic clusters in an antiferromagnetic lattice, which they concluded on the basis of kinetic and magnetic energy calculations. This made me think, well, there are three different theoretical groups talking about a phase separation of the doped holes, there might be

something to this. It would not be too surprising, after all, as we have a number of examples in physics where phase separation exists. From this arose the idea to organize a workshop to discuss these matters, and with the help of Giorgio Benedek, director of the School of Solid State Physics in Erice, we decided to hold the workshop in May 1992 [6]. In this workshop, on the experimental side, phase separation was shown to exist in oxygen-doped where the dopants are mobile interstitial oxygens. The phase separation into oxygen-poor antiferromagnetic domains and oxygen-rich metallic domains with was found by several groups using electrochemical oxidation, susceptibility experiments, and nuclear magnetic resonance (NMR). A point of discussion was whether these chemically separated phases were driven by electronic forces or not. Experimental indications for an electronic origin were given by A. Heeger using photo excitation experiments, and by the Orsay group

using Mössbauere experiments. A different phase separation due to ordering of polarons in linear arrays in the plane was presented by Bianconi using EXAFS and diffraction on This was assigned to an one-dimensional (ID) incommensurate charge density wave (ICDW) that coexists with itinerant carriers moving like waves of water on a corrugated shore. On the theoretical side, Victor Emery gave a quite convincing talk at the beginning of the workshop which invoked several experiments that could be explained by a phase separation of clusters of doped holes in the antiferromagnetic lattice, frustrated by Coulomb interactions. In Erice, it was decided to hold a follow-up workshop on the same topic, which was organized by Sigmund in Cottbus in 1993 [7]. In Cottbus there was an interesting experimental report from the group of Albert Furrer on using inelastic neutron scattering. They could show that upon doping the system with oxygen, it shows the evolution of three different local cluster types characterized by crystalline electric field (CEF) lines of Er are shown. They also presented the evolution of the fractal sizes of the clusters, derived from CEF line widths, as a function of oxygen

content. Thus, this was clear evidence of a chemical structural phase separation on a local scale. We have to accept that there are clusters and, of course, we are then interested in what happens inside the cluster, as in the high-energy physics, where, for example, first there are protons and neutrons, and then one becomes interested in what the neutrons and protons consist of. David Johnston gave a report on in the insulating phase at very low hole doping. His group measured the magnetic susceptibility as a function of temperature for various dopings, and found that the more one dopes the system, the smaller the susceptibility peak becomes. From the analysis of these measurements, they concluded that the holes were forming linear metallic lines, increasing in widths with doping. Their results were evidence

for linear (1D) domain walls separating antiferromagnetic domains. Antonio Bianconi reported on the measurement of the relative number of polaronic charge carriers contributing to the one-dimensional incommensurate CDW (ICDW), and

the itinerant carriers based from the variation of the periodicity of the ICDW with Y doping in the superconducting phase of the In his talk, he introduced the

From Phase Separation to Stripes

3

idea of a 2D electron gas in this doped superconducting cuprate, driven close to the Wigner localization limit by a large electron-lattice interaction. Therefore, the 1D ICDW could be assigned to a generalized Wigner CDW in a 2D electron gas with large Coulomb and lattice interactions. There was a report by Chris Hammel from Los Alamos on the NMR and Nuclear Quadrupole Resonance (NQR) investigation of the La ions. In his experiment, he observed two lines that were assigned to a distribution of tilted octahedra. This was an important step because until then, NMR people had always been saying that there was a single line supporting the homogeneous antiferromagnetic Fermi-liquid model of Pines et al. [8]. Now we think that his findings indicate pinned stripes; otherwise, he could not have seen them. Furthermore, these tilts were continuous, showing that in these stripes we have a distribution of lattice distortions. In Cottbus we decided to split up these topics and hold a meeting in Bled on “Anharmonic Properties of Cuprates” in 1994 [9], followed by a conference on “Stripes and Lattice Instabilities” in Rome at the end of 1996 [10]. In between there was a NATO School entitled, Superconductivity. Ten Years after the Discovery,” organized by Kaldis, Liokarpis, and me [11]. Let me mention a few highlights from the Bled meeting and then the Delphi workshop. Another workshop on phase separation was held in Erice during 1997, but, having been absent then, I am not going to speak much about it. The details on this meeting can be found in the proceedings [12]. Finally, another meeting in Erice is to follow this STRIPES98 conference, where polarons are the main topic to be discussed [13]. Dragan Mihailovic, Gianpietro Ruani, Emanuel Kaldis, and I were involved in organizing the Bled workshop. The emphasis was on the fact that the potential of the oxygen may be anharmonic, as shown on the cover of the proceedings [9]. Let me mention a point that I believe to be relevant: In the YBCO compound, it was shown that selective substitution of oxygens more than 80% of the isotope effect for small and large dopings results from the planes (for large doping it was not easy to measure). This tells us that the drama mostly takes place quite in the planes. One of the important results in this conference was that the lattice polarons are present in the planes. There was a report by Egami, who used pulsed neutron and diffuse x-ray scattering to show that there are two type of carriers—namely, heavy polarons and light carriers. Bianconi reported the width of the mesoscopic stripes to be of the order of 15 Å, as determined by condensed polarons in Bi2212. He provided evidence of the existence of a “shape resonance” for the electrons on the Fermi level, yielding an origin for the amplification. The report by the Calvani group on electron-doped showed polarons in optical conductivity measurements. Then Dragan Mihailovic reported photoexcitation measurements showing the presence of carriers with Drude-like conductivity and also a quasi-localized one (he put a question mark regarding their polaronic character). On the theoretical side, there was a remarkable report by Alexandrov on the formation of bipolarons at a temperature higher than which might be responsible for the anomalous kinetics and thermodynamics observed in the cuprates. To me, this was the first time when the bipolarons entered the picture. Next came a report by Julius Ranninger, who emphasized that itinerant valence electrons coexist with bipolarons above while condensation occurs at In Delphi, many subjects were discussed in the review talks [11]. One of them was by Wells, who summarized the results of MIT and Brookhaven on the system,

4

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doped with both Sr and oxygen. He showed that doping with Sr gives a substantially different phase diagram than doping with the oxygen does. With Sr doping, a random distribution of Sr ions at the La sites results, while doping with oxygen yields several phases that can be classified by different stagings of oxygen atoms in the LaO layers, as shown in Fig. 1, similar to what is known to occur in the intercalated graphite systems. Mihailovic reported optical spectroscopy results on the YBCO system. He observed two signals with different dynamics—one with slow dynamics, while the other one with faster dynamics. From these results it was clear that there are at least two types of charge carriers. Then came the important presentation by Bianconi, who projected the reproduced picture here (Fig. 2), which shows two alternating local site structures of the Cu ion in the plane. The first one with the so-called LTO-type tilts, with quasi-metallic stripes having a width of about 16 Å. The other environment has so-called LTT-type tilts, where two planar oxygens are shifted towards, and two away from the central copper ion. This is precisely the mode of the Jahn–Teller distortion. Here, I would like to recall briefly the two Jahn–Teller modes which are degenerate and belong to the same symmetry representation [1]: one with two oxygens moved out and two moved in while the other with four oxygens moved in and two moved out as shown in Fig. 3. The first one


5

is the same as that shown by the EXAFS results [14]. I will come again back the mode later. There was also a report by Teplov and his group at Kazan University, who performed extensive NMR and NQR measurements in the TmBaCuO compound, which also becomes superconducting in the 123 and 124 phases. Basically, they found different copper centers having orthorhombic and tetragonal distortions characterized by the NMR/NQR line widths and spin relaxation rates. The relative content was about 2/3 for the orthorhombic and 1/3

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Müller

for the tetragonal. Their interpretation was pinned stripes with antiferromagnetic ordering in the plane along the ( 1 1 ) direction. However, they were not sure about the orientation of the stripes, i.e., whether they occurred along the (11) or the (10) direction. Then came the STRIPES 96 conference in Rome. The proceedings were published as a special issue of the Journal of Superconductivity and edited by Bianconi and Saini [9]. This was a relatively large conference, even if not as large as this STRIPES 98, but still large enough to indicate that the subject of stripes was gathering momentum. There were a number of interesting contributions from several groups, so let me just give one or two examples that particularly attracted my attention. Mook et al. of Oak Ridge performed inelastic neutron scattering measurements and were able to see incommensurate fluctuations in the Bi2212 compound. They associated the results with a dynamic stripe phase. De Lozanne from Houston used atomic force microscopy YBCO superconductors to show that there is an

incommensurate plane wave with coherence along the a-direction of about three lattices, as in Mook’s experiments. John Goodenough presented results on the temperature dependence of the resistance and the thermoelectric power with hydrostatic pressure. In the proceedings paper [10], he wrote

We conclude that electron–lattice coupling is the dominant factor influencing the stabilisation of a dynamically heterogeneous, thermodynamically distinguishable phase and the transport properties exhibited by that phase. It is probable that in the copper oxides a non-retarded elastic coupling of cooper pairs replaces the BCS retarded-potential coupling of conventional superconductors. An elastic coupling, because if there are Jahn–Teller polarons, they deform the plane in a quadrupolar manner and it is clear that one cannot have an abrupt discontinuity from one stripe to the other. This issue will perhaps become clearer in the coarse of this conference, i.e., up to what extent the elastic coupling is relevant or more relevant than the Coulomb forces between the charges. I am now moving to the colossal magneto-resistance (CMR) compounds. Let me show you a figure which I modified from Khomskii and Sawatzky [15] (Fig. 4) and that is in relation to the superconductivity in the cuprates and the ferromagnetism in the CMR systems. What they have considered in their paper is the following: Normally transition metal oxides there are antiferromagnets because of the super exchange between the transition metal ion via the oxygen, as shown schematically in Fig. 4 [15]. Of course, this is the case for the copper oxides and also for many other magnetic oxides. Now, if the transition metal ion has a spin 1/2 and if a large Coloumb repulsion, U, is present, the hole’s main probability is on the oxygens. Then, the two spins can couple and form a spin 0 state, as shown in the lower part of the figure. This is the so-called Zhang–Rice singlet, which can become mobile in the antiferromagnetic lattice. This is the case for the cuprate superconductors. However, if the spin is larger than 1/2—suppose it is 1—then the magnetic moment will also couple as well to the doped hole (U being large) and ferromagnetism results (upper part of Fig. 4).

In the Khomskii–Sawatzky scheme, you do not have that. Copper is the last in the row of the 3D elements, and if one moves to the manganese, the U is smaller, and the hole resides mainly on the transition metals, and an antiferromagnetic state is realized. Finally, I want to show that in these systems the presence of stripes of polarons is now becoming clear. Figure 5 shows an example of the ordering of alternated rows of


7

8

Müller

manganite. Here, a charge-ordered state is formed, coexisting with an antiferromagnetic spin order as obtained by Radaelli et al. [16] from x-ray data. Furthermore, lines of small polarons at sites with the distortion alternate with lines of non-Jahn–Teller sites, clearly indicating the presence of the Jahn–Teller stripes in this system. ACKNOWLEDGMENTS

The author is grateful to Antonio Bianconi, Alessandra Lanzara and Naurang Saini for transcribing the recorded oral presentation into the present fine text. REFERENCES 1. K. H. Köck, H. Nickisch, and H. Thomas, Helv. Phys. Acta 56, 237 (1983). 2. J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986).

3. V. Hizhnyakov and E. Sigmund, Physica C 156, 655 (1988). 4. C. Di Castro, L. F. Feiner, and M. Grilli, Phys. Rev. Lett. 66, 3209 (1991); ibid. 75, 4650 (1995) and references therein. 5. V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993) and references therein.

6. “Phase Separation in Cuprate Superconductors” (1st Erice Meeting, Italy, 1992), edited by K. A. Müller and G. Benedek (World Scientific, Singapore, 1993). 7. “Phase Separation in Cuprate Superconductors” (Cottbus Workshop, Germany, 1993), edited by E. Sigmund and A. K. Müller (Springer Verlag, Berlin–Heidelberg, 1994). 8. See, e.g., H. Monien in ref. 5, p. 232, and references therein.

9. “Anharmonic Properties of Cuprates” (Bled Workshop, Slovania, 1994), edited by D. Miailovic, G. Ruani, E. Kaldis, and A. K. Müller (World Scientific, 1995). 10. Special issue on “Stripes, Lattice Instabilities, and Superconductivity” (1st Rome Conference, STRIPES96, 1996), edited by A. Bianconi and N. L. Saini [J. Supercond. 10, No. 4 (1997)]. 1 1. Superconductivity1996: Ten Years after the Discovery” (Delphi Workshop, 1996), edited by E. Kaldis, E. Liarokapis, and K. A. Müller (NATO ASI Series), Vol. 343 (Kluwer Academic Publishers, 1996).

12. Proceedings of Erice Conference on Polarons, 1996, edited by A. Bussmann-Holder [see special issue of Z. Phys. B 104, (1997)].

13. Special Issue: Proceedings of Erice Conference on Superconductivity and Magnetism, June, 1998, edited by A. Bussmann-Holder and V. Kresin [J. Supercond. 12(1) 1 (1999)]. 14. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996). 15. D. I. Khomskii and G. A. Sawatsky, Solid State Comm. 102, 87 (1997). 16. P. G. Radaelli, D. E. Cox, M. Marezio, and S. W. Cheong, Phys. Rev. B 55, 3015 (1997).

Lattice-Charge Stripes in the Superconductors A. Bianconi,1 S. Agrestini,1 G. Bianconi,1 D. Di Castro,1 and N. L. Saini1

We report a 2D plot

for the doped perovskites, where is the doping and

is the mismatch between the layers and the rocksalt layers. We identify for the first time the quantum critical point (QCP) at and for the onset of the polaron stripes coexisting with itinerant

carriers. The plot

for

shows the highest

at the critical

point The lattice mismatch drives the system to this QCP of the electron– lattice interaction for local lattice deformations at metallic densities. The incommensurate superlattice of charge-lattice (polaron) stripes are due to critical fluctuations near this QCP. The solution of the mystery of the anomalous normal phase of

superconductors implies a solution for the pairing mechanism:

the attractive pseudo-Jahn–Teller polaron–polaron interaction and a particular type of critical charge and spin fluctuations (forming a superlattice of quantum stripes tuned at a “shape resonance”) near this QCP play a key role in the pairing mechanism.

1. INTRODUCTION

A charge transfer Mott insulator is at

in the conventional phase diagram of

cuprate superconductors The superconductivity with appears in the range We show that the mystery of the normal phase of the superconductors is solved by introducing a second variable, the electron–lattice interaction The electron lattice interaction is driven by the compression acting on the lattice due to the lattice mismatch of the metallic plane embedded in the perovskite crystalline lattice. In 1

Dipartimento di Fisica, and Unitá INFM, Università di Roma “La Sapienza,” P. le Aldo Moro 2, 00185 Roma, Italy.


9

10

Bianconi, Agrestini, Bianconi, Di Castro, and Saini

the new phase diagram 1.

2.

3. 4.

we can identify the following:

A disordered phase in the limit of low density where the charges are in the localization limit an insulating ordered phase A where the charges are associated with vibronic cooperative pseudo-Jahn–Teller (JT) local lattice distortions (LTT-like) of radius about 5 Å. In this phase, they form a commensurate polaron crystal (CPC) at for This crystalline phase can be described as an anharmonic polaronic one-dimensional (1D) charge-density wave commensurate with the lattice, forming a superlattice of lattice-charge stripes (polaron stripes) with an associated orbital density wave (ODW). a phase B made of itinerant carriers forming a normal metallic phase. a region of coexistence of the phases A and B. In this inhomogeneous phase, a JT polaronic incommensurate charge density wave (ICDW) appears. The anharmonic ICDW forms stripes of undistorted lattice with conducting carriers intercalated by stripes of distorted lattice with trapped localized JT polarons. We can identify the quantum critical point (QCP) for the polaron stripe formation at Spin charge, and lattice fluctuations observed in the electronic properties of

superconductors

the plane

and the

superconductivity occurs in the critical region in

near this quantum critical point.

2. ON THE COEXISTENCE OF JAHN–TELLER POLARONS AND CONDUCTING CARRIERS

An anomalous metallic phase of the Jahn–Teller (JT) polarons in doped cuprate perovskites had been the driving idea for the discovery of the superconductivity by Müller and Bednorz [1]. The scenario of superconducting stripes in superconductors where “the free charges move mainly in one direction, like the water running in the grooves of a corrugated iron foil,” was first introduced in 1992 at Erice [see, e.g., p. 129 in ref. 2]. In fact, the experiments of the Rome group have shown that the anomalous metallic phase is made of conducting carriers and more localized charges forming a superlattice of polaron stripes [2–6]. It was first established that doping introduces holes in the O 2p orbital (L), leaving a single hole in the Cu site, i.e., the configuration of the antiferromagnetic lattice [7]. The experimental evidence of JT polarons was provided by the presence of two different types of doped holes in the oxygen orbital [8–12]: (1) of partial symmetry, mixed with (orbital angular momentum and (2) of symmetry mixed with (orbital angular momentum The pseudo-JT polarons have been associated with the doped holes because the lattice distortion forms molecular orbitals of mixed and angular momentum [2]. The search for the polaronic lattice distortions motivated the Rome group, in collaboration with Calestani in Parma, in 1989 to solve the incommensurate structural modulation of the plane in (Bi2212) by joint single crystal x-ray diffraction and EXAFS [3]. It was known that the superstructure in the diffraction pattern of Bi2212 was due to (i) an in-plane longitudinal compression wave in the Bi-O plane [13], and (ii) the ordering

Lattice-Charge Stripes in the

Superconductors

11

of dopants, interstitial oxygens, forming lines in the diagonal direction in the Bi2O2 layers as observed by neutron diffraction [14]. Our joint x-ray diffraction (XRD) and EXAFS works have shown a third relevant contribution to the superstructure. This is the intrinsic modulation of the lattice due to 1D ordering of polaronic lattice distortions, i.e., JT polaron stripes with an associated incommensurate and anharmonic 1D modulation of the orbital angular momentum (called also orbital density wave, ODW). These results have been confirmed by the technical advances in the collection of Cu K-edge EXAFS spectra with higher signal-to-noise ratio and using polarization effects to select different neighboring atoms. The EXAFS probes fast polaron fluctuations because it gives the Cu-O pair distribution function (PDF) with the measuring time scale of about sec. The measure of the 1D modulation of the Cu-O (apical) bond, was used as a conformational parameter to identify localized JT polaronic charges. This has allowed us to measure the width L of stripes of free carriers in Bi2212 and the size W of polaronic stripes in 1993 [15–22]. Compelling evidence for polaron stripes in the plane of Bi2212 has been obtained by extending the Cu K-edge EXAFS data to higher photoelectron momentum. This allows the measurement of the PDF of Cu-O (planar) pairs with higher resolution [23]. Cu K-edge anomalous XRD has been used to detect directly the anharmonic modulation of the copper oxide plane [24]. The Cu K-edge EXAFS experiments have been extended to other families of cuprate superconductors [25–28] and the temperature-dependent changes associated with polaron formation have been determined. Figure 1 shows the evolution of the Cu-O (planar) pair distribution function in for at optimum doping as a function of temperature. Above a temperature , the PDF shows a single peak with a width determined by thermal fluctuations. Below , the PDF shows the anomalous long bonds, larger than the amplitude of thermal fluctuations, that are a direct measure of the lattice distortion associated with the pseudo-JT polarons. The polaronic local lattice distortions detected by this fast probe below are similar in different superconducting families, as shown in Fig. 2. Although the local lattice fluctuations appear to be a generic feature of probed by a fast probe such as EXAFS, the dynamics of the polaronic stripe fluctuations are very different in different families and at different doping. Only slow fluctuating stripes, as for example in oxygen-doped are observed by probes with a long measuring time such as nuclear magnetic resonance (NMR) [29]. Polaron stripes have also been detected by slowing down their fluctuations by adding impurity centers [30]. The polaronic charge ordering in the inhomogeneous plane gives a superlattice of alternating quantum stripes in the plane with an approximate width of 10 Å in Bi2212 and La214 systems below a charge-ordering temperature If one considers two different kinds of charge carriers in the alternating stripes, i.e., polaronic-type charge carriers in stripe A and free carriers in stripe B, as shown in Fig. 3, one may explain the spin susceptibility data [31]. Compelling evidence for role of polarons in underdoped phase comes from isotope effect experiments on the in-plane penetration depth [32]. The two relaxation processes following laser excitation and the two components observed in infrared spectroscopy have provided evidence for the universal coexistence of polarons and free carriers in superconducting cuprates [33]. 3. THE

PHASE DIAGRAM OF Bi2212

After the discovery of the physics of cuprates was described by a generic phase diagram where the critical temperature is plotted as a function of doping i.e., a

12



Superconductors

13

measure of the charge density and the distance from the Mott Hubbard insulator at By increasing the system goes through quite different states. At low doping, the doped polaronic charges are pinned to impurities and form a disordered electron glass. At high doping, a normal metal phase appears. The high superconducting phase appears between these two phases. In 1993 at the Cottbus conference [16–20], we presented the phase diagram for Bi2212, shown in Fig. 4, that is becoming quite robust at this second international conference on “Stripes and High Superconductivity.”

14


We predicted the formation of an insulating commensurate JT polaron crystal at in with a charge ordering commensurate with the perovskite lattice. Therefore, the doping was measured in units of This polaron crystal is in competition with superconductivity, and is at the origin of the huge suppression of the superconducting critical temperature in at The long-range Coulomb interaction between JT polarons is expected to play a key role in pushing the JT polaron gas toward a charge-ordered phase of localized charges. Therefore, we call this phase a generalized Wigner commensurate JT polaron crystal (CPC). The CPC was not observed at in the phase diagram

shown in Fig. 4, where only a weak minimum of appears at By decreasing the temperature, below about the system forms an inhomogeneous phase where a 1D incommensurate polaronic charge density wave (ICDW) or polaronic stripes coexists with free carriers [16–22]. In the underdoped phase, the charge density of free carriers is smaller than that of JT polarons in the ICDW. In the high-doping phase, the charge density of free carriers is larger than that of JT polarons in the ICDW. This particular ICDW does not suppress but promotes the pairing of the free carriers below In fact, in this unexpected metallic phase of condensed matter, a superlattice of quantum mesoscopic stripes of width L, the chemical potential is tuned to a “shape resonance,” which occurs when the de Broglie wavelength of electrons at the Fermi level The measure of the stripe width L in 1993 established the presence of the shape resonance at optimum doping [15,17–21]. At the shape resonance, the chemical potential is tuned to the bottom of a superlattice subband and therefore to a narrow peak in the density of states (DOS). At optimum doping, a BCS-like superconductivity is observed and the highest is reached where the chemical potential is tuned to this narrow DOS peak via the calculated shape resonance effect on the superconducting gap [34]. A patent has been granted with priority date 7 Dec 1993 [15] for a method of amplification via the shape resonance effect in new materials formed by a superlattice of quantum wires. A very good agreement with experimental data has been found for the calculated critical temperature plotted in Fig. 4 assuming a pairing mechanism mediated by charge fluctuations in a superlattice of quantum wires at the shape resonance [20]. This


Superconductors

15

pairing mechanism, typical of a metal with a negative dielectric constant near the Wigner localization, provides a superconducting phase with a coherence length as short as the particle–particle distance. 4. COMMENSURATE POLARON CRYSTALS Commensurate polaron crystals (CPC) formed by polaron stripes were observed by Chen et al. [35] using electron diffraction in 1993 in the isostructural perovskite family

(nikelates, where Ni substitutes for Cu). Commensurate diffraction peaks due to charge ordering with wavevector which are the characteristic features of CPC, appear in the doping range These peaks coexist with others due to static

spin ordering detected by neutron scattering [36]. It has been well documented that in , the stripe order is particularly stable for hole concentrations 1/3, and 1/2. A mechanism of commensurate polaron ordering [37] has been suggested to be responsible for the formation of stripes in nikelates. This interpretation has been supported by further results providing consistent evidence that polaronic charge order is the driving force in these compounds. In fact, in the incommensurate charge peak occurs at significantly higher temperature than the spin peak, even though the charge order shows a shorter correlation length than the spin order. In 1989, Zaanen and Gunnarson [38] predicted that in a two-dimensional (2D) AF lattice of a Mott–Hubbard insulator the doped holes, at low doping, would not form a correlated 2D metal but would segregate in insulating 1D chains, or domain walls separating undoped AF domains, that we call here spin-charge stripes or spin stripes. In 1993, Johnston et al. [39] at Ames found experimental evidence for the predicted formation of 1D-domain walls in the doped antiferromagnetic lattice in the insulating phase of cuprates. The localized doped charges form linear domain walls separating the antiferromagnetic domains in insulating at Sr doping of the order of Emery and Kivelson [40] proposed that the holes pushed away from the undoped AF spin lattice form metallic droplets. This phase separation of hole-reach droplets and hole-depleted AF regions is frustrated by the presence of the Coulomb interaction due to the static and homogeneous distribution of negative dopants in the rocksalt layers. This frustrated phase separation gives bubbles or fluctuating droplets. Following the experimental evidence for ID polaronic stripes, which we reported in 1992 [1–6], the stability of arrays of spin-charge stripes in the AF Mott–Hubbard insulator [due to frustrated phase separation in the low doping region depending on the value of the long range Coulomb repulsion] has been calculated [41]. These arrays of spin-charge stripes are characterized by the presence of undoped AF domains. These hole-depleted regions have been observed by muon spin resonance in the low-doping range [42]. The formation of a static insulating generalized Wigner JT CPC at the critical doping was predicted to explain the drop of at The static spin and lattice modulation associated with this CPC of doped charges at in was solved by Tranquada in 1995 [43]. Following this work, the dynamic spin fluctuations detected by inelastic neutron scattering [44,45] in superconducting samples have been associated with dynamical spin-charge stripes. The suppression of the photoelectron spectral weight at hotspots of the Fermi surface of Bi2212 provides direct evidence for coupling of the electrons with the ICDW in Bi2212 [46,47].

16


The presence of insulating electronic crystals of doped charges forming a striped generalized Wigner JT CPC have also been found in doped

Here, the

JT electron lattice interaction is larger than in cuprates, and the doped charges form small polarons where the charge is localized on a single Mn ion. La doping forms JT in a lattice formed of non-JT . In this strong coupling limit, static polaronic striped CPC are observed for doping

and

forming beautiful superlattice of

mesoscopic stripes commensurate with the lattice. At small-charge densities, a disordered electronic glass of pinned charges is formed.

The strength of JT electron lattice interaction can be measured from the local lattice distortion [2]. The JT distortion of the Mn-O coordination shell has been measured by us using EXAFS. Figure 5 shows the variation of the Mn-O pair distribution function (PDF)

detected by Mn K-edge EXAFS going from the CPC to the metallic phase. The JT elongation of a Mn-O bond at is much larger than that observed in the cuprates. At high doping, , an insulator-to-metal transition is observed, and the amplitude of the polaronic distortions in the Mn-O PDF decreases in the metallic phase also if it does not go to zero. This last metallic phase, characterized by charge inhomogeneity, shows colossal magneto-resistance (CMR).

5. SUPERCONDUCTIVITY AND QUANTUM PHASE TRANSITIONS

Following the discovery of stripes in Bi2212 superconducting cuprates, the field has expanded, and there is now a consensus that 1. 2.

3.

In several families of cuprates, the superconductivity coexists with an ICDW In some families, at an insulating charge-ordered phase, which we call commensurate polaron crystal (CPC), competes with the superconductivity and is suppressed CPCs at several doping levels appear in manganites and nikelates. It is well established that superconductivity with

values in metals and alloys

appear when two conditions are satisfied: (1) the chemical potential is tuned to a peak in the density of states, and (2) the metallic phase is close to an instability that gives a large particle–particle attraction.


Superconductors

17

The chemical potential at the optimum doping is tuned to a narrow peak in the density of states formed by the superlattice of stripes, i.e., via the shape resonance effect [34].

Therefore, in the stripe scenario the first requirement for high is satisfied [34]. The second requirement can be satisfied if the cuprates are near a critical point for the SDW-to-metal or the CDW-to-metal quantum phase transition (QPT) [49]. A QPT is a zero temperature genetically continuous transition tuned by a parameter in the Hamiltonian. Near this transition, quantum fluctuations take the system between two distinct ground states. Examples of QPT include the metal-to-insulator transition in disordered alloys, the integer and fractional Quantum-Hall transitions, magnetic transitions in heavy Fermion alloys, and the superconducting-to-insulator transition in granular superconductors [50]. The presence of nonconventional pairing mechanisms in superconductivity near a QPT

is well established in several exotic materials [51]. In several organic materials and heavy Fermions, the superconducting phase appears by increasing the pressure above a critical value. The superconducting phase is in the regime of quantum fluctuations near a QPT from a metal- to an SDW-ordered phase [51]. In barium bismuthates and the exotic superconducting phase is near QPT to a charge-ordered phase

(CDW) due to valence skipping [49,51]. The superconducting phase with a short coherence length in these materials has clear similarities with A phenomenologic model for the low-energy spin dynamics in the normal state of

shows that these systems are close to a QPT [52]. It has been proposed that the critical point is due to the doping of the AF Mott-Hubbard insulator. However, in this case, the predicted maximum is expected at the critical point for disappearing of AF order in the range

in disagreement with the experiments. Other authors have

considered a case of QPT near a metal-to-insulating CDW phase transition [54–56]. There is large disagreement on the location of a CDW critical point in the

phase diagram.

Moreover, the critical point for the spin fluctuations (SDW) seems to be different from that for the charge fluctuations (ICDW).

Recently, some experiments have provided further compelling evidence for quantum critical fluctuations in

superconductors [57,58]. Therefore, the key problem to be

solved is the actual nature of the QPT and the location of the critical point present in 6. THE JT POLARON CRYSTAL IN OXYGEN-DOPED

In order to shed light on the nature of the unknown quantum critical point in we have studied the simplest doped cuprate perovskite, oxygen-doped by XRD. There is no frustrated phase-separation regime in because the dopants are mobile oxygen counterions. It shows the phase separation (below 300K) between an insulating doped AF lattice and a metallic phase in the range of doping In this range, it does not show the spin glass phase of the doped Mott insulator where the spin/charge stripes due to frustrated phase separation are expected. The model of frustrated phase separation cannot explain here the formation of the incommensurate polaronic charge ordering, or “lattice-charge stripes,” that appear in the phase-separated superconducting phase. Polaronic stripes in the plane were found to exist by several experimental techniques probing different physical parameters: NMR revealing two different Cu sites [29], and EXAFS solving the local rhombic distortions characteristic of the pseudo-Jahn-

Teller polarons [28]. The universal 1D incommensurate dynamical spin fluctuations have been observed below 60K by inelastic magnetic neutron scattering.

18


crystal was doped by electrochemical oxidation with The diffraction data were collected on the crystallography beam-line at the wiggler source on the Elettra synchrotron radiation source. We have collected the data using an imaging plate detector. Figure 6 shows a representative diffraction pattern. In our crystal, we observe the coexistence of two different charge-ordered domains. The first domains show the diffuse diffraction peaks, denoted by D in the diffraction image shown in Fig. 6, where the subscript I (II) stands for the first-second harmonic. They are assigned to incommensurate charge ordering (ICDW and/or ODW) in the plane or a superlattice

of diagonal stripes with wavevector (using the orthorhombic notation). The short-range ICDW is indicated by the large width of the diffraction line giving a short coherence length of about 350 Å. The anharmonicity of this modulation is evident from the large intensity of higher harmonics (second and third). The second harmonic of the charge modulation at has the same wavevector of the nesting vector at observed by Saini et al. [46] on the Fermi surface of Bi2212 that induces the suppression of the spectral weight at selected spots in the k space and gives a broken Fermi surface. This ICDW coexists with the superconducting phase with critical temperature 40K, measured by surface resistivity in the radio frequency region at optimum doping, The oxygen distribution in the sample is not homogeneous and we have found that about half of the volume is not superconducting. These portions are made of a second set

of domains characterized by narrow, resolution-limited diffraction peaks (denoted by E in Fig. 6) with a commensurate 2D modulation with wavevector Here, the superlattice of diagonal charge stripes has a commensurate period of 21.5 Å and each stripe has a finite length of 60 Å. The structure ofthese domains is formed by diagonal stripes with a period of four lattice units, 4b, that indicate the formation of CPC with a local doping Several experimental tests indicate that this domain is associated with an insulating charge-ordered nonsuperconducting phase with static spin ordering. The charge-ordering temperature for this commensurate polaron crystal is about 270K.


Superconductors

19

7. THE UNIVERSAL PHASE DIAGRAM OF DOPED PEROVSKITES AND THE QUANTUM CRITICAL POINT

From these studies, we have deduced a phase diagram

for the electronic phases is the charge density given by doping and defines a parameter measuring the polaronic JT electron lattice interaction [2] that characterizes different perovskite compounds. The central red region A in Fig. 7a indicates the stability of the CPC in different perovskites with variable JT electron-lattice interaction The JT polarons strongly interact with the lattice; therefore, they form CPC at the charge densities giving spatial arrays of polarons commensurate with the lattice. The key role of long-range Coulomb interaction is indicated by the actual striped structure of the CPC. In nikelates and manganites, the CPCs appear at doping The incommensurate modulations for intermediate doping levels are formed by the mixture of the two near-neighbor commensurate phases. It is well established experimentally that at low doping the CPC becomes unstable and a transition from CPC to a low-density phase of an electron glass (the yellow region in Fig. 7a) that is in the localization limit appears. in the doped perovskites (cuprates, nikelates, and manganites) shown in Fig. 7a, where

20


In the limit of high densities, the polarons overlap and the charge is separated from the local lattice JT distortion, giving the gas of itinerant carriers (the blue region B in Fig. 7a). Between these homogeneous phases there is an inhomogeneous phase, where polarons and

free carriers coexist (the green region metal transition from CPC to B, at

). For example, in manganites at the insulator-toan inhomogeneous phase of coex-

isting fluctuating polaronic domains and free carriers has been observed, and it shows CMR. Considering all families of cuprate superconductors, each one characterized by its electron-lattice interaction, in the intermediate polaronic coupling regime we can identify

the point indicated by the black dot in Fig. 7a (upper panel), where the CPC appears. This is above the critical value of the electron-lattice interaction, where at metallic density the charges are trapped by local lattice distortions. The cuprates form a CPC only for a single value

in few families where the

block layers are rocksalt fcc layers, as in which we have studied here. The doping is a critical value for the formation of the polaron commensurate crystal in cuprates [16–20] at the JT electron-lattice interaction Increasing the charge density the near-neighbor CPC, expected at does not appear in cuprates because the pseudo-JT polarons dissociate and the homogeneous metallic phase B occurs for The near-neighbor CPC at lower density, does not

form because of the competing disordered phase in the localization limit. In oxygen-doped there are two different electronic phases by increasing the doping following the upper white horizontal arrow in Fig. 7a: (1) the underdoped regime, where the system is unstable between the CPC and the disordered glassy phase; and (2) the optimum doping phase,

where the system is unstable between the CPC and phase B of itinerant carriers. Figure 7a shows that the insulating CPC appears in cuprate perovskite families only for the JT electron-lattice interaction whereas for the CPC does not show up, as in the case of Bi2212 shown in Fig. 4, because we do not cross the CPC following the lower horizontal arrow. In this material, the superconducting phase always remains in the coexistence regime The physics of cuprates is dominated by the green inhomogeneous phase, where the phases A and B coexist. There is a quantum critical point (QCP) indicated by the blue ellipse in Fig. 7a at We have found that the superconducting phase in the cuprate is near this QCP. The polaronic electron–lattice interaction in cuprate families is driven by the compression of the plane. superconductors are heterogeneous materials made of alternated layers of metallic bcc layers and insulating rocksalt fcc AO layers [59,60]. The bond-length mismatch across a block-layer interface is given by the Goldschmidt tolerance where and are the respective bond lengths in homogeneous isolated parent materials A-O and The hole-doped cuprate perovskite heterostructures are stable in the range which corresponds to a mismatch of The sheets are under compression and (AO) layers under tension. The electron–lattice interaction of the pseudo-JT-type is given by [2], where Q is the conformational parameter for the distortions of the square, like the LTT-type tilting and its rhombic distortion; is the dimpling angle that measures the displacement of the Cu ion from the plane of oxygens; and is the JT splitting that is modulated by the Cu-O (apical) bonds. Q and/or increases with the increasing mismatch Moreover, the mismatch induces also the decrease the polaronic bandwidth [61].


Superconductors

21

Therefore, the pseudo-JT polaronic electron–lattice interaction is a function For small variations of in all cuprate perovskites we can assume a proportionality between and We have measured the mismatch for each sample from the measured average bond by EXAFS and diffraction; where is the Cu-O equilibrium distance. The critical temperature of several cuprate families as a function of doping and mismatch is plotted in color picture shown in Fig. 7b (lower panel). It is clear from the 2D plot that the superconductivity appears around the point identified by the point where the CPC appears with the minimum electron–lattice interaction is given by the families of and oxygen-doped at the doping The cuprates with mismatch larger than the mismatch occupy the upper region of the rectangle, whereas those in the weak electron–lattice interaction regime show only quantum polaron fluctuations, i.e., dynamic incommensurate charge density waves, in the neighboring region around the black point The maximum appears in the circle indicated in the blue region of the lower panel at In Fig. 8a, we reported the value of the critical temperature as a function of the lattice mismatch at a fixed doping, [i.e., along the vertical arrow passing across the black dot in Fig. 7a]. The superconducting phase appears below in the green

22


area. It is evident that the critical temperature reaches a maximum value near the critical value which is the quantum critical point (QCP) shown in Fig. 7a. The experimentally observed charge-ordering temperature for the Lal24 and Bi2212 systems at are shown separating the metallic region (blue) from the coexistence region, or ICDW and free carriers, and the JT polaron commensurate crystal phase A (yellow). The critical mismatch is the critical point for the onset of the polaron stripes that is associated with the critical point for spin-ordering SDW. The quantum critical point at gives the highest superconducting transition temperature as expected. The superconducting phase is rapidly suppressed by the formation of the CPC that competes with the superconducting order. An anomalous metallic phase in a quantum critical regime is expected at (blue region) as shown by many experiments. The plot at constant doping, i.e., for optimally doped samples, follow-

ing the second vertical white arrow in Fig. 7a is shown in Fig. 8 (upper panel). The plot shows a quantum critical point In this regime, the superconducting phase extends well beyond the critical point for indicating that the particular superlattice of polaron stripes due to quantum fluctuations can coexist and amplify the critical temperature in agreement with the well-established shape resonance amplification of the critical temperature for this particular ICDW [17,18,34]. In summary, we have shown that

superconductivity occurs at the critical point

of the electron-lattice interaction for the formation of local lattice distortions at metallic density. The electron-lattice interaction is driven at this critical point by the lattice mismatch between the metallic layers and the intercalated block layers. The maximum occurs at the critical point for the transition from a homogeneous metallic phase to an inhomogeneous metallic phase with coexisting polaron stripes and free carriers at For by decreasing the temperature the materials exhibit a transition at from the homogeneous (B) to the inhomogeneous phase that can be defined as the temperature for the polaron stripe formation. We can understand now the complex phenomenology of cuprates showing quite different superconducting and normal phases in the underdoped to overdoped regime. For the superconductivity is suppressed by the formation of the CPC. On the contrary, the superconductivity coexists with the ICDW. In the underdoped regime at , the highest appears in a low-density inhomogeneous phase where the number of free carriers is smaller than that of polarons. In this regime, superconductivity is formed by local pairs with anomalous large ratio [31,32]. In the optimum doping regime, the density of itinerant carries is larger than that of localized carriers. In conclusion, we have deduced a phase diagram for the superconducting phases where depends on both doping and lattice mismatch The anomalous normal phase of cuprate superconductors is determined by an inhomogeneous phase with coexisting polaron stripes and itinerant carriers that appears for an electron lattice interaction larger than a critical value Lattice-charge stripes or polaron stripes appear in this critical fluctuation regime. The standard plots of La214 and Bi2212 do not cross the quantum critical point. The lattice mismatch drives the electron lattice interaction to a QCP of a quantum phase transition. The plot crosses the critical point and the highest superconducting critical temperature occurs at These results show that the particular spin and charge critical fluctuations in the inhomogeneous phase favor the superconducting pairing.


Superconductors

23

ACKNOWLEDGMENTS

Thanks are due to Alessandra Lanzara for help and discussions. This research has been supported by “Istituto Nazionale di Fisica della Materia” (INFM), by the “Ministero dell’Universita’ e della Ricerca Scientifica” (MURST) Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale, and by “Progetto 5% Superconduttivita of Consiglio Nazionale delle Ricerche” (CNR). REFERENCES 1. A. K. Müller, in Proc. of the 2nd Int. Conf. “Stripes and High Superconductivity” held in Rome, June 1998, Stripes and Related Phenomena, Kluwer, Plenum Press, NY, 2000. 2. A. Bianconi, in Proc. of the Erice Workshop, May 6–12, 1992, on Phase Separation in Cuprate Superconductors, edited by K. A. Muller and G. Benedek, World Scientific, Singapore (1993) at p. 352; ibid. p. 125. 3. A. Bianconi, P. Castrucci, M. De Simone, A. Di Cicco, A. Fabrizi, C. Li, M. Pompa, D. Udron, A. M. Flank, P. Lagarde, and G. Calestani, in High Temperature Superconductivity, edited by C. Ferdeghini and A. S. Siri, World Scientific Co., Singapore (1990) 144; A. Bianconi in International Conference on Superconductivity

(proc. of the meeting held in Bangalore, India, Jan 10–14, 1990), edited by S. K. Joshi, C.N.R. Rao, and S. V. Subramanyam, World Scientific, Singapore, p. 448 (1990). 4. A. Bianconi, A. M. Flank, P. Lagarde, C. Li, I. Pettiti, M. Pompa, and D. Udron, in High Temperature Superconductivity, edited by J. Ashkenazi, S. E. Barnes, F. Zuo, G. C. Vezzoli, and B. M. Klein (Plenum, New York, 1991), p. 363. 5. A. Bianconi, S. Della Longa, M. Missori, I. Pettiti, and M. Pompa, in Lattice Effects in Superconductors (proc. of the meeting held in Santa Fe, New Mexico, January 13–15, 1992), edited by Y. Bar-Yam, T. Egami, J. Mustre de Leon, and A. R. Bishop; World Scientific Publ., Singapore, (1992), p. 6; A. Bianconi,

S. Delia Longa, M. Missori, C. Li, M. Pompa, A. Soldatov, S. Turtu, and S. Pagliuca, in Proc. of Bejing Int. Conf. High-Temperature Superconductivity (BHTSC’92), edited by Z. Z. Gan, S. S. Xie, and Z. X. Zhao, World Scientific, Singapore, 1993, p. 147. 6. A. Bianconi, S. Della Longa, M. Missori, I. Pettiti, M. Pompa, and A. Soldatov, Jpn. J. Appl. Phys. 32 suppl. 32-2, 578 (1993). 7. A. Bianconi, A. Congiu Castellano, M. De Santis, P. Rudolf, P. Lagarde, A. M. Flank, and A. Marcelli, Solid

State Commun. 63, 1009 (1987); A. Bianconi, A. Clozza, A. Congiu Castellano, S. Della Longa, M. De Santis, A. Di Cicco, K. Garg, P. Delogu, A. Gargano, R. Giorgi, P. Lagarde, A. M. Flank, and A. Marcelli in Int. J. Mod. Phys. B 1, 853 (1987).

8. A. Bianconi, M. De Santis, A. Di Cicco, A. M. Flank, A. Fontaine, P. Lagarde, H. Katayama-Yoshida, A. Kotani, and A. Marcelli, Phys. Rev. B 38, 7196 (1988).

9. Y. Seino, A. Kotani, and A. Bianconi, J. Phys. Soc. Jpn. 59, 815 (1990). 10. M. Pompa, S. Turtù, A. Bianconi, F. Campanella, A. M. Flank, P. Lagarde, C. Li, I. Pettiti, and D. Udron, Physica C 185–189, 1061–1062 (1991). 11. M. Pompa, P. Castrucci, C. Li, D. Udron, A. M. Flank, P. Lagarde, H. Katayama-Yoshida, S. Della Longa, and A. Bianconi, Physica C 184, 102–112 (1991). 12. A. Bianconi, M. Pompa, S. Turtu, S. Pagliuca, P. Lagarde, A. M. Flank, and C. Li, Jpn. J. Appl. Phys. 32 (Suppl. 32-2), 581 (1993). 13. Y. Gao, P. Lee, P. Coppens, M. A. Subramanian, A. W. Sleight, Science 24, 954 (1988). 14. E. A. Hewatt, J. J. Capponi, and M. Marezio, Physica C 157, 502 (1989). 15. A. Bianconi, European Patent N. 0733271: High superconductors made by metal heterostuctures at the atomic limit (priority date 7 Dec 1993), published in European Patent Bulletin 98/22, May 27, 1998.

16. A. Bianconi and M. Missori, in Phase Separation in Cuprate Superconductors edited by E. Sigmund and K. A. Müller, Springer Verlag, Berlin (1994) 272.

17. A. Bianconi, and M. Missori, J. Phys. I (France) 4, 361 (1994). 18. A. Bianconi, Sol. State Commun. 89, 933 (1994). 19. A. Bianconi (invited talk at the Int. Conf. Materials and Mechanisms of Superconductivity, High Temperature Superconductors Grenoble, France, July 5–9, 1994), Physica C 235–240, 269 (1994). 20. A. Bianconi, Sol. State Commun. 91, 1 (1994); ibid. 91, 287 (1994).

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21. A. Bianconi, M. Missori, N. L. Saini, H. Oyanagi, H. Yamaguchi, D. H. Ha, and Y. Nishihara, J. Supercond. 8, 545 (1995).

22. A. Bianconi, M. Missori, H. Oyanagi, H. Yamaguchi, D. H. Ha, Y. Nishihara, and S. Della Longa, Europhys Lett. 31, 4 1 1 (1995).

23. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishihara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996). 24. A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, Å. Kvick, and P. Radaelli, Phys. Rev. B 54, 4310 (1996).

25. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996). 26. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55, 12759 (1997).

27. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Physica C 268, 121 (1996). 28. A. Lanzara, N. L. Saini, A. Bianconi, J. L. Hazemann, Y. Soldo, F. C. Chou, and D. C. Johnston, Phys. Rev. B 55, 9120 (1997).

29. B. W. Statt, P. C. Hammel, Z. Fisk, S. W. Cheong, F. C. Chou, D. C. Johnston, and J. E. Schirber, Phys. Rev. B 52, 15575 (1995).

30. X. L. Dong, Z. F. Dong, B. R. Zhao, Z. X. Zhao, X. F. Duan, L.-M. Peng, W. W. Huang, B. Xu, Y. Z. Zhang, S. Q. Guo, L. H. Zhao, and L. Li, Phys. Rev. Lett. 80, 2701 (1998). 31. K. A. Müller, G.-M. Zhao, K. Conder, and H. Keller, J. Phys. Condens. Matt. 10, L291 (1998). 32. G.-M. Zhao, M. B. Hunt, H. Keller, and K. A. Müller, Nature 385, 236 (1997). 33. D. Mihailovich and K. A. Müller, High

Superconductivity: Ten Years after the Discovery (Nato ASI,

Vol. 343), ed. E. Kaldis, E. Liarokapis, and K. A. Müller (Dordrecht, Kluwer) (1996), p. 243.

34. 35. 36. 37.

A. Bianconi, A. Valletta, A. Perali, and N. L. Saini, Physica C 296, 269 (1998). C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993). J. M. Tranquada, D. J. Butrey, V. Sachan, and J. E. Lorenzo, Phys. Rev. Lett. 73, 1003 (1994). J. Zaanen and P. B. Littlewood, Phys. Rev. B 50, 7222 (1994).

38. 39. 40. 41.

J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7222 (1989). J. H. Cho, F. C. Chu, and D. C. Johnston, Phys. Rev. Lett. 70, 222 (1993). V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993). U. Low, V. J. Emery, K. Fabricius, and S. A. Kivelson, in Phase Separation in Cuprate Superconductors, edited by E. Sigmund and K. A. Müller (Springer Verlag, Berlin-Heidelberg, 1994), p. 383. 42. C. Niedermayer, C. Bernhard, T. Blasius, A. Golnik, A. Moodenbaugh, and J. I. Budnick, Phys. Rev. Lett. 80, 3843 (1998).

43. J. M. Tranquada et al., Nature 375, 561 (1995).

44. H. Yoshizawa, S. Mitsuda, H. Kitazawa, and K. Katsumada, J. Phys. Soc. Jpn. 57, 3686 (1988); S.-W. Cheong, G. Aeppli, T. E. Mason, H. Mook, S. M. Hayden, P. C. Canfield, Z. Fisk, K. N. Clausen, and J. M. Martinez, Phys. Rev. Lett. 67, 1791 (1991); G. Aeppli, T. E. Meson, M. S. Hayden, H. A. Mook, and J. Kulda, Science 278, 1432 (1997); K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, Y. Kimura, S. Ueki, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, M. A. Kastner, and Y. Kim, Phys. Rev. B 57, 6165 (1998) and references therein.

45. H. A. Mook, F. Dogan, and B. C. Chakoumakos, p. 135 in this volume. 46. N. L. Saini, J. Avila, A. Bianconi, A. Lanzara, M. C. Asensio, S. Tajima, G. D. Gu, and N. Koshizuka, Phys. Rev. Lett. 79, 3464 (1997). 47. N. L. Saini, J. Avila, M. C. Asensio, S. Tajima, G. D. Gu, N. Koshizuka, A. Lanzara, and A. Bianconi, Phys. Rev. B 57, R11101 (1998).

48. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996); S. Mori, C. H. Chen, and S.-W. Cheong, Nature 392, 473 (1998); S. Mori, C. H. Chen, and S.-W. Cheong, Phys. Rev. Lett. 81, 3972 (1998); C. H. Chen, S. Mori, and S.-W. Cheong, J. Phys. IV France 9, Pr10-307 (1999). 49. R. Micnas, and S. Robaszkiewicz, in High Tc Superconductivity: Ten years after the Discovery (Nato ASI,

Vol. 343), ed. E. Kaldis, E. Liarokapis, and K. A. Müller (Dordrecht, Kluwer) (1996), p. 31. 50. S. Sachdev, Quantum Phase Transitions (Cambridge Univ. Press, New York, 1999).

51. Superconductivity in Magnetic and Exotic Materials, edited by T. Matsubara and A. Kotani, Springer Series

in Solid State Science 52, Springer-Verlag, Berlin, 1984. 52. A. J. Millis, H. Monien, and D. Pines, Phys. Rev. B 42, 167 (1990); D. Pines, Physica B 163, 78 (1990). 53. N. E. Bikers and S. R. White, Phys. Rev. B 43, 8044 (1991). 54. A. Perali, C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. B 54, 16216 (1996).


Superconductors

25

55. R. Micnas, J. Ranninger, and S. Robasziewicz, Rev. Mod. Phys. 62, 113 (1990). 56. R. Micnas and S. Robasziewicz, Phys. Rev. B 45, 9900 (1992); ibid., Phys. Rev. B 52, 6863 (1995). 57. G. Aeppli, T. E. Manson, S. M. Hayden, H. A. Mook, and J. Kulda, Science 278, 1432 (1997). 58. T. Valla et al., Science 285, 2110 (1999).

59. C. N. R. Rao and A. K. Ganguli, Chem. Soc. Rev. 24, 1 (1995). 60. P. P. Edwards, G. B. Peakok, J. P. Hodges, A. Asab, and I. Gameson, in High Tc Superconductivity: Ten years after the Discovery (Nato ASI, Vol. 343), ed. E. Kaldis, E. Liarokapis, and K. A. Müller (Dordrecht, Kluwer) (1996), p. 135. 61. J. B. Goodenough, Supercond. Sci. Tech. 3, 26 (1990); J. B. Goodenough and A. Marthiram, J. Sol. State Chem. 88, 115 (1990); J. B. Goodenough et al., p. 199 in this volume.


Stripes, Electron-Like and Polaron-Like Carriers, and in the Cuprates J. Ashkenazi1

Both “large-U” and “small-U” orbitals are used to study the electronic structure of the cuprates. A striped structure, with three types of carriers is induced, polaron-like “stripons,” which carry charge, “quasi-electrons,” which carry both charge and spin; and “svivons,” which carry spin and lattice distortion. Anomalous physical properties of the cuprates are derived, and specifically the systematic behavior of the resistivity, Hall constant, and thermoelectric power. Transitions between pair states of quasielectrons and stripons drive high-temperature superconductivity.

1. INTRODUCTION Evidence is growing [ 1 ] thatthe planes in the cuprates possess a static or dynamic striped structure. The physical properties of these materials, and specifically their transport properties, are characterized by intriguing anomalies suggesting the inadequacy of the Fermi–liquid scenario, and/or the coexistence of itinerant and almost localized (or polaron-like) carriers. First-principles electronic structure studies [2] suggest that realistic theoretical models of the electrons in the vicinity of the Fermi level should take into account both “large-U” and “small-U” orbitals [3]. Let us denote the fermion creation operator of a small-U electron in band v, spin and wave vector k by 2. AUXILIARY SPACE

Let us treat the large-U orbitals by the “slave-fermion” method [4]. A large-U electron in site i and spin is then created by if it is in the “upper-Hubbard-band,” and 1

Physics Department, University of Miami, P.O. Box 248046, Coral Gables, FL 33124, USA.


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Ashkenazi

by if it is in a Zhang–Rice-type “lower-Hubbard-band.” Here, and are (“excession” and “holon”) fermion operators, and is (“spinon”) boson operators. The constraint

should be satisfied in every site. Within an auxiliary Hilbert space a chemical-potential-like Lagrange multiplier is introduced to impose the constraint on the average. Physical observables are then projected into the physical space by taking appropriate combinations of Green’s functions of the auxiliary space. Because the time evolution of Green’s functions is determined by the Hamiltonian that obeys the constraint rigorously, effects of constraint violation may result only from the approximations applied treating these Green’s functions. For example, within the “spin-charge separation” approximation, two-particle spinon–holon Green’s functions are decoupled into products of one-(auxiliary)-particle Green’s functions. The spinon states are diagonalized by applying the Bogoliubov transformation

The Bose operators create spinon states with “bare” energies that have a V-shape zero minimum at whose value is either or Bose condensation results in antiferromagnetism (AF), and the spinon reciprocal lattice is extended by adding the wave vector The slave-fermion method is known to describe well an AF state. Because within this method AF order is obtained by the Bose condensation of spinons, the decoupling of two-particle spinon–spinon Green’s functions, relevant for physical spin processes, does

not harm the treatment of spin–spin correlations.

3. STRIPES AND QUASI-PARTICLES Theoretically, a lightly doped AF plane tends to separate into a “charged” phase and an AF phase. A preferred structure under long-range Coulomb repulsion is [5] of frustrated stripes of these phases. Experiment [6] confirms such a scenario, and neutron-scattering measurements [7] indicate at least in certain cases a structure in which narrow-charged stripes form antiphase domain walls separating wider AF stripes. Growing evidence [1] supports the assumption that such a structure exists, at least dynamically, in all the superconducting cuprates. Because the spin-charge separation approximation is valid in one dimension, it should apply for holons (excessions) within the charged stripes, and they are referred to as stripons. They carry charge, but not spin. We denote their fermion creation operators by and their bare energies by It is evident [ 1 ] that the stripes in the cuprates are far from being perfect. Even when they are not dynamic, one expects them to be defected and “frustrated,” and to consist of disconnected segments. Such a structure is fatal for itinerancy in one dimension, and it is reasonable to choose localized states as the starting point for the stripon states.

Stripes, Electron-Like and Polaron-Like Carriers, and

in the Cuprates

29

Other carriers (of both charge and spin) result from the hybridization (in the auxiliary space) of small-U electrons and coupled holon-spinons (excession-spinons) within the AF stripes. We refer to these carriers as quasi-electrons (QEs), and denote their fermion creation operators by . Their bare energies form quasi-continuous ranges of

bands crossing

over ranges of the Brillouin zone (BZ).

4. SPECTRAL FUNCTIONS

The physical observables are evaluated using electrons Green’s functions. Expressions are derived where the observables are expressed in terms of the auxiliary space spectral functions and of the QEs, spinons, and stripons, respectively. The quasi-particle fields are strongly coupled to each other due to hopping and hybridization terms of the Hamiltonian. This coupling can be expressed through an effective Hamiltonian term whose parameters can in principle be derived self-consistently from the original Hamiltonian. It has the form

The auxiliary space spectral functions are calculated through the standard diagrammatic technique where introduces a vertex connecting QE, stripon, and spinon propagators. It turns out that the stripon bandwidth is at least an order of magnitude smaller than the QE and spinon bandwidths. Thus, by a generalized Migdal theorem, one gets that “vertex corrections” are negligible, and a second-order perturbation expansion in is applicable. Applying the diagrammatic technique, self-consistent expressions are derived for the scattering rates and of the QEs, spinons, and stripons, respectively. For sufficiently doped cuprates, the self-consistent solution has the following

features. 4.1. Spinons The spinon spectral functions behave as

for small Thus, T for where is the Bose distribution function at temperature T. Namely, there is no long-range AF order (associated with the divergence in the number of spinons at ).

4.2. Stripons The coupling between the stripon field and the other fields results in the renormalization of the localized stripon energies into a very narrow range around (thus getting polaronlike states). Some hopping via QE–spinon states results is the onset of itineracy at low

temperatures, with a bandwidth of The stripon reciprocal lattice is extended by adding wave vectors corresponding to the approximate periodicity of the striped structure. The stripon scattering rates can be expressed as

30

Ashkenazi

4.3. Quasi-Electrons

The QE scattering rates, resulting from their coupling to the other fields, can be approximately expressed as

This becomesi in the limit and in agreement with “marginal Fermi liquid” phenomenology [8].

in the l i m i t

5. SOME PHYSICAL ANOMALIES 5.1. Lattice Effects (“Svivons”)

As was found by Bianconi et al. [6], the charged stripes are characterized by LTT structure, whereas the AF stripes are characterized by LTO structure. Thus, in any physical process induced by (3), in which a stripon transforms into a QE, or vice versa, followed by the emission/absorption of a spinon, phonons are also emitted/absorbed, and the stripons

have lattice features of polarons. The result is that a spinon propagator linked to the vertex is “dressed” by phonon propagators. We refer to such a phonon-dressed spinon as a svivon. Its propagator can be expressed as a spinon propagator multiplied by a power series of phonon propagators. As dressed spinons, the svivons carry spin, but not charge; however, they also “carry” lattice distortion.

5.2. Optical Conductivity The optical conductivity of the doped cuprates is characterized [9] by a Drude term and mid-IR peaks. Transitions between low energy QE states result in the Drude term. Excitations of the very low energy stripon states result in the mid-IR peaks. Such excitation can leave a stripon in the same stripe segment, exciting spinon and phonon states, or transform a stripon into a QE–svivon state. 5.3. Spectroscopic Anomalies

The electronic spectral function, measured, e.g., in photoemission experiments, includes “coherent” bands, and an “incoherent” background of a comparable integrated weight. The coherent part is due to few QE bands, and the frequently observed bandwidth is consistent with Eq. (5). The incoherent part is due to the quasi-continuum of other QE bands and to the stripon states (note that the spectroscopic signature of stripons is smeared over few tenths of an around due to the accompanying svivon excitations). The observed “shadow bands,” “extended” van Hove singularities (vHs), and normalstate pseudo-gaps result from the effect of the striped structure on the QE bands [10], due to the extension of the reciprocal lattice (discussed above). The vHs are extended to supply the spectral weight for the stripon states, and when the vHs are missing, this transferred spectral weight causes a pseudo-gap in the same place in the BZ [11].


in the Cuprates

31

6. TRANSPORT PROPERTIES

6.1. Electric Current (dc) The electric current j is expressed as a sum of QE and stripon contributions

and

respectively

As was discussed above, stripons transport occurs through transitions to intermediate QE– spinon states. Consequently, the expressions for the currents yield

where is approximately T-independent. This condition is satisfied by the formation of gradients and of the QE and stripon chemical potentials (respectively) in the presence of an electric field. The constraint on the number of electrons imposes

where and are, respectively, the contributions of QEs and stripons to the electrons density of states at

6.2. Electrical Conductivity and Hall Constant The Kubo formalism is applied to derive expressions for the dc conductivity and Hall constant in terms of Green’s functions. Such expressions are based on diagonal and nondiagonal conductivity QE terms and stripon terms and and mixed terms The currents in an electric field E are expressed as

where

Using Eqs. (8) and (10), one can express

and because

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Ashkenazi

the resistivity in the x direction

can be expressed as

Similarly, the Hall constant

can be expressed as

where

In order to find the temperature dependencies of the transport quantities, we apply and given in Eqs. (4) and (5), to which we add temperature-independent impurity scattering terms. Consequently, one can express the temperature dependencies of the conductivity terms using parameters A, B, C, D, N, and our results for the scattering rates

Z, as follows:

The transport quantities are then expressed as

These expressions reproduce the systematic behavior of the transport quantities in different cuprates, except for the effect of the pseudo-gap in underdoped cuprates, ignored here. Results for sets of parameters corresponding to data in and are presented in Figs. 1, 2, and 3, respectively. The idea that and are determined by different scattering rates was first suggested

by Anderson [15], and in his analysis, the T2 term is due to spinons. However, it has been


in the Cuprates

33

observed in ac Hall effect results [16] that the energy scale corresponding to this term is of about 120K. This energy is in agreement with the very low energies of the stripons of our analysis, and not with spinon energies (which are of tenths of an eV).

6.3. Thermoelectric Power

When both an electric field and a temperature gradient are present, one can express and

as

34

Ashkenazi

The thermoelectric power (TEP) is given by

By Eq. (7), the condition Thus, by introducing

for the evaluation of the TEP becomes


in the Cuprates

35

and using Eqs. (11), (23), and (24), we get that the TEP can be expressed as

The QE bandwidths are close to an eV. Thus, one gets

similarly to the TEP in normal metals. However, the stripon bandwidth is of order 0.02 eV.

Thus, one expects

to saturate at

to the narrow-band result [17]

36

Ashkenazi

where is the fractional occupation of the stripon band. This result for the TEP is consistent with the typical behavior observed in the cuprates.

Such a behavior has been parametrized as [18]

It was found [ 17, 19] that

(namely, the stripon band is half full) for slightly overdoped

cuprates. The effect of the doping of a cuprate is [7] both to change the density of the charged

stripes within a plane and to change the density of carriers (stripons) within a charged stripe. It is the second type of doping effect that changes Overdoping is often limited because large density carriers in the charged stripes results in an increase of Coulomb repulsion energy. 7. MECHANISM FOR The coupling Hamiltonian (3) provides a mechanism for The pairing mechanism involves transitions between pair states of QEs and stripons through the exchange of svivons. Such a mechanism has similarities to the interband pair transition mechanism proposed by Kondo [20]. The symmetry of the superconducting gap is strongly affected by the symmetry of the QE–stripon coupling through and is thus related to the symmetry of the normalstate pseudo-gap (also determined by QE-stripon coupling). Similarity between the k-space symmetries of these gaps has been observed [11].

A condition for superconductivity within the present approach is that the narrow stripon is determined by the temperature at which such coherence sets in. When the stripon band is almost empty (or almost full), it can be treated in the parabolic approximation, and it is characterized by the distance of the Fermi level from the bottom (top) of the band at Using a two-dimensional approximation, one can express as band maintains coherence between different stripe segments. Thus, an upper limit for

where in the stripons effective mass and (note that the stripons are spinless).

is their density per unit area of a

plane

Stripon coherence is energetically favorable at temperatures at which there is a clear cutoff between occupied and unoccupied stripon band states, which is of order of

and this determines an upper limit for

This result agrees with the “Uemura plots” [21] if the ratio for stripons is approximately proportional to that for the supercurrent carriers, appearing in the expression


in the Cuprates

37

for the London penetration depth (the supercurrent carriers are hybridized QE and stripon

pairs). The “boomerang-type” behavior of the Uemura plots in overdoped cuprates [22] is understood as a transition between a band-top and a band-bottom behavior.

8. SUMMARY The electronic structure of the cuprates has been studied on the basis of both large-U and small-U orbitals. A striped structure and three types of carriers are obtained: polaron-like stripons carrying charge, QEs carrying charge and spin, and svivons carrying spin and lattice distortion. Anomalous normal-state properties of the cuprates are understood, and the systematic behavior of the resistivity, Hall constant, and thermoelectric power is explained. The mechanism is based on transitions between pair states of stripons and QEs through the exchange of svivons. REFERENCES 1. 2. 3. 4.

Papers in J. Supercond. 10, 4 (1997), and in this issue. O. K. Andersen et al., J. Phys. Chem. Solids 56, 1573 (1995). J. Ashkenazi, J. Supercond. 7, 719 (1994); ibid. 8, 559 (1995); ibid. 10, 379 (1997). S. E. Barnes, Adv. Phys. 30, 801 (1980).

5. V. J. Emery, and S. A. Kivelson, Physica C 209, 597 (1993). 6. A. Bianconi et al., Phys. Rev. B 54, 12018, (1996); Phys. Rev. Lett. 76, 3412 (1996). 7. J. M. Tranquada et al., Phys. Rev. B 54, 7489, (1996); Phys. Rev. Lett. 78, 338 (1997).

8. C. M. Varma et al., Phys. Rev. Lett. 63, 1996 (1989). 9. D. B. Tanner, and T. Timusk, Physical Properties of High Temperature Superconductors III, edited by

D. M. Ginsberg (World Scientific, 1992), p. 363. 10. M. I. Salkola et al., Phys. Rev. Lett. 77, 155 (1996). 11. D. S. Marshall et al., Phys. Rev. Lett. 76, 4841 (1996). 12. T. R. Chien et al., Phys. Rev. Lett. 67, 2088 (1991).

13. Y. Kubo and T. Manako, Physica C 197, 378 (1992). 14. 15. 16. 17. 18.

H. Takagi et al., Phys. Rev. Lett. 69, 2975 (1992); H. Y. Hwang et al., ibid. 72, 2636 (1994). P. W. Anderson, Phys. Rev. Lett. 67, 2092 (1991). H. D. Drew, S. Wu, and H.-T. S. Linh, preprint. B. Fisher et al., J. Supercond. 1, 53 (1988); J. Genossar et al., Physica C 157, 320 (1989). S. Tanaka et al., J. Phys. Soc. Jpn. 61, 1271 (1992).

19. K. Matsuura et al., Phys. Rev. B 46, 11923 (1992); S. D. Obertelli et al, ibid., p. 14928; C. K. Subramaniam et al., Physica C 203, 298 (1992). 20. J. Kondo, Prog. Theor. Phys. 29, 1 (1963). 21. Y. J. Uemura et al., Phys. Rev. Lett. 62, 2317 (1989). 22. Ch. Niedermayer et al., Phys. Rev. Lett. 71, 1764 (1993).


Charge Ordering and Stripe Formation in Cuprates Annette Bussmann-Holder1

Results from recently obtained exact solutions of an anharmonic electron–phonon interaction model are used to calculate the response of the electronic subsystem to the dynamics of the lattice. Depending on the degree of anharmonicity, anomalies in the k-dependent energy gap appear that in real space result in modulated chargerich and charge-poor areas and define a new length scale. The results are related to recent experimental findings from EXAFS, PDF, and NMR techniques that have been interpreted in terms of stripe formation.

The local structure of transition metal oxides and especially high-temperature superconducting (HTSC) copper oxides deviates strongly from their average crystallographic structure [1], as for example, obtained from x-ray scattering techniques. Specifically, the oxygen ions are observed to tend to displaced positions that alternate periodically on a new length scale to form stripe and tweed patterns [2]. These experimental results have been obtained from EXAFS, PDF, and NMR techniques that use a much faster time scale than conventional scattering techniques and test local atomic positions within a range of a few lattice constants. Even though the data have been interpreted in terms of charge and spin ordering, the actual experimental information is obtained from measuring the individual interatomic distances that in turn are compared to the average structure data. The observed local ionic distortions have recently been modeled by an anhamonic electron-density–multiphonon interaction Hamiltonian [3] that has been solved exactly numerically for arbitrary wave vector q. It was found there that the degree of anharmonicity determines the “superstructure” pattern that is dynamic and obeys new time and length scales that are not accessible to conventional scattering techniques, but show up there only through e.g., line width broadening or unusual Debye–Waller factors. 1

Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 11, D-70569, Stuttgart, Germany.


39

40

Bussmann-Holder

In the following, the same model and its nonlinear solutions for the ionic displacements are used to investigate the effect of multiphonon–electron-density interactions on the electronic band energies. In addition, the induced dynamical potential associated with these solutions is investigated to seek for the origin of the unusual pseudo-periodic pattern found for the displacement coordinates.

The Hamiltonian is thought to model the planes, where anharmonic effects arise mainly from the oxygen ion nonlinear polarizability [4]. As has been verified in transition metal perovskite oxides [5], p-d nonlinear hybridization effects are crucial in understanding the lattice dynamics of these systems. The structural similarity of HTSC and compounds suggests that a similarly important role is played by them here as well, specifically because from the experimental data, unusual dynamics are mainly related to the oxygen ions. The local instability of the is best modeled by introducing onsite multiphonon-density–density interactions whereas the Cu-ion and its d-electron shell interact harmonically with the phonon bath and the p-electron bands. The corresponding Hamiltonian reads

are the momentum k-dependent energies of the electrons with creation and annihilation operators and is the p-d hopping integral, and for simplicity the d-electron Coulomb repulsion is neglected. The lattice Hamiltonian consists of the harmonic q-dependent part with energy and phonon creation and annihilation operators The degree of lattice anharmonicity is determined by the magnitude of V, which might be larger than the harmonic term. In the Fröhlich-type electron-phonon interaction is proportional to whereas the strength of density-density multiphonon in– teractions is given by . Note that acts on both bands whereas is active at the oxygen ion lattice site only. Depending on the sign of and polaron formation might occur as well as local double-well potentials can be generated. The static and dimerized ground states of the Hamiltonian have been discussed in detail [6,7], but neglecting the terms proportional to, Spin–Peierls and Jahn–Teller instabilities may result [6,7], as well as charge and spin density wave ground states. The new terms considered here, even when they are small, have important consequences for the real space dynamics because they induce anomalies in the displacement coordinates at finite q, which are absent if they are neglected. From Eq. 1, the equations of motion can easily be derived if specific energy and momentum conservation processes are considered. As has been shown in [3], the resulting

Charge Ordering and Stripe Formation in

Cuprates

41

equations of motion represent a nonlinearly coupled system that can be solved in a rather straightforward manner if a self-consistent phonon approximation scheme is used [8]. The pseudo-harmonic approach has advantages to the exact solutions when the calculation of temperature-dependent averages is carried through, as for example, the soft mode temperature dependence in ferroelectrics. It misses, of course, the huge variety of nonlinear

solutions inherent to the problem that are relevant in describing modulated structures. In order to obtain the extra dynamics on top of the phonon dynamics, the exact solutions have been calculated numerically by starting with trial frequencies to solve for the ionic displacement coordinates, reinsert these back to obtain the corresponding frequencies, and iterate until convergence is achieved. Here, a very slow convergence is typical and a signature of the metastability of the system. As has been shown previously [3], the ionic displacement coordinates develop a pseudo-periodic pattern in real space that can be identified with experimentally observed stripe formation and exhibiting the same length scale of a few lattice constants. Here, the emphasis is put on the electron–phonon-induced modulations in the electronic energies that modify as follows:

where is the Fermi energy and are the self-consistently obtained solutions of the equations of motion. The index 1 refers to the oxygen ion, 2 to the Cu ion, respectively,

and Temperature effects are implicitly incorperated in Eq. 2 through varying which determines the barrier height of the local double-well potential if and have opposite signs. For both and having the same sign, no double well exists; is a measure of the degree of anharmonicity. Inserting the solutions of [3] in Eq. 3, the induced gap is obtained, which is Fourier transformed to yield the real space modulations of the elctronic bands. The results are shown in Fig. 1 for various values of and and always have opposite signs. In all three cases that have been investigated, nearly periodic modulations of are observed with depending pseudoperiodicity but always being of the order of several lattice constants. These modulations can easily be interpreted as regions of charge-rich and charge-poor areas

correponding to charged stripes. Note that the periodicity found numerically is of the same order of magnitude as observed experimentally. Including antiferromagnetic interactions between the Cu-ions yields additional features that texture the charge-poor areas with antiferromagnetic spin–spin correlations [9]. In terms of a polaronic scenario, the charge-rich areas can be viewed as dynamic polarons that, due to the existence of charge-poor spincorrelated regimes, do not dominate the dynamics but coexist with the antiferromagnetic polarons. Preliminary results for a three-dimensional model for HTSC [9] yield convinc– ing evidence that the dynamic polarons are related to effects stemming from the c axis, which drives a self-consistent buckling of the planes and stabilizes an in-plane singlet state [10]. The extra dynamics due to anharmonicity and higher-order electron–phonon effects are accompanied by potential fluctuations. In real space (Fig. 2), these potential fluctuations are very close to zero within a few lattice constants and small anharmonicity. Increasing

the depth of the double-well potential leads to short-range repulsive areas. It is important to

42

Bussmann-Holder

Charge Ordering and Stripe Formation in

Cuprates

43

note here that a small degree of orthorhombicity is sufficient to always observe the repulsive regimes. This specifically applies to the HTSC compounds. In conclusion, it has been shown that stripe formation of the lattice is accompanied by modulations of the electronic energies that form charge-rich and charge-poor areas on the same length scale as the lattice. Although the charge-rich domains can be identified with c axis–induced polarons, the charge-poor areas are dominated by antiferromagnetic

correlations and have a strong similarity to the Zhang–Rice singlet state [10]. REFERENCES 1. See, e.g., Lattice Effects in Superconductors, Y. Bar-Yam, T. Egami, J. Mustre-de-Leon, and A. R. Bishop, eds. (World Scientific, Singapore, 1992). 2. A. Bianconi et al, Phys. Rev. Lett. 76, 3412 (1996); S. D. Conradson, D. Raistrich, and A. R. Bishop, Science 248, 1394 (1990); T. Egami et al., Proceedings of the International Workshop on Anharmonic Properties of

Cuprates, Bled 1994, D. Mihailovic, G. Ruani, E. Kaldis, and K. A. Müller, eds. (World Scientific, Singapore, 1994), p. 118; H. L. Edwards et al., Phys. Rev. Lett. 73, 1154 (1994); see also the contributions presented in this special issue. 3. A. Bussmann-Holder and A. R. Bishop, Phys. Rev. B 56,5297 (1997); A. Bussmann-Holder and A. R. Bishop,

4. 5. 6. 7.

J. Supercond. 10, 289 (1997). A. Bussmann, H. Bilz, R. Roenspiess, and K. Schwarz, Ferroelectrics 25, 343 (1980). H. Bilz, G. Benedek, and A. Bussmann-Holder, Phys. Rev. B 35, 4840 (1987). Y. Lepine, Phys. Rev. B 28, 2659 (1983). S. Kivelson, Phys. Rev. B 28, 2653 (1983).

8. E. Pytte, Phys. Rev. B 10, 4637 (1974); 5, 3758 (1972); E. Pytte and J. Feder, Phys. Rev. 187, 1077 (1969);

A. Bussmann-Holder, H. Bilz, and G. Benedek, Phys. Rev. B 39, 9214 (1998). 9. A. Bussmann-Holder, unpublished, 1998.

10. F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 (1988).


The Stripe-Phase Quantum-Critical-Point Scenario for Superconductors S. Caprara,1 C. Castellani,1 C. Di Castro,1 M. Grilli,1 A. Perali,1 and M. Sulpizi1

A summary is given of the main outcomes of the quantum-critical-point scenario for superconductors, developed in the last few years by the Rome group. Phase separation, which commonly occurs in strongly correlated electronic systems, turns into a stripe instability when Coulomb interaction is taken into account.

The stripe phase continuously connects the high-doping regime, dominated by charge degrees of freedom to the low-doping regime, where spin degrees of freedom are most relevant. Dynamical stripe fluctuations enslave antiferromagnetic fluctuations at high doping. Critical fluctuations near the stripe instability mediate a singular interaction between quasi-particles, which is responsible for the

non-Fermi liquid behavior in the metallic phase and for the Cooper pairing with d-wave symmetry in the superconducting phase.

1. THE FRAMEWORK Since the discovery of superconducting (SC) copper oxides [1], a formidable effort

has been produced to provide a unified theory for the rich phase diagram of these materials (Fig. 1). The antiferromagnetic (AFM) phase at zero and very low doping is usually described

as resulting from the strongly correlated nature of the copper-oxygen planes, within the Hubbard model or the related model. As far as the SC phase is concerned, the main points under investigation are the nature of the (strong) pairing mechanism, the unusual (d-wave) symmetry of the order parameter, and the strong dependence of the critical temperature on the doping x. 1

Dipartimento di Fisica—Università di Roma “La Sapienza” and Istituto Nazionale per la Fisica della Materia, Unità di Roma 1, P.le A. Moro 2, I-00185 Roma—Italy.


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46

Caprara, Castellani, Di Castro, Grilli, Perali, and Sulpizi

The properties of the normal state are to some extent even more challenging, the standard Fermi liquid (FL) theory appearing to be violated. The copper oxides are characterized by a low dimensionality, revealed by the strong anisotropy of the transport properties. In the metallic phase above at optimum doping, a non-FL behavior sets in, with a linear in-plane resistivity over a wide range of temperatures [2], indicating the absence of any energy scale, besides the temperature itself. In the underdoped region, two new temperature scales appear above The higher scale, marks the onset of a new regime characterized by a reduction of the quasi-particle density of states, and is mainly revealed by the presence of broad maxima in the spin susceptibility [3] and a downward deviation of the in-plane resistivity as a function of the temperature [4]. At a lower temperature, a (local) gap in the spin and charge channels appears in ARPES [5–7], NMR [8], neutron scattering [9–12], and specific heat measurements [13]. Anderson [14] proposed to extend the Luttinger liquid behavior to and explain the anomalies in the metallic phase. However, no sign of such a new quantum metallic state was found within a renormalization-group approach in Rather, a dimensional crossover drives the system to a FL state as soon as d > 1 in the presence of short-range forces [16]. When long-range forces are taken into account, a non-FL behavior may arise in the presence of a sufficiently singular interaction ' with The onset of an instability is a mechanism that provides a suitable singular scattering. Indeed, critical fluctuations mediate an effective interaction between quasi-particles where V is the strength of the static effective potential at criticality, Q is the critical wavevector, and is a mass term that is

related to the inverse of the correlation length and provides a measure of the distance from

The Stripe-Phase Quantum-Critical-Point Scenario

47

criticality. The characteristic time scale of the critical fluctuations is We point out that the static part of this effective interaction has the form of the Ornstein–Zernike critical correlator. Proposals about the nature of the relevant instability include (i) an AFM Quantum Critical Point (QCP) [ 18, 19], (ii) a charge-transfer instability [20], (iii) an as-yet-unidentified

QCP regulating a first-order phase transition between the AFM state and the SC state [21 ], and (iv) an incommensurate charge-density-wave (ICDW) QCP [22, 23]. The theory of the AFM QCP [19] is based on the hypothesis that the presence of an AFM phase at low doping is the relevant feature common to all cuprates and on the observation that strong AFM fluctuations survive at larger doping [9–12]. However, at doping as high as the optimum doping, it is likely that charge degrees of freedom play a major role, whereas spin degrees of freedom follow the charge dynamics, and are enslaved [24] by the charge instability controlled by the ICDW QCP [22,23]. The AFM fluctuations are thus extended to a region far away from the AFM QCP, due to the natural tendency of hole-poor domains toward antiferromagnetism. The strong interplay between charge and spin degrees of freedom gives rise to the “stripe phase,” which continuously connects the onset of the charge instability (ICDW QCP) at high doping, with the low-doping regime characterized by the tendency of the AFM background to expel mobile holes. Because of this, we more properly refer to the ICDW QCP as the stripe QCP. Therefore, we point out that the presence of a stripe QCP is not alternative to the presence of the AFM QCP, which is found at lower doping. The two points control the behavior of the system in different regions of doping. However, the existence of a QCP at optimum doping, where no other energy scale besides the temperature is present in transport measurements, is the natural explanation for the peculiar nature of this doping regime in the phase diagram of all SC copper oxides. There is an increasing amount of theoretical and experimental evidence in favor of the presence of a QCP near optimum doping. Indeed, the instability with respect to phase separation (PS) into hole-rich and hole-poor regions is a generic feature of models for strongly correlated electrons with short-range interactions [25], which is turned into a frustrated PS [26] or in an ICDW instability [22] when long-range Coulomb forces are taken into account to guarantee large-scale neutrality. Close to PS (or ICDW), there is always a region in parameter space where Cooper pair formation is present, pointing toward a connection between PS and superconductivity. The most compelling evidence for a QCP near optimum doping is provided by the resistivity measurements. An insulator-to-metal transition is found when the SC phase is suppressed by means of a pulsed magnetic field [27]. When extrapolated to zero temperature, such a transition takes place near optimum doping, and at too high a doping to be associated to the spin-glass region [28] characterized by the local moment formation as seen in the muon experiments [29]. The spin-glass region should instead be a signature of the coexistence of superconductivity with antiferromagnetism proper of the SO(5) theory [30]. Moreover, a clear indication that this insulator-to-metal transition [27] is driven by some spatial charge ordering is provided by its occurrence at a much higher temperature in samples near the filling 1/8. Commensurability effects near this “magic” filling have repeatedly been reported in related compounds [31]. Hints for a critical behavior of the charge susceptibility come from the study of the chemical potential shift in PES and BIS experiments [32]. A dramatic flattening of the vs curve starting at could be the signature of a divergent compressibility. Finally,

48


stripes of either statical or dynamical nature are seen in neutron scattering experiments [24], EXAFS [33], and x-ray diffraction (XRD) [34]. It must be pointed out that the characteristics of the stripe phase produced by the ICDW instability are system and model dependent. The direction of the critical wavevector is diagonal in YBCO [10]; and in nickelates [35], where one-hole filled domain walls are present; and vertical in Nd-doped LSCO [31], with half-filled domain walls. It has been shown that a strong on-site Hubbard repulsion and a long-range potential stabilize vertical half-filled stripes [36]. The stripe–QCP scenario provides therefore a scheme to interpolate between the repulsion that gives rise to the AFM state at low doping and the attraction giving rise to SC through the (local) PS or ICDW. 2. THE NORMAL STATE

To investigate the effect of the stripe QCP on the normal-state properties of the system, we are led to consider an effective interaction between quasi-particles

where which is mediated by both charge (c) and (enslaved) spin (s) fluctuations. The q dependence in Eq. (1) was taken in the cos-like form to reproduce the behavior close to the critical wavevectors and maintain the lattice periodicity near the zone boundary. We point out that Eq. (1) for the interaction mediated by charge fluctuations was found within a slave–boson approach to the Hubbard–Holstein model with long-range Coulomb interaction, close to the ICDW instability [37]. The same form mediated by spin fluctuations corresponds to the dynamic susceptibility proposed by Millis, Monien, and Pines [38] to fit NMR and neutron scattering experiments, in the limit of strong damping. We take a free-electron spectrum of the form

where nearest-neighbor (t) and next-to-nearest-neighbor hopping terms are considered, to reproduce the main features of the band dispersion and the Fermi surface (FS) observed in SC copper oxides. The chemical potential is treated self-consistently within a perturbative approach to fix the number of particles. The first-order in perturbation theory yields an electron self-energy where

is the Fermi function and is the Bose function. The real part of the self-energy is obtained by a Kramers-Krönig transformation of Eq. (3). To keep the inversion symmetry we symmetrize the self-energies


49

with respect to We assume that neglecting the possibility for a discommensuration of the spin fluctuations in a (dynamical) stripe phase [24]. This would

introduce minor changes in our results. The relevant direction of the critical wavevector is still debated and can be material dependent [39]. Here, we analyze the case which has been suggested by the analysis of ARPES experiments on Bi2212 [40]. In the absence of superconductivity, the stripe QCP, located at is the end point of two lines that divide the T vs x plane in three regions: the ordered-phase region at lower doping and low temperature, the quantum disordered region at higher doping and low temperature, and the quantum critical region around the critical doping where the only energy scale is and the maximum violation of the FL behavior in the metallic phase is found. In this region, the system is characterized by an anomalously large quasi-particle inverse scattering time at those points of the FS connected by the critical wavevectors (hot spots) and displays a linear-in-T resistivity in which is turned to a which could be a signature of quantum critical behavior for less anisotropic systems, with the change in the temperature dependence occurring over the whole temperature range by increasing doping. To study the effect of the singular interaction in Eq. (1) on the single-particle properties in the metallic phase we calculate the spectral density To allow for a comparison with ARPES experiments, we analyze the convoluted spectral density

which takes care of the absence of occupied states above the Fermi energy, through the Fermi function and of the experimental energy resolution through a resolution function We take according to numeric convenience. For the sake of definiteness, we choose our parameters in Eq. (2) to fit the band structure and FS of Bi2212; namely, t = 200 meV, and corresponding to a hole doping with respect

to half filling. The parameters appearing in the effective interaction in Eq. (1) were taken as

and The quasi-particle spectra are characterized by a coherent quasi-particle peak at an

energy and by shadow peaks at energies produced by the interaction with charge and spin fluctuations. The shadow peaks do not generally correspond to new poles in the electron Green function and are essentially incoherent, although they follow the dispersion of the shadow bands. Their intensity varies strongly with k and increases when approaches the value . In particular, at the hot spots, where and the non-FL inverse scattering time there is a suppression of the coherent spectral weight at the Fermi energy. We also study the k-distribution of low-laying spectral weight The transfer of the spectral weight from the main FS to the different branches of the shadow FS at produces features that are characteristic of the interaction with charge and spin fluctuations and of their interplay. In particular, the symmetric suppression of spectral weight at the M points of the Brillouin zone, which would be due to spin fluctuations alone, is modulated by charge fluctuations (Fig. 2). This is also the case for the (weak) hole pockets produced by spin fluctuations around the points The interference

50


with the branches of the shadow FS due to charge fluctuations enhances these pockets around and suppresses them around (Fig. 2). Experimental results on this issue are controversial. Strong shadow peaks in the diagonal directions, giving rise to hole pockets in the FS, have been reported in the literature [40,42] where other experiments found only weak (or even absent) features [5]. We point out that, because of the transfer of spectral weight to the shadow FS, the experimentally observed FS may be rather different from the theoretical FS, determined The observed evolution of the FS could, indeed, be associated with the change in the distribution of the low-laying spectral weight, without the topologic change in the quasi-particle FS that was proposed in [43]. 3. SUPERCONDUCTIVITY In the stripe-QCP scenario the dynamical precursors of the ICDW mediate an attractive interaction in the Cooper channel [44]. As a matter of simplification, we solve the BCS-like equation

where is the gap parameter, and both the charge- and spin-induced static effective interactions in the Cooper channel have been considered, corresponding


51

to an interaction in the particle-hole channel (1). A constructive interference between the small-q attraction associated to charge fluctuations and the large-q repulsion associated to spin fluctuations yields both a high critical temperature and a gap parameter with d-wave symmetry (Fig. 3). The variation of the critical temperature with doping follows the variation of the relevant energy scale in each region of the phase diagram. In the overdoped (quantum disordered) region, (at low temperatures) and decreases rapidly with increasing doping. In the quantum critical region around optimal doping, and is almost doping independent.

In the underdoped region, new scales of energy appear. At

which corresponds

to the mean-field ICDW critical temperature, precursors of the stripe phase show up in

the reduction of spectral weight near the Fermi energy, i.e., a reduction of the static spin susceptibility [3]. At the same time, the damping of the AFM fluctuations is reduced and (almost) propagating spin waves appear, with a natural crossover from a dynamical index to At lower temperatures, because Boebinger's experiment [27] suggests that superconductivity is hindering the formation of a static CDW, we must assume that the onset of (local) superconductivity introduces a cutoff for critical fluctuations. Thus, near the onset of the stripe phase we take a mass term where is the maximum over k of the (local) superconducting gap . When introduced into Eq. (5), this dependence allows the (local) gap to survive up to a temperature as high as even if the phase coherence, which is necessary for bulk superconductivity, develops at a lower temperature This produces a long pseudo-gap tail in the underdoped region, which, despite the crudeness of our approximations, displays the behavior of the analogous quantity as measured in ARPES experiments in underdoped Bi2212 [6].

52


4. CONCLUSIONS In this paper, we briefly recapitulated the stripe–QCP scenario and presented some of its consequences in the normal and SC states. Within this scenario, the occurrence of a charge-ordering instability, hindered only by the setting in of a SC phase (in this sense it would be more appropriate to speak about a “missed-QCP” scenario), provides the underlying mechanism ruling the physics of the SC cuprates. In particular, it gives rise to the formation of the observed stripe textures in these materials, to the non-Fermi liquid properties of the normal phase, to the main features found in ARPES experiments, and to the strong pairing interaction. The most natural location for this QCP is optimum doping, where the strongest violation to the FL behavior and the highest critical temperature occur. Indeed, the physical properties governed by the proximity to a QCP account for the ubiquitous universal behavior, observed near optimal doping in all SC copper oxides. This rationale is missing in the theory of the AFM QCP or in the theory of the QCP associated to the coexistence of antiferromagnetism and superconductivity, which would also be located near the AF phase at low doping or near the spin-glass transition. We conclude by remarking that the scenario of the stripe QCP near optimum doping, hidden by the occurrence of the SC phase, shares a common origin (the Coulomb-frustrated phase separation) with the scenario proposed by Emery, Kivelson, and coworkers, but relies on a distinct mechanism. In this latter proposal, the anomalous normal properties stem from the marked one-dimensional character of the metallic stripe phase, and the pairing arises from the in-and-out pair hopping from the 1D stripes into the spin-gapped AF background. A related description of the stripe phase in terms of purely one-dimensional strings has been also put forward by Zaanen [45]. In our picture, the non-FL character of the metal arises from the singular scattering by critical fluctuations near the QCP for the onset of the stripe phase. The fluctuations of the stripe texture also provide a strong pairing potential accounting for high critical temperatures. We believe that this more two-dimensional physical description in terms fluctuations of the stripe texture is closer to the reality, at least for the optimal and overdoped systems, where the substantial metallic character of the systems is difficult to reconcile with the formation of strongly one-dimensional long-living stripe structures. ACKNOWLEDGMENT Part of this work was carried out with the financial support of the INFM, PRA 1996.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986). H. Takagi et al., Phys. Rev. B 40, 2254 (1989). D. C. Johnston, Phys. Rev. Lett. 62, 957 (1993). T. Ito, K. Takenaka and S. Uchida, Phys. Rev. Lett. 70, 3995 (1993). D. S. Marshall et al., Phys. Rev. Lett. 76, 4841 (1996). J. M. Harris et al., Phys. Rev. Lett. 79, 143 (1997). H. Ding et al,. Nature 382, 51 (1996). C. Berthier et al., J. de Physique I, December 1996. J. Rossat-Mignod et al., Physica B 186–189, 1 (1993).


53

10. Pengcheng Dai, H. A. Mook and , Phys. Rev. Lett. 80, 1738 (1998). 1 1 . T. E. Mason et al., Physica B 199–200, 284 (1994); T. E. Mason et al., Phys. Rev. Lett. 77, 1604 (1996). 12. S. Petit et al., Physica B 234–236, 800 (1997); P. Bourges et al., Phys. Rev. B 53, 876 (1996). 13. J. W. Loram et al., Phys. Rev. Lett. 71, 1740 (1993).

14. P. W. Anderson, Science 235, 1196 (1987); Phys. Rev. Lett. 64, 1839 (1990); ibid. 65,2306 (1990). 15. J. Feldman, H. Knörrer, D. Lehmann, and E. Trubowitz in Constructive Physics, edited by V. Rivasseau, Springer Lecture in Physics (Springer, New York, 1995). 16. C. Castellani, C. Di Castro, and W. Metzner, Phys. Rev. Lett. 72, 316 (1994). 17. C. Castellani and C. Di Castro, Physica C 235–240, 99 (1994). 18. S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

19. P. Montoux, A. V. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992); A. Sokol and D. Pines, Phys. Rev. Lett. 71, 2813 (1993); P. Montoux and D. Pines, Phys. Rev. B 50, 16015 (1994). 20. C. M. Varma, Physica C 263, 39 (1996) and references therein. 21. R. B. Laughlin, cond-mat/9709195. 22. C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995). 23. C. Castellani, C. Di Castro, and M. Grilli, Z. Phys. B 103, 137 (1997). 24. J. M. Tranquada et al., Phys. Rev. B 56, 7689 (1996). 25. V. J. Emery, S. A. Kievelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990); M. Marder, N. Papanicolau, and G. C. Psaltakis, Phys. Rev., 41, 6920 (1990); M. Grilli et al., Phys. Rev. Lett. 67, 259 (1991); N. Cancrini et al., Europhys. Lett. 14, 597 (1991); R. Raimondi et al., Phys. Rev. B 47, 3331 (1993); S. Caprara, C. Di

26. 27. 28. 29. 30.

Castro, and M. Grilli, Phys. Rev. B 51, 9286 (1995); F. Bucci et al., Phys. Rev. B 52, 6880 (1995); M. Grilli and C. Castellani, Phys. Rev. B 50, 16880 (1994). V. J. Emery and S. A. Kievelson, Physica C 209, 597 (1993). G. S. Boebinger et al., Phys. Rev. Lett. 77, 5417 (1996). J. H. Chou et al., Phys. Rev. B 46, 3179 (1992). Ch. Niedermayer et al., Phys. Rev. Lett. 80, 3843 (1998). Shou-Cheng Zhang, Science 275, 1089 (1997).

31. J. M. Tranquada el al., Nature 375, 561 (1995).

32. A. Ino et al., Phys. Rev. Lett. 79, 2101 (1997).

33. A. Bianconi, Physica C 235–240, 269 (1994); A. Bianconi et al., Phys. Rev. Lett. 76, 3412 (1996); A. Bianconi et al., Phys. Rev. 54, 12018 (1996).

34. A. Bianconi et al, Phys. Rev. B 54, 4310 (1996). 35. J. M. Tranquada, D. J. Buttrey and V. Sachan, Phys. Rev. B 54, 12318 (1996). 36. G. Seibold et al., cond-mat/9803184. 37. F. Becca et al., Phys. Rev. B 56, 12443 (1996).

38. A. J. Millis, H. Monien and D. Pines, Phys. Rev. B 42, 167 (1990). 39. H. A. Mook, p. 315 in this volume. 40. N. L. Saini et al., Phys. Rev. Lett. 79, 3467 (1997).

41. R. Hlubina and T. M. Rice, Phys. Rev. B 51, 9253 (1995). 42. S. La Rosa et al., Solid State Commun. 104, 459 (1997).

43. A. V. Chubukov, D. K. Morr and A. Shakhnovich, Phil. Mag. B 74, 563 (1996). 44. A. Perali et al., Phys. Rev. B 54, 16216 (1996). 45. J. Zaanen, this conference.


Phase and Amplitude Fluctuation in Superconductors: Formation of Gap Stripes Due to Lack of Electron-Hole Symmetry in Cuprate Oxides B. K. Chakraverty1 and K. P. Jain2

in high-temperature superconductors seems well described by coherence of overall phase of the superconducting order parameter that is complex and fluctuates in space and time (thermal and quantum phase fluctuation). Above is

the pseudogap phase, where one has a nonzero pairing gap without global phase coherence. If one were to define a meanfield BCS temperature, where pairing susceptibility diverges—the so-called Thouless instability—then we show in this communication that if the system has no electron-hole symmetry, we obtain a nonuniform phase of stripes where the pairing gap develops a periodic

amplitude modulation at This suppresses the —i.e., the BCS instability.

mode instability at

1. INTRODUCTION

There seems a general consensus that the oxide superconductors do not obey in any way the usual BCS meanfield behavior. The observed is much bigger than the conventional value of 3.3 indicating a meanfield BCS temperature almost twice as much as the ones observed. In addition, for any given superconducting series the amplitude of the superconducting gap, is larger as the doping becomes smaller whereas the transition temperature decreases monotonically [1,2]. These and a variety of observations, the most important of which being that of the existence of a pseudo-gap [3] at points out clearly that the superconducting state has a distinctly second energy scale other than that 1

2

Lepes, C.N.R.S, 38042, Grenoble, France. Indian Institute of Technology, New Delhi, 110016, India.


55

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Chakraverty and Jain

which determines the amplitude of the gap and that this must be related to the phase proprties

of the complex order parameter. In general, we have where the phase is blocked in the superconducting state at the same value everywhere spatially, indicating infinite phase correlation or off-diagonal long-range order. It was pointed out for the first time [4] that the energy scale associated with phase fluctuation is related to Josephson plasma frequency. The resultant agrees quite well with the observed values in the cuprates [5] and also quantum fluctuation of the phase gives the right trend as to the diminution of with underdoping [6]. In this communication, we address the question of how and where the

pairing amplitude itself is lost. In particular, we indicate that the BCS meanfield transition superconducting instability at

is suppressed as a result of amplitude fluctuation if

we assume that there is no electron-hole symmetry in the cuprate oxide superconductors. This fluctuation takes the form of a spinodal decomposition or gap stripes, which are associated with charge density fluctuation (incidentally, a lattice distortion wave is expected to accompany the charge density) [7]. Thus the so-called normal nonsuperconducting phase

above

is to be envisioned as a phase of dynamic gap stripes.

We have the basic Hamiltonian given by

where cs are the electron operators, g is a local attractive interaction between the carriers with momentum labels k, arrows are the spin indices, and is the electron kinetic energy measured from the fermi level Applying a Hubbard-Stratanovitch transformation to this Hamiltonian, we can obtain [8] the free energy as a sum of two parts: a homogeneous part of the Ginzburg-Landau meanfield form and a quadratic term that contains quadratic

fluctuation at the saddle point of the superconducting gap amplitude its phase and the electronic density n. The homogeneous part is written in the standard form to the fourth order

Here,

is the superconducting gap defined by

The meanfield

gap amplitude is given by minimizing Eq. (2) to give The BCS transition temperature is signalled by the divergence of the RPA pairing susceptibility

Because the superconducting order parameter is complex, we write it as The fluctuating term to the quadratic order becomes

Here we have whereas the charge density deviation is given by The last two terms of Eq. (3) is the phase fluctuation Hamiltonian [13], the phase and n being conjugate quantities. These two terms were the object of our previous study [6], in which we showed how they govern the superconducting to nonsuperconducting phase transition at where the phase stiffness The last term of Eq. (3) is a local charge fluctuation energy, and the magnitude of U determines the quantum

Phase and Amplitude Fluctuation in

Superconductors

57

fluctuation of phase because of phase-charge conjugation, and the phase fluctuation dynamic energy is given by the larger U is, the easier it is to fluctuate the phase and destroy overall phase coherence. A critical value of U leads to a quantum phase transition from superconducting to insulating state, as we have shown in our earlier paper. The static phase stiffness energy and the dynamic phase fluctuation energy can be combined to give a quantum coupling constant of phase fluctuation where and are the London penetration depth and Pippard correlation length, respectively, such that beyond a critical coupling constant the superconducting state is phase destroyed and system becomes an insulator! At in the absence of a magnetic field, this is the lowest concentration below which the system is no longer superconducting and the insulating state that it goes to is a novel state analogous to a paired Wigner crystal [9]. In this region, addition or removal of a pair from a region costs a Coulomb energy in two-dimensional at low temperature. This superconductor to insulator phase transition can be driven to higher concentration by application of a magnetic field perpendicular to the plane (H decreases and increases and this is a perfect way to tune the quantum coupling constant g) as was done [10]. The Coulomb gap opens up at higher and higher concentration until at

and the system is globally insulating below the temperature

The

qualitative phase diagram is shown in Fig. 1, where we replaced g by carrier concentration

signifying the optimum concentration for the highest

In the figure,

represents the temperature at which pseudo-gap is seen in the ARPES experiments and

is the temperature where the meanfield BCS temperature ought to be At

going from the antiferromagnetic state to the superconducting state through

the intervening insulating state, we can characterize each one of them by three sets of energies inherent in the system—they are the two particle energy gap to add or subtract a pair of charge

the single particle energy gap to add a charge that we just considered, the energy needed and the energy needed to flip a spin, the

58


so called spin gap of charge 0,

The following table shows these energies to be

expected in the three states:

The single particle energy gap in the superconducting state is what we called earlier, whereas was denoted The spin gap observed at optimum doping by NMR or neutron scattering [11] goes smoothly to zero in the AF state as the doping is decreased. In this chapter we do not take up this issue any further, and start from Eq. (3) as it stands, omitting the fourth term because we are looking into the nonsuperconducting phase above where by construction is zero. The central question is to understand whether this homogeneous insulating state with an uniform gap amplitude all around and immersed in a dynamic phase field is a stable state of the system and whether it describes well the cuprates above Our principal concern in this communication is to unravel the importance of the third term, which couples charge fluctuation to amplitude fluctuation, being the coupling parameter and show that if it exists the homogeneous insulating phase develops a modulation of the gap amplitude or domain walls. In BCS-like theories there

is no explicit coupling between density fluctuation and the superconducting amplitude. The density couples to phase variation and gives rise to the phase mode, the AndresonBogoliubov mode, of the broken symmetry solution (which leads to the Josephson plasma frequency for long-range Coulomb interaction). This automatically leads to because in the BCS formulation, electron-hole symmetry has been assumed at the Fermi level from the very beginning. The most elementary way of seeing this is to write the BCS gap equation at

Here with respect to

where to obtain

is the chemical potential. We easily obtain the derivative

This can be written in terms of the density fluctuation as

The numerator of this expression when summed over k vanishes because is symmetric in the BCS. formulation irrespective of the charge rigidity (inverse of charge susceptibility) Thus is zero in the BCS regime! [12].


Superconductors

59

Linear response theory allows us to proceed further to investigate response of the amplitude to the density fluctuation and obtain some microscopic understanding of the coupling We write a perturbing Hamiltonian as

where the Josephson relation, with being a local scalar potential reflecting the dynamic phase fluctuation above The amplitude response to this local potential is given by

where the term within the bracket is the retarded commutator or the response function to be evaluated. In the most general case, the perturbing Hamiltonian can be written as

where the given by

are the pairing fields defined by We get the response as

and the gap is accordingly

Expression (10) is more general than before and gives the definition of the anomalous susceptibility as

This shows the divergence at the same temperature

where the pairing susceptibility

(so-called Thouless instability) or the t -matrix also diverges given by

This demonstrates that the BCS meanfield solution is not valid for very large amplitude fluctuation, which could occur due to the term. These anomalous susceptibilities have been calculated by us and will be published elsewhere [14]. A typical element of the response function is shown in the diagram of Fig. 2 and when evaluated at gives the expression

Here, the and are the usual BCS coherence factors and tation energy given by Note the factor

is the quasi-particle exciand so Eq. (13) vanishes

60


when the k sum is done for electron-hole symmetry just as in the the earlier illustrative

Eq. (6). Any breakdown of this electron-hole symmetry gives a nonzero coupling between density and amplitude fluctuation whose consequence is shown in the next section.

2. FORMATION OF GAP STRIPES Using Eqs. (2) and (3), where in (3) we omit the fourth term, we can write the

Hamiltonian density for gaussian fluctuation as

Here we note that showing the importance of the fourth-order term (with coefficient B) in stabilizing the amplitude fluctuation. Writing

The partition function may be expressed as

where

is obtained after integrating out the density fluctuation

This gives

The coefficient of the quadratic term becomes negative when and is seen in Fig. 3. In superconductors of d-wave symmetry can be particularly large because of gapless electron-hole excitations available around the Fermi point at

For


Superconductors

61

Eq. (18) has a spatially periodic solution for the the amplitude fluctuation

given

by

where

gives periodicity of the gap stripes. This leads to

The above spinodal decomposition is a spontaneous instability of the system to “a gap density wave.” The phenomenon can also be envisioned as a spontaneous breakdown into domain walls. The resultant inhomogeneous phase has striped regions alternating with no superconducting gap at all (these regions resemble a spin-liquid phase) with regions fully

gapped, but without any phase stiffness (Fig. 4). At around

this inhomogeneous

62


phase starts to form the meanfield transition temperature, where

and

is still

quite small such that can be At a lower temperature the zero gap phase, which we call a spin liquid, can stabilize itself through lattice distortion, thereby breaking locally the translational symmetry and opening up a spin gap at the Fermi level. This leads to dramatic decrease of low energy spin fluctuation from downward. It is only when this “spin jungle” disappears that overall phase coherence between gapped regions can be established at Below the domain walls are “delocalized” and are essentially low-frequency Josephson plasma waves. We end by emphasizing that the formation of gap stripes is a necessary corollary of breakdown of electron-hole symmetry—a situation pertinent to the non-BCS behavior of superconductors. REFERENCES 1. 2. 3. 4.

J. W. Loram et al., Physica C 282, 1405(1997). J. M. Harris et al., Phys. Rev. B 54, R15665 (1996). H. Ding et al., Nature 382, 51 (1996). B. K. Chakraverty et al., Physica C 235, 2323 (1994).

5. V. J. Emery and S. Kivelson, Nature 374, 434 (1995).

6. B. K. Chakraverty and T. V. Ramakrishnan, Physica C 282, 290 (1997). 7. A. Bianconi et al., Phys. Rev. Lett. 76, 3412 (1996). 8. S. Palo, Thesis—Univ. of Rome, 1996; S. Palo et al., Phys. Rev. B 60, 564 (1999). 9. K. Moulopoulos and N. W. Ashcroft, Phys. Rev. B. 42, 7885 (1990). 10. G. S. Boebinger et al., Phys. Rev. Lett. 77, 5417 (1996).

11. M.-H. Julien, Ph.D thesis, Univ. of Grenoble 1998; M. H.-Julien et al., Phys. Rev. Lett. 76, 4238 (1996); J. Rossat-Mignod et al., Physica C 185–189, 86 (1991). 12. P. B. Littlewood and C. M. Varma, Phys. Rev. B 26, 4883 (1982).

13. T. V. Ramakrishnan, Physica Scripta T27, 24 (1989). 14. K. P. Jain and B. K. Chakraverty, unpublished, 1998.

Stripe on a Lattice: Superconducting Kink/Soliton Condensate Yu. A. Dimashko1 and C. Morais Smith1

We study the transversal dynamics of a completely filled nickelate stripe (quantum string), as well as the longitudinal dynamics of a half-filled cuprate stripe (quantum spring) at Both problems belong to the same universality class as the charge motion in a 1D Josephson junction array and exhibit an insulator/

superconductor transition at the critical value where t denotes the hopping amplitude and 1 is the stripe stiffness. We suggest an experiment in order to observe the predicted behavior.

One important question concerning the charged stripes in doped antiferromagnets is whether and how they are related to superconductivity. A static stripe order was proposed to exist in the insulating phase [1]. However, in the superconducting regime, the stripes are expected to fluctuate strongly. Due to the different possibilities for the domain wall (DW) structure, the transport properties of this system can be rather distinct: for DW with one hole/site, as in the nickelates, only transversal excitations are possible, whereas for the half-filled cuprate

stripes, the charge can be carried also along the wall. The transversal dynamics of filled nickelate stripes was studied in the frame of a quan-

tum lattice string model, with the stripes being regarded as vibrating strings [2]. Later, a connection was established between this elastic and the more “microscopic” model [3]. The longitudinal transport in cuprate-stripes was considered, based on the Luttinger liquid approach [4]. Recently, a 1D model was proposed that describes both the transversal and longitudinal charge motion in a half-filled stripe [5]. It was suggested that in the long wavelength limit these modes separate. The stripe configuration in a doped antiferromagnet can be regarded as a charge density wave (CDW). The conductivity from CDW was considered initially by Fröhlich [6], who pointed out that due to the blocking of the scattering processes by the Landau criterion [7], 1

Institut für Theoretische Physik, Universität Hamburg, D-20355 Hamburg, Germany.


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Dimashko and Morais Smith

the collective (sliding) mode displays perfect conductivity. Later, Lee, Rice, and Anderson (LRA) [8] analyzed the dynamics of the sliding mode in the presence of a lattice and argued that pinning by the lattice and/or by impurities destroys the perfect conductivity. However, LRA did not consider the effect of quantum fluctuations, which can compete with pinning even at Hence, it remains to be verified whether the quantum fluctuations are able to depin the stripes (or commensurable CDW), and whether this quantum depinning restores the perfect conductivity of the sliding mode. In order to investigate this problem, we study both the transversal (and the longitudinal) dynamics of a completely (half) filled stripe, accounting for quantum fluctuations. By performing a canonical transformation in the quantum Hamiltonian, we map both systems onto a Josephson junction chain, which is known to exhibit a insulator/superconductor transition. Then, we calculate the transport properties in two limitting cases that can be solved explicitly, and clarify the meaning of the transition in the stripe contexte. We find that the conducting properties of the stripe are determined from the competition between the hopping amplitude t and the stiffness parameter J: in the weak-fluctuation limit, the ground state (GS) of the system is a kink (soliton) vacuum. In this case, the stripe is pinned by the lattice, the excitation spectrum exhibits a gap and the system is insulating. In the opposite strong-fluctuation limit, the GS of the system is a kink (soliton) condensate. Then, the excitation spectrum contains a gapless phonon-like mode and the system is superconducting. The low t/ J insulating phase is separated from the large t/ J superconducting phase by a Kosterlitz–Thouless (KT)-like phase transition at corresponding to the unbinding of the kink/antikink (soliton/antisoliton) pairs. We consider a single stripe consisting of N holes placed on a square lattice. In the case of a completely filled stripe, we analyze only the transversal motion of the holes. It is the quantum string model [2,3]. The representative states of the quantum string (flat state, kink, antikink) are shown in Fig. l(a–c). In the case of a half-filled stripe, we analyze only the longitudinal motion of the holes. It is the quantum spring model. The representative states of the quantum spring (uniform state, soliton, antisoliton) are shown in Fig. l(d–f). In both cases the holes are restricted to move in the x direction only. Assuming an elastic interaction

Stripe on a Lattice

65

between neighboring holes, one can write a common Hamiltonian for both models

The integer v assumes the values for the quantum string and for the quantum spring. The lattice constant is taken as the unit of length and the index numerates the holes. Together with the current operator and the canonical commutation relations the Hamiltonian Eq. (1) contains all the conducting properties of the system. The local excitations of the stripe play an important role in the further analysis of these properties: They are the kinks/antikinks (K-AK) for the quantum string model and the solitons/antisolitons (S-AS) for the quantum spring model (see Fig. 1). It is convenient to classify these states by the value of the topological charge K and S are states with AK and AS are states with Here, we consider periodic boundary conditions, Hence, the total topological charge of the stripe is zero, and only creation of pairs K-AK or S-AS is allowed. Next, we perform a duality transformation to new variables

These variables to not refer to separate holes, but to pairs of neighboring holes. They are restricted to the interval and can be treated as a phase. As a result, the Hamiltonian and the current operator acquire the form

which is known from the theory of superconducting chains. Equation (3) describes a Josephson junction chain (JJC), with the Coulomb interaction taken into account. The solution of this problem at was found by Bradley and Doniach [9]. Depending on the ratio t/J, the chain is either insulating (small t/ J) or superconducting (large t / J ) . At the Josephson chain undergoes a KT insulator/superconductor transition. Because both the quantum string and the quantum spring models are described by exactly the same Hamiltonian and current operator as the JJC, they also undergo an insulator/ superconductor transition at this point. Below, we analize the meaning of this transition for the single stripe. It is instructive to do it in two limiting cases, which allow for an explicit solution: Let us start considering the limit of weak fluctuations, with In the absence of hopping, the excitation spectrum of the stripe is discrete, with The ground level corresponds to the vector representing a flat state of the completely filled stripe (see Fig. la), or a uniform state of the half-filled stripe (see Fig. 1d). In other words, it is the kink vacuum or the soliton vacuum, respectively. The first elementary excitation corresponds to the creation of a pair K-AK or S-AS, i.e., to the state are local creation operators of K or and

66


AK or This state has energy All the levels are degenerated, and therefore accounting for small but nonzero hopping t can split them. A simple analysis shows that in the thermodynamical limit each vth level splits into a band of width The energy band structure of the stripe at is shown in Fig. 2. Because the ground level is not split, the excitation spectrum has a gap which is nothing but the minimal energy required to create a K-AK or S-AS pair. The existence of the gap at suggests that the GS of the stripe in this limit is insulating (the stripe is pinned by the lattice). This conclusion is in full accordance with the results for the JJC in this limit [9], where the frequency-dependent conductivity Re Nevertheless, one can consider the conductivity of the excited states caused by the presence of the kinks or solitons. Such an analysis shows that kinks play the role of current carriers, with an effective electrical charge proportional to the topological charge of the kink, Similar consideration leads to analogous results for the solitons. However, the effective electrical charge of the soliton turns out to be fractional It is worth noting that in the limit, the GS of the stripe contains only local K-AK or S-AS pairs. Indeed, a perturbative calculation of the K-AK or S-AS pair correlation function reveals only short-range correlations: The correlation length i can be treated as an average dimension of the virtual K-AK or S-AS pairs. Such a pair cannot carry current because it forms a bound state with zero topological charge. Therefore, the locality of the K-AK (or S-AS) pairs is responsible for the insulating character of the GS at small t. The abovementioned exponential behavior of the phase correlator at is well known for the JJC [9]. Now, we concentrate on the opposite limit Then, we can expand the t term in the Hamiltonian Eq. (3) up to second order, By diagonalizing the quadratic Hamiltonian, we obtain the phonon-like spectrum with

The spectrum is continuous and has no gap. Therefore, the GS is conducting. The calcula-

tions of the conductivity in this limit are straightforward and reveal the same superconducting singularity as in the case of the JJC [91. Furthermore, the K-AK (or S-AS) pair correlator exhibits quasi-long-range order, Hence, in the limit the average dimension of the K-AK (or S-AS) pairs diverges (the pairs decouple), providing the conducting GS. However,

Stripe on a Lattice

67

this does not mean absence of the correlations along the stripe: The GS exhibits the same quasi-long-range correlations of the phase, which is known for the JJC. The most important property of the GS in this limit is the presence of decoupled kinks and antikinks (solitons and antisolitons) that form a kink condensate (soliton condensate). This condensate is a gauge-invariant state, and the phonon-like modes in Eq. (4) are its elementary excitations breaking the gauge symmetry These excitations obey the Landau criterion [7], providing superflow and supercurrent for sufficiently small velocities of the condensate In this way, the 1D superconductivity of the completely filled/half-filled stripe is related to the existence of the kink/soliton condensate. Now, we can summarize our results: At the GS of the completely filled/halffilled stripe is a kink/soliton vacuum. it has only bound K-AK/S-AS pairs, exhibits no long-range phase order, and the energy spectrum is gapped. Then the system is insulating. the K-AK/S-AS pairs are already decoupled and there is no longer any gap. Then the phase is quasi-long-range ordered, the GS is the kink/soliton condensate and the system is superconducting. Based on the Josephson chain results, it follows that at the stripe undergoes a KT-transition: The gap vanishes, the K-AK/S-AS pairs decouple, and the kink/soliton condensate arises. The disappearance of the gap means quantum depinning of the stripe from the lattice. Appearance of the condensate with phonon-like excitation spectrum gives rise to a superconducting depinned phase. Hence, the quantum fluctuations are able to depin the stripe from the lattice and restore the Fröhlich’s perfect conductivity. All these results are found at At any small thermal fluctuations will “spoil” the kink/solitonic superconductivity. The problem is rooted in the 1D treatment of the dynamics. At higher dimensions (array of stripes), thermal fluctuations do not play such a destructive role anymore. Therefore, a 2D theory coupling the longitudinal and transversal dynamics is needed for obtaining the kink/solitonic superconductivity in the striped system at By assuming a 1D (either longitudinal or transversal) dynamics for the holes in a stripe, we have concluded that there is phase coherence along the stripe at However, we expect our results to remain valid at for systems that allow for a 2D dynamics, as it is the case for the half-filled cuprate stripes. Moreover, we presume that in this 2D case, coherence between the stripes should also be established. In order to verify our conclusion and presumption, we propose an experiment based on the analogy between this problem and the Josephson junction chain. Applying a finite voltage U to the sample containing stripes across/along the stripes, is equivalent to applying a small voltage to each segment of the stripe between two neighboring holes. In accordance with our mapping,

each such a segment corresponds to one Josephson contact. In the superconducting state the current, exceeding the critical value, must oscillate with frequency (the nonstationary Josephson effect). In the case of the hole stripes (without pairs) this frequency where N (number of holes in one stripe) is proportional to the size L of the sample, obtains

Taking and the lattice constant one Evidence of such a low-frequency and size-dependent response in

uniform monocrystallic samples would confirm both the phase coherence within one stripe and the phase coherence between the stripes. Together, this would mean the kink/solitonic origin of the supercurrent in the cuprates.

68


ACKNOWLEDGMENTS We are indebted to N. Hasselmann and H. Schmidt for fruitfull discussions. Yu. A. D. acknowledges financial support from the Otto Benecke-Stiftung. This work was also partially supported by the DAAD-CAPES project #415-probral/schü. REFERENCES 1. J. M. Tranquada et al., Nature 375, 561 (1995). 2. H. Eskes et al., cond-mat/9510129; cond-mat/9712316; J. Zaanen et al., Phys. Rev. B 53, 8671 (1996); J. Zaanen and W. van Saarloos, cond-mat/9702060. 3. C. Morais Smith et al., Phys. Rev. B 58, 1 (1998). 4. A. H. Castro Neto and D. Hone, Phys. Rev. Lett. 76, 2165 (1996); A. H. Castro Neto, Phys. Rev. Lett. 78,

3931 (1997). 5. J. Zaanen et al., cond-mat/9804300. 6. H. Fröhlich, Proc. Roy. Soc. A 223, 296 (1954). 7. L. D. Landau, J. Phys. USSR 5, 71 (1941). 8. P. A. Lee, T. M. Rice, and P. W. Anderson, Solid State Commun. 14, 703 (1974).

9. R. M. Bradley and S. Doniach, Phys. Rev. B 30, 1138 (1984). 10. S. Kivelson and J. R. Schrieffer, Phys. Rev. Lett. 25, 6447 (1982).

Microscopic Theory of High-Temperature Superconductivity V. J. Emery1 and S. A. Kivelson2

It is argued that the BCS many-body theory, which is outstandingly successful for conventional superconductors, does not apply to the high-temperature superconductors and that a realistic theory must take account of the local electronic structure (stripes). The spin-gap proximity effect is a mechanism by which the charge carriers on the stripes and the spins in the intervening regions acquire a

spin gap at a relatively high temperature with only strong repulsive interactions. Superconducting phase order is achieved at a lower temperature determined by the (relatively low) superfluid density of the doped insulator. This picture is consistent with the phenomenology of the high-temperature superconductors. It is shown that, in momentum space, the spin gap first arises in the neighborhood of

the points

and then spreads along arcs of the Fermi surface.

Some of the experimental consequences of this picture are discussed.

1. INTRODUCTION The high-temperature superconductors [1] are quasi-two-dimensional doped insulators, obtained by chemically introducing charge carriers into a highly correlated antiferromagnetic insulating state. There is a large “Fermi surface” containing all of the holes in the relevant Cu(3d) and O(2p) orbitals [2], but n/m* vanishes as the dopant concentration tends to zero. [3,4] (Here m* is the effective mass of a hole and n is either the superfluid

density or the density of mobile charges in the normal state.) Clearly, understanding the origin of high-temperature superconductivity and the nature of the doped insulating state are intimately related. The doped insulating state is well understood in one dimension: The added charges form extended objects, or solitons, that move through a background of spins that have distinct dynamics [5] (this is the origin of the concept of the separation of spin and 1 2

Dept. of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000. Dept. of Physics, University of California at Los Angeles, Los Angeles, CA 90095.


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charge). In two dimensions, the doped-insulating state also is characterized by a 1D array of extended objects, but they are slowly fluctuating, metallic charge stripes that separate the spins into antiphase domains. These self-organized structures are driven by the tendency of the correlated antiferromagnet to expel the doped holes, and not by specific features of the environment of the planes [6]. The evolution of these ideas and the extensive evidence for this local electronic structure of the planes is described in a companion paper at this conference. [7] Rather general and phenomenological arguments indicate that the BCS many-body theory, which is so successful for conventional superconductors, must be revised for the high-temperature superconductors (Section 2). Once this is accepted, it is clear that any new many-body theory must be based on the local electronic structure of the doped insulator, especially structure on the scale of the superconducting coherence length. In Section 3 it is shown that, locally, the stripe structure may be regarded as a quasi-1D electron gas in an active environment provided by the antiphase spin domains. For a quasi-1D system there are two routes to superconductivity— route that is analogous to BCS theory and a potentially route in which a spin gap is formed at a relatively high-temperature and is independent of the onset of phase coherence that takes place at a lower temperature governed by the superfluid density [8,9]. In a quasi-lD electron gas (1DEG), both routes require some sort of attractive interaction [5]. However, the active environment adds a new element to the picture by allowing the formation of a spin gap with purely repulsive interactions via the “spin-gap proximity effect” [10]. The driving force is a lowering of the zero-point kinetic energy of the mobile holes, and it constitutes our mechanism of high-temperature superconductivity. In this way, the stripe picture allows us to derive the phenomenology of the high-temperature superconductors. The symmetry of the order parameter emerges once these ideas are reexpressed in momentum space (Section 4). We show that d-wave symmetry gives the lowest energy if the range of the gap function in real space is one lattice spacing. However, secondand third-neighbor components of the gap function favor s-wave symmetry, and in certain circumstances they could either mix with the d-wave component (breaking time-reversal symmetry or lattice-rotational symmetry) or even become dominant. 2. BCS MANY-BODY THEORY It has been argued that the quasi-particle concept does not apply to many synthetic metals, including the high-temperature superconductors [11]. This idea is supported by angular

resolved photoemission spectroscopy (ARPES) on the high-temperature superconductors, which shows no sign of a normal-state quasi-particle peak near the points

and

where high-temperature superconductivity originates [12]. If there are no quasiparticles, there is no Fermi surface in the usual sense of a discontinuity in the occupation number

at zero temperature. This undermines the very foundation of the BCS meanfield theory, which is a Fermi surface instability that relies on the existence quasi-particles. A major problem for any mechanism of high-temperature superconductivity is how to achieve a high pairing scale in the presence of the repulsive Coulomb interaction, especially in a doped Mott insulator in which there is poor screening. In the high-temperature superconductors, the coherence length is no more than a few lattice spacings, so neither retardation nor a long-range attractive interaction is effective in overcoming the bare Coulomb repulsion. Nevertheless, ARPES experiments [13] show that the major component of the

Microscopic Theory of High-Temperature Superconductivity

71

energy gap is Because the Fourier transform of this quantity vanishes unless the distance is one lattice spacing, it follows that the gap (and hence, in BCS theory, the net pairing force) is a maximum for holes separated by one lattice spacing, where the bare

Coulomb interaction is very large allowing for atomic polarization). It is not easy to find a source of an attraction that is strong enough to overcome the Coulomb force at short distances and achieve high-temperature superconductivity by the usual Cooper pairing in a

natural way. Thus, although the outstanding success of the BCS theory for conventional superconductors tempts us to use it for the high-temperature superconductors, it is clear that we should resist the temptation and seek an alternative many-body theory. There is phenomenological support for this point of view. In the BCS meanfield theory, an estimate of is given by where is the energy gap measured at zero temperature. This is a good approximation for conventional superconductors because the classic phase-ordering temperature is very high. A rough upper bound on is obtained by considering the disordering effects of only the classic phase fluctuations as where is the zero-temperature value of the “phase stiffness” (which sets the energy scale for the spatial variation of the superconducting phase) and A is a number of order unity [14]. may be expressed in terms of the superfluid density or, equivalently, the experimentally measured penetration depth

where a is a length scale that depends on the dimensionality of the material. For a conventional superconductor such as Pb, is about which implies that phase ordering occurs very close to the temperature at which pairing is established [14]. For the high-temperature superconductors, especially underdoped materials, and it varies with doping. The ratio ranges from about 2 to 4 as a function of However, provides a quite good estimate of for the high-temperature superconductors [ 14], an estimate that can be improved by making a plausible generalization of the classic phase Hamiltonian [15]. This behavior is qualitatively consistent with the route to superconductivity in the 1DEG, as discussed above. This phenomenology led us to conclude [14] that the spin gap observed in NMR and other experiments [16J [e.g., as a peak in at a temperature where is the nuclear spin relaxation time] should be identified with a superconducting pseudogap and not with a pseudogap associated with impending antiferromagnetic order at zero doping. This identification is now supported by ARPES experiments on underdoped materials [17] that find a pseudogap above with the same shape and magnitude as the gap observed in the superconducting state. Also, in underdoped materials, the opticalconductivity in the ab plane develops a pseudo-delta function, or a narrowing of the central “Drude-like” coherent peak above [18]. Essentially, all of the spectral weight moves downward, which indicates the development of superconducting correlations. The existence of local superconducting correlations below indicates that the amplitude of the order parameter is well established, but there is no long-range phase coherence. This situation could, in principle, be realized either by increasing and elevating the pairing scale or by decreasing and depressing the phase coherence scale as the doping x is decreased below its optimal value. Experimentally, as x decreases, varies very little

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(or even increases), whereas the superfluid density tends to zero as An increase in would amount to a crossover to Bose–Einstein condensation, which also requires that the chemical potential descends into the band or that the doped holes form a separate band, both of which are contradicted by ARPES experiments [2]. In other words, the separation of temperature scales for pairing and phase coherence in underdoped high-temperature superconductors is a consequence of the fact that the high-temperature superconductors are doped insulators; it is not a crossover from BCS physics to Bose-Einstein condensation. Another way of looking at the situation is to compare the superfluid density with the number of particles involved in pairing. In BCS theory, at is of order (where is the Fermi energy) and > is given by all the particles in the Fermi sea; i.e., For Bose condensation, We shall argue that, in the hightemperature superconductors, most of the holes in the Fermi sea participate in the spin gap below but the superfluid density of the doped insulator is small. An intuitive although somewhat imprecise picture of the third possibility is provided by the hard-core dimer model in which all the holes participate in dimers, but the mobile charge density is proportional to x. 3. SPIN GAP PROXIMITY EFFECT

The existence of a charge-glass state [7] in a substantial range of doping in the hightemperature superconductors implies that the dynamics of holes along the stripe is much faster than the fluctuation dynamics of the stripe itself. Thus, on a finite length scale an individual stripe may be regarded as a 1DEG in an active environment of undoped spin regions between the stripes. Then it is appropriate to start out with a discussion of an extended 1DEC in which the singlet pair operator may be written

where creates a right-going or left-going fermion with spin One route to superconductivity in the 1DEG is similar to the BCS many-body theory. At zero temperature in a gapless phase of the 1DEG, the correlation function is a power law with an exponent where are the critical exponent parameters for the charge and spin degrees of freedom and specify the location of the system along lines of (quantum critical) fixed points [5]. For a noninteracting system, pairing correlations are enhanced and pair hopping between the different members of an array of 1DEGs leads to a BCS-like superconducting phase transition, in which pairing and phase coherence develop at essentially the same temperature. Typically, this is a low-temperature route to superconductivity, and like BCS theory, it requires an attractive interaction between the charge carriers However, there is another route that is much closer to the phenomenology of the hightemperature superconductors. The fermion operators of a 1DEG may be expressed in terms of Bose fields and their conjugate momenta corresponding to the charge and spin collective modes, respectively. In particular, the pair operator becomes [5]


73

where In other words, there is an operator relation in which the amplitude of the pairing operator depends on the spin fields only and the (superconducting) phase is a property of the charge degrees of freedom. Now, if the system acquires a spin gap, the amplitude acquires a finite expectation value, and superconductivity appears when the charge degrees of freedom become phase coherent. Below the spin-gap temperature, the critical exponent of the pairing operator is given by which can more easily fall below 2 and generate superconductivity for an array, because there is no contribution from More to the point, the spin-gap temperature can be quite high, even in a single 1DEG, and it is generically distinct from the phase-ordering temperature [5,9]. Of course, phase order can only be established in a quasi-1D system because, in a simple 1 DEG, it is destroyed by quantum fluctuations, even at zero temperature. For an array of 1DEGs, a spin gap occurs only if there is an attractive interaction in the spin degrees of freedom. However, this is no longer true if the array is in contact with an active (spin) environment, as in the stripe phases. We have shown that pair hopping between the 1DEG and the environment conveys a preexisting spin gap from the environment to the 1DEG, or generates a spin gap in both the stripe and the environment, even for purely repulsive interactions [10]. A simple intuitive picture of this process is as follows: The spin part of the singlet pair operator on a stripe is However, locally, the spins in the environment have a Neel spin configuration Then, by the exclusion principle, the amplitude for pair hopping between the stripe and the environment has a (spin) factor However, pair hopping is enhanced by a factor and the kinetic energy lowered if the spins in the environment also form singlets. Note that the sign of the singlet wave functions in the environment must be chosen to maximize

the overall hopping amplitude of the pairs, as the phase varies along a stripe. This corresponds to the composite order parameter that appears in the quantum field theory treatment of the problem [10]. In principle, this process may not lead to a gap for all of the spins in the environment in the normal state. However, once pair hopping between the stripes becomes coherent, the remaining spins acquire a gap via the spin-gap proximity effect [10].

This mechanism of high-temperature superconductivity also avoids problem of the strong Coulomb interaction because it involves pairing of neutral fermions, or spinons, that are known to exist in the 1DEG [5]. It allows a spin gap with a range of one lattice spacing in the environment and about two lattice spacings on a stripe. Not only does this route to superconductivity correspond closely to the phenomenology of the high-temperature superconductors, but it also works for a short stripe. It is well

known—for example, from an analysis of numerical calculations—that, if the length scale associated with the spin gap is short compared to the length of a stripe, then the calculation for an infinite system is a good approximation for the finite system. Furthermore, once the spin degrees of freedom are frozen in this way, the remaining Hamiltonian corresponds to a phase-number model that we have used to analyze the effects of quantum phase fluctuations [11]. Superconductivity appears when the different stripes become phase coherent, and the superconducting coherence length is given by the spacing between stripes and not by the range of the pair wavefunction as in BCS theory. A consequence is that, in the superconducting state, the radius of a vortex core should have a very weak temperature dependence, and that the core should be an essentially undoped region with a spin gap. Both of these conclusions are supported by experiment [20,21].

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4. MOMENTUM SPACE

So far, we have discussed the consequences of stripes in real space, but ARPES experiments show that the high-temperature supeconductors have a “Fermi surface” even though there are no well-defined quasi-particles. Therefore, it is appropriate to ask how this physics is realized in momentum space. We have calculated the spectral function of a simplified stripe model and have found a reasonable correspondence with the ARPES experiments [22]. The spin and charge wave vectors transverse to vertical stripes span the Fermi surface in the neighborhood of the points and give rise to regions of degenerate states. Horizontal stripes have the same effect in the neighborhood of These are indeed the regions in which high-temperature superconductivity originates [12]. In practice, these regions are connected by arcs that are approximately 45° sections of a circle. Along these arcs, stripe wave vectors span the Fermi surface at isolated points at most. Therefore, the arc must become aware of the stripes by many-body effects such as the scattering of a pair of particles with total momentum zero into the regions near the and This implies that the spin gap should spread over the arcs as the system is cooled below the spin-gap temperature, which is consistent with ARPES observations [12]. 4.1. Symmetry of the Order Parameter

The momentum space picture also has consequences for the symmetry of the order parameter. The regions near to and communicate with each other via the arcs of the Fermi surface, and the relative phase of these regions must be chosen to maximize the amplitude of the order parameter along the arcs. As mentioned above, experimentally, the range of the gap function is nearest neighbor in real space for optimal doping, corresponding to the d-wave or the extended s-wave Evidently, the amplitude of the extended s-wave vanishes at the points, so the d-wave order parameter has the greater condensation energy [23]. This view of the origin of the symmetry of the order parameter leads to a number of interesting consequences. First of all, the existence of a nearest-neighbor gap function along the arcs of the Fermi surface suggests that the arcs correspond to the regions between stripes. Second, for the second and third neighbor components of the gap function, the amplitudes of the d-waves components vanish at the points but the amplitudes of the s-wave components are maximized. In certain circumstances, these s-wave components of the order parameter could either mix with the d-wave component (breaking time-reversal symmetry or lattice-rotational symmetry) or even become dominant. There is evidence from tunneling spectroscopy that order parameter mixing is induced in surfaces of An s-wave order parameter or component of the order parameter might also appear in overdoped materials, where the stripe structure is breaking up: the increased meandering of the stripes tends to mix the shortrange gap function of the environment with the longer-range gap function on the stripes. ACKNOWLEDGMENTS We would like to acknowledge frequent discussions of the physics of high-temperature superconductors with J. Tranquada. This work was supported at UCLA by the National Science Foundation grant number DMR93-12606 and, at Brookhaven, by the Division of Materials Sciences, U.S. Department of Energy under contract No. DE-AC02-98CH10886.


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REFERENCES 1. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986). 2. Z.-X. Shen et al., Science 267, 343 (1995). 3. B. Batlogg in High Temperature Superconductivity, edited by K. S. Bedell et al. (Addison-Wesley, Redwood

City, 1990), p. 37. 4. Y. J. Uemura et al. Phys. Rev. Lett. 62, 2317 (1989); Phys. Rev. Lett. 66, 2665 (1991). 5. V. J. Emery in Highly Conducting One-Dimensional Solids, edited by J. T. Devreese, R. P. Evrard, and V. E. van Doren (Plenum, New York, 1979) p. 327; J. Solyom, Adv. Phys. 28, 201 (1979). 6. A. Bianconi et al. in Lattice Effects in Superconductors, edited by Y. Bar-Yam., T. Egami, J. Mustre-de Leon, and A. R. Bishop (World Scientific, Singapore, 1992), p. 65. 7. S. A. Kivelson and V. J. Emery, this conference. 8. A. H. Luther and V. J. Emery, Phys. Rev. Lett. 33, 589 (1974). 9. V. J. Emery, Phys. Rev. Lett. 14, 2989 (1976); G. T. Zimanyi, S. A. Kivelson, and A. Luther, Phys. Rev. Lett. 60, 2089 (1988); J. Voit, Phys. Rev. Lett. 62, 1053 (1989), Phys. Rev. Lett. 64, 323 (1990); S. A. Kivelson and G. T. Zimanyi, Molec. Cryst. Liq. Cryst. 160, 457 (1988).

10. 11. 12. 13. 14. 15. 16. 17.

V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. Lett. 56, 6120 (1997). V. J. Emery and S. A. Kivelson, Phys. Rev. Lett. 74, 3253 (1995). M. R. Norman et al., Nature 392, 157 (1998). Z-X. Shen et al., Phys. Rev. Lett. 70, 1553 (1993); H. Ding et al., Phys. Rev. Lett. 54, R9678 (1996). V. J. Emery and S. A. Kivelson, Nature 374, 4347 (1995). E. Carlson et al., Phys. Rev. Lett. 83, 612 (1999). B. Batlogg et al., J. Low Temp. Phys. 95, 23 (1994); Physica C 235–240, 130 (1994). A. G. Loeser et al., Science 273, 325 (1996). H. Ding et al., Nature 382, 51 (1996).

18. D. N. Basov et al., Phys. Rev. Lett. 77, 4090 (1997).

19. D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988). 20. J. E. Sonier et al., Phys. Rev. Lett. 79, 2875 (1997). 21. Ch. Renner et al., Phys. Rev. Lett. 80, 3606 (1998).

22. M. Salkola, V. J. Emery, and S. A. Kivelson, Phys. Rev. Lett. 77, 155 (1996).

23. V. J. Emery, Phys. Rev. Lett. 58, 2794 (1987). 24. M. Covington et al., Phys. Rev. Lett. 79, 277 (1997); M. Fogelström et al., Phys. Rev. Lett. 79, 281 (1997).


Two Reasons of Instability in Layered Cuprates I. Eremin,1,2 M. Eremin,1 and S. Varlamov1

The phase boundary of the paramagnetic phase has been calculated, taking into account the strong electron correlations. The instability wave vector gets displaced from pure antiferromagnetic wave vector if doping increases. The calculated

pictures of Lindhard response functions and complex momentum dependence of a CDW pseudogap are presented. Spin susceptibility expression for the singlet correlated band at has been deduced.

Electronic structure of layered cuprates is very complicated and far from being completely understood. Various kind of instabilities are possible in these compounds and they are in focus of present investigations. In Furukawa et al. [1], instabilities were studied in a frame of two-dimensional (2D) Fermi liquid with Fermi surface containing the saddle points Becca et al. [2] analyzed the charge instabilities using infinite-U threeband Hubbard model. Zheleznyak et al. [3] explored instabilities occuring in the electron sybsystem with flat region at the Fermi surface. Here, we present the theoretical study of instabilities in a simple frame using charge and spin response functions. In our calculation we start from the Hamiltonian:

where are Hubbard-like quasi-particle operators describing the motion of the Zhang– Rice singlets over copper spin sublattice [4], is a superexchange coupling parameter, and is a Coulomb repulsion of the doped holes In 1 2

Physics Department, Kazan State University, 420008 Kazan, Russia. Corresponding Author: I. Eremin, Physics Department, Kazan State Univeristy, 420008 Kazan, Russian Federation: E-mail: [email protected]; Tel: 007 8432 315116; Fax: 007 8432 380901.


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a bilayer compound the interplane hopping leads to the splitting of the conductivity band on the bonding and antibonding bands. Because the latter is almost empty, the spectral

weight of the bonding band depends on the doping level as and the half-filling takes place at For the bilayer, means the number of the holes per two-copper site. As one can see the half-filling holds already at 1/7 holes per one Cu(2) position. It corresponds to the optimal doping, i.e., to the maximum First, we investigate the instability of the spin subsystem. The expression for the dynamic spin susceptibility was deduced in a frame of singlet-band model, taking into account strong electron correlation effects [7].

Here,

and is a superexchange coupling parameter between the neighboring copper spins, is a well-known Fermi-liquid formula for the spin susceptibility, is due to strong electron correlation effects. The instability in the paramagnetic phase occurs when the denominator in Eq. (2) equals zero. The calculated phase boundary of the paramagnetic state is given in Fig. 1. One can see for the underdoped compounds (the left side of the diagram, that the instability wave vector Q moves away from value if the doping increases. This

Two Reasons of Instability in Layered Cuprates

79

behavior of the spin susceptibility qualitatively explains the linear correlation between the doping or and incommensurability of the spatially modulated correlations vectors as compounds [8]. Now, we turn to another instability that is motored by the topological properties of the Fermi surface and the high density of states of the singlet-correlated band. Generally,

in the singlet band there are two peaks [9]. One of them is saddle singularity peak, and the second one, placed near the bottom of the band, is so-called hybridization peak. In bilayer compounds such as the chemical potential is placed near the saddle peak [10]. Inset of Fig. 2(a) shows the evolution of the Fermi surface that is presented with decreasing of the doping level towards underdoped regime. The calculated shape of the Lindhard response function

along the Brillouin zone are given in Fig. 2(b) for the different positions of the chemical potential and respectively. At higher doping the response function has the main maximum near . When the doping level is decreased, its peak goes

down but new hills are grown. The new vectors of the instability appear. In particular, at the intermediate doping , they are around and for the hills are shifted toward , as was pointed out earlier [11]. The hills provide a hint of the possible instability wave vector.

Recently, we examined a charge density wave scenario near the optimal doping with Three kinds of the interactions (phonon-mediated, superexchange, and short-range Coulomb repulsion) were taken into account. The momentum dependence of the CDW gap function is written as

where A(T) and B(T) parameters are calculated self-consistently following to the usual CDW theory [12]. The third term, D(T), has appeared due to a superexchange and a shortrange Coulomb interactions. It is remarkable to note that both of them support each other in opening of the D(T) component. Therefore, the critical temperature in a mean-field approximation is higher than because, in the case of superconducting transition temperature, the Coulomb repulsion suppress superexchange pairing interaction. In particular, for using well-known experimental data for Cu(2) Knight shift data, we deduced about 300K and The critical temperature for A and B components is 150–180K [5]. The instabilities in the charge and spin subsystems are probably not independent from each other. There must be a compromise between them. In this respect, it would be desirable to obtain an expression for the spin susceptibility with taking into account the instability in a charge subsystem. Using as an assumption that an instability in a charge subsystem is due to a CDW interaction, the analytical expression for the dynamic spin susceptibility has

80 been derived. It looks like Eq. (2) and

Eremin, Eremin, and Varlamov can be now written as:

Two Reasons of Instability in Layered Cuprates

81

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Here,

and,

where which is given by:

and

are the coefficients of the Bogolubov’s transformation,

are the momentum distribution functions. In the case of Eqs. (5) and (6) are naturally transformed to the normal state expression [7]. In a conclusion, we have calculated the Lindhard response function at different doping level. Its maximum provides a hint of the possible instability wave vector in a charge subsystem of singlet-bands model. Using the spin susceptibility, we have found that instability wave vector gets displaced from pure antiferromagnetic of doping increases. A new spin susceptibility expression in the presence of CDW instability has been deduced. ACKNOWLEDGMENTS

Our work was supported by INTAS Grant No. 96-0393. I. Eremin is grateful for the financial support to the Swiss National Science Foundation (Grant No. 7IP/051830) and International Centre of Fundamental Physics in Moscow (Grant No. INTAS 96-457). The work of M. Eremin and S. Varlamov is partially supported by Russian Scientific Council on Superconductivity (Grant No. 98014).

REFERENCES 1. 2. 3. 4. 5.

N. Furukawa, T. M. Rice, and M. Salmhofer, Phys. Rev. Lett. 81, 3195 (1998). F. Becca, F. Bucci, and M. Grilli, Phys. Rev. B 57, 4382 (1998). A. T. Zheleznyak, V. M. Yakovenko, and I. E. Dzyaloshinskii, Phys. Rev. B 55, 3200 (1997). F. C. Zhang, and T. M. Rice, Phys. Rev. B 37, 3759 (1988). S. V. Varlamov, M. V. Eremin, and I. M. Eremin, JETP Lett. 66, 569 (1997).

6. G. V. M. Williams, J .L. Tallon, E. M. Haines et al., Phys. Rev. Lett. 78, 721 (1997). 7. I. Eremin, Physica (North-Holland) B 234–236, 792 (1997).

8. 9. 10. 11. 12.

K. Yamada, C. L. Lee, Y. Endoh et al., Physica (North-Holland) C 282–287, 85 (1997). M. V. Eremin, S. G. Solovjanov, S. V. Varlamov et al., JETP Lett. 60, 125 (1994). Z.-X. Shen, and D. S. Dessau, Phys. Rep. 253, 1 (1995). I. Eremin, M. Eremin, S. Varlamov et al., Phys. Rev. B 56, 11305 (1997). C. A. Balseiro, and L. M. Falicov, Phys. Rev. B 20, 4457 (1979).

Influence of Disorder and Lattice Potentials on the Striped Phase N. Hasselmann,1,2 A. H. Castro Neto,1 and C. Morais Smith2

The influence of disorder and lattice effects on the striped phase of the cuprates and nickelates is studied within a perturbative renormalization group (RG) approach. Three regimes are identified: the free gaussian stripe, the flat stripe pinned by the lattice, and the disorder pinned stripe. Also, the effect of the stripe fluctuations on the spin correlations are discussed where we account for weak stripe-stripe interactions. We compare our findings with recent measurements on the La-based nickelates and cuprates and find good agreement with our calculations. PACS numbers: 71.45.Lr, 74.20.Mn, 74.72.Dn, 75.30.F

Recently, a number of materials with strong electronic interactions and spin correlations have been shown to exhibit spatially inhomogeneous ground states with simultaneous charge and spin density wave (CDW, SDW) order (striped phase). Static striped phase order has been most clearly detected in the nickelate families A very similar static order was also discovered in at and at 0.15,0.2 [6]. The most interesting but also most controversial candidates for stripe-ordered materials are the superconducting cuprates. There is no evidence of static order in these compounds, but a considerable number of experiments can be interpreted as evidence of a slowly fluctuating striped phase [7–10]. Evidence for stripe order has recently also been reported in the family [11]. A scenario for the formation of stripes has been proposed some time ago by Emery and Kivelson [12], and is based on the idea of frustrated phase separation. Recent numerical simulations have confirmed this picture [13]. SDW or CDW order arising from Fermi surface instabilities represent a highly coherent state. In contrast, in a frustrated phase-separated striped phase, the coupling between neighboring stripes is much weaker and in an intermediate time and length scale regime 1 2

Dept. of Physics, University of California, Riverside, CA, 92521, USA. I. Institut für Theoretische Physik, Universität Hamburg, D-20355 Hamburg, Germany.


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the fluctuations of each stripe can be treated as independent from the fluctuations of the neighboring stripes. Here, we want to discuss the influence of disorder and lattice potentials on the dynamic and static properties of the striped phase observed in the nickelates and cuprates. Our analysis assumes weak disorder, which should therefore describe the disorder produced by dopants (e.g., Sr) that are located out of the CuO or NiO planes. As quantum fluctuations play an important role in these compounds, we use a formalism that fully accounts for quantum fluctuations and concentrate at the possible phases at We take advantage of the weak coupling between neighboring stripes and consider single stripe dynamics but account for the confinement of the stripe wandering by its neighbors. This confinement reduces the influence of disorder drastically, as we show below. Physically, this effect is easy to understand: to take advantage of the quenched disorder potential, the stripe must wander strongly; but if the wandering of the stripe is confined, the stripe cannot find the optimal path. We use a phenomenological model of a stripe on a lattice in a quenched disorder background. Although in this model the stripes are oriented along one of the simple lattice directions, the continuum limit we use below is not sensitive to the microscopic details, and our results also apply for a striped phase with a diagonal orientation. Implicitly, we assume that a particular striped-phase order is well separated in energy from other configurations, so that through the renormalization group (RG), the topology of the charge and spin order does not change. Our stripe is modeled by a directed string of holes on a square lattice with lattice constant a. Each hole is allowed to hop in the transversal direction only. To account for the stripe stiffness, we include a parabolic potential of strength K, which couples neighboring holes in the stripe. The Hamiltonian in first quantized language is then given by where (we use dimensions with

Here, t is the hopping parameter, and are canonical conjugate transversal momentum and position variables of the nth hole, respectively. The last term, describes the interaction of the stripe with an uncorrelated disorder potential, where denotes the gaussian average over the disorder ensemble with Here, c is the characteristic velocity of the stripe excitations and is the impurity scattering time. Due to the stripe repulsion where L is the average interstripe distance. Using the replica trick, in the continuum limit written as [14]

the action of the stripe can be

Influence of Disorder and Lattice Potentials on the Striped Phase

85

where given by

is

is the stripe density and i is the replica index. The gaussian action

The dimensionless fields are defined through and the dimensionless parameter

We introduced the velocity which is a measure of the strength

of quantum fluctuations. The parameter g accounts for the lattice effects. In the derivation, we used and kept only the most relevant modes The form of this reflects the confinement of the stripe by its neighbors. The action in Eq. (1) is similar to an action studied by Giamarchi and Schulz [15] in the context of disordered one-dimensional conductors, and we can use their technique to derive the the RG equations to first order in D and We find [14]

with

and In the limit of small and there are two instabilities, one at and another at For the lattice potential is relevant and the stripe is pinned by the lattice. Such a lattice-pinned phase is flat in the long wavelength limit. In contrast, for the disorder potential is relevant and the stripe wandering is very strong. Strong external electric fields are expected to lead to a depinning of such a disorder-pinned stripe [16]. If both perturbations are irrelevant and the stripe is in the free gaussian phase, which has logarithmic wandering, strong quantum fluctuations, and massless excitations. If both disorder and lattice potentials are relevant, the disorder will always win in the long wavelength limit. Whereas the flat phase is massive, the disordered phase supports gapless albeit localized excitations. However, a relevant lattice potential can still lead to lattice pinning on scales smaller than the disorder-pinning length, which gives rise to a suppression of spectral weight at small energies (pseudogap behavior) [14]. Based on the available experimental data, we can now locate the La-based nickelates and cuprates in our phase diagram. Both the Sr- and the O-doped nickelates show strong lattice commensuration effects [1,2], and we therefore conclude that for the nickelates. The two most commonly used dopants for the nickelates are Sr and O. However, whereas the Sr dopants produce a quenched-disorder potential, the interstitial O dopants have been observed to order and thus produce only annealed disorder. Hence, by comparing these two compounds we obtain direct information about the relevance of quenched disorder. Neutron scattering experiments have shown that in the Sr-doped compound, the striped phase order is short ranged, whereas in some O-doped samples the width of the incommensurate (IC) magnetic peaks were resolution limited (Wochner et al. [1]). This is a clear indication for the relevance of quenched disorder in these compounds, and we conclude that both lattice and disorder perturbations are relevant.

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In passing, we comment on the melting of the striped phase observed in a compound on which a series of papers has been published recently [3]. Rather interestingly, the melting of the charge order has been shown to belong to the 2D Ising universality class (Su et al. [3]), a yet unexplained observation. This is somewhat surprising, as the stripes form a registered phase in this material and conventional wisdom [17] would suggest 3-Potts or 3-chiral-Potts critical exponents. However, we can understand the Ising behavior quite naturally within the frustrated phase separation scenario, where the melting transition occurs at the end of a coexistence curve of two phases with different densities, the charged stripes and the undoped AF regions. The striped phase in fluctuates strongly even at very low temperatures, from which we immediately conclude that the stripes are in the free phase. This is consistent with the absence of strong lattice commensurability effects in this material—i.e., the lattice potential is not relevant. Similar to the nickelates, the relevance of quenched disorder can be inferred from a comparison of Sr-doped lanthanides with O-doped lanthanides. As the disorder is expected to be irrelevant because the stripes are free, the IC spin fluctuations of these two materials should be almost identical, which is indeed what is observed experimentally [10]. However, our phase diagram predicts a phase transition to the disorder-pinned phase by reducing the stripe density As the stripe density decreases with decreasing doping, we expect a critical doping, below which the stripes become pinned. It is possible that such a transition has already been observed by Yamada et al. [9], who found a commensurate-IC transition of the spin fluctuations around We now discuss how the stripe fluctuations affect the experimentally observable spin fluctuations. Let us first take a simplified view on the stripe dynamics, accounting also for weak interstripe interactions. As mentioned above, these interactions are only relevant in the very long wavelength limit. We ignore disorder and lattice potentials and assume that they are both irrelevant. Approximating the effective stripe–stripe coupling by a harmonic potential, the action of the stripe array can then be written as

where U is dimensionless and quantifies the strength of the interstripe coupling. The integer

m numerates the stripes and should not be confused with the replica index i used in Eq. (1). The corresponding propagator is then

with UV cutoffs and The Matsubara frequencies are given by The propagator is massless around but the modes at the Brillouin boundary show a gap For lattice-pinned stripes, an additional mass would appear in the propagator. Let us now address the question, how the fluctuations of the stripes affect the spin correlations of the cuprates. In the striped phase, the charged stripes act as domain walls that separate undoped regions with opposite (probably short-ranged)


87

staggered magnetic order. We therefore write the staggered spin density as a product of the form where describes the staggered spin density of the confined undoped regions and is a function that changes sign at the position of the domain walls. As the stripes are separated in space from the spins in the undoped region, it is a reasonable assumption that the dynamics of decouple [18] so that

It is then easy to show that we can write

where step function of domain wall width

is the Fourier transform of a broadened Heavyside Averaging with the action (3) gives

The last decomposition is only possible at (for diverges because of the Mermin–Wagner theorem). The exponential prefactor, in the last line is the quantum analog of the Debye–Waller factor, with Hence, we see that strong quantum fluctuations of the stripes and weak stripe-stripe coupling and small U) suppress We can now decompose the Fourier transform of into an elastic and an inelastic part to find

where is the Fourier transform of Using Eq. (4), we see that the inelastic pan has gapless (acoustic) modes around and gapped (optical) modes at (both with . Because of the convolution with , as implied by Eq. (5), the wave vector k is actually measured with respect to the commensurate AF positions The acoustic modes, which are excited at low energies, therefore give rise to IC scattering in neutron scattering experiments. However, increasing the energy results in peaks that disperse away from the IC positions and finally the optical modes that are located

at the commensurate positions are excited. Therefore, starting from low energies, one would first see IC peaks that, with increasing energies, merge into a broad commensurate peak. Our simple picture actually describes the experimentally observed evolution of the IC peaks

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in . In Zaanen et al. [20], a merging of the IC peaks has been explained as a result of single stripe dispersion. In our view, the single-stripe dynamics contribute to the continuum of excitations, but the lower bound of the continuum is determined by the interstripe coupling. Furthermore, as can be seen from Eq. (6), the dispersion away from the IC position is not symmetric around the IC positions, because signals at large values of are exponentially suppressed both by the finite thickness of the domain walls and the quantum fluctuation of the stripes. Such an asymmetry has also been observed experimentally [19]. We have sketched the evolution with energy schematically in Fig. 1, for a situation with long-range AF order. A finite coherence length in the AF regions would further broaden the signal and would add a gap to the spectrum, which would result in a shift of the spectrum in Fig. 1 to higher energies. The absence of any signal at very low energy transfers in the neutron scattering experiments thus indicates a finite coherence length of the spin order in the AF regions. In conclusion, we have studied the relevance of weak disorder and lattice potentials in the striped phase and identified three different phases: a free gaussian phase, in which both perturbations are irrelevant; a flat phase, which is pinned by the lattice; and a disordered phase, in which the stripe accommodates to the disorder potential. Using the available data on the striped phase of the La-based nickelates and cuprates, we could locate these materials in our phase diagram. Finally, we discussed the dynamics of a fluctuating but well-ordered striped phase and showed that our simple model could reproduce the main features of the energy and momentum dependence of the IC spin fluctuations in the lanthanides.

ACKNOWLEDGMENTS We thank G. Aeppli, A. O. Caldeira, G. Castilla, Y. D. Dimashko, M. P. A. Fisher, A.

van Otterlo, and H. Schmidt for helpful discussions. NH acknowledges support from the Gottlieb Daimler- und Karl Benz-Stiftung and the Graduiertenkolleg “Physik nanostruk-

turierter Festkörper,” Univ. Hamburg. NH and CMS received support from DAAD-CAPES


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PROBRAL project no. 415. AHCN acknowledges support from the Alfred P. Sloan Foundation and the U.S. Department of Energy. REFERENCES 1. P. Wochner et al., Phys. Rev. B 57, 1066 (1998); J. M. Tranquada et al., Phys. Rev. Lett. 79, 2133 (1997); K. Nakajima et al., J. Phys. Soc. Jpn. 66, 809 (1997); J. M. Tranquada et al., Phys. Rev. Lett. 73, 1003 (1994). 2. J. M. Tranquada et al., Phys. Rev. B 54, 12318 (1996). 3. Y. Su et al., in this conf. proc.; Y. Yoshinari et al., cond-mat/9804219; G. Blumberg et al., Phys. Rev. Lett. 80, 564 (1998); S. H. Lee and S.-W. Cheong, ibid. 79, 2514 (1997); A. P. Ramirez et al., ibid. 76, 447 (1996). 4. J. M. Tranquada et al., Nature 365, 561 (1995); J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). 5. M. von Zimmermann et al, Europhys. Lett. 41, 629 (1998). 6. J. M. Tranquada et al., Phys. Rev. Lett. 78, 338 (1997). 7. G. Aeppli et al., Science 278, 1432 (1997); T. E. Mason et al., Phys. Rev. Lett. 1414 (1992). 8. P. Dai et al., Phys. Rev. Lett. 8 0 , 1738(1998); H. A. Mook et al., cond-mat/9712326. 9. Y. Endoh et al., Physica C 263, 349 (19%); K. Yamada et al,J. Supercond. 10, 343 (1997). 10. B. O. Wells et al., Science 277, 1067 (1997). 11. N. L. Saini et al., Phys. Rev. Lett. 79, 3467 (1997); N. L. Saini et al., Phys. Rev. B 58, 11768(1998). 12. V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993). 13. Stojkovic et al., cond-mat/9805367. 14. N. Hasselmann et al., unpublished. 15. T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325 (1988). 16. C. Morais Smith et al., Phys. Rev. B 58, 453 (1998). 17. D. A. Huse and M. E. Fisher, Phys. Rev. B 29, 239 (1984). 18. J. Zaanen and W. van Saarloos, Physica C 282–287, 178 (1997). 19. S. M. Hayden et al., Phys. Rev. Lett. 76, 1344 (1996); T. E. Mason et al., ibid. 77, 1604 (1996). 20. J. Zaanen et al., Phys. Rev. B 53, 8671 (1996).


Stripe Liquid, Crystal, and Glass Phases of Doped Antiferromagnets S. A. Kivelson1 and V. J. Emery2

A largely descriptive survey is given of the ordered phases of doped antiferromagnets, and of the long wavelength properties that can be derived from an order-parameter theory. In particular, we show that the competition between the long-range Coulomb repulsion and the strong short-distance tendency of doped holes to coalesce into regions of supressed antiferromagnetism leads to a vari-

ety of self-organized charge structures on intermediate length scales, of which “stripes” are the most common, both theoretically and experimentally. These structures lead to a rich assortment of novel electronic phases and crossover phenomena, as indicated in the title. We use the high-temperature superconductors as the experimentally best-studied examples of doped antiferromagnets.

Highly correlated materials have intermediate electron densities and are frequently doped Mott insulators, so that neither the kinetic energy nor the potential energy is totally dominant, and both must be treated on equal footing. The question arises: Are there actual “intermediate” low temperature phases of matter that interpolate between the high-density

“gas” phase (usually called a Fermi liquid) and the low-density strongly insulating Wigner crystal phase? We have shown that, at least in the case of lightly doped antiferromagnets, the tendency of the antiferromagnet to expel holes always [1–4] leads to phase separation, which, when frustrated by the long-range piece of the Coulomb interaction, leads [5–7] to the formation of states that are inhomogeneous on intermediate length scales and (possibly) time scales. The most common self-organized structures that result from these competing interactions [7,8] are “stripes,” by which we generally mean dimensional antiphase domain walls across which the antiferromagnetic order changes sign, and along which the doped holes are concentrated [9]; the term stripe is, of course, 1 2

Dept. of Physics, UCLA, Los Angeles, CA 90095. Dept. of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000.


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a reference to the important two-dimensional case relevant to the high-temperature

superconductors. Even if the discussion is confined to ordered phases of doped antiferromagnets, we are left with an exceedingly complex problem. Many varieties of order have been observed in doped antiferromagnets, including spin and charge order and, of course, superconductivity. The spin and charge order can be commensurate or incommensurate, and both can be ideal or glassy. There are also various structural phases, such as the tetragonal and orthorhombic phases of that may reflect important changes in the electronic state, as the structural order can couple to various forms of “electronic liquid crystalline” order [10]. Of course, all of these types of order can compete or coexist in various ways. 1. LANDAU THEORY OF COUPLED CDW AND SDW ORDER

We begin by discussing density-wave order, and in particular the interplay between spin-density wave (SDW) and charge-density wave (CDW) order. This can be analyzed

most simply by studying the Landau theory of coupled order parameters [11]. Although it is possible to have various sorts of “spiral” spin phases, the only spin order in much of parameter space is collinear, so the discussion is confined to this region. The resulting phase diagram is shown schematically in Fig. 1. Three features of the analysis, which is discussed in detail in Ref. 1 1 , bear repeating: 1.

In order for the SDW order parameter and the CDW order parameter to couple in the lowest possible order (third), it is necessary that the ordering vectors satisfy

the relation

or in other words, the wavelength of the SDW is twice that of

the CDW; this gives precise meaning of the concept [5] of “topological doping” and implies that the charge is effectively concentrated along antiphase domain walls in the

magnetic order. 2.

It is possible to have a phase with CDW order, but no SDW order, whereas SDW order

always implies CDW order. This is important to bear in mind when thinking about the experimentally determined phase diagram of the high-temperature superconductors or any other doped antiferromagnet, because there are many good probes (such as NMR,

Stripe Liquid, Crystal, and Glass Phases of Doped Antiferromagnets

3.

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and neutron scattering) that are sensitive to spin order or fluctuations, but fewer that are sensitive to charge order. Where incommensurate spin order is detected, we can infer the existence of charge order directly, but where no magnetic order is observed, there may or may not exist as yet undetected charge order. Although Landau theory by its very character is relatively insensitive to the microscopic considerations conventionally referred to as the mechanism of ordering, an important classification of mechanisms follows directly from these considerations. If, upon lowering temperature, CDW order is encountered first and SDW order is either entirely absent or only appears at lower temperatures when the CDW order is already well developed, the density wave transition is “charge-driven,” and we can infer that the SDW order is in some sense parasitic, i.e., driven by the interaction with the CDW. However, if both CDW and SDW order develop simultaneously, but with the CDW order turning on more slowly at the transition according to then the ordering can be said to be spin driven. (Intermediate cases, in which the spin and charge ordering must be treated on an equal footing, are also possible [11].) Hartree-Fock treatments [12] of stripes lead to spin-driven ordering, whereas frustrated phase separation (and, indeed, experiments in the nickelates, manganates, and the appropriate cuprates) imply that the ordering is, in fact, charge driven.

2. CONCERNING THE MECHANISM OF STRIPE FORMATION

The charge ordering described above, which we think of as ordered arrangements of charged “stripes,” differs from more usual CDW order in metals in that it is a consequence of frustrated phase separation, not a Fermi-surface instability. There are three important consequences: (1) Whereas conventional CDW order, as observed in many charge-transfer

salts and bronzes, tends to oscillate in all directions, the stripe order that we have in mind here involves a one-dimensional modulation of the charge density; hence the name stripes. (2) There can be additional low-energy electronic degrees of freedom within a stripe, i.e.,

the stripe can be metallic and have its own spin dynamics. (3) The density of states at the Fermi level is increased [13], not decreased by the opening of a gap in the electron energy spectrum, as for a CDW driven by a Fermi-surface instability. The second point is exceedingly important in distinguishing mechanisms of stripe formation, and is unique feature of stripes produced by competing long- and short-range interactions [5]. In any model with only short-range interactions, stripes, if they form at all, have a preferred hole density that is usually commensurate and, in turn, determines the stripe concentration. Consequently, in general, even if there were a region of parameters in which a putative stripe phase were not unstable to phase separation, it always

would be insulating [ 12,14] with a substantial charge gap. By contrast, whenever the stripes arise as a consequence of a competition between a long-range Coulomb interaction and a short-range tendency to phase separation, the stripe concentration is a compromise between these two forces, so gaps in the intrastripe excitation spectrum are no longer the rule [15]. Finally, self-organized stripe structures are an intrinsic property of a doped antiferromagnet. Although they certainly couple in interesting material-specific ways to external influences, such as the lattic modulation in or the chains in [17], these extrinsic effects should be viewed as reflecting the intrinsic stripe physics rather than causing it.

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3. ELECTRONIC LIQUID CRYSTAL PHASES

Following the work in Ref. 10, we analyze the case in which intrastripe metallic degrees of freedom interact with the fluctuating “geometry” of the stripe arrangement, resulting in a set of new states of matter, which, in analogy with classical liquid crystals, we named electronic liquid crystals.

By analogy with classical liquid crystals, we can readily deduce the schematic phase diagram, as shown in Fig. 2. Here, the x axis is a quantum parameter, related to the transverse zero-point energy of the stripes, i.e., it measures the extent of quantum fluctuations of the stripe order. These phases, which can exist at either zero or finite temperature, can be classified as follows. 3.1. Stripe Crystal Phases Here translational symmetry in the direction perpendicular to the stripes is broken because the stripes are ordered, and translational symmetry along the stripe is broken because the electrons along the stripe form a CDW that is phase locked between stripes. This phase is conceptually similar to the CDW phases that occur in charge transfer salts and, in common with them, it is insulating. In the important but special cases in which the

stripes have no low-energy internal degrees of freedom (e.g., stripes that are full of holes or electrons), the insulating state is achieved without breaking translational symmetry along the

stripe direction. At zero temperature in two spatial dimensions, the stripe order is always pinned at a period commensurate with the host crystalline lattice, whereas at non-zero


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temperature it can be commensurate or incommensurate, but where it is incommensurate, the positional order is only quasi-long range.

3.2. Electronic Smectic Phases If the stripe order is maintained, but the intrastripe excitations remain unpinned, one obtains an electronic smectic phase. This phase posseses the same broken symmetries transverse to the stripe direction as the stripe crystal phases (with all the same dimension specific considerations mentioned above), but simultaneously exhibits liquid-like behavior associated with the motion of electrons within a stripe, and the tunneling of electrons between stripes. Depending on additional interactions in the problem, this phase can either remain an electron liquid down to zero temperture, or can become superconducting below a critical temperature, A major portion of the work in Ref. 10 addressed the issue of how the transverse fluctuations of the stripes controls both the transition between the stripe crystal and electronic smectic phases, and the magnitude of the superconducting within the smectic phase. Basically, in the presence of a spin gap, the intrastripe electron gas is prone to large superconducting and CDW fluctuations. In two spatial dimensions (where within a stripe the electrons form an effective one dimensional electron gas), it is known that both these susceptibilies typically will diverge as with the CDW being the more divergent. In higher dimensions, more complicated possibilities exist, but the physics is not qualitatively

different. Now, because the CDW order involves short-wavelength density oscillations along the stripe, the CDW ordering on neighboring stripes is readily dephased by transverse stripe fluctuations; through this mechanism, increasing transverse stripe fluctuations (either quan-

tum or thermal) can easily be shown to stabilize the smectic phase at the expense of the stripe crystal. Conversely, pair tunneling between neighboring stripes, and hence the transverse component of the superfluid density, is enhanced by transverse stripe fluctuations. To see this, we note that the local pair-tunneling matrix element, depends exponentially on the local separation, between stripes

To get an idea for the physics, we average this quantity over transverse stripe fluctuations, keeping terms up to second order in a cumulant expansion, with the result

where

is the variance of

Clearly, for fixed mean spacing

between stripes, the pair

tunneling is a strongly increasing function of

3.3. Electronic (Ising) Nematic When the transverese stripe fluctuations get sufficiently violent, they will certainly lead to a liquid state, with full translational symmetry. However, it is possible that the general

orientation of the stripes can persist beyond the melting transition. In this case, the electronic phase is liquid-like and translationally invariant, but it breaks the discrete rotational symmetry of the host crystal. The nematic phase in fact breaks this discrete symmetry. For instance, in a tetragonal crystal with a four-fold rotational symmetry, the nematic phase

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would be electronically orthorhombic, with only a two-fold rotational symmetry surviving [18]. Of course, there is bound to be some back coupling between electronic and lattice structure, so such a phase would also be accompanied by a locally orthorhombic distortion of the crystalline lattice. Like the smectic, the nematic phase can either remain a normal liquid down to zero temperature, or become superconducting. We have sketched a representative superconducting phase boundary as a dashed line in Fig. 2. The logic governing its shape is as follows: Superconductivity requires both pairing, which occurs below a temperature where is the zero temperature magnitude of the superconducting (spin) gap or pseudogap, and phase ordering, which sets in below a temperature which is proportional to the zero temperature superfluid density [19]. It is clear that the general trend found in the smectic, in which an increase in the transverse fluctuations increases the transverse phase stiffness, should apply in the nematic phase close to the smectic phase boundary. Thus, so long as determines it will be an increasing function of Of course, ultimately the transverse fluctuations become so violent that even the local concept of a stripe ceases to be well defined; if we take the view that stripes are an essential ingredient in the pairing mechanism [20] (equivalently, in the spin-gap formation), then must first rise and then decrease with increasing flucutations. We have shown the peak in near the nematic to isotropic phase boundary, because we imagine that this is where the stripes start to fall apart at a local level. The points marked are quantum critical points. As drawn, the superconducting phase boundary crosses the nematic phase boundary at a tetracritical point, but it is equally possible that the superconducting phase boundary could end at a bicritical point (say, roughly where appears in the figure), and that beyond this there is a (possibly weak) first-order phase boundary marking the simultaneous onset of superconducting and nematic order. Alternatively, the superconducting phase might, in some circumstances, lie entirely inside the nematic phase, if the quantum critical points and were exchanged, in which case there would be no multicritical point analagous to 3.4. Isotropic Stripe Liquid With sufficiently large transverse fluctuations, all symmetry is restored, so an isotropic liquid phase results. However, if sufficient stripe order persists on a local level, the resulting isotropic stripe liquid may be quite different from a noninteracting electron gas. Because

there is no symmetry distinction, it is possible that the evolution from a stripe liquid to a Fermi gas involves a crossover, but often, as in a liquid gas transition, a first order transition separates two qualitative different but asymptotically similar phases; for that reason, we have included such a first-order line, ending in a critical point, in Fig. 2.

4. QUENCHED DISORDER AND STRIPE-GLASS PHASES It was shown by Larkin [23] that quenched disorder is relevant in CDW systems in dimension i.e., in any physical dimension, true long-range CDW order simply does not occur in the presence of disorder! However, in high enough dimension and for weak enough disorder, there exists a “Bragg glass” phase, [24] which in the present context we refer to as a stripe-glass, which has power-law density-wave order. This is a distinct state

of matter, and hence must be separated from the high-temperature melted phase by a sharp phase transition. This phase is now more or less established in and it has been argued


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[25] that in even if an exponentially dilute concentration of free dislocations spoils the power-law decay of correlations at very long distances, this has very little practical consequence. For the materials of interest to us here, which are either three-dimensional or quasi-two-dimensional, it is safe to conclude that a true stripe-glass phase exists, with a sharply defined glass transition temperature, Where, as a function of varying material parameters, we expect a quantum critical point. However, this quantum critical point has quite different character than those invoked in various theories of high-temperature superconductivity [26], in that the glass transition is disorder driven. Rather, the quantum critical properties are more or less similar to the melting of a “Wigner glass,” which has been invoked to explain the apparent metalinsulator transition observed in Si MOSFETs [25]. Because the charged stripes in doped antiferromagnets are typically antiphase domain walls in the magnetic order, the freezing of the stripe motion at opens up the possibility of subsequent (i.e., at still lower temperature) ordering of the spins. As we discussed some time ago [6], spin ordering in a stripe glass leads to a “cluster spin-glass” phase. Because spin-glass ordering involves a broken symmetry (time reversal), it is easier to detect experimentally than stripe-glass ordering, and so more is known about the occurrence of this sort of ordering in doped antiferromagnets. We infer that, in doped antiferromagnets in which cluster spin-glass ordering is observed below a spin-glass transition temperature there also should be a stripe-glass ordering transition with Finally, it is worth making a couple of general observations concerning the effects of quenched disorder. (1) One of the most dramatic features of all glasses is the dramatic slowing down of dynamics over a broad range of temperatures as the glass transition is approached. One consequence is that is always very difficult to determine experimentally (or even to determine whether there really is a finite at all); rather, there is an apparent that depends on the frequency at which the system is probed. Thus, it is to be expected when dealing with a glass transition that the phase diagram, as determined by different experimental probes, will look rather different, with For instance, a glass phase boundary, as determined by neutrons, is a function of the energy resolution and extrapolates to the phase boundary as the energy resolution tends to zero. (2) It can be proved that disorder eliminates any first-order transitions in two dimensions [27]. Weak disorder can either convert a first-order line into a more or less sharp crossover or turn it into a continuous transition if there remains a fundamental distinction between the two phases. For instance, in the phase diagram in Fig. 2, if the tetracritical point were replaced by a bicritical point, followed by a first-order transition to a superconducting nematic phase, the first-order line would be replaced by a line of continuous phase transitions in the presence of disorder. 5. SOME COMMENTS ON PARTICULAR PHASES OBSERVED IN THE HIGH-TEMPERATURE SUPERCONDUCTORS We conclude by making some remarks about a few recent experimental discoveries in the most studied of doped antiferromagnets, the high-temperature superconductors, and their relationship to the theoretical considerations discussed above. Here, the “undoped” system is a highly insulating spin-1/2 antiferromagnet, which is made conducting and superconducting by the addition of a small concentration x of “doped” holes. Although much of what we discuss is known or presumed to be common to all of the high-temperature

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superconductors, we refer explicitly to experiments in the LSCO and YBCO families of

materials. 5.1. The Insulating “Spin Glass” For x in the range between about 2% and 5%, no long-range order of any sort has been observed, and the material is insulating at low temperatures. There is, however, a well

defined spin-glass transition [28]. Although the spin-glass phase is generally viewed as a curiosity, of no central significance, we have long taken the view [6] that, because this is the only ordered phase proximate to the high-temperature superconducting phase, it should in fact play a central role in our thinking about these materials. We proposed that it is in fact a “cluster spin-glass” phase, in which a frozen random array of stripes produces the frustration, so that the spin glass consists of patches of locally antiferromagnetically ordered spins with an axis of magnetization that varies from patch to patch. In particular, we showed that this proposal naturally accounts for the remarkable fact [29], derived from early neutron scattering measurements of the spin structure factor, that the inverse correlation length, obeys the simple composition rule

It is clear that there must be a transition at which the stripes freeze into a stripe glass. The spin-glass transtion is more readily detected experimentally because it involves symmetry breaking, whereas the stripe-glass transition involves only replica-symmetry breaking [24,25]. However, we believe that the stripe glass is the fundamental phenomenon and that the spin-glass transition is more or less parasitic. Indeed, it is likely that the stripeglass transition temperature is greater than we await experimental input on this last issue. 5.2. The Superconducting Stripe Glass

The importance of the spin glass is further substantiated by the old observation, which has recently been dramatically confirmed by Niedermayer and collaborators [28], that spinglass and superconducting order in fact coexist in underdoped high-temperature superconductors! That static spin order and superconducting order can coexist in a single-component electronic system is very surprising in terms of conventional paradigms of superconductivity. Of course, the realization that the spin glass is actually a stripe glass, which can be viewed as a slightly disordered version of the electronic smectic phase discussed above, renders this observation a key experimental confirmation of the relevance of stripe physics to high-temperature superconductivity. In a sense, self-organization into stripes generates a two-component electronic system—a localized spin component that lives between the stripes, and a metallic component that flows along the stripes. In the past few years, neutron scattering studies of have revealed

a still more dramatic and detailed aspect of the coexistence of superconducting and magnetic order. Specifically, Tranquada and collaborators [30] found a sequence of transitions that depends somewhat on Sr concentration. As the temperature decreases, there is (1) a structural

transition to a low temperature tetragonal (LTT) phase, (2) a charge-ordering transition at which the appearance of well-defined incomensurate elastic peaks in the lattice structure factor are driven by charge stripe ordering, (3) a spin-ordering transition, with ordering


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vector twice that of the charge ordering vector, and (4) a superconducting transition, with, however, reduced relative to that in at the same Sr concentration. The coexistence of superconducting and stripe order was considered surprising, and indeed it has sometimes been attributed to sample inhomogeneity. However, the evidence in favor of coexistence continues to increase [31]. That charge order sets in before spin order confirms that the density-wave ordering is “charge driven” in the sense defined above. The addition of Nd to the material stabilizes the LTT structure, which allows the oxygen tilting phonon to couple more strongly to any charge order—in terms of the schematic phase diagram in Fig. 2, this reduce the magnitude of the quantum fluctuations of the stripes, so the material should be viewed as living farther to the left than Indeed, it is tempting to relate the sequence of observed charge transitions to those on a trajectory on our phase diagram that passes from the normal state at high temperature, through a nematic phase, to a smectic phase, and finally to a superconducting smectic phase at low temperatures. This identification is made slightly less than airtight by two subtleties: (1) It is not clear to what extent the structural phase transition to the LTT phase can be viewed as electronically driven; and (2) the elastic peaks, observed in neutron scattering, have a finite width, corresponding to a long but finite correlation length for the density-wave order. As discussed above, this is to be expected in a quasi-two-dimensional system with disorder, where any ordered state must be glassy, but it makes the unambiguous identification of the various phases less secure. Still more recently, neutron scattering experiments on [16,32] underdoped and even [33] optimally oxygen-doped (with as high as 42K) have shown that static, fairly long-range stripe order and superconductivity coexist. In these materials, the tansition temperatures for spin ordering (which is all that has been detected to date) and superconducting ordering appear to be close to each other, or possibly exactly the same. This demonstrates an intimate relation between stripe ordering and superconductivity, and is an important new piece of “theory independent” evidence for the critical role played by stripe order in the mechanism of high-temperature superconductivity. ACKNOWLEDGMENTS

We would like to acknowledge frequent discussions of the physics of high-temperature superconductors with J. Tranquada and G. Aeppli. This work was supported at UCLA by the National Science Foundation grant number DMR93-12606, and at Brookhaven by the Division of Materials Sciences, U.S. Department of Energy, under contract No. DE-AC0298CH10886. REFERENCES 1. V. J. Emery, S. A. Kivelson, and H.-Q. Lin, in Proceedings of the International Conference on the Physics of Highly correlated Electron Systems. Santa Fe, New Mexico, September, 1989. Edited by J. O. Willis, J. D.

Thompson, R. P. Guertin, and J. E. Crow, Physica B 163, 306 (1990); V. J. Emery, S. A. Kivelson, and H.-Q. Lin, Phys. Rev. Leu. 64, 475 (1990). 2. M. Marder, N. Papanicolau, and G. C. Psaltakis, Phys. Rev. B 41 6920 (1990). 3. S. Hellberg and E. Manousakis, Phys. Rev. Lett. 78, 4609 (1997).

4. L. Pryadko, D. Hone, and S. A. Kivelson, Phys. Rev. Lett. 80, 5651 (1998). 5. S. A. Kivelson and V. J. Emery, Synth. Met. 80, 151 (1996). 6. V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993). 7. U. Löw et al., Phys. Rev. Lett. 72, 1918 (1994).

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8. M. Seul and D. Andelman, Science 267, 476 (1995) and references therein. 9. E. Carlson et al., Phys. Rev. B 57, 14704 (1998). 10. S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998). 1 1 . O. Zachar, S. A. Kivelson, and V. J. Emergy, Phys. Rev. B 57, 1422 (1998). 12. J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7391 (1989). H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990). 13. M. Salkola, V. J. Emery, and S. A. Kivelson, Phys. Rev. Lett. 77, 155 (1996). 14. S. R. White and D. J. Scalapino, Phys. Rev. Lett. 80, 1272 (1998). For a demonstration that the stripes found here are, in fact, insulating, see V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 59, 15641 (1999). 15. Yamada et al., plotted the spin incommensurability as a function of the doped hole concentration for the LSCO family [16]. For several points appear to lie on or close to the line which corresponds to a commensurate doping of one hole per two unit cells along a stripe, which would imply insulating stripes. However, the data points actually fall on an arc of a circle, which touches the line near to the special hole concentration at which is suppressed in the LSCO family. 16. K. Yamada et al., Phys. Rev. B 57, 6165 (1998). 17. A. Bianconi et al. in Lattice Effects in High- Superconductors, edited by Y. Bar-Yam, T. Egami, J. Mustre-de Leon, and A. R. Bishop (World Scientific, Singapore, 1992), p. 65. 18. An Abanov et al., Phys. Rev. B 51, 1023 (1995). 19. V. J. Emery, S. A. Kivelson, Nature 434, 374 (1995). 20. V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 56, 6120 (1997). 21. S. H. Lee and S. W. Cheong, Phys. Rev. Lett. 79, 2514 (1997). 22. G. Aeppli et al., Science 278, 1432 (1997). 23. A. I. Larkin, Zh. Eksp. Teor Fiz. 58, 1466 (1970) [Sov. Phys. JETP, 31, 784 (1970)]. 24. T. Giamarchi and P. Le Doussal, Phys. Rev. B 53, 15206 (1996); D. Fisher, Phys. Rev. Lett. 78, 1964 (1997); M. J. P. Gingras and D. A. Huse, Phys. Rev. B 53, 15193 (1996). For a review, see T. Giamarchi and P. Le Doussal in Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, 1997), and references therein. 25. S. Chakravarty et al., cond-mat/9805383. 26. See, for example, R. B. Laughlin, cond-mat/9709195; C. Castellani et al., Physica C, 282, 260 (1997); C. M. Varma, Phys. Rev. B55, 14554 (1997); S. Sachdev et al., Phys. Rev. B51 14874 (1995), and references therein. 27. Y. Imry and M. Wortis, Phys. Rev. B 19, 3580 (1979); M. Aizenman and J. Wehr, Phys. Rev. Lett. 62, 2503 (1989). A particularly transparent physical argument is given by A. N. Berker, Physica A 194, 72 (1993). 28. Ch. Niedermayer et al., Phys. Rev. Lett. 80, 3843 (1998). See also A. Weidinger et al., Phys. Rev. Lett. 62, 102 (1989) and F. C. Chou et al., Phys. Rev. Lett. 75, 2204 (1995). 29. B. Keimer et al., Phys. Rev. B 46, 14034 (1992). See also S. M. Hayden et al., Phys. Rev. Lett. 66, 821 (1991). 30. J. Tranquada et al., Nature 375, 561 (1995); I. M. Tranquada, Proceeding of the International Conference on Neutron Scattering Toronto, Canada (1997), Physica B 241–243, 745 (1998). 31. J. E. Ostenson et al., Phys. Rev. B 56, 2820 (1997). J. Tranquada et al., Phys. Rev. Lett. 78, 338 (1997). 32. T. Suzuki et al., Phys. Rev. B 57, R3229 (1998) and H. Kimura et al., Phys. Rev. B 61, 14366 (2000). 33. Y. S. Lee and R. J. Birgeneau, private communication.

Dynamical Mean-Field Theory of Stripe Ordering Alexander I. Lichtenstein,1 Marcus Fleck,2 Andrzej M. Oles,2,3 and Lars Hedin2

Applying the dynamical mean-field theory to the two-dimensional Hubbard model, we calculate self-consistent solutions of doped antiferromagnets with spatially varying spin density using realistic tight-binding parameters. The local self-energy of the supercell includes transverse and longitudinal spin fluctuations with an effective local potential due to short-range electron-electron correlations. It is found that metallic stripes are stabilized by a pseudogap. The stripes along (1,0) direction filled by one hole per two-domain wall unit cells change with increasing Coulomb interaction U to the more extended stripes along (1,1) direction consisting of four atoms filled by 1 /4 doped hole each. These findings agree qualitatively with the experimental observations in the superconducting cuprates, and

predict a qualitative difference between various compounds due to differences in the extended hopping parameters.

Charge localization has been observed in recent years in several doped transition metal oxides, including nickelates, cuprates, and manganites. Localized holes organize into onedimensional (1D) structures called stripes, which form commensurate patterns at a doping of holes as discovered in Theoretically, the stripe phase has been obtained by solving the Hubbard model in the Hartree–Fock (HF) approximation [2– 7], and by numerical density matrix renormalization group (DMRG) calculations [8]. Unfortunately, the observed antiferromagnetic (AF) domains [1] separated by four Cu-O-Cu spacings along the (1,0) or (0,1) direction, filled by one hole per two 4 × 1 domain wall unit cells, so-called half-filled stripes, are less stable within the HF calculations than filled 1

Forschungszentrum Jülich, D-52425 Jülich, Federal Republic of Germany. Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Federal Republic of Germany. 3 Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Kraków, Poland.

2

Stripes and Related Phenomena, edited by Bianconi and Saini.

Kluwer Academic/Plenum Publishers, New York, 2000.

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Lichtenstein, Fleck, Oles, and Hedin

stripes (with one hole per one

unit cell) [7]. In contrast, there is evidence from DMRG

calculations for a two-dimensional model with hole doping that the ground state consists of AF domains that differ by a phase and are separated by half-filled domain walls [8]. Thus it is crucial to go beyond the HF approximation and include local electron correlations due to charge and spin fluctuations.

Recently, incommensurate magnetic fluctuations along the (1,1) direction have been observed in the bilayer compound In this material, the low-frequency spin fluctuations change from commensurate to incommensurate on cooling, with the incommensurability first appearing at temperatures above At present, it remains unclear

whether spatial segregation of the charges is associated with the observed incommensurate magnetic order. HF results on the Hubbard model show periodic arrays of line defects or line solitons along the (1,1) direction [3,6]. However, there are no explicit predictions concerning the incommensurate structure in A physically satisfactory treatment of the correlation effects is possible in the limit of large spatial dimension when the fermion dynamics is described by a local self-energy in quantum impurity models, solved self-consistently in the dynamical mean-field theory (DMFT) [11]. Thereby a correct implementation of the low-energy scale due to magnetic excitations plays a prominent role [12]. Recently, we showed [13] that a formulation of the DMFT for the magnetically ordered (AF and spiral) states in the 2D Hubbard model is possible by using the spin-fluctuation (SF) exchange interaction with an effective potential due to particle–particle scattering [14]. Here we present a generalization of this approach to the static stripe phase with long-range order and discuss the obtained spin and charge distributions for the cuprates. We consider the 2D Hubbard model defined on a square lattice built of cells containing L sites

where the pair of indices {mi} labels the unit cell m and the position

of atom within the unit cell, with the summations performed over both independent coordinates. Effective parameters for the single-band model of the cuprates include hopping integrals between first-, second-, and third-nearest neighbors, , and respectively [15]. The one-particle Green function in the stripe phase is given by a matrix for all atoms of the cell, where are fermionic Matsubara frequencies. We approximate the Green function using a local cell self-energy [11,13]

Here

is a

matrix that describes the single-particle part with the self-consistent

HF potential calculated using the actual local electron density for the opposite spin

in the stripe phase

Our approach to the stripe phase makes a different approximation than the recently proposed dynamical cluster approximation (DCA) [17], as the dynamical effects within the cell are

Dynamical Mean-Field Theory of Stripe Ordering

103

treated using the local approximation. Note that we are not including 1 /d-corrections going beyond the DMFT. The local Green functions for each nonequivalent atom i are therefore

calculated from the diagonal elements of the Green function matrix Eq. (2), and the SF part of the cell self-energy becomes local but site dependent in the stripe phase. The self-consistency between the considered atom and

its surrounding is imposed using the cavity method [11]

The dynamical part of the self-energy

with was constructed by considering the Kadanoff-Baym potential containing a class of diagrams up to infinite order. In the spirit of DMFT, we substitute the functional

dependence of the self-energy

The transverse

and longitudinal

susceptibility in Eq. (5) are found in random phase approximation (RPA) with renormalized

site-dependent interaction

Here, the noninteracting susceptibilty is calculated from the

local DMFT Green function (4)

The renormalized interaction results from the screening by particle-particle diagrams [14], with the scattering kernel

The self-energy (5) expresses the SF exchange interaction [16] with an effective potential due to particle-particle scattering [14]. Equations (2), (4), (5), and (9) represent a solution for the one-particle Green functions within the DMFT. They have been solved self-consistently, and the energetically stable charge and magnetization densities were found. In the physically interesting regime of the stable configuration at low temperature is found to be always a stripe phase for

. There is a competition between the horizontal (1,0) and diagonal

104


(1,1) stripes, and the detailed density distribution and the stripe ordering depends on the ratio U/t and on the values of the next-neighbor hopping elements,

stripes (Fig. la) are more stable up to

and

The (1,0)

whereas the (1,1) stripes (Fig. 1b) take

over for As a consequence of local hole correlations, the crossover from (1,0) to (1,1) stripes occurs for U being about twice as big as in the HF calculations. The diagonal stripes are also easier realized for increased second and third neighbor hoppings, and The parameters used in Fig. 1 correspond to the effective single band model for the cuprates: (a) and (b) [15]. For intermediate temperatures, e.g., at and the homogeneous spin spiral phase [13] along the (1,1) direction becomes stable in the same regime of interaction, and hopping parameters. Therefore, the stripe structures are so difficult to detect in the Monte Carlo calculations, where one is typically limited to not sufficiently low temperatures. The doped holes concentrate mainly on vertical stripes in the (1,0) structure (Fig. la). In agreement with the experimental finding [1], our calculation converged to half-filled domain walls for the effective single-band model for The magnetic domain wall is not identical to the vertical line of atoms with increased hole density. It alternates between the left and right bond with respect to the (1,0) charge stripe, whereas the charges are identical on both sides of the wall. The zig-zag alternation of the bond-centered domain walls repeats itself, and thus the domain wall spacing remains to be four bonds in all rows. In contrast to the HF calculations [7], the atoms at domain walls have magnetic moments all parallel to each other. A similar state with parallel magnetizations of the domain walls is metastable, with a small energy difference per one-unit cell. This shows that ferromagnetic (FM) polarons might also contribute to the stability of static domain walls, particularly at dopings larger than We note that a similar distribution of charge and magnetization densities was found recently using an unrestricted Gutzwiller variational approach [18|. The low-energy physics of the self-consistent (1,0) stripe solution (Fig. 1 a) is that of doped three-leg ladders [19,8] and isolated half-filled AF chains. Both subsystems have different properties, as the screening of U depends on the local magnetization, and would be largest at nonmagnetic atoms, Physically, the bare Coulomb interaction U is only weakly screened on those sites where the two-body wave


105

function tends to vanish. To quantify that, we give in Table 1 examples for the values of the particle-particle scattering vertex calculated from Eq. (9). The small value of the particle-particle kernel for electrons scattered on the atoms of the nearly undoped AF chains, for suggests that the AF chains can indeed be considered as isolated. In contrast, the screening is most efficient at the domain walls, with for . As a result, the values of the screened Coulomb interaction depend on the hole density and local magnetization in the stripe phase, and vary between and . on different atoms in the domains shown in Fig. 1a. This

demonstrates the importance of local electron correlations in the states with decreased magnetization, playing a decisive role in the stability of the stripe structures discussed below. Interestingly, the three-leg ladder shows alternating FM-AF rung and leg interaction. FM spin correlations included in SF self-energy [16] on the three-leg ladder might be a reason for the suppression of superconductivity in The doped phase for the effective single-band model with the parameters corresponding to shows a tendency toward the formation of stripes along the ( 1 , 1 ) direction. In this diagonal stripe phase, one finds the bond-centered domain walls oriented along the (1,1) direction. Guided by the experience from the (1,0) phases at smaller U, we tried several possibilities of stripe phases with different width of AF domains separated by the domain walls with the same doped-hole density and magnetization at neighboring atoms of the wall. This latter restriction allows us to reach fast convergence

and was accepted for convenience. As a result, we have found that the most stable solutions involve M = 4 neighboring sites in each row (see Fig. 1b). This charge distribution is the manifestation of stronger correlations for the parameters of with a stronger tendency toward phase separation. The domain walls, defined as usually by the phase shift of in the magnetic order parameter of the AF background, are merged together within an extended region that can be viewed as an FM polaron. The density and magnetization distribution in the (1,1) stripe phase of Fig. 1b is shown in more detail in Table 1. These FM polarons form domains of incresed doped hole concentration and are separated by nearly undoped domains of four sites with AF order. By making AF domains separated by FM domain walls, the magnetic and the kinetic energy of the system is optimized. The doped holes within the (1,1) stripe gain kinetic energy due to their delocalization along the stripe, mainly due to the hopping of the majority spin electrons. Such a state is favored in particular by the appreciable values of the extended hopping parameters,

106


We note that the larger U value and the increased extended hopping result in a stronger FM polarization of the domain wall than in the La-compound. Therefore, the value of the particle-particle kernel at the domain walls, is strongly reduced as compared to the FM walls with larger hole density in the (1,0) stripe, This result in a weaker screening of U, being at the domain wall, and 0.85, 0.88, respectively, at the atoms within the AF domain. Also, the overall filling of the domain walls is quite different from the (1,0) phase. One finds that one doped hole is distributed over four atoms of the wall, i.e., the average density is 1/4 doped hole per one domain wall atom. Taking a different definition and distributing the doped holes over the FM bonds that separate two neighboring AF domains would result in the filling of 1/3 hole per one FM bond. For comparison, we calculated the scattering vertex in Eq. (9) for the Hubbard model with and in the homogeneous (1,0) spin spiral phase at and found This results in the screened Coulomb interaction and is in reasonable agreement with the effective interaction used as an fitting parameter for Monte Carlo results [21]. The rather small values of the scattering vertices in the stripe phase, as reported in Table 1, suggest a strange metallic behavior of the cuprates at low temperatures. The total density of states

consists of two maxima that correspond to the Hubbard subbands, separated by a large gap, whereas the Fermi energy falls within a pseudogap that results from the magnetic order (Figs. 2a and 3a). These feature agree very well with the results of exact diagonalization


107

of a cluster in the Hubbard model [22]. The overall shape of the density of states of the static stripe phase agrees very well with the homogeneous (1,1) spin spiral phase at higher temperature, as shown in Figs. 2a and 3a. As in the case of spin spirals [13], the pseudogap results from the incommensurate magnetic order and separates the majority and minority spin states. At high temperature the pseudogap disappears. The increase of the next-neighbor hopping elements, and make the spectra look more incoherent. However, as shown in Fig. 2a, upon the transition to the stripe phase, the quasi-particle in the photoemission gets more coherent and the low energy electron addition states are shifted toward the Fermi energy. The spin spiral phase has a uniform charge distribution and the density of states shows very little weight near the Fermi energy (see Fig. 3b). At lower temperatures charge ordering along the diagonal sets in, and we observe a strong increase in PES weight at energies The doped holes are predominantly localized on single atoms in the vertical rows around the domain walls, as shown in Fig. 2b by the local density of doped holes

defined for an arbitrary horizontal row of the stripe phase. In the diagonal stripe, the holes are localized on single atoms in the vertical rows as well as in the horizontal columns, as shown in Fig. 3b. The domain walls, obtained within our DMFT calculation, are centered on the bonds between two sites. Thus, spins on the adjacent sites are parallel. Such bond walls might change into oxygen-centered stripes in a more realistic three-band model [23]. It is convenient to define [8]

108


used in Figs. 2b and 3b to show the magnetic structure. The phase factors adequate for staggered magnetic structure in the sum taken over two neighboring horizontal rows eliminates both zig-zag alternation of AF order and the FM polarization of the extended (1,1) stripes. The local moments found within the DMFT are only slightly reduced from the HF values [2,7]. The reduction of due to quantum fluctuations, similar to that

in the 2D Heisenberg antiferromagnet [24], goes beyond the local self-energy as it involves spin-flips on two neighboring sites. After simulating this effect by correcting to 0.606 one finds a very good agreement with the DMRG data [8]. The moments within the AF domains are very similar to each other, but strongly reduced on the stripes. The magnetic unit cell consists of eight atoms, whereas the charge unit cell consists of

four atoms, as shown in Fig. 2b. Instead, for the

model parameters we find

an increased periodicity of 16 (8) atoms in the magnetic (charge) structure, respectively (Fig. 3b).

In summary, the obtained (1,0) and (1,1) static stripes agree with the experimental observations for La- and Y-based superconductors [1,9J. The large Mott-Hubbard gap accompanied by a small pseudogap that separates the occupied and empty states are shown to be generic features of doped Mott-Hubbard insulators. The present approach explains the stability of half-filled stripes in by the existence of a pseudogap that opens

in the electronic structure due to the magnetic order and decreases the density of states at the Fermi level. Thus, we believe that long-range Coulomb interactions that stabilize half-

filled domain walls in Gutzwiller approximation [18] are not essential for the formation of these states. The same mechanism stabilizes more extended (1,1) stripe phase at larger values of U, which coexists with FM polaronic domain walls promoted by increased values

of the extended hopping described by Our calculations make therefore a specific prediction for the doped compounds, with increased periodicity of the stripes, and lower hole density in more extended stripe structures with weak FM polarization. It would be interesting to verify these predictions experimentally, if it would be possible to pin the suggested stripes [91 in the Y-based superconductors.

ACKNOWLEDGMENTS We thank J. Zaanen for valuable discussions, and acknowledge the support by the Committee of Scientific Research (KBN) of Poland, Project No. 2 P03B 175 14.

REFERENCES 1. J. M. Tranquada et al., Nature (London) 375, 561 (1995); J. M. Tranquada et al., Phys. Rev. Lett. 78, 338 (1997). 2. J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). 3. 4. 5. 6.

D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989). M. Inui and P. B. Littlewood, Phys. Rev. B 44, 4415 (1991). K. Machida, Physica C 158, 192 (1989). M. Ichimura et al., J. Phys. Soc. Jpn. 61, 2027 (1992).

7. J. Zaanen and A. M. Oles, Ann. Phys. 6, 224 (19%). 8. S. R. White and D. J. Scalapino, Phys. Rev. Lett. 80, 1272 (1998). 9. P. Dai et al., Phys. Rev. Lett. 80, 1738 (1998). 10. W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).

1 1 . A. Georges, G. Kotliar, G. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).

Dynamical Mean-Field Theory of Stripe Ordering 12. D. E. Logan, M. D. Eastwood, and M. A. Tusch, Phys. Rev. Lett. 76, 4785 (1996). 13. M. Fleck et al., Phys. Rev. Lett. 80, 2393 (1998). 14. L. Chen et al., Phys. Rev. Lett. 66, 369 (1991). 15. O. K. Andersen et al., J. Phys. Chem. Sol. 56, 1573 (1995); private communications.

16. N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett. 17, 433 (1966). 17. 18. 19. 20.

M. H. Hettler et al., cond-mat/9803295. G. Seibold, C. Castellani, C. Di Castro, and M. Grilli, cond-mat/9803184. E. Dagotto and T. M. Rice, Science 271, 618 (1996). S. R. White and D. J. Scalapino, Phys. Rev. 57, 3031 (1998).

21. 22. 23. 24.

N. Bulut et al., Phys. Rev. B 47, 2742 (1993). E. Dagotto et al., Phys. Rev. B 46, 3183 (1992). Z. G. Yu et al., Phys. Rev. B 57, R3241 (1998). A. Singh and Phys. Rev. B41, 614 (1990); ibid. 41, 11457 (1990).

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Tunneling and Photoemission in an SO(6) Superconductor R. S. Markiewicz,1,2 C. Kusko,1,2,3 and M. T. Vaughn1

Combining the results of tunneling, photoemission, and thermodynamic studies,

the pseudogap is unambiguously demonstrated to be caused by Van Hove nesting: a splitting of the density of states peak at The fact that the splitting remains symmetric about the Fermi level over an extended doping range indicates that the Van Hove singularity is pinned to the Fermi level. Despite these

positive results, an ambiguity remains as to what instability causes the pseudogap. Charge or spin density waves, superconducting fluctuations, and flux phases all remain viable possibilities. This ambiguity arises because the instabilities of the two-dimensional Van Hove singularity are associated with an approximate SO(6) symmetry group, which contains Zhang’s SO(5) as a subgroup. It has two 6-component superspins, one of which mixes Zhang’s (spin-density wave plus d-wave superconductivity) superspin with a flux phase operator. This is the small-

est group that can explain striped phases in the cuprates. Evidence for a prefered hole density in the charged stripes is discussed.

1. ELEVEN YEARS OF THE VAN HOVE SCENARIO: REMINISCENCES BY RSM Early work on the Van Hove model of high-

superconductivity was criticized for a

number of reasons, chiefly (1) the perfect nesting of the Fermi surface would cause instabilities which would overwhelm superconductivity; and (2) the Van Hove singularity (VHS) is associated with a special doping, and hence would require “fine tuning” of the model parameters to fall at the Fermi level, whereas superconductivity exists over an extended doping range. In June 1987, one of us suggested [1] the picture that has become the heart of the 1

Physics Department, Northeastern University, Boston, MA 02115, USA. Barnett Institute, Northeastern University, Boston, MA 02115, USA. 3 On leave of absence from Inst. of Atomic Physics, Bucharest, Romania. 2


111

112

Markiewicz, Kusko, and Vaughn

generalized Van Hove scenario [2]. First, introduction of next-nearest-neighbor hopping— ) in a one (three)-band model—would push the VHS off of half-filling, leading to

imperfect nesting, with residual hole pockets and ghost Fermi surfaces [31. Second, this very sensitivity to instability could in turn lower the free energy precisely when the Fermi surface coincides with the VHS, thereby stabilizing this special doping—and that the reason superconductivity might persist over such an extended doping range is that the material is inhomogeneous, with one phase pinned at the VHS. Due to charging effects, such phase separation would be nanoscale [4]. Although a microscopic model appeared only in 1989 [5], we quickly found that the doping dependence of both resistivity and Hall effect could be understood in the context of a percolation model [3,6], and Bill Giessen and we noted that the Uemura plot [7] could be interpreted in terms of an optimal doping for each cuprate, which we identified with

the pure VHS phase [8]. At the 1988 MRS Meeting, Jim Jorgensen asked if the phase separation really had to be nanoscale, and we said it was a problem of the counterions: if

the Sr in could be made mobile, there was no reason that a macroscopic phase separation could not occur—with one phase pinned at optimal doping. We did not know then that Jim and coworkers had just discovered such a phase in

where

the doping is due to mobile interstitial oxygens [9]. One problem was that just where is the VHS. A VHS at half-filling could produce an

antiferromagnetic instability with large pseudogap [10] but no hole pockets [11], whereas a VHS at finite doping could explain hole pockets and optimal doping, but not antiferromagnetism. One of us explored this issue by putting into a slave boson model, to separate Mott and nesting instabilities [5]. We found that the Mott instability—and hence the ac-

companying antiferromagnetism—remain locked to half-filling, whereas for the VHS we found a striking surprise: correlation effects pin the Fermi level close to the VHS over an extended doping range, extending across the underdoped regime from half-filling to opti-

mal doping (close to the bare VHS). Numerous subsequent studies have confirmed the Van Hove pinning (cited on p. 1223 of Ref. [2]).

In this same paper [5], we demonstrated a viable mechanism for nanoscale phase separation: a charge-density wave (CDW) instability is strongly enhanced when the Fermi level coincides with the bare VHS, leading to a free energy minimum at optimal doping. Because the Mott instability produces a separate minimum at half-filling, the energy is

lowered by separating into the two end phases. This model was clearly distinguished from other models in that there is a highly unusual and characteristic fractional hole doping of the hole-rich phase (this doping fixed by the VHS), whereas all other models had a doped phase with one hole per Cu. Now we had an embarras de richesses: both pinning

of the VHS by correlation effects and phase separation, which amounted to a different pinning mechanism. Put another way: Were the cuprates characterized by phase separation, or by the (pseudo) gap associated with nesting? I puzzled over this issue for a number of years, convinced that somehow both viewpoints were correct—rather like the particle vs wave problem in quantum mechanics. One of us was (and is) convinced that both the lowtemperature orthorhombic (LTO) and low-temperature tetragonal (LTT) phases in LSCO was driven by this mechanism, and we introduced the concept of Van Hove-Jahn-Teller

effect to explain the LTO phase as a dynamic Jahn-Teller (JT) phase [12,13]. But there was something missing: There should have been a tetragonal dynamic JT phase at higher temperatures. In the years from 1987 to 1991, one of us was often asked where was the evidence for a CDW or other nesting instability.

Tunneling and Photoemission in an SO(6) Superconductor

113

Things changed when we realized that the “spin gap” and thermodynamic [14] data were consistent with a simple model for the pinned Van Hove phase [12,15]. New photoemission data [16] showed that the pseudogap is associated with the band dispersion near (π, 0)—i.e. the locus of the VHS, while Tranquada et al. [ 1 7 ] showed the presence of nanoscale phase separation in the form of stripes, which appeared to be suspiciously coextensive with the pseudogap regime. After writing an extensive review [2], one of us put my ideas together into a self-consistent three-band slave boson calculation, reported at the first Stripes Conference [18]. Correlation effects keep the Fermi level close to the VHS from half filling to optimal doping. Near half-filling, correlation effects drive the Cu-O hopping to zero, and the remaining dispersion due to J has a VHS at half-filling, and gains additional stability by splitting the VHS via a flux phase [18,19]. Doping restores the hopping, simultaneously introducing a strong electron-phonon coupling via modulation of the Cu-O separation, leading to a maximal CDW instability at optimal doping. The two instabilities in turn drive phase separation. Attempts to model this phase separation led to good fits to the doping dependence of the photoemission dispersion. There was an important prediction: the VHS is found in photoemission to be below the Fermi level, because it is simultaneously above the Fermi level: the pseudogap consists of a splitting of the VHS into two features at but split in energy about the Fermi level. Photoemission could not reveal the upper VHS, but recent tunneling studies [20,21] fully confirm this prediction, as well as demonstrating that the lower peak coincides with the photoemission VHS feature. At the mean-field level, the CDW has long-range order. However, when fluctuations are included in a mode-coupling scheme, there is only short-range order, with a real pseudogap opening up in the temperature range between the mean field transition temperature and a much lower transition to long-range order, driven by interlayer coupling [22]. If this interlayer coupling is absent, the CDW resembles a quantum critical point (QCP), with correlation length diverging as It is not a conventional QCP, in that in the absence of phase sepatation it is not the terminus of a finite temperature phase transition (i.e., there is no renormalized classical regime).

2. SUPERCONDUCTIVITY AND TUNNELING

The three-band model [18] revealed that the striped phases are associated with two nesting instabilities, flux phase near half-filling and CDW at optimal doping, leading to a large pseudogap near and simultaneously to nanoscale phase separation between magnetic and hole-doped stripes. In the LSCO system, adding Nd or replacing the Sr with Ba leads to long-range stripe phase order [17] while suppressing superconductivity—clearly demonstrating that three distinct phases must be involved. In the one-band model [12,15], this situation is simplified by replacing the striped phase by a uniform CDW phase. The key approximation is the Ansatz of reducing the strong correlation effects and phase separation by their most important effect: the pinning of the Fermi level to the VHS, over the full doping range from half-filling to optimal doping. In the overdoped regime, the simplest approximation is assumed: The band structure stops changing with doping, and the Fermi level shifts in a rigid band fashion. The pseudogap seen in tunneling measurements on (Bi-2212) [20,21] can be fit to this model. We have calculated the tunneling spectrum [23] (Fig. 1), and find a smooth evolution from pseudogap (CDW) phase to mixed CDW–superconductor.

114


For a pure CDW, the spectral function is of BCS form:

with

where the nesting vector and the gap and dispersion are defined in Refs. 12 and 24. The model involves three gap parameters, two and i associated with CDW order, and one

with superconductivity. Figure 1 shows the calculated phase diagram and

the net low-T tunneling gap, defined as half the peak-peak separation. The inset shows that in the mixed CDW-superconducting state, a single gap evolves in the calculated tunneling density of states (except for phonon structure). The ratio of the total gap to the CDW/superconducting onset temperature is nearly doping independent, Because we use the model of Balseiro and Falicov (BF) [24] to describe the underlying CDW-superconductivity competition, we refer to this as the pinned BF (pBF) model. At present, it involves s-wave superconductivity, but we are working on an SO(6) generalization, including d-wave superconductivity. For tunneling along the c-axis into a two-dimensional (2D) material, the tunneling density of states (dos) is an average of the in-plane quasi-particle dos [25]. In this case, there is a one-to-one correspondence between features in the electronic band dispersion, as measured in photoemission, and peaks in the tunneling dos, Fig. 2. The main tunneling peaks (A) coincide with the split electronic energy dispersion near of the Brillouin zone—and hence with the corresponding photoemission peaks, as found experimentally [21 ]. In the present BCS-like model, the slight discontinuity at the phonon energy produces a large peak in the tunneling spectrum (D). Note that at the CDW and superconducting gaps combine to form a single feature (A) in both photoemission and


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tunneling, whereas the gaps split into two separate features: B associated with superconductivity and C with the CDW, near The presence of only a single combined gap at is a consequence of an underlying SO(6) symmetry of the VHS, as discussed in the following section. In turn, this explains the smooth evolution of the pseudogap into the superconducting gap, Fig. 1 insert, which therefore need not be taken as evidence for precursor pairing in the pseudogap phase. A most exciting possibility is that by comparing the photoemission and tunneling

data, one should be able to experimentally measure the pinning of the Fermi level to the VHS. Indeed, Renner et al. [20], unaware of this prediction [2], noted that the pseudogap is centered at the Fermi level in both under- and overdoped samples. It is therefore unlikely that the pseudogap results from a band structure effect. To quantify the extent of pinning, we test the null hypothesis. We compare the presently available data with the expected tunneling spectra for a pure d-wave superconductor in the absence of pinning, as doping is varied and the Fermi level passes through the VHS, Fig. 3.

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From the inset, it can be seen that the two peaks in tunneling are symmetric when the Fermi level is exactly at the VHS (optimal doping), whereas the peak on the photoemission (inverse photoemission) side is stronger for underdoped (overdoped) samples, in accord with experiment. However, for strong enough doping away from optimal, the peak should split, which is not seen. Because the superconducting gap shifts off of we take the difference between the tunneling gap and the photoemission gap as an experimental measure of the splitting, Fig. 3. Although there is considerable error, the splitting in overdoped samples is close to what is expected, but the splitting is absent in underdoped samples—

strong evidence for Van Hove pinning. 3. SO(6) A one-dimensional (1D) metal is susceptible to a variety of instabilities, including singlet or triplet superconductivity, CDWs, and spin-density waves. These instabilities can be organized group theoretically [27], either on the basis of a symmetry group, or in terms of a larger spectrum-generating algebra (SGA), which contains the Hamiltonian as a group

element, and hence can be used to generate the full energy spectrum. A similar analysis can be applied to the Van Hove scenario, in terms of an approximate SO(6) symmetry group, or SO(8) SGA [28]. This SO(6) group contains as subgroups Zhang’s SO(5) [29], Yang and Zhang’s SO(4) [30], and Wen and Lee’s SO(3) [31] (SU(2)). It includes two 6-dimensionaI superspins, which form an “isospin” doublet [28]: one combines Zhang’s SO(5) superspin (antiferromagnetism plus d-wave superconductivity) and the flux phase; the other involves s-wave superconductivity and a CDW (as in the pBF model) with an exotic spin current phase. There is a most interesting evolution of these groups from one dimension to two, Table 1. Lin et al. [32] analyzed the group structure of a two-leg ladder. They found an SO(8) symmetry group, which involves the SO(6) group as a subgroup, plus operators that are antisymmetric for These latter operators are essential in the 1D, in which the Fermi surface consists of two points but are irrelevant for the VHSs, which are on the Brillouin zone boundary, and hence do not couple to these operators. Table 1 illustrates the evolution of a single superspin (Lin et al.’s d–Mott state) from 1D to 2D. In this table, SDW = spin-density wave; sc = superconductivity; are symbols introduced by Lin et al. [32] for two of their SO(8) operators, called band spin difference and relative band chirality; and a dash indicates that a corresponding operator is lacking. Note that the 1D CDW connects whereas the 2D CDW discussed above connects and Note further that the SO(6) structure persists down to ladders two cells wide, and hence should remain valid in describing the striped phases. In the 1D metals, the SGA aspect is more fundamental than the symmetry aspect. The various instabilities are usually not degenerate in energy, as required by a symmetry group.


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Instead, they are governed by the allowed interaction terms, gs—hence the name g-ology— and the object of research is to derive the allowed phase diagram as a function of the possible g-values. In this case, the SGA is useful in cataloging the allowed instabilities [27]. The situation is similar for the VHS. The one-band model is not itself symmetric under SO(6), but shows considerable signs of the underlying SO(8) SGA. Thus, the form of the phase

diagram in Fig. 1 is generic of any competition between a nesting operator and a pairing operator, whereas the pseudogap at has the simple form

where the are the gaps associated with the twelve components of both superspin vectors [33]. When , this vector addition of the gaps holds for the full Fermi surface. Note that for a symmetry group, all the in Eq. (2) would have equal magnitudes. The corresponding 2D g-ology phase diagram can be worked out [28,34], in analogy with the 1D case. In the Hubbard limit, the only interaction is the U term, and the phase diagram has a natural evolution from SDW at half-filling to d-wave superconductivity in the doped materials. However, this simple picture cannot account for the striped phases, that compete with superconductivity (e.g., at 1/8 doping, where there is long-ranged stripe order, superconductivity is suppressed). By adding a phonon-mediated effective electron-electron coupling, a CDW phase can be stabilized in the doped material, and competition between CDW and SDW generates a striped phase.

4. POSTSCRIPT: HOW WIDE ARE THE CHARGED STRIPES? There was considerable discussion at the conference as to whether the charged stripes were one or two cells wide (i.e., site-centered or bond-centered). This may seem like a trivial issue, but it can have profound consequences: the nature of the striped phase at optimal doping, and ultimately the relevance of the model to superconductivity. The issues may be conveniently addressed by considering the White-Scalapino [35] model of the 1/8 doped state. The overall magnetic and charge orders are consistent with the experiment [17], but our main concern here is the actual charge distribution. The charge periodicity is four cells across, with two magnetic cells having an average hole density of holes per Cu, and two hole-doped cells, each containing hole. Because the magnetic and hole-doped stripes have equal width, this is the highest doping at which a magnetic domain wall model makes sense. This is particularly true, because there is a strong tendency for the magnetic stripes to contain an even number of cells—because in that case, there can be a spin gap, as in an even-legged ladder [36]. Hence, as the doping is increased above 1/8, the magnetic cells can no longer shrink in width, and there must be a phase transition in the nature of the stripes. (If the hole-doped stripes were only one cell wide, this anomalous behavior would only arise near a doping of 1/4.) White and Scalapino [37] have indeed reported evidence for such a phase transition; here we would like to present a reinterpretation of their data, Fig. 4a of Ref. 37. Because the stripes are sensitive to boundary conditions, we prefer to look at the average hole density along each row (parallel to the stripes) of the simulation, Fig. 4. The data (open circles) fall very close to the form expected for a phase separation model (solid lines), with the same average densities as at 1/8 filling, but now the magnetic stripes retain their minimum width, whereas the hole-doped stripes get wider. Because 0.18 is close to optimal doping,

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the optimally doped materials are likely to be characterized by a set of widely separated magnetic ladders, with little residual interaction. The physics will be dominated by the physics of the hole-doped stripes at their special doping. This is bad news for the model. It was specifically designed as a highly simplified model that retained just enough physics to accurately describe the cuprates near the insulating phase at half-filling. It is highly unlikely that the neglect of the oxygens and electron-phonon interactions will continue to be valid in the new hole-doped phase.

ACKNOWLEDGMENTS

We would like to thank A. M. Gabovich for useful discussions about tunneling, and NATO for enabling him to visit us. Publication 743 of the Barnett Institute.

REFERENCES 1. R. S. Markiewicz, Mod. Phys. Lett. B 1, 187 (1987). 2. R. S. Markiewicz, J. Phys. Chem. Sol., 58, 1179 (1997). 3. R. S. Markiewicz, in High Temperature Superconductors, ed. by M. B. Brodsky et al., (Pittsburgh, MRS, 1988), p. 411. 4. E. L. Nagaev, Physics of Magnetic Semiconductors (Moscow, Mir, 1983).

5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

17. 18.

R. S. Markiewicz, Physica C 162–164, 235 (1989); J. Phys. Cond. Matt. 2, 665 (1990). R. S. Markiewicz, Physica C 153–155, 1181 (1988). Y. J. Uemura et al., Phys. Rev. Lett. 62, 2317 (1989). R. S. Markiewicz and B. C. Giessen, Physica C 160, 497 (1989). J. D. Jorgensen et al., Phys. Rev. B 38, 11337 (1988). A. Kampf and J. Schrieffer, Phys. Rev. B 42, 7967 (1990). W. Putikka et al.,to be published, J. Phys. Chem. Sol. 59 (cond-mat/9803141). R S. Markiewicz, Physica C 193, 323 (1992). R S. Markiewicz, Physica C 200, 65 (1992), 210, 235 (1993), and 207, 281 (1993). J. W. Loram et al., Phys. Rev. Lett. 71, 1740 (1993). R. S. Markiewicz, Phys. Rev. Lett. 73, 1310 (1994). D. S. Marshall et al.,Phys. Rev. Lett. 76,4841 (1996); A. G. Loeser et al., Science 273, 325 (1996); H. Ding et al., Nature 382, 51 (1996). J. M. Tranquada et al., Nature 375, 561 (1995). R. S. Markiewicz, J. Supercond. 10, 333 (1997); Phys. Rev. B 56, 9091 (1997).


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19. I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988); R. B. Laughlin, J. Phys. Chem. Sol. 56, 1627

(1995); X.-G. Wen and P. A. Lee, Phys. Rev. Lett. 76, 503 (1996). 20. Ch. Renner et al, Phys. Rev. Lett. 80, 149 (1998). 21. N. Miyakawa et al., Phys. Rev. Lett. 80, 157 (1998).

22. R. S. Markiewicz Physica C 169,63 (1990). 23. R. S. Markiewicz and C. Kusko, unpublished (cond-mat/9802079); R. S. Markiewicz, C. Kusko, and V. Kidambi, unpublished. C. A. Balseiro and L. M. Falicov, Phys. Rev. B 20, 4457 (1979). J. Y. T. Wei et al., Phys. Rev. B 57, 3650 (1998). H. Ding et. al., to be published, J. Phys. Chem. Sol. 59 (cond-mat/9712100). A. I. Solomon and J. L. Birman, J. Math. Phys. 28, 1526 (1987). R. S. Markiewicz and M. T. Vaughn, to be published, J. Phys. Chem. Sol. 59 (cond-mat/9709137), and Phys. Rev. B 57, 14052 (1998). 29. S.-C. Zhang, Science 275, 1089 (1997). 30. C. N. Yang and S.-C. Zhang, Mod. Phys. Lett. B 4, 759 (1990).

24. 25. 26. 27. 28.

31. X.-G. Wen and P. A. Lee, Phys. Rev. Lett. 80, 2193 (1998). 32. H.-H. Lin et al., unpublished (cond-mat/9801285). 33. C. Kusko, R. S. Markiewicz, and M. T. Vaughn, unpublished.

34. H. J. Schulz, Phys. Rev. B 39, 2940 (1987). 35. S. R. White and D. I. Scalapino, Phys. Rev. Lett. 80, 1272 (1998). 36. H. Tsunetsugu et al., Phys. Rev. B 51, 16456 (1995). 37. S. R. White and D. J. Scalapino, unpublished (cond-mat/9801274).


Spin, Charge, and Orbital Ordering in 3d Transition-Metal Oxides Studied by Model Hartree–Fock Calculation T. Mizokawa1,2 and A. Fujimori1

We have spin, charge, and orbital ordered states in the perovskite-type lattice models for by means of unrestricted Hartree-Fock calculations. Present calculations show that, although the vertical charge stripes along the (1,0) direction of the Cu-O square lattice are favored in the cuprates with the charge-transfer energy the diagonal stripes along the (1,1) direction are stable in the nickelates with For the manganites with it has been found that the (l,l,0)-type

charge-ordered state with the orbital ordering at the the Jahn–Teller distortion at the

sites are stabilized by

sites.

1. INTRODUCTION Charge-ordering phenomena have widely been observed in hole-doped perovskite-type

3d transition-metal oxides and have attracted broad interest. Neutron and electron diffraction studies have shown that charge-ordered (CO) states with various types of domain walls

are realized in these compounds [ 1–4]. Whereas in states with charge stripes along the (1,1) direction or along the Ni-Ni direction have been found, a neutron diffraction measurement of is consistent with charge stripes along the (1,0) direction or along the Cu-O direction. As for three-dimensional (3D) perovskites, in

served, and in

charge domain walls perpendicular to the (1,1,0) direction have been obthe domain walls are perpendicular to the (1,1,1) direction.

1

Department of Physics & Department of Complexity Science and Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan. 2 Solid State Physics Laboratory, Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands.


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The existence of the CO state in the doped antiferromagnetic (AFM) systems has been predicted by Hartree-Fock (HF) calculations on single-band Hubbard models [5]. However, because the parent compounds of these perovskite-type doped Mott insulators—namely, and —are of the charge-transfer type [6], it is possible that domain walls of the doped holes are centered at the oxygen sites rather than at the metal site. In order to study the stability of the oxygen-centered domain walls, the oxygen 2p orbitals should be taken into account explicitly. In addition, it is important to include full degeneracy of the transition-metal 3d orbitals to study the CO states in the manganites in which the orbital degree of freedom may play a role. In this paper, we present unrestricted HF calculations on multiband models for the cuprates, the nickelates, and the manganites in which the full degeneracy of the transition-metal 3d and oxygen 2p orbitals is included. We have investigated the relative stability of the various spin- and charge-ordered states as a function of hole concentration x as well as the Jahn–Teller distortion.

2. METHOD

We used the multiband model for the perovskite-type lattice, where the 10-fold degeneracy of the transition-metal 3d orbitals and the six-fold degeneracy of the oxygen 2p orbitals are taken into account [7]. The intra-atomic Coulomb interaction is expressed using Kanamori parameters, The charge-transfer energy is defined by where are the energies of the bare metal 3d and oxygen 2p orbitals and is the multiplet-averaged Coulomb interaction. The transfer integrals between the transition-metal 3d and oxygen 2p orbitals are given in

terms of Slater-Koster parameters energies for and

The charge-transfer are 4,2, and 4 eV, respectively [6].

3. RESULTS AND DISCUSSION 3.1.

In the HF calculations for the various CO states with the diagonal stripes running along the (1,1) direction and with the vertical stripes along the (1,0) direction have lower energy than the ferromagnetic (FM) and AFM metallic states. Spin and charge arrangements of these CO states for are shown in Fig. 1. The energy difference between the AFM metallic state and the CO states becomes larger in going from to to However, for the FM metallic state is the lowest in energy, indicating that the CO state is not so stable at. as at and Among these CO states, the solution with the diagonal metal-centered stripes, in which is ferromagnetically coupled with the neighboring and the AFM ordering in is not disturbed, is lowest in energy for and 1/3. The metal-centered stripes with the FM coupling between and can be viewed as magnetic polaron stripes in which forms stripes of magnetic polaron. The solution with the diagonal stripes centered at the oxygen sites has the second lowest energy. As shown in Fig. 1, the CO state with the diagonal magnetic polaron stripes and that with the diagonal oxygen-centered stripes are almost degenerate in energy. Although in these magnetic polaron stripes the neighboring two domains are in phase because of the FM coupling between and the neighboring two domains are in antiphase in the CO state with the oxygen-centered stripes. Although

Spin, Charge, and Orbital Ordering in 3d Transition-Metal Oxides

the CO state with the oxygen-centered stripes is AFM for

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because each domain

has four rows of , those for and are ferrimagnetic. However, the CO state with the magnetic polaron stripes is ferrimagnetic for and those for and

are AFM. When is decreased to 2.0 eV, the vertical magnetic polaron stripes become lower in energy than the diagonal ones, indicating that the larger value of in the nickelates than in the cuprates plays an important role to realize the diagonal stripes in the doped system 3.2.

In the HF calculations for the cuprates, the CO solutions with the vertical charge stripes running along the (1,0) direction are stable for various hole concentrations, whereas the CO

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states with the diagonal stripes are unstable. For and 1/3, the vertical stripes or domain walls, which are centered at the metal sites, and those centered at the oxygen sites are almost degenerate and are the lowest in energy. The energy difference between the CO states with the vertical stripes and the AFM metallic state are and 22 meV for and 1 /3, respectively. The CO states with the diagonal stripes are unstable for the charge-transfer energy of 2 eV. However, when is increased to 6 eV, the diagonal stripes become stable and are lower in energy than the vertical stripes, indicating that the small charge-transfer energy in the cuprates is essential to stabilize the vertical stripes compared to the diagonal ones. These domain walls have one hole per metal site and are so-called filled domain walls [9], in which the period of the stripes is a/x, where a is the in-plane lattice constant. Zaanen and Oles [9] investigated the half-filled domain walls or stripes by using HF calculations and found that the metal-centered half-filled stripes are moderately stabilized by charge- and spin-ordering along the stripes, but are still higher in energy than the filled stripes. In the metal-centered half-filled stripes, the charge- and spin-ordering along the stripes causes quadrupling of the period along the stripes [9]. We studied the stability of the half-filled stripes for and 1/8. For only a CO state with the oxygencentered half-filled stripes is obtained. However, for both the metal-centered and oxygen-centered half-filled stripes are obtained. While the metal-centered stripes have the quadrupling of the period along the stripes as predicted by Zaanen and Oles [9], the oxygencentered ones are not accompanied by the quadrupling (Fig. 2). In the present calculation for the oxygen-centered half-filled stripes are by higher in energy than the filled vertical stripes, as shown in Fig. 2. The magnitude of the band gap of the half-filled stripes is calculated to be less than which is much smaller than that of the filled stripes In this model calculation, the energy difference between the CO states with the filled and half-filled stripes becomes smaller in going from to indicating that the filled stripes become preferred to the half-filled ones with hole doping. Very recently, the density matrix renormalization group calculation on the t-J model has shown that the ground state has the half-filled vertical stripes for and the filled ones for It has also been reported that the intersite Coulomb interaction stabilizes the half-filled stripes compared to the filled ones [11]. In order to fully understand the nature of the charge stripe, it is required to study more realistic models, including the

intersite Coulomb interaction and the electron-lattice interaction. In particular, the stripe superstructure along the diagonal direction was observed in

suggesting that the coupling between the charge stripe and the structural modulation is important.

3.3. In order to study how the stability of the CO states is affected by the orbital ordering in the manganites, we have performed model HF calculations for Without the JT distortion, the FM state is the lowest in energy and the A-type AFM state is the second

lowest. The FM and A-type AFM states are metallic and are not accompanied by charge ordering. The CE-type AFM CO solution with a band gap of is obtained, but is higher in energy than the FM and A-type AFM state. In the CE-type AFM CO state, the and are interlaced like a checkerboard within the c plane as shown in Fig. 3a. Along the c axis, the same in-plane arrangement of is stacked and the neighboring planes are antiferromagnetically coupled. The sites are accompanied by the orbital ordering even without the JT distortion [13]. This is because


125

the orbital of the site tends to point the neighboring site in order to gain the kinetic exchange energy. It is interesting to note that the orbital ordering in is contrasted with that in which is a mixture of the -type and the -type when the JT distortion is absent [13]. We have calculated the energies of the A-type and CE-type AFM states relative to the FM state as functions of the JT distortion that is consistent with the orbital arrangement.

The present calculation shows that the CE-type AFM state is stabilized by the JT distortion. However, the tilting of the octaherda does not reduce the energy difference between the AFM states and the FM state very much. The checkerboard-type charge ordering couples with the in-plane breathing-type lattice distortion in which the ion is expanded and the ion is contracted within the ab plane, keeping the Mn-O bond distance along the c axis the same. The A-type AFM CO state, the orbital ordering as shown in Fig. 3b is expected to be favored. Actually, with the

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breathing-type distortion, the A -type AFM CO state with the x2 – y2-type orbital ordering

is calculated to be the lowest in energy and to have a band gap of ~0.35 eV without the JT distortion. The present model HF calculations indicate that the JT-type and breathing-type distortions control the relative stability of the FM state and the CE- type and A -type AFM states. 4. CONCLUDING REMARKS

Spin and charge ordered states with domain walls in two- and three-dimensional perovskite-type 3d transition-metal oxides have been investigated using model HF calculations. It has been found that the metal-centered and oxygen-centered stripes are nearly degenerate in energy both in the cuprates and in the nickelates. The present HF calculations show that the magnitude of the charge-transfer energy controles the relative stability of the

domain walls along the (1,0) and (1,1) directions. For the manganites, it has been found that the (1,1,0)-type charge-ordered state with the orbital ordering at the sites are stabilized by the JT distortion at the sites. In the CO states with the oxygen-centered domain walls, the superexchange coupling between the domain edges is affected by the doped holes sitting at the oxygen sites. The coupling between the domain edges becomes FM, and therefore the neighboring AFM domains become in antiphase. Let us denote the energy gain per metal-oxygen-metal bond by the FM coupling between the domain edges as and that in the AFM domains as K.. In the CO states with the metal-centered domain walls, the domain walls are formed by metal ions with a different formal valence from that of the parent insulator. If the domain wall has local magnetic moments, the superexchange coupling between the domain edge and the domain wall and that within the domain wall (K ) are important. In this case, the neighboring domains are in phase—namely, the spin arrangement of the parent AFM Mott insulator is not disturbed. In the two-dimensional lattice, the domain walls along (1,0) are favored for K and those along (1,1) are favored for as shown in Fig. 4a. Although the


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layered perovskite-type cuprates have large K because the charge-transfer energy is small K is relatively small in the nickelates with of However, or is less sensitive to the change in than K. This is consistent with the fact that the domain walls along (1,0) are observed in the layered cuprates and that those along (1,1) are found in the layered nickelates [1,2]. In the three-dimensional lattice, although the domain walls perpendicular to (1,0,0)

are favored for those perpendicular to (1,1,1) are favored for as shown in Fig. 4b. Because of the orbital ordering at the sites, the superexchange coupling in the manganites is highly anisotropic and the system should be viewed as twodimensional one rather than three-dimensional [3,13]. Because the manganites have the charge-transfer energy of 4 eV, the domain walls along (1,1) is favored in the c plane, as shown in Fig. 3. The same in-plane arrangement is stacked along the c axis because of the interplane coupling through the shift of the A-site ions [14]. As a result, the (l,l)-type stripes forms the domain walls perpendicular to (1,1,0) in However, in the CO state of the formal valences of the Fe ions are and [4]. Because both and have no orbital degrees of freedom, is an ideal three-dimensional system. Because the charge-transfer energy of is and the domain walls perpendicular to (1,1,1) are realized in ACKNOWLEDGMENTS

The authors would like to thank J. Zaanen, D. I. Khomskii, and G. A. Sawatzky for useful discussions. The present work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture and by the New Energy and Industrial Technology Development Organization (NEDO). REFERENCES I . C. H. Chen, S-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993); S. M. Hayden, G. H. Lander,

J. Zarestky, P. J. Brown, C. Stassis. P. Metcalf, and J. M. Honig, Phys. Rev. Lett. 68, 1061 (1992); V. Sachan, D. J. Buttrey, J. M. Tranquada, J. E. Lorenzo, and G. Shirane, Phys. Rev. B 51, 12742 (1993).

128


2. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995); J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, Phys. Rev. B 54, 7489 (1996).

3. Z. Jirak, S. Krupicka, Z. Simsa, M. Dlouha, and S. Vratislav, J. Mag. Mag. Mat. 53, 153 (1985); Y. Tomioka, A. Asamitsu, Y. Moritomo, H. Kuwahara, and Y. Tokura, Phys. Rev. Lett. 74, 5108 (1995); P. G. Radaelli, D. E. Cox, M. Marezio, and S-W. Cheong, Phys. Rev. B 55, 3015 (1997). 4. M. Takano, J. Kawachi, N. Nakanishi, and Y. Takeda, J. Sol. State Chem. 39, 75 (1981); P. D. Battle, T. C. Gibb, and P. Lightfoot, J. Sol. State Chem. 84, 271 (1990); Phys. Rev. Lett. 54, 7489 (1996); J. Q. Li, Y. Matsui,

S.-K. Park, and Y. Tokura, Phys. Rev. Lett. 79, 297 (1997). 5. J. Zaanen and O.Gunnarsson,Phys. Rev. B 40, 7391 (1989); H. J. Schulz, J. Phys. 50, 2833 (1989); D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989); M. Kato, K. Machida, H. Nakanishi, and M. Fujita, J. Phys. Soc. Jpn. 59, 1047 (1990); M. Inui and P. B. Littlewood, Phys. Rev. B 44, 4415 (1991); J. Zaanen and P. B. Littlewood, Phys. Rev. B 50, 7222 (1994). 6. A. E. Bocqeut, T. Mizokawa, A. Fujimori, and H. Namatame, Phys. Rev. B 46, 3771 (1992); H. Eisaki,

S. Uchida, T. Mizokawa, H. Namatame, A. Fujimori, J. van Elp, P. Kuiper, G. A. Sawatzky, S. Hosoya, and H. Katayama-Yoshida, Phys. Rev. B 45, 12513 (1992). 7. T. Mizokawa and A. Fujimori, Phys. Rev. B 51, 12880 (1995); T. Mizokawa and A. Fujimori, Phys. Rev. B 54, 5368 (1996).

8. J. Kanamori, Prog. Theor. Phys. 30, 275 (1963). 9. J. Zaanen and A. M. Oles, Ann. Phys. (Leipzig) 5, 224 (1996). 10. S. R. White and D. J. Scalapino, Phys. Rev. Lett. 81, 3227 (1998).

1 1 . G. Seibold, C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. B 58, 13506 (1998). 12. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rosseti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito,

Phys. Rev. Lett. 76, 3412 (1996). 13. V. I. Anisimov. I. S. Elfimov, M. A. Korotin, and K. Terakura, Phys. Rev. B 55, 15494 (1997); T. Mizokawa and A. Fujimori, Phys. Rev. B 56, R493 (1997).

14. T. Mizokawa, D. 1. Khomskii, and G. A. Sawatzky, unpublished.

Sliding Stripes in 2D Antiferromagnets C. Morais Smith,1 Yu. A. Dimashko,1 N. Hasselmann,1,2 and A.O. Caldeira3

The stripe dynamics is studied within the framework of elastic manifolds in disordered media. By applying an external electrical field perpendicular to the

stripes, a sliding behavior arises, due to the competition between the pinning potential (produced by randomly distributed impurities) and the bias field. We

consider the problem at low doping, when the interaction between stripes can be neglected. The characteristics is determined, and a nonlinear conductivity with a glassy phase at low fields is predicted. Besides, we study the dynamics of a gas of holes, and show that in this case the low field response is ohmic. Therefore, measurements of the characteristics at vanishingly small bias can serve as a tool for verifying the existence of the striped phase. PACS numbers: 74.20.Mn, 74.20.-z, 71.45.Lr

1. INTRODUCTION There has been a crescent deal of interest concerning the charge and spin distribution in doped antiferromagnets. Previously, the problem was addressed by assuming that the system could be described by a gas of holes with uniform density. However, recent calculations [1] and measurements [2–5] suggest that the holes cluster along lines (stripes) that separate undoped antiferromagnetic domains. The existence of stripes has been confirmed from several distinct experimental techniques, like elastic [2] and inelastic [3] neutron scattering, muon spin resonance, nuclear quadrupole resonance [4], and x-ray diffraction (XRD) [5].

¹I Institut für Theoretische Physik, Universität Hamburg, D-20355 Hamburg, Germany. 2

Dept. of Physics, University of California, Riverside, CA, 92521, USA. Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, CP 6165, 13085-970 Campinas SP, Brasil.

3


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Here, we suggest another simple experiment that can serve as a check for verifying the stripe formation. By applying a small electrical bias field E to a sample, one can determine

the characteristics, which exhibit a different dependence on the bias, depending if the system is a gas or a line of holes.

2. THE MODEL

We consider the limit of low doping, where the interaction between the domain walls (chain of holes) can be neglected and we can concentrate on a single-stripe dynamics. Moreover, we study a completely filled domain wall (DW), as found in the the nickelate materials. For the sake of simplicity, we concentrate on the vertical configuration, with the stripe located along the y direction, and allow for jumps of the holes only along the perpendicular x direction (Fig. 1). The DW separates the antiferromagnetic plane into two Néel phases, which we consider to be frozen. The problem is treated in the frame of the t – J model,

where t is the hopping parameter and J is the exchange energy. The displacement of the nth hole along the x direction is denoted by In order to describe the holes dynamics, we map the t — J Hamitonian onto a quantum spin chain one: we denote the relative displacement of two neighboring holes by and treat this value as the component of some effective local spin This spin is related to the nth segment of the stripe. We then obtain the spin-chain Hamiltonian [6]

Here,

is the maximal relative displacement between two neighboring holes, and it can take any integer value. A similar model accounting only for displacements, but allowing for overhangs, was recently presented [7]. The configuration of the stripe is determined by the projection of the spin, Let us first calculate the quantum equation governing the motion of the operator, Then, we consider the classical and the long wavelength

Sliding Stripes in 2D Antiferromagnets (discrete form,

131

continuous y) limits and find that the equation of motion acquires a wavelike

Here, denotes the time and a is the lattice spacing. Equation (3) implies that the DW can be regarded as an elastic string, with linear mass

density and elastic tension coefficient the long wavelength excitations of the string are gapless, is rough [8].

. It is important to notice that indicating that the string

3. RESULTS Next, we account for the presence of randomly distributed impurities, which act to pin the stripe. The pinning energy provided by an ensemble of quenched impurities was shown to be where the pinning parameter Here, denotes the Coulomb energy scale, is the doping concentration and L is the stripe length. In order to reduce the pinning barrier provided by the random impurities, we apply and external electrical field perpendicular to the stripe formation. The holes dynamics then arise from the competition between the pinning potential and the bias field. The threshold field Ec is defined as the value for which the barriers vanish and the holes can freely flow through the sample: this is the free flow (FF) or metallic regime, with ohmic dissipation. Below the threshold field, a finite pinning barrier prevents the charge motion. However, the holes can still jump over the barrier due to thermal activation. This is the thermally activated flow (TAF) regime. The thermal decay rate for a “particle” trapped into a metastable state is given by the Arrhenius law e x p ( — U / T ) , where U is determined within the semiclassical approximation by the free energy evaluated at the saddle point configuration This rate corresponds to the escape of trapped charges, i.e., it is proportional to a current, Hence, the conductivity in the thermally activated flow regime is exponentially reduced in comparison with the conductivity in the free flow regime i.e., It is important to notice that the value of U and depend strongly on the considered system. We denote the activation barrier and the conductivity for the gas of holes as and

respectively, whereas for the stripe configuration we use and Next, we study each case in more detail. Let us start analyzing the stripe configuration. The free energy describing an elastic line along the y direction, which tends to move due to an external electrical field E competing against the pinning potential is

with Due to the sublinear growth of the pinning energy, a length of the string is collectively pinned by the impurities within this region, as indicated by the

132


Collective Pinning Theory of Larkin and Ovchinnikov [9]. Deep in the insulating phase, with we find that this length is of order and the threshold electrical field In the following, we apply the results that are known for describing the dynamics of elastic manifolds in disordered media in order to predict the characteristics for the stripe configuration.

The collective pinning barrier for an elastic line was estimated to be [6]

At low applied fields, the minimal barrier for creep displays a glassy behavior,

whereas close to the critical field

it exhibits a power-law behavior,

The conductivity in the thermally activated stripe flow (TASF) regime then reads

with given by Eqs. (5)–(7). Hence, the characteristics are highly nonlinear, i.e., the barrier diverges and consequently the current vanishes in the limit of small applied bias field (Fig. 2). However, the gas of holes exhibits a different behavior. In this case, the threshold field is still the same, whereas the pinning barrier felt by each hole is substantially smaller, in agreement with experimental data [11]. As a consequence, the holes can be thermally activated easier over the barriers. The low field behavior is then the thermally activated hole flow regime (TAHF), which is ohmic, i.e.,

Sliding Stripes in 2D Antiferromagnets

133

In contrast to the stripe case [Eq. (8)], here does not depend on the bias field E[10]. The characteristics then will be rather distinct than in the stripe case (Fig. 3).

4. CONCLUSIONS We studied the sliding behavior of a stripe within a phenomenological elastic model, which can be related to the t – J model. By accounting for the presence of randomly distributed impurities, we evaluated the threshold electrical field that must be applied to the system in order to release the stripes. We also considered the analogous problem for a gas of holes. It was found that the behavior exhibited by the gas and the line of holes is fundamentally different: In the limit of vanishingly small applied fields, the gas of holes displays an ohmic behavior (non-zero slope of the current field dependence), whereas the

elastic line is in a glassy state characterized by a diverging barrier (zero slope). Therefore, measurements of the characteristics of a doped antiferromagnet can serve as a test to check the existence of the striped phase.

ACKNOWLEDGMENTS We are indebted to H. Schmidt for fruitfull discussions. This work has been supported by the DAAD-CAPES PROBRAL project number 415. NH acknowledges financial support from the Gottlieb Daimler- und Karl Benz- Striftung and the Graduiertenkolleg “Physik nanostrukturierter Festkörper,” Universität Hamburg. YD acknowledges financial support from the Otto Benecke-Stiftung. REFERENCES 1. J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989); H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990); D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989); M. Inui and P. B. Littlewood, Phys. Rev. B 44, 4415 (1991); K. Machida, Physica C 158, 192 (1989); M. Kato et al., J. Phys. Soc. Jpn. 59, 1047 (1990); P. Prelovšek and X. Zotos, Phys. Rev. B 47, 5984 (1993); T. Giamarchi and C. L. Lhuilier, Phys. Rev. B 42, 10641 (1990); S. R. White and D. J. Scalapino, cond-mat/9705128. 2. J. M. Tranquada et al., Phys. Rev. Lett. 73, 1003 (1994); Phys. Rev. B. 52, 3581 (1995); Nature 375, 561 (1995); Phys. Rev. B 54,7489 (1996); Phys. Rev. Lett. 78, 338 (1997); V. Sachan et al., Phys. Rev. B 51, 12742 (1995). 3. K. Yamada et al., preprint; G. Aepli el al., Science 278, 1432 (1997); T. E. Mason et al., Phys. Rev. Lett. 68, 1414 (1992) and 77, 1604 (1996).

4. F. Borsa et al., Phys. Rev. B 52, 7334 (1995).

134 5. 6. 7. 8. 9.


M. von Zimmermann et al., preprint. C. Morais Smith et al., Phys. Rev. B 58, I (1998). H. Eskes et al., cond-mat/9510129; cond-mat/9712316. N. Hasselmann et al., preprint. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teoi. Fiz. 65, 1704 [Sov. Phys. JETP 38, 854 (1974)]; J. Low Temp. Phys. 34, 409(1979). 10. If we assume the impurities to do not have all exactly the same pinning strength, but instead, to have different and gaussian distributed strengths, we obtain see O.S. Wagner et al., condmat/9803119. 1 1 . J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986).

Quantum Interference Mechanism of the Stripe-Phase Ordering S. I. Mukhin1

A microscopic mechanism of the multimode spin-charge ordering (“stripe phase”) in the weakly coupled repulsive quasi-one-dimensional electron system is found. As a consequence, a new phase is predicted that is composed of the long wavelength charge density wave (CDW) with the wave vector mediating between the two umklapp-related incommensurate spin density waves (SDWs) with the

wave vectors vector, and

Here,

is a commensurate antiferromagnetic wave

measures a deviation from the half-filling of the bare conduction

band. Relationship with the stripe phases observed in cuprates and nickelates is discussed.

Recent experimental discovery of the coupled spin and charge ordering phase transitions (“stripe phase”) and/or dynamic stripe correlations in the various layered transition metal oxides [1] poses a challenging problem for the theory to discover the nature of the multimode instabilities in the interacting electron system. Inelastic neutron scattering experiments reveal a general property of the stripe spin-charge order. Namely,

the spin density wave-vectors are equal to in the twodimensional (2D) Brillouin zone while simultaneously the charge density wave-vectors are equal to and with Here and below, the units of are used, where a is the lattice constant. Geometrically, this means that spin orders nearly antiferromagnetically with the wave vector , except for some slow modulation (“stripes”) with the period which is twice the period of the charge density

variation, Another basic observation is a dependence of on the (hole) doping concentration which measures deviation of the bare conduction band from the half-filling. In particular, experimentally, for small enough and developes a shoulder at greater ¹Moscow Institute for Steel and Alloys, Theoretical Physics Dept., Leninskii prospect 4, 117936 Moscow, Russia.


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Mukhin

Stripe-phase state was predicted in a variety of theoretical approaches [2,3], based on the numerical simulations in the strong coupling regime at zero temperature. The present work is an analytical study of the mechanism of the spin and charge mode coupling in the weakly interacting regime. The main atribute of this theory is an electron system on a periodic crystal lattice with the nested pieces of the Fermi surface. In order to make a description of the new effect most transparent, a (quasi) one-dimensional (1D) system is considered, while an extension to the quasi-2D systems (more relevant, e.g., for the layered oxides) is straightforward. In 1D case, close to half-filling of the conduction

band, the opposite Fermi-points are separated by a momentum deviating from the one-half of the inverse lattice period. The difference is expressed via the hole (electron) doping concentration as: This whole construction is replicated in the neighboring Brillouin zone of the unit width in the periodic zone scheme. Now introduce an interelectron repulsion, , which is considered here as “weak,” i.e., where 2t is the bandwidth. Then, an availability of the nested parts of the Fermi surface in a quasi-1D system is known [4] to cause the Stoner instability and a subsequent condensation (on the mean-field level of accuracy)

of the static SDW with the incommensurate wave vector The related spin density variation is, for example, However, the wave vector also connects the nested parts of the Fermi surface in the periodic zone scheme. Therefore, condensation of an SDW with the wave vector and related spin density variation is a rightful alternative. As is shown below, a constructive quantum interference of the scattering amplitudes of electron by the periodic potentials due to SDWs makes energetically favorable a linear combination state: This is a weak coupling analog of the spin component part of the stripe phase structure [1,2]. The further decrease of the

(free) energy of the system comes now in a strikingly straightforward way, being caused by the charge component of the stripe-phase structure. Namely, one easily notices that scattering of an electron by the

SDW potential is transformed into the scattering by

the SDW potential if there is an extra—e.g., a CDW—scattering potential with the “matching” [5] wave As is shown below, this CDW causes a decrease of the total energy of the system due to additional constructive interference between and electron scattering amplitudes. This happens when the phase shift between the CDW and the spin modulation is such that, depending on the nature of the CDW and a character of doping (hole or electron), either the minima or the maxima of

the charge density coincide with the nodes of the spin density. Hence, the resulting stripe phase potential seen by the electrons might be regarded as a frozen “instanton,” with the CDW mediating between the two umklapp-symmetric and SDWs. Summarizing, we present here theory recovers in the weak coupling limit the main features of the

stripe phase, which is experimentally observed in the underdoped transition metal oxides and was predicted from the numerical simulations in the strong coupling limit. Besides that, another most important results are (i) the stripe-phase transition proves to be first (second) order in both spin and charge components at low (higher) doping concentrations (ii) the transition temperature

value

decreases with the increase of

which is exponentially lower than

at

reaching the limiting

(iii) the CDW amplitude

decreases linear with and (iv) the lowest transition temperature, incides with the temperature of condensation of the CDW-free phase with the two SDWs.

Quantum Interference Mechanism of the Stripe-Phase Ordering

137

The simplest model Hamiltonian of the electron system in the presence of the spin and charge mean fields, and has the following most general form

where and are electronic creation and annihilation fields, is the chemical potential of electrons, 2t is the bare bandwidth, is the spin index and summation over repeated indices is implied. A short-range repulsion between electrons is introduced that is responsible for the SDW instability in the absence of a CDW. In general, the charge density terms include both electron–electron and electron–lattice coupling energies as well as the

lattice deformation energy. The latter is incorporated in the K-term (compare, e.g., Ref. 6). If the electron-lattice interaction dominates interelectron repulsion energy, then is the density of the “lattice-charge,” g is the electron–lattice coupling, and the electronic density deviation from the homogeneous value follows as: . In the opposite case of a very weak electron–lattice coupling, the interelectron repulsion terms are obtained in Eq. (1) after the changes: and + —in front of the term. Then, the meaning of see for example, Ref. 2. The single-particle eigenstates of the Hamiltonian Eq. (1) can be written in the left and right movers representation as

where the inverse lattice wave vector is

Simultaneously, so that the fast variation of m(x) with the wave vector is factored out. The slowly varying functions and obey the Bogoliubov-de Gennes equations, which according to Eq. (1) are (compare Ref. 6):

Here, the unit length is the lattice spacing a, and 2t is substituted below with which has the meaning of the Fermi velocity of electrons. A single CDW corresponds to A modulated spin density could be decomposed as the two incommensurate SDWs: which in turn defines Fixing of the mutual phase between the CDW and the SDWs is crucial for the quantum interference phenomenon described below. Now an explicit separation of the WKB part of the eigenfunction

where than

in the slow CDW potential

gives (compare Ref. 7)

is the Bessel function of an integer order n, and the terms of the higher order are neglected provided that After a substitution of Eq. (3) into

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Mukhin

Eq. (2), and with the “matching” condition finds for the case of the “hole doping,”

for the CDW wave vector fulfilled, one

The terms are neglected in Eq. (4) provided the doping concentration is not too small, i.e., Solving Eq. (4), one finds the single-particle spectrum

where

In the “electron doping” case, the sign in front of and in Eqs. (5) and (6) should be changed. The physical implication of Eqs. (5) and (6) is remarkable. Namely, the coupling

strength U, which causes SDW condensation, is renormalized and equals in the presence of the CDW. Depending on the sign of the factor might be less or greater than 1, which in turn means that coupling strength, , is either enhanced or suppressed with respect to the bare value, U. This is a manifestation of the quantum interference of the scattering amplitudes of electron in, for example, the and in the combined plus CDW periodic potentials. Using Eqs. (1) and (5), it is straightforward to derive the free energy of the system (per unit length), at a finite temperature T:

where is the upper cutoff of the electron energy, and (see also Ref. 8). A detailed description of the phase diagram following from Eq. (7) is given elsewhere. Here we merely list the main results (see also Figs. 2-4). 1.

Starting from the high temperature limit, the stripe phase condenses first with depending on the sign of for a hole (electron) doping . Thus, the spin density behaves as:

in the case and as : case. While the CDW density is the same in the both cases: Hence, the nodes of the spin density coincide with the maxima (minima) of the charge density in the case of the hole (electron) doping. In the both cases where This function of reaches its maximum

Quantum Interference Mechanism of the Stripe-Phase Ordering

0.83, where 2.

139

whereas in the absence of the CDW

The stripe phase condensation temperature at small doping concentrations, is higher than the bare and reaches as doping concentration increases beyond see Fig. 1. Simultaneously, the character of the phase transition changes from the first order to the second order as increases, see Figs. 2 and 3. In the first-order transition regime, the doping dependence of is:

where as:

Simultaneously, the CDW- and SDW amplitudes behave

Quantum Interference Mechanism of the Stripe-Phase Ordering 3.

In the second-order phase transition regime, parameters close to behave as:

141 and the order

where 4.

In the low temperature limit, eter, depends on

, the saturation value of the spin-order param-

The dependences of the order parameters on the doping concentration at a low temperature are presented in Fig. 4. In conclusion, a microscopic theory of the stripe-phase

ordering in the weak coupling limit is presented. It demonstrates the quantum interference mechanism of the spin-charge coupling in the correlated electron system, which leads to the stripe-phase instability. The stripe-ordering temperature in the underdoped

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Mukhin regime is enhanced with respect to the bare spin-ordering temperature. The mechanism of a high-temperature condensation of the “matching” long-wavelength CDW in the system, which is unstable with respect to the incommensurate antiferromagnetic spin ordering, provides an important hint for the construction of the theory of the high-temperature superconductivity in cuprates, where the role of the CDW may play a space-modulated density of the Cooper-pairs condensate. A detailed analysis of this possibility is now in progress.

ACKNOWLEDGMENTS

Useful discussions with Jan Zaanen, A. A. Abrikosov, and Wim van Saarloos are highly appreciated. The work was supported in part by NWO and FOM (Dutch Foundation for Fundamental Research) during the author’s stay at the Lorentz Institute in Leiden in the January 1998.

REFERENCES 1. J. M. Tranquada et al., Nature (London) 375, 561 (1995); J. M. Tranquada, D. J. Buttrey, and V. Sachan, Phys. Rev. B 54, 12318 (1996);C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993). 2. J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7391 (1989); H. J. Schulz, J. Phys. (Paris) 50, 2833 (1989).

3. V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993); ibid., 235–240, 189 (1994); C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995). 4. B. Horovitz, H. Gutfreund, and M. Weger, Phys. Rev. B 12, 3174 (1975). 5. S.I. Mukhin, Self-Matching Property of Correlated Electrons: Charge-Density Wave Enhances Spin Ordering, in the Proceedings of the Conference Spectroscopies in Novel Superconductors, Cape Cod, MA, September

14-18, 1997, to be published in J. Phys. Chem. Solids, 1998; S. I. Mukhin and Jan Zaanen, 1998 (unpublished). 6. H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990).

7. L. P. Gor’kov and A. G. Lebed’, J. Phys. Lett. 45, L-433 (1984). 8. It is important to note here that even in the CDW-free case a modulated spin-density state has lower (free) energy than a single incommensurate state,

The reason is that despite the appearance of a factor 1/2 in front of the term in Eq. (7) in the single-SDW case, the gap amplitude should be simultaneously devided by 2 in Eq. (5), which finally makes the single SDW state energetically less favorable than the modulated spin-density state.

Spontaneous Orientation of a Quantum Lattice String Osman Y. Osman,1 Wim van Saarloos,1 and Jan Zaanen1

Using exact diagonalization and quantum Monte Carlo techniques, we study a quantum lattice string model introduced as a model for a single cuprate stripe. We focus on the ground state properties of the string. Our result shows that in the physically relevant region of the parameter space, a zero-temperature spontaneous symmetry breaking occurs. The string spontaneously orients itself along one direction in space and becomes directed. We introduce an order parameter for the directedness and show that at zero temperature, this order parameter reaches its saturation value.

Since the experimental discovery of the stripe phase [1] interest in this field has grown rapidly. Many issues concerning stripes are discussed, ranging from their origin to their relation to superconductivity, including the dynamical properties of the stripes. In this contribution, we are concerned with the last subject. We focus on the problem of a single-stripe/single-charged domain wall. We consider the domain wall to be a connected trajectory (string) of particles, communicating with the lattice, whereas the precise nature of these particles is not further specified: the quantum lattice string (QLS) model [2]. Studying this model numerically, we discovered a zero-temperature symmetry breaking: Although the string can be quantum delocalized, it spontaneously picks a direction in space. This symmetry breaking happens always in the part of parameter space that is of physical relevance. At first sight, one might expect that the quantum fluctuations (kinetic energy) would tend to disorder the string (i.e., to decrease the tendency for the string to be directed). That the opposite effect happens can be seen as follows. A first intuition can be obtained by considering the analogy with surface statistical mechanics. The quantum string problem

can be formulated as a classic problem of a two dimensional surface (worldsheet) in 2 + 1 dimension, where the third direction is the imaginary time direction. The larger the kinetic 1

Instituut-Lorentz, Universiteit Leiden, P.O.B. 9506, 2300 RA Leiden, The Netherlands.


143

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Osman, van Saarloos, and Jan Zaanen

term or the smaller the temperature, the further the worldsheet stretches out in the time direction. At zero temperature, the worldsheet becomes infinite in this direction as well. The statistical physics of a string is then equivalent to that of a fluctuating sheet in three dimensions. Now, it is well known from studies of classical interfaces [5] that although a one-dimensional (1D) classical interface in two dimensions does not stay directed due to the strong fluctuations, for a two-dimensional (2D) sheet, the entropic fluctuations are so small that interfaces can stay macroscopically flat in the presence of a lattice [6,7]. In other words, even if microscopic configurations with overhangs are allowed, a classical interface on a lattice in three dimensions can stay macroscopically flat or “directed.” In the present context, we show that the directedness is a caused by an order-out-of-disorder mechanism: In order to maximize the fluctuations transversal to the local string directions, overhangs should be avoided on the worldsheet. It remains to be seen if this mechanism is of a more general application. In the QLS model the string configurations are specified by the position of the particles Two consecutive particles i and l + 1 should either be nearest or next-nearest neighbors, The set of all such configurations is the string Hilbert space. The Hamiltonian consists of a classical energy term and a quantum (hopping) term. The classical energy is a sum of local interactions between nearest and next-nearest particles in the string.

with

The quantum term allows the particles to hop to nearest-neighbor lattice positions, giving rise to the meandering of the whole string. These hops should respect the string constraint. To enforce the constraint, a projection operator is introduced that ensures that the motion of particle l keeps the string intact. The string is quantized by introducing conjugate momenta and the hopping

Spontaneous Orientation of a Quantum Lattice String is described by

145

The kinetic energy becomes (Fig. 1)

The above string model is invariant under rotation of the string in space. As is discussed below, we find that for physical choices of the parameters, the invariance under symmetry operations of the lattice is broken. The string acquires a sense of direction in space. This occurs even when the string is critical (delocalized in space). The string’s trajectories, on average, are such that they move forward in one direction while the string might delocalize in the other direction. The relation of the string problem to surface models is established by using SuzukiTrotter mapping, which maps a 2D quantum problem to a 2 + 1 D classical problem. A classical model of two coupled RSOS (restricted solid-on-solid) surfaces results. These are classical models for surface roughening [3] in which overhangs are not allowed. For the quantum string case, the two RSOS surfaces describe the motion of the string in the x and y spatial directions. Skipping detailed calculation, the partition function of the quantum string can be mapped to the following classical problem [2],

where k is the trotter index and with the constraint The above classical model can be viewed as two coupled RSOS surfaces, The x coordinate of particle l at the trotter height k is the height at position (l, k) in the first surface, and similarly the y coordinates define a second RSOS surface, coupled strongly to

the first by the above classical interactions. Let us first discuss the numerical results. It is clear that the directedness property is a

global quantity. For a string living in 2D lattice with open boundary conditions, directedness means that if it start at, say, the left boundary it must end at the right boundary and will never end at the top or the bottom boundaries of the lattice. Although in the above model one can introduce a local order parameter to measure the directedness of a string, a more general quantitative measure for this global property can be constructed. This measure is not easily evaluated analytically, but it can easily be calculated numerically; most importantly, it illustrates clearly and effectively the directedness phenomenon. Every string configuration s defines a curve in the 2D space [x(t), y(t)], where t could, for instance, be the discrete label of the successive particles along the string. When this curve can be parametrized by a single-valued function x(y) or y ( x ) , we call the string configuration

146


directed. The quantum string vacuum is a linear superposition of many string configurations. When all configurations in the vacuum correspond to single-valued functions x(y) or y(x), the string vacuum is directed. At zero temperature, the ground-state wave function of the string is where every state in string configuration space corresponds to a trajectory [ x ( t ) , y(t)]. Consider first the case of a continuous string. For every configuration, the total string arclength is given by

Consider now an indicator function that equals 1 when the string is single-valued when projected onto the x axis and zero otherwise, and analogously a function for the y axis (Fig. 2). The total directed lengths in the x and y directions are defined as

The measure of directedness is then defined as the larger of

where

and On the lattice, one measures the directedness in analogy with the above definition, except that we just count the number of directed bonds, irrespective of whether they

Spontaneous Orientation of a Quantum Lattice String

147

are oriented diagonally or horizontally. The finite temperature measure of the directedness density is simply given by thermally averaging the above definition.

where is the directedness density of an excited string with energy To study the directedness property, we performed exact diagonalization and quantum Monte Carlo studies. Although the quantum Monte Carlo study is the more extensive, we start by discussing the exact diagonalization results, as it gives a clear indication for the symmetry breaking directly at zero temperature. Here, we consider an lattice. We think of a string living in such a finite lattice as part of an infinite one, and therefore the ends of the string should live on the boundaries of the cluster. To fix the length of the string inside the cluster, we take as a criterion that the energy per particle be minimum. We plot the energy per particle versus the number of particles in the string. The minimum defines the optimal length of a string in the cluster. Upon setting the parameter to zero and investigating different points in the parameter space, we found that the optimal length one should consider is the linear dimension of the lattice. Therefore, in an lattice, we consider a string of length N. Such a string can be directed along the x (horizontal) or y (vertical) direction. If the directedness assumption is fulfilled, the Hilbert space effectively splits into two subspaces: strings directed along the x direction and strings directed along the y direction. If nondirected strings are present, there should be a non-zero tunneling probability between the two sectors. By measuring the probability to tunnel from the x to the y sectors as a function of the linear dimension of the system, it should be possible to see the tendency toward spontaneous directedness symmetry breaking in the thermodynamic limit. Table 1 gives this tunneling probability for different points in the parameter space. For all cases, we set The choice of these points was motivated by the directed string problem [2]. The data are shown for lattices up to For a lattice, the tunneling probability turns out to be less than the accuracy of our numerical technique. These results clearly indicate that in the thermodynamic limit there is no tunneling between the two sectors and the string should be directed either along the x or the y direction. We then used quantum Monte Carlo to calculate the directedness density as a function of temperature, Our results are displayed in Fig. 3. Four points in parameter space were considered. These points are representative for phases with a varying strength of the quantum fluctuations and serve to substantiate our conclusion. In Fig. 3a, the dashed

148


line is the result when all classical energies are zero (i.e., for optimally quantum string). The dashed-dotted and dotted lines correspond to all potential energy parameters set to zero except that and 1.8, respectively, corresponding to a string localized in the (1,0) or (0,1) directions. Decreasing the parameter causes stronger local fluctuations. The full line is the result for a classical string , where only flat segments and corners are allowed (no diagonal segments). The same classical result is shown again in Fig. 3b, together with the result at the point corresponding to a free (critical) string. The classical string would be flat at zero temperature, directed along (say) a (1,0) direction. A local "corner" (Fig. la) would be an excitation with energy (alternatively, one could consider two kinks). Clearly, a single corner suffices to destroy the directedness of the classical ground state. At any finite temperature, the probability of the occurrence of at least one corner is finite: Hence, directedness order

Spontaneous Orientation of a Quantum Lattice String

149

cannot exist at any non-zero temperature, for the same reason that any long-range order is destroyed at any finite temperature in one dimension. In the simulations the string is of finite

length, and the infinite temperature limit of is therefore not zero, but rather a small but non-zero value for a domain wall of length 50). is already close to this value for all temperatures of order and larger. For an infinitely long domain wall, drops very fast to zero with increasing temperature. For low T where grows rapidly to 1. Again, because the string is of finite length, it becomes directed already at a finite temperature: For all temperatures such that the string configurations in our simulations are typically completely directed. An infinitely long classical string becomes directed only at , of course, because at any non-zero temperature some corners always occur in a sufficiently long string. The results for the quantum string always look similar to the classical one. For temperatures higher than the kinetic scale, all curves approach each other and the classical limit is reached. At low grows rapidly to 1. As in the classical case, it reaches this value at a finite temperature for a finite length string. This is even valid for the pure quantum string, where all classical energies are zero (dashed line in Fig. 3a). Again, this can be understood in terms of an effective corner or bend energy that is produced by the quantum fluctuations. In analogy to the classical case, the probability for the occurrence of a bend is proportional to At zero temperature, no bend is present and the string becomes directed. A finite length string effectively becomes directed already at a temperature such that At intermediate temperatures, where the temperature is of the order of the kinetic term, the situation is less clear. Especially in this region, all the various energies may play a role, and the interplay of these on the

directedness is rather complicated. Nevertheless, as is clear from the data of Fig. 3a, this region connects the high and low temperature limits smoothly. Finally, by comparing the results for the three quantum strings in this figure, it is also clear that when the string is more quantum, mechanical is higher. The spontaneous directedness of the quantum string for can be understood by the following argument. The bends in strings block the propagation of links along the chain. Close to the bend itself the particles in the chain cannot move as freely as in the rest of the chain. This effect is shown in Fig. 4. In space-time, the bend is like a straight rod in time. Therefore, the presence of such kinks increase the kinetic energy. For the argument, it makes little difference whether the bend consists of a single corner or two corners. This confirms that it is the kinetic energy that keeps the strings oriented along one particular direction. In terms of a

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directedness order parameter, this result implies that such a quantity is always finite, except when or when the hopping term vanishes (it is easy to see that in the classical case, in many regions of parameter space the problem becomes that of a self-avoiding walk on a lattice in the limit For the two equivalent RSOS surfaces, this means that one of the two surfaces spontaneously orders while the other RSOS sheet can be either ordered or disordered. Our general conclusion, based also on Monte Carlo studies of the behavior in many other points in the parameter space, is that apart from some extreme classical limits, the general lattice string model at zero temperatures is a directed string. The qualitative picture of corners blocking the propagation of kinks appears to be a natural explanation for these numerical findings. ACKNOWLEDGMENT We are grateful to Henk Eskes for a collaboration from which this work is an outgrowth.

REFERENCES 1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995); J. M. Tranquada, Physica B, 241–243, 745 (1998). 2. H. Eskes, R. Grimberg, W. van Saarloos, and J. Zaanen, Phys. Rev. B 54, 724 (1996); H. Eskes, Osman Yousif Osman, R. Grimberg, W. van Saarloos, and J. Zaanen, to appear in Phys. Rev. B; C. Morais Smith et al., Phys. Rev. B 58, 1 (1998). 3. M. den Nijs, in Phase Transitions and Critical Phenomena, Vol. 12, Eds. C. Domb and J. L. Lebowitz, Academic

Press, London, 1988. 4. In the high temperature limit, the string becomes a self-avoiding walk on a two-dimensional lattice. For such a walk of length N, the radius of gyration grows as where in This implies that the directedness defined in this paper should go to zero as for in the high temperature limit. 5. J. D. Weeks, J. Chem. Phys. 67, 3106 (1977); J. D. Weeks, Phys. Rev. Lett. 52, 2160 (1984). 6. J. D. Weeks, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste, Plenum, New

York, 1980, p. 293. 7. H. van Beijeren and I. Nolten, in Structure and Dynamics of Surfaces II, eds. W. Schommers and P. von

Blanckenhagen (Springer, Berlin, 1987).

Domain Wall Structures in the Two-Dimensional Hubbard Model with Long-Range Coulomb Interaction G. Seibold,1 C. Castellani,2 C. Di Castro,2 and M. Grilli2

The structure and stability of partially filled domain walls is investigated in the two-dimensional Hubbard model supplemented with a long-range Coulomb interaction. Using an unrestricted Gutzwiller variational approach, we show that the strong local interaction favors charge segregation in stripe domain walls. This

approach supports the stabilization of half-filled walls for commensurate doping and large values of U even without the long-range part in contrast to results of Hartree–Fock calculations. Inclusion of the long-range interaction favors the formation of half-filled vertical stripes also in the intermediate U regime. These walls are characterized by a period doubling due to the charge and a period quadrupling due to the spins along the wall. We find that, as well as the underlying lattice structure, there is also an electronic stabilization mechanism for half-filled

vertical domain walls in

1. INTRODUCTION The observation of charged domain walls in the high-

superconductors presently

attracts a lot of interest also with regard to possible pairing scenarios [1–3]. Incommensurate spin correlations have first been observed in (LSCO) by neutron scattering

experiments [4–6]. More recently, it was found that the incommensurate spin fluctuations are pinned in Nd-doped LSCO and nickel oxide compounds [7–9], leading to spin- and charge-stripe order in these materials [10]. However, whereas the domain walls in the hole-doped system are oriented along the diagonals of the lattice, it turns 1 2

Institut fur Physik, BTU Cottbus, PBox 101344, 03013 Cottbus, Germany. Istituto Nazionale di Fisica della Materia e Dipartimento di Fisica, Università di Roma “La Sapienza,” Piazzale A. Moro 2, 00185 Roma, Italy.


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out that in the orientation of the stripes is along the Cu-O bond direction. Moreover, the hole concentration in the domain walls is one hole per Ni site in the nickelates and one hole every second Cu site in the Nd-doped cuprates. A comparison of the

low-temperature orthorhombic and the low-temperature tetragonal structure suggests that the vertical stripes in are pinned by the tetragonal lattice potential, whereas the orthorombic phase in the nickel oxides favors a diagonal orientation. We show

here that the electronic interactions may also play an important role in establishing different domain wall structures. The stripe instability for doped antiferromagnets was predicted theoretically in [ 1 1 ] within Hartree-Fock (HF) theory applied to the extended Hubbard model and confirmed by a number of subsequent investigations [12,13]. For small values of the Hubbard onsite repulsion U (generally smaller than 3t-4t) these calculations result in a striped phase

oriented along the (10) or (01) direction, whereas for higher values of U the orientation is along the diagonals. Within HF theory the stripe solutions become unstable for toward the formation of isolated spin polarons. However, all stripe calculations performed so far within the HF approximation of the Hubbard model predict one hole per site along the domain wall (filled stripe). This contrasts the observation of half-filled stripes (i.e., with half a hole per site) in the system, which is a yet unresolved problem of mean-field theory. Zaanen and Oles [114] addressed the question of whether the inclusion of an additional nearest-neighbor repulsion V in the Hubbard-Hamiltonian may favor the formation of partially filled stripes. According to Ref. [14J, the half-filled stripe solution is stabilized by a quadrupling of the charge or spin period along the stripe. However, although the nearest-neighbor repulsion slightly enhances the stability of the half-filled wall, this never corresponds to the HF ground state for realistic parameter values. Instead, the main effect of V is to shift the crossover to isolated spin polarons to lower values of U. In the present paper we show that a proper treatment of the strong local repulsion U plays an indirect but crucial role in stabilizing half-filled vertical domain walls. Specifically, we apply a slave-boson version of the Gutzwiller approach within an unrestricted variational scheme. Contrary to the pure HF approach, which underestimates heavily the effective attraction between the charge carriers and predicts repulsion for it was recently shown that within the slave-boson scheme the attraction persists up to very large U [15]. As a consequence of this more suitable treatment of the strong coupling limit, U greatly favors the charge segregation in striped domains as opposed to spin polarons. In the absence of long-range (LR) forces and values of (which is believed to be the physically relevant regime for copper oxides [17]), completely filled diagonal stripes stay more stable than half-filled vertical ones. However, due to their increased stability with respect to isolated polarons, the stripe solutions now allow for a less disruptive introduction of stronger LR forces, which affect the completely filled stripes more than the half-filled ones. Then, for a sizable but still realistic LR repulsion, the half-filled vertical stripe may become the ground-state configuration. In the following we consider the two-dimensional (2D) Hubbard model on a square lattice, with hopping restricted to nearest neighbors (indicated by the bracket ) and an additional LR interaction:

Domain Wall Structures in the Two-Dimensional Hubbard Model

153

where destroys an electron with spin at site i, and U is the on-site Hubbard repulsion and t the transfer parameter. For all calculations we take For the Coulomb potential, we assume an interaction of the form where the parameters and α are specified through the on-site repulsion U and the nearest-neighbor interaction = and Because we consider a finite lattice with periodic boundary conditions, it is also necessary to restrict the LR part to half of the lattice dimension both in x and y direction. To proceed, we treat the strong on-site correlation term in the slave-boson version of the Gutzwiller approximation proposed by Kotliar and Ruckenstein [18]. The long-range

part is linearized via the HF decoupling. In contrast to previous calculations [13] we do not restrict the solution to a specific functional form, but allow for an unrestricted variation of the bosonic and fermionic fields (for further details of this approach, see Ref. 15). 2. RESULTS WITHOUT LONG-RANGE FORCES To illustrate the importance of a proper treatment of correlation effects for the stability of domain walls, we present in Fig. 1 the binding energy per hole of various stripe phases for commensurate doping 1 /8. Also shown is the binding energy for isolated polarons. The energy per hole of each configuration was calculated with respect to the reference state of

a uniform AF lattice with one particle per site: In particular, the energy of an isolated spin-polaron is given by Figure 1 differs in various important aspects from corresponding results obtained via the HF approximation [12]. First, there does not occur a crossover from the stripe to the polaronic (Wigner) phase for all considered values of U. Within the HF approach, this

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happens for Moreover in the slave-boson approximation the stability of half-filled walls is significantly enhanced for large values of U because within HF theory these solutions have higher energy than isolated polarons (see, e.g., Ref. 14). Upon further increasing the onsite repulsion, we finally obtain the crossover to half-filled vertical walls at This

behavior is in agreement with the results in Ref. [19], where a density matrix renormalization group approach has been applied to the 2D t–J model. As a consequence, also in the intermediate U range the present approach allows for the stabilization of half-filled walls by including the repulsive effect of a LR potential. This, together with the detailed structure of the half-filled stripes, is discussed in the next paragraph.

3. INCLUSION OF LONG-RANGE FORCES; SINGLE-STRIPE CALCULATION In the presence of LR interactions a single stripe always results unstable with respect

to isolated polarons by increasing the length of the stripe. Indeed, the Coulomb energy per hole of a charged wall of length L increases as log L and it is not compensated by the coupling to a uniform distribution of background charges of opposite sign, as it is the case for a regular array of stripes. Nevertheless, the analysis of the single-stripe case allows us to illustrate the stabilization effect of intrastripe LR repulsive forces and find out the most stable stripe configuration within each class of stripes. In Fig. 2 we sketch the charge and spin structures of the two types of half-filled vertical domain walls that have the lowest energy among the different realizations. Figure 2a is of the CDW type, in which the main charge modulation is with along the wall, whereas the spin varies with We find that among the half-filled vertical walls this configuration is lowest in energy, in the range

repulsion

However, for strong on-site

the charge and spin realization with the lowest energy corresponds


155

to the staggered structure indicated in Fig. 2b, although the difference in energy to the CDW-type of Fig. 2a is rather small (the difference in energy per hole for is To compare the stability of the various stripe textures, we report in Fig. 3 the energy differences E(config) — E(polaron) as a function of the nearest-neighbor value of the Coulomb repulsion for in Fig. 3a and for in Fig. 3b. Because we are dealing with a finite size system, we cut the LR forces at “half-minus-one” the size of the supercell in order to avoid double counting in the Coulomb interaction energies. In case of half-filled vertical stripes the curves in Figs. 3a and 3b correspond to the structures in Figs. 2a and 2b, respectively. Disregarding the eventual above-mentioned instability of the single stripes with respect to isolated polarons by increasing length, various features are worth noting. First, completely filled stripes increase their energy more rapidly (have a larger slope) than half-filled ones on increasing the LR repulsion. Therefore, LR forces favor half-filled stripes, which eventually become the most favorable wall textures. However, the most relevant effect to be noticed here is the role of a large local repulsion U affecting the energies of the various textures (cf. Fig. 1). In particular, a comparison between Figs. 3a and Fig. 3b shows that U strongly reduces the energy of the half-filled stripes with respect to the filled ones already at By combining this reduction with the effects of LR forces on the stripes, it follows that increasing U makes the half-filled vertical stripe the most stable among the wall solutions at smaller values of 4. INTERSTRIPE INTERACTION AND SPIN-POLARON LATTICE To assess the actual ground state we now consider the interaction between stripes or between spin polarons. Indeed, one cannot neglect the interstripe repulsion because, for

example, in

the stripe separation is four times the Cu-Cu distance

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only. To incorporate the repulsion between stripes, we calculated the energy of vertically

oriented stripes on a lattice. For a concentration of 1/8, this results in an array of 4 completely filled or 8 half-filled stripes. The energy of these arrays with respect to the polaronic Wigner lattice are depicted in Fig. 4 (the Wigner lattice now corresponds to 16 spin polarons with maximum distance). In this case we obtain a crossover to half-filled vertical domain walls for It is interesting to observe that now the stripe solutions gain in energy on switching on the LR part with respect to the polaron lattice. This enhances the parameter range of stability for the stripes, which no longer become unstable toward the decay into isolated polarons. To assess the absolute stability of the half-filled vertical stripes, we should also compare their energy with the filled diagonal stripes, which, in the single-stripe analysis, result to be more stable than the filled vertical stripes. However, filled diagonal stripes are strongly destabilized by the elongated shape of the supercell so that we do not included their energy in Fig. 4. As an alternative to the direct calculations on the elongated cluster, to extract informations about diagonal stripe configurations, we analyzed a 2D regular array of charged wires with fixed global charge density. We found that the electrostatic potential energy is lower for wires with higher linear charge density at a larger distance than for less-charged wires more closely spaced. Therefore, diagonal stripes, which at given planar density are closer by but less densely charged by the same factor, are less favorable than the vertical stripes as far as the electrostatic Coulombic energy is concerned. However, our single-stripe investigation already demonstrated that a proper treatment of the strong local repulsion U opens the way to a stabilization of half-filled vertical stripes with respect to the filled diagonal stripes. From the above purely electrostatic analysis and from the results of Fig. 4, we can therefore safely conclude that half-filled vertical stripes are the ground-state configuration for at large enough U. To summarize, we have shown that a LR Coulomb interaction added to the 2D Hubbard model gives rise to half-filled vertically oriented domain walls when treated within an unrestricted Gutzwiller approach. This feature does not appear in semiclassical HF approximations in which the effective attraction between spin polarons is underestimated. Depending on the value of the on-site repulsion U, we expect the domain wall structure


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in the Nd-doped LaCuO system to be of the type shown in Figs. 2a and 2b, respectively. Moreover, the competition investigated here between completely filled diagonal and halffilled vertical stripes can explain the different hole orderings in nickelates and in Nd cuprates even without invoking a relevant role of lattice interactions. Specifically, our findings suggest that nickelates could be characterized by a smaller and/or a smaller V accounting for their filled diagonal stripes. However, although the half-filled vertical stripes in the Ndenriched LaSrCuO systems are to some extent likely fixed along the (1,0) direction by the underlying lattice structure, we showed here that, if large values of U and sizable values of V characterize these systems, then electronic correlations would also contribute to give rise to vertical half-filled stripes. In general, the filling and orientation of the stripes depend on the specificity of the electronic forces and structures, and therefore inside the various oxide families different textures of the stripe phase may prevail. ACKNOWLEDGMENTS GS acknowledges financial support from the Deutsche Forschungsgemeinschaft as well as hospitality and support from the Dipartimento di Fisica of Università di Roma “La Sapienza,” where part of this work was carried out. This work was partially supported by INFM-PRA(1996). REFERENCES 1. For a review, see, e.g., Proceedings of the IV International MMS-HTSC Conference, February 28–March 4, 1997, Beijing, China. 2. C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995); C. Castellani, C. Di Castro, and M. Grilli, Z. fur Physik, 103, 137 (1997). 3. V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 56, 6120 (1997). 4. S.-W. Cheong et al., Phys. Rev. Lett. 67, 1791 (1991). 5. T. E. Mason, G. Aeppli, and H. A. Mook, Phys. Rev. Lett. 65, 2466 (1990). 6. T. R. Thurston et al., Phys. Rev. B 46, 9128 (1992). 7. J. M. Tranquada, D. J. Buttrey, V. Sachan, and J. E. Lorenzo, Phys. Rev. Lett. 73, 1003 (1994). 8. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995). 9. J. M. Tranquada, J. D. Axe, N. Ichikawa, A. R. Moodenbaugh, Y. Nakamura, and S. Uchida, Phys. Rev. Lett. 78,338(1997). 10. The existence of charge modulation in the CuO 2 planes of Bi22I2 was proposed early on by Bianconi and coworkers: A. Bianconi, Proceedings of the workshop on Phase Separation in Cuprate Superconductors, edited by K. A. Muller and G. Benedek (World Scientific, Singapore, 1993). 11. J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). 12. D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989); M. Inui and P. B. Littlewood, Phys. Rev. B 44, 4415 (1991); H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990); J. Zaanen and P. B. Littlewood, Phys. Rev. B 50, 7222(1994). 13. T. Giamarchi and C. Lhuillier, Phys. Rev. B 42, 10641 (1990). 14. J. Zaanen and M. Ann. Physik 5, 224 (1996). 15. G. Seibold, E. Sigmund, and V. Hizhnyakov, Phys. Rev. B 57, 6937 (1998); G. Seibold, C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. B 58, 13506 (1998). 16. The EAF was calculated for an 8 × 8 cluster and rescaled by a factor (9 x 8)/(8 x 8). 17. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). 18. G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986). 19. Steven R. White and D. J. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).


Boson-Fermion Mixtures, d-Wave Condensate, and Tunneling in Cuprates A. S. Alexandrov1

We argue that local pairs (bipolarons), formed by any short-range attraction are localized and cannot give rise to no matter whether they are hybridized with Fermions. However, a long-range Fröhlich electron-phonon interaction can provide mobile intersite bipolarons in the plane condensing at high The ground state of cuprates is thought to be a charged Bose-liquid of intersite bipolarons with single polarons existing only as thermal excitations. We show that some bipolaron configurations lead to a d wave charged Bose-Einstein condensate in cuprates. It is the bipolaron energy dispersion rather than a particular pairing interaction that is responsible for the d wave symmetry. Single-particle spectral density is derived, taking into account realistic band structure and disorder. The tunneling and photoemission (PES) spectra of cuprates are described.

PACS numbers: 74.20.-z, 74.65.+n, 74.60.Mj

1. INTRODUCTION There is a fundamental problem with any theory involving real-space pairs (bosons) tightly bound by a field of a pure electronic origin. As stressed by Emery et al. [1], such theories are a priori implausible due to the strong short-range Coulomb repulsion between two carriers. A direct (density–density) repulsion is usually much stronger than any exchange interaction. The attraction potential generated by the electron-phonon interaction of the

Holstein model may overcome the short-range Coulomb repulsion, but inevitably involves a huge carrier mass enhancement otherwise the phonon frequency would be extremely high [2], Although we do not exclude a coexistence of degenerate Fermi and Bose carriers in some systems (in fact, we discussed their mixture in 1986 [3]), we present reasoned

arguments [4] ruling out any role that their hybridization could play. Our study of localized 1

Department of Physics, Loughborough University, Loughborough LE 11 3TU, U.K.



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Alexandrov

bosons hybridized with propagating fermions [i.e., a boson–fermion model (BFM)] beyond the mean-field approximation shows that: 1. 2. 3.

there is no Cooper pairing of fermions without a condensation of real bosons; there is no condensation of bosons and therefore no pairing of fermions at any temperature in two dimensions (fermions in planes), the critical temperature of the Bose-Einstein condensation of the 3D BFM is very low, less than 1 K, and the inverse lifetime of the long-wave bosons is of the order of their energy;

4.

the Cooper pairing is possible if and only if bosons are condensed. Statements 1, 2, and 4, derived with the most divergent “ladder” approximation for the

boson self-energy and for the fermion vertex, are, in fact, exact.

However, we have shown [5] that the Fröhlich electron–phonon interaction can provide intrinsically mobile intersite small bipolarons, which are condensed at high of the order of 100 K. We believe that this interaction operating on a scale of the order of 1 eV can compensate the intersite Coulomb repulsion allowing the deformation potential (together with an exchange interaction of any origin) to bind two holes into an intersite mobile bipolaron in the

plane. The bipolaron mass renormalization appears to be smaller by several orders of magnitude than in the Holstein model with the same value of the attraction potential. Although the charged Bose liquid of bipolarons describes well the anomalous thermodynamic and kinetic properties of cuprates [2], finite frequency/momentum response functions of bipolaronic superconductors remain to be established. Moreover, it was claimed [6] that an experimental evidence for the d-wave order parameter and a large Fermi-surface in several cuprates are incompatible with bipolarons. In this paper, we establish a d-wave symmetry of the bipolaronic condensate and a single-particle spectral function that quantitatively describes the tunneling spectra and some photoemission features of cuprates ([7-9], and references therein).

2. d-WAVE BOSE-EINSTEIN CONDENSATION OF “PEROXY” BIPOLARONS

The evidence for a d-like order parameter (changing sign when the plane is rotated by has been reviewed by Annett, Goldenfeld, and Legget [10] and more recently by Brandow [ 1 1 ] as well as by several other authors. Although a number of phase-sensitive experiments [12] provide unambiguous evidence for the d-wave case, some SIN tunneling studies show more usual s-like shape of the gap function. Under certain conditions, the d wave pairing occurs both for the spin fluctuation [13] and the electron–phonon pairing mechanism [14] within the BCS approach. One can reach a compromise between conflicting experimental results by mixing s and d order parameters (sometimes violating the time reversal symmetry). However, the observation of the normal state pseudogap in tunneling and photoemission spectra (PES), non-Fermi-liquid normal state kinetics and thermodynamics, and unusual critical phenomena tell us that many highcuprates are not BCS superconductors. Therefore, an explanation for the d-like order parameter should be found beyond the BCS gap equation. Here we argue that the symmetry of the Bose-Einstein condensate in the bipolaronic superconductor should be distinguished from that of the “internal” wave function of a single intersite bipolaron, depending on the polaron–polaron

Boson-Fermion Mixtures, d-Wave Condensate, and Tunneling in Cuprates

161

interaction and also from the symmetry of the excitation spectrum, which depends on the bipolaron–bipolaron repulsion and the polaronic band dispersion. We show that the BoseEinstein condensate in cuprates is d wave owing to the bipolaron energy band structure rather than to a particular pairing interaction. The existence of the “parent” Mott insulators allows us to consider cuprates as doped semiconductors with narrow electron bands. Therefore, different bipolaron configurations can be found with computer simulation techniques based on the minimization of the ground-state energy of an ionic lattice with two holes fully taking into account the

lattice deformation and the Coulomb repulsion [15,16]. The intersite pairing of the in-plane oxygen hole with the apex one is energetically favorable in the layered perovskite structures as established by Catlow et al. [17]. The apex or peroxy-like bipolaron can tunnel from one cell to another via a direct single polaron tunneling from one apex oxygen to its apex neighbor as shown in Fig. 1. The bipolaron band structure is derived by the use of the generic Hamiltonian including the oxygen–oxygen and oxygen-copper hopping integrals, the coupling of holes with phonons and their Coulomb repulsion [5]. The hole bipolaron energy spectrum in the tight binding approximation consists of two bands formed by the overlap of and apex polaron orbitals, respectively,

Here and below we take the in-plane lattice constant

t is twice of the renormalized

bipolaron hopping integral between p orbitals of the same symmetry elongated in the

direction of the hopping perpendicular direction boundary, of the and

and is twice of the renormalized hopping integral in the The energy band minima are found at the Brillouin zone

and rather than at point owing to the opposite sign hopping integrals. Only their relative sign is important, so we choose

If the bipolaron atomic density is low, the bipolaron Hamiltonian can be mapped onto the charged Bose-gas [2]. Charged bosons are condensed below into the states of the Brillouin zone with the lowest energy, which are and for the x and y bipolarons, respectively. These four states are degenerate, so the order parameter (the condensate wave function) in the real (site) space

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Alexandrov

given by

where

is the bipolaron (boson) annihilation operator in the k space that is a c number for

the condensate and N is the number of cells. Other combinations of four degenerate states do not respect time reversal and (or) parity symmetry. Two solutions, Eq. (3), are physically identical because one of them is expressed through another by the use of the translation, They have d-wave symmetry, changing sign when the plane is rotated by around (0,0) or around (0,1) for and respectively (Fig. 2). We notice that the continuous (r) real space representation of the order parameter is irrelevant for cuprates because of a very small coherence volume compared with the unit cell one. The d-wave symmetry is entirely due to the bipolaron energy dispersion with four minima at When the minima located at the point of the Brillouin zone the condensate is s-like. 3. SINGLE-PARTICLE TUNNELING INTO AND FROM CUPRATES

Single polarons exist only as excitations with the energy

or larger, if they are

included in the Hilbert space of the generic Hamiltonian. A single-particle gap, is the difference between the bottom of polaronic and bipolaronic bands. It is temperature

independent, differently from the BCS gap. Hence, there is no other phase transition except a superfluid one at There is, however, a characteristic temperature of the normal phase, which is a crossover temperature of the order of in which the population of the upper polaronic band becomes comparable with the bipolaron density. Along this line

the theory of tunneling in the bipolaronic superconductors was developed both for twoparticle [18] and one-particle [19] tunneling through a dielectric contact. To derive the tunneling conductance, one can apply a single-particle tunneling Hamiltonian describing injection of an electron into a single hole polaronic state P with


the matrix element

163

and into a paired hole (bipolaronic) state B with the matrix element

Here, and are the quantum numbers describing an electron in the tip (Fig. 3) (annihilation operator a hole polaron and a hole bipolaron in the plane

in the random field, respectively. If the eigenstates are known, the matrix elements and are derived by the use of the site representation and the canonical polaronic transformation as discussed in detail in Ref. [18]. They are almost independent of in a wide voltage and binding energy range, In general, B and P are different because the second hole in a small coherence volume changes a potential barrier of the contact for the tunneling B compared with P. Calculating the injection and emission rates with the Fermi Golden Rule, one can obtain the expression for the tunneling and PES spectra in the voltage (energy) region, in which the high-frequency phonon shakeoff is forbidden by the energy conservation. In particular, we find for

where is the density of states of a single polaron (unoccupied) band. The p-hole polaron in cuprates is almost one-dimensional (1D) due to a large difference in the and hopping integrals and an effective 1D localization by the random potential as described in Ref. [5]. This is confirmed by the angle-resolved photoemission (ARPES) [20] with no dispersion along certain directions of the 2D Brillouin zone. Because the amount of disorder

is high and the screening radius is about the lattice constant, we can describe the effect of disorder and of the thermal fluctuations as “white Gaussian noise.” The relevant spectral

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Alexandrov

density A(k, E) for a one dimensional particle in a random Gaussian potential was derived by Halperin [21] and the density of states, by Frish and Lloyd [22]. The result is

with the Airy functions. A constant depends on the density of states in the tip and the scattering parameters, and . The constant D describes the second moment of the Gaussian potential

comprising thermal and random fluctuations as where is the amplitude of the deformation potential, M is the elastic modulus, is the impurity density, and is the coefficient of the function impurity potential. The “asymmetry” value depends on doping x. We compare the conductance, Eq. (5), with the scanning tunneling microscope (STM) [7] and point-contact tunneling (PCT) [8] measurements in an overdoped and optimally doped in Figs. 4 and 5, respectively. The bipolaron theory describes quantitatively the spectra in the gap region, including the zero-bias conductance at the asymmetry, and the decreasing background at higher voltages that are inconsistent with the classic BCS theory, no matter s or d wave. The zero-bias conductance at is explained by the presence of the impurity tails of the polaronic DOS inside the gap, while the decreasing background, proportional to for is explained by the 1D band dispersion of polarons. The peak amplitudes and the zero-bias conductance are determined by the ratio of the bipolaron binding energy to the characteristic scattering rate The position of two peaks and their amplitudes relative to the zero-bias conductance allows us to determine the relevant parameters and the asymmetry A with the error bar . The doping dependence of agrees with that found from the uniform magnetic susceptibility and explained by us [23] (see also [24]) as the result of the screening of the Fröhlich interaction by free carriers.


165

An essential feature of the bipolaron theory is the temperature-independent gap with the ratio

which might be quite different from the BCS one,

We

obtain a very large ratio, for both overdoped Fig. 4 and optimally doped Fig. 5 samples. Such a large ratio is difficult to understand in the framework of the BCS theory,

including its canonical strong-coupling extension. Renner et al. [7] emphasized that the evolution of the STM spectra with temperature was very different from the classic BCS behavior, both s and d wave. The “superconducting” gap was found temperature independent evolving into the “normal” gap above In our theory, this is one and the same gap, which is the bound energy of real-space bipolarons. It does not disappear at any temperature, neither at nor at The theoretical evolution of the spectrum with temperature, Fig. 5 (insert), reproduces the experimental features well. In particular, the zero-bias conductance increases with temperature and there is no sign that the gap closes at a given temperature, in agreement with the experiment [7]. The latter observation rules out any role of superconducting phase or spin fluctuations in the normal gap. We notice that the theoretical peaks shift to higher energies above Fig. 5 (insert), as observed (see Fig. 2 in Ref. [7]). There is some characteristic voltage (binding energy) (Figs. 4 and 5), above which the experimental STM and PCT conductance deviate from the theoretical one. We believe that a hump observed above is due to a polaronic cloud, as discussed in Ref. [2]. The highfrequency phonons and magnetic fluctuations contribute to the excess spectral weight with a maximum around twice of the Franck-Condon shift. A loss of the spectral weight at is explained by the electron-collective excitation coupling as suggested by Shen and Schrieffer [9] for PES and discussed by DeWilde et al. [81 for STM and PCT. We have shown earlier that a similar dip structure appears in the electronic DOS as a result of the electron-Einstein phonon interaction [25]. The present theory of tunneling and PES can be generalized to describe SIS junctions, c axis current at high voltage and the angle-resolved photoemission (ARPES). In particular,

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ARPES can be described with the spectral function A(k, E) determined numerically [21]. Although such a feature of ARPES as the normal state gap is understood within the present analysis, the k dispersion is presented elsewhere.

4. CONCLUSION

In summary, we would like to outline the main results: First, A simple estimate of the polaronic level shift shows that the electron-phonon coupling is more than sufficient to bind two polarons into a small mobile bipolaron. For the Fröhlich interaction, one estimates the polaron level shift as The hole-hole coupling via phonons is much stronger than the magnetic coupling, 2. The ground state of superconducting cuprates is the Bose-Einstein condensate of peroxy bipolarons with the d-wave symmetry owing to the bipolaron band dispersion. 3. The single-particle spectral function is derived for cuprates that describes the spectral features observed in tunneling and photoemission. In particular, the temperature independent gap and the anomalous ratio, injection/emission asymmetry both in magnitude and shape, zero-bias conductance at zero temperature, the spectral shape inside and outside the gap region, temperature/doping dependence, and dip-hump structure of the tunneling

conductance are described.

ACKNOWLEDGMENTS The author greatly appreciates enlightening discussions with A. R. Bishop, B. Brandow,

J. T. Devreese, O. Fisher, V. V. Kabanov, H. Kamimura, P. E. Kornilovitch, G. J. Kaye, W. Y. Liang, Ch. Renner, S. G. Rubin, J. R. Schrieffer, A. Simon, Z.-X. Shen, G. Zhao, and K. R. A. Ziebeck. REFERENCES 1. V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B56, 6120 (1997).

2. A. S. Alexandrov and N. F. Mott, Rep. Prog. Phys. 57 1197; “High Temperature Superconductors and Other Superfluids,” Taylor and Francis, London (1994); “Polarons and Bipolarons.” World Scientific, Singapore (1995). 3. A. S. Aleksandrov and A. B. Khmelinin, Fiz. Tverd. Tela (Leningrad) 28, 3403 (1986) (Sov. Phys. Solid State

4. 5. 6. 7. 8.

28, 1915(1986)). A. S. Alexandrov, J. Phys.: Condens. Matt. 8, 6923 (1996); Physica C 274, 237 (1997). A. S. Alexandrov, Phys. Rev. B 53, 2863 (1996). P. W. Anderson, Phys. World 9, 16 (1997); N. P. Ong and P. W. Anderson, Phys. Rev. Lett. 79, 4718 (1997). Ch. Renner et al., Phys. Rev. Lett. 80, 149 (1998). Y. DeWilde et al., Phys. Rev. Lett. 80, 153 (1998).

9. Z.-X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997).

10. J. Annett, N. Goldenfeld, and A. J. Legget, in Physical Properties of High Temperature Superconductors, Vol. 5, edited by D. M. Ginsberg World Scientific, Singapore, 375 (1996). 1 1 . B. Brandow, Phys. Repts. 296, 1 (1998). 12. D. A. Wollman et al., Phys. Rev. Lett. 71, 2134 (1993); C. C. Tsuei et al., Phys. Rev. Lett. 73, 593 (1994);

J. R. Kirtley et al., Nature 373, 225 (1995); C. C. Tsuei et al.. Science 272, 329 (1996). 13. D. J. Scalapino, Phys. Repts. 250, 331 (1994) and references therein. 14. H. Kamimura et al., Phys. Rev. Lett. 77, 723 (1996).

15. X. Zhang and C. R. A. Catlow, J. Mater. Chem. 1, 233 (1991).


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16. N. L. Allan and W. C. Mackrodt, Advances in Solid-State Chemistry, Vol. 3, ed. C. R. A. Catlow (London: JAI Press) (1993). 17. C. R. A. Catlow, M. S. Islam and X. Zhang, J. Phys.: Condens. Matter 10, L49 (1998). 18. A. S. Alexandrov, M. P. Kazeko, and S. G. Rubin, Zh. Eksp. Teor. Fiz. 98, 1656 (1990) (JETP 71, 1656 (1990)). 19. A. S. Alexandrov, invited talk at the Workshop on “Strongly Correlated Electrons” (Tellahassee, Florida,

11–14 March, 1998). 20. D. M. King et al., Phys. Rev. Lett. 73, 3298 (1994); K. Gofron et al., ibid. 3302 (1994). 21. B. I. Halperin, Phys. Rev. 139, A104 (1965).

22. 23. 24. 25.

H. L. Frisch and S. P. Lloyd, Phys. Rev. 120, 1175 (1960). A. S. Alexandrov, V. V. Kabanov, and N. F. Mott, Phys. Rev. Lett. 77, 4796 (1996). K. A. Müller et al., J. Phys.: Condens. Matter 10, L291 (1998). A. S. Aleksandrov (Alexandrov), V. N. Grebenev, and E. A. Mazur, Pis’ma Zh. Eksp Teor. Fiz. 45, 357 (1987) (JETP Lett. 45, 455 (1987)).


The Small Polaron Crossover: Role of Dimensionality M. Capone,1 S. Ciuchi,2 and C. Grimaldi3

The crossover from quasi–free electron to small polaron in the Holstein model for a

single electron is studied by means of both exact and self-consistent calculations in one dimension (1D) and on an infinite coordination lattice in order to understand the role of dimensionality in such a crossover. We show that a small polaron

ground-state occurs when both strong coupling and multiphonon 1) conditions are fulfilled, leading to different relevant coupling constants in adiabatic and anti-adiabatic region of the parameter’s space. We also show that the self-consistent calculations obtained by including the first electron– phonon vertex correction give accurate results in a sizeable region of the phase diagram well separated from the polaronic crossover.

1. INTRODUCTION

Recent optical measurements of the insulating parent compounds of the hightemperature superconductors show the presence of polaronic carriers [1], and evidence for intermediate and strong lattice distortions has been given also for the colossal mag-

netoresistance manganites [2] and nickel compounds [3]. The recent observation of onedimensional (1D) stripes in the high-temperature superconductors [4] and in manganites suggests a comprehensive study of the role of dimensionality in the polaronic crossover. A detailed study of the small polaron crossover is demanded also by the recent experimental results on manganites [5]. 1

I.N.F.M. and International School for Advanced Studies, SISSA-ISAS, Trieste, Italy 34013. Dipartimento di Fisica, Universitá de L’Aquila, via Vetoio, 67100 Coppito-L’Aquila, Italy and I.N.F.M., Unitá de L’Aquila. 3 I.N.F.M., Unitá di Roma 1, Dipartimento di Fisica, Universitá di Roma “La Sapienza.” P.le A. Moro 2, 00185

2

Roma, Italy. Present address: École Polytechnique Fédérale de Lausanne, DMT-IPM, CH-1015 Lausanne,

Switzerland.


169

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Capone, Ciuchi, and Grimaldi

The polaronic state is characterized by strong local electron–lattice correlation and is a nonperturbative phenomenon. It therefore cannot be described by simple summation of the perturbative series such as the one that defines the Migdal–Eliashberg (ME) theory. Here, we provide a detailed study of the crossover that occurs at intermediate electron– lattice couplings from quasi–free electron to small polaron ground-state, with a particular emphasis on the role of system dimensionality. We consider the simple Holstein molecular-crystal Hamiltonian for a single electron, which reads:

where and are, respectively, the destruction (creation) operators for an electron and for a dispersionless phonon of frequency on site i. The Hamiltonian (1) represents a nontrivial many-body problem, and has been already studied in recent years by means of numeric [6–9] and analytic [10–12] techniques. Two dimensionless parameters are introduced to measure the strenght of electron– phonon (el-ph) interaction: and where is the halfbandwidth for the free electron and d is the system dimensionality. is originally introduced in the weak coupling pertubation theory and is the coupling parameter of a ME approach in the case of one electron. It can also be viewed as the ratio between the small polaron energy and the free-electron energy The parameter α is the relevant coupling in the atomic limit measures the lattice displacement associated to the polaron and

In this limit, α is the average num-

ber of phonons bound to the electron. According to the Lang–Firsov results followed by the Holstein approximation, α also rules the reduction of the effective hopping [9,13]. Besides and α , the el-ph system described by Eq. (1) is governed also by another

dimensionless parameter: It measures the degree of adiabaticy of the lattice motion (lattice kinetic energy compared to the electron one (electron kinetic energy In the adiabatic regime is a condition sufficient to give a polaronic state because the electron is bound to the slowly moving lattice giving rise to a strong enhancement of effective mass. In the antiadiabatic regime such a picture is no longer true due to the fast lattice motion. In this case, polaronic features such as strong

local electron–lattice correlations arise only when the electron is bound to a large number of phonons To summarize, in both adiabatic and antiadiabatic regimes, a polaronic state is formed when both and inequalities are fulfilled [9]. This conclusion

is in contrast with ref. [10], in which it is argued that

is the only condition for small

polaron formation.

The parameter influences also the dependence of the behavior of the el-ph coupled system on the system dimensionality. We shall show that in the antiadiabatic regime the small polaron formation does not depend on the system dimensionality. However, dimensionality plays a crucial role in the adiabatic regime This can be traced back to the adiabatic limit In fact, in the ground-state is localized for any finite value of and a crossover occurs between large and small polaron at whereas for it has been shown that a localization transition occurs at finite from free electron to small polaron [14]. The different adiabatic behaviors between 1d and 2d systems could be relevant

The Small Polaron Crossover

171

to describe the motion of polarons as defects on top of 1d charge-striped structures, such as those observed in cuprates [4] and manganites [2]. 2. RESULTS We study the relevance of

and of the lattice dimensionality d by using two

alternative exact calculations: exact diagonalization of small, 1D clusters (ED-1d) and dynamical mean field theory (DMFT-3d). In the ED-1d approach, the infinite phonon Hilbert space must be truncated to allow for a given maximum number of phonons per site In

order to properly describe the multiphonon regime (expecially in the adiabatic regime, in which a large number of low-energy phonons can be excited), we chose a cutoff of 20. This high value forced us to restrict our analysis to a four-site cluster in the strongcoupling adiabatic regime. In the weak-coupling regime and for larger phonon frequencies, a lower value of is needed, allowing us to consider larger clusters up to 10 to 12 sites. We checked that finite-size effects do not affect the crossover coupling because small-polaron formation is a local, high-energy process. The dynamical mean field theory approach can be seen as the exact solution of the

small polaron problem on an infinite coordination lattice [11]. The formulation of the DMFT requires the knowledge of the free-particle DOS. A semicircular DOS can mimic a three-dimensional (3D) case: In the following, we therefore refer to this approach to as DMFT-3d.

We calculate the exact ground state energy

obtained by means of ED-1d and

DMFT-3d and we compare the results with the self-consistent noncrossing (NCA) and

vertex corrected approximations (VCA). These two approximations are defined by the selfconsistent calculation of the electronic zero-temperature self-energy given below:

)

where

is the retarded fully renormalized single-electron Green’s function:

which will be determined self-consistently. From Eqs. (2,3), the ground-state energy

is

given by the lowest energy solution of Re The NCA approach amounts to compute by retaining only the 1 in the square brackets of Eq. (2). The VCA is instead given by the inclusion also of the second term in square brackets of Eq. (2), which represents the first vertex correction. This approach is formally similar to the approximation scheme used in the formulation of the nonadiabatic theory of superconductivity [15], and a comparison with exact results therefore provides also a test of reliability of such an approach for the

one-electron case. In Fig. 1 we compare the ground-state energy

obtained by ED-1d with the NCA

and VCA results. The same quantities evaluated in the DMFT-3d case are shown in Fig. 2. We have chosen the same half-bandwidth D in both DMFT-3d and ED-1d. In the adiabatic

172


The Small Polaron Crossover

173

regime, the agreement of both approximations with exact results strongly depends on the system dimensionality as a result of the different low-energy behavior of the DOS. In fact, moving from before the crossover the agreement of the selfconsistent calculations with the exact results is improved for the 1d case (Fig. 1), whereas it becomes poorer for the 3d case (Fig. 2). However, the VCA approach represents a significant improvement with respect to the NCA for every system dimensionality and over a significant

range of parameters. As is seen from the comparison of Figs. 1 and 2, for large both approximate and exact results tend to become independent of dimensionality. This can be understood by realizing that in this regime, the system can be thought as a flat-band “atomic” system in interaction with high-energy phonons. It is also clear from Figs. 1 and 2 that both the selfconsistent NCA and VCA calculations deviate from the exact results when the crossover toward the small polaron regime is approached. A complete comparison between the exact results and the VCA approach in the parameter space is shown in Fig. 3. We explicitly evaluated both in 1d (Fig. 3a) and

3d (Fig. 3b) the relative difference

, where

and are the ground-state energies evaluated by exact techniques and the VCA, respectively. To analyze the region in the parameter space in which the VCA agrees within a

given accuracy with the exact results, in Fig. 3 we report lines of constant

174


As already mentioned, the agreement between self-consistent approximations and exact results is sensible to dimensionality. For , approaching the adiabatic limit and for small couplings the electron tends to be free. For this reason, self-consistent approximations work well. On the contrary, in the adiabatic limit and for , the ground-state is a localized large polaron and self-consistent approximations fail to predict its energy. In general, VCA (and so NCA) works well outside the polaron region whatever polarons are, either small or large. This can be seen directly from Fig. 3, in which the critical coupling of the crossover to small polaron is depicted as a dotted line. The critical coupling is defined as the value at which has maximum slope. In the same figures, we provide also an estimate of the width of the crossover (shaded areas) obtained by looking at the maximum slope of We checked that different criteria—for example, the effective mass enhancement [11]—provide the same qualitative results.

3. CONCLUSIONS In conclusion, we have shown that the crossover towards the small polaron state depends strongly on the adiabaticity parameter In the antiadiabatic regime, the crossover is ruled by and is independent of the system dimensionality. In the adiabatic regime the relevant coupling is and the crossover occurs from large to small polaron in 1d, whereas in 3d the crossover is from quasi–free electrons to small polarons. In the latter case, self-consistent approximations work better than in 1d systems. We have also shown that self-consistent calculations provide ground-state energies that agree well with exact results outside the small and large polaron region of the phase diagram and that such an agreement is increased when vertex corrections are taken into account. ACKNOWLEDGMENTS

We thank M. Grilli, F. de Pasquale, D. Feinberg and L. Pietronero for stimulating discussions. C. G. acknowledges the support of a I.N.F.M. PRA project. REFERENCES 1. P. Calvani, M. Capizzi, S. Lupi, P. Maselli, A. Paolone, and P. Roy, Phys. Rev. B 53, 2756 (1996).

2. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996); P. G. Radaelli, D. E. Cox, M. Marezio, and S-W. Cheong, Phys. Rev. B 55, 3015 (1997). 3. P. Wochner, J. M. Tranquada, D. J. Buttrey, and V. Sachan, Phys. Rev. B 57, 1066 (1998). 4. A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, A. Kvick, and P. G. Radaelli, Phys. Rev. B 54, 4310

(1996); A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishitara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996). 5. A. Lanzara, N. L. Saini, M. Brunelli, F. Natali, A. Bianconi, P. G. Radaelli, and S.-W. Cheong, Phys. Rev. Lett. 81, 878 (1998). 6. F. Marsiglio, Physica C 244, 21 (1995). 7. H. De Raedt and A. Lagendjick, Phys. Rev. B 27, 6097 (1983); ibid. 30, 1671 (1984). 8. G. Wellein and H. Fehske, Phys. Rev. B 56, 4513 (1997).

9. M. Capone, W. Stephan, and M. Grilli, Phys. Rev. B 56, 4484 (1997). 10. A. S. Alexandrov and V. V. Kabanov, Phys. Rev. B 54, 1 (1996). 11. S. Ciuchi, F. de Pasquale, S. Fratini, and D. Feinberg, Phys. Rev. B 56, 4494 (1997). 12. Y. Zhao, D. W. Brown, and K. Lindenberg, J. Chem. Phys. 100, 2335 (1994). 13. D. Feinberg, S. Ciuchi, and F. de Pasquale, Int. J. Mod. Phys B 4, 1317 (1990).

14. V. V. Kabanov and O. Yu. Mashtakov, Phys. Rev. B 47, 6060 (1993). 15. C. Grimaldi, L. Pietronero, and S. Strässler, Phys. Rev. Lett. 75, 1158 (1995).

CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas V. Cataudella,1 G. De Filippis,2 and G. Iadonisi1

A simple approach to the many-polaron problem for both weak and intermediate electron–phonon coupling and valid for densities much smaller than those typical of metals is presented. Within the model, the collective excitation spectrum is studied pointing out the presence of charge density wave instabilities in a finite range of the electron densities. Finally, preliminary results on the optical

absorption of an interacting large polaron gas are presented.

Several experiments on the optical response of the cuprates have been reported since the late 1980s, both in the normal and in the superconducting phases [1]. The large amount of conductivity data in the normal phase has shown an important infrared absorption that has been interpreted as an indication of the presence of small and/or large polarons, both in electron-doped and in hole-doped compounds, with features depending strongly on the doping and the temperature [2]. From a theoretic point of view, the formation of large polarons and bipolarons in polar materials has been studied quite extensively [3]. However, a large amount of work has been

devoted to the simpler single-polaron and bipolaron problems neglecting the effects of the polaron–polaron interaction that, instead, are expected to play an important role in highsuperconductors . In the present paper, we focus our attention on a model of interacting Fröhlich large polarons, showing the presence of charge density wave (CDW) instabilities in a finite range of particle density when the electron–phonon (e-ph) interaction is sufficiently large [4]. However, for low-charge carrier densities and/or small values of the e-ph coupling constant a gas of large polarons describes correctly the interacting electron–phonon system. In this region, we present preliminary results for the normal state conductivity [5], 1 2

Dipartimento di Scienze Fisiche, Università di Napoli I-80125 Napoli, Italy. Dipartimento di Scienze Fisiche, Università di Salerno I-84081 Baronissi (Salerno), Italy.



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Cataudella, De Filippis, and Iadonisi

including the many-body effects, and we show that

exhibits features similar to those

observed in the infrared spectra of the cuprates. THE MODEL We consider a system made of electrons interacting with nondispersive LO phonons and repelling each other through the Coulomb potential screened by the background high frequency dielectric constant [6]. The e-ph interaction is assumed to be

where

is the

e-ph matrix element [7].

In the above expressions, indicate, respectively, the annihilation (creation) operators for electrons and phonons, V is the system’s volume, is the Fröhlich

e-ph coupling constant, is the polaron radius, and is the LO phonon frequency. In order to study the many polaron effects for weak and intermediate values of the e-ph coupling constant, we must go beyond the perturbative approach [6]. To this aim, we adopt the following model. First we consider the Hamiltonian of two electrons interacting with the longitudinal optical phonons via the Fröhlich coupling and repelling each other through the Coulomb force

where are the position and momentum of the center of mass of the pair and of the relative particle and M and are the total and reduced masses, respectively. Then we obtain an effective potential for the two electrons of Eq. (3) eliminating, as we review briefly, the phonon degrees of freedom. Finally, the many-body effects are taken into account, considering many electrons interacting with each other through the obtained effective potential.

The problem described by Eq. (3) has been studied extensively [9]. A variational solution and the effective potential can be obtained as follows. Because the Hamiltonian in

Eq. (3) commutes with the total momentum of the system

CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas 177 we can eliminate the electronic variable r through the unitary transformation

where is the eigenvalue of Following Lee–Low–Pines [8] (LLP), we choose the variational trial ground state for the transformed Hamiltonian

where

is the vacuum of

The envelope function

and the operator

is given by

is chosen to be an hydrogenic-like radial wave function

The phonon distribution functions are determined in a self-consistent way from a functional variational procedure [9,10]. In particular, the Eulero–Lagrange equations for the functions can be solved exactly and the parameters and of the pair envelope function fixed by imposing the total energy to be at a minimum. Within this variational approach, it is possible to obtain an effective e-e potential due to the exchange of virtual phonons [10]

It contains a short-range attractive term and a long-range repulsive term screened at large distances by the static dielectric constant and, in the opposite limit, by the background high-frequency dielectric constant [9,10]. In particular, when tends to the self-energy of two free polarons in the LLP approximation [8]. It is worthwhile to note

that the proposed approach can be, in principle, improved if one chooses better and better estimates for the effective potential. This procedure allows us to eliminate the phonon degrees of freedom from the system, simplifying the treatment of many electron effects, and to investigate larger values of the e-ph coupling constant with respect to the perturbative approach proposed by Mahan [6]. This happens because the LLP transformation gives rise to phonon corrections to the bare e-ph vertex [11], corrections that can be neglected according to the Migdal theorem only when the Fermi energy is much larger than (normal metals).

178


The model proposed is studied within the R.P. A. [12] and Hubbard [13] approximation at T = 0 . Within these approximations the effective interaction between the electrons takes the form

where is the lowest order proper polarization propagator. The function f ( q ) takes the value 0 in the R.P.A. and

in the Hubbard approximation. The collective excitation frequencies of this system are determined by the poles of the retarded density correlation function, which occur at the solutions equation

of the

In Fig. 1, we present the numeric results for the collective energy mode in the Hubbard

approximation. goes from to the roots of an electron gas screened by the background high-frequency dielectric constant for large values of q and softens for a critical wave vector indicating strong correlation between the electrons. If the attractive potential is sufficiently strong, the collective energy softens completely. This

softening, which is present in a finite range of densities indicates that the system becomes unstable with respect to the formation of the CDW. The parameters chosen are and It is well known that the retarded dielectric function is analytic in the upper half of the complex plane provided There are examples of physical systems for which the linear response function violates the causal requirement: In this case, the linear retarded dielectric function can no longer describe the behavior of the system correctly. It has been shown that the temperature-dependent correlation function in the R.P.A. for an interacting many-particle system, when the interaction is sufficiently attractive, has forbidden zeros on the imaginary axis of the complex frequency plane [15,16]. In our model, if the value of the coupling constant α is sufficiently large, the linear response function possesses a pair of imaginary poles. It is interesting to note that the same type of instability has been suggested by Di Castro et al. [17] for a large class of systems of interacting electrons and phonons as due to very ineffective electron corrections to the e-ph vertex. In Fig. 2, we report the charge density and the strength of the attractive term of at which we observe the softening of the collective energy mode We see that there is a wide region where CDW instability sets in. Therefore, a gas of large polarons

CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas

179

provides a useful description of a system of interacting electrons and phonons only for small values of the e-ph coupling constant and/or low-charge carrier densities.

OPTICAL PROPERTIES In the range of values of α where a gas of large polarons is well defined, it is possible to study the normal-state conductivity within the R.P.A. approximation, describing the polarons through the spectral weight function of the Feynman model [18]

where and are the Bessel functions of complex argument. The dimensionless parameters and are related to the mass and the elastic constant of the model, in which the electron is coupled via a harmonic force to a fictitious second particle simulating the phonon degrees of freedom. The values of can be obtained by the variational approach described by Feynman [18] and Schultz [19].

180


We note that the model proposed in this paper restores, when the well-known results of the optical absorption of a single large polaron [20,21] and allows to introduce within the R.P.A. approximation the effects of the polaron–polaron interaction.

RESULTS

In Fig. 3 is reported the optical absorption per polaron as a function of the frequency for different values of the charge carrier density at Three different structures appear in the normal state conductivity: (a) a zero-frequency delta function contribution; (b) a strong band starting at that is the overlap of two components: a contribution from the

intraband process and a peak due to the polaron transition from the ground-state to the first relaxed excited state; and (c) a smaller band at higher frequency due to the Frank–Condon transition of the polaron.

Increasing the charge carrier density, we find that the large polaronic band due to the excitation involving the relaxed states (b contribution) tends to move toward lower frequencies, whereas its intensity decreases in favor of the rise of a Drude-like term around This behavior is in agreement with the experimental data on the normal-state conductivity

of many cuprates both in the insulating and in the metallic phases [22,23].

CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas 181

REFERENCES 1. T. Timusk and D, B. Tanner, in Physical Properties of High Temperature Superconductors, edited by D. M. Ginsberg (World Scientific, Singapore, 1989) and references therein. 2. C. Taliani et al., High- Superconductors, edited by A. Bianconi and A. Marcelli (Pergamon Press, Oxford), 1989. 3. J. T. Devreese, Polarons, in Encyclopedia of Applied Physics, edited by G. L. Trigg (New York: VCH, vol. 14,

p. 383), and references therein, 1996. 4. G. De Filippis, V. Cataudella, G. Iadonisi, unpublished. 1998. 5. V. Cataudella, G. De Filippis, G. Iadonisi, Phys. Rev. B 60, 15163 (1998); Phys. Rev. B 62, 1496 (2000). 6. G. D. Mahan, Many-Particle Physics (Plenum, New York, 1981), Chap. 6, p. 545. 7. H. Fröhlich et al., Philos. Mag. 41,221 (1950); H. Fröhlich, in Polarons and Excitons, edited by C. G. Kuper and G. A. Whitfield (Oliver and Boyd, Edinburg, 1963), p. 1.

8. T. D. Lee, F. Low, and D. Pines, Phys. Rev. 90, 297 (1953). 9. V. Cataudella, G. Iadonisi, and D. Ninno, Physica Scripta. T39, 71 (1991); F. Bassani, M. Geddo, G. Iadonisi, and D. Ninno, Phys. Rev. B 43, 5296 (1991). 10. G. Iadonisi, M. Chiofalo, V. Cataudella, and D. Ninno, Phys. Rev B 48, 12966 (1993); G. Capone, V. Cataudella, G. Iadonisi, and D. Ninno, Il Nuovo Cimento 17D, 143 (1995). 11. W. van Haeringen, Phys. Rev. 137, 1902 (1965). 12. J. K. Lindhard, Dan. Vidensk. Selsk. Mat. Fys. Medd. 28, 8 (1954). 13. J. Hubbard, Proc. R. Soc. London Ser. A 243, 336 (1957).

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14. D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1963).

15. N.D. Mermin, Ann. Phys. 18, 421, 454(1962). 16. L. P. Kadanoff and P. C. Martin, Phys. Rev. 124, 670 (1961). 17. M. Grilli and C. Castellani, Phys. Rev. B 50, 16880 (1994); C. Castellani, C. Di Castro, and M. Grilli, J. Supercond. 9, 413 (1996).

18. R. P. Feynman, Phys. Rev. 97, 660 (1955). 19. T. D. Schultz, Phys. Rev. 116, 526 (1959); T. D. Schultz, in Polarons and Excitons, edited by C. G. Kuper and

G. A. Whitfield (Oliver and Boyd, Edinburg, 1963), p. 71. 20. R. P. Feynman et al., Phys. Rev. B 127, 1004 (1962).

21. F. M. Peeters and J. T. Devreese, Phys. Rev. B 28, 6051 (1983). 22. P. Calvani et al., Phys. Rev. B 53, 2756 (1996). 23. S. Uchida et al., Phys. Rev. B 43, 7942 (1991).

The Charge-Ordered State from Weak to Strong Coupling S. Ciuchi1 and F. de Pasquale2

We apply the dynamical mean field theory to the problem of charge ordering. In the normal state as well as in the charge-ordered (CO) state, the existence of polarons (i.e., electrons strongly coupled to local lattice deformation) is associated to the qualitative properties of the lattice polarization distribution function (LPDF). At intermediate and strong coupling, a CO state characterized by a certain

amount of thermally activated defects arises from the spatial ordering of preexisting, randomly distributed polarons. Properties of this particular CO state give a qualitative understanding of the low frequency behavior of optical conductivity of Ni perovskites.

1. INTRODUCTION There has been a renewed interest for the charge-ordering transition that has been found associated to lattice displacements in cuprates nickelates and manganites.

Charge-stripes order has been detected in neodimium-doped cuprates [1] and conjectured in LASCO [2]. X-ray studies of BISCO also shown a modulated structure of CuO planes [3]. Commensurate charge order appears in doped nickelates [2] also related with peculiar magnetic properties [4], and finally, large lattice distortions have been found in manganites, which can be associated to either commensurate or incommensurate charge ordering [5]. The amplitude of such a distortion increases from cuprates to manganites, a fact that may support the hypothesis of an increasing charge-lattice

interaction. Another important observation supporting the presence of polaronic carrier,

1

Dipartimento di Fisica and INFM UdR l’Aquila, Università de L’Aquila, via Vetoio, I-67100 Coppito-L’Aquila, Italy. 2 Dipartimento di Fisica and INFM UdR Roma 1, Università di Roma “La Sapienza,” piazzale Aldo Moro 5, I-00185 Roma, Italy.


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and therefore an intermediate–strong local charge-lattice interaction, is the presence of a polaronic peak in the MIR band of doped Ni compound [6, 7]. The aim of this paper is to show how nonperturbative results obtained in the framework of the dynamical mean field theory (DMFT) can be helpful to understand the low-energy behavior of the optical conductivity in Ni perovskites. We first summarize the results of our theory of the charge-ordered (CO) state, then we show a calculation of the optical conductivity that is in qualitative agreement with experimental observation in charge-ordered Ni perovskites. In the weak coupling case, the CO transition is the well-known charge density

wave (CDW) instability of the Fermi liquid. Whereas in strong coupling there have been several attempts to understand the CO state based on mean field approach on strong coupling

effective Hamiltonians [8] and studies of the ground-state [9]. To discuss the CO state in a nonperturbative fashion, we consider the simplest model of local electron–lattice interaction, i.e., the Holstein molecular crystal model [10]

where

creates (destroys) an electron at site i, and

are the local oscillators

displacements and momentum. Electrons and phonons are coupled via the density fluctuations (n being the average electron density). The electrons move on a bipartite lattice of connectivity z and have a band of half-bandwidth t. The main approximation we consider is the adiabatic approximation which can be obtained in the limit In this limit we neglect the first term in Eq. (1), therefore are constant of motion and can be replaced by c numbers. This approximation turns out to be valid if the following two conditions hold: 1. 2.

As far as thermodynamic properties are concerned, temperatures must be greater than the typical phonon energy scale As far as spectral properties are concerned, energies must be greater than the typical phonon energy scale

However, adiabatic limit allows us to solve the model with a little amount of numerics giving the spectral properties of electrons in real frequencies and statistical properties of the lattice. We consider also spinless electrons to account for a polaronic rather than a bipolaronic ground-state at large couplings. This restriction, even if at a very rough level, mimics the action of an on-site Coulomb repulsion.

We apply the machinery of the DMFT, which is the exact solution of local-type interaction on an infinite coordination lattice (infinite dimensions) [14]. To have a nontrivial limit, a scaling of the hopping, as in Eq. (1), t with the number of neighbors is required. The DMFT approach maps the problem of locally interacting fermions on a lattice into a single site equivalent problem [14]. A detailed study of the Holstein model based on the Monte Carlo solution of the single site problem has been first carried out in Ref. [15]. This analysis has been extended to the spectral properties of the normal state in Ref. [ 1 1 ] by using the adiabatic limit to obtain an analytical solution of the single site model. We extend this analytical approach to the study of the CO state. We consider here alternate charge ordering in two interpenetrating sublattices, A and B. The quantity to be determined self-consistently is the lattice polarization distribution function (LPDF) P(X).

The Charge-Ordered State from Weak to Strong Coupling

185

Different regimes are related to qualitative changes in the shape of P(X). Our main results can be summarized by the self-consistent equations that determine the LPDF and the local electron Green function in each sublattice

Equations (2) and (3) are obtained in the simple but nontrivial case of Bethe lattice of bandwidth 2t. From Eq. (3), we see that the Green function is that of a particle propagating

in a randomly distorted sublattice, and sublattice A is coupled to B and vice versa. The real frequency representation of the (retarded) Green function is simply obtained by substituting in Eq. (3); therefore, the adiabatic limit allows us to obtain the spectral properties in real frequencies. 2. RESULTS We have solved the self-consistent scheme introduced in the previous section by numeric iteration procedure. We consider the spinless electron half filled case, i.e., one electron for each two sites . We start with an ansatz for the sublattice Green function, then we get the function P at discrete points through Eq. (2) and through a numeric integration

[Eq. (3)], we obtain the new G.

In the adiabatic limit, only one relevant coupling parameter measures the electron– lattice interaction It can be expressed as the ratio of self-trapping energy (polaron energy to the electron kinetic energy energy (t). A crossover from strong to weak coupling behavior is expected around 1 [12, 13]. These expectations are confirmed by the the phase diagram at half-filling shown in Fig. 1. The continuous curve represents the CO critical temperature as a function of the coupling strength. The dashed line marks the normal to polaron crossover in the normal phase [11] and the crossover from weak

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coupling CO (A) to strong coupling CO (B). In both normal and ordered states, a crossover

line separates the monomodal and the bimodal behavior of the LPDF. In the ordered state, bimodality appears in the sublattice LPDF. Results are summarized in Fig. 2. The typical weak coupling behavior of LPDF across the transition temperature is shown in Fig. 2a. Upon decreasing the temperature, a uniform polarization of a given sublattice arises. The other sublattice, whose LPDF is not shown in the figure, develops an opposite polarization so that the net total polarization is zero as should be for the Hamiltonian Eq. (1), which couples deformation and density fluctuations. The variance decreases with decreasing temperature.

Moving to polaronic nonordered state, the LPDF is clearly bimodal and symmetric at half-filling, as seen from Fig. 2c (dashed lines). Upon decreasing the temperature below

the sublattice LPDF unbalances in favor of a net sublattice polarization but still remains bimodal. The weight of the secondary peak decreases by a further decreasing of temperature. We explain this secondary peak as due to temperature-activated defects in the CO state [9]. The bimodal behavior of LPDF, which is clear at very large coupling, becomes less pronounced at intermediate coupling (see Fig. 2b). In this region in both the nonordered and the ordered states near we observe that a nonnegligible amount of sites are nearly undistorted. In this intermediate region of the coupling we observe a strong non-Gaussian behavior of LPDF. Even if a secondary peak is present in the normal


187

phase, it is not well pronounced in this region. As the temperature is lowered below it may happen that this secondary peak is washed out in the ordered phase but a pronounced

shoulder remains in the LPDF. It eventually develops again a secondary peak upon a further decrease in temperature. The optical conductivity is obtained once the local Green functions of the two sublattices are known by a generalization of the Kubo formula (details of the calculation are presented elsewhere). We show in Fig. 3 the results obtained at half-filling for three different values of the coupling constant characteristic of small, intermediate, and large couplings. We see that at small coupling (Fig. 3a) the optical conductivity of the normal state shows a peak

at

reminiscent of the classical Drude behavior. This peak is shifted in the CO state at where is the CDW gap. The position of the peak depends on temperature and is shifted toward higher energies as the temperature decreases following the enhancing of the order parameter. As the coupling is increased (Fig. 3b), a peak at is still present also at indicating the presence of polarons in the disordered phase. We notice, however, a shift of the spectral weight from low to high energies as the temperature is decreased. This effect is less evident at stronger couplings (Fig. 3c) because in this case almost all sites are polarized (see. Fig. 2c), and consequently we have no appreciable spectral weight at low energy. In any case, whenever polarons are present in the nonordered state, a shift toward

larger energies of the spectral weight and a change in the temperature dependence of the amplitude of the peak is observed in the ordered state.


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A strong coupling approximation, equivalent to Reik’s approximation [16] at high temperatures [ 17], gives the interpretation of this spectral weight shift in terms of the number of defects in the CO state. In this approximation the ratio of the optical conductivity in the CO state to the same quantity in the normal state is

In Eq.(4) is number of defects in the CO state that depends on is a function of T only.

whereas

From this formula, it is easy to obtain a spectral weight shift proportional to from low to high energy. It is worthwhile to note that the shift of the spectral weight from low to high energy is a phenomenon actually observed in the Ni oxides together with an anomalous ratio Both observations are consistent with an intermediate coupling CO of polarons.

3. CONCLUSIONS The crossover from weak to strong coupling CO have been studied in details by introducing an LPDF. The qualitative change from monomodal to bimodal behavior of this function is interpreted as existence of defects in the ordered phase. In terms of the defects activation, we obtain a qualitative understanding of the shift from low to high frequency in the spectral weight observed below the CO transition in Ni perovskites. ACKNOWLEDGMENTS

The authors acknowledge useful discussions with D. Feinberg and P. Calvani. REFERENCES 1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 563 (1995). 2. J. M. Tranquada, preprint cond-mat/9802043. 3. A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, A. Kvick, and P. G. Radaelli, Phys. Rev. B 54, 4310 (1996); A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishitara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996).

4. P. Worchner, J. M. Tranquada, D. J. Buttrey, and V. Sachan, Phys. Rev. B 57, 1066 (1998). 5. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996); P. G. Radaelli, D. E. Cox, M. Marezio, and S.-W. Cheong, Phys. Rev. B 55, 3015 (1997).

6. P. Calvani, P. Dore, S. Lupi, A. Paolone, P. Maselli, P. Giura, B. Ruzicka, S.-W. Cheong, and W. Sadowsky, J. Supercond. 10 (1997) 193; P. Calvani, A. Paolone, P. Dore, S. Lupi, P. Maselli, P. C. Medaglia, and S.-W. Cheong, Phys.. Rev. B 54 R9592 (1996). 7. T. Katsufuji, T. Tanabe, T. Ishigawa, Y. Fukuda, T. Arima, and Y. Tokura, Phys. Rev. B 54 R 14230 (1996).

8. R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. 62 113 (1990).


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9. S. Aubry and P. Quemerais, “Breaking of analiticity in charge-density wave systems physical interpretation and consequences,” in Low Dimensional Electronic Properties of Molybdenium Bronzes and Oxides, edited by C. Schlenker (Kluwer Academic Publishers, Dordrecht, Boston, London, 1987). 10. T. Holstein, Ann. Phys. 8, 325 (1959), ibid. 343 (1959). 11. A. J. Millis, R. Mueller, and B. I. Shraiman, Phys. Rev. B 54, 5389 (1996). 12. S. Ciuchi, F. de Pasquale, S. Fratini, and D. Feinberg, Phys. Rev. B 56, 4418 (1997). 13. M. Capone, S. Ciuchi, and C. Grimaldi, Europhys. Lett. 42, 523 (1998). 14. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 15. J. K. Freericks, M. Jarrel, and D. J. Scalapino, Phys. Rev. B 48, 6302 (1993). 16. H. G. Reik, Z. Phys. B 203, 346 (1967). 17. D. Emin, Phys. Rev. B 48, 13692 (1993).


Low-Temperature Phonon Anomalies in Cuprates T. Egami,1 R. J. McQueeney,2 Y. Petrov,3 G. Shirane,4 and Y. Endoh5

Our inelastic neutron scattering measurement on single crystals shows that the in-plane LO phonon dispersion at low temperature is incompatible with the current view on the dynamic charge stripes, which for this composition should have the periodicity of 4a. Instead, the results are consistent with the dynamic stripes with the periodicity of 2a, half of what is expected and a quarter of the magnetic periodicity. Calculations with the two-band model suggest that such 2a stripe charge ordering may help hole pairing.

1. INTRODUCTION For some time, the electron–lattice coupling has been considered to play a negligible role in the superconductivity of cuprates, and most theories have focused on magnetic mechanisms. However, the observation of the static spin-charge stripes in nonsuperconducting opened up the possibility that the role of the lattice is more important than has been previously thought [1]. Currently, the prevailing thought is that the charge-spin stripes exist even in the superconducting systems, but are dynamic and short range. The well-known dynamic incommensurate magnetic peaks observed by inelastic neutron scattering are supposed to be the signature of magnetic stripes, but the corresponding lattice signature of charge stripes has not been observed. The initial aim of this work has been to observe the dynamic charge-spin stripes by studying high-energy 1

Department of Materials Science and Engineering and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104. 2 Los Alamos National Laboratory, Los Alamos, NM 87545. 3 Department of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104. 4

Physics Department, Brookhaven National Laboratory, Upton, NY 11973.

5

Department of Physics, Tohoku University, Sendai, 980 Japan.


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phonons, because for such phonons the stripes may look frozen in time. As it turned out,

we observed something related but decidedly different. Our data are not consistent with the current idea of the charge-spin stripes, but suggest that the periodicity of the dynamic charge stripes is half of what is currently believed. For the periodicity of the magnetic correlation determined from the dynamic magnetic satellites is

implies the periodicity of the charge stripes the charge periodicity of 2a.

This whereas our data are more consistent with

2. EXPERIMENTAL RESULTS We carried out inelastic neutron scattering measurements of the high-frequency bondstretching LO phonons of This phonon branch is known to show strong softening with hole doping [2]. Two single crystals were comounted in an aluminum can filled with He exchange gas. Both samples were grown by the floating zone method and were obtained from the same batch. The total size of the sample is approximately Previous measurements and characterizations attest to the high quality of this sample [3].

Low-Temperature Phonon Anomalies in Cuprates

193

The experiments were made on the HB-3 triple axis spectrometer at the high flux isotope

reactor (HFIR) at the Oak Ridge National Laboratory. The spectrometer configuration used the beryllium (002) and pyrolitic graphite (002) reflections as the monochromator and analyzer, respectively. The analyzer angle was fixed when performing inelastic scans, giving a fixed final energy of 14.87 meV. Soller collimators of angular divergence

were placed along the flight path from source to detector. To reduce higher-order Bragg scattering contamination from the analyzer, a pyrolitic graphite filter was placed before the analyzer. Some of the results of constant-Q energy scans taken in the

Brillouin

zone, in tetragonal notation are shown in Fig. 1 for and 0.2. Because of the large incident energy required for measurements up to 90 meV, the flux was low, and the count rates were The measurement in the Brillouin zone around (5, 0, 0) [2, 4] suffers from spurious scattering consisting of the (6, 2, 0) Bragg reflection from the sample scattering incoherently from the analyzer that obscured the main phonon branch. Thus, we stayed in the zone, even though the intensity here is significantly weaker than in the zone. In Fig. 1, the large peak at 58 meV is due to the oxygen in-plane Cu-O bond-bending mode. The 70 meV peak is associated with the oxygen Cu-O bond-stretching mode also in the plane. The energy scans at

various values of show that the frequency of the 70 meV branch remains constant from (3.5, 0, 0) to (3.25, 0, 0), below which its intensity diminishes rapidly. At the same time, some intensity appears at 85 meV, and becomes a strong peak below (3.25, 0, 0) down to (3,0,0). The peak positions are shown in Fig. 2. Thus, it appears that bond-stretching phonon branch has split into two nearly dispersionless subbranches, with the intensity crossover at (3.25, 0, 0).

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3. DISCUSSION The experimental results at T = 10 Kshown here are in conflict with the previous measurement of the bond-stretching branch in at room temperature [2]. The previous result shows a strongly dispersing, but continuous, single branch from (0, 0, 0) to (0.5, 0, 0), in contrast to the two remarkably dispersionless subbranches shown in the data presented here. The difference in these two measurements most likely arises from the difference in temperature, according to our preliminary measurements of temperature dependence. However, our results are similar to the ones observed for and The two peaks seen in Fig. 1 for resemble those observed for that provoked a dispute on the “extra” phonon branch [2, 6]. The claim of the extra phonon branch was later withdrawn, citing the possibility of compositional inhomogeneity, but such inhomogeneity is highly unlikely for the present sample, which

shows a very clean spin gap [3]. Thus, our observation revives the controversy over the extra branch. The dynamic magnetic satellites in this sample were observed at and with indicating the wavelength of magnetic periodicity is 8a

[3]. Thus, we expected the lattice signature of the charge-spin stripes at

and

Such dynamic superstructures create pseudo-Brillouin zone boundaries for

high-energy phonons at

and

The observed dispersion shown in Fig. 2 is consistent


195

with the pseudo-Brillouin zone boundary at but not at and Indeed a spring model created assuming the charge-spin stripes with the period of 4a shows a dispersion which is split at and and is qualitatively very different from the observed one as shown in Fig. 3. However, if we assume the charge-spin stripes with the periodicity of 2a (Fig. 4) instead of 4a, the calculated dispersion is in good agreement with the observation as shown in Fig. 5. It is interesting to note that in this case the charge must be on oxygen ions. The model with the charge on Cu showed poor agreement. Because we assumed a periodic structure in the simulation, the branches have some dispersion. If we introduced disorder and localization, the branches would have shown less dispersion. From the flatness of dispersion, we estimate the correlation length of charge ordering to be about 20 Å (5a) along the stripes and 8 Å (2a) across them. These correlation lengths define an area that contains just about one hole, because the linear charge density in the stripe is 1/4 per unit cell. Our earlier search for superlattice diffraction (elastic) at (0.5, 0, 0) was negative. Thus, we conjecture that the charge periodicity with is dynamic and short range. Our results cast a serious doubt on the conventional picture of charge-spin stripes obtained merely by extrapolating from the static stripe structure of nonsuperconducting compounds. Instead, our results suggest that although the dynamic charge stripes do indeed exist, they have the periodicity of 2a and are centered on oxygen ions. Such charge stripes interact strongly with the Cu bond-stretching half-breathing mode at (0.5, 0, 0). It is interesting to note that Harashina et al. [7] reported strange phonon behavior in for this mode in the vicinity of the (0.5, 0, 0) point around 31–33 meV. We observed that the

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corresponding mode in the present sample at 29 meV also shows anomalous temperature

dependence. This provides more evidence of the presence of the 2a charge periodicity. It is still possible that our data are consistent with the stripe periodicity of 4a, if we assume that the 2a charge periodicity exists within the stripe. Because the linear charge density is 1/2 in the conventional stripe, the Peierls distortion in the spin-polarized state produces the lattice distortion with the periodicity of 2a, and charges are localized. However, our simulations for such a case failed to reproduce the strong crossover at and the dispersion appeared similar to the result in Fig. 3. Furthermore, this picture is inconsistent with the high conductivity of the system and the widely held assumption that spins are unpolarized within the stripes. Thus, in our view, this possibility is very remote.

The static stripes observed earlier were associated with the antiferromagnetic (AFM) domain boundaries [1, 8]. In order for the charge periodicity of 2a to be compatible with the magnetic periodicity of 8a, the spin rotation through the charge stripe must be rather than as in the static stripe, and the magnetic structure must be in a chiral AFM state with a phase slip of at every other Cu-Cu bond. The average linear charge density is about

1/4 per unit cell, rather than 1/2 as in the static stripes. This provokes interesting thoughts about the relationship among charge, spin of the charge, the magnitude of spin rotation, and chirality. This subject deserves a very detailed study. We expanded the model of Emery and Reiter [9] with the holes on oxygen ions,

and studied the effect of Cu half-breathing mode using the exact diagonalization method [ 10] . For a perfectly periodic

plane, the strongest hole-hole pairing occurs for oxygen


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ions separated by 2a along the Cu-O chain, followed by the pairs separated by a across the square. This is already suggestive of the possible relation between the stripes and

pairing. If a static half-breathing Cu mode was introduced by the frozen phonon approach, the 2a charge stripes are induced. Also, the strengths of the two kinds of pairs above are exchanged, making the intrastripe pairing (separated by a across the square) more favored compared to the interstripe pairing (separated by 2a along the Cu-O chain). These results suggest that the presence of the 2a charge stripes may enhance superconductivity in the plane.

ACKNOWLEDGMENTS

The authors are grateful to J. B. Goodenough, A. R. Bishop, J. Tranquada, V. Emery, J. Zaanen, H. Mook, E. Mele, H. Kamimura and M. Tachiki for useful discussions. The work at the University of Pennsylvania was supported by the National Science Foundation through DMR96-28134. HFIR is operated by the U.S. Department of Energy. REFERENCES 1. 2. 3. 4. 5. 6.

J. M. Tranquada et al., Nature 375, 561 (1995). L. Pintschovius et al., Physica C 185–189, 156(1991). K. Yamada et al., Phys. Rev. Lett. 75, 1626 (1995). R. J. McQueeney et al., Phys. Rev. B 54, R9689 (1996). M. Braden et al., J. Supercond. 8, 1 (1995). L. Pintschovius and W. Reichardt, in Physical Properties ofHigh Temperature Superconductors IV, ed. D. M.

Ginsberg, World Scientific, Singapore (1994) p. 295. 7. H. Harashina et al., J. Phys. Soc. Jpn. 63, 1386 (1994).

8. J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7391 (1989). 9. V. J. Emery and G. Reiter, Phys. Rev. B 38, 4547 (1988). 10. Y. Petrov and T. Egami, unpublished, 1998.


Enhanced Thermoelectric Power and Stripes in Cuprate Superconductors J. B. Goodenough1 and J.-S. Zhou1

Magnetic, pressure, and angle-resolved photoemission spectroscopy experiments on several perovskite-related transition-metal oxides at the crossover from localized to itinerant electronic behavior reveal strong electron coupling not only to

static, but also to dynamic oxygen displacements. They indicate a breakdown of the Brinkman–Rice strong-correlation model before long-range magnetic order is stabilized. The superconductive copper oxides appear to stabilize a distinguishable thermodynamic state below a where a multicenter polaron gas

condenses into a polaron liquid that progressively orders into mobile stripes as the temperature is lowered to It is suggested that itinerant vibronic states are formed as a result of coupling of itinerant electrons to dynamic oxygen displacements along the Cu-O bond axes in the mobile stripes. Such vibronic states would be responsible for a remarkable transfer of spectral weight to states propagating

parallel to the bond axes. The transfer of spectral weight is responsible for the

anisotropy of the superconductive gap and an enhancement of the thermoelectric power in the interval that is uniquely associated with the superconductive phenomenon.

1. INTRODUCTION Transition-metal oxides with perovskite-related structures provide an opportunity to study the transition from localized to itinerant electronic behavior in a three-dimensional

array or a 2D sheet of corner shared octahedra in which the dominant interactions of interest are M-O-M interactions between configurations on the transition-metal atoms M. Transitions in single-valent systems must be distinguished from those occurring in mixed-valent arrays or sheets. In this brief paper, attention is focused on evidence for dynamic electron–lattice interactions that give rise to isotropic 1

Texas Materials Institute. ETC 9.102, University of Texas at Austin, Austin, TX 78712.


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ferromagnetic superexchange spin-spin coupling, “correlation fluctuations,” or the formation of “itinerant vibronic states.”

2. PEROVSKITE STRUCTURE

The ideal structures of

perovskites, and related intergrowth oxides such as

are, respectively, cubic and tetragonal. In this paper, the larger A

cation is a lanthanide, yttrium, or an alkaline earth, and M is a first-long-period transitionmetal atom. In each structure, the deviation from unity of a tolerance factor

is a measure of the mismatch between equilibrium bond lengths (A - O) and (M - O) in the ideal structures. Equilibrium (A - O) and (M - O) bond lengths at ambient temperature

and pressure may be calculated from the sums of tabulated empirical ionic radii obtained from x-ray diffraction (XRD) data [1,2]. A larger thermal-expansion coefficient of the equilibrium (A - O) bond makes Normally, the equilibrium (A - O) bond is more compressible, which makes but, as discussed below, a is found at a crossover from localized to itinerant electronic behavior. A places the M - O bonds under a compressive stress, the A - O bonds under a

tensile stress. These stresses are relieved by a cooperative rotation of the octahedra that bends the M-O-M bond angle from A cooperative rotation about the cubic [110] axis reduces the symmetry to orthorhombic (Pbnm) with

A

cooperative rotation about the [ 111] axis would give rhombohedral R3c symmetry. Cooperative oxygen displacements may be superimposed on the cooperative rotations. In for example, cooperative oxygen displacements along the Mn-O-Mn bond axes within (001) planes reflect a Jahn–Teller orbital ordering that changes the or-

thorhombic axial ratio to

Cooperative oxygen displacements may also create

metal atoms with long M-O bonds alternating with metal atoms having short M-O bonds as occurs with a disproportionation reaction of the type which

is commonly referred to as a “negative-U” charge-density wave (CDW) Static, cooperative oxygen displacements are directly measurable with XRD and neutron diffraction and have been identified in ferroelectric, Jahn–Teller, and disproportionation transitions as well as

in CDW “stripes.” However, direct measurement of dynamic, cooperative oxygen displacements requires a fast experimental probe; dynamic displacements normally are not revealed

by a conventional diffraction experiment.

3. ELECTRONIC CONSIDERATIONS Localized configurations are described by crystal-field theory; itinerant 3d electrons by tight-binding band theory. If the lowest unoccupied orbitals of a configuration

lie at an energy

above the top of the

orbitals of an ionic model that is large enough

for electronic back transfer from the oxygen to the M atoms to be treated in second-order

perturbation theory, the basis d orbital wave functions of an M cation in an octahedral site

Enhanced Thermoelectric Power and Stripes in Cuprate Superconductors

201

may be formulated as

where and that only

are, respectively, the ionic three-fold degenerate xy, orbitals with the oxygen ligands and the two-fold degenerate orbitals that only with the oxygen ligands. The and are appropriately symmetrized and orbitals, and the covalent mixing

parameters

contain the cation–anion resonance integral and the energy corresponding to an or electron transfer. The cubic-field splitting

contain a small electrostatic contribution

The electron on-site electrostatic energy may be enhanced by exchange splitting nance integrals.

If the

the primary contribution comes from

between

and

configurations

or, at half-filling of a manifold at by the Hund intraatomic The interatomic M-O-M interactions involve spin-dependent reso-

configurations retain a localized spin, spin-dependent integrals

must be used, where is the angle between localized spins on neighboring atoms The tight-binding bandwidth is

where z is the number of like nearest-neighbor M atoms. A

configurations, a

and

leaves localized

transforms the localized electrons to itinerant electrons

occupying narrow or bands. Zaanen, Sawatzky, and Alien [4] emphasized that a distinction should be made between

systems where the energy gap is

between

and

configurations and those where

the gap is between the top of the bands and the empty redox energy in an ionic model. This distinction has aided interpretation of optical and photoelectron spectra, but it

has been misleading to the discussion of ground-state properties, as oxidation does not lower the Fermi energy into a bonding valence band, but into antibonding states of d-orbital symmetry consisting of strongly hybridized O-2p and M-3d states. At the crossover from a to a gap in the ionic model, the perturbation expansion defining the covalent-mixing

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parameters is no longer applicable to the basis wave functions associated with the holes; the holes need to be treated in molecular-orbital theory either within a polaronic complex

or within a second phase. The system appears to represent a transition from localized to itinerant electronic behavior where the parent compound has a gap. The superconductive copper oxides, on the other hand, have sheets with a gap in the ionic model of the antiferromagnetic parent compounds, but oxidation introduces holes into molecular orbitals that, in the overdoped compositions, become transformed into itinerant-electron states of an antibonding band. Because and it is possible to have localized configurations in the presence of itinerant electrons. This situation arises in mixed-valent, ferromagnetic perovskites where the Hund intraatomic exchange field couples the itinerant electron spins parallel to the localized

configurations to give a tight-binding bandwidth

We have discussed elsewhere [5] the origin of the colossal magnetoresistance (CMR) found in the manganese oxides; here, we draw attention to the consequences of strong, dynamic electron–lattice interactions in some other oxides with perovskite-related structures. 4. SINGLE-VALENT SYSTEMS

4.1. Dynamic Jahn–Teller Effect Stoichiometric undergoes a static, cooperative Jahn–Teller deformation to the structure below a below it becomes a type A antiferromagnet with ferromagnetic superexchange in the ab planes and antiferromagnetic coupling between planes [6]. An antisymmetric superexchange term cants the spins from

collinear to give a weak ferromagnetic component. In 1961, one of us asked what would happen if were substituted for in until the concentration of ions became too small for a static, cooperative Jahn–Teller deformation to symmetry [7]. At that time, and again subsequently [8], it was shown that at the to O transition, the Mn-O-Mn interactions became isotropically ferromagnetic. According to the rules for the sign of the superexchange interactions, this finding demonstrates that a dynamic Jahn– Teller coupling of the -bonding e electrons to the two optical-mode lattice vibrations of symmetry creates local “vibronic” states that correlate a dominant electron transfer from half-filled e orbitals on one atom to an empty e orbital on the neighboring atoms. clustering to minimize the elastic energy associated with dynamic Jahn–Teller interactions was also appreciated at about that time [9].

4.2. Strongly Correlated The

Systems

and ions each contain a single configuration, and would represent spin-1/2 systems with a gap were the

and configurations

localized. In fact, each system has strongly correlated, itinerant electrons at the threshold of a transition from localized to itinerant electronic behavior [10]. In these single-valent systems, the bands are narrowed by substitution of a smaller, more acidic A cation. Inoue et al. [11] reported photoemission spectroscopy (PES) data showing the coexistence of coherent and incoherent states with a continuous shift of spectral weight from


203

coherent to incoherent states as the band narrows with increasing x in Nevertheless, the system remains an enhanced Pauli paramagnet for all x. The PES data also indicated a maximum in the effective mass m* at an intermediate value of x. In order to determine whether the maximum in m* is due to a perturbation of the periodic potential by dissimilar A cations or to correlation fluctuations implicit in the PES data, we measured the temperature dependence of the thermoelectric power for under different hydrostatic pressures [12]. Our data showed an increase in ) with pressure,

whereas Pt showed the expected decrease in Moreover, pressure increased the low-temperature phonon-drag component of but left that of Pt unchanged. These results confirmed that m* increases with as a result of the coexistence of incoherent localized-state fluctuations coexisting with coherent itinerant-electron states. We concluded that, as illustrated in Fig. 1, the Brinkman–Rice relationship

for strongly correlated itinerant electrons breaks down with the onset of incoherent-state fluctuations. As the on-site electrostatic energy U approaches the critical value the evolution of the electronic state in a perovskite-related structure is not continuous, as envisaged by Hubbard, but undergoes a first-order transition from itinerant to localized electronic behavior. Stoichiometric is a Pauli paramagnetic metal above a and becomes a type G antiferromagnet below without exhibiting the cooperative Jahn–Teller deformation to be expected for a localized-electron collinear-spin antiferromagnet [14]. PES data [15] have shown for this system also the coexistence of incoherent and coherent electronic states, which has suggested to us [16] that the measured [17] pressure dependence does not signal a localized-spin configuration [18], as was originally inferred [17], but a transfer of spectral weight from incoherent to coherent states and a that increases with the density of states at the Fermi energy. This interpretation is consistent with an observed decrease in with decreasing width of the band as the smaller ion is substituted for in Significantly, as decreases to zero with increasing x, a ferromagnetic Curie temperature increases with x, reaching a

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Goodenough and Zhou

in the ferromagnetic insulator redox energies [19].

having a

gap between the

and

4.3.

The metallic perovskite system

contains a half-filled

band. In the

absence of localized spins, and the factor becomes unity in Eq. (8). Introduction of a smaller ion increases which decreases and narrows whereas pressure increases without introducing any additional perturbation of the periodic potential. We have measured under different hydrostatic pressures for 0.25, and 0.50 [20]. In exhibited a negative phonon-drag component with a maximum at a typical of a conventional metal. However, the phonon-drag component was enhanced by the application of pressure as in The sample showed a stronger suppression of the phonon-drag component, undoubtedly partly due to

the perturbation of the periodic potential by different A cations, but it was enhanced by the application of pressure. Moreover, this sample showed an increase with pressure of as in and opposite to Pt, which is indicative of an increase in m* with pressure. A

similar situation was found for the

sample, but the phonon-drag component was

more strongly suppressed. These results indicate that the

perovskite system

has a narrow o band with an electronic heterogeneity similar to that observed in 4.4. The

Perovskites

The perovskite family has been prepared from to Lu and Y. Whereas can be prepared easily at ambient pressure, the other members of the family were first obtained only with a high-pressure preparation. Moreover, has a lowspin configuration and is a bad metal exhibiting an enhanced Pauli paramagnetism down to lowest temperatures [21]. however, undergoes a first-order transition to

an antiferromagnetic insulator below a and as the size of the lanthanide ion decreases, increases to more than 450 K in The antiferromagnetic phase has the larger volume, and pressure decreases sharply being reported [23] for The antiferromagnetic order is unusual; ferromagnetic ( 1 1 1 ) Ni planes having a nickel-atom moment are alternately coupled ferromagnetically and antiferromagnetically [24]. This order can be rationalized if a CDW is also stabilized in which ferromagnetic itinerant-electron Ni-O-Ni ( 1 1 1 ) blocks are coupled antiferromagnetically across (111) plane of via Ni-O-Ni superexchange [25]. Medarde et al. [26] have reported a decrease of 10 K in the of on the exchange of for Pressure reduces the Ni-O bond length and hence the bond angle whereas exchange reduces the frequency of the axial vibration of an oxygen atom of mass so we conclude that varies sensitively with an The stabilization of an antiferromagnetic phase below a in indicates that may, like and be in the transitional region of Fig. 1 with strong-correlation fluctuations giving rise to an electronic heterogeneity. Preliminary measurements show that the phonon-drag component of for is suppressed, which supports this conjecture. Within this picture, stabilization of the CDW in occurs where the concentration of strong-correlation fluctuations becomes large enough for them to condense into an ordered array.


205

The low-spin ions would be strong Jahn–Teller ions like the high-spin ions were the single e electron per ion localized. Therefore, it is interesting to compare the isotope shift found in with that found for the Curie temperature in the mixed-valent, ferromagnetic manganese oxides exhibiting a CMR [27,28]. In our experiments [28], we chose an sample that was at the threshold of the to O-orthorhombic transition and applied pressure to see how the shift changes on going from the static to the dynamic Jahn–Teller regime. We found a change from a second-order magnetic transition with no isotope shift of in the phase to a first-order transition in the O phase. In the sample, the to O transition occurs at a whereas in the sample it is shifted to a Moreover, the isotope-shift parameter has a maximum value of about 5 at 10.5 kbar and decreases with further increase in pressure. Zhao et al. [27] also noted a decrease in with increasing in the O-orthorhombic phase. Here also is evidence for strong electron coupling to dynamic, cooperative oxygen displacements associated with a first-order magnetic transition and a tolerance factor that depends sensitively on

of

5. EQUILIBRIUM M-O BOND LENGTHS

Thus far, we have shown by both magnetic and isotope measurements that orbital degeneracies at strong Jahn–Teller ions result in electron coupling to dynamic, cooperative oxygen displacements. Moreover, the existence of an electronically heterogeneous region in Fig. 1 in which there is a transfer of spectral weight from incoherent to coherent states with increasing bandwidth implies a segregation into two phases via strong electron coupling to

dynamic, cooperative oxygen displacements. In order to demonstrate that a first-order phase change, and hence phase segregation, may occur at the crossover from localized to itinerant electronic behavior, we turn to the virial theorem for central force fields, which states

If the mean kinetic energy of the electronic system increases discontinuously at a crossover from localized to itinerant electronic behavior, then the mean potential energy must decrease discontinuously. For antibonding electrons, a disontinuous decrease in means a discontinuous shortening of the equilibrium M-O bond length, which leads to

a double-well potential at crossover [29]. A double-well potential would be manifest by the following observations: (1) a first-order phase change at crossover that results in a dynamic phase segregation into coherent and incoherent electronic states or the stabilization of a static CDW; and (2) an anomalously high compressibility of the localized electron M-O bond at crossover in which pressure transfers spectral weight from incoherent to coherent states. In the closing section, we apply to mixed-valent systems the consequences of the virial theorem and electron coupling to dynamic oxygen displacements as illustrated by the superconductive copper oxides.

6. COPPER-OXIDE SUPERCONDUCTORS The copper oxides have intergrowth structures in which sheets have a partially occupied, antibonding band. The simplest superconductive copper-oxide system is

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Goodenough and Zhou

The parent compound, is an antiferromagnetic insulator, but drops precipitously to zero with increasing x, and superconductivity appears for Three compositional ranges are generally recognized: an underdoped range, in which the superconductive critical temperature increases with x and the hole-

rich superconductive phase segregates from the antiferromagnetic parent phase below about 240 K; an optimally doped range, in which reaches a maximum near and an overdoped range, in which drops, probably stepwise, to zero with increasing x and the charge carriers change from holes to electrons [30]. The phase segregation in the underdoped region occurs via clustering of oxygen interstitials in

in which the interstitial oxygen are mobile to well below room temperature [31], but in it can only occur by cooperative oxygen displacements. Investigation of stoichimetric has proved instructive. Because this structure contains no charge reservoir, the occupancy of the

band of a superconductive

sheet is

unambiguously determined by the value of x, and x can be varied over the entire range of superconductive compositions.

6.1. Underdoped

Sheets

At low carrier concentrations, holes introduced into the band are polaronic despite the fact that they occupy orbitals that can no longer be described by crystalfield theory. The holes are confined by the localized-spin matrix within which they are introduced. Normally, nonadiabatic polarons are confined to a single metal site even where the lowest unoccupied states of the cluster must be described by molecular-orbital rather than crystal-field theory. However, in the e-orbital degeneracy of the octa-

hedral site

configuration is not strongly removed by the tetragonal component

of the crystalline field, so a pseudo-Jahn–Teller deformation may create at a low-spin

two shorter and two longer Cu-O bonds within a sheet. The elastic energy of the local deformation is reduced by enlarging the polaron to include, for example, the four nearest-neighbor copper centers. Calculation [32] has predicted a single-hole

polaron containing thalpy

pseudo-Jahn–Teller copper centers with a small motional enbecause the multicenter polaron moves one center at a time via

dynamic, cooperative oxygen-atom displacements. Experimentally, we have reported [33] a temperature-independent thermoelectric power above 240 K, which signals the presence

of a nonadiabatic polaron gas described by the statistical term

that includes the spin degeneracy. The parameter k1 corresponds to the mean number of copper centers in a polaron; the fit in Fig. 2 is to Below 240 K, we anticipate a dynamic segregation into antiferromagnetic and superconductive phases in the absence of a magnetic field. However, Boebinger et al. [34] showed an increase in the resistivity with decreasing temperature below 50 K in single crystals of with in which superconductivity was suppressed by a magnetic field It is tempting to interpret this result as a manifestation of polaronic motion in these high fields with a only at lowest temperatures. The multicenter polarons can be considered a segregated hole-rich phase in which the holes occupy molecular orbitals within a matrix of localized electrons at


207

6.2. Optimum Carrier Concentration at

Phase segregation in the underdoped composition into antiferromagnetic and superconductive rather than antiferromagnetic and overdoped phases indicates that the superconductive phase is a thermodynamically distinguishable phase [33]. Moreover, the pressure dependence of the tolerance factor, which is manifest by an orthorhombic to tetragonal transition at a critical pressure in the superconductive compositional range [30], indicates that the superconductive phase is stabilized at the crossover from localized to itinerant electronic behavior. in the orthorhombic phase

becomes in the tetragonal phase, which shows that is suppressed by bending the Cu-O-Cu bond angle from 180°. Ambient pressure thermoelectric power obtained [35] for the sample shows two features are to be noted: at temperatures is nearly temperature independent, which indicates the charge carriers form a polaron gas; in the interval exhibits an enhancement with a which is too high for an acoustic-mode phonon drag. We [35] interpreted this curve to signal the condensation of the polaron gas into a polaron liquid below but we did not specify the nature of the polaron liquid. However, it was pointed out [32] that the elastic energy of a polaronic liquid would be minimized by the formation of stripes at the time Bianconi et al. [36] were first presenting evidence for stripes in their XAFS data. Tranquada et al. [37] identified static stripes parallel to [100] and [010] axes in

alternate

sheets of a nonsuperconductive low-temperature-tetragonal (LTT) phase In this phase, the stripes are pinned by cooperative rotations of the octahedra alternately about [100] and [010] axes of successive sheets. In the orthorhombic phase, rotations about [110] axes at 45° to the stripes do not pin the stripes.

Pinning the stripes suppresses superconductivity; superconductivity is regained even in the LTT phase if the stripes become mobile [38].

We have shown [39] that the enhancement of

below

with a

is

associated exclusively with the superconductive phase, which by inference means it is associated exclusively with the existence of mobile stripes ordered parallel to [100] and [010]

axes in alternate

sheets. Why this should be so is clarified by recent angle-resolved PES

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Goodenough and Zhou

measurements by Norman et al. [40] of the temperature dependence of the Fermi surface. As Coleman [41] illustrated, the data show a dramatic transfer of spectral weight into directions of the Cu-O bonds in the sheets. Such a transfer of spectral eight requires a stabilization of itinerant-electron states propagating along the [100] and [010] axes. A strong

coupling of itinerant-electron states to the optical phonon modes traveling parallel to the propagation vector Q of a mobile stripe would increase with greater ordering of the stripes on lowering the temperature below The electron–phonon coupling would create itinerant vibronic states having a stabilization energy proportional to or, taking account of

perpendicular stripes in alternate planes, to where is the angle between k and Q in a particular plane. An exceptional flatness of near the M point of the Brillouin zone was already noted by Dessau et al. [42] in 1993, and we emphasized that the enhancement in the below indicates an increase in the asymmetry of the curve about The increasing transfer of spectral weight with decreasing temperature accounts well for the increase in with decreasing temperature in the range We therefore conjecture that the decrease in with decreasing temperature in the range is due to the onset of vibronic Cooper-pairs from itinerant vibronic states. ACKNOWLEDGMENT

We wish to thank the NSF and the Robert A. Welch Foundation, Houston, Texas, for financial assistance.

REFERENCES 1. R. D. Shannon and C. T. Prewitt, Acta Crystallogr. B 25, 725 (1969). 2. R. D. Shannon and C. T. Prewitt, Acta Crystallogr. B 26, 1046 (1970). 3. J. B. Goodenough, J. A. Kafalas, and J. M. Longo, in Preparative Methods in Solid State Chemistry, P. Hagenmuller, ed.. (Academic Press, New York, 1972) Chap. 1.

4. J. Zaanen. G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 (1985). 5. 6. 7. 8.

J. B. Goodenough and J. S. Zhou, MRS Symp. Proc. 474, (1998). E. O. Wollan and W. C. Koehler, Phys. Rev. 100, 545 (1955). J. B. Goodenough, A. Wold, R. J. Arnott, and N. Menyuk, Phys. Rev. 124, 373 (1961). J. Topfer and J. B. Goodenough, unpublished, 1998.

9. J. B. Goodenough, J. Appl. Phys. 36, 2342 (1965). 10. J. B. Goodenough, Appl. Phys. 39, 403 (1968).

1 1 . I. H. Inoue et al., Phys. Rev. Lett. 74, 2539 (1995). 12. J.-S. Zhou and J. B. Goodenough, Phys. Rev. B 54, 13393 (1996). 13. Y. Okimoto et al., Phys. Rev. B 51, 9581 (1995). 14. 15. 16. 17.

J. B. Goodenough, Phys. Rev. 171, 466 (1968). A. Fujimori et al., Phys. Rev. Lett. 69, 1796 (1992). J. B. Goodenough and J.-S. Zhou, unpublished, 1998. Y. Okada et al., Phys. Rev. B 48, 9677 (1993).

18. J. B. Goodenough, Prog. Sol. State Chem. 5, 145 (1971). 19. K. Kumagai et al., Phys. Rev. B 48, 7636(1993).

20. J.-S. Zhou, W. B. Archibald, and J. B. Goodenough, Phys. Rev. B 57, R2017 (1998). 21. J. B. Goodenough and P.M. Raccah, J. Appl. Phys. 36, 1031 (1965).

22. J. B. Torrance et al., Phys. Rev. B 45, 8209 (1992). 23. X. Obradors et al., Phys. Rev. B 47, 12353 (1993); P. C. Cranfield, J. D. Thompson, S. W. Cheong, and L. W.

Rupp, Phys. Rev. B 47, 12357 (1997).


209

24. J. L. Garcia-Munoz, P. Lacorre, and R. Cywinski, Phys. Rev. B 51, 15197 (1995); J. Rodriguez Carvajal et al., Phys. Rev. B 57, 456(1998). 25. J. B. Goodenough, J. Solid State Chem. 127, 126 (1996). 26. M. Medarde et al., Phys. Rev. Lett. 80, 2397 (1998). 27. G. M. Zhao, K. Konder, H. Keller, and K. A. Muller, Nature 381, 676 (1996). 28. J.-S. Zhou and J. B. Goodenough, Phys. Rev. Lett. 80, 2665 (1998). 29. J. B. Goodenough, Ferroelectrics 130, 77 (1992). 30. J. S. Zhou, H. Chen, and J. B. Goodenough, Phys. Rev. B 49, 9084 (1994). 31. J. C. Grenier et al., Physica C 202, 209 (1992).

32. G. I. Bersuker and J. B. Goodenough, Physica C 274, 267 (1997). 33. J. B. Goodenough, J.-S. Zhou, and J. Chan, Phys. Rev. B 47, 5275 (1993). 34. G. S. Boebinger et al., Phys. Rev. Lett. 77, 5417 (1996). 35. J. B. Goodenough and J.-S. Zhou, Phys. Rev. B 49, 4251 (1994). 36. A. Bianconi et al., Phys. Rev. Lett. 76, 3412 (1996); Phys. Rev. B 54, 4310, 12018 (1996). 37. J. M. Tranguada et al., Nature 375, 561 (1995); Phys. Rev. B 54, 7489 (1996). 38. J.-S. Zhou and J. B. Goodenough, Phys. Rev. B 56, 6288 (1997). 39. J.-S. Zhou and J. B. Goodenough, Phys. Rev. B 51, 3104 (1995); Phys. Rev. Lett. 77, 151, 1190 (1996). 40. M. R. Norman et al., Nature 392, 157 (1998). 41. P. Coleman, Nature 392, 134 (1998). 42. D. S. Dessau et al., Phys. Rev. Lett. 71, 2781 (1993).


A Refined Picture of the Structure: Sequence of Dimpling-Chain Superstructures, 1D-Modulation of the Planes, Phase Separation Phenomena E. Kaldis,1 E. Liarokapis,2 N. Poulakis,2 D. Palles,2 and K. Conder1

High resolution volumetric oxygen determination of over hundred equilibrium samples allowed to find different properties with mesoscopic and macroscopic methods. Thus, whereas X-ray and neutron high resolution structure refinements do show homogeneous samples, Raman scattering shows an intrinsic inhomogeneity of In the underdoped region a sequence of superstructures exists with stepwise softening of the O2/O3 in-phase mode (A1g) corresponding to decreasing dimpling (Cu-O2/O3 distance) with increasing oxygen doping. These superstructures have

oxygen contents corresponding to the oxygen vacancies chainsuperstructures, found in the past with electron microscopy. This relationship

between the 1D-vacancies ordering of the chains and the distance between the oxygens (O2,O3) and the copper (Cu2) of the planes, implies a 1-D modulation

of the planes along the c-axis, possibly via the apical bond. At the onset of the overdoped range neutron diffraction shows the existence of a phase transition and EXAFS illustrate its mechanism (displacive transformation). In the overdoped phase Raman scattering supports the existence of two phases, as shown in the past from the splitting of the diamagnetic and resistive transitions, as well as of

the peak of the specific heat.

1. INTRODUCTION

It is well known that O-doping at the normal state changes the occupancy of the chains and the carriers concentration in the planes. In the past, high-resolution electron microscopy 1

2

Laboratorium für Festkörperphysik ETH, CH-8093 Zurich, Switzerland. National Techn. University of Athens, Physics Department, Athens, Greece.


211

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Kaldis, Liarokapis, Poulakis, Palles, and Conder

(HREM) investigations with nearly atomic resolution have shown in microcrystallites the existence of oxygen–vacancy chain superstructures with increasing O-doping: 2ao (Ortho

II, every second chain empty), 3ao (every third chain empty), etc. [1a]. Some other superstructures have been also observed by x-ray diffraction (XRD) in some single crystals [2]. The importance of the chain superstructures for superconductivity was manifested by the plateau of the curve, which was considered to be the transition from the Ortho I to the Ortho II superconducting phase [1b]. This plateau does not appear as expected at but is smeared between 6.60 and 6.80. This idea of the transition has been unanimously adopted without any clear understanding how this takes place structurally and thermodynamically. Not so many possibilities are existing from the point of view of the thermodynamics of the Gibbs phase rule: In temperature-composition (T-x) or pressure-composition (P-x) phase diagrams of nonstoichiometric systems, variation of the nonstoichiometry (x) leads to a sequence of phases of the same components with different structures, separated by miscibility gaps (two-phase regions). A simple phase diagram of this type has been discussed in the past for 123 [3]. In two-phase regions, the intensive properties (e.g., lattice parameters, magnetic moment) of each phase remain constant, and only their ratio changes with the overall composition. The mechanism is diffusive and the transformation (miscibility gap) is second order. The coexistence of two phases has never been proved for underdroped with any structural method except for the indications of HREM. The latter did not show a two-phase region with sharp concentration boundaries but always mixtures of phases with the ordered superstructure of the matrix. Therefore, from the point of view of the phase rule, the I transformation leaves several open questions. In the case that diffusion between the two phases of the miscibility is not possible due to lattice strains, the alternative is a displacive first-order phase transition (e.g., a martensitic transformation that very often appears in LT superconducting phases [4]). Two such transitions have been found for 123: at . (tetragonal to orthorhombic) [4] and recently at (underdoped to overdoped) [5]. No transition has been found for with increasing doping More ambiguous was the information about the structure of the planes in which a dimpling of the O2 and O3 has been observed very early [6]. Triggered from our observation of the nonlinearity of the lattice parameters as a function of the carrier’s concentration (O-doping) and an anomaly of the c lattice parameter at we have performed a systemic study of a great number of oxygen compositions. Neutron diffraction confirmed the minimum of the c lattice parameter and showed that it is due to a phase transition in the dimpling at the onset of the overdoped range [3,10]. Raman scattering as a function of doping showed a strong softening of the in-phase O2/O3 phonon at the same composition [8]. As we discuss later in this paper, magnetization measurements show a splitting of the diamagnetic transition in the overdoped range 2. EXPERIMENTAL The oxygen concentration of more than 100 polycrystalline samples in the range of has been determined with an extremely accurate volumetric method at the third decimal point 10–20 times more accurate [11] than

A Refined Picture of the

Structure

213

all other methods discussed in the literature. Thus, a unique opportunity is given to study with

very high resolution the trends of structure and physical properties with increasing carriers concentration induced by oxygen doping. The samples analyzed for oxygen with the above method have been, therefore, characterized with dc magnetization and neutron diffraction [3,9,10], EXAFS [12], and micro-Raman scattering [8,10] as a function of doping.

We note that the samples used in these investigations are near to the thermodynamic equilibrium (very slowly cooled) [9], in contrast to the usually used quenched samples

[1b]. Quenching of the samples was used to avoid the freezing of the oxygen mobility that was assumed to take place at Investigations of the diffusion coefficient of 123 showed that thermodynamic equilibrium is reached at least down to and kinetics controlled mobility exists at much lower temperatures. Earlier experiments have shown that

even at much lower temperaturers, oxygen is still mobile [14]. In La 214, it has been shown that the mobility of oxygen is kept down to 200 K! [15]. The selection of polycrystalline samples instead of single crystals is for two reasons. The determination of oxygen needs 100 mg of the material, a rather large weight for single crystals of cuprates, and the homogeneity of the polycrystalline material is much better than that of single crystals. Generally, in single crystals in addition to the incorporation of traces of the flux and impurities from the crucible walls, the long diffusion paths for oxygen hinder the attainment of equilibrium. Experimental details about the micro-Raman measurements have been given elsewhere [8].

3. RESULTS AND DISCUSSION

The main conclusions obtained from the macroscopic (length scale of experimental methods briefly discussed in this introduction were two: transition in dimpling at the onset of the overdoped range and splitting of the diamagnetic transition in the overdoped range. The mesoscopic picture (length scale was derived from micro-Raman measurements on individual microcrystallites and EXAFS on the same polycrystalline samples [5,12]. The extreme sensitivity of the Raman scattering is due to the fact that only a few atoms are necessary in order to form a new phonon EXAFS showed the mechanism of the dimpling transformations. Details of the change of the bondlengths along the c axis as a function of doping: Cu2 nearing Cu1 with doping (complex movement twoard Ba along the 111 plane) and increasing its distance to O2/O3, which remains at constant distance to Cu2. As the movement of Cu2 takes place at almost constant Y-Cu2 distance, an antiferro distortion of the pyramids takes place (pseudo-Jahn–Teller distortion) [12]. At a displacive phase transition (martensitic transformation) takes place at the boundary between the optimally doped and the overdoped region, the O2/O3 atoms jumping suddenly toward Y. Further O-doping (overdoped region) leads to a movement of the O2/O3 oxygens toward and decreasing of dimpling. Plotting the micro-Raman shifts (frequency of the O2/O3 in-phase mode) of the individual crystallites (not their statistic average) vs. the O-doping, a new picture is revealed (Fig. 1) [5]. Several dimpling phases appear in the underdoped range (F, E, D, C) separated by two-phase regions (miscibility gaps). The oxygen contents of these phases coincide with the ideal compositions of some of the superstructures discovered long ago by electron

214

Kaldis, Liarokapis, Poulakis, Palles, and Conder

microscopy ( and probably ). Near the optimal doping the picture becomes less clear, due to the coexistence of several phases. In the overdoped range two phases appear (A, B). Thus, we can conclude that instead of a two-phase sequence as the macroscopic structural evidence indicates, a sequence of at least four phases

exists in mesoscopic scale, separated by miscibility gaps (phase separation). These results show for the first time the correspondence existing between the dimpling in the planes and the one-dimensional ordering of the chains. It seems, therefore, reasonable to assume that a ID-ordering could also exist in the superconducting planes of The interaction between dimpling and chains could possibly take place via the apical bond. Characteristic of these superstructures is the stepwise increase of the distance between the oxygens (O2, O3) and the copper (Cu2) of the planes with doping, as the Cu2 is moving away of the O2/O3 with doping. The above results show that this change cannot take place continuously, but a “quantisation” along the c axis exists, associated with the changes of the chain ordering. We note that in the underdoped region with increasing distance of the

oxygen (O2,O3) from the copper (Cu2) of the planes (dimpling) and distortion of the pyramids, the also increases. This picture of 123 (near equilibrium samples) has some interesting similarities with the phase diagram of proposed recently [15]. In this compound the O-atoms are mobile down to 200 K, in contrast to where the Sr atoms freeze at the melting point (1400 K). Thus, the O-doped 214 system is much nearer to the thermodynamic equilibrium, and we expect that it will show much more structural details. Indeed, similar to a series of miscibility gaps is found, dividing superconducting 214 phases with different stages (superstructures) of oxygen intercalation. Therefore, the question arises that if a pattern of superconducting regions divided by phase separated regions is a general one, necessary for superconductivity. An interesting difference between the two-phase diagrams is that the phase separation in underdoped Yappears in mesoscopic scale, whereas in La-124 in macroscopic scale. The authors


Structure

215

[15] investigated single crystals with elastic and inelastic neutron diffraction. From the elastic scattering stripe superstructures, appears to be similar to some degree to those of the Tranquada model [16]. From the incommensurate spin fluctuations they measure, they conclude that possibly two kinds of stripes exists: (a) Charge and spin stripes fluctuating in time and therefore giving inelastic but not elastic neutron scattering peaks. These stripes are oriented along the Cu-O bond, (b) The static stripes resulting from the in-plane ordering of the 214 stages, taking place at temperatures above These form a 45° angle to the Cu-O direction, and cannot be responsible for the measured spin modulations [15]. We note that an important difference between the stripes in the La-cuprates and the 1D modulation that could be expected in

is that the former appears in the ab planes, whereas the later appears along the c axis and probably influences the apical bond. The importance of this bond for superconductivity has been discussed very early after

its discovery [17]. Recently, this discussion has been revived after the finding of near-edge x-ray absorption fine structure (NEXAFS) investigations [18], that the apical site plays an important role for the superconductivity in This is supported also from the increasing consensus that infinite layer compounds do not so bulk superconductivity [19], except if some apical bonds are formed by doping [20]. We note that in underdoped 123 the dimpling/chain superstructure phases appear only in mesoscopic scale, invoking phase-separation scenario. The sequence of phases separated by miscibility gaps seems to obey the Gibbs phase rule as in macroscopic phase diagrams (chemical phase separation). However, the mesoscopic length scale indicates a picture of physical phase separation as could be resulting from the hole doping of the antiferromagnetic insulator. Near optimal doping and in the overdoped region, several phases coexist

in mesoscopic scale. A splitting of the diamagnetic transition [3,5,10] supports the phase separation in the overdoped region.

4. CONCLUSIONS The main messages emanating from the above investigations are: •

The intrinsic inhomogeneity of 123 and the existence in mesoscopic scale of a staircase of dimpling/chain superstructures separated by minute miscibility gaps. With increas-

•

ing unit length of these superstructures, the Cu2 of the planes move away (along the c axis) from the oxygens (O2, O3) of the planes and increases. SQUID measurements show a splitting of the diamagnetic transition at indicating the coexistence of two superconducting phases in the overdoped region, the one having a near that of the optimally doped and the other a few degrees lower. Unfortunately, the 123 overdoped range is not wide enough to reveal possibly existing trends. Doping with Ca to extend the overdoped range shows more than one phase and

a strong splitting of the diamagnetic transition [21]. •

•

The correspondence between distortion of the planes (dimpling) and the superstructures of the chain leads to the assumption that the distortion of the planes may have 1Dsymmetry. This 1D distortion of the planes would be significantly different from the stripes found

in the La-124 compounds. Whereas the later are in the ab plane, the former is a distortion along the c axis. It is not clear at present how this vertical distortion can influence the copper–oxygen bonds in the ab plane.

216 •

Kaldis, Liarokapis, Poulakis, Palles, and Conder Phase separation appears in the sequence of chemical phase separation—i.e., phases separated by miscibility gaps—but can be observed only in mesoscopic scale, which

• •

could be characteristic for the physical phase separation resulting from the insulator– superconductor transition. The individual superstructures are not miscible with each other, and are separated by regions of lattice instabilities. Pseudo-Jahn–Teller distortion of the pyramids, leading to a displacive transformation at the transition between optimally doped and overdoped range. These results support the idea that the lattice geometry (superstructures) is important for superconductivity.

ACKNOWLEDGMENT

Many thanks are due to the NFP30 program of the Swiss Nat. Fonds for supporting the work at ETH. REFERENCES 1. (a) R. Beyers, B. T. Ahn, G. Gorman, V. Y. Lee, S. S. P. Parkin, G. Lim, M. L. Ramirez, K. P. Roche, J. E.

Vasquez, T. M. Gür, and R. A. Huggins, Nature 340, 619 (1990). 1. (b) See e.g., R. J. Cava, A. W. Hewat, E. A. Hewat, B. Batlogg, M. Marezio, K. M. Rabe, J. J. Krajewski, W. F. Peck, and L. W. Rupp, Physica C 165, 419 (1990).

2. J. Grybos, D. Hohlwein, Th. Zeiske, R. Sonntag, F. Kubanek, K. Eichhorn, and Th. Wolf, Physica C 220, 138 (1994). 3. K. Conder, D. Zech, Ch. Krüger, E. Kaldis, H. Keller, A. W. Hewat, and E. Jilek, in “Phase Separation in

Cuprate Superconductors,” E. Sigmund and K. A. Müller, eds. (Springer, 1994) p. 210. 4. J. E. Krumhansl, in “Lattice Effects in Superconductors,” Y. Bar-Yam, T. Egami, J. Mustre-de-Leon, and A. R. Bishop, eds. (World Scientific, 1992); S. Kartha, J. A. Krumhansl, J. P. Sethna, and L. K. Wickam, Phys. Rev. B 52, 803 (1995). 5. E. Kaldis, J. Röhler, E. Liarokapis, N. Poulakis, K. Conder, and P. W. Loeffen, Phys. Rev. Lett. 79, 4894, (1997). 6. J. J. Capponi, C. Chaillout, A. W. Hewat, P. Lejay, M. Marezio, N. Nguyen, B. Raveau, J. L. Soubeyroux, J. L. Tholence, and R. Tournier, Europhys. Lett. 3, 1301 (1987); S. J. Billinge, P. K. Davies, T. Egami, and C. R. A. Catlow, Phys. Rev. B 43, 10340(1991).

7. S. Rusiecki, E. Kaldis, E. Jilek, C. Rossel, and J. Less, Comm. Met. 164, 31 (1990). 8. N. Poulakis, D. Palles, E. Liarokapis, K. Conder, E. Kaldis, and K. A. Muller, Phys. Rev. B 53, R534,

(1996); E. Liarokapis p. 447 in “Workshop on Superconductivity 1996; Ten Years after the Discovery” E. Kaldis, E. Liarokapis, and K. A. Müller, eds., in NATO ASI Series (Kluwer Academic Publishers, Dordrecht/Boston/London, 1997). 9. D. Zech, K. Conder, H. Keller, E. Kaldis, E. Liarokapis, N. Poulakis, and K. A. Müller, in Anharmonic Properties of Cuprates, D. Mihailovic, G. Ruani, E. Kaldis, and K. A. Müller, eds. (World Scientific, Singapore, 1995), p. 18. 10. E. Kaldis, in Workshop on

Superconductivity 1996: Ten Years after the Discovery, E. Kaldis,

E. Liarokapis, and K. A. Müller, eds. in NATO ASI Series (Kluwer Academic Publishers, Dordrecht/ Boston/London. 1997), p. 411. 11. K. Conder, S. Rusietski, and E. Kaldis, Mat. Res. Bull. 24, 581 (1989). 12. J. Röhler, P. W. Loeffen, S. Müllender, K. Conder, and E. Kaldis, in Workshop on Superconductivity 1996: Ten Years after the Discovery, E. Kaldis, E. Liarokapis, and K. A. Müller, eds., in NATO ASI Series

(Kluwer Academic Publishers, Dordrecht/Boston/London, 1997), p. 469. 13. K. Conder, Ch. Krüger, E. Kaldis, D. Zech, and H. Keller, Physica C 225, 13 (1994). 14. B. W. Veal, H. You, P. Paulikas, H. Shi, Y. Fang, and J. W. Downey, Phys. Rev. B 42, 4770 (1990).

15. B. O. Wells, Y. S. Lee, M. A. Kastner, R. J. Christianson, F. C. Chou, R. J. Birgeneau, K. Yamada, K. Yamada, Y. Endoh, and G. Shirane, in Workshop on

Superconductivity 1996:

Ten Years after the


Structure

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Discovery, E. Kaldis, E. Liarokapis, and K. A. Müller, eds., in NATO ASI Series (Kluwer Academic Publishers, Dordrecht/Boston/London, 1997), p. 349.

16. J. M. Tranquada, Y. Kong, J. E. Lorenzo, D. J. Buttrey, D. E. Rice, and V. Sachan, Phys. Rev. B. 50, 6340 (1994). 17. A. R. Bishop, R. L. Martin, K. A. Müller, and Z. Tesanovic, Z. Phys. B—Cond. Matt. 76, 17 (1989). 18. N. Nücker et al., J. Supercond. 12 (1999), Proc. Euroconference “Polarons: Condensation, Pairing, Magnetism,” Erice, June 1998. 19. N. Nucker, Forschungszentrum Karlsruhe, INFP (private communication); M. Merz, N. Nücker, P. Schweiss, S. Schuppler, C. T. Chen, V. Chakarian, J. Freeland, Y. U. Itzverda, M. Kläser, G. Müller-Vogt, and Th. Wolf, Phys. Rev. Lett. 80, 5192 (1998). 20. H. Zhang, Y. Y. Wang, H. Zhang, V. P. Dravid, L. D. Marks, P. D. Han, D. A. Payne, P. G. Radaelli, and J. D. Jorgensen, Nature 370 (1994). 21. E. Kaldis, G. Böttger, E. Liarokapis, J. Röhler, D. Palles, and Ch. Rossel, unpublished, 1998.

Note Added in Proof (May 17, 1999) Long after finishing the above manuscript (September 1998), we had the opportunity to read the preprint of the very interesting paper, “Oxygen Ordering Superstructures and Structural Phase Diagram of Studied by Hard X-ray Diffraction,” by M.V.

Zimmermann, T. Frello, N. H. Andersen, J. Madsen, M. Käll, O. Schmidt, T. Niemöller, J. R. Schneider, H. F. Poulsen, Th. Wolf, R. Liang, P. Dosanjh, and W. N. Hardy. This paper shows with single crystal studies on very slowly cooled crystals the existence of 2ao, 3ao, 5ao, and 8ao superstructures supporting the findings of the above Raman investigations. The intrinsic inhomogeneity of is therefore supported also from completely different studies. The fact that these superstructures appear in single crystals at different compositions than the ideal ones assigned in our polycrystalline samples [5] indicates the

strong dependence of these effects on the thermal and chemical history of the samples. E.K. is indebted to Dr. Niels H. Andersen (Risoe Nat. Lab., Denmark) for sending this preprint.


Evolution of the Gap Structure from Underdoped to Optimally Doped from Femtosecond Optical Spectroscopy D. Mihailovic,1 J. Demsar,1 and B. Podobnik1

The evolution of the gap structure with doping in

is investigated with real-time optical time-resolved spectroscopy measurements of quasiparticle (QP) recombination dynamics. In the underdoped phase the existence of a T-independent gap for charge excitations is found. Its presence becomes apparent when where in good agreement with many other experimental techniques. Approaching optimum doping another T-dependent gap becomes apparent which opens at Careful examination of the data near optimum doping reveals that the T-independent and T-dependent gaps coexist and have approximately the same value. Both gaps were found to be nodeless, whereas a d wave gap was found to be incompatible with the data. In the underdoped phase, where no

change in DOS is observed at we conclude that phase coherence is established at with no change in pairing amplitude at this temperature.

1. INTRODUCTION

Femtosecond photo-induced optical modulation experiments of high-temperature superconductors can give very detailed information about quasi-particle (QP) recombination dynamics across the energy gap. With the aid of a recently developed theoretical model, the data on photo-induced absorption can give direct information about the temperature dependence of the gap and its symmetry and magnitude.

After photo-excitation of a metal or superconductor by a short laser pulse, an electron and a hole are created with a relative kinetic energy, which is equal to the energy of the 1

Jozef Stefan Institute, Jamova 39, 1001 Ljubljana, Slovenia.



219

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Mihailovic, Demsar, and Podobnik

incident photon. These charge carriers loose their excess energy very rapidly, settling within 10–30 fs of photo-excitation into QP states near the Fermi energy. In a superconductor, further energy relaxation of the QP into pairs is much slower because the energy gap limits the total number of final states available and further energy loss is inhibited. The recombination time across the gap is determined by high-energy phonon emission whose energy is greater that the gap and is in the range of 400–3000 fs, depending on the size of the gap and temperature. The QPs therefore form a quasi-steady-state distribution due to this relaxation bottleneck and can be probed by a suitably delayed weak laser pulse. The initial states for the optical transitions of the probe pulse are the occupied photo-populated QP states, whereas the final states are in a band well above the plasma frequency. The form of the temperature dependence is determined by the balance between phonon emission and their reabsorption across the gap. Measurement of the temperature-dependence of the photoinduced absorption, transmission, or reflection can thus give direct information on the QP population as a function of time and temperature, and consequently also a great deal of information about the gap itself. The model calculation for the temperature dependence of the QP population recently proposed [1] gives very distinct predictions regarding the temperature dependence of the photoinduced change in optical constants for different gap symmetries and for different temperature dependences of the gap. Here we show photoinduced transmission data through thin films of for a large range of doping and as a function of temperature, from which we deduce the temperature and doping dependence of the gap as well its symmetry.

2. EXPERIMENTAL DATA

For these measurements thin films of substrates were used with O concentration adjusted by annealing at high temperatures. The experimental set-up consists of a Ti: sapphire laser giving 100 fs pulses at 800 nm. The details of the experimental setup as well as time-domain data have been shown in detail in ref. [1].

The photoinduced transmission amplitude,

as a function of temperature is

shown in Fig. 1 for three different doping levels. In the underdoped sample data, the fall of the photoinduced transmission amplitude is asymptotic and occurs at progressively higher

temperatures as increases. The temperature T* where

falls to approximately 10%

of its maximum value is shown by the arrows for three different in Fig. 1. For small however, the amplitude of the photoinduced transmission falls much more rapidly and close

to

(Although many samples with different were measured, only three such temperature

dependences are shown in Fig. 1 for reasons of clarity.)

3. DISCUSSION

The theoretical model for the photoinduced transmission amplitude [1] predicts very different temperature dependences, depending on the temperature dependence of the gap itself. In the case of a temperature independent gap,

Evolution of the Gap Structure

221

where is the gap, is the effective number of phonons per unit cell emitted in the QP recombination process, is the typical phonon frequency, and N ( 0 ) is the density of states at A fit to the data for the two samples with using this formula is shown in Fig. 1. The predicted photoinduced transmission amplitude falls to zero exponentially at

high temperatures in good agreement with the data, and one clearly cannot speak of a gap opening at some specific temperature. The criterion for choosing an onset temperature T* is therefore somewhat arbitrary and signifies the point when (Here we have chosen T* at the point where the amplitude of the signal falls to 10% of maximum.) If the gap closes at a well-defined temperature due to a collective effect—i.e., as as in the BCS scenario—the relaxation bottleneck dissappears at this temperature and the formula is somewhat modified [1], and one obtains a sharp drop of the photoinduced transmission amplitude at as shown by the fit in Fig. 1 for The temperature dependence data clearly imply a distinction between the underdoped case with and samples near optimum doping with We emphasize that the two cases are qualitatively different. A T-dependent BCS-like collective gap, which closes at cannot be used to describe the asymptotic behaviour at high temperatures, whereas in the

T-independent gap model, falls far too slowly at high T to be able to describe the data near optimum doping. The fact that no change in is observed at in the underdoped phase implies that the DOS and gap structure are also unchanged at from which we can conclude that phase coherence is established at with no change in pairing amplitude (via the sum rule). An important aspect of the phase diagram is the crossover region between the two regions of doping. As already discussed, falls very rapidly near and the BCS

222


gap model fits the data very well (including the slight maximum, which is observed in the data below However, close examination of the plot for in Fig. 2 above shows that some photoinduced transmission is present up to temperatures as high as 130 K. This cannot be described by either model, and is well beyond the temperature at which pair amplitude fluctuations are expected to play a role. The fit to the data over the entire temperature range is now made using a two-component model, with both gaps present

simultaneously. The values of the two gaps are found to be very similar in this doping region. This observation is consistent with the fact that only a single exponential recombination time is observed in the 0–3 ps timescale in the time-resolved data. A further independent signature of the coexistence of two gaps near optimum doping comes from the anomaly in the QP recombination time

observed in the time-evolution

of the photoinduced transmission in our experiments. The recombination time has been shown to behave as [1,2]. For a BCS-like collective gap, so is expected to diverge at We observe no such divergence for which is consistent with a T-independent gap. However, near optimum doping a divergence at

temperatures has been observed by a number of authors [1,3,4], corroborating the evidence from the temperature dependence of In Fig. 3, we plot and from an exponential fit to our time-resolved data for different clearly showing the evolution of two gaps with doping near A somewhat unexpected feature of the data is that over the whole region of doping, the gap appears to be described very accurately using an isotropic gap [1]. It is easy to see

qualitatively why the data are not consistent with a d-wave gap model. Because a d-wave gap implies the existence of QP excitations down to the lowest temperatures, we would expect

to keep increasing down to the lowest temperatures with no real bottleneck

in the QP recombination above

The model calculation [1] for the QP recombination


223

dynamics in the case of a d-wave gap (calculated for 2D and 3D) gives a very distinct form of temperature dependence, as shown in Fig. 4. The data, which fits very well to the isotropic model, clearly cannot be fitted by the d-wave model. A comparison of the predictions for the T-dependence of the induced transmission recombination time intensity

224

dependence, and time-dependence of


between the models with the data for s and d gap symmetries is shown in Table 1. It should be noted that the experimental evidence for d-wave behavior is strong in where many of the tunneling and photoemission measurements have been performed. YBCO, however, has an orthorhombic structure in which the d-wave representation is—strictly speaking—not allowed by symmetry and an s-wave component is necessary to satisfy symmetry requirements. One possible reason for the different gap symmetry found here compared to other experiments is that the order parameter may be different in the bulk of the sample than on the surface. The possibility that the order parameter varies as where s and d are the amplitudes of the s- and d-wave components and and describe the changing magnitude of the two components at a distance z away from the surface was suggested by K. A. Müller in connection with the results of muon penetration depth measurements [5]. The present optical measurements are performed in transmission, which means that we are mainly probing the bulk of the sample. (The absorption length of light in YBCO is In contrast, many of the techniques that have shown the presence of a d-wave order parameter—for example, tri-crystal tunneling experiments—probe the order parameter on the surface [6]. It is possible that the surface has an intrinsically different electronic structure from the bulk. Surface effects become important when the characteristic length of the experimental probe is comparable to the unit cell, as in tunneling experiments [6] or photoemission [7]. Another possible experimental problem is that in some experiments, YBCO samples are cleaved and that these surfaces are not the same as the bulk. We have found that photoinduced optical reflection experiments (which are also surface sensitive, albeit much less so than photoemission or tunneling) on cleaved YBCO crystals often show rather irreproducible behavior, from which we deduce that possibly cleavage occurs in regions where the stoichiometry is different from the bulk. In contrast, we have found that polished YBCO samples show sample-independent and reproducible behavior, which is essentially identical to the thin-film transmission experiments. Finally, we mention the important possibility that the d-like gap behavior arises because of the existence of localized states in the gap in YBCO and is not, in fact, intrinsic. Experiments like tunneling and photoemission cannot distinguish between such localized states and QP states because only a spectral density is obtained. However, time-domain measurements by Stevens et al. [8] show that in YBCO temperature-activated excitations exist whose activation energy is significantly smaller than the QP gap In fact, the values found are consistent with activation from intragap states. However their time-dynamics is qualitatively different than the QP recombination


225

dynamics. It is very slow and has a very different T-dependence, consistent with localized state relaxation. Further time-resolved investigations of the systematics of the localized intragap states will hopefully elucidate this issue.

4. CONCLUSION The presented data show that at low doping the energy “pseudogap” in

is

large and T-independent. Upon further doping it decreases, according to an inverse law 1/ p , where p is the hole concentration [ 1 ]. Near optimum doping, another gap with a meanfield-like temperature dependence becomes apparent and coexists with the “pseudogap,” the two being comparable in magnitude. Its temperature dependence is suggestive of a collective excitation (e.g., Bogolyubov mode) or collective BCS-like gap. Apart from exhibiting a BCS-like temperature dependence the T-dependent gap has one further characteristic of BCS phenomenology, namely that it appears at In contrast, the underdoped state exhibits a gap, which shows no change at and is better viewed as a splitting between energy levels [9].

Whereas it is clear that the coexistence of the two gaps near optimum doping is difficult to understand in a homogeneous medium, the data can be easily understood by invoking the two-component paradigm with the existence of stripes or clusters as shown schematically in Fig. 5. In this scenario, the two gap regions are separated in real space and correspond to high carrier density and low carrier density in adjacent regions a few unit cells apart. The seemingly alternative viewpoint—where the QPs are separated in momentum space, in which one part of the Fermi surface (FS) has a collective gap whereas another has a

T-independent gap—is in fact quite compatible with the stripe picture, provided the momentum relaxation time between such regions in k-space is much shorter than the QP recombination time. This is indeed the case, because the measured QP recombination rates of

are slow compared to the momentum scattering rates of

226


inferred from infrared reflectivity measurements [10]. In other words, the excited QPs do not travel very far in real space before they are scattered elastically, but show recombination dynamics according to the region in space where they find themselves. This explains the common FS in ARPES and shows why its appearance is consistent with the stripe scenario. ACKNOWLEDGMENTS

We acknowledge V. V. Kabanov for valuable discussions, funding from the EC Ultrafast network, and also J. E. Evetts, G. A. Wagner, and L. Mechin for sending us YBCO samples

used in the present investigation. REFERENCES 1. 2. 3. 4. 5.

V. V. Kabanov, J. Demsar, B. Podobnik, and D. Mihailovic, Phys. Rev. B 59, 1497 (1998). A. Rothwarf and B. N. Taylor, Phys. Rev. Lett. 19, 27 (1967). D. Mihailovic, B. Podobnik, J. Demsar, G. Wagner, and J. Evetts, J. Phys. Chem. Sol. 59, 1937 (1998). S. G. Han, Z. V. Vardeny, K. S. Wong, O. G. Symco, and G. Koren, Phys. Rev. Lett. 65, 2708 (1990). K. A. Müller, private communication.

6. J. H. Miller Jr., Q. Y. Ying, Z. G. Zou, N. Q. Fan, J. H. Xu, M. F. Davis, and J. C. Wolfe, Phys. Rev. Lett. 74,

2347(1995). 7. See, for example, M. C. Schabel et al., Phys. Rev. B 57, 6090 (1998). 8. C. J. Stevens et al., Phys. Rev. Lett. 78, 2212 (1997). 9. A. S. Alexandrov, V. V. Kabanov, and N. F. Mott, Phys. Rev. Lett. 77, 4796 (1996). 10. D. Basov et al., Phys. Rev. Lett. 77, 4090 (1996).

Local Lattice Distortions in Doping Dependence H. Oyanagi,1 J. Zegenhagen,2 and T. Haage3

Doping dependence of the local lattice distortions in thin films epitaxially grown on has been studied by polarized extended x-ray absorption fine structure (EXAFS). We find that the in-plane Cu-O bond distribution becomes asymmetric and broadens below a characteristic temperature T*. The

results indicate that distorted domains are characterized by an elongated in-plane Cu2-O2/O3 bond. The observed Cu-O relative displacement shows signatures of local anomalies at and a few tens of Kelvin above i.e., a sharp increase of the

magnitude of Fourier transform

a maximum at

and a minimum

at The outcome suggests that temperature and doping dependent anomalies in the local lattice dynamics, below T* in the normal state, are universal to the superconductors.

1. INTRODUCTION

A theory on the basis of spin-charge separation has been proposed as a possible mechanism of superconductivity in the early years of superconductivity (HTSC) research [1]; however, the lattice was assumed to be homogeneous on the basis of available crystallographic studies. Recently, a number of experimental techniques have shown that the lattice is rather inhomogeneous. In fact, the new models, based on phase separation in carrier-rich and carrier-poor regions are coming up to review the situation [2]. At the experimental front, a combination of extended x-ray absorption fine struc-

ture (EXAFS) and x-ray diffraction (XRD) has demonstrated that there are lattice stripes of undistorted and distorted local structure [3] alternating with a mesoscopic length scale comparable to the coherence length in HTSC. Structural evidence for two-component electronic system was also provided by neutron pair distribution function (see, e.g., Egami et al. 1 2

Electrotechnical Laboratory, Umezono 1-1-4 Tsukuba Ibaraki 305, Japan. Max-Planck Institute fur Festkorperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany.


227

228

Oyanagi, Zegenhagen, and Haage

in Ref. [4]), and other techniques have also indicated a temperature T* [5], below which the planes may have ordered stripes of carrier-rich and carrier-poor domains [6]. As the inhomegeneous structure became evident, renewal of interests in an alternative mechanism,

i.e., phonon scattering, the lattice effects on

superconductivity have attracted much

attention recently [7–9].

Spin susceptibility obtained from the NMR relaxation rate

[10] and inelastic

neutron scattering [ 11] have shown anomalies well above Including EXAFS, these are the techniques that provide snapshot of lattice. Among these, EXAFS is an ideal means that

reflects the radial distribution of atoms with a time scale of sec. In-plane lattice anomalies observed above are one of the general features of HTSC materials: lattice distortions have been observed for and Bianconi et al. [15] attributed the local distortion to the charge ordering into stripes of locally distorted low temperature tetragonal (LTT) and undistorted low temperature orthorhombic (LTO) regions. The stripe structure has been proposed to a mechanism of enhancement. In case of much attention has been paid to out-of-plane oxygen motions, e.g., EXAFS studies observed a large apical oxygen displacement around attributed to the bipolaron tunneling [16]. However, the origin of the out-of-plane phonon anomaly is still a

controversial problem, because the frequency shift in infrared (IR) and Raman experiments are interpreted as an anomalous change in phonon self-energy [17] or fluctuations associated with superconductivity [18] is smaller than the reported change in tunneling frequency [16]. However, our polarized EXAFS study of optimally doped have shown the in-plane local lattice anomaly below a characteristic temperature that lies above and close to the characteristic temperature for opening of a spin gap T*. Whether the in-plane lattice anomaly is related to the charge stripe or spin–phonon interaction is an interesting problem. Spin gap observed in various experiments such as NMR [10], neutron scattering [11], and transport properties [19,20] have been related to short-range ordering

of spin singlets [21]. In case of

however, the T* is found to coincide with

the in an optimally doped sample [22]. In this communication, we report the doping dependence of in-plane lattice anomalies in over a wide range of oxygen doping

and discuss the results in relation to the lattice-charge stripes [3] and spin and charge excitations.

2. EXPERIMENTAL

Highly oriented

thin-film samples were prepared by a pulsed laser ablation

technique using a KrF excimer laser [23] onto single crystal substrates of at 750°C. Oxygen stoichiometry (y) was carefully controlled by an oxygen partial

pressure after the growth; samples were slowly cooled down from the growth temperature under oxygen pressure of 0.1–200 mbar, yielding samples with ranging from 31.3 to 90 K. Temperature dependence of resistivity was measured prior to the experiment for all samples. As-grown samples (100-nm thick) showed a sharp superconducting transition with a typical transition width of about 5 K indicating a high degree of oxygen ordering along the Cu1-O1 chain and homogeniety. Three samples with a sharp transition width have been chosen to study doping dependence of local distortions in over a wide range in In this work, we focus our attention on two samples

Local Lattice Distortions in

229

Polarized EXAFS experiments were performed in a fluorescence mode at the 27-pole wiggler station BL13B1 of the Photon Factory using synchrotron radiation from a 2.5 GeV storage ring. For polarized fluorescence detection at low temperature down to 15 K, a closedcycle He refrigerator was mounted on a high-precision two-axis Huber 422 goniometer [24]. For an in-plane polarization geometry, a sample was mounted on the horizontal and a grazing-incidence angle was optimized for monitoring a thin layer of 100 nm (1.6 degree). The stability of temperature achieved by a precision temperature controller CONDUCTUS LT20 was within The fluorescence signal was collected by a solid-state x-ray detector array consisting of 19 pure Ge elements [25 ]. The output of each detector was energy-analyzed in order to remove channels affected by strong diffractions from substrates. EXAFS scans were made by a sagittaly focusing Si(111) double crystal monochromator. A typical energy resolution was 2 eV at 9 keV with a photon flux of about photons/sec when the storage ring was operated at 350 mA.

3. RESULTS Normalized EXAFS oscillations are Fourier-transformed (FT) after multiplying k over the range Upper and lower columuns of Fig. 1 compare the magnitude and imaginary part of the Fourier transform (FT) for the EXAFS data for taken with the electrical field vector parallel to the ab plane at and 40 K. For comparison, the results for undistorted phase are also shown. Shaded area indicates the difference in the FT magnitude between the two data. In Fig. 2, the magnitude and imaginary part of the FT for the data for thin-film measured at and 20 K are shown where the FT results for are also compared. In Figs. 1 and 2, a prominent peak observed at around consists of square planar oxygen atoms (O2, O3) of the pyramid and a small contribution from pair correlation. Theoretical Cu-O phase shift functions obtained by FEFF6 [26] were corrected in the FT. As can be seen in Figs. 1 and 2, the site-averaged Cu-O pair correlation around 2.0 A shows asymmetry and broadens on decreasing temperature. In Fig. 3, the magnitude of the FT for is plotted as a function of T over a wide temperature range (20–300 K) for the two samples and 55 K). Figure 4 shows the same plot as a function of normalized T, i.e., A constant increase of FT magnitude with the decrease of T is due to the thermal vibration term. At however, the magnitude increases its intensity rapidly and a sharp drop giving rise to a minimum at

4. DISCUSSION In the unit cell of optimally doped copper atoms (Cu2) in the plane are coordinated by square-planar oxygen atoms (O2 and O3) and an apical oxygen atom (O4), whereas Cu1 atoms form a linear chain (Cu1-O1) along the b axis. The in-plane polarized Cu K-EXAFS probes the Cu-O pair correlation averaged over all Cu-O pairs parallel with the ab plane. The contribution of Cu1-O1 to the average coordination (2y — 4)/3 in a twinned sample is for sample and for sample As can be seen in Figs. 1 and 2, the Cu-O peak in the FT magnitude becomes asymmetric with the decrease of T. Inspecting the imaginary part that

230


grows only at the large R side, one can find that the asymmetry is caused by an additional component with a slightly longer Cu-O bond, which indicates that the local distortion occurs at low temperature. One can imagine various models of local lattice distortions involving in-plane Cu-O bonds. One of such models is a pseudo-Jahn–Teller distortion in which two opposing Cu-O bonds are shortened while the other two perpendicular bonds are elongated [27]. This model would result in the symmetric variation (decrease) of the first peak because of the interference between the two closely separated Cu-O bond lengths, which is in disagreement with the observation. Other possible distortions are a LTO-like and LTTlike tilts of pyramids. In the former distortion, where all square-planar Cu-O bonds are elongated, the site-averaged Cu-O peak would be symmetric and shift to a large R

direction. In the latter case, in which two Cu-O square-planar bonds (Cu2-O3) are elongated while other two bonds (Cu2-O2) are kept constant in length, an asymmetric and broadened pair correlation is expected. Thus we conclude that the observed asymmetric

pair distribution indicates the phase separation into the distorted domain with an elongated Cu-O bond and undistorted domains. Here we omitted the effect of Cu1-O1 linear


231

chain because the Cu1-O1 distance (1.94 Å) is close to the Cu2-O2/O3 distances (1.93– 1.96 Å). In the following, we describe the qualitative results of FT as a function of T. In Fig. 1, in the upper column, we show the highest temperature data which shows an asymmetry in the FT peak Comparing the low temperature data for the two samples it seems that the tilt angle is almost the same. A detail study on this is an object of future publications [28]. The FT magnitude is a sensitive measure of the change in the pair correlation function either due to local distortions and/or phonons. As illustrated in Figs. 3 and 4, the FT magnitude for both samples show anomalous T-dependence above that can be described as a universal behavior if T is normalized to i.e., onset around mamimumat and a minimum at We define as an onset temperature of local lattice anomaly. A remarkable feature is that as seen in Fig. 4, the maximum around 1.3 exactly coincides for the two samples. Two major factors which affect the FT magnitude are considered, i.e., the degree and fraction of distortion and magnitude of relative displacement of the Cu-O distance. Because the lattice distortion evidenced by an asymmetric FT peak is observed

232

over a wide T-range below


, the anomalous variation of FT peak magnitude indicates

local phonon anomalies observed by other techniques. Arai et al. [11] reported that T-

dependence of S(Q, E) obtained by an inelastic neutron scattering technique for optimally doped shows an anomalous increase around 120 K close to (123 K) for A sharp increase of S(Q, E) maximizing at is ascribed to an expansion of dynamical correlation length associated with a local structural distortion [11]. A sharp increase in the FT magnitude at is explained by the increased correlation in the Cu-O stretching vibration, which would decrease the mean-square relative displacement (MSRD) and thus sharpen the radial distribution as observed as a maximum at Similar anomalies have been reported in internal friction at The present observation that is consistent with Raman experiments reported by Ruani and Ricci [30] who observed anomalies in the electronic peak at 1.6 It is interesting to note that the T-range for a possible increased correlation in Cu-O stretching coincides with the anomalous S(Q, E) variation around As shown in Fig. 5, for slightly underdoped agree well with defined as an onset of deviation from T-linear resistivity associated with a pseudogap opening of a spin excitation [19]. Because of insufficient number of data points, for underdoped the relation between the local lattice distortion and pseudogap opening is not clear but the formation of distorted domain is consistent with the phase diagram of pseudogap [31]. Thus we conclude that the signatures


233

234


of local phonon anomalies below

occur after the opening of pseudogap in spin and charge excitation. Mihailovic et al. [31] reported that below T*, in-plane optical conductivity changes from a single-component to two-component carrier regime. Such a two-component carrier picture below T* is consistent with a charge stripe of a distorted (localized) and undistorted (itinerant) domains [3]. A variety of interpretations are possible for T*; in the Bose–Einstein condensation picture, T* is interpreted as the onset of pairing which can be separate with condensation [31]. However, T* observed in and Raman frequency shifts is related to the onset of short range order of Zhang–Rice singlets [21]. Bianconi et al. [15] proposed a amplification mechanism due to the shape resonance of charge stripes. It is still a long way from distinguishing the mechanism of HTSC, but the structural phase separation into distorted and undistorted domains below T* and local phonon anomalies below seem to be a universal feature of underdoped HTSC materials. 5. CONCLUSION

We have reported the doping dependence of the local lattice distortions in epitaxial thin films using ab plane polarized EXAFS. The results indicate that the inplane Cu-O pair correlation becomes asymmetric on lowering T associated with the local distortion of units below which can be explained by LTT-like tilting of

units. The FT magnitude shows a universal feature, i.e., maximum and minimum, at certain characteristic temperature normalized by These anomalous changes in the FT magnitude above suggest that there exist local phonon anomalies in ab plane Cu-O bonds. Comparison of the present results with c axis polarized experiments suggests that the MSRD of apical oxygen correlates with the in-plane lattice anomalies. Future experiments on an untwinned single crystal would provide us details of the anomalies in local lattice dynamics and hopefully the role of lattice dynamics and charge stripes in HTSC mechanism. ACKNOWLEDGMENTS

The authors are thankful to A. Bianconi, N. L. Saini, A. Lanzara, D. Mihailovic, M. Arai, T. Ito, C. H. Lee, J. Ranninger, and K. Yamaji for valuable discussions. REFERENCES 1. P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett. 58, 2790 (1987). 2. V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990); C. Di Castro and M. Grilli, in Phase Separation in Cuprate Superconductors, ed. by K. A. Muller and G. Bendek (World Scientific, Singapore, 1992), p. 85. 3. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, and Y. Nishihara, Phys. Rev. B 54, 12018 (1996); A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, A. Kvick,

and P. G. Radaelli, Phys. Rev. B 54, 4310 (1996). 4. T. Egami and S. J. L. Billinge, Prog. Mater. Sci. 38, 359 (1994). 5. T* is defined as an onset temperature of pseudogap opening in spin or charge excitation spectra, whereas is defined as an onset of local phonon anomalies and The onset temperature of local lattice distortion is close to T* rather than 6. C. Berthier et al., Physica C 235–240, 67 (1994); M. A. Teplov et al., in ., Superconductivity 1996: Ten Years after the Discovery, ed. by E. Kaldis, E. Liarokapis, and K. A. Muller (Kluwer Academic Publishers,

Dordecht, 1997) p. 531; Y. Wu, S. Pradhan, and P. Boolchand, Phys. Rev. Lett. 67, 3184 (1991).

Local Lattice Distortions in 7. Lattice Effects in

235

Superconductors, ed. by Y. Bar-Yam, T. Egami, J. Mustre-de Leon, and A. R.

Bishop (World Scientific Publishing, Singapore, 1992). 8. T. Egami and S. J. L. Billinge, Prog. Mater. Sci. 38, 359 (1994). 9. Phase Separation in Cuprate Superconductors, ed. by K. A. Muller and G. Benedek (World Scientific, Singapore, 1993); Phase Separation in Cuprate Superconductors, ed. by E. Sigmund and K. A. Muller (Springer-Verlag, Berlin, Heidelberg, 1994).

10. M. Takigawa, A. P. Reyes, P. C. Hammel, J. D. Thompson, R. H. Heffner, Z. Fisk, and K. C. Ott, Phys. Rev. B 43, 247(1991). 11. M. Arai, K. Yamada, Y. Hidaka, S. Itoh, Z. A. Bowden, A. D. Taylor, and Y. Endoh, Phys. Rev. Lett. 69, 359 (1992).

12. H. Yamaguchi, S. Nakajima, Y. Kuwahara, H. Oyanagi, and Y. Shono, Physica C 213, 375 (1993). 13. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996); N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55, 12759 (1996); N. L. Saini, A. Lanzara, A. Bianconi, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Physica C 268, 121 (1996), 14. A. Lanzara, N. L. Saini, A. Bianconi, Y. Soldo, F. C. Chou, and D. C. Johnson, Phys. Rev. B 55, 9120 (1997). 15. A. Bianconi, M. Missori, N. L. Saini, H. Oyanagi, H. Yamaguchi, D. Ha, Y. Nishihara, and S. Delia Longa, J. Supercond. 8, 545 (1995). 16. J. Mustre de Leon, S. D. Conradson, I. Batistic, A. R. Bishop, I. D. Raistrick, M. C. Aronson, and F. H. Garzon, Phys. Rev. B 45, 2447 (1992).

17. C. Thomsen, B. Friedl, and M. Cardona, Sot. State Commun. 75, 2447 (1992). 18. H. S. Obhi and E. K. H. Salje, J. Phys. Condens. Matt. 4, 195 (1992).

19. T. Ito, K. Takenaka, and S. Uchida, Phys. Rev. Lett. 70, 3995 (1993). 20. C. Ludwig, Q. Jiang, J. Kuhl, and J. Zegenhagen, Physica C 269, 249 (1996). 21. T. Tanamoto, K. Kohno, and H. Fukuyama, J. Phys. Soc. Jpn. 61, 1886 (1992); T. Tanamoto, K. Kohno, and H. Fukuyama, J. Phys. Soc. Jpn. 62, 717 (1993). 22. H. Kimura, H. Oyanagi, T. Terashima, H. Yamaguchi, Y. Bando, and J. Mizuki, Jpn. J. Appl. Phys. 32, 584 (1993); H. Oyanagi and J. Zegenhagen, J. Supercond. 10, 415 (1997). 23. T. Haage, J. Q. Li, B. Leibold, M. Cardona, J. Zegenhagen, H.-U. Habermeier, A. Forkl, Ch. Joos,

R. Warthmann, and H. Kronmuller, Sol. State Commun. 99, 553 (1996). 24. H. Oyanagi, R. Shioda, Y. Kuwahara, and K. Haga, J. Synchrotron. Rad. 2, 99 (1995); H. Oyanagi, J. Synchrotron. Rad. 5, 48 (1998).

25. H. Oyanagi, M. Saito, and M. Martini, Nucl. Instrum. Meth. A403, 58 (1998). 26. J. J. Rehr, S. I. Zabinsky, and R. C. Albers, Phys. Rev. Lett. 69, 3397 (1992). 27. M. Bacci, Jpn. J. Appl. Phys. 27, L1699 (1988). 28. H. Oyanagi et al., unpublished, 1998. 29. G. Cannelli, R. Cantelli, F. Cordero, G. A. Costa, M. Ferretti, and G. L. Olcese, Europhysics Lett. 6, 271 (1988).

30. G. Ruani and P. Ricci, Phys. Rev. B 55, 93 (1997). 31. D. Miailovic, T. Mertelj, and K. A. Muller, Phys. Rev. B 57, 6116 (1998).


Fermi Surface of Superconductor by Angle-Scanning Photoemission M. C. Asensio,1 J. Avila,1 N. L. Saini,2 A. Lanzara,2 A. Bianconi,2 S. Tajima,3 G. D. Gu,3 and N. Koshizuka3

We made a complete k space mapping of the Fermi surface of a

(Bi2212)

superconductor at the optimum doping

by angle

scanning photoemission using high-intensity synchrotron radiation. We measured the Fermi surface of the Bi2212 in the even and odd symmetry to discriminate different initial states by matrix element effects. The observed features are discussed in terms of a strongly interacting electron gas forming a superlattice of quantum stripes in the plane.

1. INTRODUCTION

Angle-resolved photoemission has been a useful tool to study single particle properties of superconducting materials [1]. There are two main approaches of angle-resolved photoemission (ARPES) that are used to study the Fermi surface features of the superconductors. The standard method is based on the measurement of energy distribution curves (EDC) in all high-symmetry directions of the Brillouin zone for determination of the points in which the quasi-particle peaks cross the Fermi level. The second approach is based on measuring the photointensity within a narrow energy window at the Fermi energy defined by the spectrometer resolution to get the distribution of spectral weight near the

Fermi level in the k space [2]. The second approach has an advantage over the standard EDC method because it provides a global view of the Fermi surface, whereas the standard EDC 1

LURE, Bat 209D Universite Paris-Sud, F-91405 Orsay, France, & Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain. 2 U n i t à INFM and Dipartimento di Fisica, Università di Roma “La Sapienza” P. A. Moro 2, 00185 Roma, Italy. 3 Superconductivity Research Laboratory, ISTEC, Shinonome 1-10-13 Koto-ku, Tokyo 135, Japan.


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method may suffer from the extrapolation in the process of constructing the Fermi surface image using dispersion curves. However, the EDC method has an advantage over the new angle-scanning approach because the EDC method also provides important information on

the identification of the quasi-particle bands and their dispersion below the Fermi level. In addition, the angle-scanning method is constrained by the spectrometer resolution, and it may not be straightforward to differentiate the states representing the true Fermi surface and the occupied electronic states, below the Fermi surface, lying within the resolution. Moreover, the measured Fermi surface image is affected by the matrix element effects depending

on the angles between the polarization vector of the photon beam and the wavevectors of the initial state and the final state [3]. In this paper, we report the Fermi surface of Bi2212 system measured by angle-scanning photoemission using high intensity of polarized synchrotron radiation to address the mentioned points. We have made the measurements in two different polarization geometries to identify the matrix element effects.

2. EXPERIMENTAL

A single crystal of size

grown by floating zone method was used for the

experiments. The crystal was well characterized for its transport and structural properties. It is as grown at optimum doping with a sharp superconducting transition of 91 K [4]. Its structure, studied by synchrotron radiation diffraction, shows the characteristic features of superconducting samples with satellites due to the incommensurate modulations of both BiO and plane [5]. The crystal was aligned by standard method using specular laser reflection from the crystal surface. The clean and flat surface was obtained by cleaving the crystal at room temperature. The experiment was repeated with different cleavage. Each cleavage gave the same Fermi surface, proving the high quality of the crystal, and the surface

was stable for several days. The experiments were carried out at the Laboratoire pour l’Utilisation du Rayonnement Electromagnetique (LURE) (Orsay-France) on the SU6 beamline. The experiments were performed in an ultra-high vacuum (UHV) chamber equipped with an angle-resolved hemispherical analyzer and a high-precision manipulator that permits rotation in the full 360° azimuth emission angle and 90° polar emission

angle relative to the surface normal [6]. The photoelectron intensity at the Fermi level and bellow was recorded along a series of azimuth scans. The sample was rotated around its normal and the intensity was recorded every 1.5° at fixed theta, with an absolute angular precision better than 0.5°. The Fermi surface map was obtained by centering the electron energy window at the and collecting the electrons within an energy window of the order of spectrometer resolution using a photon energy of 32 eV. The polarization vector of the synchrotron light, the direction of the photon beam, and the surface normal were kept in the same horizontal plane, called the scattering plane, for all the measurements. The mirror plane is defined by

the sample normal and the direction of the emitted photoelectron selected by the detector position. The detector is moved in the fixed mirror plane by changing the polar angle in order to select different values of the wave vector in the superconducting plane The direction of the initial state is selected by rotating the sample around its normal, which is collinear with the crystallographic c axis.

Fermi Surface of

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239

We mapped the Fermi surface in two different experimental geometries, shown picto-

rially in Fig. 1. The upper picture shows the geometry for the “even” symmetry, whereas the lower picture represents the geometry used for the “odd” symmetry. In the even symmetry, the scattering plane is coplanar with the mirror plane, whereas in the odd symmetry the mirror plane is orthogonal to the scattering plane. The initial states of symmetry (formed by a mixing of and orbitals) are even with respect to the mirror plane for parallel to the Cu-O-Cu bonds whereas they are odd with respect to the mirror plane for at 45° to the Cu-O-Cu bonds [1,7,8]. In the even experimental geometry, the transition from these states forming the conduction band is fully allowed for in the and equivalent directions and forbidden for

in the

and equivalent directions. On the contrary, in the odd geometry, the

transition from these states is fully allowed for

in the and equivalent directions and forbidden for in the and equivalent directions. All the experiments were repeated with different cleavage in several experimental runs and a perfect reproducibility of the experimentally observed Fermi surface features was achieved ascertaining intrinsic nature of the features and high quality of the crystal used.

3. RESULTS AND DISCUSSION The Fermi surface maps of Bi2212 superconductor are shown in Fig. 2. The upper panel shows the Fermi surface measured in the even experimental geometry, whereas the lower

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panel shows the one measured in the odd experimental geometry. The brighter regions indicate higher intensity of emitted photoelectrons excited from the initial state having constant energy and in-plane wavevector spanned over reciprocal space of the two-dimensional (2D) plane. In the Fermi surface map recorded using the experimental even geometry (upper panel), the photointensity along the direction in the first Brillouin zone is absent due to matrix element effects. In this experimental geometry, the transitions from initial states of (Cu symmetry are allowed having even symmetry with respect to the mirror plane in the direction, and hence the Fermi surface features around the and directions can be clearly identified in this map. However, in the Fermi surface map recorded using the experimental odd geometry (lower panel of Fig. 2), the photointensity along the and direction in the first

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Superconductor

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Brillouin zone is absent due to matrix element effects. In this experimental geometry, the transitions are allowed from initial states having odd symmetry with respect to the mirror plane in the and direction, and hence the Fermi surface features around these

directions can be clearly identified in the map. Thus the combination of the two maps completes the information on the Fermi surface features of the Bi2212 superconductor. One can obtain precise Fermi surface parameters by using the fact that, within the Fermi liquid framework, the sum rule for ARPES is that relates the energy-integrated ARPES intensity to the momentum distribution The n(k) shows discontinuities at each Fermi wave-vector These discontinuities are smeared out by the finite angular resolution. The discontinuities at show singularities in the modules of the gradient corresponding to the Fermi surface crossings. Thus the gradient of the Fermi surface measured in angle-scanning mode is a useful method to define precise Fermi surface parameters [9]. We have used this approach and calculated the gradient of the two Fermi surface maps. The modules of the Fermi surface gradient is shown in Fig. 3. All Fermi surface features could be seen clearly in the gradient however with photointensity doublets. The

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outer part of the gradient map represents the real Fermi surface and unaffected by the spectral photointensity below the Fermi level [9]. A careful observation of the photointensity maps (Fig. 2) and their gradients reveals global information about the topological features on the Fermi surface of the Bi2212 system. The even symmetry image reproduces most of the features observed in the earlier studies (e.g., the umklapp bands in the direction due to superstructure in the Bi-O plane), the small pockets around and expected in lightly doped Mott insulators, absence of photointensity around the and and asymmetry of the and directions [10–14]. The shadow bands associated with the formation of small pockets could be seen clearly in the odd symmetry image (lower panel of Fig. 2) in which the transition along the are fully allowed by the selection rules. The Fermi surface in Fig. 2 clearly shows the shadow features around locations. It should be mentioned that the observation of the shadow band features has been controversial due to the fact that presence of umklapp reflections along the direction complicates their identification on the Fermi surface. However, the present measurements made in the even and odd symmetries allow us to identify the presence of these bands along the and directions ascertaining their presence. The present Fermi surface maps confirm previous results obtained at high k values that have shown

shadow band features to appear with even higher photointensity around locations [15]. The shadow features can be associated with coupling of electrons with spin density waves in the plane of superconductors [13, 16, 17] or to structural origin [18,19]. We observe a strong suppression of photointensity around the M points. The suppression of spectral weight has been argued to be due to spin density wave background [21], quasi-particle decay into holons and spinons [20], and several other related reasons. We argue that a phase segregation giving quasi-one-dimensional charge ordering in stripes along direction [5,22] is the reason for the same. It is now being established that the charge segregation in stripes within the plane plays important role in the electronic

structure of the cuprate superconductors [23]. In the Bi2212 system, the charge ordering in stripes within the plane along the direction has been shown by Cu K-edge extended x-ray absorption fine structure (EXAFS) and anomalous diffraction measurements [5,22]. In fact, the photointensity suppression around the M points is asymmetric with a well-defined nesting vector in the diagonal direction that is the second harmonic of the main lattice modulation. In summary, a complete k space mapping of the Fermi surface features of Bi2212 superconductor at the optimum doping is done by angle-scanning photoemission in the even and odd symmetry, allowing us to provide a clear identification to the Fermi surface features. The results show absence of photointensity at the Fermi surface around the M points. In conclusion, the Fermi surface reported in the present work indicate that electrons moving in a superlattice of quantum stripes are strongly interacting with spin and charge collective excitations.

ACKNOWLEDGMENTS The experiment was done at the Spanish–French beam line of LURE with the support of the large-scale Installation Program and the Spanish agency DGICYT under grant

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Superconductor

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PB-94-0022-C02-01. This work was partially funded by Istituto Nazionale di Fisica della Materia (INFM) and Consiglio Nazionale delle Ricerche (CNR) of Italy.

REFERENCES 1. Z.-X. Shen, W. E. Spicer, D. M. King, D. S. Dessau, and B. O. Wells, Science 267, 343 (1995); Z.-X. Shen and D. S. Dessau, Phys. Rep. 253, 1 (1995) and references therein. 2. A. Santoni, L. J. Terminello, F. J. Himpsel, and T. Takahashi, Appl. Phys. A 52, 229 (1991). 3. See, e.g., M. Lindroos and A. Bansil, Phys. Rev. Lett. 77, 2985 (1996) and references therein. 4. G. D. Gu, K. Takamuka, N. Koshizuku, and S. Tanaka, J. Crystal Growth 130, 325 (1993); S. Tajima, G. D.

Gu, S. Miyamoto, A. Odagawa, and N. Koshizuka, Phys. Rev. B 48, 16164 (1993). 5. A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, Å. Kvick, and P. G. Radaelli, Phys. Rev. B 54, 4310 (1996); A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishihara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996).

6. J. Avila, C. Casado, M. C. Asensio, J. L. Munoz, and F. Soria, J. Vac. Sci. Tech. A 13, 1501 (1995). 7. J. Ma, C. Quitmann, R. J. Kelley, P. Alméras, H. Berger, G. Margaritondo, and M. Onellion, Phys. Rev. B 51,

3832 (1995); R. J. Kelley, J. Ma, G. Margaritondo, and M. Onellion, Phys. Rev. Lett. 71, 4051 (1993); J. Ma, P. Almeras, R. J. Kelley, H. Berger, G. Magaritondo, X. Y. Cai, Y. Feng, and M. Onellion, Phys. Rev. B 51, 9271 (1995). 8. H. Ding, A. F. Bellman, J. C. Campuzano, M. Randeria, M. R. Norman, T. Yokoya, T. Takahashi, H. KatayamaYoshida, T. Mochiku, K. Kadowaki, G. Jennings, and G. P. Brivio, Phys. Rev. Lett. 76, 1533 (1996); H. Ding, J. C. Campuzano, A. F. Bellman, T. Yokoya, M. R. Norman, M. Randeria, T. Takahashi, H. Katayama-Yoshida, T. Mochiku, K. Kadowaki, and G. Jennings, Phys. Rev. Lett. 74, 2784 (1995); M. R. Norman, M. Randeria. H. Ding, and J. C. Campuzano, Phys. Rev. B 52, 615 (1995); M. R. Norman, M. Randeria, H. Ding, J. C. Campuzano, and A. F. Bellman, Phys. Rev. B 52, 15107 (1995). 9. Th. Straub, R. Claessen, P. Steiner, S. Hüfner, V. Eyert, K. Friemelt, and E. Bucher, Phys. Rev. B 55, 13473 (1996). 10. D. Marshall, D. S. Dessau, A. G. Loeser, C. H. Park, A. Y. Matsuura, J. N. Eckstein, I. Bozovic, P. Fournier, A. Kapitulnik, W. E. Spicer, and Z. X. Shen, Phys. Rev. Lett. 76, 4841 (1996).

11. H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria, M. R. Norman, T. Mochiku, K. Kadowaki, and J. Giapintzaki, Nature 382, 51 (1996). 12. A. G. Loeser, Z. X. Shen, D. S. Dessau, D. S. Marshall, C. H. Park, P. Fournier, and A. Kapitulnik, Science

273, 325 (1996); A. G. Loeser, D. S. Dessau, and Z. X. Shen, Physica C 263, 208 (1996). 13. J. Osterwalder, P. Aebi, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and K. Kadowaki, Appl. Phys. A 60, 247 (1995); P. Aebi, J. Osterwalder, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and K. Kadowaki, Phys. Rev. Lett. 72, 2757 (1994).

14. N. L. Saini, J. Avila, M. C. Asensio, S. Tajima, G. D. Gu, N. Koshizuka, A. Lanzara, and A. Bianconi, Phys. Rev. B 57, R11101 (1998). 15. N. L. Saini, J. Avila, A. Bianconi, A. Lanzara, M. C. Asensio, S. Tajima, G. D. Gu, and N. Koshizuka, Phys. Rev. Lett. 79, 3464 (1997).

16. H. Ushio and H. Kamimura, J. Phys. Soc. Jpn. 64, 2585 (1995). 17. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994); E. Dagotto and T. M. Rice, Science 271, 618 (1996) and references therein. 18. O. K. Andersen, O. Jepsen, A. 1. Liechtenstein, and I. I. Mazin, Phys. Rev. B 49, 4145 (1994). 19. D. J. Singh and W. E. Pickett, J. Supercond. 8, 583 (1995); Phys. Rev. B 51, 3128 (1995). 20. R. B. Laughlin, Phys. Rev. Lett. 79, 1726 (1997). 21. J. Schmalian, D. Pines, and B. Stojkovic, Phys. Rev. Lett. 80, 3839 (1998).

22. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996). 23. See, for example, the special issue on Stripe Lattice Instabilities and

A. Bianconi and N. L. Saini [J. Supercond. 10, No. 4 (1997)].

Superconductivity, edited by


Local Lattice Fluctuations and the Incoherent ARPES Background J. Ranninger1 and A. Romano2

We consider high-temperature superconductors to be composed of molecular

units that can undergo local charge fluctuations, alternating between quasi-free electrons (uncoupled to the lattice) and bipolaronic electron pairs (strongly coupled to local lattice deformations). On the basis of a generalized Boson–Fermion model, in which the internal degrees of freedom of the bosonic bipolarons are explicitly taken account, we show that the single-particle spectrum in the normal

state displays a pseudogap together with a broad incoherent background arising from phonon shake-off effects, compatible with the features seen by ARPES. The behavior of pair distribution function, measuring the intracluster deformations, is characterized by a change-over from a single-peak to a double-peak structure as the temperature is increased, representing a hallmark of the lattice fluctuation

origin of this incoherent background.

Copper-based high-temperature superconductors (HTS) can be seen as being composed of molecular clusters, made out of basic units of the planes and the adjacent dielectric layers. Intracluster lattice deformations trigger fluctuations between two distinct charge carriers, namely, small bipolarons that strongly couple to these deformations and quasifree electrons that do not. With an effective half bandwidth D of about 0.5 eV and a characteristic cluster vibrational frequency of about 0.05 eV, HTS are in the adiabatic

rather than antiadiabatic regime, and hence the polaronic features observed in these materials [1–3] require strong electron-phonon coupling with a dimensionless coupling constant Under such conditions, bipolarons cannot exist in form of itinerant band states, but can decay into pairs of itinerant electrons as a consequence of local lattice ¹Centre de Recherches sur les Très Basses Temperatures, Laboratoire Associé á l’ Université Joseph Fourier, Centre National de la Recherche Scientifique, BP 166, 38042, Grenoble Cédex 9, France. 2

Dipartimento di Scienze Fisiche “E. R. Caianiello,” Università di Salerno, I-84081 Baronissi (Salerno), Italy-Unità I.N.F.M. di Salerno.


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fluctuations. Such a situation can be described in terms of the Boson–Fermion model (BFM), which was proposed [6] well before the discovery of HTS, in order to overcome the inherent difficulty of the scenario of a bipolaronic superconductivity [7]. In this article, we introduce a generalization of this model [8], which takes into account the internal structure of the

localized bipolarons. Introducing deformed harmonic oscillator states equilibrium positions shifted by

having

(M is the mass of the oscillators), a

bipolaron localized on a site i is represented in terms of the state

where are hard-core boson operators associated with self-trapped electron pairs, are phonon operators describing local lattice deformations, and are undisplaced oscillator states. Because of the small overlap of the electron and bipolaron wave functions, we may, to within a first approximation, consider the boson and fermion operators as commuting with each other. We are then led to a generalization of the original BFM, in which only the bosonic component is coupled to the lattice fluctuations. The corresponding Hamiltonian is

where are fermionic operators describing itinerant electrons with spin and t, and v denote the bare hopping integral for the electrons, their half bandwidth, the boson

energy level, and the boson–fermion pair-exchange coupling constant, respectively. The chemical potential is taken to be common to fermions and bosons (with a factor 2 for the bosons, which are made out of two charge carriers) in order to guarantee charge neutrality in the system. The indices i denote effective sites corresponding to the molecular clusters susceptible of local lattice deformations. As far as the low-energy properties of the above model are concerned, we can consider the internal degrees of freedom of the bipolarons to be frozen out. This leads us back to the original BFM, which is recovered upon neglecting the coupling to the phonons and replacing the boson–fermion coupling constant v by some renormalized value. Our previous studies of the original BFM permitted us to predict the temperature variation of the pseudogap and the corresponding manifestations of it in thermodynamic, transport, and magnetic properties [9]. In order to study the high-energy sector associated with the Hamiltonian (2), as required for the interpretation of the angle-resolved photo-emission spectroscopy (ARPES) results, we must deal with the indirect polaronic nature of the electrons that couple to the phonons only via a charge exchange with the bipolarons. ARPES has now clearly established that a pseudogap opens up in the normal state below some characteristic temperature T* that, depending on doping, can be well above the superconducting transition temperature This pseudogap develops predominantly near the of the Brillouin zone and is accompanied by a large incoherent background in the quasi-particle spectrum, extending over a regime in frequency of typically half the

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bandwidth [10]. The pseudogap smoothly evolves into the true superconducting gap upon either decreasing the temperature below or crossing the boundary of the vortex cores in the mixed phase at a fixed temperature below This strongly suggests that the pseudogap and the superconducting gap are of the same origin, both being due to strong pair fluctuations rather than antiferromagnetic fluctuations. This does not mean, however, that the pseudogap is due to preformed pairs, what is clearly ruled out by its non-s-wave symmetry [13]. The presence of preformed pairs is relevant here essentially because it leads, via charge exchange with the conduction electrons, to pair fluctuations in the electronic subsystem, which in general are anisotropic in the Brillouin zone. The opening of the pseudogap is thus due to the fact that, close to the Fermi surface, the single-particle states are diminished in favor of two-particle states. It has invariably been observed that in the regions of the Brillouin zone where the pseudogap opens, ARPES shows the appearance of a strong incoherent background in the single-particle spectrum. It has been suggested earlier that this feature might come from a coupling of the electrons to some collective modes such as spin fluctuations [14]. The explanation for the incoherent spectrum that we shall put forward here is different from that, and is directly linked to the excitations of the quantum coherent deformed oscillators in terms of which small bipolarons are described. In an ARPES experiment, such coherent states are broken up, leading to multiphonon shake-off processes that cover a wide regime of frequencies. Contrary to the pseudogap phenomenon, the incoherent part of the spectrum, extending over an energy region of about 0.5 eV, is a high-frequency phenomenon for which a study of the Hamiltonian in Eq. (2) in the atomic limit makes sense. In this case, the eigenstates are given by

with the corresponding eigenvalues

with and In the above set of eigenstates, a site occupied by an electron with spin is denoted by and a site occupied by a pair of electrons with spin up and down by and denote, respectively, a site unoccupied and occupied by a boson. denotes the l-th excited oscillator state and the l-th excited shifted oscillator state. The states contain the mechanism by which the polaronic character is transferred from the bipolarons onto the intrinsically nonpolaronic electrons. Because photoemission only couples to the latter, it is via this mechanism that photoemission spec-

tra acquire features that are reminiscent of polaronic quasi-particles. The oscillator states and involved in this context have to be determined via numeric diagonalization of the above Hamiltonian in the atomic limit, by expanding them in a set of excited harmonic oscillator states in the form and

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Ranninger and Romano The above eigenstates of the local BFM problem fully determine its spectral function,

which is given by

with denoting the partition function. consists of a coherent nonbonding part with spectral weight describing single-electron states uncoupled to the phonons, and an incoherent part stemming from the electron–phonon coupling introduced via the bonding and antibonding two-electron states and In order to evaluate the temperature dependence of the

density of states and, in particular, of the pseudogap, we must fix the various parameters so as to describe as realistically as possible the situation encountered in the HTS. For a system such as, for instance, YBCO, the number of doping-induced bipolarons [approximately given by half the number of dopant ions in the chains] varies between 0 and 0.5 per effective site, and the number of fermions is equal to 1 if the boson–fermion exchange coupling were absent. We thus obtain a total

number of charge carriers close to 2 for optimally doped systems. We further require that the coupling of the phonons to the electron pairs is such as to yield a bipolaronic level with energy close to the band center, the bipolaron binding energy being given by For a typical half-bandwidth of the order of 0.5 e V and a phonon frequency of the order of 0.05 e V, we require an for bipolaron formation to occur [4, 5]. As in our previous studies, we take such that the resulting value for T* i s of the order of a few hundred degrees K. In order to obtain an

we adjust the precise position of this level by putting with Given these parameters, in Fig. 1 we plot the temperature evolution of the density of states We notice that for high values of T, a central peak accompanied by two side

shoulders is clearly distinguishable. As the temperature is lowered, this central peak, which corresponds to the single-electron nonbonding contribution, is gradually reduced in height

and finally gives rise to the appearance of the pseudogap close to the Fermi level. The intensity of this peak is shifted in an asymmetric way toward the bonding and antibonding contributions that, because of their strong coupling to multiple phonon states, broaden considerably the incoherent part of the spectrum. The experimental way to render these features visible is by photoemission electron spectroscopy (PES). Its intensity is given by , where represents the emission part of [corresponding to the term proportional to in expression (4)] and denotes the Fermi distribution function. In Fig. 2 we illustrate the behavior of as a function of frequency for several temperatures. We notice that the decrease of T leads to the formation of a pseudogap,

measured as the distance in energy between the chemical potential at and the point at half height of the leading edge of the spectrum. This pseudogap turns out to have a zero temperature limit of meV and closes up at a characteristic temperature which are reasonable numbers. Concomitantly with the opening of the pseudogap, the incoherent part of the quasi-particle spectrum increases and broadens. Considering that only bosons couple to phonons, this result is explained by the fact that


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decreasing temperatures give rise to an increase in the number of bosons relative to that of fermions (Fig. 3).

A crucial test whether the incoherent part of the quasi-particle spectrum is indeed related to phonon shake-off effects would be an independent verification of the existence

of coherent state excitations of polaronic nature. One such possible test could be the investigation of the temperature dependence of the local intracluster deformations, which can be measured by EXAFS and pulsed neutron scattering techniques. The measured quantity is

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the so-called pair distribution function (PDF), given by

The behavior of the PDF, illustrated in Fig. 4 for different temperatures, is characterized by a single sharp peak at low T due to the predominance of sites occupied by bipolarons (see Fig. 3). As T* is approached from below, we observe a distinct splitting of this feature into two well-separated peaks that, upon further increase of temperature, get broadened such


251

that at high temperature, the PDF is again characterized by a single-peak structure, although now very smeared. The two peak positions characterize the two deformations of the local

lattice environment in which a given site is occupied alternatively by a pair of electrons or by a bipolaron with comparable proability, as expected on the basis of Fig. 3. Recent EXAFS [15, 16] and pulsed neutron scattering [17] experiments give direct evidence for such dynamical local lattice fluctuations. In conclusion, we have shown that a scenario of fluctuating deformable molecular clusters, triggering alternate occupation by quasi-free electrons and bipolarons, can account for the anomalous quasi-particle features of the electrons seen in ARPES. In particular, the occurrence of a broad incoherent background associated with the formation of a pseudogap in the normal phase is ascribed to temperature-dependent phonon shake-off effects. An independent check for the lattice-driven origin of the background is represented by the behavior of the pair distribution function, which allows for the detection of the existence of such dynamical local lattice fluctuations. Taking into account long-range Coulomb forces between the various charge carriers (so far neglected in our studies) would lead to instabilities such as stripe formation of the deformable clusters. The essential physics reported here, however, would remain essentially the same, except, of course, for specific momentumdependent features, such as those recently examined in experimental studies [18]. REFERENCES 1. C. Taliani et al., in Electronic Properties of

Superconductors and Related Compounds, eds. H.

Kuzmany, M. Mehring, and J. Fink (Springer 1990), p. 280.

2. Xiang-Xin Bi and P. Ecklund, Phys. Rev. Lett. 70, 2625 (1993). 3. P. Calvani et al., Phys. Rev. B 53, 2756 (1996). 4. B. K. Chakraverty, J. Ranninger, and D. Feinberg, Phys. Rev. Lett. 81, issue 1 (6 July 1998).

5. E. V. L. de Mello and J. Ranninger, preprint (1998). 6. J. Ranninger and S. Robaszkiewicz, Physica B 135, 468 (1985).

7. A. S. Alexandrov and J. Ranninger, Phys. Rev. B 23, 1796 (1981). 8. J. Ranninger and A. Romano, Phys. Rev. Lett. 80, issue 25 (22 June 1998). 9. J. Ranninger, J.-M. Robin, and M. Eschrig, Phys. Rev. Lett. 74, 4027 (1995); J. Ranninger and J.-M. Robin, Phys. Rev. B 53, R l1961 (1996). 10. D. S. Marshall et al., Phys. Rev. Lett. 76, 4841 (1996). 11. H. Ding et al., Nature 382, 51 (1996).

12. Ch. Renner et al., Phys. Rev. Lett. 80, 3606 (1998). 13. A. J. Leggett, J. Phys. (Paris) Colloq. 41, C7-19 (1980); M. Randeria, J. Duan, and L. Shieh, Phys. Rev. B 47, 327 (1990). 14. Z.-X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997). 15. N. L. Saini et al., Physica C 268, 121 (1996); Phys. Rev. B 55, 12759(1997). 16. J. Röhler et al., in Superconductivity 1996: Ten Years after the Discovery, E. Kaldis et al., eds.

(Kluwer Academic Publishers, Dordrecht, 1997), p. 469. 17. T. Egami and S. J. L. Billinge, in Physical Properties of High-Temperature Superconductors, ed. V. D. M. Ginsberg (World Scientific, 1996), p. 265. 18. Z.-X. Shen et al., Science 280, 259 (1998).


Evidence for Strongly Interacting Electrons with Collective Modes at and in the Normal Phase of Superconductors N. L. Saini,1 A. Lanzara,1 A. Bianconi,1 J. Avila,2 M. C. Asensio,2 S. Tajima,3 G. D. Gu,3 and N. Koshizuka3

Here we report the k distribution of the photoelectron spectral weight in an

energy window of 50 meV around the chemical potential giving a so-called marginal Fermi surface (MFS) of a representative superconductor, Combining two modes of angle-resolved photoemission (ARPES), namely the constant initial-state angle scanning (ASP) mode and conventional energy distribution curve (EDC) mode, we determined the wavevectors of the collective modes strongly interacting with the conduction electrons: a first

one at assigned to a charge density wave (CDW) that is second harmonic of the underlying incommensurate and anharmonic lattice fluctuation forming a superlattice of quantum stripes; and a second one at associated with spin density waves (SDW).

1. INTRODUCTION Anomalous electronic properties of cuprates have been a point of wide discussion in recent years. Angle resolved photoemission (ARPES) is one of the few experimental tools that has been used to take up this task as the technique has the advantage of being resolved both in energy and momentum space [1]. Study of the evolution of anomalous single-particle properties of superconductors could be possible due to availability 1

2

3

Unita’ INFM and Dipartimento di Fisica, Università di Roma “La Sapienza” P. A. Moro 2, 00185 Roma, Italy.

LURE, Bat 209D Universite Paris-Sud, F-91405 Orsay, France, and LURE and Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain. Superconductivity Research Laboratory, ISTEC, Shinonome 1-10-13 Koto-ku, Tokyo 135, Japan.


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of better quality samples with a wide range of doping and new developments of infrastructure for performing ARPES experiments.

There is growing experimental evidence for the breakdown of the one-electron picture in

superconductors. The breakdown occurs for a strongly interacting electron gas

where the Fermi liquid model of quasi-particles is not valid. In this situation, one expects that the energy indetermination of charge carriers in the conduction band is of the order of the bandwidth, i.e., in the photoemission spectra, the uncertainty in the energy determination is of the same order of magnitude as its energy dispersion. The well-known example of

a strongly interacting electron gas, where the Fermi liquid model breaks down, is a onedimensional conductor where charge current can flow through collective excitations with

separation of spin and charge modes. Recent experiments are uncovering the fact that the doped perovskites are complex materials with segregation of localized and itinerant charge carriers in stripes [2]. It has been found by combined analysis of EXAFS and diffraction studies [3–7] that superconductivity coexists with an incommensurate and anharmonic lattice modulation, forming stripes in the plane along the diagonal direction. These lattice fluctuations coexist with incommensurate spin fluctuations with slow dynamics in the horizontal direction

(the direction of Cu-O-Cu bonds) [8]. Based on the magnetic and charge superstructure peaks observed in insulating doped nikelates and 1/8 doped cuprates, a model of charge and spin fluctuating horizontal stripes has been proposed in inelastic neutron scattering experiments [9]. In the stripe scenario the dimensionality of the system is lower than two due to reduced hopping between the stripes. Therefore, the Fermi liquid picture for a quasi-particle is not valid anymore to describe the electronic structure of these materials, that is close to a onedimensional Luttinger liquid or a marginal Fermi liquid [10]. To investigate these aspects we have measured the k distribution of the electronic states near the Fermi level in an energy range of 50 meV, which is of the order of expected energy cutoff for the interactions involved

in the superconducting pairing mechanism. This is done by an unconventional mode ARPES based on constant initial state angle scanning [11]. We have selected Bi2212 system at the optimum doping as representative material due to its good suitability for such measurements.

Because there is not a two-dimensional Fermi surface for this strongly interacting electron gas, we call the measured constant energy contour marginal Fermi surface map. The resulting marginal Fermi surface (MFS) provides a clear identification to the key features related with the collective excitations in the superconductors. The measurements are combined with conventional mode of ARPES based on energy distribution curves (EDC) [12]. In this contribution, we discuss two main features of the MFS: (1) asymmetric suppression of spectral weight around the M points and (2) identification of one-dimensional set of

electronic states. We argue that the asymmetric suppression of the spectral weight around the M points is due to coupling of electrons with dynamical charge fluctuations along the diagonal direction with a wavevector whereas the one-dimensional set of states in the direction might be at the origin of the dynamical incommensurate spin fluctuations [8]. 2. EXPERIMENTAL

The ARPES measurements were carried out at the Laboratoire pour l’Utilisation du Rayonnement Electromagnetique (LURE) (Orsay-France) on the SU6 undulator beamline.

Evidence for Strongly Interacting Electrons with Collective Modes A Bi2212 stoichiometric crystal

255

of size

grown by floating zone method [13], was used for the measurements. We used the constant

initial energy angle-scanning photoemission accompanied with the standard approach, exploiting the high intensity of the synchrotron radiation emitted by an undulator source. The experiments on the well-aligned and clean Bi2212 surface were performed in an ultra-high vacuum (UHV) chamber equipped with an angle-resolving hemispherical analyzer and a high-precision manipulator permitting an azimuthal sample

rotation

of 360° and polar emission angle relative to the surface normal

of 90° [14].

The angular resolution used for the MFS is 1.5°, whereas for the energy distribution curves (EDC) it is 1°. The spectrometer energy resolution was about 50 meV. The measurements were performed using the linearly polarized synchrotron light with photon energy of 32 eV in the even symmetry. In this geometry, the polarization vector of the synchrotron light, the wave vector of the emitted photoelectron and the surface normal were in the same horizontal plane (where the horizontal plane plays the role of a mirror plane) and the transition from the initial states with even symmetry is fully allowed [1,15]. This

geometry has been used to have maximum emission intensity along the and directions. The experiments were repeated with different cleavage in several experimental runs, and each cleave gave the same MFS map, proving the high quality of the crystal as well as reproducibility of the experimentally observed MFS features.

3. RESULTS In Fig. 1 we plot the well-known and well-established band dispersion (E(k) curves) in the and directions obtained by plotting the energy positions of the dispersing peaks of EDC. Similar band dispersions on the Bi2212 system have been measured by many groups and interpreted in term of dispersing quasi-particles and fitted with band structure models. The ratio of the width and the binding energy of the peaks (obtained by fitting the EDC curve with an asymmetric Gaussian function and a Fermi step function) is shown as insets. The ratio is close to 1 indicating the breakdown of the single particle picture. The measured EDC are shown in Fig. 2 as a two-dimensional picture, where the intensity is plotted in a two-dimensional plane (E,k) [16]. In the picture, it could be seen that the EDCs are quite complex. The dispersing maxima (peaks) disappear for in the and in the direction and the energy dispersion for is of the order of 300 meV. Apart from a large peak width that is of the same order of magnitude as the total dispersion, the EDC curves show long tails extending up to about 0.6 eV energy below the chemical potential

This kind of

tail is observed in the one-dimensional ladder systems and interpreted in term of dispersion of spinons and holons [17]. The shape of these curves indicate directly that the electrons in the conduction band are strongly interacting. Figure 3 shows the global view of the MFS of the Bi2212 superconductor. We now come directly to the point of missing segments of the MFS around the M points. To have a better view of the suppression of the spectral weight, we enlarged the parts around the M and and they are shown in Fig. 3. It is clear that the photointensity is suppressed asymmetrically [16]. The asymmetric topology of the missing segments could be seen in Fig. 4, which shows measured photointensity along azimuthal curvature from

direction to

direction across the

direction on the MFS. From this figure, it

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is clear that the suppression of spectral weight around the M points is asymmetric. Moving

away from the M direction to the direction we find no evidence of a peak (the so-called quasi-particle peak), whereas along the direction we clearly observe such a peak, as shown in Fig. 4b.

Evidence for Strongly Interacting Electrons with Collective Modes

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After a careful analysis, we identified two well-defined wavevectors on the MFS connecting the points where a suppression of the spectral weight occurs. These points in the first Brillouin zone are and They are connected by the vectors and as shown in Fig. 5. These vectors are the wavevectors of collective excitations that suppress the spectral weight of the quasi-particles at We notice that the wavevector is diagonal and is the second harmonic of the incommensurate and anharmonic lattice fluctuation; therefore, it can be associated with diagonal charge density waves CDW. However, the vector is the antiferromagnetic wavevector and it is the wavevector of SDW that have been observed in the superconducting phase [18]. We now turn to the next observation. We identified a new set of electronic states with a one-dimension-like dispersion in the to direction crossing the Fermi level at one point The observed electronic states are beyond the expected one electron-like Fermi surface, and we could identify these states by combining the angle-scanning mode with the conventional EDC mode. Figure 6 compares the energy distribution curves measured in the (solid line) and (dotted line) directions at the same k locations. There are dispersing features in both directions;


259

however, the EDC in the two directions show clear differences in their line shapes. The

main spectral band is clearly visible in both directions for polar angles above 5° off the point and disperses toward the Fermi energy. Although the main band appears quite similar in the two orthogonal directions, the EDC along shows a second dispersive spectral feature at lower angles with a smaller intensity. This feature crosses the Fermi level at around with a total energy dispersion However, we do not see this new set of states in the EDCs measured along the orthogonal direction. The direct spectral differences are plotted in Fig. 6 (right panel), showing clearly the new set of electronic states. The second band, appearing only in one direction, was reproducibly observed in different runs performed on different cleaved surfaces ascertaining its intrinsic nature. Thus this result not only shows the anisotropy of the MFS, but also provides a direct evidence for a set of electronic states at

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the MFS having one-dimensional character along the Cu-O-Cu direction. The differences

in the and were further ascertained by measuring phtotointensity at the MFS using scanning the polar angle along and The direct measurement of the photointensity scans at the MFS demonstrated that the observed band exists only along the direction, whereas it is absent along the direction [19].

4. DISCUSSION AND CONCLUSIONS

Having established experimentally the presence of a new feature on the MFS the question to be asked in on the origin of this feature. There are two points that should be considered here: (1) The energy dispersion of this new set of states is of the order of antiferromagnetic exchange interaction There are incommensurate spin fluctuations along the Cu-O-Cu direction observed in several inelastic neutron scattering experiments [8,18], and the wavevector of the observed band is of the same order of magnitude if charge fluctuation occurs due to these incommensurate spin fluctuations in the plane. These facts encourage us to correlate the new states to the incommensurate spin fluctuations in the system. We now turn to discuss the suppression of spectral weight around the M points. The missing parts of the MFS around is one of the anomalous features that has been discussed widely during recent years. Preformed pairs without coherence, quasi-particle decay into spinons and holons [20], precursor effect of antiferromagnetic correlations [21] are some of the arguments that have been put forward to understand the suppression of the spectral weight. Pairing of quasi-particles (coherent) with a collective mode (incoherent) [22] could also be a reason for the missing segments around the points as argued by Shen and Shrieffer. We assigned the suppression of photointensity on the MFS to charge fluctuations associated with stripe ordering in the plane along the diagonal direction, i.e., direction. The wavevector of this along the diagonal direction. Comparing this wavevector with the charge modulation in plane [4], measured by anomalous diffraction in the same system, we find that it is the second harmonic of the modulation that destroys the MFS around the M points. It should be recalled that, due to strong contribution of second harmonic, the modulation in the Bi2212 system forms striped phase where the potential barrier between the stripes is controlled by the contribution of the second harmonic [4]. In summary, the combination of two modes of ARPES allowed us to identify several anomalous features on the MFS of the Bi2212 superconductor at the optimum doping. We have identified a wavevector around M points along the diagonal direction. It is shown that the asymmetric suppression of spectral weight around the M points is due to a charge ordering in stripes of the plane along the diagonal direction with a well-defined wavevector of We have found a new set of electronic states along the direction with a small dispersion and small beyond the main MFS. The one-dimensional nature of these states has been confirmed by angle scanning of the MFS as well by recording energy distribution curves. We think that the new set of electronic states along the direction is related to the low-energy spin fluctuations observed in inelastic neutron scattering experiments. Thus the present results suggest the way to reconcile the magnetic fluctuations along the Cu-O-Cu direction observed in the inelastic magnetic scattering experiments [8,18] and charge ordering along the 45° from the Cu-O-Cu direction observed by EXAFS and


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anomalous diffraction studies [3–7]. It should be mentioned that a simultaneous ordering of spin along the Cu-O-Cu direction [23] and charge along the Cu-Cu direction [24] has been found to take place in In conclusion, we show direct evidence of stripes in the electronic structure of superconductors. The experimental observations suggest that there are charge stripes along the direction with spin fluctuations along direction.

ACKNOWLEDGMENTS

The authors are happy to acknowledge stimulating and useful discussions with A. Perali. This work was partially funded by Istituto Nazionale di Fisica della Materia (INFM) and Consiglio Nazionale delle Ricerche (CNR) of Italy. The experiment was done at the Spanish–French beam line of LURE with the support of the large-scale Installation Program and the Spanish agency DGICYT under grant PB-94-0022-C02-01.

REFERENCES 1. Z.-X. Shen and D. S. Dessau, Phys. Rep. 253, 1 (1995), and references therein. 2. See, for example, the special issue on Stripe Lattice Instabilities and Superconductivity, edited by

A. Bianconi and N. L. Saini [J. Supercond. 10, No. 4 (1997)]. 3. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguci, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996). 4. A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, Å. Kvick, and P. G. Radaelli, Phys. Rev. B 54, 4310 (1996).

5. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishihara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996). 6. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55, 12759 (1997). 7. A. Lanzara, N. L. Saini, A. Bianconi, J. L. Hazemann, Y. Soldo, F. C. Chou, and D. C. Johnston, Phys. Rev. B 55, 9120 (1997). 8. K. Yamada, C. H. Lee, J. Wada, K. Kurahashi, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, and M. A. Kastner, J. Supercond. 10, 343 (1997); S. M. Hayden, G. Aeppli, H. A. Mook, T. G. Perring, T. E.

9. 10. 12.

11.

Mason, S.-W. Cheong, and Z. Fisk, Phys. Rev. Lett. 76, 1344 (1996) and references therein; E. D. Isaacs, G. Aeppli, P. Zschack, S.-W. Cheong, H. Williams, and D. J. Buttrey, Phys. Rev. Lett. 72, 3421 (1994). J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995). C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996 (1989); J. Ashkenazi, J. Supercond. 10, 379 (1997). The conventional method of Fermi surface measurements is based on measurement of energy distribution curves (EDC) in all high-symmetry directions of the Brillouin zone and find the locations of the Fermi surface by following the dispersion of peaks in the EDC (see, e.g., Ref. 1). J. Osterwalder, P. Aebi, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and K. Kadowaki, Appl. Phys. A 60, 247 (1995); P. Aebi, J. Osterwalder, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and K. Kadowaki, Phys. Rev. Lett. 72, 2757 (1994); M. Lindroos and A. Bansil, Phys. Rev. Lett. 77, 2985 (1996).

13. G. D. Gu, K. Takamuka, N. Koshizuku, and S. Tanaka, J. Cryst. Gr. 130, 325 (1993). 14. J. Avila, C. Casado, M. C. Asensio, J. L. Munoz, and F. Soria, J. Vac. Sci. Tech. A 13, 1501 (1995). 15. M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phys. Rev. B 52, 615 (1995); M. R. Norman, M. Randeria, H. Ding, J. C. Campuzano, and A. F. Bellman, Phys. Rev. B 52, 15107 (1995). 16. N. L. Saini, J. Avila, A. Bianconi, A. Lanzara, M. C. Asensio, S. Tajima, G. D. Gu, and N. Koshizuka, Phys. Rev. Lett. 79, 3467(1997).

17. C. Kim, Z. X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa, Phys. Rev. B 56, 15589 (1997). 18. G. Aeppli, T. E. Meson, M. S. Hayden, H. A. Mook, and J. Kulda, Science 278, 1432 (1997) and references therein.

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19. N. L. Saini, J. Avila, M. C. Asensio, S. Tajima, G. D. Gu, N. Koshizuka, A. Lanzara, and A. Bianconi, Phys. Rev. B 57, R11101 (1998). 20. R. B. Laughlin, Phys. Rev. Lett. 79, 1726 (1997).

21. J. Schmalian, D. Pines, and B. Stojkovic, Phys. Rev. Lett. 80, 3839 (1998). 22. Z. X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997). 23. B. O. Wells, Y. S. Lee, M. A. Kastner, R. J. Christianson, R. J. Birgeneau, K. Yamada, Y. Endoh, and G. Shirane, Science 277, 1067 (1997) and references therein. 24. A. Bianconi et al., p. 9 in this volume.

Angle-Resolved Photoemission Study of 1D Chain and Two-Leg Ladder T. Sato,1 T. Yokoya,1 H. Fujisawa,1 T. Takahashi,1 M. Uehara,2 T. Nagata,2 J. Akimitsu,2 S. Miyasaka,3 M. Kibune,3 and H. Takagi3

We performed angle-resolved photoemission spectroscopy (ARPES) on and to study the difference of electronic structure between the cornersharing one-dimensional (1D) chain and the two-leg ladder cuprate. ARPES spectra of show two independent dispersive bands near the Fermi level which are ascribed to the spinon and the holon band due to the spin-charge separation in the 1D chain. In contrast, ARPES spectra of exhibit only one dispersive band near with the periodicity of ladder sublattice, which is assigned to a quasi-particle band where holon and spinon are confined together owing to the

opening of the spin gap on the ladder. The present results show a clear difference in the nature of the electronic structure near

liquid

between the Tomonaga–Luttinger

and the Luther–Emery liquid

1. INTRODUCTION

The interacting one-dimensional (1D) system referred as Tomonaga–Luttinger (TL) liquid shows many anomalous physical properties different from those of a 3D system

(Fermi liquid). It has been predicted theoretically that spin and charge degrees of freedom are separated in TL liquid and behave as if they are two independent quasi particles called spinon and holon [1]. Being stimulated by the discovery of high-temperature superconductivity in 2D cuprates, many theoretical and experimental studies have also been directed to 1D systems because the spin-charge separation has been regarded as one of

possible driving forces in 2D curprates [2]. Theoretically, possible superconductivity has been predicted in a quasi-1D spin ladder system with even legs [3] and recently 1

Department of Physics, Tohoku University, Sendai 980-8578, Japan. Department of Physics, Aoyama–Gakuin University, Tokyo 157-0071, Japan. 3 Institute for Solid State Physics, University of Tokyo, Tokyo 106-0032, Japan. 2



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superconductivity of has actually been observed in which consists of edge-sharing 1D chains and two-leg ladders. In contrast to the 1D chain, spin

and charge degrees of freedom are expected to be confined in the two-leg ladder owing to opening of a finite spin gap [Luther-Emery (LE) liquid] [5]. In this study, we performed angle-resolved photoemission spectroscopy (ARPES) of and single crystals. consists of corner-sharing single Cu-O chains (Fig. 1 a) [6], whereas has two-leg ladders as well as edge-sharing chains (Fig. 1 b) [7]. ARPES spectra of show to independent dispersive bands near the Fermi level that are ascribed to the spinon and the holon band due to the spincharge separation in the 1D chain. In contrast, spectra of show only one dispersive band with the periodicity of ladder sublattice that is assigned to a quasi particle band where holon and spinon are confined together owing to the opening of the spin gap on the ladder. The present results give a direct experimental evidence for the difference in the nature of the electronic structure near between the TL liquid and the LE liquid

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2. EXPERIMENT Single crystals of and were prepared with the traveling solvent floating zone method. ARPES measurements have been performed at Tohoku University using a home-built ARPES spectrometer equipped with a He discharging lamp with total energy and angular resolutions of 100 meV and and kept at 300 K or 130 K

respectively. The samples were cleaved during photoemission measurements.

3. ARPES RESULT OF 1D CHAIN:

We show ARPES spectra near of measured along the chain direction (b axis) in Fig. 2. The wave vector in the unit of 1/a is shown on each spectrum. Intensity of spectra is normalized with the incident photon flux. The ARPES spectrum at point has a steplike structure at about 1 eV binding energy, which is followed by a strong main peak located at about 2.6 eV (not shown in Fig. 2). As increases, a small structure appears between the strong peak at 2.6 eV and the steplike structure

then gradually grows up on approaching

to

and

finally forms a distinct, well-resolved peak at 1 eV binding energy The steplike structure observed in the spectrum at appears to merge into this new peak

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around These spectral changes suggest the existence of two different bands near that have different energy dispersions but have a common maximum point closest

to is further increased from to the 1 eV peak at disperses to higher binding energy and at the same time loses its intensity. In contrast to the first half of the Brillouin zone, we do not find a steplike structure in the spectrum at the zone boundary in the second half of Brillouin zone. Instead, the ARPES spectra near the zone boundary have a weaker and broader structure on the tail from the main peak. We

found that this small structure around is extrinsic because this structure showed a gradual growth with time. It is noted here that the present experimental result on is essentially the same as that observed for for the relative spectral intensity at

having double Cu-O chains [8,9], except

In order to map out the “band dispersion,” we took the second derivative of ARPES spectra in Fig. 2 and plotted the contour map of intensity with gradual shading as a function of

wavevector and binding energy. The result is shown in Fig. 3a. The dark parts correspond to “bands.” Broken lines are used as a guide to the eyes. Taking the second derivative of spectra diminishes the effect of background and determines the peak position more accurately. This method is particularly effective in the present case, in which the strong main peak located around 2.6 eV has a wide and strong tail spreading to forming a sizable background

to the small structures near

All the characteristic features of band dispersions observed

in the raw ARPES spectra are more clearly visible in Fig. 3a. We find again that there are two dispersive bands in the first half of the Brillouin zone and they have different energy dispersions with a common maximum point closest to One of new findings in Fig. 3a is that there is a continuous distribution of finite intensity between

these two bands, which is not clear in the raw ARPES spectra. It is also clear again in Fig. 3a that one of the two bands with a larger energy dispersion looks symmetric with respect to but has a much stronger intensity in the second half of the Brillouin zone.


267

To compare the obtained band dispersion with the theoretical calculation, we show the

spectral function for different momenta k calculated with a 1D Hubbard model at in Fig. 3b. We find that is not symmetric with respect to According to the theory, the flat band located at appearing only in the first half of the Brillouin zone represents the spinon, whereas the symmetric dispersive band corresponds to the holon. The finite intensity distributing between these two bands in the first half of the Brillouin zone indicates a mixture of spinon and holon excitations. We find that all these characteristic features indicative of the spin-charge separation are clearly observed in the experiment (Fig. 3a). However, the “shadow band” resulting from spin fluctuation [11,12], which is theoretically predicted to appear in the energy range of to (not shown in Fig. 3b), was not observed in the present study, probably due to overlapping from a large main peak located at 2.6 eV binding energy. In the t-J model with a

finite U, the dispersive feature of spinons and holons is scaled with J (exchange coupling) and t (hopping parameter), the band width being calculated to be and 2t, respectively [13]. Because the experimental spinon and holon bands show the band width of 0.2– 0.25 eV and 1 eV, respectively, we obtain and although there remains ambiguity due to the broad feature of bands, in particular for spinon. The obtained value of J seems consistent with the value (0.13–0.2 eV) reported by the magnetic susceptibility measurement [14,15], but is smaller than the value (0.26 eV) from the optical measurement [16]. Although there is no direct experimental estimate of t to be compared with the present value the ratio t / J obtained in the present study (3–4) is comparable to the value of cuprate high-temperature superconductors.

4. ARPES RESULT OF TWO-LEG LADDER:

We turn our attention to the result of the two-leg ladder. ARPES spectra near of measured along the ladder direction (c axis) are shown in Fig. 4. Polar angle measured with respect to the surface normal is indicated on the spectra. The intensity of

each spectra is normalized with the integrated area. We find a broad structure around 1 eV binding energy, which is followed by a prominent peak located at 2.5 eV (not shown in Fig. 4). We also notice that there is a small feature around 0.5 eV that appears periodically at certain polar angles

To map out the band structures near we used the same procedure as in Fig. 3. The result is shown in Fig. 5, in which the dark shading corresponds to bands and the white broken lines are used as a guide to the eyes. The size of the Brillouin zone of each ladder and chain sublattices is indicated below the figure. The procedure also works well in this case, because Fig. 5 more clearly shows the two structures found in the raw spectra. One is an almost dispersionless band located at 1 eV and the other is relatively dispersive, but appears in a certain portion of the Brillouin zone. One important observation in Fig. 5 is that the dispersive band has the periodicity matching well with the ladder sublattice but not with the chain sublattice; the band has the lowest binding energy of about and in the reciprocal lattice space of the ladder sublattice. This experimental observation indicates that the dispersive band appearing near originates in the ladder. We also find that this dispersive band has the lowest binding energy at as if the band is folded at However, the intensity of band is not symmetric; it is weaker in the region of As for the flat band observed

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269

at 1 eV, it may originate in the chain because it is expected that the holes are localized on the chain due to negligible transfer integral (t) between two nearest-neighbor Cu atoms via nearly 90° Cu-O-Cu bond angle. Next we compare the present ARPES result of with the t-J calculations [17–19]. The calculation predicts that the one-particle excitation spectrum at a half filling and posses a quasi-particle band near which consists of two degenerated bands due to two Cu-O chains (legs) in the two-leg ladder. The quasi-particle band is roughly symmetric with respect to where the band has the lowest binding energy and the spectral intensity of the quasi-particle band is weaker in the second half of the Brillouin zone than in the first half of the Brillouin zone This prediction is qualitatively consistent with the present ARPES result in Fig. 5 showing highest energy at and intensity reduction in the region from to This suggests that the observed band is a quasi-particle band in which spinon and holon are confined due to the existence of finite spin gap. 5. COMPARISON OF

AND

Finally, we compare the electronic structure between a 1D chain and two-leg ladder. For we observe two dispersive bands near one exhibits the dispersion only from and the other is symmetric with respect to (Fig. 3a). Because these features show a good agreement with the Hubbard model as well as the t-J model calculation they are ascribed to the spinon and holon bands, respectively. However, for only one band with the periodicity of ladder sublattice is observed near

(Fig. 5). Because

the observed band with ladder origin is qualitatively understood within the result of t-J

calculations, the band is assigned to the quasi-particle band. These observations clearly illustrate the difference in character of the electronic states between the two compounds. Electrons in are regarded to form a TL liquid. In contrast, the existence of finite spin gap in prevents the spin-charge separation and tends to confine the both into one quasi-particle located near as shown by the exact diagonalization of the t-J ladder.

6. CONCLUSIONS We have performed ARPES on

the electronic structure near

and

to study the difference in

between the 1D chain cuprate and the two-leg ladder. We

observed two dispersive bands near ascribable to spinon and holon due to the spincharge separation in whereas we found only one dispersive band in that is ascribed to a quasi-particle in which spin and charge freedoms are confined. These

results indicate a clear difference in the electronic structure between two different quasi-1D

cuprates. ACKNOWLEDGMENT This work was supported by grants from Core Research for Evolutional Science and Technology Corporation (CREST) and the Ministry of Education, Science and Culture of Japan.

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REFERENCES 1. S. Tomonaga, Prog. Theor. Phys. 5, 349 (1950); J. M Luttinger, J. Math. Phys. 4, 1154 (1963). 2. P. W. Anderson, Phys. Rev. Lett. 64, 1839 (1987). 3. T. M. Rice, S. Gopalan, and M. Sigrist, Europhys. Lett. 23, 445 (1993); E. Dagotto and T. M. Rice, Science 271, 618 (1996), and references therein. 4. M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996). 5. H. Tsunetsugu, M. Troyer, and T. M. Rice, Phys. Rev. B 51, 16456 (1995). 6. E. M. McCarron, M. A. Subramanian, J. C. Calabrese, and R. L. Harlow, Mat. Res. Bull. 23, 1355 (1988); T. Siegrist, L. F. Schneemeyer, S. A. Sunshine, J. V. Waszczak, and R. S. Roth, Mat. Res. Bull. 23, 1429 (1988). 7. Von Chr. U. Teske and Hk. Muller-Bouschbaum, Z. Anorg. Allg. Chem. 371, 325 (1969). 8. C. Kim, A. Y. Matsuura, Z.-X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa, Phys. Rev. Lett. 77, 4054 (1996); C. Kirn, Z.-X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa, Phys. Rev. B 56, 15589 (1997). 9. H. Hujisawa, T. Yokoya, T. Takahashi, M. Tanaka, M. Hasegawa, and H. Takei, J. Elec. Spectros. Rel. Phenom. 88–91,461 (1998). 10. S. Sorella and A. Parola, J. Phys.: Condens. Matt. 4, 3589 (1992). 11. K. Penc, K. Hallberg, F. Mila, and H. Shiba, Phys. Rev. Lett. 77, 1390 (1996). 12. J. Favand, S. Mass, K. Penc, F. Mila, and E. Dagotto, Phys. Rev. B 55, R4859 (1997). 13. P.-A. Bares, G. Blatter, and M. Ogata, Phys. Rev. B 44, 130 (1991); M. Ogata, M. U. Luchini, S. Sorella, and

F. F. Assaad, Phys. Rev. Lett. 66, 2388 (1991). 14. T. Ami, M. K. Crawfbrd, R. L. Harlow, Z. R. Wang, and D. C. Johnston, Phys. Rev. B 51, 5994 (1995). 15. S. Eggert, Phys. Rev. B 53, 5116 (1996). 16. H. Suzuura, H. Yashuara, A. Furusaki, N. Nagaosa, and Y. Tokura, Phys. Rev. Lett. 76, 2579 (1996).

17. H. Tsunetsugu, M. Troyer, and T. M. Rice, Phys. Rev. B 49, 16078 (1994). 18. M. Troyer, H. Tsunetsugu, and T. M. Rice, Phys. Rev. B 53, 251 (1996). 19. S. Haas and E. Dagotto, Phys. Rev. B 54, R3718 (1996).

Optical Study of Spin/Charge Stripe Order Phase in S. Tajima,1 N. L. Wang,1 M. Takaba,2 N. Ichikawa,2 H. Eisaki,2 S. Uchida,2 H. Kitano,3 and A. Maeda3

Optical spectra of the spin/charge stripe order phase,

have been investigated for tra of Nd-free

and 0.20, compared with the specFor E//c, a sharp plasma edge for superconducting

carriers cannot be observed in LNSC within our covering frequency range down to This implies that the stripe charge order strongly suppresses the superfluid density and/or the Josephson coupling strength between the layers. For optical conductivity is substantially suppressed over a wide frequency range even at room temperature, while no appreciable spectral change takes place at the structural (LTO-LTT) phase transition temperature The former fact is indicative of a strong stripe fluctuation well above The metallic and

T-dependencies of optical conductivity in LNSC are in contrast to the low conductivity at dc and in the microwave region, which exhibits a weak semiconducting T-dependence below This suggests that some disorder effect dominates in the stripe charge dynamics in the very low region.

1. INTRODUCTION

The spin and charge stripe-ordering discovered by neutron scattering [1,2] has become a hot issue in the research of doped antiferromagnetic (AF) Mott insulators, such as Mn-, Ni-, and Cu-oxides. Although there are some common properties in these perovskite oxides, the nature of stripes in the cuprates seems to be considerably different from that in the other two materials. For example, in resistivity in the static ordered state is not large [2], whereas in the manganites and nickelates it becomes extremely large 1

2

3

Superconductivity Research Laboratory, ISTEC, Tokyo 135-0062, Japan. Dept. of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan.

Dept. of Basic Science, The University of Tokyo, Tokyo 153-0041, Japan.


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below the phase transition temperature [3]. Therefore, it is of great importance to investigate the charge dynamics of stripe-ordered state peculiar to the curates. However, so far there has been little direct study of charge excitation in LNSC, although a charge ordering was observed indirectly as a lattice distortion in the neutron diffraction experiments [4]. In this work, we investigated charge dynamics of LNSC by measuring optical reflectivity spectra and microwave conductivity. Comparing the spectra of LNSC and Nd-free

we found a dramatic effect of Nd substitution on the in-plane optical conductivity at room temperature. In this high-temperature phase, a strong stripe fluctuation seems to exist, and T- and -dependencies of optical conductivity are metallic, which persists even below the phase transition temperature. In the superconducting state, a substantial reduction of superfluid density and/or reduction of Josephson coupling was observed in the out-of-plane spectra of Nd substituted crystals. 2. EXPERIMENT

Large single crystals of and with various Sr contents (x) were grown by a traveling solvent floating zone method. The measurement samples were cut out along the c axis from as grown crystals. The ac surfaces were polished

by using at

powder for optical measurements. LNSC crystals show the phase transition from the low-T orthorhombic (LTO) to the low-T tetragonal (LTT)

phase. In the LTT phase, the static stripe order has been observed by the neutron diffraction measurements [4]. Optical reflectivity spectra were measured by using a standard FTIR spectrometer for and grating-type spectrometer for with the light polarization and The samples were mounted in a He-gas flow-type cryostat together with Au-evaporated mirror, which enables us to measure reflectivity accurately from 300 to 6 K. Conductivity in the microwave region was determined by measuring surface impedance of the samples in a microwave cavity. 3. IN-PLANE CHARGE DYNAMICS

Figure la shows the in-plane reflectivity spectra of LSC and LNSC for Sr content and 0.15 at room temperature. In all the spectra, reflectivity increases with decreasing frequency indicating a metallic charge response. As x increases, the well-known spectral growth is observed in the Nd-free LSC crystals. The spectral change with Nd substitution is

not pronounced at whereas reflectivity of LNSC is substantially depressed over a wide frequency range at This dramatic effect of Nd substitution is also seen in the conductivity spectra, as shown in Fig. 1b. The conductivity is strongly suppressed up to for whereas for almost no spectral change can be seen in this frequency scale. The spectral weight up to is reduced by 26% for and 7% for whereas it is less than 1 % for It should be noted that the Nd substitution effect is most pronounced at the composition where the static stripeorder is clearly observed. The result, that a stripe effect on the spectrum is almost negligible for is consistent with the recent neutron result that charge stripes cannot be seen clearly for whereas the spin ordering can be observed [5].

It is surprising is that the Nd effect, namely, the stripe-order effect, manifests itself at room temperature that is well above the phase transition temperature. This implies that

Optical Study of Spin/Charge Stripe Order Phase

273

the observed spectral change is a result of strong fluctuation of the stripe order far above the static ordering temperature. The reduced conductivity by the Nd substitution up to high of order of 1 eV is reminiscent of the effect of lowering the doping level. This may give a support for the 1D charge stripe model proposed by Tranquada et al. [4].

Assuming that a stripe direction rotates 90° in the adjacent layers, an optical spectrum is expected to be a mixture of the two components. One is a metallic spectrum along the stripes and the other is a presumably nonmetallic component perpendicular to the stripes. If the carriers could not hop between stripes or the spectrum were a completely insulating one for perpendicular polarization like that for the resulting spectrum would have shown much lower reflectivity or conductivity for to Therefore, the present result indicates that the hopping between stripes is not necessarily prohibited, or that carriers with 2D character may coexist with 1D carriers confined to the stripes. Next, temperature dependence of the reflectivity spectrum is shown for LNSC with in Fig. 2. Because of the strong stripe fluctuation effect at high temperatures, there is no dramatic effect on the spectrum when we cool the sample across the phase

transition temperature With lowering temperature, the far-infrared reflectivity increases monotonously, which results from a reduction of the carrier damping. This metallic temperature dependence as well as the metallic

dependence of reflectivity does not change

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remarkably, even below This is in contrast to the dc resistivity behavior, which shows a small jump at and a following upturn, reflecting a static charge ordering. In order to know at which frequency the crossover from a semiconducting to a metallic behavior takes place, we investigated the in-plane conductivity in the microwave region by measuring surface impedance. Figure 3 shows the temperature dependence of resistivity for LNSC at 50 GHz, that was calculated from the surface impedance. The observed temperature dependence is in agreement with the dc behavior. We also measured the resistivity at 100 GHz Its T dependence is qualitatively the same as the result in Fig. 3. Therefore, we can expect a crossover at frequency between 3 and One of the plausible explanations for the difference between the low- and the behaviors is that the charge stripes are not ideally long straight lines but are broken into short segments due to defects or something else. In such a case, the low conductivity would be affected by the disorder, whereas conductivity becomes metallic at frequencies higher than the hopping energy between the segments.


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4. OUT-OF-PLANE SPECTRA

As is similar to the spectra of the Nd-free LSC, the c axis spectrum of LNSC is dominated by phonons. As temperature decreases, the phonon peaks become sharp, but there is no remarkable change at except for an appearance of a small peak around This is in a sharp contrast to another typical stripe phase in where a clear phonon split due to the lattice distortion is observed [6]. In the case of LNSC, the lattice distortion accompanied by the charge ordering is very small. As is well known, below a sharp plasma edge for the superconducting carriers

appears in the c axis spectrum of LSC, which indicates that a superconducting gap energy is larger than a screened plasma energy [7]. Because the temperature limit in our measurement

system is about 6 K, we could not examine the superconducting state in LNSC for and 0.12 with lower than 4 K. For and 0.20, no reflectivity edge was observed, even at the lowest temperature (Fig. 4). The values are 12 K and 16 K for and 0.20, respectively. There are some possible explanations for this phenomenon. One

possibility is that a plasma frequency becomes lower than our frequency limit

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by the Nd substitution, which originates from a reduction in the superconducting carrier density and/or in the interlayer Josephson coupling. Assuming the model of the alternating stack of stripes with 90° different directions, it is understandable that the superconducting

phase coherence is desturbed between the layers. The other possibility is that a number of unpaired carriers are created by the pair breaking, which smears out a plasma edge feature. A small reduction of reflectivity below for is a signature of superconducting gap. A slightly increasing reflectivity curve with reducing suggests that a screened plasma frequency is not far from the observed frequency range. Therefore, in this case, the reduction in the superfluid density is caused predominantly by the appearance of a huge amount of residual carriers. An increase in residual conductivity within the gap is characteristic feature of pair breaking in a d wave superconductor [8]. If we consider the stripe ordering as a kind of disorder in superconductors, it could be an origin of pair breaking.

5. SUMMARY

The optical conductivity and the microwave conductivity of LNSC with various Sr contents have been investigated, compared with the conductivity of Nd-free LSC. For the Nd substitution substantially suppresses conductivity spectral weight below 1 eV at all

measurement temperatures. This is suggestive of a strong fluctuation of the stripe order that makes the CuO plane insulating for polarization perpendicular to the stripes, resulting in a relative enhancement of the mid-infrared absorption. Compared with the spectra of other typical materials, such as and with the same doping level, the conductivity spectrum of LSC is enhanced in the mid-infrared region, as shown in Fig. 5. It may indicate that even the Nd-free LSC is affected by the stripe fluctuation, which could be an origin of the disordered nature of the electronic state in LSC. The and the T-dependencies of conductivity are dominated by the metallic component, which presumably corresponds to the conductivity along the stripes. Being inconsistent with the behavior in the microwave region and at dc, the metallic behavior at


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high remains even below This supports a segment model of charge stripes in which the disorder affects the charge dynamics up to the frequency of the order of millimeter

wavelength. In the c axis spectra, a striking effect of the Nd substitution on the superconducting plasma is observed, namely, a sharp reflectivity edge for the superconducting plasma disappears in the studied LNSC crystals. It indicates that the stripe-order fluctuation substantially reduces the number of superconducting carriers and/or the interlayer Josephson coupling. This can be understood if the stripe order destroys the superconducting phase coherence between the layers. ACKNOWLEDGMENT This work was partially supported by NEDO for the Research and Development of Industrial Science and Technology Frontier Program.

REFERENCES 1. C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993); J. M. Tranquuda et al., Phys. Rev. Lett. 73, 1003 (1994); B. J. Sternlieb et al., Phys. Rev. Lett. 76, 2169(1996).

2. J. M. Tranquada et al., Nature 375, 561 (1995).

3. Y. Nakamura and S. Uchida, Phys. Rev. B 46, 5841 (1992). 4. S.-W. Cheong et al., Phys. Rev. B 49, 7088 (1994). 5. J. Tranquada et al., provate communication. 6. T. Katsufuji et al., Phys. Rev. B 54, R14230 (1996).

7. K. Tamasaku, Y. Nakamura, and S. Uchida, Phys. Rev. Lett. 69, 1455 (1992). 8. N. L. Wang, S. Tajima, A. I. Rykov, and K. Tomimoto, Phys. Rev. B 57, R 1 1 0 8 1 (1998). 9. S. L. Cooper et al., Phys. Rev. B 47, 8233 (1993).


Vibrational Pseudo-Diffusive Motion of the Oxygen Octahedra in from Anelastic and Quadrupolar Relaxation F. Cordero,1 C. R. Grandini,2 R. Cantelli,3 M. Corti,4 A. Campana,4 and A. Rigamonti1

The vibrational dynamics in nearly stoichiometric is studied by means of anelastic and NQR relaxation. After lowering to values of the order of 0.001, a relaxational peak remains in both the anelastic and NQR spectra with a mean relaxation time

Such a process

is attributed to the motion of octahedra in doublewell potentials whose cooperative character increases the effective energy barrier to the observed value. It is speculated that the freezing of the motion of the octahedra may correspond

to the locking of the stripes.

1. INTRODUCTION

Evidence is accumulating that the local structure of many, if not all, the cuprate superconductors differs from the average structure that is extracted from traditional diffraction experiments. In some cases, the local inhomogeneities can be put in close relationship with the separation of the carriers into antiferromagnetic insulating domains and conducting domains, often observed as parallel stripes [1]. In spite of the extensive investigation on this subject, it is only recently that the issue of the dynamics of such inhomogeneities has been addressed by showing that the anelastic relaxation spectrum of nearly stoichiometric exhibits intense peaks of thermally activated type that are due to intrinsic 1

CNR, Area di Ricerca di Tor Vergata, Istituto di Acustica “O.M. Corbino,” Via del Fosso del Cavaliere 100, I-00133 Roma, and INFM, Italy. 2 Universidade Estadual Paulista, Departamento de Fisica, 17.033-360, Bauru, SP, Brazil. 3 Università di Roma “La Sapienza,” Dipartimento di Fisica, P.le A. Moro 2, I-00185 Roma, and INFM, Italy. 4 Dipartimento di Fisica “A. Volta,” Università di Pavia, Pavia, and INFM, Italy.


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fluctuations of the lattice [2]. Here, we present combined anelastic and NQR relaxation measurements of undoped which we interpret in terms of the collective dynamics of the tilts of the octahedra in the LTO phase.

2. EXPERIMENTAL AND RESULTS The samples were sintered bars of with dimensions No impurity or amorphous phases were detected by the powder x-ray diffraction (XRD) spectra taken on some of the samples in the as-prepared state and by resistivity measurements. The nonstoichiometric oxygen was extracted by outgassing at 700–750°C down to an partial pressure of the order of torr. It was checked that after several outgassing and oxygenation treatments no decomposition of the sample occurred by measuring a superconducting transition at 34 K and in the oxygenated state Further details on the preparation, oxygenation, and deoxygenation are reported in Ref. [2]. The elastic energy loss coefficient, measured by exciting the flexural vibrations (0.46 and 6.9 kHz for sample 2), is given by where is the complex dynamic compliance, with unrelaxed value The contribution to the imaginary susceptibility from a relaxation process with characteristic time

is

where x is the atomic fraction of relaxing entities, each producing a change of the strain when its state changes; is times the measuring frequency. The curve has a maximum at the temperature where so allowing the determination of the microscopic relaxation rate at the peak temperature. Figure 1 shows two spectra of sample 2 in the as-prepared state and after extracting excess oxygen by annealing in vacuum at 750°C for 1 hr. All the peaks are shifted to higher temperature at the higher frequency, indicating that they are due to thermally activated relaxation processes with decreasing with temperature. The peaks in the as-prepared state are due to the diffusive hopping of interstitial oxygen atoms (peak O 1 with and activation energy of 5600 K) and to pairs or other complexes of excess oxygen (peak O2) [2]. The outgassing treatment apparently did not completely remove the excess oxygen because a trace of peak O1 is still present both in the anelastic and NQR spectra; from its residual intensity, we estimate that has been lowered of at least 20 times, and therefore In this condition, the curve is not flat, as expected from a defect-free stoichiometric lattice, but develops an intense peak with an apparent activation energy of 2800 K, labeled T, which we attribute to relaxational dynamics of the octahedra [2]. In the same sample 2, relaxation measurements have been carried out, and from the recovery plots of the NQR echo signals at and the relaxation mechanism has been identified [3,4]. It was found that, at least for the relaxation mechanism is quadrupolar, namely due to the time-dependent electric field gradient at the La site (Fig. 2a, inset). Two main contributions to the quadrupolar relaxation are present. One contribution, corresponding to peaks O1 and O2, is due to the diffusion of the extra stoichiometric oxygen [5], and for small it is sizeable only for

Vibrational Pseudo-Diffusive Motion

281

For relatively large (sample 5 oxygenated by slowly cooling from 620°C to 310°C in 820 torr O2), the contribution from the diffusion of the oxygen becomes dominant also at T 200 K (Fig. 2a). The relaxation mechanism related to oxygen diffusion is discussed elsewhere [6]. Here, we focus our attention only on the results that refer to the contribution from the phonon-like motions of the oxygen octahedra (T < 400 K) for sample 2. 3. DISCUSSION AND CONCLUSIONS Let us first discuss the quadrupolar contribution to the relaxation due to the motions of octahedra. First, in the light of the experimental findings (see later), one must consider a direct relaxation process due to the spectral component of the functions at Furthermore, for an order of magnitude estimate of the relaxation rate, one can assume uncorrelated motions of the four octahedra with apical oxygens in the same plane of La ions (LaO plane) and of the one with apical oxygen directly above La ion along the c axis. From a linear expansion of the EFG in terms of the displacements s(t) of the ions from the equilibrium lattice positions, the quadrupolar relaxation rate for the nuclei close to the octahedra involved in the pseudo-diffusive motion can be written [71

where a is a constant of the order of and is the correlation function for the motion of the apical oxygen (for details in the calculation see Ref. [8]).

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One should remark that, for harmonic local potential, namely underdamped phonon modes, Eq. (2) would give a negligible contribution to the relaxation. For purely phonon modes in fact, only the second-order Raman process, with no maxima as a function of T and no frequency dependence in the relaxation rate, would be present [7]. The mere observation of the maxima in (Fig. 3) and in the anelastic relaxation spectrum implies that a strongly anharmonic local potential characterizes the oxygen motions. Such motions can be described in terms of a 1D model of interacting atoms in a potential of


283

the form where each atom moves in a double well with an energy barrier and the coupling to the neighbors is bilinear, with taking into account a cluster average over configurations. It can be proved [9] that the spectral density in Eq. (2), besides a resonant peak at the typical frequency displays a central peak of pseudo-diffusional character with characteristic frequency with The effective potential barrier depends on the local potential and on the term describing the coupling between the octahedra, whereas

where v is the average velocity of propagation of the nonlinear soliton-like excitation through the atoms spaced by d. Then, Eq. (2) is rewritten

where

is of the order of the distance between the two minima in the double well local

potential.

The imaginary part of the elastic compliance the strain correlation function

V being the sample volume it is averaged). The strain

can also be expressed in terms of

is inversely proportional to the volume over which is directly related to the displacements s (t) of the ions

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of the octahedra, although not in a trivial way, and therefore Eq. (4) differs from that for the NQR relaxation rate, Eq. (2) below, only for a factor and the constants expressing the dependence of strain and NQR frequency on the atomic displacements [compare also Eqs. (3) and (1)]. The interstitial oxygen atoms prevent the relaxational dynamics of the neighbour-

ing octahedra, as demonstrated by the fact that the anelastic relaxation peak T is readily suppressed by the introduction of small amounts of excess oxygen (Fig. 1 and Ref. [2]). Therefore, the clusters of octahedra free to build up the cooperative relaxational dynamics are limited by the interstitial oxygen atoms, and this implies in turn that the cluster average of the interaction strength depends on the cluster size and shape. For this reason, we distributed . according to a gaussian, which results in a distribution of effective energy barriers (the effect on was included but is negligible). A feature of the anelastic peak that cannot be accounted for by the above formulas is the increase of its height at higher temperature instead of a decrease as 1/T. Such a temperature dependence is observed in the case of relaxation among states that differ in energy by it can be shown [11] that the relaxation strength between the two states 1 and 2 differing in energy by must be multiplied by a factor containing the product of their equilibrium occupation numbers, In addition, by writing the rate equation for the relaxation between the inequivalent states, one obtains

Although the correction to the rate does not affect much the relaxation curves, the correction to the intensity produces a maximum at in the relaxation strength, which again falls off as

at higher temperature. These corrections are valid for relaxation

between definite levels without cooperative effects, and their extension to the above model

of cluster dynamics is not obvious; nonetheless, the temperature dependence of the anelastic relaxation strength imposes the consideration of relaxation occurring among states that are somehow energetically inequivalent. Figure 3 presents a fit of both the anelastic and NQR relaxation curves with the above expressions, namely,

The values of the potential parameters were chosen in order to obtain a single particle energy barrier with (as theoretically estimated from a self-consistent analysis of the temperature dependence of the soft mode and of the elastic constant [12]). The mean value of the coupling constant was and its distribution width was and 0.25 for the NQR and anelastic data (a temperature-dependent width may result from the ordering of interstitial O); such values of result in a mean effective energy barrier with a distribution width of The mean value of is The asymmetry energy is 10 times smaller than and therefore does not change the overall picture much.


285

The fit value of the strength of the relaxation mechanism is Because is the La-O distance), for one would derive a strength of the relaxation mechanism of the order of 8.6 MHz, namely, about a factor 100 larger than the experimental data. However, one must take into account that only a fraction x of the octahedra is involved in the tilts. Then a configurational average of a “local” relaxation rate should be considered. This average is roughly equivalent [13] to multiply Eq. (2) by a factor of the order of so requiring This factor ensures the consistency of the experimental result with the theoretical estimate of . for of the order of as derived in Ref. [12]. Summarizing, the relaxation due to the motions of the octahedra, coupled with the anelastic relaxation, allows one to conclude: (i) the tilts of the oxygen octahedra occur in a double well local potential; (ii) the mechanical and quadrupolar relaxation are related to the low-frequency central component of the spectral density with an effective width given by (iii) the effective barrier appears much larger than the barrier height of the doublewell evidencing the cooperative character of the pseudo-diffusional motion of the oxygen octahedra; (iv) both the experimental findings are compatible with only a fraction of oxygen octahedra involved in the cooperative tilting motions; (v) one could speculate that the freezing of the tilts occurring below about 100 K corresponds to the pinning of the stripes. In this respect, we note that the formation of domains as stripes or tweed patterns is common to several types of structural phase transitions in which one has the coexistence of domains with different in-plane shear strain [14]. The tilts of the octahedra in the double-well potentials are clearly related with the charge-lattice stripe fluctuations in the same system at higher doping [15,16]. The present work provides important information on the dynamics of such tilt waves at very low doping. REFERENCES 1. Proc. Int. Conf. on Stripes, Lattice Instabilities and Superconductivity, eds. A. Bianconi and N. L. Saini, J. Supercond. 10, (1997). 2. F. Cordero, C. R. Grandini, G. Cannelli, R. Cantelli, F. Trequattrini, and M. Ferretti, Phys. Rev. B 57, 8580 (1998). 3. T. Rega, J. Phys.: Condens. Matter 3, 1871 (1991).

4. 5. 6. 7.

I. Watanabe, J. Phys. Soc. Jpn. 63, 1560 (1994). S. Rubini, F. Borsa, A. Lascialfari, and A. Rigamonti, Il Nuovo Cimento 16 D, 1799 (1994). F. Cordero, R. Cantelli, M. Corti, M. Campana, and A. Rigamonti, Phys. Rev. B 59, 12078 (1999). A. Rigamonti, Adv. Phys. 33, 115 (1984).

8. A. Campana, Thesis, University of Pavia, 1998. 9. S. Torre and A. Rigamonti, Phys. Rev. B 36, 8274 (1987). 10. H. Wipf and B. Kappesser, J. Phys.: Condens. Matt. 8, 7293 (1996). 1 1 . F. Cordero, Phys. Rev. B 47, 7674 (1993). 12. A. Bussmann-Holder, A. Migliori, Z. Fisk, J. L. Sarrao, R. G. Leisure, and S. W. Cheong, Phys. Rev. Lett. 67, 512(1991).

13. M. Corti, A. Rigamonti, F. Tabak, P. Carretta, F. Licci, and L. Raffo, Phys. Rev. B 52, 4226 (1995). 14. S. Kartha, J. A. Krumhansl, J. P. Sethna, and L. K. Wickham, Phys. Rev. B 52, 803 (1995). 15. A. Lanzara, N. L. Saini, A. Bianconi, J. L. Hazemann, Y. Soldo, F. C. Chou, and D. C. Johnston, Phys. Rev. B 55, 9120(1997). 16. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996).


Charge and Spin Dynamics of Cu-O Chains in Cuprates An NMR/NQR Study B. Grévin,1 Y. Berthier,1 G. Collin,2 and P. Mendels3

Spin and charge fluctuations are investigated along the three lattice directions on the Cu(1) site in a nonsuperconducting sample using NMR/NQR. Our data give evidence for charge instability (CDW) involving charge fluctuations along the O(4)-Cu(l)-O(4) apical axis. In view of these findings, we reanalyzed the situation of chains and planes in using Cu( 1) and Cu(2) NQR. We confirm the important role of charge/lattice effects in these cuprates

and we discuss the consequences of these mechanisms on the interaction between Cu-O chains and

planes.

1. INTRODUCTION There is an increasing interest in the physics of Cu-O chains in rare earth (RE)-123 systems, as they may play an important role in the normal and superconducting states of these compounds. Although many NMR results are available for the properties ofthe planes, little attention has been paid to the intermediate CuO layers. The study of the intrinsic physics of these chains is not straightforward, as they are surrounded by superconducting planes in most of the RE-123 cuprates. An exception occurs with the insulating compound in which antiferromagnetic order in the planes is present below 285 K. The holes are localized in these planes [1], whereas the chains are expected to remain metallic on a local scale as in In a previous study [2], we show that a charge instability occurred at 120 K preceded by charge fluctuations in the temperature range 120–180 K. The 1

Laboratoire de Spectrométrie Physique UMR 5588 CNRS, Université Joseph-Fourier Grenoble-1, B.P. 87, 38402

Saint Martin d’Hères, France. 2

Laboratoire Léon Brillouin, CEA-Saclay, 91191 Gif sur Yvette, France. LPS URA2 CNRS, Université Paris Sud, 91405 Orsay Cedex, France.

3


287

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Grévin, Berthier, Collin, and Mendels

key problems, therefore, are the origin of this transition, the range of the order parameter, and the possible interaction with the planes when they are superconducting, as in In order to give some insight into these fundamental questions, we present a comparative NMR/NQR study of the copper sites in these two compounds. 2. CHARGE INSTABILITY IN THE

CHAINS

In a previous work [2] we show that some kind of charge instability arises in the chains. Here we give a summary of the main results and discuss the nature of this transition.

2.1 NMR and NQR of the Cu(1)Site in The samples used in these experiments were carefully synthesized in order to prevent any substitution on the barium site and to minimize the amount of oxygen vacancies in

the chains. The procedure is described in detail elsewhere [2]. NMR measurements were performed on powder oriented along the c axis in an epoxy resin under an applied field of 6 T. spectra at 300 K and 4.2 K are shown in Fig. 1. At low temperature, an anomalously broadened line is observed, but the spectrum is remarkably well resolved at high temperature, with a line shape similar to but broader than that obtained

in We deduce a value very near to that reported for the chains in which confirms that the hole-doping of chains is similar in Pr-123 and Y-123. The full width at half maximum (FWHM) of the line, shown on Fig. 2a, starts to increase below 170 K and the same behavior is observed for the spin–lattice and the spin–spin relaxation rates (see Fig. 2b). Below 120 K, the increase of

prevents the detection of any NQR signal owing to the dead time of the spectrometer. Fortunately, the NMR of copper on the Cu( 1) site allows us to cover the whole temper-

ature range between 300 K and 4.2 K as shown on Fig. 3a,b. Analysis of measured on the two copper isotopes at 8.5 and 4.5 T shows that charge fluctuations contribute to

Charge and Spin Dynamics of Cu-O Chains

289

the relaxation between 200 K and 120 K. Below this temperature, a gap opens in the low energy excitation. Furthermore, no discontinuity occurs between 200 K and 100 K on the spin part of the magnetic hyperfine shift measured on Cu(1), as shown in Fig. 3c. All these results clearly indicate that charge fluctuations are involved in the mechanisms of this transition. 2.2 Analysis and Discussion One of the key problems raised by our results concerns the nature of the mechanisms responsible for the charge instability observed at The comparison between the relaxation rates measured in NQR and NMR with parallel to the c axis strongly suggests that the relaxation mechanism involves charge fluctuations along the c axis. Indeed NMR probes only fluctuations perpendicular to the c axis, whereas NQR—in the particular case of the Cu(1) site [where the EFG is asymmetric ]—is sensitive to thefluctuations in the three directions [3,4]. This accounts very well for the stronger critical fluctuations observed in NQR. The temperature dependence of the NMR spin–lattice relaxation rate suggests a simultaneous condensation of charge and spin degrees of freedom in the same energy gap below a temperature that is quite high. This indicates that the electronic correlations are in a moderately strong regime (i.e., no spin–charge separation). Another feature of the transition is revealed by the large value These two points are in favor of a CDW driven by a rather strong electron–phonon coupling. The change in the NQR lineshape between high and low temperatures is well simulated with the hypothesis of an EFG modulation induced by a CDW transition. Details of this simulation are the subject of a later paper. A last question concerning the range of the low-temperature order (long range or short range) is beyond the possibility of our NMR/NQR investigations alone. At this stage, it

290


should be noted that our results agree quite well with some aspects of the microscopic model proposed by Fehrenbacher [5]. Following this author, a short-range CDW order is expected if one assumes polaron–polaron interactions in the chains. This assumption is consistent with our preceding conclusion. We also emphasize that experiments on Y-123 and Y-124 compounds have shown the absence of long-range order in the chains at low temperatures [6,7]. In particular, pair distribution function (PDF) analysis of pulsed neutron data in Y-124 [8] has been interpreted in terms of polarized microdomains linked to the pseudogap phenomena observed in underdoped cuprates. 3. SPIN AND CHARGE FLUCTUATIONS IN

Since the early NQR experiments in and Cu(2), the evidence for a discontinuous change of

on the two copper sites Cu(1) (T) near the critical temperature


291

is still controversial [11,12]. This issue, which implies a lattice effect accompanying the superconducting transition, underlines the consequences of such a result. We have revisited this crucial issue by making accurate measurements. 3.1 NQR on the Cu(1) and Cu(2) Sites in

The powder sample used for NQR experiments is overdoped

, as confirmed by the temperature variation of the Cu(2) relaxation The small amount of oxygen vacancies in the sample provides well-resolved NQR spectra on the two copper sites. Figure 4 shows the temperature dependence of on the plane and chain copper sites. In the normal state, the change of EFG at the two sites with decreasing temperature has been interpreted as resulting from a lattice compression effect [10]. Moreover, a crossover occurs below 170 K in the NQR frequencies of the two sites. At anomalies affect the two frequencies, with a larger dip for Cu1(1). Cu(1). This This unambiguously confirms the existence of a lattice/charge effect occurring at We wish to stress here that great attention was paid to the thermal equilibrium of the sample during our experiments. We now turn to the T dependence of the spin–lattice relaxation rate on the Cu(1) site. Generally, our results (see Fig. 5) agree well with those of Imai et al. [13]. The characteristic behavior of the relaxation rate, which is higher than that of the Cu(2), emphasizes the exceptional mechanism responsible for the relaxation of Cu(1) chain site. This situation is different from that found in Hg- and Tl- based HTSC, where the relaxation of the Hg/Tl nuclei is dominated by fluctuations transferred from Cu(2) sites. Some authors attribute this

difference to the ID character of the Cu-O chains, but it should also be remembered that

and nuclei have a spin and consequently are insensitive to charge fluctuations. By contrast, the copper chain site is sensitive to both relaxation channels, magnetic and charge fluctuations. In fact, the comparison we made between

292


measured on the two copper isotopes showed that, for temperatures between 300 and 170 K, charge fluctuations partially contribute to the relaxation of Cu(1) site. In addition to the overall temperature dependence of on the Cu(1) site, new features come out from our measurement both in the normal state and near the critical temperature. A marked discontinuity occurs at 240 K and a crossover starts below 170 K. A linear T dependence is observed between 80 and 140 K and, contrary to previous reports, the opening of the low-temperature gap seems to occur below This important point is discussed in detail in a forthcoming paper. 3.2 The Interaction between Chains and Planes In a recent NMR/NQR study in the underdoped double chain compound Suter et al. [14] showed that an electronic crossover occurs below implying charge fluctuations in the planes and the chains. These results were interpreted as the reflection of a charge density wave driven by the strongly correlated planes. As is close to the spin pseudogap opening temperature the results were taken as favoring a model by Eremin et al. [15] that links the pseudogap phenomenon to a CDW occuring in the planes. These findings raise the question of the interaction mechanism between planes and chains, and more precisely the origin of the charge instability reported by Suter et al. for two reasons: (i) PDF neutron data have shown that short-range correlated oxygen displacements originate from the chains and (ii) our results in show that CDW correlations develop inside the chains. The comparison between and give some evidence for a complex interaction between planes and chains beyond the simple charge-transfer mechanism. If one assumes that there is no coupling between these two subsystems, the temperature dependence of low-energy excitations on Cu(1) would be similar in these two cuprates. Obviously, this is not the case. A possible scenario for the coexistence of superconducting planes and quasi-1D chains with CDW correlation has been proposed by Hertel et al. [16]. In the normal state, the scattering on the chain induced by the plane carriers prevents the development of a long-range


293

CDW order parameter. Such an ordered state is possible only below the superconducting transition, as the plane carriers condense in the superconducting gap. In this context, the

crossovers detected below 170 K and 180 K in and respectively, could be attributed to short-range CDW fluctuations in the chains. In this case, long-range CDW order should be present at low temperatures. Such a situation does not appear clearly from experimental results, as PDF and tunneling microscopy (STM) [6,7] only indicate short-range correlation, even at low temperatures. The presence of a crossover in the underdoped compound as well as in overdoped place a severe constraint on the models assuming a link between this mechanism and the pseudogap phenomenon. If we assume that the crossover in is effectively connected to the pseudogap, can we imagine the same link in It is important to point out that the existence of a pseudogap at in overdoped opening at very near is still controversial. Moreover, in Bi-2212 sysregime. Furthermore, the oxygen order in the chains strongly affects

in underdoped

4. CONCLUSION

In summary, our NMR/NQR study has shown that charge fluctuations along the O(4)Cu(1)-O(4) axis play an important role in the physics of chains in cuprates The comparison between

and

clearly demonstrates that a

complex interaction exists between chains and planes in in the normal state and at In particular, the dips observed on the quadrupolar frequencies on Cu(1) and Cu(2) give some arguments for charge/lattice effects occurring above and at the superconducting transition.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

R. Fehrenbacher and T. M. Rice, Phys. Rev. Lett. 70, 3471 (1993). B. Grévin, Y. Berthier, G. Collin, and P. Mendels, Phys. Rev. Lett. 80, 2405 (1998). J. Chepin and J. H. Ross, J. Phys.: Cond. Matt. 3, 8103 (1991). R. Kind and J. Seliger in Proceedings of Vth Ampère International Summer School on Nuclear Resonance in Solids (1978). R. Fehrenbacher, Phys. Rev. B 49, 12230 (1993). H. L. Edwards, A. L Barr, J. T. Markert, and A. L. Lozanne, Phys. Rev. Lett. 73, 1154 (1994). H. L. Edwards et al., Phys. Rev. Lett. 75, 1387 (1995). T. R. Sendyka et al., Phys. Rev. B 51, 6747 (1995). H. Riesmeier et al., Solid State Commun. 64, 309 (1987). M. Mali et al., Phys. Lett. A 124, 112 (1987). D. Brinkmann, Appl. Magn. Res. 3, 483 (1992). H. Riesmeier, S. Gärtner, V. Müller, and K. Lüders, Appl. Magn. Reson. 3, 641 (1992). T. Imai et al., J. Phys. Soc. Jpn. 57, 2280 (1988). A. Suter et al., Phys. Rev. B 56, 5542 (1997). I. Eremin et al., Phys. Rev. B 56, 11305 (1997). P. Hertel, J. Appel, and J. C. Swihart, Phys. Rev. B 39, 6708 (1989). Ch. Renner et al., Phys. Rev. Lett. 80, 149(1998). P. Carretta, Physica C 292, 286 (1997).


Mobile Antiphase Domains in Lightly Doped Lanthanum Cuprate P. C. Hammel,1 B. J. Suh,1 J. L. Sarrao,1 and Z. Fisk2

Light hole doping of lanthanum cuprate strongly suppresses the onset of antiferromagnetic (AF) order. Surprisingly, it simultaneously suppresses the extrapolated zero temperature sublattice magnetization. results in-lightly doped demonstrate that these effects are independent of the details of the mobility of the added holes. We propose a model in which doped holes phase separate into charged domain walls that surround “antiphase” domains. These domains are mobile down to at which point they either become pinned to the lattice or evaporate as their constituent holes become pinned to dopant impurities.

1. INTRODUCTION A fundamental issue in the normal state of the superconducting cuprates is the behavior of holes doped into a two-dimensional (2D) lattice of spins with strong antiferromagnetic (AF) interactions. Even for lightly doped, single-layer lanthanum cuprate many important issues remain poorly understood. Long-range AF order occurs at in undoped lanthanum cuprate, but is rapidly suppressed by the addition of a small density, p of holes per Cu. This rapid suppression is clearly related to the disruptive effects of mobile holes: is sufficient to suppress to zero, whereas isovalent substitution of Zn or Mg for Cu is required [1] to produce the same effect. A range of studies [2] including measurements [3] in lightly doped have demonstrated that the suppression of and in fact, all the magnetic properties of lightly doped lanthanum cuprate are essentially invariant without regard for the means of hole doping and consequent variations in hole mobility. It is unlikely that a collection of individual holes can lead to magnetic behavior that is entirely independent of compositional variation that leads to substantial variations in 1

Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545.

2

National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306.


295

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resistivity (at constant doping). We argue, instead, that this is strong evidence that holes form collective structures. An important and well-documented aspect of doped cuprates is their tendency toward inhomogeneous charge distribution [4]. Segregation of doped holes into charged stripes separating hole-free domains has been predicted [5–11] and recently observed directly in lanthanum cuprate [12]. It was proposed earlier that phase segregation of holes could be responsible for the unusual magnetic properties of lightly Sr-doped lanthanum cuprate [13–15]. We make a related proposal that holes form charged domain walls that form closed loops with the important differences that these walls form antiphase domain walls (so the phase of the AF order inside these domains is reversed) and that the walls and hence the enclosed domains are mobile, and the charged walls have the density of 1 hole per 2 Cu sites in agreement with neutron scattering results [12]. The antiphase character means that mobile (above 30 K) domains suppress the time-averaged static moment, thus suppressing as well as These domain structures have contrasting interactions with in-plane vs. out-of-plane dopants (e.g., stronger scattering by in-plane impurities), which explains the different transport behaviors, whereas the universal magnetic properties can be understood as long as the domains are sufficiently mobile that they move across a given site rapidly compared to a measurement time.

2. LIGHTLY DOPED LANTHANUM CUPRATE A systematic study of the temperature T and doping dependence of the static susceptibility in lightly doped lanthanum cuprate by Cho et al. [13] provided evidence that the added holes are inhomogeneously distributed. The development of long-range AF order is signaled by a peak in the static susceptibility; they showed the rapid increase in the width of this peak with increasing hole density could be understood as arising from finite-size effects. They proposed that doped holes form hole-rich domain walls that bound hole-free domains, thus cutting off spin interactions across the boundary and truncating the growth of the spin–spin correlation length with decreasing temperature above They deduced the doping dependence of the dimension of the hole free-regions, and found this suggests that the density of holes within the boundary stripe is very low, hole per 5 or 10 Cu sites. measurements in lightly Sr-doped lanthanum cuprate by Chou et al. provided a detailed picture of the T and p dependence of its magnetic properties [14]. They found that for the sublattice magnetization , is strongly suppressed as p increases. However, below 30 K, recovers to its value. The low temperature spin dynamics are also unusual; the nuclear spin-lattice relaxation rate has a strong peak at a doping dependent temperature in the vicinity of 10–15 K. To explain these unusual features, they extended the finite size model of Cho et al. [13], and proposed that the suppression of could be understood in the context of the restricted set of spin wave modes accessible in the confined AF domains [15]. The low temperature peak in 2W is clearly associated with freezing of Cu spin degrees of freedom; they interpreted this in terms of freezing out of hole motion within the domain walls surrounding hole-free regions. Adding holes by means of in-plane substitution of for introduces impurities into the planes that strongly alter charge transport properties. We used nuclear quadrupole resonance (NQR) measurements to microscopically examine the effects of doped holes on the AF spin correlations (Fig. 1), in this case where the hole mobility

Mobile Antiphase Domains

297

is much reduced compared to the Sr-doping case [3]. Comparing La2–ySryCuO4 (LSCO) and at one finds that the room temperature resistivity of LCLO [2,17] exceeds that of LSCO [18] by over an order of magnitude. Furthermore, unlike LSCO, the resistivity of LCLO always increases monotonically with

decreasing temperature. With increasing doping, the contrast becomes more dramatic as LSCO becomes metallic and superconducting whereas LCLO becomes ever more insulating with doping above In spite of this, we find that the magnetic behavior of the two materials is essentially identical [3]. In addition to the similarly strong suppression of by doping [2], we find that is also suppressed, and the correspondence between the suppression of and by doping is identical to that observed in LSCO [15]. In Fig. 2, for both LCLO [3] and LSCO [15] is plotted against Here, is the value of obtained by extrapolating the data for i.e., the value of

298


the solid lines shown in Fig. 1a. The solid line through the data is due to a theory of Neto and Hone [16] (see also van Duin and Zaanen [19]). The strong peak in 2W occurs at the same temperature and exhibits the same binding energy (as extracted from the T dependence on the high temperature side of the peak) [3], Finally, the temperature dependence of the low-energy dynamical susceptibility [obtained from measurements of 2W(T)] exhibits the same finite-size effects [3] as were observed in the static susceptibility by Cho et al. [13]. 3. MOBILE ANTIPHASE DOMAINS

There is clear evidence for stripe formation in 2D doped AF. In (isostructural to lanthanum cuprate) static stripes have been observed in several cases [20–24]. The recent observation of similar elastic superlattice peaks in demonstrates the existence of static charged stripes in the cuprates, and supports the idea that stripes are universally present in lanthanum cuprate [25,26] but that they are observable as static only under special conditions that pin the stripes to the lattice [12]. Similarities between elastic superlattice peaks associated with static stripes and the incommensurate peaks observed in inelastic neutron studies of have been noted, and these incommensurate peaks are being reconsidered as possible evidence for the presence of dynamic charged stripes in the cuprate [28]. The density of holes in the charged domain walls depends on the material: in the nickelates it is 1 hole per stripe Ni site; in the cuprate, the density is 1/2 hole per stripe Cu site. The neutron diffraction studies have demonstrated that spin–spin interactions are not cut off by the charged domain walls; rather, interactions across them are strong: it is universally observed that they serve as antiphase domain walls between the hole-free regions they separate. Thus the sign of the spin correlations is reversed on crossing the domain wall. The formation of domain walls into loops as opposed to parallel stripes has been observed in Hartree–Fock calculations [5]. Using density matrix renormalization group techniques to calculate the energy of a domain wall in the 2D t-J model, White and


299

Scalapino [29] observed that charged domain walls form loops. They point out that this is favorable at low doping in the case in which the coupling between planes is significant. Because the walls constitute antiphase domain walls, the coupling between two planes is disrupted, in general, by domain walls. Hence, interplane coupling would favor domain walls forming closed loops so that most of each plane would be in the dominant AF phase. In the event that these antiphase domains are mobile, passage of such a domain over a

given site reverses the orientation of a particular ordered Cu moment. The splitting of the line (shown in Fig. 1) is proportional to local hyperfine field due to the ordered

moment on the neighboring Cu site. If this moment is time varying, the splitting will be proportional to the time-averaged local moment. In the absence of antiphase domains, the hyperfine field will be constant, giving the value of observed in undoped lanthanum cuprate. If the motion of the antiphase domains is rapid compared to the NQR measurement time, the net local hyperfine field will be proportional to the fraction of time the moment is in the dominant AF phase minus the time it is in an antiphase domain, and hence proportional to the area of the dominant phase minus the area of the antiphase domain. We can estimate the doping dependence of the size and spacing of the antiphase domains from the known behavior of If we define the data [15] for is well described by with For simplicity, we assume that a (1,0) or (0,1) domain wall orientation is preferred, and so consider square domains. If a region of size L contains, on average, one antiphase domain of size l (Fig. 3; all lengths are in units of the lattice parameter), then

Here,

where

is the number of sites in the antiphase domain

300


and is the number of sites in the dominant AF phase. The number of holes in the region of size L is the domain wall that bounds the antiphase domain contains 1 hole per 2 Cu sites, so From Eq. 1,

and

hence,

The variation of l with p based on the experimentally determined variation

of R(p) is shown in Fig. 4a. It should be noted that the behavior found here is particularly

simple as a consequence of the parameterization of R(p) chosen; this parameterization is not uniquely determined by the data. This simple model has several appealing features. The model described in Section 2, which relies on static domain walls implies a very low hole density in the wall 0.2 holes/Cu site), which must nonetheless maintain its integrity as a charged stripe and entirely cut off AF interactions across the stripe. Our model posits a density of 0.5 holes per Cu site such as is observed in neutron scattering and predicted by calculations [29]. The recovery of below 30 K is straightforwardly understandable because once motion of


301

the antiphase domains becomes slow compared to the NQR time scale time averaging of the reversed spin directions ceases and the full ordered moment is observed. This could arise either from pinning of the antiphase domain to the lattice or evaporation of the domain walls due pinning of the constituent holes to the charged donor impurities; in either case, the coincidence of the recovery of and the freezing of spin degrees of freedom evidenced by the low T peak in 2 W is naturally explained. The correspondence between suppression of and is natural in this case because interlayer coupling is hampered wherever an antiphase domain is present, thus impeding the development 3D AF ordering (see the discussion in Ref. [29] in this regard). This model also explains the finite-size effects revealed by the susceptibility analysis of Cho et al. [13] if we consider that the appropriate length scale between domain walls is The variation of with p is shown in Fig. 4b and compared with the variation of the square of the characteristic length scale obtained by Cho et al. [13] (scaled vertically to obtain the best agreement). Finally, we note from Fig. 4a that L and l converge with increasing p, and we expect that loops will cease to be stable when L approaches l. For the parameterization of R(p), we have chosen, when near the doping at which the metal-insulator transition and spin–glass behavior are found. We speculate, then, that these are related to the transition in the configuration of the charged domain walls from loops to parallel stripes. In conclusion, we presented a model that explains the range of unusual magnetic phenomena observed in lightly doped lanthanum cuprate. In particular, we can understand the insensitivity of magnetic properties to materials variations that substantially increase the resistivity. This indicates that mobile antiphase domains play a central role in determining the magnetic properties of lightly doped lanthanum cuprate. It may point to an explanation of the poorly understood “spin–glass” regime of the phase diagram in terms of a crossover in domain wall topology from loops to parallel stripes. More generally, it suggests that the development of stripe order may play a determining role in the phase diagram of the cuprates (see, e.g., Ref. [30]). Rather than requiring mobile domain walls, superconductivity may more sensitively depend on the nature of the ordering of the walls into parallel stripes. ACKNOWLEDGMENTS We gratefully acknowledge stimulating conversations with J. Zaanen, who suggested the idea behind the model presented here. Work at Los Alamos was performed under the auspices of the U.S. Department of Energy. The NHMFL is supported by the NSF and the state of Florida through cooperative agreement DMR 95-27035. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

S. W. Cheong et al., Phys. Rev. B 44, 9739 (1991). J. L. Sarrao et al., Phys. Rev. B 54, 12014 (1996). B. J. Suh et al., cond-mat/9804200 (unpublished). Proceedings of the Workshop on Phase Separation in Cuprate Superconductors, edited by K. A. Müller and G. Benedek (World Scientific, Singapore, 1993). J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989). H. J. Schulz, J. Phys. (Paris) 50, 2833 (1989). V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990).

302


9. H. E. Viertiö and T. M. Rice, J. Phys. Condens. Matt. 6, 7091 (1994).

10. M. Kato, K. Machida, H. Nakanish, and M. Fujita, J. Phys. Sac. Jpn. 59, 1047 (1990). 11. J. Zaanen, J. Phys. Chem. Solids 59, 1769 (1998). 12. J.M. Tranquada et al.., Nature 375, 561 (1995). 13. 14. 15. 16. 17. 18.

J. H. Cho, F. C. Chou, and D. C. Johnston, Phys. Rev. Lett. 70, 222 (1993). F. C. Chou et al., Phys. Rev. Lett. 71, 2323 (1993). F. Borsa et al., Phys. Rev. B 52, 7334 (1995). A. H. C. Neto and D. Hone, Phys. Rev. Lett. 76, 2165 (1996). M. A. Kastner et al., Phys. Rev. B 37, 111 (1988). H. Takagi et al., in Proceedings of the Workshop on Phase Separation in Cuprate Superconductors, edited

by K. A. Müller and G. Benedek (World Scientific, Singapore, 1993), pp. 165–176. See Ref. [4]. 19. C. N. A. van Duin and J. Zaanen, Phys. Rev. Lett. 78, 3019 (1997).

20. S. M. Hayden et al., Phys. Rev. Lett. 68, 1061 (1992). 21. 22. 23. 24.

C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2187 (1993). J. M. Tranquada, D. J. Buttrey, V. Sachan, and J. E. Lorenzo, Phys. Rev. Lett. 73, 1003 (1994). J. M. Tranquada, D. J. Buttrey, and V. Sachan, Phys. Rev. B 54, 12318 (1996). S.-H. Lee and S. Cheong, Phys. Rev. Lett. 79, 2514 (1997).

25. V. J. Emery and S. A. Kivelson, Phys. Rev. Lett. 74, 3253 (1995). 26. J. Zaanen, M. L. Horbach, and W. Vansaarloos, Phys. Rev. B 53, 8671 (1996).

27. S. W. Cheong et al., Phys. Rev. Lett. 67, 1791 (1991). 28. J. M. Tranquada, Physica C 282, 166 (1997). 29. S. R. White and D. J. Scalapino, cond-mat/9801274 (unpublished).

30. S. A. Kivelson, E. Fradkin, and V. Emery, cond-mat/9707327 (unpublished).

On the Structure of the Cu B Site in J. Haase,1 R. Stern,1 D. G. Hinks,2 and C. P. Slichter1,3

By using NQR and new NMR methods, we isolated the A and B site signals in for at 300 K. This enabled us to measure lineshapes, relaxation rates, and the magnetic shifts separately for the Cu A and B sites. Trapped holes cause a substantial magnetic linewidth that is experienced by both Cu sites in a similar way. An axially symmetric lattice modulation is responsible for the quadrupolar broadening, but also affects both sites. The B sites do not cluster, and differ from the A sites mainly by a small change in the quadrupole frequency. Combining our results with literature data, we conclude that the B sites represent regular Cu sites in slightly contracted octahedra.

1. INTRODUCTION

The understanding of the normal state properties of the Sr or O doped compounds remains a challenge. More evidence accumulates [1–4] that local deviations from the average structure may be of importance in understanding the properties of these materials. In this context, the structural assignment of the secondary Cu site, which was detected some years ago with NQR [5,6], must be accomplished. This so-called B site appears on doping of by Sr or Ba. A similar site also occurs for O doping. NMR and NQR studies [7,8] on revealed that the number of B-site Cu atoms increases approximately linearly with the doping level x above Various models have been used to explain the B site in connection with particular Sr environments 1

Department of Physics and Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801-3080. 2 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439. 3

Author to whom correspondence should be addressed: Charles P. Slichter, Department of Physics, University of Illinois at Urbana–Champaign, 1110 W. Green Street, Urbana, IL 61801-3080; Tel.: (217) 333 3834; Fax: (217) 333-9819; email: [email protected].



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near the Cu atoms. Such approaches were questioned by Hammel et al. [9], who argued that because the oxygen-doped materials also show a similar B line, it might be a consequence

of the doped holes rather than the dopant itself. This idea is supported by quantum chemical cluster calculations [10]. In order to account for the observed number of B sites, which is much larger for the O-doped material, Hammel [1] suggests that holes become localized at Cu sites by a local charge of two (one doped O, or two doped Sr in close proximity). He proposes that one O or two neighboring Sr localize one hole, which stays centered on a certain copper site and its four neighbor oxygens. The NMR of this central Cu has not yet been observed. Hammel assigns the B site to the 4 Cu sites next to the trapped hole. With this model, one can explain the experimental intensities for the B site. For a random Sr distribution on La sites and for this results in 0.02 localized holes per Cu.

2. METHODS AND TECHNIQUES

We applied and NMR methods to optimally Sr doped ( ), magnetically aligned powders of at 300 K and fields of 8.3 T, 9.4 T, or zero field with home-built single- and double-resonance probes [11] and spectrometers. The sample material is described in Ref. [12]. In addition to the ordinary NQR/NMR, we used a new NMR double-resonance technique [13] that is based on population transfer between quadrupolar split Zeeman levels. Because neighboring transitions share one Zeeman level, a pulse on one transition always increases the population difference for a neighboring transition. Assume we had recorded a spin echo from transition ( ). After several times in a second experiment, we apply a selective to the transition ( ), which shares level 2 with ( ). Then, shortly after the (within ), we again excite a spin echo from ( ), which is now bigger (by a factor of 2 for a ). Subtracting the first signal from the second (signal averaging in add/subtract mode), we obtain a signal that is due only to the transferred (labeled) spins. We show applications later on in the text. 3. RESULTS Our ordinary Cu NQR line positions

and widths (data not shown) agree with

reported literature data [8,14,15]; decomposing the spectra with 4 Gaussian lines (two isotopes and two sites), we find for for the A site and 38.3 MHz for the B site. The linewidths are about 1.9 MHz for the A site and 1.5 MHz for the B site. The intensity of the B line is about of the total intensity. The ordinary NMR spectrum of the central transition for both isotopes and alignments consists of overlapped A and B lines, and yields for both sites an equal magnetic shift of 1.23% for and 0.70% for (c is the crystal c axis and B the external magnetic field). The NMR satellite transition spectrum for Fig. 1a, was recorded and fitted with 4 Gaussians using the measured magnetic shifts, with the two widths and the B site intensity as variables. The fit results are and Because there is a slight discrepancy between the NMR and NQR linewidths for both lines, we used our new technique to transfer spins from the central transition into the

satellite transitions. With the central transition transfer pulse fixed, we recorded the satellite transition lineshape. The results, single isotope satellite lineshapes, are shown in Fig. 1b.

On the Structure of the Cu B Site

305

For we fit linewidths of 2.0 MHz and 1.7 MHz for the A and B line, and conclude that NQR and NMR linewidths agree quite well. The isotopic dependence of the satellite

linewidths shows that it arises from a spread in the electric field gradient. Comparison of Fig. 1a with Fig. 1b shows that the low frequency tail as well as part of the linewidths are

due to misaligned grains. NMR lineshapes for are extremely sensitive to the size of the asymmetry parameter because for this orientation the crystal a and b axis are at random angles with respect to the magnetic field. The 300 K data are shown in Fig. 2. We estimate for both sites that Thus, both the A and the B sites are axially symmetric about the c axis. From the values for and we expect for very small linewidths of the central transitions; however, experimentally one observes a rather large total width [16]. To clarify this point, we again used our new technique for the A and B site separation in the central

transition: The central transitions of the different isotopes are well separated, whereas for a given isotope the central transitions of A and B are not resolved. However, because the satellites of the two sites A and B of a given isotope do not overlap (cf. Fig. 1) we can

306

transfer spins from

Haase, Stern, Hinks, and Slichter

and

separately into the central

transition, and thus obtain

their NMR parameters one at a time. Holding the transfer pulse at fixed frequency at the

satellite, we plotted out the central transition lineshape (in add/subtract mode). The result for orientation is shown in Fig. 3a. We find that both sites have very similar shifts and linewidths, at From isotope comparisons, we conclude that the linebreadth is magnetic in origin. In Fig. 3b, we show the spectra for which confirm the similarity of their shapes. The discrepancy in the apparent shifts for this orientation arises from different quadrupolar couplings. We only briefly mention our data for the spin–lattice relaxation time and the Gaussian component of the spin echo decay By employing the new technique, we measured for both sites selectively and found agreement with NQR results, in that the B site relaxation is slower but has a similar anisotropy as the A line. Next, we determined With the knowledge of all we corrected the raw data for effects [17]. We find at the line maxima for the (for A and B, for the central an satellite transitions, for both isotopes, for NMR and NQR), when normalized to the a unique value of at

300 K. Thus, the

of both sites are the same.

4. DISCUSSION First, we would like to discuss the possible origin of the observed magnetic width. It is known from experiment [18] that Ni substitution for Cu causes a magnetic width that is well understood by theory [19]. Magnetic impurities like Ni couple to distant nuclei via the electron spin susceptibility. Estimates based on the same mechanism for our material [20] show that we would need on the order of a few percent of a spin–1/2 impurity to induce our observed width. Because such impurity levels are well above any level of foreign atoms in our material, we suggest that localized holes are responsible for the magnetic line broadening. Randomly distributed trapped holes broaden our NMR lines effectively. Assuming a striplike ordering would increase the necessary amount of trapped holes substantially.

If the B sites were close to a trapped hole, their magnetic linewidth should be substantially bigger than that of the A sites, and one would expect that their differ

On the Structure of the Cu B Site

307

from that of the A site. Also, it seems not very likely that the B sites would maintain

an axially symmetric field gradient in such a position. Finally, one would not expect the NMR satellite and NQR linewidths of the B line to be even smaller than those for the A site. It is known from NQR (see, e.g., Ref. [21]) that small doping levels cause a drastic increase in the Cu NQR linewidths However, above the NQR line widths remain nearly constant as x is increased. Such a nonlinear x dependence cannot be understood in terms of local distortions induced by the Sr atoms alone. More likely, additional static lattice modulations of axial symmetry appear already above and affect both A and B sites similarly. Recent XAFS and EXAFS data [2,3,22,23] show two Sr—apical oxygen and Cu-apical oxygen distances, respectively, at all temperatures. This observation suggests that the Cu B sites could be formed by contracted octahedra that have an apical oxygen connected to a Sr atom. Such a conclusion would be consistent with our results. 5. CONCLUSIONS

We have resolved and studied the A and B Cu NMR/NQR sites in Our results show that the A and B site Cu nuclei are very similar, at least in the optimally doped material. This is inconsistent with the current interpretation as A being the main, undisturbed Cu site whereas B arises from Cu neighbors to trapped holes. We find that neither of the two sites shows a particular relation with the lattice modulations or the location of the trapped holes. Rather, the subtle differences between A and B are caused by local lattice distortions of axial symmetry that create a contracted B site octahedron.

ACKNOWLEDGMENTS

This work was supported by the Science and Technology Center for Superconductivity under NSF Grant No. DMR 91-200000 and the U.S. DOE Division of Materials Research under Grant No. DEFG 02-91ER45439. J.H. acknowledges support from the Deutsche Forschungsgemeinschaft.

REFERENCES 1. P.C. Hammel, Phys. Rev. B 57, R712 (1998).

2. D. Haskel, E. A. Stern, D. G. Hinks, A. W. Mitchell, and J. D. Jorgensen, Phys. Rev. B 56, R521 (1997). 3. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996). 4. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55, 12759

5. 6. 7. 8. 9. 10. 11. 12.

(1997). K. Yoshimura, T. Imai, T. Shimizu, Y. Ueda, K. Kosuge, and H. Yasuoka, J. Phys. Soc. Jpn. 58, 3057 (1989). K. Kumagai and Y. Nakamura, Physica C 157, 307 (1989). K. Yoshimura, T. Uemura, M. Kato, K. Kosuge, T. Imai, and H. Yasuoka, Hyper. Interact. 79, 867 (1993). M. A. Kennard, Y. Song, K. R. Poeppelmeier, and W. P. Halperin, Chem. Mater. 3, 672 (1991). P. C. Hammel, A. P. Reyes, S.-W. Cheong, and Z. Fisk, Phys. Rev. Lett. 71, 440 (1993). R. L. Martin, Phys. Rev. Lett. 75, 744 (1995). J. Haase, N. J. Curro, and C. P. Slichter, unpublished, 1998. P. G. Radaelli, D. G. Hinks, A. W. Mitchell, B. A. Hunter, J. L. Wagner, B. Dabrowski, K. G. Vandervoort, H. K. Viswanathan, and J. D. Jorgensen, Phys. Rev. B 49, 4163 (1994).

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13. J. Haase, N. J. Curro, R. Stern, and C. P. Slichter, preprint.

14. S. Ohsugi, J. Phys. Soc. Jpn. 64, 3656 (1995). 15. S. Fujiyama, Y. Itho, H. Yasuoka, and Y. Ueda, J. Phys. Soc. Jpn. 66, 2864 (1997). 16. Y. Itoh, M. Matsumura, and H. Yamagata, J. Phys. Soc. Jpn. 65, 3747 (1996).

17. N. J. Curro, T. Imai, C. P. Slichter, and B. Dabrowski, Phys. Rev. B 56, 877 (1997). 18. J. Bobroff, H. Alloul, Y. Yoshinari, A. Keven, P. Mendels, N. Blanchard, G. Collin, and J. F. Marucco, Phys. Rev. Lett. 79, 2117(1997). 19. D. Morr, J. Schmalian, R. Stern and C. P. Slichter, preprint. 20. D. Morr and J. Schmalian, private communication. 21. S. Ohsugi, Y. Kitaoka, K. Ishida, G. Zheng, and K. Asayama, J. Phys. Soc. Jpn. 63, 700 (1994). 22. D. Haskel, E. A. Stern, D. G. Hinks, A. W. Mitchell, J. D. Jorgensen, and J. I. Budnick, Phys. Rev. Lett. 76, 349 (1996). 23. N. L. Saini, A. Lanzara, A. Bianconi, D. Law, A. Menovsky, K. B. Garg, and H. Oyanagi, J. Phys. Soc. Jpn. 67, 393 (1998).

On the Estimate of the Spin-Gap in Quasi-1D Heisenberg Antiferromagnets from Nuclear Spin–Lattice Relaxation R. Melzi1 and P. Carretta1

We present a careful analysis of the temperature dependence of the nuclear spinlattice relaxation rate in gapped quasi-1D Heisenberg antiferromagnets. It is found that in order to estimate the value of the gap correctly from the peculiar features of the dispersion curve for the triplet excitations must be taken into account. The temperature dependence of due to two-magnon processes is reported for different values of the ratio between the superexchange constants in a 2-leg ladder. As an illustrative example, we compare our results to the experimental findings for in the dimerized chains and 2-leg ladders contained in PACS numbers: 76.60.Es, 75.40.Gb, 74.72.Jt

The many peculiar aspects of quasi-1D quantum Heisenberg antiferromagnets (1DQHAF) have stimulated an intense research activity since the late 1980s [1]. Moreover,

the recent observation of superconductivity in the 2-leg ladder compound [2] and the occurrence of a phase separation in high-temperature superconductors (HTSC) in hole-rich and hole-depleted regions analogous to spin-ladders [3] have brought to a renewed interest on 1DQHAF. One of the relevant issues is wether the spin-gap observed in some of these 1DQHAF is related to the one observed in the normal state of HTSC [4]. For these reasons, many NMR groups working on HTSC have focused their attention on these systems and on the determination of the spin-gap values in pure and hole-doped compounds [5–16]. However, since the early measurements, a clear discrepancy between the values for the gap estimated by means of nuclear spin–lattice relaxation and

susceptibility (or Knight shift) measurements has emerged [5]. In many compounds the 1

Department of Physics “A. Volta,” Unitá INFM di Pavia, 27100 Pavia, Italy.


309

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Melzi and Carretta

gap estimated by means of using the activated form derived by Troyer et al. [17], turned out to be times larger than the one estimated by using susceptibility or inelastic neutron scattering measurements (Table 1). Many attempt models,

theoretical [18] or phenomenological [14], have tried to explain these differences; however, although they were able to describe the findings for some compounds, they were not able

to explain the results obtained in other gapped 1DQHAF. In fact, as can be observed in Table 1, whereas for certain 2-leg ladders [15] an agreement between the gap estimated from and through other techniques is found, in several other systems it is not [5,7–12]. It is interesting to observe that the 1DQHAF where the agreement is observed are the ones in the strong coupling limit; namely, either dimerized chains or 2-leg ladders with a superexchange coupling along the rungs much larger than the one along the chains. Therefore, one can conclude that the disagreement is not always present and must be associated with

the peculiar properties of the spin excitations in each system (i.e., with the form of the dispersion curve for the triplet excitations). In this article we show that the discrepancy relies essentially on the use for of an expression that is valid in general only at very low temperatures and its application to higher temperatures depends on the

form of the dispersion curve for the triplet spin excitations. In particular, for dimerized chains the validity of a simple activated expression extends to higher temperatures than for a 2-leg ladder. As an illustrative example, we analyze the temperature dependence of for the nuclei in the dimer chains [Cu(l)] and in the 2-leg ladders [Cu(2)] contained in In the following, we consider the contribution to nuclear relaxation arising from 2magnon Raman processes only. Namely, we assume that although the system is not in the

very low temperature limit

the temperature is low enough

so that 3-

magnon processes as well as the spin damping can be neglected. If the large value of the

gap derived by means of was due to these contributions, which are proportional to one should observe some discrepancy also for the 1 DQHAF in the strong coupling limit, at variance with the experimental findings (see Table 1). The approach we use follows exactly the same steps outlined in the paper by Troyer et al. [17], in which, by

assuming a quadratic dispersion for the triplet excitations

namely

On the Estimate of the Spin-Gap in units of

311

they found that

with the resonance frequency and the hyperfine coupling constant. We remark that there is a factor 4 difference with respect to the equation reported by Troyer et al. [17] that is related to a different definition of the hyperfine Hamiltonian and of the dispersion curve. The values of the hyperfine constants are for for In the case of a general form for the dispersion relation, by considering that the low-energy processes are the ones corresponding to an exchanged momentum and one can write the contribution related to 2-magnon Raman processes in the form [17]

where is the dispersion relation for the triplet spin excitations, normalized to the gap value, whereas For a 2-leg ladder, a general form describing is

which is strongly dependent on the ratio

between the superexchange coupling

along the rungs and along the legs. We have taken the dispersion curves derived by Oitmaa et al. [20] from an extensive series studies and estimated the parameters and accordingly. Then, starting from Eqs. 3 and 4, by means of a numerical integration one can derive directly for a 2-leg ladder for different values of r. It should be remarked that for r of the order of unity, the dispersion curve for the triplet excitations has a maximum around a wave-vector (Fig. 1) and also low-energy processes from could contribute to the relaxation. However, this processes should become relevant only at where also 3-magnon processes and the damping of the spin excitations become relevant.

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Melzi and Carretta

In Fig. 2 we report the results obtained on the basis of Eqs. 3 and 4 for for different values of the superexchange anisotropy r. One observes that whereas for the dimerized chains, corresponding to the limit follows an activated behavior as the one given in Eq. 2, for the 2-leg ladders with r ~ 1 one observes some differences with respect to the simple activated behavior already at temperatures This analysis points out that for a 2-leg ladder with r of the order of unity it is not correct to estimate the gap from by using Eq. 2, at least for In fact, it is noticed that the quadratic approximation for the dispersion curve becomes valid for a more restricted range of around as r decreases (see Fig. 1). This seems to contradict the results reported in Fig. 2a, where the departure from the quadratic approximation is found more pronounced for than for However, this artifact is related to the choice of the horizontal scale, namely to have reported vs. because increases with In fact, if we report (Fig. 2b), with independent of r, one immediately notices that the deviation from the quadratic approximation starts at lower temperatures for the lowest value of r. One can then analyze the experimental data on the basis of Eq. 3 by taking the value for the gap estimated by other techniques and check if there is an agreement. We have fit

On the Estimate of the Spin-Gap

313

the experimental data for (Fig. 3b) and (Fig. 3a) in by taking and respectively, as estimated from susceptibility or NMR shift data [7, 10]. In both cases, we find a good agreement between theory and experiment by taking for the ladder site and for the chain site. If the data for were fitted according to Eq. 2 one would derive a value for the gap around 650 K, a factor 1.5 larger than the actual value (see Table 1). For also a quantitative agreement with the experimental data for is found. However, this fact seems to be at variance with the estimates by Johnston [21] based on the analysis of DC susceptibility data and with the recent findings by Imai et al. [14] based on the study of NMR shift anisotropy, in which a value for was derived. If we take this value for r we find that the experimental data are a factor larger than expected. This disagreement could originate, at least partially, from having considered for the processes the values for the matrix elements estimated by Troyer et al. [17] for the case One must also mention that the estimate

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of the hyperfine coupling constants could suffer from some uncertainties, particularly the contribution from the transferred hyperfine interaction with the neighboring spins. This contribution should be particularly relevant for the nuclei, whereas it should

be small for However, it must be recalled that because depends quadratically on the hyperfine coupling constant even for sizeable corrections can be exepected. Finally, it must be observed that in these systems the low-frequency divergence of is cut because of the finite coupling among the ladders (or chains), introducing another correction to the absolute value of The low-frequency divergence of was found to follow the logarithmic behavior reported by Troyer et al. [17] (see also Eq. 2) and does not change on varying the anisotropy factor r, for . In fact, the form of this divergence is related to the shape of the dispersion curve close to where it is always correctly approximated by a quadratic form for In conclusion, we presented a careful analysis of the problem of estimating the spin-gap from nuclear spin–lattice relaxation measurements in 1DQHAF. It is found that in order to estimate the gap correctly, one should either perform the experiments at temperatures where in many cases other contributions to the relaxation process emerge [5,7,10], or use an appropriate expression for that takes into account the form of the dispersion curve for the triplet excitations. Then a good agreement for the gap value estimated by means of and other techniques is found, allowing also to derive information on the anisotropy of the superexchange constants. ACKNOWLEDGMENTS We would like to thank D. C. Johnston for useful discussions. The research was carried out with the financial support of INFM and of INFN.

REFERENCES 1. E. Dagotto and T. M. Rice, Science 271, 619 (1995). 2. M. Uehara et al., J. Phys. Soc. Jpn. 65, 2764 (1996). 3. J. M. Tranquadae et al., Nature 375, 561 (1995). 4. P. Carretta, Physica C 292, 286 (1997). 5. M. Azuma et al., Phys. Rev. Lett. 73, 3463 (1994). 6. M. Takigawa et al., Phys. Rev. Lett. 76, 2173 (1996). 7. M. Takigawa et al., Phys. Rev. B 57, 1124 (1998). 8. K. Kumagai et al., Phys. Rev. Lett. 78, 1992 (1997). 9. K. Magishi et al., Phys. Rev. B 57, 11533 (1998). 10. P. Carretta et al., Phys. Rev. B 56, 14587 (1997). 11. P. Carretta, A. Vietkin, and A. Revcolevschi, Phys. Rev. B 57, R5606 (1998). 12. H. Mayaftre et al., Science 279, 345 (1998). 13. E Tedoldi et al., J. App. Phys. 83, 6605 (1998). 14. T. Imai et al., Phys. Rev. Lett. 81, 220 (1998). 15. G. Chaboussant et al., Phys. Rev. Lett. 79, 925 (1997). 16. Y. Furukawa et al., J. Phys. Soc. Jpn. 65, 2393 (1996). 17. M. Troyer, H. Tsunetsugu, and D. Würtz, Phys. Rev. B 50, 13515 (1994). 18. S. Sachdev and K. Damle, Phys. Rev. Lett. 78, 943 (1997). 19. E. M. McCarron et al., Mater. Res. Bull. 23, 1355(1988). 20. J. Oitmaa, R. R. P. Singh, and Z. Weihong, Phys. Rev. B 54, 1009 (1996). 21. D. C. Johnston, Phys. Rev. B 54, 13009 (1996).

Magnetic and Charge Fluctuations in Superconductors H. A. Mook,1 F. Dogan,2 and B. C. Chakoumakos1

Neutron scattering has been used to study the spin fluctuations in the and

materials. Evidence is found for both incommensurate fluc-

tuations and a commensurate resonance excitation. Measurements on the lattice

dynamics for

show incommensurate structure that appears to stem

from charge fluctuations that are associated with the spin fluctuations.

1. INTRODUCTION

Neutron scattering measurements continue to provide information of direct relevance to some of the most important issues in the cuprate superconductors. The magnetic excitations of these materials are the spin fluctuations, and recent measurements have shown that the low-energy spin fluctuations in are incommensurate in nature whereas a commensurate excitation that is relatively sharp in energy called a resonance is found at about 35 meV [1]. The incommensurability was originally discovered by the filter integration technique [2] that integrates over the outgoing neutron energy in a direction along and thus provides a high data collection rate for the study of lower dimensional excitations. The disadvantage of the technique is that no discrete energy information is available. Thus when a discovery is made by the integration technique, further measurements are made by triple-axis or time-of-flight techniques to determine the energy spectrum. Figure 1a shows the direction of the integrating scan that is made through the point to observe the incommensurate fluctuations shown by the dots at the position from the commensurate position. Such a scan uses high resolution along the scan direction but coarse resolution perpendicular to the scan direction, and thus cannot determine the exact wave-vector position of the incommensurate peaks. The result of the scan 1 2

Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393.

Department of Materials Sciences and Engineering, University of Washington, Seattle, Washington 98195, USA.



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Mook, Dogan, and Chakoumakos

for YBCO6.6 is shown in Fig. 1b. The position is at (0.5, 0.5) in reciprocal lattice units (rlu). A recent measurement using a pulsed spallation source with time-of-flight energy determination has used a two-dimensional (2D) position-sensitive detector bank to determine the wave-vector and energy of the YBCO6.6 incommensurate peaks [3]. The peaks are

found to be along the and directions as shown in Fig. 1a. The peaks are therefore not exactly on the scan direction shown in Fig. 1a, but are above and below it, being observed in the integrated scan through the relaxed vertical resolution. The peaks in the scan in Fig. 1b are found at about 0.055 on either side of the commensurate point. The wave-vector of the incommensurate scattering is then times 0.055 from the geometry in Fig. 1, and this value must be multiplied by again as the scan is in units of (h, h), giving The number determined from the time-of-flight measurement is This incommensurability is essentially identical to that observed in similarly doped [4] It was found that the intensities and the correlation lengths are also very similar in the (214) single-layer and YBCO (123) bilayer materials, thus the low-energy spin fluctuations appear to be universal for the cuprate materials measured to date.

2. MAGNETIC FLUCTUATIONS IN BSSCO

is a high-temperature superconductor of considerable importance as one can cleave the material easily to obtain a good surface for photoemission measurements and other surface-sensitive techniques. Unfortunately, it is difficult to obtain the large single crystals needed for neutron scattering for the BSSCO composition. We were successful in growing rods of the material that have a [110] reciprocal lattice direction along the rod direction (using the same definition of a and b as for YBCO). It was then possible to align a number of these rods together to form a sample of 35 g for neutron studies. We then have a crystal with one set of the [110] directions aligned, and this permits some information to be obtained. A difficulty with the sample for the study of magnetic excitations is that the material has bilayers similar to those in YBCO, and thus we expect that the magnetic fluctuations are coupled in a similar way. In this case, the low energy magnetic fluctuations have no structure factor unless a finite value of is used and is randomly orientated perpendicular to the

Magnetic and Charge Fluctuations

317

aligned direction of the crystal. One must then set the spectrometer to sample a point off the [110J direction, which means that only some of the values desired fall within the resolution volume of the spectrometer, considerably reducing the magnetic signal.

One might expect that a resonance excitation might exist in BSSCO in a similar way as for YBCO, and one can search for it in the same way [5] using a triple-axis spectrometer.

Our sample of BSSCO has an oxygen composition near optimum doping and we expect that the resonance will not be observable much above in this case. The simplest experiment is thus to scan energy at the momentum value where the resonance is expected and take the difference between data taken well below and data taken above This works in YBCO,

but fails in BSCCO for two related reasons. The first is that the signal is small relative to the phonons as we cannot achieve the full magnetic intensity that would be available at a fixed value for The second problem is the phonons are much more temperature sensitive in BSSCO than they are in YBCO, so that the difference in data at high and low temperatures strongly reflects the phonon differences. The only way to circumvent these difficulties is to use a polarized neutron beam to isolate the magnetic scattering. Unfortunately, this results in even less intensity because polarized beams are much weaker then unpolarized ones. Nevertheless, after considerable counting, reasonable results were obtained. Momentum

values near (0.5, 0.5) and (1.5, 1.5,) were both tried, and similar patterns were with the magnetic scattering near (1.5, 1.5) being considerably weaker because of the magnetic form factor. The results are shown in Fig. 2. A peak in energy about 10 meV wide,

which is equal to the energy resolution of the experiment, is observed at 10 K, whereas the result at 100 K appears rather featureless. The peak is only found in the spin flip channel guaranteeing that it is magnetic. The peak is observed at about 37 meV, which is near the

value expected if the of the BSSCO of 84 K is scaled to that of YBCO for the same doping level. The results strongly suggest that BSSCO has a resonance excitation rather similar to that of YBCO. However, the results should be checked with single crystals when they become available. The same BSSCO sample was used to search for magnetic incommensurate fluctuations. The integration technique was used in the same way as for YBCO6.6. The experiment works in a similar manner except that the integration now takes place over the directions perpendicular to the [1,1,0] direction and thus is only partly along For 2D scattering from bilayers, this results in an intensity loss in the magnetic signal. However, we see from Fig. 1 that the magnetic signal in the integration technique is substantial so that an intensity

loss may be tolerated. The results of the measurement are shown in Fig. 3, which shows data presented in the same way as for the YBCO6.6 in Fig. 1. The results suggest the possibility of small incommensurate peaks, although the counting errors are larger than desired. The

data shown are from a number of runs averaged together. Fig. 3b–d show one of the satellite peaks measured at different temperatures. In this case, the background was obtained by a

30° rotation of the sample relative to the position where the scan is performed. The magnetic signal is expected to be small in the 30° rotation case, which samples reciprocal space well removed from (0.5, 0.5). The signal decreases with temperature, as would be expected for a magnetic excitation. The peak is broad so the center is hard to determine accurately with

the errors involved; however, the peak appears to be centered at about 0.42 rlu or 0.08 rlu units from the (0.5,0.5) position. If the magnetic satellites are arranged as in Fig. 1, would be about 0.32. The value of for fully doped 214 materials is about 0.25, so the value for BSSCO appears to be somewhat larger than for the 214 materials assuming the same type of incommensurability. However, the BSCCO measurement has sizable counting errors and

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we do not know the actual pattern of the incommensurability. A measurement on a single crystal is needed for a more accurate determination. Until such an experiment is performed, the present result suggests that BSSCO has low-energy incommensurate fluctuations that may be similar to those found in 123 and 214 materials. 3. CHARGE FLUCTUATIONS IN YBCO

Neutrons cannot measure charge directly, but can observe a change in the mass density that is either static or dynamic. One can assume that static mass displacements reflect static charge ordering, and such effects have been observed in the 214 cuprate materials in special cases [6]. Dynamic charge ordering has not been observed so far and the present results serve as an indication that this occurs in the YBCO materials. We have made measurements on the same YBCO6.6 sample in which the incommensurate magnetic fluctuations are observed. Again, we start with the integration technique, except we examine the region around the peaks of the reciprocal lattice stemming from the atoms rather than from the magnetism. Figure 4 shows scans around the (1,0) reciprocal lattice peak. Data are again shown that use a


319

measurement at 300 K as a background. As the sample is cooled, distinct peaks form on both sides of the (1,0) peak that we assume reflects a dynamic incommensurate mass fluctuation

that can be considered to stem from an incommensurate charge fluctuation. However, we have not completely ruled out magnetic effects. We note the peaks are small, being an order of magnitude smaller than the magnetic satellite peaks shown in Fig. 1. The scan is along the (h, 0) reciprocal lattice or the direction, and thus is along the direction of the magnetic incommensurate scattering. The charge fluctuation peaks are about 0.22 in rlu units from the commensurate position so that the value for them is 0.22, or twice the wave-vector of the incommensurate magnetic satellites. However, the absolute direction of

the incommensurate wave-vector cannot be determined with the integration technique, and the peaks could be at a wave-vector off the direction. Figure 4d shows an identical measurement for YBCO6.35 that has only commensurate magnetic order. No indications of incommensurate charge fluctuations are found for this material. Work has been underway with triple-axis spectrometry to determine the energy spectra of the charge fluctuations, but that work is still incomplete. It has been noted, however, that

certain phonon branches show anomalies at the wave-vector of the charge fluctuations. The origin of the charge fluctuations is not clear. It would seem extremely likely that the

320


magnetic and charge fluctuations stem from the same source. Obviously, the observation of charge fluctuations strongly suggests a dynamic striped phase in YBCO6.6. However, other possibilities exist, including Fermi surface effects or dynamic charge density waves (CDW). The next step is to determine the energy spectra and absolute wave-vector of the incommensurate charge scattering.

4. CONCLUSION

We have shown new neutron scattering results for the cuprate superconductors. Measurements on a BSCCO sample of crystals with a [110] direction aligned show strong evidence for a resonance excitation and indications of incommensurate magnetic fluctuations. It would be good to have these results confirmed by a high-quality single crystal. For YBCO6.6, clear dynamic incommensurate peaks are observed at low temperatues on either side of the (1, 0) reciprocal lattice peak. Because these are found at positions relative to


321

the crystal reciprocal lattice, they are assumed to stem from mass fluctuations driven by charge fluctuations. No such peaks are found for a YBCO6.35 sample. The wave-vector

of the charge fluctuation peaks is twice that of the magnetic fluctuations if we assume the charge peaks are on the direction. It would seem likely the magnetic and charge excitations are related. The results give support to a dynamic striped phase model for the cuprate superconductors.

ACKNOWLEDGMENTS The submitted manuscript has been authored by a contractor of the U.S. Government under contract No. DE-AC05-96OR22464. Accordingly, the U.S. Government retains a

nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Prepared by Solid State Division Oak Ridge National Laboratory; Managed by Lockheed Martin Energy Research Corp. under Contract No. DE-AC05-96OR22464 with the U.S. Department of Energy Oak Ridge, Tennessee, September 1998. REFERENCES 1. P. Dai, H. A. Mook, and F. Dogan, Phys. Rev. Lett. 80, 1738 (1998).

2. H. A. Mook, P. Dai, K. Salama, D. Lee, F. Dogan, G. Aeppli, A. T. Boothroyd, and M. Mostoller, Phys. Rev. Lett. 77, 370 (1996).

3. H. A. Mook, Pengcheng Dai, S. M. Hayden, G. Aeppli, T. G. Perring, and F. Dogan, Nature 395, 580 (1998). 4. K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, M. Greven, M. A. Kaster, and Y. J. Kim, Phys. Rev. B 57, 6165 (1998).

5. H. A. Mook, M. Yethiraj, G. Aeppli, T. E. Mason, and T. Armstrong, Phys. Rev. Lett. 70, 3490 (1993). 6. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995).


Neutron Scattering Study of the Incommensurate Magnetic Fluctuation in T. Nishijima,1 M. Arai,1 Y. Endoh,2 S. M. Bennington,3 R. S. Eccleston,3 and S. Tajima4

Inelastic neutron scattering experiments have been performed in order to study

the wave vector dependence of the magnetic fluctuation in and Incommensurability of spin dynamics was clearly observed for the underdoped compound. The incommensurate peaks

appear at

with the same symmetry of

that of The incommensurability rlu for the lowenergy region, and the peaks merge at about 41 meV. However, the optimum-

doped compound also seems to have an incommensurate structure, although the incommensurability is not well defined due to the intense resonance peak at the commensurate position.

1. INTRODUCTION The persistence of antiferromagnetic fluctuations in the metallic state of cuprates probably has an important role for the superconductivity mechanism. In order

to elucidate the details of the magnetic fluctuations, intensive experimental works on have been performed by using neutron scattering techniques, which have an crucial role in characterizing the wavevector and energy dependence of the imaginary part of the dynamical susceptibility. In LSCO, the magnetic scattering is incommensurate, and four sharp magnetic rods locate 1

Institute of Materials Structure Science, KEK, 1-1 Oho, Tsukuba 305, Japan.

2

Department of Physics, Tohoku University, Aoba, Sendai, 980, Japan.

3

Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK.

4

Superconductivity Research Laboratory, ISTEC, Koto-ku, Tokyo 135, Japan.



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around the 2D antiferromagnetic (AF) zone center at

The incommensurability is observed with a weak energy dependence. Recently in YBCO, incommensurate magnetic fluctuation was clearly observed below [6]. However, it is ambiguous whether the symmetry of the imcommensurability is the same as that of YBCO or a 45° rotated structure from the former case. Because of the resolution effect, the reported results on YBCO was too broad to give a convincing conclusion. Hence, we have performed neutron scattering for underdoped and optimum-doped YBCO in order to elucidate the q dependence of magnetic fluctuation with better instrumental resolution on high-quality

crystals. 2. EXPERIMENTAL Experiments were performed on the chopper spectrometers MARI and HET at the

ISIS pulsed spalation neutron source of Rutherford Appleton Laboratory. The sample used in the present experiments was an assembly of single crystals of in total amount. The samples were synthesized by SRL-CP method at the Superconductivity Research Laboratory [7]. The optimum-doped sample was prepared by annealing at for 39 days and susceptibility measurements revealed with the transition width of Underdoped samples were prepared with the same crystals annealed at 640°C for 3 weeks after the neutron scattering experiment on HET, and susceptibility

measurements revealed

with a somewhat broad transition of

The c*

axis was aligned to the incident neutron beam so that the (HHL) plane was the scattering

plane. The large number of detectors can cover a wide range of the energy—momentum space in the (HHL) plane simultaneously. 3. RESULTS Figure 1 shows the intensity contour map of the dynamical structure factor on the plane for the underdoped sample observed on

the MARI spectrometer, where stands for the 2D momentum transfer and E the energy transfer. As shown in Fig. 1, two magnetic rods were clearly observed from about 15 meV to 40 meV, and these merge at 41 meV above the AF zone center of (1/2, 1 /2, 0),

corresponding to locate at

It can be interpreted that the magnetic incommensurate peaks with the same symmetry of that of LSCO;

the counter length along (–h, h, 0) of the spectrometer can cover the two incommensurate peaks simultaneously, as shown in Fig. 2. The constant-energy slices of at 24, 32, 40, and 52 meV show q dependence of the magnetic spectrum in Fig. 3. The q dependence at 24 and 32 meV shows by two peaks around the AF zone center at . The magnetic signal was fitted by double gaussians with the incoherent phonon background of

Although the profile at 40 meV looks like a single peak, it can be fit by double gaussians. The peak separation of two peaks seems to make a shorter distance at 40 meV. Actually, just below 40 meV the peaks make a flat-top structure and are well fitted by double

gaussians. The energy dependence of the peak separation, which is defined by

in Fig. 2,

is shown in Fig. 4. In the low-energy region, the peak separation, incommensurability, is about rlu and decreases with increasing energy up to 40 meV. Above this energy

the peaks split again, with a somewhat longer distance and a much broader peak profile.

Incommensurate Magnetic Fluctuation in

325

For the optimum-doped sample, the magnetic resonance peak was observed at 41 meV, resonance energy at the (1/2, 1/2, 0) position by using HET spectrometer, as was already

reported [5]. The spectrum at 10 K and 100 K around the resonance energy was shown in Fig. 5a and the subtracted data (10–100 K) was shown in Fig. 5b. As shown in Fig. 5b,

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the large enhancement of magnetic signal at the commensurate position was observed. There seems to be shoulder structures around commensurate resonance peak. Hence the peak profile was fit by three gaussians, although the physical meaning is unknown, with a constraint that the two gaussians for the shoulders should have the same width and the

Incommensurate Magnetic Fluctuation in

327

same splitting from (1/2, 1/2, 0). The fitting results indicate that the incommensurability

is about

4. DISCUSSION

We could confirm that the magnetic excitation spectrum in the underdoped YBCO has incommensurate peaks with the symmetry as the same as that of LSCO. One theoretical explanation about incommensurate magnetic peaks in LSCO suggests that the dynamical spin susceptibility is enhanced at incommensurate position, which is related to a nesting

wave-vector of the Fermi surface [8,9]. However, the theory does not predict any incommensurability in YBCO due to the lack of Fermi-surface nesting. Hence this scenario should be reconciled by our experimental results. Another viewpoint on the incommensulability is stripe domain structure based on the recent neutron scattering experiments on

by Tranquada et al. [10, 11]—i.e., dopant-induced holes segregate into periodically spaced stripe structure that separates AF domains with double periodicity of the former. Emery et al. [12] pointed out a possible emergence of the dynamical microphase separation in the plane by taking into account the long-range Coulomb force in the t-J model, which expects that the incommensurate magnetic fluctuation can be a common feature in cuprate superconductor.

Recently, Yamada et al. [13] reported a linear relation between and up to the optimum doping regime revealed by systematic neutron scattering studies on Incommensurability scaled by where (max) stands for transition temperature at optimum doping in each system, is depicted in Fig. 6 together with LSCO and YBCO. It is clearly recognized that there is a similarity in the diagram for LSCO and YBCO. However, it is noted that the incommensurability has a strong energy dependence and makes a sudden dip at around 40 meV, which has never been observed in LSCO. Therefore, further analysis and additional experiments are needed to discuss the

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details of incommensurability and the resonance peak of the magnetic fluctuation in cuprate superconductors. ACKNOWLEDGMENT

Quite recently, we were informed that H. Mook et al. also observed the incommensurate magnetic peaks in YBCO with the same symmetry as ours. This work was supported by a Grant-in-Aid on Scientific Research on Priority Areas “Anomalous Metallic State near the Mott Transition” (07237102) of the Ministry of Education, Science, Sports and Culture,

Japan, and done under collaboration with NEDO. The authors acknowledge J. W. Jang, A. I. Rykov, M. Kusao, S. Koyama, and K. Tomimoto for preparing the sample. REFERENCES 1. S. M. Hayden et al., Phys. Rev. Lett. 76,1344(1996).

2. K. Yamada et al., Phys. Soc. Japan 64, 2742 (1996). 3. H. A. Mook et al., Phys. Rev. Lett. 70, 3490 (1993).

4. 5. 6. 7. 8. 9. 10. 11. 12.

P. Dai et al., Phys. Rev. Lett. 77, 5425 (1996). H. F. Fong et al., Phys. Rev. Lett. 78, 713 (1997). P. Dai et al., Science 284, 1344 (1999). Y. Yamada et al., Physica C 217, 182 (1993). Q. Si et al., Phys. Rev. B 47, 9055 (1993). T. Tanamoto et al., J. Phys. Soc. Jpn. 63, 2739 (1994). J. M. Tranquada et al., Nature 375, 561 (1995). J. M. Tranquada et al., Phys. Rev. Lett. 73, 338 (1997). V. J. Emery et al., Physica C 209, 597 (1993).

13. K. Yamada et al., Phys. Rev. B 57, 6165 (1998).

Rare Earth Spin Dynamics in the Nd-Doped Superconductor M. Roepke,1 E. Holland-Moritz,1 B. Büchner,1 R. Borowski,1 R. Kahn,2 R. E. Lechner,3 S. Longeville,3 and J. Fitter3

Inelastic magnetic neutron scattering experiments on the superconductor were carried out. We investigated the relaxation behavior of the 4f moments for different Sr and Nd concentrations. In all our samples, the high-temperature behavior of the magnetic response of the 4f moments is similar and can be described in the context of a two-phonon–Orbach relaxation process, i.e., a coupling between phonons and CEF excitations. At lower temperatures, the rare earth (RE) spin dynamics is correlated with the electronic properties of the planes. In an inelastic excitation (INE) occurs below 80 K, which can be interpreted as splitting of the groundstate doublet. In the quasi-elastic line (QE) broadens below In contrast to the inhomogeneity due to the formation of stripe order of holes and spins causes a distribution of various energy excitations over different Nd sites. neither an INE nor a QE line broadening was found.

1. INTRODUCTION The investigation of magnetic correlations in superconducting and related materials is essential to understand the mechanisms leading to high-temperature superconductivity.

In this paper, we present inelastic magnetic neutron scattering experiments on Nd-doped at various temperatures. In LSCO, the rare earth (RE) doping causes a further structural phase transition from the low-temperature orthorhombic (LTO) to the low-temperature tetragonal (LTT) phase [1]. In a certain composition range, this LTT phase 1 2

II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, D-50937 Köln, Germany.

Laboratoire Leon Brillouin, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France.

3

Hahn-Meitner-Institut, Glienicker Str. 100, D-14109 Berlin.


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is not superconducting, but antiferromagnetic (AF) order occurs at finite Sr, i.e., charge carrier concentration. Tranquada et al. [2] show from elastic neutron diffraction experiments on that the influence of the structural transition on the electronic

properties is due to a pinning of stripe correlations of spins and holes, i.e., due to a formation of static antiphase AF domains in the planes that are separated by quasi-1D stripes containing the doped charge carriers. We investigated the magnetic response of the RE 4f moments at low-energy transfers (typical neutron incident energy ) in order to obtain information about the Cu magnetism in the layers via the RE–Cu interaction. Such an interaction was suggested from previous experiments on that show an anomalous behavior of the 4f magnetic scattering response (see, e.g., Ref. [3]). RE–Cu interaction is also well established in the electron-doped compound

[4].

2. EXPERIMENTAL

We performed temperature-dependent studies on and using the time-of-flight (TOP) spectrometers V3 NEAT [5] (HMI, Berlin) and G6.2 MIBEMOL (LLB, Saclay) with a cold neutron source. We chose incoming energies between

and

resulting in energy resolutions between and respectively. The experiments were performed on well-characterized powder samples [6]. For details concerning the data analysis, see Ref. [7].


In all samples, a quasi-elastic (QE) line is observed above a certain temperature (Figs. 1 and 2). At high temperatures, the QE line width increases almost linearly with increasing

Rare Earth Spin Dynamics

331

temperature and is similar for all samples, no matter the Sr or Nd concentration Note that the same behavior occurs for insulators without charge carriers and compounds that are doped with holes This temperature dependence of was often described by the Korringa law even in the absence of charge carriers [8,9]. In contrast to that, Staub et al. [10] reported a different temperature dependence of In this compound, Tb has a quasi-doublet ground state, i.e., two singlets separated by It was found that obeys a power law Conclusively, they claimed that the RE exchange interaction might be the dominant process for the 4f–spin relaxation and not the interaction with charge carriers. They obtained similar results for Tb doped in YBCO [11]. However, we find that the relaxation of the RE moments can be described within a two-phonon–Orbach process [12], i.e., a coupling between phonons and CEF excitations. Such a relaxation process has been observed in ESR measurements of REdoped LSCO [13] and YBCO [14]. The line width then follows with being the first excited crystal electric field state. The constant factor c considers the strength of the orbit–lattice coupling between the ground and the first excited state. A coupling between phonons and crystal field excitations has already been reported in Raman scattering studies on and with

Figure 2 shows a fit of the data for process described above. A residual line width of The analysis reveals reasonable values

assuming the relaxation was also taken into account. The latter is in fair

and agreement with the first excited CEF state of The line width is related to the spin fluctuation frequency via Thus, the decrease of the line width with decreasing

temperature is a direct evidence for the lowering of the 4f–spin fluctuation frequency. In the following, we focus on the low-temperature behavior. As a function of the Srdoping x, i.e, the charge concentration, three qualitative differences in the low-temperature

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Roepke et al.

spectra are observed that correlate with the electronic properties of the layers. In the insulating compounds with long-range AF order of the Cu moments occurs below In both samples, the QE line vanishes below 80 K, and

instead an inelastic excitation is observed [19] (Fig. 3). This excitation can be interpreted as lifting of the degeneracy of the ground-state Kramers doublet due to the Nd-Cu– exchange interaction, in quantitative agreement with the observed Schottky anomaly of the

low-temperature specific heat [20]. The increase of the energy excitation from meV at 80 K to meV at 3.3 K (see inset, Fig. 3) indicates an increase of the staggered magnetization in the layers. The fact that the INE in occurs in both samples at the same temperature (independent of the Nd concentration) shows that the Cu spin reorientation [21] at does not influence the exchange field at the Nd site markedly.

The findings for the Sr-doped compounds differ from those for the sample due to the change of the electronic properties. In with superconductivity is strongly suppressed in the LTT phase [1] and a formation of AF domains separated by stripes containing the charge carriers is expected [2]. In both samples we observe a strongly enhanced line width of the QE line (Fig. 4) below In contrast to the high-temperature region (where the QE line is a Lorentzian), the spectra are consistent with a gaussian QE line shape. The line width is meV and meV for 30 K and 20 K, respectively, and remains roughly constant for lower temperatures. Because there is a coupling between the Cu and Nd moments see above), these INE neutron data are most probably due to a freezing of the Cu spin fluctuations [22] in the samples with The differences in the data for and i.e., the observation of a strongly enhanced QE line width instead of a well-defined INE line, suggests a distribution of different energy splittings over different Nd sites in the

with

Rare Earth Spin Dynamics

333

Sr-doped compound. It is obvious that the inhomogeneity due to pattern of stripe correlation of spins and holes [2] is the reason for this observation. Finally, comparing the width of the gaussian line at with the energy of the INE line reveals a reduced (average) splitting, which is related to a reduced zero temperature staggered magnetization in the planes.

No difference of the temperature behavior of the QE line widths is observable in with (Fig. 2). In both samples the line width decreases with decreasing temperature, as expected by the Orbach relaxation process. No hints for an INE excitation or a QE broadening are found at lowest temperature (2.8 K and 1.6 K for and respectively). This is not surprising for which is (bulk) superconducting below In contrast, the tilt angle of the octahedra exceeds the critical value and thus superconductivity is strongly suppressed [1]. Therefore, magnetic order of the Cu moments and hence a broadening of the QE line below is expected. The absence of such

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a broadening in in the planes.

indicates a further-reduced staggered magnetization

4. CONCLUSION

To summarize, we presented INE magnetic neutron scattering experiments on Nddoped LSCO. In all samples at high temperatures, a QE line is observed, with a line width that decreases with decreasing temperature. In the absence of RE–Cu interaction, the relaxation of the 4f moments is dominated by the Orbach relaxation process via the coupling of phonons and CEF excitations. The low-temperature behavior clearly correlates with the electronic properties. For the undoped samples below about 80 K an INE excitation occurs that shows the splitting of the Kramers ground-state due to the exchange field at the Nd site. For the samples with stripe order of spins and holes a broad QE line infers a distribution of different energy splittings over different Nd sites. In

no indication for a RE–Cu interaction has been found, i.e., follows the temperature dependence as expected by the Orbach relaxation process down to the lowest temperature. ACKNOWLEDGMENT Our work is supported by the BMBF under contract number 03-HO4KOE.

REFERENCES 1. B. Büchner et al., Phys. Rev. Lett. 73, 1841 (1994). 2. J. M. Tranquada et al., Nature 375, 561 (1995); J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996); J. M. Tranquada et al., Phys. Rev. Lett. 78, 338 (1997). 3. H. Drössler et al., Z. Phys. B 100, 1 (1996). 4. W. Henggeler et al., Phys. Rev. B 55, 1269 (1997). 5. R. E. Lechner, Neutron News 7, 4, 9–11 (1996). 6. M. Breuer et al., Physica C 208, 217 (1993). 7. E. Holland-Moritz and G. Lander, Handbook of the Chemistry and Physics of Rare Earths 19, 1 (1994). 8. M. Loewenhaupt et al., J. Magn. Magn. Mater. 140–144, 1293 (1995). 9. P. Allenspach, A. Furrer, and F. Hulliger, Phys. Rev. B 39, 2226 (1989). 10. U. Staub et al., Europhys. Lett. 34, 447 (1996). 1 1 . U. Staub et al., Physica B 234–236, 841 (1997). 12. A. Abragam and B. Bleany, Electron Paramagnetic Resonance of Transition Ions (Clarendon, Oxford, ch. 10, 1970). 13. L. Kan, S. Elschner, and B. Elschner, Solid State Comm. 79, 61 (1991). 14. H. Shimizu et al., Physica C 288, 190 (1997). 15. E. T. Heyen, R. Wegerer, and M. Cardona, Phys. Rev. Lett. 67, 144 (1991). 16. S. Jandl et al., Solid State Comm. 87, 609 (1993). 17. J. A. Sanjurjo et al., Phys. Rev. B 49, 4391 (1994). 18. C.-K. Loong and L. Soderholm, Phys. Rev. B 48, 14001 (1993). 19. M. Roepke et al., Physica B 234-236, 723 (1997); M. Roepke et al., J. Phys. Chem. Solids 59, 2233 (1998). 20. B. Büchner, private communication. 21. B. Keimer et al., Z. Phys. B 91, 373 (1993). 22. V. Kataev et al., Phys. Rev. B 55, R3394 (1997).

Static Incommensurate Magnetic Order in the Superconducting State of K. Yamada,1 R. J. Birgeneau,2 Y. Endoh,3 M. Fujita,1 K. Hirota,3 H. Kimura,3 C. H. Lee,3 S. H. Lee,4 H. Matsushita,3 G. Shirane,5 S. Ueki,3 and S. Wakimoto3

Spin correlations in the superconducting state of hole-doped

were stud-

ied by neutron scattering. In the underdoped superconducting phase, a longrange spatially modulated magnetic order was found to develop below For at and oxygenated with similar hole doping,

the onset temperature of the magnetic order almost coincides with

Deviation

from the hole doping at 0.12 degrades the magnetic order. The incommensurability as well as the peak position of the elastic magnetic peaks around are the

same as those in the low-energy inelastic peaks. For

at

1.2% substitution of Zn ions at Cu sites induces a quasi-static incommensurate

magnetic order. However, 3% substitution of Zn degrades the long-range magnetic order in the Zn-free sample. In the spin-glass phase at near the lower critical doping of superconductivity, both commensurate and incommensurate spin correlations coexist.

1. INTRODUCTION Although the mechanism of pairing interaction of superconductivity is not fully understood at present, the interplay between the magnetic correlation and superconductivity is one of the central issues in the basic physics of the lamellar

1

Institute for Chemical Research, Kyoto University, Uji 611 -0011, Japan. Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 3 Department of Physics Tohoku University, Aramaki Aoba, Sendai 980-77, Japan. 2

4

Center for Neutron Research, NIST, Gaithersberg, Washington, USA.

5

Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA.



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materials [ 1 ] . Particularly, recent observation of a long-range incommensurate static magnetic order in the superconducting states of [2] and [3] around has evoked a new problem: Does the incommensurate static magnetic order microscopically coexist with the superconductivity, and if it does, are they concordant or competing with each other? To answer this question, comprehensive neutron scattering

studies have been performed and several unexpected new details have been discovered. The purpose of this paper is to introduce such recent results of neutron scattering and discuss the interplay between the spin correlation and superconductivity. 2. PREVIOUS STUDY ON DYNAMICAL SPIN CORRELATION

Before describing the static spin correlation, we summarize our previous neutron scattering study on the dynamical spin fluctuations in The incommensurate spin density modulation along the Cu-O bonds becomes noticeable with doping close to the lower critical concentration of the superconductivity, The incommensurability increases with increasing the doping rate in the underdoped region. Beyond the starts to saturate, with the maximum value of around 1/8. The spatial spin modulation is

robust against the pair breaking, such as Zn substitution of Cu site or oxygen reduction [4]. Except the doping region close to

the

is simply determined by the effective doping

rate on plane calculated for dence of is approximately the same as that of the superconducting

Doping depen-

There, the crystal structure is the so-called low temperature tetragonal (LTT) phase different from the low temperature orthorhombic (LTO) one for the Sr- or oxygen-doped Moreover, recent observation by Mook et al. [5] of incommensurate spin fluctuations in the superconducting Y123 and Bi2212 systems revealed the same direction of spin modulation as well as the similar doping dependence of compared to La214. Therefore, the spin density modulation along the Cu-O bonds is considered to be a common nature of the su-

perconducting phase of cuprates irrespective of the details of crystal structure. The incommensurability is simply determined the effective doping rate on the planes. We remark that the spin density modulation or a striped hole order in Sr- or oxygen-doped insulating is observed along the diagonal direction in the square lattice of plane, which differs by 45° from the direction in cuprates. In addition, the incommensurability in does not saturate by doping, at least up to around [6]. The spatial coherency of the spin correlation at low energies as well as temperatures is degraded beyond the optimum doping by keeping the constant [4,7]. The coherence length shows a maximum around Such an elongation of the coherence length may correspond to the appearance of static long-range magnetic order (described in the next section). Beyond the upper critical doping of superconductivity, no well-defined spin correlation exists [8]. These results suggest a concordant relation between the dynamical spin fluctuations and superconductivity. We note that around the optimum doping, many transport properties exhibit a crossover from holelike to electron-like behavior [9], and a change in the topology of Fermi surface is observed by angle-resolved photoemission [10]. Therefore, we expect a change in the orbital or site character of the doped holes around the optimum doping. A linear relation between and was obtained for the Sr-doped samples [4]. This relation is completely free from any uncertainties connected with the doping level in each sample and demonstrates directly a concordant relation between the incommensurate dynamical spin correlation and superconductivity.

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We studied the doping dependence of energy spectra of the incommensurate spin fluctuations. A well-defined energy gap of opens for the optimally doped or slightly overdoped phases in the superconducting states. In contrast, no welldefined energy gap can be seen for the underdoped and heavily overdoped superconducting phases. An analysis by Lee et al. [7] using a phenomenological gap function proposed a possible reason for the missing of the energy gap: a smearing of the gap edge corresponding to a shortening of lifetime of quasi-particles for the underdoped sample and a substantial decrease in the gap size for the heavily doped sample. For gaplike behavior of still remains, even at in contrast to the conventional superconducting gap.

3. STATIC SPIN CORRELATION Quite recently, besides the dynamical spin fluctuations summarized above, a static longrange magnetic order was observed in the superconducting state of La214 system around 1/8 doping. Although such magnetic ordering was predicted by [11,12] and NMR [13] measurements for both Ba- and Sr-doped La214 systems, details of the spatial correlation and the relation with the superconductivity were first presented by Tranquada et al. [2] based on neutron scattering study on In the sample, they found a charge segregation that preceded the magnetic order with incommensurate spin density modulation. Furthermore, they showed a competitive relation between the static magnetic order and the superconductivity: the onset temperature of the magnetic order is lower for samples with higher Neutron scattering study to search for similar magnetic order in LTO system was conducted by Suzuki et al. [3] on They found sharp incommensurate peaks at the same positions around as those of the dynamical spin fluctuations. The onset temperature of the peak was similar to After this measurement, Kimura et al. [14] performed more comprehensive neutron scattering measurement by using cold neutrons for both and nonsuperconducting The resolutionlimited peak width for the former indicates the static magnetic correlation length exceeds 200 Å isotropically on the planes. The 3% Zn substitution, however, degrades the long-range magnetic order; the peak appears at lower temperature (17 K) and the correlation length is shorter (77 Å) compared to the Zn-free sample (Figs. 1 and 2). Such well-defined incommensurate elastic peaks were observed up to and down t o The onset temperatures of these peaks, however, are lower than that for and than the corresponding Therefore, the long range magnetic order seems to be an intrinsic phenomenon around Before the measurement made by Suzuki et al. [3], Hirota et al. [15] observed incommensurate quasi-elastic peaks for Zn-doped sample, in a similar temperature range as (Fig. 3). The quasi-static magnetic correlation length of about 80 Å is also similar to the Zn-doped sample at Therefore, the Zn substitution not only degrades an existing long-range magnetic order, but also induces a new quasi-static magnetic order in the dynamically fluctuating incommensurate phase, possibly by pinning the dynamical spin fluctuations. The latter fact may correspond to the enhancement of low-energy spin fluctuations by Zn doping, which hinders out the energy gap [16]. Therefore, phenomenologically, the reason for the missing of

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energy gap in the underdoped superconducting state can be connected with the Zn-doping effect. Stimulated by the observation of incommensurate elastic peak in the Sr-doped Lee et al. [17] searched for the elastic peak in the electrochemically oxygen-doped superconducting As a result, they found similar incommensurate peaks. In this case, the onset of the magnetic order almost coincides with which is higher than that

for the optimally Sr-doped sample. More surprisingly, the doping rate of the oxygenated sample was estimated to be about 0.12, where an anomalous • suppression appears for the Sr-doped system. In order to elucidate the interplay of such long-range magnetic order with the superconductivity, extension of the doping region is quite important. Wakimoto et al. [18] observed a mixture of commensurate and incommensurate elastic spin correlation in the sample at exhibiting no superconductivity but a spin–glass behavior in the uniform magnetic susceptibility. We note that a recent measurement suggests a coexistence of the spin–glass and superconducting phases [19]. However, the momentum-sensitive neutron scattering revealed the more surprising fact that the observed peak position differs by 45° from that of the superconducting phase [18]. We remark that such a change in the direction of spin–charge density modulation was previously predicted by theoretical calculations based on the simple Hubbard model on a 2D square lattice [20]. Due to the sharp q spectrum of the elastic peak, the precise peak position around was determined by Lee et al. [17] for the oxygen-doped Originally, the four peaks were believed to locate on the corner of a square as shown in Fig. 4. However, they found the peaks are on a rectangule. Although the reason for such shift of peak position is

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not understood, same deviation is also observed in the Sr-doped sample with much smaller orthorhombicity. Therefore, the origin of the shift may not relate with the orthorhombic distortion or twinned structure.

4. DISCUSSION AND SUMMARY We observed a new static incommensurate magnetic order in the underdoped superconducting phases of both Sr-and oxygen-doped At present, it has not been determined whether the magnetic order in the superconducting state coexists microscopically with the superconducting phase. In contrast to the case of we observed a coincidence of the onset of the magnetic order to that of the superconductivity near 1/8 doping. It is important to clarify whether this coincidence is just accidental or relates with the onset of superconductivity. In order to elucidate this phenomenon, we must study the effect of crystal structure on the stability of the magnetic order. Concerning this point, recent measurement of sound velocity of both Sr- and oxygen-doped superconducting samples is very interesting. They observed a remarkable hardening of mode in the superconducting states for both systems near 1/8 doping that is absent in the optimally doped sample [21]. Therefore, the lattice hardening may couple with the longrange spin order. If the spatial spin modulation is induced by the hole order, as is predicted by the so-called stripe model, the anomaly in the sound velocity may be caused by the hole order. However, we have so far observed no superlattice Bragg peak in LTO systems. It is not understood if the missing of superlattice peak is due to the experimental limitation of sensitivity. Because the anomalous suppression is widely believed to be connected by 1/8 doping, it is quite unexpected that the oxygenated sample with the doping close to 1/8 exhibits of around 42 K close to the highest in this system. More detailed study on the doping dependence of for the oxygen-doped sample is required.


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At just below we observed a mixture of a broad commensurate peak and sharp incommensurate peaks with the spin modulation diagonal to the square lattice. It is hoped that this new type of spin modulation triggers a reinvestigation of the peak shape of the commensurate spin correlation for in more detail. Now, we have a more precise criterion of the superconducting phase from the point of spin correlation:

the superconducting phase is accompanied with the long period spin density modulation

along Cu-O bonds.

ACKNOWLEDGMENT

The neutron experiments have been dominantly performed by using thermal and cold neutron 3-axis spectrometers of both reactors in JAERI and NIST. Single crystals were grown in Tohoku University and Yamanashi University and Kyoto University by using lamp-image furnaces for TSFZ method. The authors acknowledge K. Nemoto and M. Onodera for their technical assistance at JAERI and Tohoku University. We also thank Y. Kojima, I. Tanaka, and S. Hosoya in Yamanashi University for their helpful discussions of

crystal growth. We wish to thank M. Greven, M. A. Kastner, Y. M. Kim, Y. S. Lee, T. Suzuki, and T. Fukase for their valuable discussions. Work at Brookhaven National Laboratory was carried out under contract no. DE-AC02-98-CH10886, Division of Material Science, U.S. Department of Energy. The research at Massachusetts Institute of Technology was supported by the National Science Foundation under grant no. DMR97-04532 and by MRSEC Program of the National Science Foundation under award no. DMR94-00334. The present work in NIST and Brookhaven National Laboratory was supported by a US–Japan collaboration

program on neutron scattering. The present work in part was also supported by a grant-in-aid for scientific research from the Ministry of Education, Science, Culture and Sports of Japan and by a grant for the promotion of science from the Science and Technology Agency and by CREST.

REFERENCES 1. See, e.g., M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).

2. J. M. Tranquada, J. D. Axe, N. Ichikawa, A. R. Moodenbaugh, Y. Nakamura, and S. Uchida, Phys. Rev. Lett. 78, 338 (1997). 3. T. Suzuki, T. Goto, T. Shinoda, T. Fukase, H. Kimura, K. Yamada, M. Ohashi, and Y. Yamaguchi, Phys. Rev. B 57, R3229 (1998).

4. K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, Y. Kimura, S. Ueki, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, M. Greven, M. A. Kastner, and Y. Kim, Phys. Rev. B 57 6165 (1998). 5. H. A. Mook, P. Dai, S. M. Hayden, G. Aeppli, T. G. Perring, and F. Dogan, Nature 395, 580 (1998).

6. H. Yoshizawa, T. Kakeshita, R. Kajimoto, T. Tanabe, T. Katsufuji, and Y. Tokura, Physica B 241–243, 880 (1998). 7. C.-H. Lee, K. Yamada, Y. Endoh, G. Shirane, R. J. Birgeneau, M. A. Kastner, M. Greven, and Y. M. Kim, J. Phys. Soc. Jpn. 69, 1170 (2000). 8. K. Yamada, Advances in Superconductivity X (1998) 37.

9. For a review, see N. P. Ong, In Ginsberg, D. M. (ed.) Physical Properties of High-Temperature Superconductors II(World Scientific, Singapore, 1990, pp. 459). 10. A. Fujimori, A. Ino, T. Mizokawa, C. Kim, Z.-X. Shen, T. Sasagawa, T. Kimura, K. Kishio, M. Takaba,

K. Tamasaku, H. Eisaki, and S. Uchida, 1997 International Conference on Spectroscopies in Novel Superconductors. 1 1 . K. Kumagai, K. Kawano, I. Watanabe, K. Nishiyama, and K. Nagamine, J. Supercond. 7, 63 (1994).

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12. G. M. Luke, L. P. Le, B. J. Sternlieb, W. D. Wu, Y. J. Uemura, J. H. Brewer, T. N. Riseman, S. Ishibashi, and S. Uchida, Physica C 185–189, 1175 (1991). 13. T. Goto, S. Kazama, K. Miyagawa, and T. Fukase, J. Phys. Soc. Jpn. 63, 3494 (1994). 14. H. Kimura, K. Hirota, H. Matsushita, K. Yamada, Y. Endoh, S. H. Lee, C. H. Majkrzak, R. Erwin, G. Shirane, M. Greven, Y S. Lee, M. A. Kastner, and R. J. Birgeneau, Phys. Rev. B. 59, 6517 (1999). 15. K. Hirota, K. Yamada, I. Tanaka, and H. Kojima, Physica B 241–243, 880 (1998). 16. M. Matsuda, R. J. Birgeneau, H. Chou, Y. Endoh, M. A. Kastner, H. Kojima, K. Kuroda, G. Shirane, I. Tanaka, and K. Yamada, J. Phys. Soc. Jpn. 62, 443 (1993). 17. Y. S. Lee et al., unpublished, 1999.

18. S. Wakimoto, S. Ueki, K. Yamada, and Y. Endoh, in preparation. 19. Ch. Niedermayer, C. Bernhard, T. Blasius, A. Golnik, A. Moodenbaugh, and J. I. Budnick, Phys. Rev. Lett. 80, 3843 (1998). 20. H. J. Schulz, J. Phys. France 50, 2833 (1989); M. Kato, K. Machida, H. Nakanishi, and M. Fujita, J. Phys. Soc. Jpn. 59, 1047 (1990). 21. T. Suzuki and T. Fukase et al., unpublished data.

Marginal Stability of d-Wave Superconductor: Spontaneous P and T Violation in the Presence of Magnetic Impurities A. V. Balatsky1 and R. Movshovich1

We argue that the wave superconductor is marginally stable in the presence of external perturbations. Subjected to the external perturbations by magnetic impurities, it develops a secondary component of the gap, complex to maximize the coupling to impurities and lower the total energy. The secondary component exists at high temperatures and produces the full gap in the single particle spectrum around each impurity apart from impurity-induced broadening. At low temperatures, the phase-ordering transition into global state occurs.

The point of this note is to emphasize the recently recognized new aspect of the hightemperature superconductors: a marginal stability of the wave superconductor toward secondary ordering in the presence of the symmetry perturbing field. Namely, in the presence of the perturbing field the wave superconductor generates the secondary superconducting component of the order parameter, likely to be in our case, to maximize the coupling to this field and hence lower the total energy. This instability can occur in many different ways. Recently, the surface scatteringinduced s wave component in materials has been observed [1] and the model explaining the effect was proposed [2]. The existence of the secondary gap in the external magnetic field was suggested to explain the anomalies in thermal transport in Bi2212 [3,4]. In both of these cases the superconductor was subjected to the perturbing fields: the surface scattering or the external magnetic field. The above examples can be thought of as a specific

realizations of the general phenomena of marginal stability of 1

wave superconductor.

T-Div and MST-Div, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


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Specifically, we investigated the role of magnetic and nonmagnetic impurities on the superconductors. We find that in the vicinity of each magnetic impurity, in the presence of the spin-orbit coupling, there is a patch of local complex gap generated from impurity scattering. This is the first example, to the best of our knowledge, when impurity scattering produces the coherent component (i.e., secondary gap), as shown in Fig. 1. We suggest that the secondary phase transition occurs spontaneously at lower temperatures with simultaneous impurity spin ordering. Below, we present the summary of the results using mostly qualitative description. For a more technical approach, the reader is advised to look at the original papers [5,6]. 1. SINGLE MAGNETIC IMPURITY The essence of the argument is to consider the single magnetic impurity with large spin S in the superconductor. Locally the time reversal (T) and parity (P) symmetries are violated as the direction of the spin is fixed. Consequently, in the presence

of the symmetry perturbing field it is favorable for superconducting state to generate the secondary component (i.e., the so that the superconducting condensate couples to the impurity spin and lowers the total energy. Consider scattering of a pair off the single impurity site. Interaction Hamiltonian is is the angular momentum operator, is the angle on the cylindrical 2D Fermi surface, is the out-of-plane component of the impurity spin and g is the spin-orbit coupling constant. There is a finite scattering amplitude in the vicinity of impurity:

where

and

order parameter amplitudes are

and

Marginal Stability of d-Wave Superconductor

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respectively. This scattering amplitude does imply the existence of the finite gap near each Ni site. The global second phase grows out of these

patches at lower temperatures. The precursors of the ordered phase (i.e., a finite quasi-particle gap near each impurity site) should be seen even at temperatures above the second transition into state. In the presence of the single impurity scattering potential: where are the pure system propagators, is the correction due to impurity scattering, are the magnitude and angle of the momentum k on the cylindrical Fermi surface, is Matsubara frequency, and is the quasi-particle energy, counted from the Fermi surface. To linear order in small is the density of states at the Fermi surface), one finds [5]

Where

is the function of incoming and outgoing momenta because of broken

translational symmetry. The first nontrivial correction to the homogeneous solution, after

integration over

and

is the xy component:

The finite induced xy component of the order parameter also leads to the xy gap:

Where

is the arbitrary sign interaction in the xy channel, assumed to be of strength. The finite minimal gap on the Fermi surface near impurity site is determined by:

Experimental prediction following from this picture is that the pseudogapped particle spectrum with minimal gap on the order of 20 K should be seen in scanning tunneling microscope experiments near each impurity site.

The usual impurity-induced broadening of the states will be present as well. At low concentrations the broadening, being a function of impurity concentration, will be small

compared to the induced

gap (see also Fig. 3).

2. FINITE IMPURITY CONCENTRATION

Recently Movshovich et al. [6] measured thermal transport in high-temperature superconductors with magnetic impurity (Bi2212 with Ni). The surprising outcome of these experiments is the observed sharp reduction in thermal conductivity at 0.2 K in the samples with 1–2%% Ni impurity concentration (Fig. 2). The observed feature in thermal

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conductivity is consistent with the second superconducting transition into state as described. The secondary phase in many respects resembles the superfluid this is a chiral state that violates P and T. The superconducting condensate has a nonzero

orbital moment L. The free energy admits the linear coupling between the Eq.(l):

and

channels, see

which can be thought of as a spin-assisted Josephson coupling between orthogonal

and xy channels. Because all other relevant terms are quadratic and higher powers in and this linear coupling is driving the transition into state. Impurities, in addition to the component produce the finite lifetime for quasiparticles. Standard arguments of Abrikosov–Gorkov theory imply that the transition temperature into is suppressed. Moreover, the same impurity scattering suppresses the

secondary transition into state. We expect there is a finite impurity concentration window where induced phase can exist (Fig. 3). The idea of marginal stability of high-temperature superconductors and of the secondary superconducting phase might have a broader application for other unconventional superconductors such as heavy Fermion compounds. It implies that the superconducting phase diagram in many of these compounds might be richer than we thought previously. Observation of such a state would represent a significant new development in the field of high-temperature superconductivity.

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ACKNOWLEDGMENTS

This work was done in collaboration with M. A. Hubbard (UIUC), M. B. Salamon (UIUC), R. Yoshizaki (Univ. of Tsukuba), J. Sarrao (LANL), and M. Jaime (LANL). The useful discussions with E. Abrahams, L. Greene, R. Laughlin, D. H. Lee, M. Salkola, and

J. Sauls are gratefully acknowledged. This work was supported by the U.S. Department of Energy. REFERENCES 1. M. Covington et al., Phys. Rev. Lett. 79, 277 (1997). 2. M. Fogelstrom et al., Phys. Rev. Lett. 79, 281 (1997).

3. K. Krishana et al., Science 277, 83 (1997). See also H. Aubin, K. Behnia, S. Ooi, T. Tamegai, K. Krishana, N. P. Ong, Q. Li, G. Gu, and N. Koshizuka, Science 280, 5360:11 (1998) (in Technical Comments). 4. R. B. Laughlin, preprint, cond-mat/9709004. 5. A. V. Balatsky, Phys. Rev. Lett. 80, 1972 (1998). Also cond-mat/9710323. 6. R. Movshovich et al., Phys. Rev. Lett. 80, 1968 (1998). Also cond-mat/9709061.


Skyrmions in 2D Quantum Heisenberg Antiferromagnet Static Magnetic Susceptibility S. I. Belov1 and B. I. Kochelaev1

The static magnetic susceptibility is calculated for a two-dimensional (2D) quantum Heisenberg antiferromagnet (AF) with thermally excited skyrmions at temperatures where J is the nearest-neighbor exchange constant. It is

found that is linear function at comparison with calculations for a nonlinear spin-wave results for is given.

and quadratic one at A model without skyrmions and

Two-dimensional (2D) antiferromagnet (AF) has been the subject of intensive research for the last few years. It is well established that the properties of superconducting cuprates are strongly influenced by 2D critical fluctuations and the undoped materials can be modeled by a nearest neighbor quantum Heisenberg antiferromagnet (QHAF) on a square lattice with a large isotropic exchange constant (J = 1580 K in La2CuO4). In this connection, numerous efforts have been made to study 2D QHAF Most of them are based on some modifications of the perturbation expansion near a homogeneous ground state. Chakravarty, Halperin, and Nelson (CHN) succeeded in a renormalization group analysis of a nonlinear -model which is assumed to represent QHAF in a continuum limit [1,2]. They were able to obtain the correlation length, local order parameter, and static magnetic susceptibility for using no adjustable parameters. Their results were later improved by Hasenfratz and Niedermayer [3] with the chiral perturbation theory. Chubukov et al. [4] developed a theory of QHAF using the 1/N-expansion method on AF with an N-component order parameter. They were able to consider both renormalized classical and quantum critical regimes. However, their results for and do not agree with each other in the intermediate interval of temperatures. In recent papers [5,6] 1

Department of Physics, Kazan State University, 420008, Kazan, Russian Federation.


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there has been proposed another method based on the picture of thermally excited skyrmions

and antiskyrmions. The local order parameter, energy spectrum of elementary spin excitations above the skyrmion background, the skyrmion-averaged radius and renormalized by quantum fluctuations energy were calculated by the Green function method [5]. It has been found that the temperature dependence of the skyrmion radius maps very well the corresponding results for the spin correlation length of 2D QHAF calculated by CHN and Chubukov et al. [4] in the absence of skyrmions. In [6] the nuclear spin relaxation rate was obtained in the temperature region Its behavior at is almost identical with the results based on the renormalizazation group approach and 1/N—expansion method. To make possible a comparative analysis of all proposed theories and reveal peculiarities of properties of 2D QHAF with skyrmions, it is desirable to calculate their static and dynamical characteristics that could be measured by well-developed experimental methods and easily compared. In this report we present our study of uniform magnetic susceptibility of 2D QHAF using the results obtained in [5]. We have calculated directly all components of susceptibility tensor and found that it is almost isotropic as it should be for the rotationally invariant system without long range order. The temperature dependence of susceptibility is close to linear in RC regime and quadratic in QC one. At low temperatures our result qualitatively agrees with Chubukov et al. [4], although the slope of the line is different. Besides, the skyrmion approach allows to extrapolate the function to giving the remarkably well agreement with rather complicated two-loop spin-wave calculations [7]. It was found previously that the number of thermally excited topological excitations is large at temperatures where is the renormalized skyrmion energy, is the local order parameter, and L/a is a linear size of a sample in lattice units [5]. We expect that the nearest neighbors of every skyrmion are antiskyrmions and vice versa, and the total staggered magnetizations equal to zero. Although the long-range order is absent, the local order inside the skyrmion (antiskyrmion) still takes place. The spin excitations above the inhomogeneous ground state are described by the Hamiltonian presented in the local coordinate axes:

Here ij means the sum over the nearest neighbors of two sublattices, Coefficients appeared due to the unitary transformation of the spin operators in the global coordinate axes to the local ones S and determined by the shape of the skyrmion (antiskyrmion). In particular, for a single skyrmion we have

where r and are the polar coordinates, is the skyrmion radius. For a skyrmion system the response to external magnetic field has some peculiarities. In

the theories examining homogeneous systems, magnetic field affects only spin fluctuations. In our case, it affects topological excitations too, resulting in deformation of inhomogeneous background. The Hamiltonian of skyrmion system in presence of magnetic field has the form:

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351

Here

is magnetic field, i runs over two sublattices. Later, we use the system of units where Following [5], we find stable inhomogeneous configuration separating the nonfluctuating part from Eq. (3) and minimizing it over In a continuum limit, the result is a set of equations

Here, describe the inhomogeneous background at are linear corrections due to the magnetic field, For a system of isolated skyrmion–antiskyrmion pairs it is enough to find a solution of Eq. (4) inside a single skyrmion (antiskyrmion):

where

Using Eq. (5) it is easy to calculate the longitudinal and transverse magnetization of the system:

where

are the corresponding components of susceptibility tensor.

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The local order parameter set of nonlinear equations [5]:

and the averaged skyrmion radius

are defined by the

with As one can see, the longitudinal and transverse components of susceptibility tensor are close to each other so that the system is almost isotropic. This result is obtained by the straightforward calculations as against one usually should do. To find one must consider the local nonzero magnetization and define the susceptibility tensor as a response to

magnetic field in the direction perpendicular to magnetization vector (see, for example [2,4]). A response in parallel direction is put to zero with the following rotational averaging of the susceptibility tensor components. Our calculation gives an isotropic susceptibility with no

auxiliary ideas. Using Eq. (7), we can obtain the expressions for

in RC and QC regimes.

It is very interesting that our quantity of is very close to derived by means of the two-loop spin-wave expansion [7]. It suggests that the skyrmion approach

allows the calculation of constants that could be found in the other theories only by using results of sophisticated calculations at At low temperatures, the expression (8) is an almost linear function (neglecting the corrections due to the logarithmic factor), whereas at we have quadratic temperature dependence. The corresponding result of Chubukov et al. [4] is linear in both RC and QC regimes (although the slopes of the lines are different from each other). The difference between the theoretical predictions could serve as a test of proposed theory validity if comparing with experiments. However, we have considered only background deformations in the magnetic field, not examining its influence on elementary spin excitations. The more detailed investigation should take into account the possible modification of their energy spectrum with magnetic field.

ACKNOWLEDGMENTS

We are grateful to Profs. K.-A. Müller and H. Keller for valuable discussions. We thank also Zürich University, where the part of this work was carried out, for its hospitality. The work has been supported by Swiss National Science Foundation (under grant no. 7 IP 051830) and Russian Foundation for Basic Research (under grant no. 98-02-17974).

Skyrmions in 2D Quantum Heisenberg Antiferromagnet REFERENCES 1. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. Lett. 60, 1057 (1988). 2. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. B 39, 2344 (1989). 3. P. Hasenfratz and F. Niedermayer, Phys. Lett. B 268, 231 (1991). 4. A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994). 5. S. I. Belov and B. I. Kochelaev, Solid State Commun. 103, 249 (1997). 6. S. I. Belov and B. I. Kochelaev, Solid State Commun. 106, 207 (1998).

7. J. Igarashi, Phys. Rev. B 39, 9760 (1989).

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Spin Peierls Order and d-Wave Superconductivity Partha Bhattacharyya1

We show that a spin Peierls (SP) state in which a Peierls order Q is modulated as is a magnetic localization wave number is manifestly scale invariant when the fermions maintaining single-site occupancy are creation

operators for fermions forming Cooper pairs. A charge-ordered stripes state is a state of modulated chirality being evidence for superconductivity. Then a quantum mode of gauge symmetry restoration, like the Josephson mode in SNS junctions, is a possibility.

1. INTRODUCTION

Among the most important ideas in the theory of electronic correlations in low dimensional systems is the possibility of d-wave superconductivity. An electronic mechanism of superconductivity implies the possibility of a symmetry restoring Josephson effect by a single electron transition (SET) in a manner similar to superfluidity in Our proposal is that in these circumstances the ground state has an additional quantum number, chirality, similar to the kinematic quantum number that determines whether the electron is on the left or right side of a normal-state barrier. This can be ensured in a spin Peierls (SP) state, in which the Peierls order Q is modulated to is a magnetic localization wave number. The SP order parameter for the chiral electrons having chirality are charges placed on the diameter of a unit sphere corresponding to single electron occupancy on a unit cell connected by a Dirac string. A longitudinal mode measures a Peierls charge discommensuration at preferred regions on the Fermi surface. A transition occurs when a monopole changes chirality, allowing a Josephson transition to take place amongst chirally equivalent configurations. Charge ordering is seen in superconductors [1] with Ising 1

Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India.


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symmetry as modulated chirality corresponding to excitations among certain symmetry directions. We expect dynamical spin correlations at lower temperatures when SP modulation becomes relevant in the superconducting state. We propose a modulated SP state as a stripes-stated spin ordering is seen in CMR compounds. The SP state has an order parameter

and conjugate), where are Majorana fermions ensuring single electron occupancy at each site, The idea is that coherent scattering from localized spins gives quantum interference among time-reversed paths, causing a Peierls gap to open up (Fig. 1) that is maximum along the directions and is minimum along as manifested in the SP state. This is manifested as symmetry order parameter and in superconductors as a distortion of the rhombohedral unit cell. This is a manifestation of a paramagnetic Meissner effect along with the Raman spin-gap or NMR anomalies. Our proposal is different from Anderson and Sarker [2], who consider tunneling diagonal in k space. 2. KOHN ANOMALY AND PARAMAGNETIC MEISSNER EFFECT

We have seen that quantum interference among time-reversed paths implies a trajectory such that a displacement across a normal domain is elastic and accompanied by a Brillouin or localization wave number K or Q where for single parameter scaling. Charge conjugation symmetry implies that a fermionic gauge invariant trajectory at transition across a line or disclination has a hidden gauge group where the electronic phase topological excitation valued in allows a quantum of excitation being a localized magnon mode mediating a transition that is normally not allowed. This is similar to the situation in superfluid where a first-order longitudinal NMR mode is seen when a weak dipolar term breaks spin-orbit symmetry. We see that the action of a finite displacement operator is to produce a phase factor exp(iQa) or, in general, where a is the displacement.


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This implies that a finite displacement is dual to an infinitesimal scale transformation on a quasi-crystalline lattice, We see that a scale invariant SPordernecessarily maintains modular S and T invariance. Invariance with respect to modular S invariance is the requirement that if an orbit in gauge field A is in a representation of SL(2, R), the gauge field A and its dual transform according as such that the elementary excitation is a fermionic oscillator. In systems with order parameter, this duality implies a longitudinal Goldstone mode corresponding to a magnetic anomaly cancellation in a superconductor. This is a paramagnetic Meissner effect, gauge invariance requiring is the Larmor torque and c is the spin-wave velocity. This is an explanation for modulated spin and charge order in dynamical magnetic correlations [1] seen in cuprates. An electron acquires a phase given by where on completing a gauge invariant trajectory and releasing a Goldstone mode [3]. In a state with duality, a rotation of the coordinates causes the creation of a quantum spin-wave at Q, a dynamically obtained localization mode, canceling the phase. A coherent SP state maintaining symmetry of duality is described by a BCS-like wave function given by Eq. (1). corresponds to half-filled normal state. is such that step operator connecting charge conjugate Cu sites and is a step operator connecting states, as shown in Fig. 1. Acting on the electronic state, a gauge field couples separately with electronic states of left- and right-hand chirality. A finite displacement operator acting on engender a nontrivial phase at sites corresponding to a localized monopole or a Josephson dissipationless mode across a line of disclination. A Kohn anomaly is seen at Q, which is an order parameter for a chiral-ordered stripes state. A Raman mode is now obtained in a manner similar to the Josephson longitudinal mode—namely in first order in gauge fields, not in second-order polarization fluctuations. According to the symmetry of the SP state, where when a voltage V is on the a-b plane, as for an LA mode along charge conjugate sites Conformal invariance ensures that the gauge field induces a spin-wave, not density fluctuations whose response is at the Coulomb plasma frequency. A localized spinon mode is created at an energy At short distances, a paramagnetic Meissner effect is seen as a manifestation of a conformal anomaly cancellation and is at once an explanation at short distances for a Raman mode in quasi-two-dimensional conductors, a nested Fermi surface and singularities in the DOS.

3. WARD IDENTITY, TOPOLOGICAL SUSCEPTIBILITY A Ward identity obtained by other means in CDW superconductors for the Raman susceptibility is obtained in the quantum limit by considering the dual of the relation connecting the magnetization with the generator of finite displacements Because a Galilean transformation results in an energy shift we seek to evaluate the pair susceptibility as the response of the local equilibrium magnetization to a discrete change in flux. This spin-wave susceptibility is different from that considered by Klein and Dierker [4]. Consider the situation, similar to spin relaxation in superfluid when a voltage V induces a charge imbalance and the quasi-particle are not in equilibrium. The

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Bhattacharyya

steady-state distribution is where being the equilibrium Fermi distribution. Because the quasi-particle charge the steady-state distribution is where the quasi-particles are in equilibrium at a chemical potential A dissipationless transition is, therefore, a

transition along a single state at the Fermi energy such that a single electron excitation is at an energy corresponding to a Mott state. This means that the topological susceptibility is obtained as the response of a single charge to an infinitesimal gauge transformation that is connected to a finite scale transformation obtained by replacing where V is the voltage bias on the a-b plane. Thus,

n(k) is Gaussian, a boson propagator, a free-electron propagator. The requirements of modular S invariance—namely that and the dual field is taken into account by replacing a single particle energy A single electron transition manifested as a scat-

and

tering state at Fermi surface at long-length scale is induced by a transition of the magnetic

monopole state. The Raman susceptibility has collective modes at Fig. 2

Scale invariance requires that a single electron transition maintaining a steady state has a relaxation rate 3.1 NMRandNQRin Asayama et al. [5] give an excellent survey of lot of the work done on knight shifts and

the relaxation rates

and

in cuprate superconductors.

is related to the transverse


359

susceptibility Eq. (5), and the temperature dependence in the normal state is easily seen as

where is a spinon mode characteristic wave number. If const. in the normal state, is a universal behavior for cuprate superconductors. It is also right in the La cuprates. When behavior is seen in in the superconducting state.

3.2 Electromagnetic Response A finite density of states at the Fermi level induced by impurities treated as unitary scatters is given as the explanation for the low-temperature behavior of the superfluid density. In the local limit, where impurity scattering is relevant, the factors are replace The correlation among the initial and final states is simply a Gaussian Impurity scattering implies such that

The electromagnetic penetration length is obtained from the transverse response, where In that limit, where implies

giving a quadratic behavior with T at low temperature. Furthermore, because in three dimensions. Because the energy given to a mode is the same, ensured by charge quantization, where the dimensionality or 3. REFERENCES 1. D. Wollmann, D. J. Van Harlingen, W. Lee, D. M. Guisberg, and A. J. Leggett, Phys. Rev. Lett. 11, 2134 (1993); J. M. Tranquada, Physica B 241–243, (1998), for modulated spin and charge ordering in cuprates. 2. S. K. Sarker and P. W. Anderson, cond-mat/9704123. 3. C. Itzykson, Int. J. Mod. Phys. A 1, 65 (1986). 4. M. Klein and R. Dierker, Phys. Rev. B 29, 4976 (1994). Raman scattering cross-sections calculated in second order vector potentials is shown in T. P. Deverena and D. Einzel, cond-mat/9408019. 5. K. Asayama, Y. Kitaoka, G.-q. Zheng, and K. Ishida, Prog. NMR Sped. 28, 221 (1996). 6. D. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994). 7. M. Franz, C. Kallin, A. J. Berlinsky, and M. I. Salkola, Phys. Rev. B 56, 7882 (1997-I); A. Andreone et al.

Phys. Rev. B 56, 7874 (1997-I).


On Localization Effects in Underdoped Cuprates C. Castellani,1 P. Schwab,1 and M. Grilli1

We comment on transport experiments in underdoped LaSrCuO in the nonsuperconducting phase. The temperature dependence of the resistance strongly resem-

bles what is expected from standard localization theory. However, this theory fails when comparing with experiments in more detail.

1. INTRODUCTION

It is generally believed that the understanding of the anomalous normal-state properties would shed light on the understanding of the pairing mechanism of high superconductors. In particular, low-temperature transport experiments in the normal state should provide valuable information on the physical mechanisms acting in

materials. The normal state

is usually inaccessible at low temperature due to the onset of superconductivity. However, superconductivity can be suppressed down to lowest temperatures by applying a sufficiently high magnetic field. In this paper, we discuss the low temperature resistance in underdoped (LSCO) at high magnetic fields, where insulating behavior with a typical log T dependence of the resistance has been observed [1]. Various proposals have been made for the origin of

this behavior. Anderson et al. [2] interpreted the data in the framework of Luttinger liquid transport theory, which predicts a power law in the c axis and in the in-plane conductivity Although it is not unreasonable to fit the experiments by a power law, much better fits are achieved assuming the logarithmic behavior. Alexandrov [3] reported a logarithmically divergent resistivity in a bipolaron model in presence of disorder; however, this model has been questioned repeatedly [4]. Varma [5] argued that in a non-Fermi liquid even small disorder drives the density of states to zero and thus drives the system to an insulator. 1

Istituto di Fisica della Materia e Dipartimento di Fisica, Università “La Sapienza,” piazzale A. Moro 2, 00185 Roma, Italy.


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Castellani, Schwab, and Grilli

Standard localization effects have been discussed as source of the increasing resistance since the first experiments [1,6]. Evidence against this interpretation has arised from measurements of the Hall resistance [7], which is nearly temperature independent. However, because the mechanism that dominates the Hall effect in the cuprates is not clear, it is hard to make conclusions. So far, a detailed analysis of the temperature dependence of the resistivity versus the predictions of localization theory has not been given. This is subject of the present paper. In the next section we briefly recall standard localization theory in two-dimensional (2D) systems, discussing both coherent backscattering and interaction effects. Then we apply localization theory to LSCO. We demonstrate that the crossover from 3d to 2d localization in LSCO is expected near optimal doping. However, the amplitude of the increase of resistance in that material in the underdoped region does not fully agree with standard 2d localization theory. We argue that anomalous localization effects are indeed to be expected in that region in the presence of a disordered stripe phase. 2. LOCALIZATION IN 2D SYSTEMS

In two dimensions, arbitrarily weak disorder can localize all electronic states. This famous result follows from the single parameter scaling theory of localization [8]. This theory is justified when interactions are negligible. In the weak disorder limit the conductivity at low temperature is given by

where is the “classical” Drude conductivity, and the logarithmic term is due to quantum corrections, with the (elastic) scattering time and the dephasing time. These “weak localization” corrections are due to quantum interferences for electrons that are diffusing along paths containing closed loops (coherent backscattering). Dephasing is due to inelastic processes, with leading to a correction to conductivity, which is logarithmic in temperature, However, single-parameter scaling fails in presence of interactions, where a scaling theory including electron interactions is needed. Such a theory has been put forward by Finkel’stein [9]. In perturbation theory, new singular contributions to the conductivity are found, which are—in 2d—proportional to log T. These singular corrections to the conductivity are due to the interplay between disorder and interaction, and arise because on distances that are larger than the mean free path, electrons move slowly and have more time to interact with each other.

The correction to the resistivity due to this mechanism is

where 1 is due to interactions in the singlet channel and 3[· · ·] are due to the triplet channels. The universal value of the singlet amplitude is due to the long-range nature of the Coulomb interaction because after screening the dimensionless interaction equals in the long

On Localization Effects in Underdoped Cuprates

363

wavelength limit always is an interaction parameter that is related to the Landau parameter Analyzing the renormalization group flow, a metal insulator transition has been found, and a phase with finite resistance at zero temperature exists [9,10]. The interacting system avoids localization, because the triplet amplitude becomes relevant under scaling The disordered Fermi liquid tends to form ferromagnetic polarons (local moments). Experimental evidence for enhanced spin fluctuations has been found near the metal-insulator transition in 3d Si: P. Experimental proof for large-spin fluctuations near the recently discovered metal-insulator transition in 2d MOSFETs is still lacking, but the effects are currently discussed. In the context of underdoped LSCO, it is important that a magnetic field drastically modifies the above scenario. A magnetic field, besides suppressing coherent backscattering due to orbital effects, also suppresses the triplet due to Zeeman splitting. As a consequence, spin fluctuations remain small and under renormalization. Thus the singlet term dominates, leading to an insulating behavior as seen in experiments in the low temperature regime [1]. We argue below that an analogous result can be envisaged in the extreme underdoped regime even in the absence of magnetic field.

3. APPLICATION TO CUPRATES In standard studies of localization the magnetoresistance is a main probe for extracting important informations on both backscattering and interaction effects. Unfortunately,

in cuprates many different contributions to the magnetoresistance have been observed related to the superconducting fluctuations [11] and to the peculiar behavior of the Hall conductance [12]. These effects may mask the localization contributions and thus make

an interpretation difficult. Therefore, we concentrate in the following on the temperature dependence of the resistance under conditions, where possibly the above complications are

not present. We refer to two types of experiments. Extremely underdoped LSCO which is nonsuperconducting even in the absence of magnetic fields, and the system near-optimum doping, but large magnetic field In both cases log T behavior in the resistivity or conductivity has been observed, suggesting that the physics of disorder and interaction in two dimensions is relevant. However, despite a substantial anisotropy, the LSCO materials are bulk systems of weakly coupled layers. This raises the relevant issue of the effective dimensionality of LSCO with respect to localization. Weak localization in a nearly 2D metal has been considered by Abrikosov [13]. We performed a similar calculation for the interaction contribution. We generalized the model of c-axis transport of Ref. [13], incorporating interplanar disorder as discussed in Ref. [14]. We found that the system behaves in a 2D fashion when the tunneling time between layers is larger than the time of the slowest processes that are contributing to localization. For weak localization, the relevant time scale is the phase coherence time whereas the time scale for the interaction contribution, which is the relevant one in high magnetic field, is Therefore, the crossover from 3d to 2d is defined by The tunneling rate is hard to estimate directly because it is not clear if processes that conserve in-plane momentum dominate, or momentum nonconserving processes dominate, for which are tunneling amplitudes, the quasi-particle

364


lifetime, and N(0) is the density of states. More conveniently, the tunneling rate is determined from the c conductivity, because Inserting the free-electron

value for the 2d density of states with we find for cm a tunneling rate of the c resistivity at 20 K was between and , By comparing to the temperature at 20 K, we conclude that the samples near optimal doping at are near the dimensional crossover from 3d to 2d. The underdoped samples are presumably still 2D at 20 K, whereas the overdoped samples may be three dimensional. In this case, a correction to the conductivity at low temperature instead of log T is expected. Preyer et al. [6] reported the conductivity in highly underdoped For a sample with they found a logarithmic correction to the conductivity

with

over more than a decade of temperature, from down to where a crossover to variable range hopping was

between

observed. This seems to be in good agreement with standard localization theory. The magnetoresistance is negative and isotropic. Assuming that weak localization is relevant, the

isotropic magnetoresistance is not consistent with conventional theory. However, conventional theory is built for a nonmagnetic Fermi liquid, whereas here a theory in a doped AF is needed, with peculiar quasi-particles (hole pockets). Also, in the absence of long-range order, conventional theory must be modified. A short magnetic correlation length makes the

system similar to a spin-glass. The random magnetic field in a spin-glass is assumed to suppress quantum corrections to the conductivity from coherent backscattering and interactions in the triplet channels. Quantum corrections in the remaining interaction singlet channel are field independent, which suggests that the experimentally seen magnetoresistance is due to a different mechanism. Although a full theory of localization in a doped AF is lacking, the

presence of a log T of the correct magnitude and the correct scale of the cross-over to strong localization strongly suggests that the physics of disorder and interactions is relevant here. Ando et al. [1] studied LSCO for higher doping, suppressing superconductivity by strong magnetic fields with pulses of up to 60 T. There is practically no magnetoresistance in the normal state (i.e., once superconductivity is destroyed, the resistance saturates). Below they found an insulating behavior at low temperature Both in ab and c direction a log T in the resistance was found,

where

and

are sample dependent. Analyzing the amplitude of the log T near the onset

calculating the “interaction constant”

we found from experimental data of Ref. [1]

according to

Standard localization theory


365

predicts in high magnetic field

which is of order 1, but never larger than 2, because stability of the Fermi liquid requires

Apparently,

are of the same order of magnitude.

To the first view the experiments seem to be in reasonable agreement with theory. There

are, however, a number of problems: (a) Although theory predicts log T in the conductivity, it is seen experimentally in a large range of resistivity. (b) The ratio does not depend on temperature (i.e., the function in ab and c direction is the same), This is predicted for anisotropic, but 3D localization [15]. In the temperature region of 2D localization a logarithmic correction to the c conductivity is expected due to the corrections to the tunneling density of states, N. Explicitly working out the theory, we found which in general differs from the correction to the resistivity in ab direction, (c) A third problem arises from a quantitative analysis of the amplitude of the log T, which is of the right order of magnitude, but nevertheless is too large. Further investigating problem (c), we found an intriguing relation between the experimentally measured amplitude of the log corrections and the amount of disorder as obtained from the absolute value of resistivity at some fixed temperature In Table 1, we report for a number of samples, comparing “clean” and “dirty” samples of the same material and dopant concentration [1,7,16]. Whereas is independent of disorder, the experimentally determined value decreases with increasing dirtiness: As shown in the table, the product is nearly disorder independent. Moreover, seems to decrease with increasing doping. For the four LSCO samples we report in Table 1, the product is roughly independent from disorder and doping. A second observation is that various features of the anomalous localization can be described phenomenologically by using the Drude formula for the conductivity and taking the scattering rate from the ansatz

This logarithmically enhanced scattering rate appears directly in the resistivity, not in the

366


conductivity, and is therefore consistent with the property outlined in point (a). Moreover the constant ratio of can be reconciled with a log T correction that is typical of 2D systems [problem (b) outlined above] by the assumption in Eq. (7) if tunneling between planes is dominated by momentum conserving processes. Finally, if dirtiness only affects but not the logarithmic term, the amplitude of the log T as a function of disorder behaves according to the experimental observation discussed above,

4. DISCUSSION We discussed some features of transport experiments in LSCO compounds in the normal state. On the one hand, the log T correction in strongly underdoped LSCO appears to be consistent with standard localization theory, although an explanation for the magnetoresistance is still lacking. On the other hand, for higher doping, standard theory is not able to explain the experiments.

There are several reasons why the conventional “old-fashioned” localization theory is not expected to work well in strongly correlated anisotropic systems like the cuprates. One first possibility is that the cuprates cannot be described by the Fermi liquid theory as already pointed out in the introduction [17]. If this is the case, a new localization theory starting from a clean non-Fermi liquid system should be devised [2,5]. Alternatively, a singular interaction could be responsible for both the disruption of the Fermi liquid and of the anomalous localization effects. Mirlin and Wölfle [18] reported anomalous localization effects within a gauge field theory [19], where particles interact via a singular transverse gauge field propagator At low temperature, a log T correction has been found with an amplitude that depends on resistance itself, In the quantum critical point (QCP) scenario of superconductivity, a QCP exists near optimum doping, with an “ordered” stripe phase in the underdoped regime [20–22]. In this context, possible sources of singular interactions are soft modes from dynamical stripes, or critical fluctuations near the QCP. Specifically, the interaction near the stripe critical wave-vector may be parameterized as [20]

where Quantum corrections to the conductivity due to exchange of these fluctuations are, to first order in proportional to the Fermi-surface average

of the static interaction,

i. The interaction is attractive,

leading to an increase of localization. For large quantum corrections must be calculated beyond first order. However, standard theory [9] does not apply here, due to the strong frequency dependence of the interaction. We have preliminary results indicating that quantum and classical fluctuations contribute differently to localization. Finally, one should also consider the possibility of a nonconventional source of disorder.

In particular, if a disordered, nearly static stripe phase is realized in these systems, one should, as well as the conventional impurity disorder, also consider the disorder coming from domain boundaries and other topological defects of the striped textures. An appealing possibility is that a disordered stripe phase could be responsible for the anomalous localization behavior both by introducing a singular scattering between the


367

electrons and by providing topological disorder via domain boundaries. As a consequence, any mechanism such as impurity or lattice pinning of (static or slow) stripe fluctuations is expected to reduce the amplitude of the effective interaction and the log T corrections, in agreement with the central observation of the present work that is nearly constant. This expectation is supported by the observation that at where stripes are more ordered, the log T is less strong and the resistance saturates at low temperature [1,23], as in The very speculative character of these considerations calls for a detailed theory, which is not available at the moment.

REFERENCES 1. Y. Ando, G. S. Boebinger, A. Passner, T. Kimura and K. Kishio, Phys. Rev. Lett. 75, 4662 (1995); G. S. Boebinger et al., Phys. Rev. Lett. 77, 5417 (1996). 2. P. W. Anderson, T. V. Ramakrishnan, S. Strong, and D. G. Clarke, Phys. Rev. Lett. 77, 4241 (1996). 3. A. S. Alexandrov, preprint, cond-mat/9610065. 4. B. K. Chakraverty, J. Ranninger, and D. Feinberg, Phys. Rev. Lett. 81, 433 (1998); A. S. Alexandrov, preprint, cond-mat/9807185 (comment). 5. C. M. Varma, Phys. Rev. Lett. 79, 1535 (1997). 6. N. W. Preyer, M. A. Kastner, C. Y. Chen, R. J. Birgenau, and Y. Hikada, Phys. Rev. B 44, 407 (1991); B. Keimer et al., Phys. Rev. B 46, 14034 (1992). 7. Y. Ando et al., Phys. Rev. B 56, 8530 (1997). 8. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). 9. A. M. Finkel’stein, JETP 57, 97 (1983); C. Castellani, C. Di Castro, P. A. Lee, and M. Ma, Phys. Rev. B 30, 527 (1984). For a review, see: P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 787 (1985); D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994). 10. C. Castellani, C. Di Castro, and P. A. Lee, Phys. Rev. B 57, 9381 (1998). 11. A. Varlamov et al., Adv. Phys. 48, 655 (1999).

12. 13. 14. 15.

A. W. Tyler et al., Phys. Rev. B 57, 728 (1998). A. A. Abrikosov, Phys. Rev. B 50 1415 (1994). A. G. Rojo and K. Levin, Phys. Rev. B 48 16861 (1993). P. Wölfle and R. N. Bhatt, Phys. Rev. B 30, 3542 (1984); R. N. Bhatt, P. Wölfle, and T. V. Ramakrishnan, Phys. Rev. B 32, 569 (1985).

16. T. W. Jing, N. P. Ong, T. V. Ramakrishnan, J. M. Tarascon, and K. Remschnig, Phys. Rev. Lett. 67,761 (1991). 17. P. W. Anderson, Science 235, 1196 (1987); Phys. Rev. Lett. 64, 1839 (1990); 65, 2306 (1990). 18. A. Mirlin and P. Wölfle, Phys. Rev. B 55, 5141 (1997). 19. N. Nagaosa and P. A. Lee, Phys. Rev. Lett. 64, 2450 (1990); P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621

(1992); 20. C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995); A. Perali, C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. B 54, 16216 (1996). 21. For experimental evidence of a stripe phase in the cuprates, see the contributions to this conference by A. Bianconi and by J. M. Tranquada. For theoretical aspects, see instead the contributions by C. Di Castro, V. J. Emery, and J. Zaanen.

22. Different QCPs have been suggested by C. M. Varma, Physica C 263, 39 (1996); P. Montoux and D. Pines, Phys. Rev. B 50, 16015 (1994), and references therein. 23. C. Castellani, C. Di Castro, and M. Grilli, cond-mat/9709278. 24. J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, Phys. Rev. B 54, 7489 (1996).


Interpolative Self-Energy Calculation for the Doped Emery Model in the Antiferromagnetic and in the Paramagnetic State J. Fritzenkötter1 and K. Dichtel1

Electronic properties of the doped Emery model for the planes in HTSC materials, including magnetic phases, are calculated. The self-energy of the problem is approximated by an interpolation between the weak coupling second-order perturbation theory limit and the strong coupling atomic limit. The self-consistent calculation of the grand canonical potential for zero temperature and all magnetic phases gives a purely antiferromagnetic (AF) phase in the neutral and nearly neutral case. Upon doping phase separation shows up between an AF and a doped paramagnetic (PM) regime, whereas for larger doping only the PM phase exists.

1. INTRODUCTION

The discovery of the high-temperature superconductivity (HTSC) enforced the interest in the highly correlated models of electronic structure, especially for the superconducting copper oxygen planes common to all HTSC materials. An adequate description of the electronic state needs at least three orbitals in the plane, the copper dx2– y2 and the oxygen and orbitals (Fig. 1). For sake of simplicity, this three-band model is mostly mapped to simpler models as the one-band Hubbard model or the t-J model, respectively. As experiments on oxygen isotope effects in the last years showed the importance of the oxygen orbitals for the coupling to the lattice, we believe the full three-band model to give a more adequate description of the electronic structure. The large values of the onsite Coulomb repulsion on the copper sites requires some treatment that covers strong coupling aspects, and the antiferromagnetic (AF) state in the neutral case must be included for a realistic picture of the interesting electronic state at low doping. We apply some 1

Institut für Theoretische Physik, Universität Kiel, 24098 Kiel, Germany.


369

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Fritzenkötter and Dichtel

interpolation method to the self-energy that we previously used for the paramagnetic (PM) state only [1]. Comparison with available QMC results for the one-band Hubbard model

proved the method to give appropriate results in this low-doping regime.

2. MODEL

We formulate the Emery (or three-band Hubbard) Hamiltonian for the A-B unit cell with sublattices A and B

pointing to the A-B unit cells. is a vector to label the corresponding sublattice (see Fig. 1) and being the paramagnetic unit vectors. The Fermi operators denote holes in the corresponding copper and the and oxygen orbitals with spin The parameters in this model are the hopping energy the charge transfer energy between copper and oxygen orbitals, and the copper on-site Coulomb repulsion U. As the on-site correlation acts only on the copper sites, it is convenient not to diagonalize to obtain the three bands but to express all quantities in the local orbital basis:

where these Fermi operators (in short notation) are related to the real space Fermi operators

Interpolative Self-Energy Calculation for the Doped Emery Model

371

above via the following Fourier transforms

N being an arbitrarily large number of paramagnetic unit cells in the problem. The unperturbed Hamilton operator then appears as a matrix

which depends on the spin index

and on the wave vector

of the A-B Brillouin zone

(ABBZ) via the quantities

3. METHOD

We repartition the Hamiltonian into two parts by introducing four effective atomic such that perturbational treatment of the second order in

levels

is performed around the Hartree–Fock-like Hamiltionian

These effective levels, of which at least two are independent due to the para-, ferro- and antiferromagnetic symmetries, are to be fixed later. First we have to determine the self-energy in second order perturbation theory, which, using the only relevant Feynman graph in this order, gives thefollowing standard expression

372

Fritzenkotter and Dichtel

with the Fermi functions and the sublattice indices r and

the A-B reciprocal lattice vectors labelling both copper orbitals per AF unit cell. Thus only

the copper elements of the self-energy matrix represented in our basis (2) are nonzero:

The copper spectral density matrices denoted turbed Hamiltonian matrix

are determined by the unper-

expressed in our representation. Of course the para- and ferromagnetic case with identical A and B lattice are included in the formulae. The interpolation of the self-energy to the strong coupling limit is performed as in Refs. 2 and 3 and was studied later [4] for a different self-consistency condition. The interpolation is applied only to the diagonal parts of the self-energy matrix

with

and

because the off-diagonal spectral densities already vanish as for vanishing hopping amplitude [see Eq. (3)], whence the off-diagonal part of the self-energy


373

yields

The copper occupancy

and the “fictitious” [4] copper occupancy

are determined by

the corresponding unperturbed

and the full spectral density

Assuming that the chemical potential is given, for a complete determination of all free parameters a sufficient number of conditions must be chosen in order to fix the effective atomic levels (i.e., one and two additional conditions for the para- and nonparamagnetic cases, respectively). Reference 5 discusses several conditions. We choose the condition from Refs. 2 and 3

as the self-consistency condition. A complete solution of the problem consists of a set of occupancies and effective levels [i.e., Eqs. (8–10)] must be solved for two and four parameters in the PM and non-PM cases, respectively. This means root finding in a two- or four-dimensional parameter space, which we perform using a modified Powell hybrid method from the numerical library SLATEC. The most time-consuming part is the calculation of the copper occupation numbers, which implies the calculation of the self-energy for every trial set of parameters. The momentum summations in Eq. (4) can be circumvented by expanding the momentumdependent self-energy in a Fourier series on a real space lattice and writing the energy numerator as a Laplace transform [6] so that extensive use of FFT can be made. We then restrict our present calculations on the local part of the self-energy (i.e., the ABBZ average of the self-energy), so that the self-energy becomes momentum independent. We believe that the momentum dependence of the self-energy has a greater impact on spectral densities and the Fermi surface than on the particle density and internal energy, which are presented in this article. With the so obtained self-consistent solution (which typically takes 1 hr on a SUN Ultra 1), the internal energy can be calculated from a standard Greens function formula, expressed in our matrix representation

374


with the full spectral density and normalized per paramagnetic unit cell and spin direction. For the internal energy is equal to the free energy, which allows us to write down the grand canonical potential

along with the total particle number

normalized as the internal energy, so that at half filling, This method is practically restricted to zero temperature when internal and free energy coincide. For finite temperature, more time-consuming procedures (e.g., coupling constant integration) would be necessary to calculate the entropy. 4. RESULTS

We solve the problem for several couplings and chemical potentials with a charge transfer energy

Figure 2 shows the grand canonical potentials for

and


varying chemical potential. The PM and AF potential intersect at slopes leading to phase separation in the interval

375

with different

that is, the low-doped AF and high-doped PM phase coexist in the system. The FM phase

is always thermodynamically unstable, and so it disappears from the phase diagram as it should and can be seen from Fig. 3. This must be compared with pure Hartree–Fock calculations where a FM phase still exists [7]. Up to now, we could solve the problem only for relatively large couplings because for lower values of Coulomb interaction the region of phase separation becomes more difficult to determine numerically. The PM/AF phase separation is mainly due to the correlation effects in the PM phase only, because the spectral density of the self-energy matrix is positive semidefinite and for the FM and AF phase turns out to have a small trace near half filling, so that the correlation is predominantly of Hartree–Fock type in the non-PM phases. The densities of states (DOS) in Fig. 4 finally show the characteristic behavior of the chemical potential in the paramagnetic state, passing a correlation-induced van Hove peak with increasing carrier concentration from half filling (A) up to large doping (F). From the phase separation at however, we conclude that the PM states with are metastable and not of real relevance. In the interval the experimentally relevant DOS is a superposition with doping-dependent weight of both the DOS of the AF insulator (D) and the PM state at (E). Upon further doping, the DOS remains purely paramagnetic [see (F)]. For the parameters used in (E), the chemical potential happens to lie very near to the peak maximum; for other values, this is also approximatively the case.

376


5. CONCLUSION

An approximative calculation of the DOS covering both weak and strong coupling situations and different magnetic phases shows that in the interesting doping regime the phase diagram differs qualitatively from the Hartree-Fock result. Contrary to the former, no ferromagnetic phase occurs and phase separation between an AF nearly neutral phase and a PM phase at slightly overdoped carrier concentration takes place. Electronic phase separation is a well-known result in models with large on-site Coulomb correlations. Whether this effect supports the formation of electronically inhomogeneous regions, say stripes or clusters of carriers, or is suppressed by the neglected long-range Coulomb effects remains an open question. We obtained the result in a realistic DOS calculation of the three-band model. Thus, we show that it is an effect to bring the Fermi level already for small doping values to the van Hove singularity and pin it near this singularity over a quite large doping interval.

REFERENCES 1. J. Fritzenkötter and K. Dichtel, J Supercond. 9, 449 (1996).

2. 3. 4. 5.

A. Martin-Rodero and F. Flores, Sol. Slate Commun. 44, 911 (1982). J. Ferrer, A. Martin-Rodero, and F. Flores, Phys. Rev. B 36, 6149 (1987). H. Kajueter and G. Kotliar, J. Phys. Chem. Solids 56, 1615 (1995). M. Potthoff, T. Wegner, and W. Nolting, Phys. Rev. B 55, 16132 (1997).

6. H. Schweitzer and G. Cz.ycholl, Sol. State Commun. 69, 171 (1989). 7. X. Y. Chen, W. P. Su, C. S. Ting, and D. Y. Ying, Sol. State Comm. 67, 349 (1988).

The Quasi-Particle Density of States of Optimally Doped Bi 2212: Break-Junction vs. Vacuum-Tunneling Measurements R. S. Gonnelli,1 G. A. Ummarino,1 and V. A. Stepanov2

In the present article we show that a strong-coupling density of states (DOS) with a symmetry, determined by direct solution of the d-wave Eliashberg equations in the real energy axis formulation, can represent very well our best break-junction optimally-doped Bi 2212 tunneling data and explain their if we use an electron-boson spectral function determined by the inverse solution of the same equations in s-wave symmetry. In contrast, even by using spectral functions of different kind, it is impossible to obtain a good fit of the DOS and, at the same time, of in the best Bi 2212 STM measures by means of the d-wave Eliashberg theory. It is also clear that the low-temperature zero-bias DOS is not fully consistent with a pure d-wave symmetry and other mechanisms, as a modest

impurity scattering in the unitary limit, have to be considered in order to explain the low-energy data.

PACS numbers:

1. INTRODUCTION The importance of tunnel spectroscopy in giving essential indications on the microscopical pairing mechanism in high superconductors is clear. Until a short time ago, all the best measures of tunnel spectroscopy in single crystals of

gave a gap around 20–25 me V, and therefore a ratio of the order of 6 [1–4]. Among the most recent and reproducible measures on optimally doped we remember our break-junction ones that allowed us to reproducibly determine the spectral function of electron–boson interaction by means of the solution of Eliashberg equations in 1

2

INFM—Dipartimento di Fisica, Politecnico di Torino, 10129 Torino, Italy.

P.N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia.


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s-wave symmetry and in presence of a normal density of states (DOS), depending on the energy [5–7]. More recently, some remarkable measures in optimally doped Bi 2212 (especially of STM type) appeared [8–11] that have shown more peculiar features: peaks of the conductance at approximately 35–40 meV, values of the ratio of the order of 8–10, and presence of conductance peaks also at (quasi-particle pseudogap). These characteristics have never been observed before also in the best crystals and, therefore, it comes spontaneous to ask for which reason the various measures on apparently similar samples turn out to be in apparent contrast. In this article, we discuss and compare the

most recent tunnel measures in optimally doped Bi 2212 crystals in order to understand the origin of the differences. Due to the rather strong evidence of the d-wave symmetry of the wave function in quasi-2D layered superconductors [12], we analyze the experimental curves by using the Eliashberg equations [13–15] for the strong electron–boson interaction in such a symmetry and compare the results to those we obtained previously in s-wave. 2. EXPERIMENTAL AND MODEL

In Fig. 1, a comparison of three normalized conductances obtained at 4.2 K in different tunneling experiments on optimally doped Bi 2212 single crystals is shown. The full circles represent our break-junction data after an unfolding procedure necessary to convert the SIS conductance at 4.2 K to the SIN conductance at the same temperature. The experimental technique for the break-junction creation at 4.2 K and details concerning the single crystals and the reproducibility of these data are described elsewhere [5,6,16]. Full triangles and open circles represent the recent STM tunneling data on optimally doped Bi 2212 by DeWilde et al. and Renner et al. respectively. The curves have been normalized in such a way that the number of states at positive bias is preserved. We want to emphasize that, in order to be confident that we are comparing the results of true optimally doped samples, only the tunneling curves of recent experiments in single crystals having between 92 and 94 K have been used.

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The conductances of Fig. 1 show some similarities but also large differences. The most striking one is the position of the conductance peak that varies between 25 mV of our break-junction data to 35 mV of STM data of Ref. [9], to 41 mV of STM data of Ref. [8]. The ratios calculated from these values are between 6.25 and 10.3. These differences are really difficult to understand. All the three experiments are done on high-quality single crystals with similar critical temperatures that are practically at the top of the curve of

as function of doping, so it is difficult to believe that doping

differences can be the cause of the different values. One could argue that in the breakjunction technique, the operation of breaking the crystal at low temperature can produce a structural modification of the surfaces of fracture (due to the mechanical stress) with a

consequent reduction of of the junction. This is contradicted by the fact that we have observed very broaden conductance peaks up to 92.8 K in these junctions, as it should occur in a superconductor with in presence of strong fluctuation phenomena [17].

A question now arises: Is a DOS with compatible with the known models for a strong boson-mediated coupling both in s- and in d-wave pair symmetry? In order to investigate the previous point we tried to fit

both our break-junction and STM data by Renner et al. [8] by means of a direct solution of the equations for the retarded electron–boson coupling (Eliashberg equations), both in pure s- and d-wave pair symmetries. As far as concern our recent Bi 2212 data, we reproducibly obtained the electron–boson spectral function by inverse solution of the s-wave Eliashberg equations. Subsequently, we numerically calculated the direct solution of these equations in presence of a normal DOS depending on energy, finding a very good

agreement with experiments as far as is concerned, the shape of the and the critical temperature The extension of these results to the d-wave case is obtained by numerically solving the generalized Eliashberg equations for the order parameter and the renormalization function in a single-band approximation, having expanded in terms of basisfunctions where is the azimuthal angle of the wave-vector k in the ab plane. The expansion is truncated to the first-order term and therefore we have

where and on one side and and on the other are the isotropic and anisotropic parts of the electron–boson spectral function and of the Coulomb pseudopotential, respectively. Here, for simplicity, we pose and where is a constant [18,19]. As a consequence, two coupling constants and are defined and the final symmetry of the solution [i.e., the symmetry of the gap function is determined by their relative values, or if by a specific constraint on the initial values of the isotropic gap. In practice, if the solution has a pure d-wave symmetry in the sense that the gap function keeps only its anisotropic part whereas, at the same time, the renormalization function shows only its isotropic one The role of this isotropic part of the renormalization function is very important in the comparison with experimental data because it has a broadening effect on the quasi-particle DOS similar

380


to that one produced by a lifetime broadening: the DOS peaks become less sharp and thus very similar to the low-temperature conductance curves in Bi 2212. Details on this d-wave strong coupling model and its direct solution in the real energy axis formulation can be found elsewhere [20,21].


We solved the real-axis d-wave Eliashberg equations numerically under the hypotheses described in the previous section by using an iterative procedure and special numerical treatments in order to remove various singularities in the integrals [21]. We want to stress here the importance of the full real-axis solution that is much more complex from the numerical point of view but allows a precise determination of the quasi-particle DOS that is not guaranteed at all the temperatures by the analytical continuation of the imaginary energy axis solution. The electron–boson spectral function of

previously determined in s-wave symmetry, was used as model spectral function for this dwave solution. It means that have been obtained by appropriate scaling of the [α2(Ω)F(Ω )]BSCCO . As a consequence, remain the only free parameters of the model in order to fit the experimental curves.

Figure 2a shows the results of this fit in the case of our break-junction tunneling data at 4.2 K. The continuous line represents the experimental curve, whereas the long dashed line is the theoretical tunneling conductance at 4.2 K obtained by the d-wave strong coupling fit for In this case, the agreement of the theoretical curve with the experiments is really very good in all the energy range but particularly between 10 and 50 mV, where the calculated curve has the same peak (both position and intensity) and the same shape of the experimental one. Some discrepancies still remain at very low bias and at where the experimental conductance shows a more pronounced dip. As far as concern the zero-bias conductance, it can be reproduced by simply adding a small amount of impurities in the unitary limit to the d-wave strongcoupling solution. The short dashed line of Fig. 2a shows the results we obtained in this case by using the same parameters of the long dashed curve and for what concerns the impurities. For the meaning of parameters, please refer to the literature [21–22]. The conductance at zero bias is correctly fitted, even if the peak is slightly reduced in amplitude, whereas the conductance at meV is not modified by the presence of impurities. One of the most striking features of these results is that the critical temperatures calculated by the sum over the Matsubara frequencies are practically the same as the experimental value being 93 K in presence of impurities and 97 K in absence. The results of the same d-wave fit applied to the STM tunneling data of Ref. [8] are shown in Fig. 2b. As in the previous case, the long dash curve is the best fit of the experimental data (continuous line) obtained by solving the d-wave Eliashberg equations with The agreement with the experimental conductance is less impressive than in the previous case, but still good. The peak of the conductance has the proper position and amplitude and the theoretical curve reproduces very well the experimental one at mV, included the famous dip largely discussed in literature. The conductance at is less properly reproduced but, as in the break-junction case, the zero-bias conductance can be fitted by including a small amount of impurities, corresponding to as it is shown by the short dashed curve of

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Fig. 2b. The great surprise occurs when we go to calculate the theoretical of these fitting curves as the temperature where In the case of no impurities, we have

whereas in presence of the impurities we obtain These values are completely different from the critical temperature of the junction mentioned in Ref. [8], In pure s-wave symmetry and by using the same spectral function, we obtain quite similar results as far as is concerned. In this case, the proper fit of is obtained for which corresponds to The impossibility to contemporary fit the gap value and the critical temperature of these STM data on Bi 2212 in the framework of an electron–boson strong-coupling model both in s- and d-wave symmetry is confirmed by the calculation of the ratios

382

as function of of


or

respectively. The ratios have been obtained by separate calculations and are shown in Fig. 3a. The d-wave behavior as function of

for different values (full symbols) is quite different from the s-wave behavior (open circles): In the first case, a maximum of the ratio is present at whereas the classical behavior proportional to occurs in s wave. This maximum of the d-wave ratio increases at the increasing of and consequently of as shown in Fig. 3a. Another interesting observation concerns the values that the ratio assumes at They are lower than the BCS s-wave value 3.53, as already predicted in literature [23]. It is clear from Fig. 3a that the ratio remains under 5.5 for reasonable values of both in s- and d-wave symmetry. This result should be compared to the ratio deduced from the best fit results of Fig. 2b and the experimental

The Quasi-Particle Density of States of Optimally-Doped The conclusion is that, according to our model, the experimental

383 is too low to be

consistent with a conductance peak at 41 mV (or vice versa). One can argue that the present results are strongly dependent on the particular form of the spectral function used in the numerical calculations. This is not the case, as shown in Fig. 3b, where the same d-wave ratio of Fig. 3a as function of for various values is shown, for a spectral function made by a single Einstein peak at meV, which corresponds roughly to the equivalent phonon energy of the The behavior of the ratio as function of is practically unchanged, and the modest increase of its maximum value as function of cannot justify the observed . and values. In conclusion, provided that an appropriate spectral function is used, the real-axis direct solution of the equations for the strong electron–boson coupling in d-wave symmetry demonstrated effectively to fit well the tunneling conductance in all the energy range and the critical temperature of Bi 2212 break-junction tunneling experiments. The same thing does not occur for more recent STM data on the same material that exhibit features (particularly quite incompatible with the d-wave strong coupling model. The reasons for these differences are still under investigation, but due to the generality of the strong electron– boson approach, we believe that explanations related to intrinsic properties of the Bi 2212 samples are rather improbable. In the near future, this d-wave strong electron–boson model will be used to investigate the effect on the tunneling conductance of a peak in the normal DOS around the Fermi energy that has been predicted as a consequence of the presence of stripes in Bi 2212 [24]. REFERENCES 1. D. Mandrus et al., Nature 351, 460 (1991).

2. R. S. Gonnelli et al., in Advances in High-Temperature Superconductivity, eds. D. Andreone, R. S. Gonnelli, and E. Mezzetti (World Scientific, Singapore, 1992); R. S. Gonnelli et al., Physica C 235–240, 1861 (1994).

3. 4. 5. 6. 7. 8.

S. I. Vedeneev et al., Physica C 235–240, 1851 (1994). D. Shimada et at., Phys. Rev. B 51, 16495 (1995). R. S. Gonnelli et al., Physica C 275, 162 (1997). R. S. Gonnelli et al., Physica C 282–287, 1473 (1997). G. A. Ummarino et al., Physica C 282–287, 1501 (1997). Ch. Renner et al., Phys. Rev. Lett. 80, 149 (1998).

9. Y. DeWilde et al., Phys. Rev. Lett. 80, 153 (1998). 10. M. Oda et al., Physica C 281, 135 (1997). 11. H. Hancotte et al., Phys. Rev. B 55, R3410 (1997).

12. D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995). 13. G. M. Eliashberg, Sov. Phys. JETP 3, 696 (1963). 14. J. P. Carbotte, Rev. Mod. Phys. 62, 1028 (1990).

15. P. B. Allen and B. Mitrovich, Theory ofSuperconducting Tc , in Solid State Physics 37 (Academic Press, New York, 1982). 16. R. S. Gonnelli et al., J. Phys. Chem. Solids 59, 10–12, 2058 (1998). 17. A. A. Varlamov et al., cond-mat 9710175. 18. K. Sakai et al., Physica C 279, 127 (1997). 19. H. J. Kaufmann et al., cond-mat 9805108. 20. G. A. Ummarino and R. S. Gonnelli, p. 407 in this volume. 21. G. A. Ummarino and R. S. Gonnelli, unpublished, 1998. 22. C. Jang et al., Phys. Rev. B 47, 5325 (1993).

23. H. Chi and J. P. Carbotte, Phys. Rev. B 49, 6143 (1994). 24. A. Bianconi et al., Physica C 296, 269 (1997).


Long-Range Terms in the Dynamically Screened Potential of R. Grassme1 and P. Seidel1

We show that for almost all directions, the dynamically screened potential contains contributions of longer range than ordinary Friedel oscillations instead of , if an unusually strong Kohn anomaly of the dielectric function appears. We show that such anomalies exist for the theoretical band structure of if the Van Hove singularity lies near the Fermi level (i.e., in the vicinity of an electronic topological transition).

1. INTRODUCTION The dielectric function of metals has singularities or “anomalies” due to the inhomogeneous distribution of electrons in the momentum space. At such anomalies, is continuous but behaves nonanalytically. These anomalies cause some effects in metals like anomalies in the phonon dispersion relation (Kohn effect [1]) and long-range oscillating terms in the screened potential of impurity ions (Friedel oscillations, [2]). We discuss here the influence of such anomalies for high superconductors because even if there are discussions about existence of a real Fermi-liquid, experimental results on band structure [3] support such studies. 2. ANOMALIES OF THE DIELECTRIC FUNCTION We consider screening within the framework of the RPA dielectric function [4]

1

Institut für Festkörperphysik, Friedrich–Schiller–Universität Jena, Helmholtzweg 5, D-07743 Jena, Germany.


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where the formfactor is assumed to be 1. Anomalies of appear for if the pole surface of the integrand in Eq. (1) touches the integraton area, what is equivalent to the following geometrical criterion: behaves nonanalytically for critical wavevectors if one of the shifted isoenergetic surfaces

which are also denoted as auxiliary surfaces, touches the Fermi surface This construction is shown in Fig. 1. In general, these critical wave-vectors form a frequencydependent surface in the space, denoted as surface of anomalies (SOA). The evaluation of the integral in Eq. (1) similar to [7] gives the singular part of the dielectric function

There is the coordinate perpendicular to the SOA, is only a cutoff parameter, is the Heavyside function and is a sign that does not depend on The weight function of the anomaly

contains

the Fermi velocities and the second derivatives of the isoenergetic surfaces at the endpoints of the critical wave-vector via the effective curvature

Long-Range Terms in the Dynamically Screened Potential

387

The matrix in Eq. (5) determines the type of the anomalies, O type: sgn and X type: sgn and furthermore the sign of the O type via Therefore, the anomaly in Eq. (3) depends on local charakteristics of the endpoints of the critical wavevector, lying at the auxiliary and the Fermi surface. 3. STRONGER ANOMALIES: POLE LINES

Independent of the types X and O, there exists a further classification of the anomalies of the dielectric function into Kohn type antiparallel), and Taylor type and parallel), [6]. For Taylor anomalies, there can appear zeros of the denominator in Eq. causing a stronger anomaly. The points with lie at lines, which we denote as pole lines, separating the areas and if there appears a sign reversal at a connected sheet of the SOA. In the static case, pole lines cannot have any influence because the Taylor anomalies cancel each other completely due to symmetry reasons [6]. Also in the case of low frequencies,

the anomalies are considerably weakened. This is due to the fact that Taylor SOA appear in pairs (Fig. 2) that distance tends to zero near the pole line This behavior is changed near an electronic topological transition (ETT), where the Fermi surface changes its topology because a Van Hove singularity (VHS) crosses the

Fermi level. superconductors are near such a transition (VHS scenario of superconductivity [3]). In this case, the relations are no longer valid, and SOA sheets of these pairs can be topologically different. Therefore, an isolated pole line can appear, which is shown for the theoretical bandstructure of A VHS of this compound is the saddlepoint between lies in the vicinity of the Fermi level because the isoenergetic surfaces in Fig. 3a are topologically different for frequencies lower than the Debye frequency of HTSC or Figure 3 shows the construction of a pair of and

form a pair similar to

It [11], which are which would

far from the ETT in Fig. 1. From the Fermi velocities at

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the Fermi surfaces 2 and 4 (Table 1), one can conclude that there lies an isolated pole line between at For the critical wave-vector there is at its top (near the line greater than

at its origin (see Fig. 3). However, the Fermi velocity at the top of can be arbitrarily small (by convenient choice of near the topological transition of For such frequencies is and between these wave-vectors lies a pole line at (see Fig. 3b). Because is far from in Fig. 3c, it must appear isolated. From these considerations, we can conclude that the dielectric function of has extraordinarily strong anomalies for a convenient choice of frequency and doping (what determines , see [3,12]).

Long-Range Terms in the Dynamically Screened Potential

389

4. SCREENING AND POLE LINES Dynamical screening is similar to static screening because the approximation

(6)

using only the scalar potential is allowed for nonrelativistically moving particles with energies up to MeV [13]. Now we consider the charge distribution of an oscillating dipole

which can be used as a model for a vibrating ion in the crystal latice (as a sum of a point charge and a time-dependent dipole). The asymptotic solution (6) with the charge distribution in Eq. (7) is similar to the static case [7]. For a given direction of one gets a sum about “stationary points” in the space (lying at the dynamic SOA), which tangential plane is perpendicular to

The factor is

The amplitude

and the phase shift of these Friedel oscillations depend similar to Eq. (4) on local characteristics (Fermi velocities and curvatures) of the isoenergetic surfaces at the endpoints of the stationary wavevectors For isolated pole lines with the solution in Eq. (8) is not valid. Developing in the vicinity of the pole line (t and are coordinates at the SOA), one gets from Eq. (6) asymptotic potential contributions with a longer range

with

The solution is similar to Eq. (8), but the potential behaves and the sum runs about stationary points at the pole line. It can be shown that such potential terms exist

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for almost all directions (i.e., they fill out a 3D volume area). These potential contributions also have a longer range than that of parabolic lines [14]. Therefore, they can dominate the screened potential. 5. DISCUSSION AND CONCLUSIONS

Near an electronic topological transition there can appear extraordinarily strong anomalies of the dielectric function. These anomalies cause long-range Friedel-oscillation terms for almost all directions in the dynamically screened potential. It behaves like instead of for ordinary metals. An example for such pole lines is demonstrated on the theoretical band structure of where one can expect that vibrating ions in the crystal cause long-range and very anisotropic interactions.

In such cases, the usual approximations in the theory of superconductivity for the interaction of electrons and phonons [using the static dielectric function or short-range screened interaction ] could be invalid and should be reconsidered. Taking into account the periodicity of the lattice, one gets similar results. REFERENCES 1. E. J. Woll and W. Kohn, Phys. Rev. 126, 1693 (1962); B. N. Brookhouse, K. R. Rao, and A. D. B. Woods, Phys. Rev. Lett. 7, 93 (1961); B. N. Brookhouse, T. Arase, G. Cagliotti, K. R. Rao, and A. D. B. Woods, Phys. Rev. 128, 1099 (1962). 2. J. Friedel, Phil. Mag. 43, 153 (1952); J. S. Langer and S. H. Vosko, J. Phys. Chem. Solids 12, 196 (1960); T. J Rowland, Phys. Rev. 119, 900 (1960); M. C. M. M. van der Wielen, A. J. A. van Roji, and H. van Kempen, Phys. Rev. Lett. 76, 1075 (1996). 3. R. S. Markiewicz, J. Phys. Chem. Solids 58, 1179 (1997). 4. L. V. Keldysh, D. A. Kirzhnitz, and A. A. Maradudin, The Dielectric Function of Condensed Systems (Elsevier Science Publishers, 1989). 5. M. I. Kaganov and A. G. Plyavenek, Zh. Éksp. Teor. Fiz. 88, 249 (1985). 6. M. I. Kaganov, A. G. Plyavenek, and M. Hietschold, Zh. Éksp. Teor. Fiz. 82, 2030 (1982). 7. L. M. Roth, H. J. Zeiger, and T. A. Kaplan, Phys. Rev. 149, 519 (1966); A. Blandin, J. Phys. Rad. 22, 507 (1961). 8. W. E. Pickett, R. E. Cohen, and H. Krakauer, Phys. Rev. B 42, 8764 (1990). 9. C. O. Rodriguez, A. 1. Liechtenstein, O. Jepsen, I. Mazin, and O. K. Andersen, Comp. Mat. Sci. 2, 39 (1994). 10. K. A. Müller, Z. Phys. B 80, 193 (1990); B. Renker, F. Gompf, E. Gering, G. Roth, W. Reichardt, D. Ewert, H. Rietschel, and H. Mutka, Z. Phys. B 80, 193 (1988). 11. O. K. Andersen, O. Jepsen, A. I. Liechtenstein, and I. I. Mazin, Phys. Rev. B 49, 4145 (1994); O. K. Andersen, A. I. Liechtenstein, O. Rodriguez, I. I. Mazin, O. Jepsen, V. P. Antropov, O. Gunnarsson, and S. Gopalan, Physica C 185–189, 147 (1991). 12. W. Pickett, Rev. Mod. Phys. 61, 433 (1989). 13. P. M. Echenique, F. J. García de Abajo, V. H. Ponce, and M. E. Uranga, Nucl. Instr. Meth. B 96, 583 (1995); J. L. Mincholé et al., Nucl. Instr. Meth. B 48, 21 (1990). 14. D. I. Golosov and M. I. Kaganov, J. phys.: Cond. Matt. 5, 1481 (1993).

Chemical Analysis of the Superconducting Cuprates by Means of Theory Itai Panas1

An atomistic quantum chemical understanding of superconductivity in the cuprates is articulated. A cluster model is formulated and evaluated by means of the regularized complete active space self-consistent field (reg-CASSCF) method. We quantify charge carriers pairing in one band, local antiferromagnetic (AF) order in a second-band and spin-mediated, nonadiabatic coupling between the two disjoint metallic and magnetic degrees of freedom. The latter resonance is pair-breaking, and becomes spin and symmetry allowed by the “simultaneous” spin excitation in the disjoint local magnetic system. If embedded in a (fluctuating) AF background (i.e., below ), long-range phase coherence necessarily results by means of the local magnetic response initiating virtual spin excitations, the absorption of which elsewhere in the material ensures delocalization of the pair amplitudes.

1. INTRODUCTION A traditional approach to properties of macroscopic objects has been to focus on assumed basic macroscopic key features, from which the phenomenology has been deduced.

We believe that such approaches will never, in a satisfactory way, articulate the origin of

superconductivity in the cuprates. Rather, the cause must be sought in the chemical composition, crystal structure, and unique local spectroscopic properties of the material. As crucial properties of an extremely inhomogeneous gap field are sought, we propose the satisfactory approach to comprise an atomistic cluster model, which displays essences of all key properties of the plane simultaneously. Such an unbiased tool is currently used to extract information regarding which bands provide the charge carrier channels; what is the role of a fluctuating AF order; how phonons affect the electronic and magnetic 1

Department of Inorganic Environmental Chemistry, Chalmers University of Technology, S-412 96 Göteborg, Sweden; E-mail: [email protected]; Fax:


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properties; what is the origin of holes segregation and stripes formation; what causes the order parameter symmetry; what is the mechanism for Cooper pairs formation; and how to understand off-diagonal long-range order (ODLRO) in the cuprates. The nonlocal form of BCS theory is the main source of inspiration for any effort that seeks to connect chemistry and superconductivity in the cuprates. A cluster model is sought, which is able to mimic a resulting self-consistent gap quality [1] of the form

and explicit wave function based ab initio quantum chemistry is used similar to the Bogoliubov equations, only we are able to address the chemical morphology of the local gap field. Presently, increasingly more experimental observations of electronic and lattice instabilities are made in the cuprate materials [2]. Neutron diffraction data are analyzed in terms of dynamic and static stripe phases [3], and theories are developed based on these features being of crucial importance to superconductivity [4]. In this context, the objective of this presentation is twofold as it seeks to demonstrate how a quantum chemical model Hamiltonian for the cuprate superconductors is constructed, how it is evaluated, and outline the microscopic results that emerge from a stripes perspective.

2. CONCEPTUAL BASIS Modeling in quantum chemistry involves choosing electron correlation descriptions and cluster models. These two components define the adequate quantum chemical model Hamiltonian and are described below. The essences of all the sought qualities necessary for successful modeling are included simultaneously and treated at the same level of theory by the model Hamiltonian. The reg-CASSCF method (vide infra) is chosen because near-degeneracy effects are expected to play crucial roles in a system that displays Cooper instability in the charge-carrier channel and superexchange coupling in the magnetic degree of freedom. Similarly, a cluster model is chosen with structure based on the unit cell because crystal symmetry and strong correlation are believed to be essential features in the phenomenology of the cuprates. The particular choice of cluster is a key element in the modeling because the gap field in the cuprates is believed to be strongly inhomogeneous.

2.1. What Is Reg-CASSCF?

The explicit variational wave function–based method, reg-CASSCF, to calculate cluster properties is outlined, following closely Ref. [5]. We start out by writing a sought clusterwave function, ground-state, or any excited state on the form

where each

and

represents a vector in a Fock space, i.e.,

are the occupation numbers of the spin orbitals

The total energy expression for

Chemical Analysis of the Superconducting Cuprates the state

393

thus takes the form

where and are the spin-orbital representations of the one-electron operator (kinetic energy and nuclear attraction) and two-electron (electron repulsion) operators, respectively, i.e.,

and are the reduced one-particle and two-particle density matrices where the spin and space parts of the spin-orbitals are assumed to factorize

are spin-summed excitation operators, defined as

and

The electron configuration basis is chosen to include all near-degenerate independent particle states explicitly in the correlated many-body wave function, and the amplitudes are solved for variationally. Thus, the nonlocal contribution to the electron correlation energy is accounted for. The local electron correlation description uses a Coulomb hole formulation [6], which effectively scales the Coulombic Green’s function by the two-particle density. This is accomplished in a most straightforward manner by expanding the orbitals in Eq. (6) in a gaussian basis

and the scaled electron repulsion operator becomes

where is a specific simple increasing function of the four exponents of the primitive gaussians that span Eq. (6). Particularly, it is noted that

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2.2 Balanced Modeling of Charge Carriers and Magnetism The choice of cluster model displays the chemical and physical understandings of the local connectivities in a compound. Also, if transport of oxygen holes are important, then nearest and next-nearest oxygen neighbors should be included in the cluster model. This description should suffice if low-dispersive bands are addressed [7,8]. Similarly, if antiferromagnetism (AF) is understood to emerge from superexchange interaction via the bridging oxygen ions, then nearest and next-nearest neighbors ions are sufficient for describing the magnetic degree of freedom [9]. If synergism between the charge carrier

and magnetic degrees of freedom is present [10], this effect should emerge spontaneously in the coupled variational calculations employing the reg-CASSCF method described in Section 2.1. The Fock space that corresponds to this understanding is spanned by occupation number vectors in tetragonal symmetry orbitals space (Fig. 1)

where

and

Cluster orbitals

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Effects of local ligand fields from the closest ions are accounted for by including such ions explicitly in the model. 3. ANALYSIS AND CONCLUSIONS The eight-electrons scenario indicated in Eq. (13) emerges from exploratory calculations [7–12] and reflects local signatures of an holes “metal” [7,8] coexisting with a AF in disjoint bands [9] of local and symmetry, respectively (see Fig. 1). The detailed basic signatures of the orbitals that produce the local charge carrier and magnetism physics are described elsewhere [10, 11]. Here, it is simply stated that normal state can be described by a Kondo lattice in the Anderson localization limit. Stability of local antiferromagnetic order in the range of 0.1 eV is calculated and destabilization in the presence of holes quantified. AF stability is recovered by singlet pairing of charge carriers, as oxygen holes can be understood as magnetic impurities vis à vis AF order. The essential physics of the local AF state can be understood in terms of two-electron states spanned by independent particle states of tetragonal symmetry as

where on

and

and

are the probability amplitudes for the two contributions. The dependencies are not shown.

Pairing of charge carriers can be understood as holes cluster states formed by hybridization of and bands (see Fig. 1 and [7,8]). Again, the pair-state is easiest seen if expressed in independent particle states with molecular orbitals symmetrized in accord with the crystal structure

where and are the probability amplitudes for the two contributions. Thus the Cooper instability is realized by mixing near-degenerate independent-particle states of anisotropic S and D symmetries. The dependencies on and are not shown in Eq. (15). Local crystal fields from buffer ions are sensitive parameters controlling the pairing stability (0–60 meV), and this stability increases with increasing lattice field from buffer ions. It is emphasized that only pairing resonances in the energy range of the AF fluctuations contribute to the superconducting ground-state, and that fields that are too strong localize the holes in the system [12], producing the Zhang–Rice scenario [13]. Local signatures of nonadiabaticity in both spin and space symmetry descriptors of the quantum mechanical ground-state are found in the charge carrier channel and magnetic degree of freedom. This results from excitations in the former contributing to the correlated many-body ground state when phase-coherent excitations in the a priori disjoint local magnetic degree of freedom are made. The depairing resonances are of symmetry in Eq. (15), and the magnetic excitation is of symmetry (14). Coupling to the remaining electrons in each of the and subspaces is necessary for

396

Panas

these excitations to become resonances that contribute to the correlated ground-state. The local wave function that emphasizes the additional nonadiabatic coupling can be written schematically as

The “vacuum” state is and comprises all adiabatic contributions—i.e., independent particle states that preserve (i) spin, (ii) space symmetry, and (iii) particle number for each of the factors. and are the probability amplitudes for the adiabatic and nonadiabatic contributions, and and are the local probabilities for paired and depaired charge carriers, respectively. Although holes cluster resonances of the form in Eq. (16) could be interpreted to develop with the opening of the pseudo-gap at it is evident that the local flexibility necessary for the nonadiabatic coupling is not automatically satisfied below the spin-gap temperature at which AF fluctuations have been proposed to develop. In fact, a necessary

requirement for the accessibility of the local virtual pair-breaking excitation in the charge carrier channel below

is the phase-coherent formation of a pair elsewhere in the material,

as conventionally introduced into Eq. (16) by multiplication of by This would occur at and be orchestrated by the emission and absorption of virtual spin excitations in the AF background. This is the physics of the order parameter and the mechanism for achieving ODLRO.

In conclusion, the physical considerations involved in the construction of a detailed microscopic quantum chemical model Hamiltonian for addressing spectroscopic properties of the superconducting cuprates were highlighted. The validity of the approach relies on the gap field being inhomogeneous on the chemical length scale. It was demonstrated (i) how pairing and nonadiabatic depairing contributions mix by coupling to the magnetic background; (ii) how the local accessibility of the nonadiabatic contributions can be understood to require ODLRO; (iii) how the origin of the dominant symmetry of the order parameter emerges from the symmetry of the virtual pair-breaking excitation, in the charge carrier channel, as this resonance can be absorbed in the local correlated ground-state only if use is made of the local flexibility of the disjoint magnetic background; and (iv) how structural instabilities such as stripes [14] become coherent with the existence

of holes clustering resonances. It is gratifying to note how the conventional physical understandings of superconductivity can be articulated from a quantum chemistry perspective. The essential Cooper instability requirement for pairing prevails, although phonon mediated long-range phase coherence is replaced by nonadiabaticity between a priori disjoint local charge carrier and nonlocal AF degrees of freedom. ODLRO results from phasecoherent delocalization of the depairing excitations of symmetry. In the overdoped regime, the stabilities of both the fluctuating AF background and the charge carriers pairing are reduced due to destructive magnetic interband interactions. However, in this limit are deviations from the conventional electron–phonon based phenomenology expected if superconductivity is understood to emerge in materials that display nonadiabatic coupling between two disjoint near-degenerate fermionic subspaces. The formulated holes clustering physics on the chemical length scale becomes consistent with reported mesoscopic stripe structures [14] in the cuprate superconductors.

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ACKNOWLEDGMENTS Support by grants from the Swedish Natural Science Research Council and the Swedish Consortium for Superconductivity are gratefully acknowledged. REFERENCES 1. A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Method of Quantum Field Theory in Statistical Physics (Pergamon, 1963). 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12.

Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375 (1995) 561. See special proceeding on stripes, J. Supercond. 10 (1997). O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. B 57 (1998) 1422. B. O. Roos, Lecture Notes in Quantum Chemistry, ed. B. O. Roos (Springer-Verlag, 1992). I. Panas and A. Snis, Theor. Chim. Acta 97 (1997) 232. I. Panas and R. Gatt, Chem. Phys. Lett. 259 (1996) 241.

I. Panas and R. Gatt, Chem. Phys. Lett. 259 (1996) 247. I. Panas and R. Gatt, Chem. Phys. Lett. 266 (1997) 410. I. Panas and R. Gatt, Chem. Phys. Lett. 270 (1997) 178. I. Panas, unpublished, 1998. I. Panas, T. Johnson, and A. Snis, Intern. J. Mod. Phys. (1998), accepted.

13. F. C. Zhang and T. M. Rice, Phys. Rev. B 37 (1988) 3759. 14. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55 (1997) 12759.


Superconductivity with Antiferromagnetic Background in a Hubbard Model S. Saito,1 S. Kurihara,1 and Y. Y. Suzuki2

We show that in an infinite dimensional

Hubbard model a supercon-

ducting phase exists in the vicinity of the Mott transition. Analyzing the antiferromagnetic (AF) phase using a Gutzwiller-type variational wave function, we show that the compressibility takes negative which, in a naive interpretation, would lead to phase separation. However, this instability should be taken

as a Cooper instability due to the strong attractive interactions between the quasiparticles. We construct a phenomenological theory of the AF Fermi liquid and

determine the corresponding Landau parameters using a microscopic approach. These results indicate the existence of spin waves whose dispersion is given by a linear spectrum which is absent in a Brinkman–Rice Fermi liquid. In addition, one of the Landau parameters becomes negative, indicating that the true ground-state is superconducting. As a result, the compressibility restores a positive value and the phase separation in a normal phase turns out to be an artifact.

1. INTRODUCTION Many superconducting materials, cuprates, bismuthates, organics, and heavy fermion systems have been discovered that might be dominated by strong correlations between electrons. However, on the theoretical side, we still have no consensus about mechanisms for these systems. In particular, we do not have a definite solution as to whether the Hubbard model in higher dimensions, one of the simplest models, has a superconducting phase [1]. Difficulties arise in taking into account various possible instabilities of spin and charge, coming from thermal and/or quantum fluctuations.

1 2

Department of Physics, Waseda University, 3-Okubo, Shinjuku-ku, Tokyo 169-8555, Japan. NTT Basic Research Laboratories, Atsugi 243-0198, Japan.


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Saito, Kurihara, and Suzuki

Significant progress is achieved by taking the infinite dimensional limit lowing Metzner and Vollhardt [2–4]. We present a variational analysis in a model on AB-bipartite hypercubic lattice,

folHubbard

which shows that a superconducting phase exists in the vicinity of the Mott transition. The

first and second terms describe the nearest and next-nearest neighbor hopping, respectively. We take the average band width as an energy unit [2,3]. We assume in numerical evaluations. In describing a Fermi-liquid phase, one of the most successful wave functions is the Gutzwiller wave function [5]. Using it, Brinkman and Rice [6] succeeded in qualitatively understanding the mass enhancement by approaching the metal-insulator transition. Although improved wave function shows that the Brinkman–Rice metal-insulator transition is absent for the perfect nesting case [2,7], it does not lose its classical meanings in the understanding of various experimental systems such as vanadates [8], a normal liquid 3He [9,10], and titanates [11]. However, it does not describe kinetic exchange processes that lead the system to an antiferromagnetic (AF) state in the strong coupling.

2. VARIATIONAL WAVE FUNCTION

We consider the following simple variational wave function in order to describe the quantum phase transition between a Fermi liquid and antiferromagnets,

where g and are variational parameters, and is a noninteracting groundstate. The basic idea of the Gutzwiller wave function is to regard the Hubbard on-site

interaction as a cause of a reduction of the density of the doubly occupied sites, described by the variational parameter g. In addition to it, we adjust the density of the singly occupied sites by the variational parameters In particular, if we take or B sublattice) and it describes the commensurate antiferromagnetism, i.e., these variational parameters act as the molecular field for each lattice site. 3. MOTT TRANSITION

The variational calculations are carried out by using the Metnzer–Vollhardt method [2], which becomes exact in limit. The staggered magnetic moment at half-filling is shown in Fig. 1. Other characteristic results are as follows: (i) near half-filling,

the system exhibits a second-order quantum phase transition from a Fermi liquid to an AF metal; (ii) at half-filling it becomes an insulator, where the Hubbard gap opens up; (iii) there exists the critical interaction strength for ordering and (iv) electron effective mass

Superconductivity with Antiferromagnetic Background

401

remains finite, in contrast to the Brinkman–Rice theory. These results seem to be consistent with recent experiments on titanates by the Tokura et al. [11] and Katsufuji et al. [12]. However, a linear stability analysis on the AF metal shows that the compressibility takes negative which, in a naive interpretation, would lead to phase separation [7].

4. ANTIFERROMAGNETIC FERMI LIQUID In order to understand this instability, we should extend the Landau–Fermi liquid theory to the AF metal, which we call antiferromagnetic Fermi liquid theory. Here, we must pay attention to the absence of adiabatic continuity from a free-electron state, due to the occurrence of the quantum phase transition, i.e., the ordinary Fermi liquid theory has broken down. However, examining an electron-distribution function, a jump at the Fermi surface does exist even in the AF phase, which is consistent with the Luttinger sum rule. It severely restricts movements of quasi-particles within a thin shell of the Fermi surface, and we expect that the ideas of Landau–Fermi liquid theory are still valid. In addition, the up(down) spin quasi-particles move mainly within A(B) sublattice in order to avoid disturbing the AF background so that the resulting quasi-particles are spinless fermions [13]. We have confirmed this fact from changes of the effective mass, as shown in Fig. 2. In general, the sublattice dependent Landau parameters are needed, but owing to spin independence and symmetry, only two of them, and are independent. Then, the change of the free energy due to the response to the external fields can be written as

This is the same as that of Landau, if we regard sublattice indices as spin indices. We phenomenologically obtain the staggered spin susceptibility

and the charge susceptibility

402


as

where and the refers to the corresponding value in the noninteracting state. We can determine these Landau parameters by comparing them to values obtained by the previous variational method. This technique is the same as that of Vollhardt et al. [9,10], who applied the Gutzwiller wave function to a normal liquid 3He, and got results in quantitative agreement with experiments.

5. SPIN WAVES AND SOFTENING The asymmetric Landau parameter thus obtained is shown in Fig. 3. It shows that spin waves can propagate in the AF metal, whose dispersion is given by a linear spectrum which is overdamped in a Brinkman–Rice Fermi liquid. We also note the softening of the spin waves by doping, which is shown in Fig. 4. Physically, we expect an attractive interaction due to the existence of a Fermi surface and the propagation of spin waves, which makes an AF Fermi liquid unstable toward superconductivity.

6. PHASE SEPARATION VS. COOPER INSTABILITY

In Fig. 5, we show the symmetric Landau parameter It assumes a large negative value in the AF metal, indicating a Cooper instability. In fact, the variational wave function


403

404


has a lower energy than phase-separated normal state, where

is a creation operator of

a quasi-particle for X sublattice. It describes a coexistence phase of superconductivity and commensurate antiferromagnetism. The superconducting order parameter depends on the sign of wave for and d wave for By taking into account the superconductivity, we have calculated the charge susceptibility χ c, as shown in Fig. 6. As a result, the compressibility restores a positive value and the phase separation in a normal phase turns out to be an artifact. This kind of restoration was first pointed out by Nozières et al. [14] in the context of a

crossover from a BCS state to a Bose–Einstein condensed state of tightly bound pairs, bipolarons. 7. COEXISTENCE PHASE

In Fig. 7, we show the phase diagram thus obtained. Near half-filling, we find a coexistence phase of superconductivity and commensurate antiferromagnetism. At exactly half-fiiling or for stronger U, the superconductivity is absent, and we have also found the


405

quantum phase transition between a pure AF phase and a coexistence phase. The coexistence of superconductivity and AF order such as stripes or incommensurate magnetic order seems to be recently observed in cuprate superconductors [15,16]. Our phase diagram may be qualitatively related to the organic and/or cuprate superconductors [15,16]. 8. SUMMARY In summary, we made a variational analysis of a Hubbard model and found a strong evidence for the coexistence of superconductivity with antiferromagnetism. We show that the phase separation in the normal phase turns out to be an artifact by ignoring

superconductivity. ACKNOWLEDGMENT

We are grateful to Prof. I. Terasaki for various useful discussions. In particular, we appreciate his pointing out a possible relevance to organic systems. REFERENCES 1. 2. 3. 4.

E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). E. Müller-Hartmann, Z. Phys. B 74, 507 (1989). A. Georges et al., Rev. Mod. Phys. 68, 13 (1996).

5. M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).

6. W. R Brinkman et al., Phys. Rev. B 2, 4302 (1970).

7. 8. 9. 10. 11.

P. Fazekas et al., Z. Phys. B 78, 69 (1990). D. B. McWhan et al., Phys. Rev. Lett. 27, 941 (1971). D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984). D. Vollhardt et al., Phys. Rev. B 35, 6703 (1987). Y. Tokura et al., Phys. Rev. Lett. 70, 2126 (1993).

12. T. Katsufuji et al., Phys. Rev. B 56, 10145 (1997).

13. 14. 15. 16.

E. Dagotto et al., Phys. Rev. Lett. 74, 310 (1995). P. Nozières et al., J. Low Temp. Phys. 59, 195 (1985). J. M. Tranquada et al., Phys. Rev. Lett. 78, 338 (1997). K. Yamada et al., Phys. Rev. B 57, 6165 (1998).


d-Wave Solution of Eliashberg Equations and Tunneling Density of States in Optimally Doped Superconductors G. A. Ummarino1 and R. S. Gonnelli

In this work we discuss the results of the direct solution of equations for the retarded electron–boson interaction (Eliashberg equations) in the case of d-wave symmetry for the pair-wave function and in presence of scattering from impurities.

In order to obtain these results, we used the Eliashberg theory and a spectral function containing an isotropic part and an anisotropic one: For appropriate values of the isotropic

electron–boson coupling constant and the anisotropic one solutions are obtained with only d-wave symmetry for the order parameter and only s-wave symmetry for the renormalization function. The results of our numerical simulations are able to explain the shape of the density of states, the value of the gap, and the critical temperature of optimally doped superconductor

as recently determined in our break-junction tunneling

experiments.

1. INTRODUCTION Since the work of Allen and Dynes [ 1 ], it is well known that there is no formal limitation to the critical temperature one can reach in the framework of Migdal–Eliashberg theory [2–4] provided the electron–boson coupling strength is sufficiently strong. In the past few years, many attempts have been made to fit some of the superconducting properties using this theory, but the great most of the works discussed only the solution in s-wave

pair symmetry. However, there are now many experimental results that strongly suggest the presence of d-wave pair symmetry, at least in layered cuprates.

1

INFM-Dipartimento di Fisica, Politecnico di Torino, 10129 Torino, Italy.


407

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Ummarino and Gonnelli Recent

tunneling measurements on the layered superconductor (Bi 2212) [5–7] yielded high or very high values of the low-temperature gap, gapless features, and a finite value of the quasi-particle density of states (DOS) at zero bias. These facts could be explained, at least partially, by assuming a d-wave pair state that has nodes within the superconducting gap and an appropriate form for the electron–boson spectral function to be used in the direct solution of the Eliashberg equations [6]. In our measurements, the superconducting DOS of optimally doped Bi 2212 single crystals with has been reproducibly obtained by applying an unfolding procedure to the SIS tunneling dI/dV data of break junctions made at 4.2 K [6]. The presence of very thin single crystals (about ) and the technique of the junctions’ realization strongly suggest the presence of a tunneling mainly along the ab planes of the crystals. The DOS showed characteristic features: rather sharp peaks at the energy low values at zero bias, and an almost constant behavior at are present. The tunneling DOS was very symmetric, and reproducible structures were present at The average value of the peak of the DOS was me V. By using these data, we were able to reproducibly determine the electron–boson spectral function of Bi 2212 by inversion of the s-wave Eliashberg equations. In the present paper, we seek to justify these experimental data in the framework of d-wave theory for strong electron–boson coupling and discuss the main features of the direct solution of the Eliashberg equations in this symmetry.

2. MODEL Our starting point are the well-known generalized Eliashberg equations for the renormalization function and the order parameter whose kernels depend on the

retarded interaction the Coulomb interaction and the effective [8–10]. Here we assume for simplicity that k and lie in the ab plane ( plane). Thus we neglect the relatively small band dispersion and the gap in the c direction. The solutions of the full equations for a tight-binding band show that the Green’s functions are sharply peaked at the Fermi surface. Thus it is a good approximation to integrate over

band

normal to the Fermi surface from

We use a single-band approximation where

the Fermi line is nearly a circle and is the azimuthal angle of k in the ab plane. We expand and in terms of basis functions the first few functions of lowest order are and In this paper, we study the following model interaction:

and we try an

solution with

where

d-Wave Solution of Eliashberg Equations

409

In this case the Eliashberg equations in presence of impurities [11] become:

where and In the previous formulas, is the impurity concentration, N(0) is the value of the normal density of states at the Fermi energy, and is the scattering phase shift. Furthermore, we know that

and the quasi-particle DOS is

Obviously‚

where and are the Fermi and Bose functions, respectively, whereas is a cutoff energy. If we want to obtain an solution, we need only to replace the denominator of and The equation for

with

which we have not written, is homogeneous in In the weak-coupling case its only solution is In principle, in the strong-coupling case there is a chance that above some threshold a solution exists with a nonzero We do not consider this rather exotic possibility and instead assume that the stable solution corresponds to for all couplings [9]. In our numerical analysis, we put for simplicity and where is a constant [12]. As a consequence, we define and

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We solved the generalized, real-axis, Eliashberg equations (3–5) in a direct way by using an iterative procedure that continues until all the real and imaginary values of the functions and at a new iteration show differences less than with respect to the values at the previous iteration. Usually, at low temperature, the convergence occurs after a number of iterations between 10 and 15; however, for if the initial values are arbitrary, the convergence is very slow. By using this approach, we found that the symmetry of the final solution depends on the values of the coupling constants and but when the final result depends on the initial values of the gap function.

3. RESULTS By solving in a direct way the real-axis Eliashberg equations we can now check if a couple of and values exist that, together with a proper electron–boson spectral function can reproduce the experimental density of states For doing this, we compare the to the quantity where is calculated from Eq. (6) by using Here, is the electron–boson spectral function we previously determined by the

inverse solution of the s-wave Eliashberg equations applied to the same Bi 2212 breakjunction tunneling data [6] and is the corresponding coupling constant. Of course, this is not a fully consistent procedure, but in absence of a program for the numerical inversion of the d-wave Eliashberg equations, it can be regarded as a first-order approach to the d-wave modelization of tunneling curves. Figure 1 shows our tunneling experimental data (open circles) and the best fit curve at 4 K (solid line) that is obtained for and and yields almost the exact With a small amount of impurities in the unitary limit ( and

d-Wave Solution of Eliashberg Equations

411

) also the zero-bias anomaly is reproduced and It is necessary to emphasize that there are different pairs of and values with and all giving and producing a rather good fit of the experimental data, but the best remains that one shown in Fig. 1. In this case, the imaginary-axis solution of our theoretical model at

that gives a ratio Real

produces

but the values of the gap edge and the DOS peak are rather different: In Fig. 1 we also show the theoretical normalized conductance at different temperatures obtained with the parameters of the best fit. We can see that the peak of the normalized conductance is almost temperature independent; this is a typical strong-coupling effect [13,14]. In Fig. 2a, the plots of and are shown: the last one grows with the temperature and then goes to zero at very abruptly, whereas the first one, in contrast with is very close to the BCS behavior (dashed line). For high values of exists [14] for which at (in our case ) there is an energy value that verifies the expression: Real In this case, the quasi-particle approximation is no more valid and it is

412

Ummarino and Gonnelli

possible to have more than one solution of the equation This fact suggests that, in this situation, the gap edge is no more a relevant physical quantity. Therefore the “gap” seen experimentally is not the gap edge but some average over the frequency of the function with a weight factor depending on the property being measured [14]. As a consequence, we calculated as the temperature where In Fig. 2b, we show the temperature dependencies of the two solutions of the previous equation and of It is remarkable to note that the bifurcation and, consequently, and disappear at The equation instead, always has only one solution. In summary, we have shown that the real-axis direct solution of a simple d-wave model in the framework of the Eliashberg theory for the strong electron–boson coupling reproduces very well our optimally doped Bi 2212 break-junction tunneling curves and their even with an electron–boson spectral function determined by the inversion of the s-wave Eliashberg equations. The next step is the real-axis direct solution of these d-wave equations in presence of an energy-dependent normal density of states. This point could have a great importance in the debate on the role of stripes in superconductivity: It has been predicted, in fact, that the presence of stripes in Bi 2212 would give rise to a peak in the normal DOS around the Fermi energy [15]. ACKNOWLEDGMENTS

We deeply acknowledge the very useful discussions with O. V. Dolgov and S. V. Shulga. REFERENCES 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13.

14. 15.

P. B. Allen and R. C. Dynes, Phys. Rev. B 12 (1975) 905. G. M. Eliashberg, Sov. Phys. JETP 3 (1963) 696. J. P. Carbotte, Rev. Mod. Phys. 62 (1990) 1028. P. B. Allen and B. Mitrovich, Theory of Superconducting in Solid State Physics, Vol. 37 (Academic Press, New York, 1982). D. Shimada et al., Phys. Rev. B 51 (1995) 16495. R. S. Gonnelli et al., Physica C 275 (1997) 162; R. S. Gonnelli et al., Physica C 282–287 (1997) 1473; G. A. Ummarino et al., Physica C 282–287 (1997) 1501. Ch. Renner et al., Phys. Rev. Lett. 80 (1998) 149; Y. DeWilde et al., Phys. Rev. Lett. 80 (1998) 153; H. Hancotte et al., Phys. Rev. B 55 (1997) R3410. A. J. Millis et al., Phys. Rev. B 37 (1988) 4975. C. T. Rieck et al., Phys. Rev. B 41 (1989) 7289; K. A. Musaelian et al., Phys. Rev. B 53 (1996) 3598. O. V. Dolgov and A. A. Golubov, Int. J. Mod. Phys. B 1 (1988) 1089. C. Jiang et al., Phys. Rev. B 47 (1993) 5325. K. Sakai et al., Physica C 279 (1997) 127; H. J. Kaufmann et al., cond-mat 9805108. J. P. Carbotte et al., Phys. Rev. B 33 (1986) 6135; P. B. Allen and D. Rainer, Nature 349 (1991) 396; A. E. Karakazov et al., Solid State Comm. 79 (1991) 329; G. Varelogiannis, Z. Phys. B 104 (1997) 411. C. R. Leavens, Phys. Rev. B 29 (1984) 5178. A. Bianconi et al., Physica C 296 (1997) 269.

Enhancement of Electron–Phonon Coupling in Exotic Superconductors near a Ferroelectric Transition M. Weger1 and M. Peter2

We suggest that the stripe wave-vector can be identified with the electronic Debye screening parameter As a result, the electron– ion potential is greatly enhanced for small q values. This gives rise to an extremely strong coupling scenario that differs from standard strong-coupling theory by having a frequency-dependent coupling, arising from the frequency dependence of The solutions of the Eliashberg equations for this situation are similar to the “standard” solutions along the imaginary (Matsubara) frequency axis, but differ significantly along the real frequency axis, possessing states in the gap, anomalously large values of and other interesting features.

is enhanced significantly, giving a maximum

“typical” values

of about 200 K for

of the unrenormalized coupling constant.

1. INTRODUCTION Conventional superconductivity is attributed to a pairing interaction originating from a phonon-mediated interaction. The motivation for the search of high-temperature supercon-

ductivity in the cuprates was this mechanism, based on estimates of a particularly strong electron–phonon interaction [1]. Nevertheless, almost all the current work in this field attributes the pairing to other interactions, such as Coulomb interactions, mediated by paramagnons, for example [2]. Some of the reasons for the rejection of the phonon mechanism are:

1. The d-wave symmetry of the superconducting gap parameter. 2. The maximum value of due to the phonon-mediated mechanism was estimated at about 30 K [3], and 30 years of research seem to confirm McMillan’s estimate. 1

Racah Institute of Physics, Hebrew University, Jerusalem, Israel. Department of Physics, Geneva University, Geneva, Switzerland.

2


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Weger and Peter

3. The electronic relaxation rate determined from Drude fits, is approximately at frequencies higher than the maximum phonon frequency. This shows that the interactions at this energy range are not phonon mediated. 4. There is almost no isotope effect in optimally doped samples [4]. 5. Obtaining by a phonon-mediated interaction requires abnormally large coupling constants for Electronic band-structure calculations do not substantiate such large values of 6. Large values of would cause large pullings of phonon frequencies that are not observed experimentally [6]. 7. The normal-state properties—namely, transport properties (conductivity, Hall effect,

thermo-electric power, etc.) as well as magnetic ones (NMR Knight shift, etc.)—are highly anomalous. 8. There is a pseudogap observed above

in particular in underdoped samples. The magnitude of the pseudogap is close to that of the superconducting gap at 9. An extended Van Hove singularity (EVHS) is observed in some cuprates [7]. It is generally attributed to a very strong electron–electron interaction. 10. Action at a distance: The superconductvity is due to the planes. Yet replacement of an atom at a large distance from the plane such as replacement of Hg, Tl, or Cu by Bi, Zn, Cd, etc. can reduce

from about 100 K to about 10 K without a

significant effect on the resistivity or Hall constant. Thus, the doping in the plane does not change. A phonon-mediated interaction (as well as a magnetic one) is local. This feature is emphasized in particular by Anderson [8]. 11. is observed in 2D systems. The phonon mechanism is not specific to 2D; Van Hove singularities favor 1D systems (like Bechgaard salts) over 2D ones. 12. The phonon-mediated mechanism is described by the well-known Eliashberg equations in a complete and precise way. Various features of the superconductivity of the cuprates are not in accord with the solutions of the Eliashberg equations. 13. “Smoking gun.” There is no dramatic striking effect pointing out the role of the phonons, like the isotope effect in “normal” superconductivity. Although any individual argument separately is probably not decisive, the accumulated weight of all these arguments seems to preclude the possibility that the phonon-mediated interaction is the primary pairing mechanism.

2. THE NEAR FERROELECTRICITY OF THE PEROVSKITES‚ AND THE STRIPES The perovskites are nearly ferroelectric. The ionic dielectric constant is found to be 30–60 [9]. This dielectric constant screens out the electron–electron Coulomb interactions, and therefore the bare Debye screening parameter becomes: The electron–ion potential is therefore given by: Because of the large value of is enhanced over the “normal” for small q values

At

This causes the electron–phonon interaction to be enhanced and possess a sharp maximum at We attribute the stripes to this maximum in I(q), i.e., claim that The maximum of I(q) is thus given by It thus diverges like when gets large. As a result, the electron–phonon coupling diverges in

Enhancement of Electron-Phonon Coupling

415

but not in (A more detailed calculation is presented in Ref. [11]). This assignment can account for a maximum of of about 200 K for the electron– phonon mechanism [11]. The small value of causes the electron–phonon interaction to cause forward scattering of the electron when it emits or absorbs a phonon; the scattering angle is radians, in contrast to “normal” metals, where and the scattering is nearly isotropic. This forward scattering accounts for the d-wave symmetry of the superconducting gap parameter. It was pointed out previously that a q-dependent I(q) can give rise to d-wave pairing by the phonon-mediated mechanism [12], but in the present

case, the small value of

achieves this without a reduction in the value of

[13].

3. THE WEIGHTED PHONON PROPAGATOR

Because V(q) at small q values is enhanced to so is I(q), and therefore the electron–phonon coupling, which is proportional to is enhanced approximately by Therefore, we replace the phonon propagator in Eliashberg theory by the weighted phonon propagator Here, is the phonon frequency (we take here for simplicity an Einstein spectrum), and is the unrenormalized McMillan coupling parameter that is of order 1 [5]. For we use the Lyddane–Sachs–Teller form [14]: Here, is the frequency of the longitudinal optical phonons, which in the cuprates is about 70–80 me V, and is the frequency of transverse optical phonons, which in the cuprates is in the range 10–20 meV.

is about 2 to 3. Thus, the large observed value of

[9] is

accounted for.

We solved the Eliashberg equations with replaced by Along the imaginary axis, is shown in Fig. 1. It is seen that the shape of is very close to Lorentzian. It is greatly enhanced (over ) at and its width is approximately in contrast with the width of Because the shape of is so close to a Lorentzian, the transition temperature that we calculate numerically is close to the value obtained for the Lorentzian that approximates it, and that can be calculated analytically. An approximate expression, with the electron–ion potential described in Section 2, and for a 2D electron gas, is given by [11]:

Here, is the Bohr radius in the c direction (perpendicular to the plane of the 2D electron gas), m is the band mass of the electrons (assumed constant), and M is the mass of the ion.

For the cuprates, is about 200 K. This expression for takes into account the screening of the very strong electron– electron interaction by the ionic dielectric constant; Empirically, a negligible is derived from tunneling measurements on BKBO [15]. Conceptually, this derivation differs from more “conventional” models in that a very strong effective coupling is considered. By BCS–McMillan theory, For the phonon-mediated mechanism, the cutoff frequency is and . This gives a maximum of about 30 K. To obtain higher values, mechanisms with a higher cutoff frequency are considered, such as excitons, paramagnons, or even the Fermi

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energy. With such a high cu toff, can be moderate, or even smaller than 1. We consider here an opposite scenario; the cutoff frequency is very small whereas the effective coupling constant is very large As a result, is increased by

about a factor of 2.5. The elimination of provides another factor of 3, thus we get a total increase from about 30 K to about 200 K. 4. SOLUTION OF THE ELIASHBERG EQUATIONS ALONG THE

AXIS

Solution of the Eliashberg equations along the axis (Matsubara formalism) is straightforward. This solution provides the value of . However, many physical properties depend on the behavior of the gap function and the renormalization function along the axis. The calculation of these functions is not simple at all. The reason is that has poles very close to the axis. We use a method proposed by Carbotte et al. [16] to extend the solutions analytically from the imaginary axis to the

real one. Here, we have in addition to the poles at (and appropriate extensions in the complex plane), also second-order poles at (and their appropriate extensions). The calculation is somewhat involved and we describe it elsewhere [17]. Here, we just present results for one set of parameters. In Fig. 2, we show the functions


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the quasi-particle density-of-states N(E), and the spectral and the “exotic metal” case The solutions for the normal case are well known [16]. The value of here is small and also the value of is not very large Already for these small values of and the deviations from the normal situation are very large. This is in sharp contrast to the behavior along the imaginary axis, where the shape of density function

for both the “normal” case

is so close to Lorentzian, and also the shapes of

and

are rather normal.

Therefore, it is not surprising that the behavior of the physical properties that depend on the real-axis behavior of like spectroscopic properties, are so unusual. Some specific features of this solution are the ratio is abnormally large; for the case of Fig. 2, The gap here is the solution ofthe equation: This is not the value of at i.e., For the conventional case, is close to because is quite flat for small values of Here it is definitely not, therefore

the difference between and is very large. Although the ratio for strongcoupling exceeds the weak-coupling BCS value of 3.53, it saturates around 11 for an “infinite” value of and for “reasonable” values of it is about 5–6. Here, we obtain for values exceeding the saturation value of the normal case! Experimentally, a ratio was measured in organic superconductors [18]. There are states in the gap, at energies of about and multiples thereof. The presence of such states is a new and unexpected result. It follows from the fact that the analytic structure

of differs essentially from that of Physically, an admixture of phonon states with superconducting-gap amplitude modes was considered by Balseiro and Falicov [19], but not within the framework of the Eliashberg equations. Zeyher and Zwicknagl [6] considered this phenomenon within the framework of the Eliashberg equations. Because here we use a modified phonon propagator, it is not surprising that our results are different from the previously reported ones. Pintschovius et al. [20] find a large shift (8%) of a phonon frequency in the organic superconductor in the superconducting state. Because the gap is and the phonon frequency is 2.2 meV, the shift should have been down, but experimentally the shift is up. Tunneling data [18] suggest the existence of states in the gap at about 2 meV, in accordance with the present calculation. Such low-lying states shift the phonon frequency up, as observed. The zero-bias anomaly sometimes observed in the cuprates [21] may be due to such states; also, a variety of other possibilities exists as well. The value of is extremely large for small values of We find values of about 20. The value of Z is known to renormalize the electronic group velocity: Thus, the group velocity is reduced by this factor, i.e., the renormalized band is flat. The extent in k space, where this renormalization takes place, is given approximately by is the size of the Brillouin zone. For we find i.e., the band is flat over an extent of about half the Brillouin zone, as observed experimentally [7]. The conventional interpretation of this EVHS is a renormalization due to electron–electron interactions [22]. The present calculation shows that an interpretation in terms of the electron–phonon interaction is also possible. 5. DISCUSSION

The discovery of stripes [23], their very wide occurrence, and the near constancy of the stripe wavelength

tor

gives some weight to our conjecture that the stripe wavevec-

is the dressed Debye screening vector, and is a single parameter


419

responsible for many of the unusual properties of the cuprates. There are a dozen or so

serious objections to the validity of the phonon pairing mechanism in the cuprates, involving many diverse physical phenomena (Section 1). If the causes for the objections were independent, the phonon mechanism would be a very unlikely candidate to account for high-temperature superconductivity. However, this is not the case. We see that replacing removes the various objections simultaneously. The various objections depend on one factor: The screening length being the smallest length in the problem, as it is in normal superconductivity. When the screening length is larger than the Bohr radius the inverse Fermi momentum and the interatomic spacing radical qualitative changes occur in the physical properties. In the present work, we briefly discuss some changes in the physical properties—the increase

in

the d-wave symmetry of the gap parameter, the EVHS, the change in the analytic

properties of the Eliashberg equations, etc.—and hint at some other properties—the cutoff

in at causing a pseudogap, for example. Therefore, the objections to the phonon mechanism as the source of high-temperature superconductivity are much weaker than is generally believed. In this interpretation, the stripes are not the cause of superconductivity. A high

is observed in optimally doped samples, where no stripes are seen. Rather, the stripes are an indicator of some unusual behavior in the normal state; and this unusual behavior is (in our opinion) the cause of high The replacement of by is a frequency-dependent renormalization. It causes the coupling constant to be renormalized as well, This is a new feature that does not exist in conventional Eliashberg theory. Transition from weak-coupling BCS theory to strong-coupling Eliashberg–McMillan theory involves the replacement of the bare mass m by the renormalized frequency-dependent mass We claim that the transition from the well-known strong-coupling theory to an extremely strong-coupling scenario requires us to introduce a renormalization of the coupling constant for empirical physical (rather than mathematical) reasons. The failure of the phonon-mediated mechanism to gain serious consideration is probably due to neglect of this factor. A point emphasized by Pietronero et al. [24] is the inapplicability of Migdal’s theorem because the phonon frequency is relatively high and the effective Fermi energy in the neighborhood of the Van Hove singularity is small. Here, the cutoff frequency enters in an essential way into the electronic properties, which are therefore highly anomalous at very low energies. Therefore the condition for the applicability of Migdal’s theorem does not hold, and the applicability of the Eliashberg equations is somewhat questionable. Nevertheless, it is illuminating to see how the Eliashberg formalism works when the dielectricity is introduced into this formalism, admittedly in an ad-hoc way. ACKNOWLEDGMENTS We benefited greatly from discussions with D. J. Scalapino, W. Kohn, M. Onellion, and V. Z. Kresin. REFERENCES 1. J. G. Bednorz and K. A. Muller, Rev. Mod. Phys. 60, 685 (1988). 2. P. Monthoux, A. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992); D. J. Scalapino, Phys. Rep. 250, 329

(1995). 3. W. L. McMillan, Phys. Rev. 167, 331 (1968).

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4. J. P. Franck, in Physical Properties of High Temperature Superconductors IV (World Scientific, 1994), p. 189. 5. S. Massidda et al., Physica C 176, 159 (1991); C. O. Rodriguez et al., Phys. Rev. B 42, 2692 (1990). 6. R. Zeyher and G. Zwicknagl, Z. Phys. B 78, 2692 (1990). 7. J. Ma et al., Phys. Rev. B 51, 3832 (1995).

8. P. W. Anderson, Theory of Superconductivity inthe High Tc Cuprates (Princeton University Press, 1997). 9. D. Reagor et al., Phys. Rev. Lett. 62, 2048 (1989); C. Y. Chen et al., Phys. Rev. B 43, 392 (1991); J. Humlicek

et al., Physica C 206, 345 (1993). 10. 11. 12. 13. 14. 15. 16. 17. 18.

M. Weger, J. Supercond. 10, 435 (1997). M. Weger, M. Peter, and L. P. Pitaevskii, Z. Phys. B 101, 573 (1996); J. Low Temp. Phys. 107, 533 (1997). N. Bulut and D. J. Scalapino, Phys. Rev. B 54, 14971 (1996). G. Santi et al., Physica C 259, 253 (1996). N. W. Ashcroft and N. B. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976), pp. 547–548. Q. Huang et al., Nature 347, 369 (1990). F. Marsiglio, M. Schossmann, and J. P. Carbotte, Phys. Rev. B 37, 4965 (1988). M. Peter, M. Weger, and L. P. Pitaevskii, Ann. Physik 7, 174 (1998). M. Weger, A. Nowack, and D. Schweitzer, Synth. Met. 42, 1885 (1992); G. Ernst et al., Europhys. Lett. 31, 411 (1995).

19. 20. 21. 22.

C. A. Balseiro and L. M. Falicov, Phys. Rev. Lett. 45, 662 (1980). L. Pintchovius et al., Europhys. Lett. 37, 627 (1997). M. Gurvitch et al., Phys. Rev. Lett. 63, 1008 (1989). A. V. Chubukov, Phys. Rev. B 52, R3840 (1995).

23. A. Bianconi, Phys. Rev. Lett. 76, 3412 (1996).

24. L. Pietronero, S. Strassler, and C. Grimaldi, Phys. Rev. B 52, 10516 (1995).

Features of the Transitions in

Structural Phase

M. Arao,1 S. Miyazaki,1 Y. Inoue,2 and Y. Koyama1

Features of the crystal structure in was investigated by means of transmission electron microscopy. The crystal structure in room temperature was

determined to change with increasing the La content x as follows; Pm3m and There coexist the and Pbnm phases in It was also found that the structural phase transition takes place around

1. INTRODUCTION

It is known that (M: 3d transition metal) with the simple perovskite structure exhibits a metal-insulator (M-I) transition and its electronic state is characterized by the highly correlated-electron system. Origins of the M-I transition, including the formation of a charge stripe, have attracted attention. Very recently, the stripe ordering related to the charge and orbital orderings was found in [1]. Among is a paramagnetic metal in and an antiferromagnetic (AF) insulator in [2,3]. That is, the M-I transition occurs around in this oxide. As for the crystal structure in only a crystal system was reported to be cubic in and tetragonal in from the analysis of x-ray powder diffraction profiles [2]. We so far examined crystal structures in by means of electron diffraction and found that the structures at room temperature are divided into three groups [4]. That is, the cubic structure with the space group of Pm3m appears only in and two different structures characterized by the rotational displacements of the oxygen octahedron 1

2

Kagami Memorial Laboratory for Materials Science and Technology, and Department of Materials Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.

NISSAN ARC, Ltd., Yokosuka, Kanagawa 237, Japan.


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Arao, Miyazaki, Inoue, and Koyama

exist in

and in

A final determination of these two structures

unfortunately could not be made in our previous work because of the lack of experimental data. It is obvious that the rotational displacement results in the change in V-O-V bond angle. Because the change in the bond angle leads to a highly correlated-electron state, the structural phase transition should have some relation to the M-I transition. In addition, if two inequivalent V sites are produced in the highly correlated state, the stripe ordering may occur. Then, we have determined the crystal structures in and examined features of a structural phase transition between these structures by transmission electron microscopy. On the basis of experimentally obtained data, we discuss features of the structural transition in

2. EXPERIMENTAL PROCEDURE samples in a whole composition range were prepared from initial powders of and by means of an arc-melting method in an Ar/20% atmosphere. A x-ray powder diffraction profile from each sample made in the present work confirmed that no impurity phase is involved in the sample. Features of its crystal

structure were examined by taking electron diffraction patterns and bright- and dark-field images from a single-crystal region. The observation was made by using both H-700H and H-800 transmission electron microscopes. In particular, the latter microscope with a helium-cooling holder was used for in-situ observation. Specimens for the observation were flakes obtained by crushing the sample.


As pointed out in our previous work [4], three types of the crystal structures in

exist at room temperature. Figures 1a–c show electron diffraction patterns at room temperature that were taken from samples with and 1.0, respectively. An electron incidence of all patterns is parallel to the [301] direction. A reason of the adoption of the [301] incidence is that the pattern with this incidence exhibits typical features of each type of the crystal structure. Diffraction spots are indexed in terms of the cubic perovskite structure with the space group of Pm3m. In the pattern of Fig. 1a, there are only diffraction spots with strong intensities. The spots are obviously due to the simple cubic perovskite structure of the space group of Pm3m. The rotational

displacement is not involved in the structure. Figure 1b is the pattern taken from an sample. The pattern exhibits superlattice reflection spots at as indicated by an arrow A, in addition to the fundamental spots due to the Pm3m structure. Note that the symmetry point at is the R point in the simple cubic Brillouin zone. That is, diffraction patterns obtained from samples with have the same features as those shown in Fig. 1b. The electron diffraction pattern of with is shown in Fig. 1c. As is seen in the pattern, superlattice reflection spots exist at

and

as marked by arrows B and C, respectively. As the M symmetry

point is specified by the superlattice spot at is called the Mtype spot in the present work. The features shown in Fig. 1 c are common in samples with

Features of the


423

In order to understand the details of diffraction patterns in these types of the crystal structure, we examined an extinction rule of superlattice spots in each structure by taking diffraction patterns with a lot of different incidences. As a result, there is no superlattice spot in whereas the superlattice reflection spots exist only at in and at and in From a careful examination of the extinction rule for these superlattice reflection spots, a space group was eventually determined to be Pm3m for

0.6, and Pbnm for in

indicates that the

In addition, there coexist the

for

and Pbnm structures

Both the existence of these superlattice spots and their extinction rule

and Pbnm structures involve the rotational displacement, as pointed

out in our previous paper. From the determined crystal structures mentioned just above, the structural phase transition is expected around in Then we examined a change in the crystal structure in a sample with on cooling from room temperature. Figure 2a and b are, respectively, two electron diffraction patterns at room temperature and 85 K that were taken from the sample. An electron incidence of both patterns is parallel to the [301] direction. In the pattern at room temperature, Fig. 2a, the R-type

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superlattice reflection spots indicating the structure are observed, in addition to the fundamental spots due to the Pm3m structure. When the sample is cooled to 85 K, the

M-type superlattice spots appears in the pattern of Fig. 2b. In order to understand features of a microstructure just after the transition, we then took darkfield images by using the M-type superlattice spot. The darkfield images at 85 K are shown in Fig. 3. An electron incidence is almost parallel to the [301] direction. Note that no change in a microstructure could be detected in brightfield images. In the image, we can see dark-line contrasts in a bright-contrast region that are perpendicular to the [010] direction. Because the contrast is due to diffraction one, the bright-contrast region should have the Pbnm structure. The dark-line contrasts can be then identified as an antiphase boundary with respect to the rotational displacement of the octahedron. From the existence of the large number of the antiphase boundaries, it is further understood that a large number of the Pbnm-structure regions are nucleated and grow in the transition. Eventually, actually undergoes the structural phase transition, just as in other 3d transition metal oxides such as and

4. CONCLUSION The present experimental data shows that when the La content increases, the crystal structure in changes as follows: Pm3m in in 0.6, and Pbnm in The and Pbnm phases coexist in In particular, the and Pbnm structures were confirmed to be characterized by the rotational displacement of the octahedron. In addition, the structural phase transition was found to occur in

on cooling.

ACKNOWLEDGMENTS

The present work was supported by grant-in-aid for Research on Specific Subject from Waseda University (No. 97A-312).

Features of the


425

REFERENCES 1. S. Mori, C. H. Chen, and C.-W. Cheong, Nature 392, 473 (1998). 2. A. V. Mahajan, D. C. Johnston, D. R. Torgeson, and F. Borsa, Phys. Rev. B 46, 10973 (1992). 3. F. Inaba, T. Arima, T. Ishikawa, T. Katsufuji, and Y. Tokura, Phys. Rev. B 52, R2221 (1995). 4. M. Arao, S. Miyazaki, Y. Inoue, and Y. Koyama, in Proceedings of the 10th International Symposium on Superconductivity (ISS ‘97), A. Tanaka and M. Kojima, eds. (Springer-Verlag, Tokyo, 1998), pp. 219–222.


Infrared Signatures of Charge Density Waves in Manganites P. Calvani,1 P. Dore,1 G. De Marzi,1 S. Lupi,1 I. Fedorov,1 P. Maselli,1 and S.-W. Cheong2

Three polycrystalline samples of with 1/2, and 2/3 have been studied in the infrared to investigate the low-energy dynamics of the carriers. At the filling of a gap in the mid-infrared below the Curie temperature monitors the metallization of the sample. In contrast, at a charge density wave (CDW) gap opens below reaches its full value at a finite temperature T0 < TN , when the sample is in its AFM phase. A comparison with the spectra collected for allows reconciliation with the double exchange model the unexpected coexistence of ferromagnetism and

charge ordering at

1. INTRODUCTION The substitution of ions by ions in produces several related effects: (i) x holes per formula unit are injected into the lattice, leading to conversion of x ions into ones and providing carriers for magnetic double exchange; (ii) the oxygen octahedra around the ions lose their Jahn–Teller distortion; and (iii) the hybridization of the Mn and O orbitals changes with x due to the smaller size of Ca ions with respect to that of La ions. The combination of these effects results in a remarkably complicated phase diagram for this manganite [1]. Although the high-temperature state is paramagnetic (PM) at any x, the ground-state of the system changes from an antiferromagnetic (AF) insulator (at ) to a ferromagnetic (FM) metal (at ). According to a widely accepted point of view [2], this transition (as well as the related “colossal 1

Istituto Nazionale di Fisica della Materia and Dipartimento di Fisica, Università di Roma “La Sapienza,” Piazzale A. Moro 2, I-00185 Roma, Italy. 2 Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974, USA, and Department of Physics, Rutgers University, Piscataway, New Jersey 08855, USA.


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Calvani, Dore, De Marzi, Lupi, Fedorov, Maselli, and Cheong

magnetoresistance” phenomenon) is due to the onset of the magnetic double-exchange mechanism, which involves a transfer of Jahn–Teller polarons along paths. At large doping a different combination of the above-mentioned effects again produces an AF ground-state, but with charge ordering. Peculiar phenomena are observed at is PM at room temperature, becomes FM at and, by further cooling (C), turns to an AF state at a Néel temperature [3]. Upon heating (H) the sample, the FM-AF transition is instead observed at X-ray, neutron [4], and electron diffraction [5] provide evidence for quasi-commensurate charge and orbital ordering in the AF phase. The incommensurability increases with temperature and follows the hysteretic behavior of the AF-FM transition, until any trace of ordered superlattices disappears slightly above [5]. In the present paper, we report on the infrared spectra of at 1/2, and 2/3. This allows us to investigate the low-energy dynamics of the carriers in different zones of the phase diagram. In particular, we address the unusual coexistence of ferromagnetism and incommensurate charge ordering in which seems to violate the close relation predicted by the double-exchange model between carrier mobility and FM order. 2. EXPERIMENTAL PROCEDURE

The infrared spectra have been collected on polycrystalline

prepared

as described in Ref. [ 1]. The material has been finely milled, diluted in CsI (1:100 in weight),

and pressed into pellets under a vacuum, as discussed in Ref. [6]. The infrared intensity

transmitted by the pellet containing the oxide and that transmitted by a pure CsI pellet have been measured at the same T. One thus obtains a normalized optical density that, as shown in Refs. [6,7], is proportional to the optical conductivity of the pure perovskite over the frequency range of interest here:

The magnetic susceptibility of the pure powders has been measured in zero static field at the frequency of 127 Hz in a commercial device based on the Hartshorn method.

3. RESULTS AND DISCUSSION The optical density

of

at different temperatures between

300 K and 20 K is shown in the inset of Fig. 1. At the “optimum” doping

the

system reaches its maximum Curie temperature [1]. The susceptibility data reported for the same powder in Fig. 2b give The monotonic decrease in the

absorption at low

is due to the fact that penetration depth

of the infrared

radiation becomes comparable with the average size of perovskite grains dissolved in CsI. This frequency-dependent background can be eliminated by extrapolating to zero frequency

the 300 K absorption spectrum, and by subtracting from the spectra at all temperatures the curve thus obtained. Once this operation is performed, one obtains the optical densities in Fig. 1. The absence of a Drude contribution typical of a free-carrier response is an

artifact due to the small size of the grains [8]. The broad peak in the mid-infrared, even

Infrared Signatures of Charge Density Waves in Manganites

429

if its shape is influenced by the random orientation of the Mn-O conducting planes [9], confirms the polaronic behavior of the carriers pointed out earlier on the basis of reflectivity measurements [10]. One may also notice that, as the temperature is lowered below the mid-infrared absorption increases and the phonon peaks are increasingly shielded by the mobile charges. Both these observations show increasing metallization of the sample. The spectral weight transfer in Fig. 1 can be evaluated by considering the effective number

of carriers

430


where is the real part of the optical conductivity and is a suitable cutoff frequency. By taking into account Eq. (1), one can here define a similar quantity

By using and one selects a range where the scattering effect is negligible and our data are reliable. as obtained from the data of Fig. 1, is plotted in Fig. 2a. The same was chosen in Ref. [ 10 ] to extract from the reflectivity of bulk polycrystallyne below 260 K. Those data, multiplied by a constant factor, are compared with in Fig. 2a. Good agreement between two sets of data confirms the reliability of the present experimental procedure. However, a comparison of with the behavior of in Fig. 2b shows that the infrared insulating gap of disappears only gradually as T is lowered. This is in contrast with the fact that FM order is established in the whole sample just below A similar behavior was observed in the reflectivity spectra [11] of a single crystal of Those spectra, which extend up to 3 eV, showed that the insulating gap was gradually filled below by transfering spectral weight from the high-energy

charge-transfer transitions. The optical density of is reported in Fig. 3 as measured by decreasing monotonically the temperature from 300 K to 20 K. At 300 K the phonon peaks are shielded by the broad background of the mobile holes introduced by Ca doping. Indeed, the dc conductivity of is rather high, at 300 K [1]. The drop in the absorption below is due to the above-mentioned increase in the penetration depth of the radiation at low Between 245 K and 215 K, a minimum appears in the free-carrier background. The minimum gradually deepens with further lowering of the temperature. Moreover, as T is lowered, phonon peaks similar to those in Fig. 1 become

more and more evident, showing that the screening action of the mobile holes weakens. Both this effect and the formation of an infrared gap provide evidence for increasing localization of the carriers. The spectral weight lost in the gap region is transfered to higher frequencies,


431

as shown by the observation that all the absorption curves cross each other at

Indeed, the absence of any coherent peak at is confirmed by the monotonic decrease in the dc conductivity (0) [ 1 ] of between room temperature and The evolution with temperature of the spectral weight in the gap region is shown in Fig. 4, where obtained by Eq. (3) is compared with the real part of the magnetic susceptibility . When cooling the sample, starts decreasing around whereas no abrupt change is observed at the Néel temperature Moreover, this process is completed well below when has reached its minimum value and an AF phase is established in the whole sample. Like the incommensurability parameter and the dc resistivity is also sensitive to the thermal history of the sample. When heating the sample, starts increasing around 120 K, again well below The difference between the infrared absorption spectra measured at 160 K when cooling and heating the sample can be evaluated in the inset of Fig. 4a. The behavior of the infrared absorption of in Fig. 3 at temperatures where incommensurate charge ordering has been reported [5] suggests the formation of a charge density wave (CDW). One can then extract from Fig. 3 the optical gap and compare its behavior with that expected for a CDW. This can be done by considering that at is similar to that of a semiconductor in the presence of direct band-toband transitions [12]. By remembering again Eq. (1), one can then fit to the experimental

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Calvani, Dore, De Marzi, Lupi, Fedorov, Maselli, and Cheong for

the expression:

The curves thus obtained are reported as dashed lines in Fig. 3. They describe well the resolved part of the gap profile at all temperatures with the same The resulting values for are plotted in Fig. 4c. BCS-like fits for have been successfully applied to the optical behavior of CDW in polar systems [13], even if the resulting values for are much higher than that (3.52) expected in the weak coupling approximation. An analytic expression for between and [14], which also holds under moderately strong coupling [15], can be written as

Unlike for an ordinary CDW, here the AF background fully localizes the charges at a finite

which moreover depends on the thermal history of the sample. We find that these effects can be taken into account by simply rescaling the temperature in Eq. (5)

As shown in Fig. 4c, Eq. (6) fits well both series of data by using the same values and One thus obtains and are introduced into Eq. (6) for the cooling and heating cycles, respectively.

An interesting comparison can be done between the gap value measured here and the dc reported for Therein, it follows the exponential behavior The activation energy (marked by the asterisk in Fig. 4c) is in excellent agreement with the present determination of However, a refinement of the present infrared data would allow the investigation of the nature of the gap (or possibly pseudogap) that appears slightly above The close correspondence between charge ordering and antiferromagnetism, pointed out in Fig. 4, is predicted by the double-exchange model. The present confirmation that the

resistivity

charges are partially localized in the FM phase is rather surprising. The results of Fig. 4, in connection with those of electron diffraction in the same powder, may help to find an explanation. As already mentioned, the behavior of the incommensurability parameter (reported in Fig. 2 of Ref. [5]) is quite similar to that of in Fig. 4a. At

240 K, where starts decreasing, weak peaks from a charged superlattice appear in the electron diffraction spectra, corresponding to As T is lowered, decreases to 0.01 at and vanishes below If a closed thermal cycle is performed, exhibits a hysteretic behavior that is quite similar to that of in Fig. 4a. The above-reported observations in the FM phase can point either to an incommensurate and homogeneous CDW or to discommensurations suggestive of phase separation in the charge system. The former possibility seems to be unlikely because at the


433

charge density is intrinsically commensurate with the lattice and because the electron–lattice coupling here is strong enough to create small polarons. We remain therefore with the latter

assumption, which implies a phase separation scenario. One may expect that in the CDW regions an AF phase is established by the superexchange interaction (evidence for such coexistence is provided below for the sample). By assuming that is due to discommensurations between charge-ordered domains, their average size should be [5] where nm is the lattice parameter in the Mn-O planes and is the order of commensuration. This gives at at If this is true, the FM observed below could only take place in the disordered regions that separate the AF clusters. In this context, the similar behavior of and between and can be explained in a simple way. Indeed, all those quantities should be sensitive to the average size of the AF clusters, which increases at the expense of the FM regions as T approaches It should be stressed that the above phase-separation scenario for at is consistent with the present optical data. Indeed, the infrared waves average out any inhomogeneity in the sample on a scale so that the mobile holes in the FM regions may appear as excited states of the ordered charges confined within the AF clusters. In Fig. 3, the mobile holes may produce the Drude-like absorption that partially fills the gap of the CDW condensate. However, in order to confirm that interpretation, one must show that the regions with commensurate charge ordering may produce an infrared gap that behaves according to Eq. (6). For this purpose, we show in Fig. 5 the mid-infrared spectra of This sample exhibits commensurate charge ordering below with wave-vector The spectra of Fig. 5 show the formation of a gap in the infrared background, starting around By extracting as done for the sample with one obtains the plot of Fig. 6. No appreciable effects around the PM-AF transition at are found to influence the gap. is again well fitted by Eq. (6) with (in good agreement with the above value of and

434


One thus obtains

a value close to that observed

in the FM phase of 4. CONCLUSION

The following conclusions can now be drawn. First, the present spectra of show the optical response of CDWs interacting with a magnetic background. Such effects can hardly be observed in low-dimensional systems where CDWs are usually detected. As a consequence, we deal here with the optical response of a CDW that undergoes a hysteretic transition. The temperature dependence of the gap is the same as predicted for a CDW under moderately strong coupling, provided that one renormalizes the temperature scale by introducing a finite

at which the AF order has “frozen in”

all the charges. This represents a novel generalization of the ordinary CDW model and provides information on the strength of the couplings among the three systems involved:

charges, lattice, and spins. In the carriers are strongly coupled with the lattice, as suggested by a good fit to Eq. (6) and by the result However, a weak coupling between the CDW and the spin system is suggested by the behavior of the infrared

gap through the FM-AF transition. Both when cooling and heating the sample through the FM-AF transition, continues to follow a BCS-like law as expected for an ordinary CDW. However, the magnetic hysteresis reflects into the existence of one and of two different “freezing temperatures” Second, the present data confirm that the intriguing coexistence of FM and charge localization observed in at intermediate temperatures does not contradict a (polaronic) double-exchange mechanism, provided that one introduces a phase-separation scenario. AF clusters (where the CDW is commensurate as in the whole sample) are expected to coexist with disordered FM domains. Even in these latters, however, most

charges are confined within a few cells by the Hubbard repulsion at the sites. At the long wavelengths typical of infrared radiation, these poorly mobile holes appear as the excited states of the CDW condensate of the AF clusters.


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ACKNOWLEDGMENTS

We are indebted to Denis Feinberg and Marco Grilli for many helpful discussions.

REFERENCES 1. P. Schiffer, A. P. Ramirez, W. Bao, and S.-W. Cheong, Phys. Rev. Lett. 75, 3336 (1996). 2. A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett. 74, 5144 (1995), and references therein. 3. P. G. Radaelli, D. E. Cox, M. Marezio, S.-W. Cheong, P. E. Schiffer, and A. P. Ramirez, Phys. Rev. Lett. 75, 3336(1996).

4. P. G. Radaelli, D. E. Cox, M. Marezio, and S.-W. Cheong, Phys. Rev. B 55, 3015 (1997). 5. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996). 6. P. Calvani, A. Paolone, P. Dore, S. Lupi, P. Maselli, G. P. Medaglia, and S.-W. Cheong, Phys. Rev. B 54, R9592(1996). 7. A. Paolone, P. Giura, P. Calvani, S. Lupi, and P. Maselli, Physica B 244, 33 (1998).

8. C. H. Rüscher, M. Götte, B. Schmidt, C. Quitmann, and G. Güntherhodt, Physica C 204, 30 (1992). 9. J. Orenstein and D. H. Rapkine, Phys. Rev. Lett. 60, 968 (1988). 10. K. H. Kim, J. H. Jung, and T. W. Noh, preprint. 11. Y. Okimoto, T. Katsufuji, T. Ishikawa, A. Urushibara, T. Arima, and Y. Tokura, Phys. Rev. Lett. 75, 109 (1995). 12. P. A. Lee, T. M. Rice, and P. W. Anderson, Solid State Commun. 14, 703 (1974). 13. T. Katsufuji, T. Tanabe, T. Ishikawa, Y. Fukuda, T. Arima, and Y. Tokura, Phys. Rev. B 54, R14230 (1996).

14. G. Burns, Solid State Physics (Academic Press, London, 1985), p. 649. 15. D. J. Thouless, Phys. Rev. 117, 1256 (1960).

16. A. P. Ramirez, P. Schiffer, S.-W. Cheong, C. H. Chen, W. Bao, T. T. M. Palstra, P. L. Gammel, D. J. Bishop, and B. Zegarski, Phys. Rev. Lett. 76, 3188 (1996).


Recent Results in the Context of Models for Ladders Elbio Dagotto,1 George Martins,1 Claudio Gazza,1 and André Malvezzi1

Recent calculations in the context of ladder systems, but with implications also for two-dimensional (2D) systems, are described here. In particular, efforts are concentrated on the influence of a hole–hole electrostatic repulsion on hole– pair binding in the t–J model on ladders. It is concluded that the pairs are very robust because a nearest-neighbor hole–hole repulsion ofstrength is needed to break the pairs. In the second part of the paper, the spectral function of holes is provided on up to clusters using a new numerical technique. A gap in the spectrum caused by hole binding is observed, as well as flat regions near Implications for studies of materials are discussed.

1. INTRODUCTION The purpose of this paper is to summarize the recent efforts of our group in the study of electronic models for copper–oxide ladder systems. Electrons moving on ladder geometries present very interesting properties that have attracted the attention of the condensed matter community [1]. In addition, several analogies with the two-dimensional (2D) cuprates exist, and in many respects learning about ladders contribute to our understanding of In Section 2, recent work going beyond the simple t–J model introducing hole–hole repulsions is described. This is of importance for theories of ladders and _ because calculations in this context are usually carried out without such a repulsion. In Section 3, the spectral function of holes in doped ladders is shown using clusters as large as A gap is observed in the spectrum as well as flat bands near in close analogy with results for the 2D cuprates. The large clusters can be studied using a novel technique briefly described here. 1

Department of Physics, and National High Magnetic Field Lab, Florida State University, Tallahassee, FL 32306, USA.


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2. INFLUENCE OF HOLE REPULSION ON t–J MODEL HOLE BOUND STATES The discovery of superconductivity under high pressure in Sr14 – xCaxCu24O41 [2] has triggered a considerable effort to understand the physics of ladder materials. Early theoretical studies predicted the existence of superconductivity based on a purely electronic mechanism [3]. The rationale for the hole–hole attraction is that each individual hole distorts the spin–liquid ground state of the undoped ladder, increasing the energy density locally. The damage to the spin arrangement is thus minimized if holes share a common distortion. A variety of calculations [4–6] have confirmed that an effective attraction between holes exists in environments with short-range antiferromagnetic (AF) correlations (both ladders and planes). However, studies of hole pairing in spin liquids are usually performed without the introduction of an intersite hole–hole Coulomb repulsion. The accuracy of such an approximation has been much debated. The neglect of electrostatic interactions is particularly important for values of J / t , such as 0.3–0.4, presumed to be realistic because in this regime, the pair size is small, roughly 2a (where a is the lattice spacing) [5,6]. Nevertheless, note that in the real materials, polarization effects can effectively reduce the strength of the electrostatic repulsion. This effect has been analyzed for the on-site repulsions and [7]. Although the bare couplings are substantially reduced by polarization, their strength remains the dominant scale, and models with no doubly occupied sites capture this effect properly. However, it is unclear if the additional neglect of the hole–hole repulsion at distance a remains a good approximation. Unfortunately, the calculation of the strength of screened Coulombic interactions is difficult. Using results for 2D copper oxides as guidance, a density-functional approach in which the LDA bands are identified with a mean-field solution of the Hubbard model reports a repulsion between a pair of holes at the Cu and O ions [8]. Because the bare value is 7.8 eV, a reduction of a factor 6.5 is effectively achieved by polarization effects. If the trend continues, then the repulsion V at distance a will be (with ). However, other estimations comparing Auger spectroscopy results with cluster calculations using a 3-band Hubbard model report In addition, for two holes in neighboring oxygens was observed [7], suggesting that the hole–hole correlations decay rapidly with distance. For this reason, the above estimation of 0.6 eV should be considered as an upper bound for V. Other studies based on the interaction of two neighboring Zhang–Rice singlets report Finally, approaches in which the dielectric constant is used straightforwardly even at short distances provide Then, current estimations locate as the realistic range for the hole–hole repulsion in a one-band model. These results suggest that the nearest-neighbor (NN) repulsion cannot be neglected in the t–J model. The importance of Coulomb interactions in cuprates has also been remarked before. In particular, they are a central ingredient of the “striped” scenarios [10]. The purpose of our recent publication [11], which is summarized in part in this section, is to discuss a computationally intensive calculation of the effect of intersite hole–hole

repulsion on the hole bound states of the t–J model on ladders. Because the size of the two holes pair on planes and ladders is comparable for the same J / t , our results have consequences also for studies in 2D. The calculation is performed with the DMRG technique [12] as well as using an optimized diagonalization technique in a reduced Hilbert

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space recently introduced [13]. The Hamiltonian used is the t–J model at a realistic coupling supplemented by a hole–hole repulsion where is the hole number operator at site i. The range R of the repulsion was restricted to 1

and lattice spacings in Ref. [11] because the speed of convergence of the numerical techniques (both variational) decreases as R grows. References to previous results in this context can be found in Ref. [11]. Figure 1 contains the hole–hole correlation obtained on clusters with open-boundary conditions (OBC) using DMRG (with up to states per block and truncation error for the case where acts only at a distance of one lattice

spacing (i.e., 0 is a site at the center of the cluster, and the figure shows the hole–hole correlation along the leg opposite where 0 is located (results for the other leg are similar). C(j) intuitively is related with the probability of finding one hole at site j for the case where there is already one at 0. Figures 1a-b corresponds to 2J, and 4J. Here, the results change only slightly as the lattice grows, and the two holes remain close to each other, indicating the existence of a bound state. Apparently a NN repulsion already larger than J is not enough to destroy the bound state, although it weakens it. This repulsion does not seem to cover the full spatial range of the effective attraction regulated by On the other hand, Fig. 1c-d shows similar results, but now for and 10J where a substantial change in the hole distribution is observed as the cluster grows. The spreading of the hole over the whole lattice suggests either the breaking of the pair or a weak bound state. In the

large-V regime, one of the holes may act as a sharp “wall” to the other, which tries to spread its wave function in an effective square-well potential. A better estimation of the critical value at which pairs are no longer formed can be found using the binding energy defined as where E(nh) is the lowest energy in the subspace of n holes. A negative implies that a bound state exists. In the absence of intersite Coulomb repulsion and at there is pairing of

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holes on 2-leg ladders, and we want to analyze what happens as V grows. Figure 2a shows that for an NN repulsion, the binding effect continues even up to large values of V/ J. Actually the region corresponds to a weakly bound state. The same figure shows results for range repulsion. In this case, the critical coupling at which the bound state is lost is between 3 and 4. Figure 2b shows the critical coupling for two ranges of Coulomb interactions [11].

Supplementing this information with the known existence of hole pairs in the pure t–J model at . as well as when independently of the range, allow us to obtain a rough estimation of the region of hole pair formation by simple interpolation.

Figure 2b is a qualitative plot that summarizes the main result of the paper—namely, the stability of the hole-bound states in t–J-like models depends on the value and range of the electrostatic Coulomb interaction between holes. When the Coulombic term is restricted to a realistic range a repulsion as large as weakens but does not destroy the pair, implying that the effective range of attraction caused by spin polarization is larger than one lattice spacing. Retardation effects (fully considered in the present calculation)

due to the different energy scales of spin and charge excitations (J vs t) likely contribute to the strong stability of the bound states [ 1 1 ] . Note also that using, for example, the pair size is This result is close to estimations of the coherence length and for optimally doped La – 214 and YBCO, respectively [14] (using To the extent that are similar on planes and ladders, apparently a realistic NN hole–hole repulsion can actually improve quantitatively the predictions of the t–J model, without destroying the pairs. Summarizing, in this section and Ref. [ 1 1 ] it has been shown that the bound-state of two holes in ladders is more stable than naively expected upon the introduction of a NN Coulomb repulsion among holes. This result provides support to theories of ladders that predict hole pairing based on electronic mechanisms that describe holes as immersed in spin-liquid


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backgrounds [3] (although certainly an explicit calculation of pair–pair correlations with and at finite hole density is needed to fully confirm these theories). It also reinforces striped scenarios for cuprates in which pairing is produced by carriers moving from the fluctuating stripes to the ladders between them [10]. Regarding 2D systems, our results show that the effect of a Coulomb interaction is a subtle quantitative problem. Actually, the

binding energy of two holes on a 4-leg ladder with has also been estimated by our group. The result is (using a cluster with states and truncation error which is similar to previous estimations in two dimensions (Fig. 24 of Ref. [4]). Then, the influence of the NN Coulomb interaction on planes stronger than on ladders. In this respect, it is imperative to obtain either experimental or theoretical information about the range and strength of the Coulomb interaction in effective one-band models for cuprates, specially in two dimensions, refining existing techniques [7,8]. Figure 2b shows that this information is crucial to support electronic hole-pairing mechanisms based on the existence of short-range AF fluctuations.

3. DENSITY EVOLUTION OF THE QUASI-PARTICLE BAND IN DOPED LADDERS

Recently, the first angle-resolved photoemission (ARPES) studies of ladder materials were reported. Both doped and undoped has been analyzed, finding onedimensional (1D) metallic characteristics [15]. Studies of another ladder compound, found similarities with LSCO, including a Fermi edge [16]. Core-level photoemission experiments for documented its chemical shift against the

hole concentration [17]. Note that the importance of ARPES studies for other materials such as the cuprates is by now clearly established. Using this technique, the evolution with doping of the Fermi surface has been discussed, including the existence of flat bands near momenta These plethora of experimental results for the cuprates should be compared against theoretical predictions. However, the calculation of the ARPES response even for simple models is difficult. The most reliable computational tools for these calculations are the exact diagonalization (ED) method, restricted to small clusters; and the Quantum Monte Carlo (QMC) technique supplemented by maximum entropy, limited in doped systems to high temperatures due to the sign problem. A list of references on the application of these techniques to ladder systems can be found in Ref. [19]. Due to the limitations of these algorithms, an important issue still unclear is the evolution of the quasi-particle dispersion between the undoped limit, dominated by strong AF fluctuations both on ladders and planes,

and the high hole-density regime in which magnetic fluctuations are negligible. Motivated by this challenging problem, the density evolution of the spectral function of 2-leg t–J ladders was presented in Ref. [19]. The calculation is carried out at zero temperature on clusters with up to sites, increasing by a substantial factor the current resolution of the ED techniques. These intermediate-size clusters were reached by working with a small fraction of the total Hilbert space of the system. The method is variational, although accurate, as shown below. The improvement over previous efforts discussed here lies in the procedure used to select the basis states of the problem. The idea is in the same spirit as any technique of the renormalization-group (RG) family. The details are as follows: if the standard basis is used (3 states per site), experience shows that a large number of states is needed to reproduce qualitatively the spin-liquid characteristics of the undoped

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ladders. The reason is that in the basis, one of the states with the highest weight in the ground-state is still the Néel state, in spite of the existence of a short AF correlation length A small basis built up around the Néel state incorrectly favors long-range spin order. However, if the Hamiltonian of the problem is exactly rewritten in, for example, the rung-basis (9 states/rung for the t–J model) before the expansion of the Hilbert space is performed, then the tendency to favor a small is natural because one of the dominant states in this basis for the undoped case corresponds to the direct product of singlets in each rung, which has along the chains. Fluctuations of the resonant-valence-bond (RVB) variety around appear naturally in this new representation of the Hamiltonian, leading to a finite Note that is just one state of the rung basis, whereas in the basis it is represented by states, with the number of rungs of the 2-leg ladder. In general, a few states in the rung basis are equivalent to a large number of states in the basis. Expanding the Hilbert space [20] in the new representation is equivalent to working in the basis with a number of states larger than can be reached directly with presentday computers. For simplicity, this technique is referred to as the optimized reduced basis approximation (ORBA). In Ref. [19] it was shown that by calculating equal-time correlations (in particular, the energy), a good agreement was found between DMRG results and those coming from ORBA (details can be found in that reference). Then, we concentrate directly on the main results. An interesting advantage of the method proposed here is that having a good approximation to the ground-state expressed in a simple enough basis allows us to obtain dynamical information without major complications. The actual procedure is simple: Consider that an operator (which could be a spin, charge, or current operator) is applied to the groundstate in the reduced basis denoted by If all states of the subspace generated by

the operation

were kept in the process, typically one would exceed the memory

capabilities of present-day workstations if the truncated ground-state has about states. Then, it is convenient to work with just a fraction of say keeping about 10% of the states. In this way the subspace under investigation typically has a similar size as the original reduced-basis ground-state, namely approximately rung-basis states. The

state constructed by this procedure is now used as the starting configuration for a standard continued fraction expansion generation of the dynamical response associated to Test of this approach are documented in Refs. [13,19]. In Figs. 3 and 4, the main results of Ref. [19] are reproduced. Only data for the bonding band are discussed here. The -functions are given a width 0.1t throughout the paper. Figure 3a corresponds to the undoped limit. A sharp quasi-particle (qp) peak is observed at the top of the PES spectra, maximized at momenta that is, close to the Fermi momentum for noninteracting electrons The qp has a very small bandwidth, because it occurs in 2D models, due to the interaction of the injected holes with the spin background [4]. Note that all peaks at momenta carry a similar weight and the dispersion in this regime is almost negligible. This unusual result is caused by strong correlation effects. The PES weight above (e.g., at is induced by the finite but robust and its existence resembles the antiferromagnetically induced features discussed before in 2D models. Figure 3b contains results at a low but finite hole density. Several interesting details are observed: (i) the PES band near continues being very flat; (ii) PES has lost (gained) weight compared with (iii) the total PES qp bandwidth has increased; and (iv) the IPES band is intense near and it is separated from the PES band by a gap. The observed gap is and is caused by


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hole pairing. Actually, the binding energy calculated with DMRG/PBC for the same cluster and density is truncation error In the overall energy scale of the ARPES spectra, this difference is small and does not affect the study of the evolution

of the quasi-particle dispersion shown here. Note that the results of Fig. 3b are similar to those observed experimentally near in the 2D cuprates’ normal state using ARPES methods [18].

Figure 4a contains the weight of the qp peaks in the PES band vs density. Size effects are small. The weight at diminishes rapidly with x, following the strength of the spin correlations of Fig. 1c. This result and those in previous figures clearly show that the region affected the most by spin correlations is approximately Figure 4b summarizes the main result of Ref. [19], providing to the reader the evolution of the ladder qp band with x. The areas of the circles are proportional to the peak intensities. At small x a hole pairing– induced gap centered at is present in the spectrum, both the PES and IPES spectra are flat near and the qp band is narrow. The PES flat regions at high momenta exist also

in the undoped limit, where they are caused by the short-range spin correlations. Actually, the undoped and lightly doped regimes are smoothly connected. As x grows to the flat regions rapidly loose intensity near and the gap collapses. The similarities between ladders and planes imply that our results are also of relevance for 2D systems along the line For instance, the abnormally flat regions near induced by hole pairing (Fig. 3b) are in good agreement with ARPES experiments for the 2D cuprates [18], and they should appear in high-resolution photoemission experiments for ladders as well. Note that in the regime studied here with pairs in the ground-state, the flat bands do not cross with doping but simply melt. When x is between 0.3 and 0.4, a quasi-free dispersion is recovered. The results of Fig. 4b resemble a Fermi level crossing at

and beyond, whereas at small hole density no crossing is observed. It is remarkable that these same qualitative behavior appeared in the ARPES results observed recently in underdoped and overdoped LSCO [21]. These common trends on ladders and planes suggest

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that the pseudogap of the latter may be caused by long-lived tight hole pairs in the normal state (as for the doped ladders studied here). However, note that the hole pairs themselves are caused by the spin–liquid RVB character of the ladder ground state. Then, this idea brings together the “preformed-pair” and “magnetic” scenarios for the high- pseudogap. We propose that the presence of d-wave hole-pairs in the normal state induces an ARPES gap at but these pairs exist as long as the is nonnegligible. In this mixed scenario, the pseudogap and the short-range spin fluctuations are correlated. Summarizing this section, the predicted ladder ARPES spectra along are remarkably similar to experimental results for the 2D cuprates along the same line. A common explanation for these features was proposed. ARPES experiments for ladders should observe flat bands and gap features near in the normal state. Finally, note that the novel numerical method discussed in Refs. [13,19] introduces a new way to calculate dynamical properties of spin and hole models on intermediate-size clusters.

ACKNOWLEDGMENTS E.D. is supported by NSF under grant DMR-95-20776. Additional support is provided by the National High Magnetic Field Lab and MARTECH. REFERENCES 1. B. Levy, Phys. Today, October 1996, p. 17. See also “Physics News in 1996,” supplement to APS News,

May 1997, p. 13. 2. M. Uehara et al., J. Phys. Soc. Jpn. 65, 2764 (1996).

3. E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B 45, 5744 (1992); T. Barnes et al., Phys. Rev. B 47, 3196 (1993); E. Dagotto and T. M. Rice, Science 271, 618 (1996), and references therein.


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4. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994); and references therein. 5. D. Poilblanc et al., Phys. Rev. B 49, 12318 (1994), and references therein; H. Tsunetsugu, M. Troyer, and T. M. Rice, Phys. Rev. B 49, 16078 (1994).

6. S. White and D. Scalapino, Phys. Rev. B 55, 6504 (1997), and references therein. 7. L. H. Tjeng, H. Eskes, and G. A. Sawatzky, in Strong Correlation in Superconductivity (Vol. 89), ed. H. Fukuyama, S. Maekawa, and A. P. Malozemoff, Springer Series in Solid-State Sciences, (1989). See also D. K. G. de Boer et al., Phys. Rev. B 29, 4401 (1984). 8. M. S. Hybertsen et al., Phys. Rev. B 39, 9028 (1989). 9. F. Barriquand and G. A. Sawatzky, Phys. Rev. B 50, 16649 (1994).

10. 11. 12. 13.

V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 56, 6120 (1997), and references therein. C. Gazza, G. Martins, J. Riera, and E. Dagotto, preprint, cond-mat/9803314. S. R. White, Phys. Rev. Lett. 69, 2863 (1992). E. Dagotto, G. Martins, J. Riera, A. Malvezzi, and C. Gazza, preprint.

14. M. Cyrot and D. Pavuna, Introduction to Superconductivity and High- Materials (World Scientific, 1992). 15. T. Takahashi et al., Phys. Rev. B 56, 7870 (1997); T. Sato et al., to appear in J. Phys. Chem. Solids 59, 1918 (1998). 16. T. Mizokawa et al., Phys. Rev. B 55, R13373 (1997). 17. T. Mizokawa et al., preprint. 18. D. S. Dessau et al., Phys. Rev. Lett. 71, 2781 (1993); K.Gofron et al., J. Phys. Chem. Solids 54, 1193 (1993). 19. G. Martins, C. Gazza, and E. Dagotto, preprint.

20. J. Riera and E. Dagotto, Phys. Rev. B 47, 15346 (1993); ibid. B 48, 9515(1993); and references therein. 21. A. Ino, C. Kim, T. Mizokawa, Z.-X. Shen, A. Fujimori, M. Takaba, K. Tamasaku, H. Eisaki,and S. Uchida, preprint.


Charge-Ordered States in Doped AFMs: Long-Range “Casimir” Attraction and Instability Daniel W. Hone,1 Steven Kivelson,2 and Leonid P. Pryadko2

We analyze the induced interactions between localized holes in weakly doped Heisenberg antiferromagnets due to the modification of the quantum zero-point spin–wave energy (i.e., the analog of the Casimir effect). We show that this interaction is uniformly attractive and falls off asymptotically as in d dimensions. For “stripes” (i.e., parallel d – 1 domensional hypersurfaces of localized holes), the interaction energy per unit hyperarea is attractive and falls, generically, like We argue that, in the absence of a long-range Coulomb repulsion between holes, these interactions lead to an instability of any charge-ordered state in the dilute doping limit.

1. INTRODUCTION

The extensive recent interest in doped quantum antiferromagnets (AF), particularly in two dimensions, has been provoked in large part by the drive to understand the evolution with hole doping of the layered oxide materials from AF Mott insulators to high-temperature superconductors [1]. However, the behavior of these and other antiferromagnets, including coupled-chain materials, are also of intrinsic interest, apart from their potential for understanding high- superconductivity. One class of proposals [2–4] for the ground-state of the doped antiferromagnets involves spatially inhomogeneous “charge ordering.” Unfortunately, numerical analysis of the stability of such states is often inconclusive because the typical energy differences between states are small and the Goldstone modes (spin waves) produce finite-size effects that decrease slowly with system size. The necessary limitation 1 2

Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106. Department of Physics & Astronomy, University of California, Los Angeles, CA 90095.



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of numerical studies to relatively small systems is especially problematic in the region of low hole doping, which is of special interest to us here. The goal of this paper is to investigate the stability of static charge-ordered groundstates in AFs with short-range exchange in the dilute doping limit. Clustering of spin defects (e.g., holes) minimizes the loss of AF exchange energy, but this must compete with

the gain by mobility of delocalization (kinetic) energy and with the Coulomb repulsion between charged defects. We are concerned here with situations in which the kinetic effects

are ultimately overcome by impurity potentials or by the self-consistent fields resulting from the short-range interactions in the system (as in Hartree–Fock solutions [2] or due to polaronic effects), so that the holes are indeed localized. We emphasize that this does not limit us to models in which there is little or no hopping of holes within the underlying Hamiltonian (such as a t–J model with very small ratio t / J ) . We also limit ourselves to Coulomb interactions that are at most short ranged from screening effects, as in the

commonly studied Hubbard model, for example. We calculate the induced interaction between localized holes due to their modification of the spin-wave spectrum, and find that it is uniformly attractive. Specifically, the asymptotic long-distance interaction between two isolated clusters of holes (Fig. 1) or that between extended clusters of holes, such as stripes, falls off as a power of r so that, in the absence of long-range Coulomb repulsion between holes, all charge-ordered states with sufficiently small hole concentration are unstable to phase separation. At smaller separations between clusters, various repulsive effects, including both Coulomb and kinetically induced interactions, may well establish stable charge-ordered configurations, suggesting the potential stability of charge-ordered ground-states for sufficiently high hole concentrations [3,4]. 2. MODEL

The number of spin degrees of freedom changes with doping, and therefore the Hilbert spaces appropriate to the doped and undoped AF are different. For mathematical convenience and to make possible a simple perturbation Hamiltonian formalism, it is preferable to treat a model with a spin S operator on every site and treat any system with localized holes as a limiting case in which the coupling between a set of “impurity” sites and its neighbors goes to zero. Effects of virtual hole hopping can be included with a larger set of modified exchange interactions in the neighborhood of the holes. Then, the spin Hamil-

tonian of the doped system differs from that of the pure AF only in the strength of some

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exchange couplings:

where is the Hamiltonian for the perfect antiferromagnet, which, for concreteness, we take to have only nearest-neighbor interactions on a hypercubic lattice,

and the perturbation Hamiltonian specifies a set of pairwise exchange interactions such that, in the limit a spin near which the hole is localized is disconnected from the rest of the system. Clearly, the interaction energy between hole clusters is obtained correctly

in this limit, although the cluster self-energy could depend on the interactions between the fictitious disconnected spins. One exceptional geometry that we treat differently is a stripe that is simultaneously an antiphase domain wall in the AF order. Such a stripe can be treated,

as shown in Fig. 2, effectively as a wall of bonds with altered exchange coupling so that we work in the proper Hilbert space from the beginning.

3. CASIMIR ATTRACTION BETWEEN STRIPES The simplest situation to treat is the interaction between parallel walls of localized holes (i.e., codimension 1 hypersurfaces, which with the case of in mind, we refer to as stripes). This is the direct analog of the Casimir effect [5], the attraction between metal

plates due to modified quantum fluctuations of the electromagnetic field between them. The distance dependence in general dimension d can be ascertained by dimensional analysis. For photons, or for any field whose quanta are described by a linear dispersion the only relevant energy scale (at for walls separated by a distance r is and the relevant energy per unit wall hyperarea (which therefore must be proportional to Although the dispersion relation for AF magnons is more complicated than this, the asymptotic long-distance interaction is governed by the long wavelength magnons whose dispersion is linear and for which we then expect to find the above general Casimir result, A specific analytic calculation simply sums the zero point energies over all modes. The stripes of holes restrict wave-vector components perpendicular to the walls to

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integer multiples of

so the distance dependent part of the energy per unit hyperarea is

which is proportional to the spin-wave velocity, c (in linear spin wave approximation [6]

4. INTERACTIONS BETWEEN ISOLATED CLUSTERS

For more general geometries of isolated hole clusters, we can come to no general conclusions based on dimensional analysis (the energy scale is set by the exchange J, but

there are now two lengths, the separation r and the size of the cluster, of order the lattice constant). In general, we can formally calculate the change in ground-state energy due to two hole clusters via the Feynman–Hellman formula, as the integral over coupling constant

of the ground-state expectation value at each

of the perturbation Hamiltonian

where the subscript on the expectation value is a reminder that it is to be taken in the full

ground-state appropriate to that value of the coupling constant. However, that is practical for giving analytic results only for an independent magnon Hamiltonian. The simplest of these is the linear spin wave (LSW) approximation, which is quantitatively accurate for

large S, but which we also expect to be a reliable method for extracting the long-distance physics for even for because already in AF order is very robust [7]. Within this approximation, the perturbation Hamiltonian is quadratic in the magnon creation and destruction operators, so the coupling constant integrand can be expressed directly

in terms of one-magnon Green’s functions. These can, in turn, be calculated exactly by standard methods for spatially localized perturbations. Symbolically, for a perturbation V with nonzero matrix elements in a spatial basis for a limited number of sites, the Dyson equation for suitable magnon Green’s functions G is

where m, n are summed only over those sites for which V has finite matrix elements, so

that G can be found by the inversion of the finite matrix over this limited space. Clearly, the simplest example algebraically is that of two isolated holes at a separation r with finite matrix elements of the perturbation only on the space spanned by each of the two hole sites and its z nearest neighbors. Then the matrix inversion must be done in a subspace of size The Dyson Eq. (5) describes propagation between sites i and j as a series of terms of the form free propagation to one of the two perturbation centers, followed


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by a series of multiple scatterings from that and the other such center, with free propagation between them and finally to the site j. For the multiple scattering from one of the centers, the problem block diagonalizes in a representation based on the point symmetry of that center (generalization of a “partial wave” analysis) [8], in this case the point symmetry of the lattice. The totally symmetric representation (“s-wave scattering”) plays a special role in that. The isolated artificial spin at the hole site is free to rotate, a zero energy bound state that implies that the determinant vanishes at zero frequency. By continuity, there is a scattering resonance at zero energy in the only channel with amplitude on the hole site, namely the s-wave. This term dominates the long-distance Casimir attraction, for which we find, in arbitrary dimension d,

Although the general finite cluster of holes does not share the full high symmetry of this special case, we assert that the asymptotic distance dependence of the Casimir attraction will be of the same form. The simplest argument is that the interaction should have the same form as the sum of the interactions between each pair of holes that would obtain if that pair were isolated as above. We point out that the same reasoning does correctly lead to the Casimir form for the interaction between stripes in d dimensions. However, it is both more rigorous and also suitable for making generalizations to less-restrictive models and approximations to generate this result by the following perturbative calculation. The artificial spins on the isolated cluster are again free to rotate, there are zero energy bound states, and there will be magnon-scattering resonances from the cluster at zero energy. However, we can remove that resonant behavior by adding a large magnetic field only to the fictitious spins on the hole sites, removing them dynamically from the problem at the start. Because these spins are fictitious, physically they must have no dynamical impact on the results; it cannot matter how we impose these local fields on them. Then the multiple scattering from a single cluster has no singular behavior at low energies. It is useful to resum the Dyson equation partially so as to display explicitly the full scattering matrix from each cluster. Within the two perturbation clusters (labeled 1 and 2), we denote the various Green’s functions by the obvious notation between two sites in cluster 1, between sites one of which lies in each cluster, etc. The ground-state energy shift is given by Eq. (4) as an integral over correlation, or Green’s, functions within a single cluster only, for which we write the Dyson equation in the form

where we have suppressed all specific site indices and the sums over them, and is the full multiple scattering matrix of the single cluster The pure crystal Green’s functions fall off rapidly with distance r between the clusters, so the leading r-dependent term for clusters separated by a large distance r is the one involving only two such factors, giving as the leading term in the interaction (“Casimir”) energy

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coupling constant integrals over the following sort of Green’s functions:

In the last line, we have taken advantage of the assumption that the single cluster scattering matrix is nonsingular, and in fact featureless at low frequency. (Note also that we have taken advantage of an imaginary time formalism to keep the denominators of the free propagators positive definite.) These arguments continue to hold when some of our simplifying assumptions are relaxed. Beyond the LSW approximation, the magnons are renormalized, but that only modifies the magnon velocity c for the long wavelength excitations that determine the above behavior. We have also considered the effects of virtual hopping of the spins to unoccupied (hole) sites. If, for example, in Fig. 3 the hole at the center virtually hops to the site above it and back, the displaced spin in the intermediate state interacts antiferromagnetically with the three remaining near neighbors to the center. This effectively introduces additional AF exchange coupling between the “moving” spin and each of those three neighbors. Those are indicated by two diagonal and one vertical dashed bond lines in the figure. Other dashed lines indicate some of the other bonds in the neighborhood of the hole that are modified. In all cases, the tendency is to weaken the AF order. Again, however, the asymptotic distance dependence of the intercluster attraction is unchanged. We discussed above the case of insulating stripes that isolate the AF region between them. However, another situation of possible relevance is unsaturated extended hole configurations, like stripes, along which the holes are mobile. Their effect may be modeled by a reduced but nonvanishing AF exchange between the spins on either side, as shown in Fig. 2. Within the above model, this amounts to a final value of coupling constant less than unity. In the general case (where, for example, the exchange extends beyond nearest neighbors) the above analysis leads again to the asymptotic behavior (see also the discussion in Ref. [2]).


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5. CONCLUSIONS

We have considered Casimir interactions between well-separated hole clusters in AFs. For hole clusters or stripes in a uniform AF, this energy is uniformly attractive and generally falls off with distance as and respectively. The interaction is quantitatively weak; for two holes in the AF, the interaction between next-nearest neighbor holes is less than However, because the interaction falls slowly with distance, it is important for an analysis of the stability of static charge-ordered structures in systems lacking long-range Coulomb repulsion. It has been conjectured [3,10] that phase separation is a ubiquitous feature of lightly doped antiferromagnets and that consequently there is always a first-order transition separating the undoped and doped states. Evidence in support [11] of and in conflict [4,12] with this conjecture has been obtained from numerical studies of small-size systems. Phase separation has been shown to occur [13,14] in the large d limit of the Hubbard and t–J models and in the mean-field spiral states of the large N t–J model [15]. The present results offer strong additional support for the validity of this conjecture. Specifically, we claim that because of this Casimir-like interaction, any staticordered state of neutral holes will be thermodynamically unstable with respect to phase separation at small-enough doping. ACKNOWLEDGMENTS We thank A. H. Castro Neto for informative conversations. This work was supported in part by NSF grants DMR93-12606 at UCLA and PHY94-07194 at ITP-UCSB. REFERENCES 1. Experiments and theories are reviewed in Ref. [3 ]. 2. J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989); H. Schulz, Phys. Rev. Lett. 64, 1445 (1990). 3. S. A. Kivelson and V. J. Emery, in The Los Alamos Symposium—1993: Strongly Correlated Electronic

Materials, ed. K. S. Bedell et al. (Addison-Wesley, NY, 1994), p. 619. 4. S. R. White and D. Scalapino, cond-mat/9705128 (unpublished). 5. H. B. G. Casimir, Proc. Kon. Ned. Akad. Wet. 51, 793 (1948). 6. A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, NY, 1994).

7. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. B 39, 2344 (1989). 8. N. Bulut, D. Hone, D. Scalapino, and E. Loh, Phys. Rev. Lett. 62, 2192 (1989). 9. A. H. Castro Neto and D. Hone, Phys. Rev. Lett. 76, 2165 (1996). 10. G. Baskaran and P. W. Anderson, cond-mat/9706076 (unpublished).

11. V. Emery, S. Kivelson, and H. Lin, Phys. Rev. Lett. 64, 475 (1990); C. Hellberg and E. Manousakis, Phys. Rev. Lett. 78, 4609 (1997), and references therein. 12. H. Viertio and T. Rice, J. Phys. (Cond. Matt.) 6, 7091 (1994).

13. P. van Dongen, Phys. Rev. Lett. 74, 182 (1995). 14. E. W. Carlson, S. A. Kivelson, Z. Nussinov, and V. J. Emery, cond-mat/9709112 (unpublished). 15. A. Auerbach and B. E. Larson, Phys. Rev. B 43, 7800 (1991).


Features of the Modulated Structure in the Layered Perovskite Manganate

Y. Horibe,1 N. Komine,1 Y. Koyama,1,2 and Y. Inoue3

Features of a modulated structure in in which the La ions in are partially replaced by the Sr ions, have been investigated by transmission electron microscopy with the temperature range between room temperature and 85 K. Diffraction patterns of exhibits satellite reflection spots at around a fundamental spot below about 250 K. The modulated structure was also found to have the same features as those in

except for the shorter period.

1. INTRODUCTION

In with the -type structure, there exists a modulated structure around and its features have been investigated so far both experimentally and theoretically [1,2]. According to the previous work on the neutron-diffraction experiment, superlattice reflection spots characterizing the modulated structure appear at and which were due to the nuclear and magnetic reflections, respectively. Based on these experimental data, they interpreted that the modulated structure is ascribed to the static ordering of doped carriers. However, Bao et al. [3] indicated on the basis of their electron-diffraction data that the superlattice reflection at can be identified as the nuclear scattering and the reflection at is just the second-order harmonic. 1

Department of Materials Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan. Kagami Memorial Laboratory for Materials Science and Technology, and Advanced Research Center for Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan. 3 Structural Analysis Section, Research Department, NISSAN ARC LTD., 1 Natsushima-cho, Yokosuka, Kanagawa 2

237, Japan. Stripes and Related Phenomena, edited by Bianconi and Saini.


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In addition to the modulated structure in Moritomo et al. [4] examined both features of the modulated structure and related physical properties in with They pointed out that the ordering temperature of the doped carriers decreases with decreasing the size of the A-site ion. As was mentioned above, there is discrepancy between the experimental data obtained by means of the neutron and electron diffractions, so we cannot but say that an physical origin of the modulated structure is still an open question. Then, we investigated features of the modulated structure in by transmission electron microscopy. In particular, the crystal structure in was examined in the present work to check both the existence and features of the modulated structure and compare them with those in

2. EXPERIMENTAL PROCEDURE

Ceramic samples used in the present work were prepared by standard procedure as mentioned in the previous work [5]. The features of the modulated structure were examined by taking electron-diffraction patterns. The observation was carried out in the temperature range between room temperature and 85 K by an H-800 type transmission electron microscope equipped with a cooling stage with a liquid He reservoir. The He stage provides stability of temperature during the observation. Specimens for the observation were prepared by an Ar-ion thinning technique.


A specimen at room temperature has the -type structure with the tetragonal symmetry. When the temperature was lowered from room temperature, the modulated structure was found to appear below about 250 K. It was also confirmed that the microstructure exhibits a domain structure composed of two variants of the modulated

structure. Figure 1a,b are, respectively, electron-diffraction patterns at room temperature and at 85 K that were taken from a single-variant region. An electron incidence of both patterns is parallel to the [001] direction. In the pattern at room temperature, Fig. 1a, there are only fundamental spots due to the I4/mmm structure, whereas satellite reflection spots appear at an incommensurate position of

in Fig. 1b, as indicated by an arrow in A. Diffraction spots identified as the second harmonic are also detected, as marked by an arrow in B. In other words, the modulated structure can be characterized as the 1q incommensurate state. In order to check double diffraction for the satellite reflection spots indicating the 1q state, we then rotated the specimen and took electron diffraction patterns. In Fig. 2, as an example, we show the fact that the satellite spots along the direction through the origin 000 are due to double diffraction. An electron-diffraction pattern with the incidence is first shown in Fig. 2a. The satellite spots are clearly observed along the direction in the pattern. When the sample is rotated, the satellite spot actually disappears, as shown in Fig. 2b. That is, there is no satellite spots on the plane in reciprocal space. This fact clearly indicates that the modulation mode of the 1q-modulated structure is just a transverse wave with These features found in is the same as those in Note that the wave-vector of the modulated structure in the latter is specified by

Features of the Modulated Structure in the Layered Perovskite Manganate

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The origin of modulated structure is simply discussed on the basis of the present experimented data. From the existence of the second-order harmonics, the modulation

mode is identified as a phase modulation. Then the modulated structure should involve an transverse atomic displacement in the crystal structure. Because a transverse wave with a large wavelength is not basically interacted with an electron, the charge-ordering model seems to be inappropriate for the origin of the structure. As a simple interpretation of these experimental data, we believe that a polarization wave due to the metal ions would be responsible for the 1q modulated structure.

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4. CONCLUSION

In the present work, the features of the modulated structure in

were

examined by transmission electron microscopy with the temperature range between room temperature and 85 K. It was found that the modulated structure is characterized as 1qincommensurate structure and its modulation mode is simply due to the transverse wave with the incommensurate wave-vector of REFERENCES 1. Y. Moritomo, Y. Tomioka, A. Asamitu, Y. Tokura, and Y. Matsui, Phys. Rev. B 51, 3297 (1995). 2. B. J. Sternlieb, J. P. Hill, U. C. Wildgruber, G. M. Luke, B. Nachumi, Y. Moritomo, and Y. Tokura, Phys. Rev. Lett. 76, 2149(1996).

3. W. Bao, C. H. Chen, S. A. Carter, and S.-W. Cheong, Solid State Comm. 98, 55 (1996). 4. Y. Moritomo, A. Nakamura, S. Mori, N. Yamamoto, K. Ohoyama, and M. Ohashi, Phys. Rev. B 56, 14879 (1997).

5. Y. Horibe, N. Komine, Y. Koyama, and Y. Inoue, unpublished, 1998.

Numerical Studies of Models for Manganites Adriana Moreo1 and Seiji Yunoki1

The Kondo lattice Hamiltonian with ferromagnetic Hund coupling and antiferromagnetic (AF) interaction between the localized spins is investigated as a model

for manganites. The phase diagram has been obtained using Monte Carlo simulations in the limit where the localized spins are classical. At low temperatures, three dominant regions were found: (i) a ferromagnetic phase, (ii) phase separation between hole-poor AF and hole-rich ferromagnetic (FM) domains, and (iii) a phase with incommensurate spin correlations. An AF interaction between classical spins enhances the tendency to phase separation. Comparisons of these results with recent neutron scattering experiments are made, including phononic

degrees of freedom phase separation between two phases with spin-ferro order is observed. One has staggered-orbital order, and the other is orbitally disordered.

The phenomenon of colossal magnetoresistance in metallic oxides has recently attracted considerable attention [1] due to its potential technological applications. A variety of experiments have revealed that oxide manganites have a rich phase diagram [2] with regions of antiferromagnetic (AF) and ferromagnetic (FM) order, as well as charge ordering, and a peculiar insulating state above the FM critical temperature, Recently, layered manganite compounds have also been synthesized [3] with properties similar to those of their 3D counterparts. Ferromagnetism at low temperatures can be explained with the double exchange (DE) mechanism. [4,5]. However, the DE model is incomplete to describe the entire phase diagram observeds experimentally. For instance, the coupling of the electrons with the lattice may be crucial to account for the insulating properties above [6]. The presence of a Berry phase in the large Hund coupling limit also challenges predictions from the DE model [7]. In this paper and Ref. [8], we remark that another phenomena occurring in manganites that is not included in the DE description, namely the charge-ordering effect, may be contained in a 1

Department of Physics and National High Magnetic Field Lab, Florida State University, Tallahassee, FL 32306, USA.


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more fundamental Kondo model in which the (localized) electrons are ferromagnetically (Hund) coupled with the (mobile) electrons. More precisely, the presence of phase separation between hole-poor AF and hole-rich FM regions in the low temperature phase diagram of the FM Kondo model has recently been observed [8]. Upon the inclusion of longrange Coulombic repulsion, charge ordering in the form of nontrivial extended structures may be stabilized. The FM Kondo Hamiltonian [4,9] is defined as

where are destruction operators for one species of -fermions at site i with spin and is the total spin of the electrons, assumed localized. The first term is the electron transfer between nearest-neighbor Mn ions, is the Hund coupling, the number of sites is L, and the rest of the notation is standard. The density is adjusted using a chemical

potential

In this paper, the spin

is considered classical (with

tum mechanical, unless otherwise stated. Phenomenologically,

rather than quanwas

but here

considered an arbitrary parameter (i.e., both large and small values for were studied). The results of Ref. [8] are summarized in the phase diagram shown in Fig. 1. The diagram in the figure corresponds to 1D but similar results were obtained for 2D and infinite dimension [8]. At low temperatures, clear indications of (i) strong ferromagnetism, (ii) incommensurate (IC) correlations, and (iii) a regime of phase separation were identified.

For and 2, finite size effects were found to be small for the lattice sizes used in this study, although the PS-IC boundary in 2D was difficult to identify accurately. Results are also available in small 3D clusters and qualitatively they agree with those in Fig. 1. In the small region, IC correlations were observed, but in a region of parameter space not realized in the manganites. The addition of an AF coupling between the localized spins produces interesting modifications to the phase diagram [10]. At large Hund coupling, phase separation continues


461

to be observed at high electronic densities but it also appears between electron-undoped AF and electron-rich FM regions in the limit of low electronic density. This can be seen in Fig. 2, where the phase diagram in the plane for is presented. At intermediate densities, a metallic FM phase was found for small

separated from an

insulating regime with a transition located at The insulating phase has very strong incommensurate magnetic correlations with a structure factor that peaks at for quarter-filling. At

the quarter-filled ground-state becomes AF, and beyond only the half-filled and empty AF phases are stable. is possibly a good value to compare with experimental results. We have observed FM and AF spin correlations coexisting for at (see Fig. 3).

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Moreo and Yunoki As in neutron scattering [11], the FM peak increases as the temperature decreases,

whereas the AF excitations vanish in the FM phase. In the present model, this behavior is due to the coexistence of FM and AF domains at high temperature, which are a consequence

of the tendency of the system to phase separate [8,10]. The model presented in Eq. (1) considers only one orbital for the itinerant electron (i.e., it is being assumed that the other orbital has a much higher energy). The one orbital model has traditionally been studied in the context of manganites but new experimental results, particularly the ones related to possible charge and orbital ordering [12], make necessary the introduction of two orbitals and vibrational degrees of freedom. The two orbital electronic Hamiltonian is given by

where is the spin of the mobile electrons. The rest of the notation is as in Eq. (1) and the indices a, b indicate the orbitals and take values 1 and 2. The electron–phonon interaction is given by

where g is the electron–phonon coupling, k is the phonon stiffness, and

The variables Q1 and Q3 are classical and allowed to vary between In Fig. 4, typical results are shown at a density of -electrons equal in average to one per site and using S(0) and are the Fourier transforms at momentums 0 and respectively,


463

of the real-space correlation functions among the classical spins. At a clear change from spin-ferro to spin-antiferro order is observed. This is reasonable because at large it is expected that symmetry breaking will occur and the one orbital results of the single orbital Kondo model will be recovered. T(0) and are Fourier transform at momenta 0 and respectively, of the real-space orbital–orbital correlations. For details, the reader can consult

Ref. [13] and references therein. Here, it is sufficient to know that when T(0) is large, the orbitals are uniformly ordered (i.e., the same orbital dominates in all sites), whereas if is large, then the orbital with the lowest energy (dynamical effect) alternates from site to site. A transition from one to the other is observed at , together with the change in the character of the spin order. At a transition from an orbital disordered regime to a staggered orbital ground-state is observed. Both sides of are spin FM. It is very important to remark that phase separation is also present in the model with Jahn–Teller phonons. It appears in three regimes: 1.

At low electronic density phase separation is observed between electron-poor AF and electron-rich FM regions, similar to the way it occurs in the one-orbital Kondo model after the addition of a direct coupling between the classical spins.

2.

At large

3.

near phase separation occurs in the regime where at the spin AF, order is stabilized. This region, again, is the analog of the phase separation observed in the one orbital model. A novel regime of phase separation has been observed near and at intermediate values of In this case, the phase separation occurs between two spin-ferro phases. One has antiferro orbital order and the other has weak orbital correlations (similar to the way it occurs on both sides of in Fig. 4).

In Fig. 5, the density vs chemical potential obtained with a Monte Carlo simulation on a 10-site cluster is presented at At close to both 1 and 0, a region of unstable densities is observed. This regime may be of relevance for the real manganites, and its importance is discussed in Ref. [13].

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SUMMARY

In this paper and Refs. [8,10, and 13], the phase diagrams of models for manganites have been presented using computational techniques. Some of these models were formulated decades ago, but only now they can be analyzed using a variety of numerical many-body algorithms. Contrary to previous expectations, no indications of canting order have been observed. This tendency is replaced instead by phase separation, which occurs for a variety of manganites models at both low- and high-electronic density. The experimental implications of this new regime have been discussed in previous publications. It is reasonable to assume that including Coulomb interactions explicitly in the problem will transform the phase separation regime between electron-rich and electron-poor regions (or hole-rich and holepoor) into clusters of one phase immersed into the other. The stabilization of more exotic structures such as stripes is a clear possibility in this context. It may occur that the chargeordering regimes observed experimentally in manganites may be related with the phase separation discussed here, again, when proper Coulomb interactions are incorporated into the problem. ACKNOWLEDGMENTS A.M. is supported by NSF under grant DMR-95-20776. Additional support is provided by the National High Magnetic Field Lab and MARTECH.

REFERENCES 1. S. Jin et al., Science 264, 413 (1994). 2. P. E. Schiffer, A. P. Ramirez, W. Bao, and S.-W. Cheong, Phys. Rev. Lett. 75, 3336 (1995); C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996). 3. Y. Moritomo et al., Nature 380, 141 (1996).

4. C. Zener, Phys. Rev. 82, 403 (1951); P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 (1955). 5. P. G. de Gennes, Phys. Rev. 118, 141 (1960). 6. A. J. Millis et al., Phys. Rev. Lett. 74, 5144 (1995); H. Röder, J. Zang, and A. R. Bishop, Phys. Rev. Lett. 76, 1356 (1996).

7. E. Müller-Hartmann and E. Dagotto, Phys. Rev. B 54, R68I9 (1996). 8. S. Yunoki, J. Hu, A. Malvezzi, A. Moreo, N. Furukawa, and E. Dagotto, Phys. Rev. Lett. 80, 845 (1998); E. Dagotto, S. Yunoki, A. Malvezzi, A. Moreo, J. Hu, S. Capponi, D. Poilblanc, and N. Furukawa, preprint, cond-mat/9709029.

9. N. Furukawa, J. Phys. Soc. Jpn. 63, 3214 (1994); ibid. 64, 2754 (1995). 10. S. Yunoki and A. Moreo, preprint, cond-mat/9712152. 1 1 . T. G. Perring, G. Aeppli, Y. Moritomo, and Y. Tokura, Phys. Rev. Lett. 78, 3197 (1997); ibid. 80, 4359 (1998).

12. S. Mori, C. H. Chen, and S.-W. Cheong, Nature 392, 473 (1998); Y. Murakami, H. Kawada, H. Kawata, M. Tanaka, T. Arima, Y. Moritomo, and Y. Tokura, Phys. Rev. Lett. 80, 1932 (1998). 13. S. Yunoki, A. Moreo, and E. Dagotto, Phys. Rev. Lett. 81, 5612 (1998).

Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds with S. Pachot,1 P. Bordet,1 C. Chaillout,1 C. Darie,1 R. J. Cava,2 M. Hanfland,3 M. Marezio,4 and H. Takagi5

The evolution at high pressure of the title compounds cell parameters has been investigated up to GPa by synchrotron x-ray diffraction (XRD) using a diamond anvil cell and an imaging plate 2D detector. The compressibility is found to be highly anisotropic, with easy compressibility along the stacking axis of the

structure. The effect of applied pressure appears to be similar to the internal pressure effect brought about by Ca substitution. A strong anomaly in the a parameter variation is observed at pressures increasing with Ca content in the 6–8 GPa range. This anomaly could be related to the disappearance of superconductivity observed in the same pressure range.

1. INTRODUCTION Following the discovery of high- superconducting cuprates, the theoretical investigation of the magnetic and transport properties in low-dimensional copper oxide systems has become increasingly active. Recently, the so-called spin-ladder systems have drawn much of the attention due to their predicted striking properties. For example, Dagotto and Rice [1] predicted the existence of a spin gap when the antiferromagnetic (AF) coupling along rungs is stronger than along legs, and Rice et al. [2] showed later that this property would exist

1

Laboratoire de Cristallographie CNRS, BP166, 38042 Grenoble Cedex 9, France. Department of Chemistry and Materials Institute, Princeton University Princeton NJ 08540, USA. 3 ESRF, BP220, 38043 Grenoble, France. 4 MASPEC-CNR, via Chiavari 18/A, 43100 Parma, Italy. 2

5

Institute for Solid State Physics, University of Tokyo, Roppongi Minato-ku, Tokyo 106, Japan.


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for even-leg ladders only. They also predicted the appearance of superconductivity when holes are introduced in this system. Superconductivity was detected for the first time in a compound containing spinladders by Uehara et al. [3], who measured the resistivity as a function of pressure and temperature in the compounds. They reported the appearance of a superconducting transition in the 3–5 GPa range, with a maximum transition temperature of about 12 K in samples with a large calcium concentration The structure of can be described as the alternate stacking of layers having the two-leg ladder arrangement and composition and layers made of isolated chains of edge-sharing squares separated by planes containing the alkaline earth cations. The ladder planes are formed by the corner-sharing connection of double chains of edge-sharing squares. Within the double chains, the Cu cations are linked through 90° Cu-O-Cu bonds, leading to a very weak magnetic coupling, and the ladders may be considered to be magnetically quasi-isolated objects. The squares belonging to the chains above and below are oriented at 45° with respect to the squares belonging to the ladders. The incommensurate ratio between the Cu-Cu separation in both types of layers resulting from this arrangement leads to a composite type of structure, which in first approximation can be described by using two orthorhombic unit cells with parameters and for the ladder-containing slabs, and and for the chain layers. The a direction is in the plane of and

perpendicular to the chains and ladders, the b direction is perpendicular to the stacking, and the c direction is along the chains and ladders. However, this schematic description of the structure may be oversimplified, and thorough crystallographic studies [4,5] pointed out the presence of buckling in both ladder and chain layers, and of interlayer Cu-O bonds that may play an important role in the doping mechanism. Optical measurements have shown [6] that the intrinsic hole doping due to the 2.25+ average copper valence is gradually transferred from the chains to the ladders on increasing calcium concentration. Because this substitution is isovalent, the doping modification is induced by altering the average size of the (Sr,Ca) mixed site, which leads to changes of the Cu-O bond lengths in both chains and ladders. Due to the smaller ionic size of Ca cations, the effect of Ca substituting for Sr can be viewed as a chemical pressure effect. Because the superconducting properties of the compound appear only at high pressure and for specific calcium concentrations, it is of the highest relevance to investigate the evolution of the structure with pressure and composition. Therefore, we have carried out a high-pressure synchrotron diffraction experiment at room temperature in the 0–10 GPa range for a set of samples with and x between 0 and 13.6 in order to investigate the combined effects of applied pressure and modification of the average ionic size and/or valence of the (Sr,M) mixed site.

2. EXPERIMENTAL Five different samples of general formula were used for this experiment, hereafter denoted as (superconducting composition); The first four samples were used to investigate the effect of pressure as a function of the average size of the mixed site, whereas the data comparison between and allows us to

Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds 467 compare the effect of changing only the average valence of this site and cations have very close ionic sizes). They were previously characterized for cationic stoichiometry and phase purity by EDS analysis and laboratory powder x-ray diffraction (XRD). The high-pressure synchrotron diffraction experiment was carried out at the ID9 beamline of the European Synchrotron Radiation Facility (ESRF) using angle-dispersive powder diffraction with image plates as detectors. Samples were loaded into a diameter and high gasket hole of a membrane diamond anvil cell (MDAC). Pressures were measured using the ruby fluorescence method [7]. Silicon oil was used as a pressure transmitting medium. Although it is known to be a rather poorly hydrostatic medium above 6 GPa, it was chosen because of its inert character with respect to the samples. In order to check the effect of pressure gradients induced by this medium at higher pressures, two additional experiments were carried out on samples and by using a 4:1 methanol/ethanol mixture known to remain a better pressure transmitter to 10 GPa. The results obtained were similar for the sample, but the sample started to decompose above 3 GPa in the methanol/ethanol mixture, leading to the appearance of calcium- and copper-containing phases. Nevertheless, the similar results obtained for the sample with both transmitting media indicate that the observed effects described below are not due to pressure gradients brought about by silicon oil at high pressures. The monochromatic beam was selected by a horizontally focusing asymmetrically cut bent Si(111) Laue monochromator [8]. It was vertically focused by a curved Pt-coated Si mirror. The beam size on the sample was Diffraction rings from the powder samples were recorded on an A3-size Fuji image plate located 441 mm from the sample. The plates were scanned on a molecular dynamics image plate reader,

and processed by using the Fit2D software developed at the ESRF [9]. The images were corrected for spatial distortion effects. Corrections for the image plate tilt with respect to the direct beam were applied by using images from a standard silicon powder, which were also used to calibrate the wavelength and the sample-to-detector distance. The corrected images were averaged over 360° about the direct beam position, yielding spectra similar to those obtained by classical diffraction techniques. Due to the complexity of the structure, we concentrated our analysis on the evolution of the cell parameters with pressure. For this purpose, we selected for each sample a set of diffraction lines (generally at low angles) that were well isolated in the whole pressure range, and obtained their angular positions by least-square refinement using a pseudo-Voigt line-shape function. The cell parameters were obtained by transforming the positions into d values using the Bragg's law. Note that in most cases, it was not possible to identify isolated peaks from the “chain” unit cell, so only the parameters and from the “ladder” unit cell could be obtained.

3. CELL PARAMETERS AT ROOM PRESSURE The relative evolution of the cell parameters at GPa is shown in Fig. 1 as a function of the average ionic radius of the mixed site. The normalization was made with respect to the sample. The cell parameter increase with increasing average ionic radius of the mixed site is strongly anisotropic, with a much larger effect in the b direction and the smallest effect along the c direction. This is most probably due to the weakness of ionic bonds along the stacking direction with respect to the strong in-plane Cu-O bonds

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Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds 469

inside the ladder and chain layers. A similar anisotropy is to be expected for the effect of

pressure. It is interesting to point out that the a- and b-cell parameters of the sample follow the regular evolution due to size effects, whereas the c parameter value is markedly higher. This indicates that the charge effect brought about by the replacement of

by

cations leads to modifications of the Cu-O bonding scheme mainly along the chain and ladder directions. 4. CELL PARAMETERS AT HIGH PRESSURE

The evolution of the cell parameters and volume as a function of pressure is shown for each sample in Fig. 2. At first glance, all samples except exhibit a similar behavior, with roughly parallel decreases of the cell parameters and volume with increasing applied pressure. Axis compressibilities are shown in Fig. 3.

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The compressibility is the smallest in the a axis direction, with values ranging from to The compressibility along c is about twice as large as that along a, with values ranging from to These values are close to those reported for the in-plane compressibilities of superconducting cuprates [10]. The b axis compressibility is much larger than the in-plane ones, with values ranging between 6.3 and These values are twice as large as the largest c axis compressibilities

reported for the superconducting cuprates.Moreover, the b axis compressibility markedly increases with increasing average ionic radius of the mixed cation site. This could indicate an increase of interlayer interactions when the size of the mixed site is decreased (i.e., when the ladder and chain layers come closer to each other). In Fig. 4, we present the relative evolution of the a cell parameters as a function of

pressure. The most striking feature observed is the presence of a lattice anomaly consisting in a strong increase of the a cell parameter above 6–8 GPa, depending on sample composition. The pressure value at which the anomaly takes place seems to increase with increasing Ca content, going from GPa for to GPa for and to GPa for

For and the anomaly appears at GPa. For the latter sample, above the a axis anomaly pressure value the c parameter seems to become stable and the b axis compressibility starts to increase again. The former effect seems to exist also for the other samples, although less pronounced. The diffraction spectra recordered at pressures above the lattice anomaly do not present noticeable differences, such as peak splitting or superlattice reflections, from those recordered below, indicating that no major structural rearrangement is occuring at the transition. Although the present data do not allow us to draw definitive conclusions about the

nature of this lattice anomaly, a possible origin could be the abrupt decrease of the ladder and chain layers buckling induced by the applied pressure. Such an effect could indeed lead to an increase or stabilization of the in-plane cell parameters a and c and allow the restoration

Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds 471

of the compressibility in the b axis stacking direction. The effect of applied pressure and lattice anomaly on the physical and superconducting properties of the compounds may be discussed on the basis of the present results and already reported structural and physical measurements [4–6]. The main effect of the substitution of Ca for Sr and of applied pressure is a strong decrease of the b-cell parameter, which corresponds to an increased coupling between the ladder and chain layers and to a charge transfer from the chains to the ladders. The similarity between the substitution and pressure effects on the cell parameter indicates that the cation substitution acts as an internal chemical pressure. However, the substitution effect alone is not sufficient to induce superconductivity at low temperature, and an additional external pressure must be applied, with appearance of the superconducting state at for the compound, which corresponds to a cell parameter The disappearance of superconductivity at might be related to the lattice anomaly, even though it seems to appear at higher pressures for the sample) at room temperature. Low temperature and high-pressure diffraction experiments are needed to confirm this model. REFERENCES 1. E. Dagotto and T. M. Rice, Science 271, 618 (1995). 2. T. M. Rice, S. Gopalan, and M. Sigrist, Europhys. Lett. 23, 445 (1993). 3. M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996).

4. A. Frost-Jensen, B. B. Iversen, V. Petricek, T. M. Schultz, and Y. Gao, Acta Cryst. B 53, 113 (1997). 5. T. Ohta, F. Izumi, M. Onoda, M. Isobe, E. Takayama-Muromachi, and A. W. Hewat, J. Phys. Soc. Jpn. 66, 3107(1997).

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6. T. Osafune, N. Motoyama, H. Eisaki, and S. Uchida, Phys. Rev. Lett. 78, 1980 (1997). 7. H. K. Mao, J. Xu, and P. M. Bell, J. Geophys. Res. 91, 4673 (1986).

8. C. Schulze, U. Lienert, M. Hanfland, M. Lorenzen, and F. Zontone, J. Synchro. Rad. to be published. 9. A. P. Hammersley. S. O. Svensson, M. Hanfland, A. N. Fitch, and D. Häusermann, High Pres. Res. 14, 235 (1996). 10. H. Takahashi and N. Mori, in Studies of High Temperature Superconductors, ed. A. Narlikar (Nova Science

Publishers, Inc., 1996), vol. 16, p. 1.

X-Ray Scattering Studies of Charge Stripes in Manganites and Nickelates Y. Su,1 C.-H. Du,1 B. K. Tanner,1 P. D. Hatton,1 S. P. Collins,2 S. Brown,3 D. F. Paul,3 and S.-W. Cheong4

Single crystals of both

and

have been stud-

ied extensively using both laboratory and synchrotron radiation x-ray scattering. Using x-ray techniques, we have detected directly the formation of charge stripes. Weak satellite peaks with a modulation wave-vector were observed below the transition temperature in the perovskite Measurements of the temperature dependence of the intensity and width of the

charge-ordering satellites over the temperature range 10–240 K demonstrate that the correlation length of the charge-order stripes is long range at all temperatures and in all directions. A first-order structural phase transition accompanying the charge ordering was also studied. We also undertook SR x-ray measurements

on and found quasi-2D-like charge stripes with a modulation wave-vector Detailed critical scattering measurements of the melting of the charge stripes gave critical exponents compatible with those expected of a 2D-Ising system.

1. INTRODUCTION Charge and spin ordering into stripes in direct space have recently attracted intensively attention due to their role in manganite colossal magnetoresistence (CMR) [ 1 ] and

cuprate superconductivity [2]. Since the discovery of CMR in the compounds 1

Department of Physics, University of Durham, Durham, DH1 3LE, UK. Daresbury Laboratory, CCLRC, Warrington, WA4 4AD, UK. 3 XMaS CRG, ESRF, 38043 Grenoble Cedex, France. 4 Department of Physics & Astronomy, Rutgers University, Piscataway, NJ 08854, USA, and Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974-0636, USA. 2


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there has been extensive experimental and theoretical efforts to understand the origin of the anomalous magnetotransport behavior [3]. Hole doping in the antiferromagnetic (AF) insulators induces an insulator to ferromagnetic (FM) metal transition and for doping levels of leads to a new type of collective state: a charge-stripe state. At these doping states, the charge ordering can compete with the FM ground-state, leading to a complex electronic phase behavior as the chemical stoichiometry is varied. In many cases, the charge ordering is accompanied by magnetic ordering, suggesting that the interplay between charge and spin ordering is important in explaining the anomalous physical properties. Recently, a charge-ordered state with charge stripes has been observed in single crystal for by neutron diffraction [1]. This has lead to considerable discussion about the relationship between charge ordering and high- superconductivity. Charge ordering into charge stripes is thus an important area of research applicable to a wide range of phenomenology in transition metal oxides. Charge ordering in the maganites is generally characterized by the direct-space ordering of and ions. At room temperature, these ions are randomly distributed in the sample planes. Upon cooling, the ions become regularly sited. The chargeordered state is expected to become stable when the repulsive Coulomb interaction between carriers dominates over the kinetic energy of carriers. The electron lattice of the CO state therefore acts as a Wigner crystal. The stability of the CO state is very sensitive to the commensurability of the hole concentration with the lattice periodicity, and hence most enhanced at nominal hole concentrations of 1/2, 1/3, etc. Furthermore, the carriers needed to form the CO state are believed to be polarons formed from the very strong electronphonon interaction in these materials [3]. Since the first report of charge ordering and spin ordering in by electron-diffraction [4] and neutron-scattering measurements [5], mechanisms such as Fermi-surface nesting, frustrated phase separation, and polaron ordering have been suggested as being responsible for the formation of such stripes [4,6]. However, experimental

results reported so far are conflicting. Both

and

are two dimensional

(2D) AF insulators with layered or planes. When holes are doped into the planes, either by substitution of Sr or excess oxygen, the cuprate becomes superconducting at surprisingly high temperatures with a hole concentration per Cu, but the nickelate, however, remains insulating, even when is increased to 0.5. In the case of the nickelate, once the hole concentration is large enough and spin-density modulations start to form within the planes.

coupled charge

To date, however, studies of charge stripe in these systems has generally relied on electron diffraction and neutron scattering. Electron scattering is very sensitive to weak

superlattice reflections caused by charge ordering. The only disadvantage is the relatively low wave-vector resolution and that the intensities are difficult to study quantitatively due to multiple scattering. Neutron diffraction has been the major technique, suitable for study of both the spin and charge stripes [5]. An unfortunate condition for neutron-scattering techniques is the need for relatively large sample sizes, which are difficult to grow in many manganites and nickelates. In addition, neutron diffraction is not directly sensitive to the charge ordering, but rather the charge ordering is detected indirectly through displacements of the nuclear positions because of the associated strain deforming the lattice. In this paper,

we briefly report on the results of experiments on charge stripes in manganites and nickelates using x-ray scattering. Surprisingly, there have been very few reports of charge ordering using x-ray scattering. X-ray techniques have the benefit of relatively small single crystal

X-Ray Scattering Studies of Charge Stripes

475

sample sizes and high wave-vector resolution. By using laboratory high-brilliance rotating anode sources, and particularly synchrotron radiation, it is possible to obtain count rates similar to those obtained by neutron techniques on far larger samples. This is important not only to avoid the difficulties in growing large crystals but also because the quality (as evidenced by the inverse FWHM of Bragg reflections) is often higher in smaller crystals. This means that x-ray scattering is a useful probe of long-range correlations of charge ordering.

2. EXPERIMENTAL PROCEDURE Single crystals of and were grown by the flux method at Bell Laboratories. These samples had been previously characterized by a number of techniques [7]. The in-house x-ray experiments were performed at the University of Durham. The crystal was mounted on the cold finger of a Displex closed-circle cryostat, where the temperature was monitored with an Si diode to an accuracy of The whole cryostat was mounted on a four-circle triple-axis diffractometer, which employed a high-brilliance rotating anode generator operated at 2.8 kW with a Cu anode. The Cu x-ray beam was selected and collimated by two flat (0001) pyrolytic graphite crystals used as the monochromator and analyzer. Such an arrangement gives a relatively poor resolution but very high intensity, and avoids any multiple scattering from higher energy harmonics. The synchrotron experiments were performed at beamline 16.3 at the SRS, Daresbury Laboratory, and at the XMaS CRG (BM28), ESRF. Double-axis geometry without using a crystal analyzer was employed on both beamlines. Both antiscatter and detector slits were

carefully adjusted to the minimum possible size to reduce the background and increase the wave-vector resolution. 3. RESULT AND DISCUSSION In

cooling below 241 K resulted in additional satellite peaks at around the (400) Bragg Peak (Fig. 1). These satellite peaks have an intensity of that of the Bragg peak. Longitudinal and transverse scans through some of the charge-ordering peaks were fitted with a Voigt function. The instrumental lengths evaluated in both direction had a similar value of about Satellite peaks with modulation wavevector around other Bragg peaks such as (800), (6,2,0), and (6, –2,0), etc., were also observed, confirming the origin of these to be a superstructure of charge stripes induced by charge ordering of and in the sheet. The temperature dependence of the integrated intensities of the charge-ordering scattering was also measured. The intensity rapidly increases around 241 K, the charge-ordering temperature, which occurs simultaneously with the structural phase transition. No temperature dependence of the FWHM of the charge-ordering peaks was observed. This implies that the melting of the charge stripes has a strong first-order transition behavior. In order to obtain higher resolution and count rate for charge-ordering scattering, measurements using synchrotron radiation on same single crystal were carried out at beamline 16.3 at the SRS. A wavelength of 1.0 Å and a small beamsize of 0.5 × 0.5 mm were used. Very long-range order in both longitudinal scan and transverse scan, more than 1000 Å, was obtained by peak fitting to a Voigt function. This result strongly demonstrates the 3D nature of charge

476


stripes in manganites. With the higher resolution and count rate achieved with SR, the charge-ordering peaks were checked on warming up and cooling down, and hysteresis of

the integrated intensity was clearly observed as Fig. 2a. A first-order structural phase transition in the vicinity of was observed as well, as shown in Fig. 2b. The Bragg peak (800) in the low-temperature monoclinic phase was monitored on warming up across the

region when, at 243 K, the intensity abruptly dropped and eventually disappeared because of the formation of the high-temperature orthorhombic phase. This can be seen as clear evidence for the strong interaction between charge stripes and the structural phase transition. Synchrotron radiation measurements were also undertaken on a single crystal at station 16.3 at the SRS. A solid-state detector was used for detecting chargestripe scattering. A wavelength of 0.8Å was chosen to optimize the beam intensity from the Wiggler beamline, increase the number of observable Bragg and charge-stripe peaks, and reduce harmonic interference from and The face-centered tetragonal notation was used, and all measurements were done in the (hhl) zone in reciprocal space. At 20 K, a systematic search for charge-stripe reflections was done by scanning along four different directions. Relatively sharp and strong charge-stripe peaks could be indexed as (3.33, 3.33, 1), (3.67, 3.67, 2), and (4.33, 4.33, 2), etc. These peaks all had an intensity approximately that of Bragg reflections. A consistent wave-vector relationship in the (hhl) zone is shown in Fig. 3, in which charge-ordered satellites are related to neighboring Bragg peaks by a modulation wave-vector (2/3, 2/3, 1). We also observed a distinct broadening of the charge-ordered satellites above 200 K, well below the charge-ordering temperature of 240 K. More detailed measurements on the melting of the charge stripes were undertaken


477

478


using XMaS at the ESRF, which has a peak flux almost 500 times higher than station 16.3

at the SRS. Strong and very clean singlet charge-ordering peaks were observed that could be well fitted by a Voigt function, and accurate FWHM can be evaluated by considering the convolution of instrumental resolution function. The correlation length along the longitudinal direction was more than 7 times longer than the value of 250 Å observed along the transverse direction. This is strong evidence that the charge stripes have a much longer correlation within the Ni-O plane compared to that of out of the plane [i.e., the charge stripes in have a quasi-2D-like nature]. Further evidence for 2D behavior was obtained by critical scattering measurements. Measurements of the intensity and width of the charge-stripes satellites were undertaken in the high-temperature regime measuring the melting of the charge stripes under increasing temperature. The following power laws, as shown in Eqs. (1) and (2), were chosen to fit the order parameter (integrated intensity) and the inverse correlation length in both longitudinal and transverse directions:

where Fig. 4a and b] were

The critical exponents obtained [fitting along (00L) shown in


479

480


which are very close to those expected exponents of a 2D-Ising model, These exponents were obtained with a fitted of 224 K, suggesting that above

this temperature, the charge ordering observed is due to critical fluctuations of (unstable) charge stripes. Figure 4c demonstrates the reduction of the correlation length as a function of temperature. This result not only clearly demonstrates the 2D feature of charge stripes in this system, but is also important for future theoretical considerations.

ACKNOWLEDGMENTS We thank the Director of Daresbury Laboratory for access to facilities at Daresbury, and to Profs. W. G. Stirling and M. J. Cooper for access to XMaS during the early stages of

commissioning. We acknowledge the help received from Dr. I. Pape, Mr. B. D. Fulthorpe, and Mr. J. Clarke during the experiment at the ESRF. This work was supported by a grant from the Engineering and Physical Sciences Research Council. Y. S. would like to thank the CVCP for the award of an ORS studentship and the Department of Physics at the University of Durham for financial support during his doctoral studies. REFERENCES 1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995); J. M. Tranquada, J. D. Axe, N. Ichikawa, A. R. Moodenbaugh, Y. Nakamura, and S. Uchida, Phys. Rev. Lett. 78, 338 (1997). 2. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996). 3. A. R. Ramirez, J. Phys. Cond. Matt. 9, 8171 (1997).

4. C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993). 5. J. M. Tranquada, D. J. Buttrey, V. Sachan, and J. E. Lorenzo, Phys. Rev. Lett. 73, 1003 (1994).

6. J. Zannen, Phys. Rev. B 40, 7391 (1989); V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993). 7. W. Bao, J. D. Axe, C. H. Chen, and S.-W. Cheong, Phys. Rev. Lett. 78, 543 (1997); S.-H. Lee and S.-W. Cheong,

Phys. Rev. Lett. 79, 2514 (1997).

Colossal Negative Magnetoresistivity of Films in Fields up to 50 T P. Wagner,1 I. Gordon,1 L. Trappeniers,1 V. V. Moshchalkov,1 and Y. Bruynseraede1

The low carrier mobility found in Mn perovskites implies that the dominant conductivity mechanism is related to Mott hopping. As a modification to the original model, we take into account that the hopping barrier is influenced by the relative spin orientation at the two Mn ions involved in an elemental hopping process. From this, we deduce a scaling behavior of the colossal negative magnetoresistivity according to the Brillouin function in the ferromagnetic–quasi-metallic state and proportional to a squared Brillouin function in the paramagnetic–semiconducting phase. Both predictions could be verified by pulsed magnetic field measurements up to 50 T.

1. INTRODUCTION

Rare earth manganites with divalent substitution at the rare-earth site are characterized by a transition from a paramagnetic–semiconducting to a ferromagnetic–quasi-metallic

state at the Curie temperature Conductivity arises from the simultaneous presence of and ions and the transfer of the electrons between them. The charge transfer in the ferromagnetic (FM) phase is usually ascribed to the double-exchange mechanism [1], and it is essential to note that the apparently metallic conductivity is orders of magnitude lower than in case of real metals. The charge carriers are therefore characterized by strong localization, suggesting that a type of hopping transport in the framework of

Mott’s formalism should be taken into account [2]. Hopping of Jahn–Teller-type polarons is meanwhile widely accepted to explain the conductivity on the paramagnetic side of the phase diagram [3,4]. Both magnetic phases exhibit a colossal negative magnetoresistance 1

Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven.


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Wagner, Gordon, Trappeniers, Moshchalkov, and Bruynseraede

(CMR) effect [5], indicating that the charge transfer can be strongly activated by decreasing the misorientation between the local magnetization vectors at the different Mn sites. The origin of CMR is indeed a field-induced mobility enhancement, because Hall measurements on magnetic perovskites gave no indication for any field dependence of the charge-carrier density [6]. This article tries to extend the hopping description of conductivity, established for the paramagnetic phase, also to the FM regime by taking into account the misorientation angle between neighboring magnetic moments as the main difference between the two magnetic states. As a side effect, this description also predicts the correct field-scaling of the CMR effect, which is phenomenologically parabolic for the paramagnetic phase and hyperbolic-like for the ferromagnet. 2. EXPERIMENTAL

films (thickness 300 nm) were prepared by de-magnetron sputtering onto heated substrates [7]. The epitaxial growth was confirmed by x-ray diffraction (XRD) and the magnetic properties were investigated by a SQUID magnetometer, yielding a Curie temperature The samples remain ferromagnetic–quasi-metallic down to the lowest temperatures and the transition to an antiferromagnetic (AF), charge-ordered state, as found for single crystals [8], could not be observed. The reason is most probably the slight excess of compared to resulting in a deviation from the exact 1:1 ratio of and The composition analysis on the films was performed by Rutherford backscattering, which is accurate within 1 % of the relative concentration. After characterization, the samples were cut into stripes and gold pads, evaporated and annealed, served as low-ohmic current and voltage probes. Resistive measurements were performed with the magnetic field orientation parallel to the film surface and the current direction. The temperature dependence of the resistivity at fixed external fields was studied in a superconducting coil (up to 12 T) with a variable temperature insert. The

negative magnetoresistivity at fixed temperatures was investigated in a pulsed magnetic fields installation [9], reaching up to 50 T with a pulse duration of 20 ms. Heating effects on the sample were found to be negligible. 3. RESULTS AND DISCUSSION

Figure 1 shows the temperature variation of resistivity in various fields, showing a crossover from semiconducting to quasi-metallic behavior close to the Curie temperature.

External fields result in a global lowering of the resistivity and a shift of the resistance maximum to higher temperatures. The data points at 50 T, which were extracted from the pulsed fields measurements discussed below, suggest a temperature-independent resistivity

value in the limit of magnetic saturation. The transport on the microscopic scale is therefore mainly controlled by the local spin configuration (i.e., the carrier mobility is highest for FM, spin aligment), which can be artificially induced by sufficiently strong external fields even in the nominally paramagnetic state. Figure 2 shows the field induced resistivity decrease for various temperatures in the paramagnetic (part a) and in the FM regime (part b). In the FM case, saturation can be readily achieved, whereas this is not the case for the paramagnet, even


Films

483

in external fields of 50 T. The solid lines for the FM part of Fig. 2 are fit curves based on the Brillouin function B and in case of the paramagnetic part we used fit functions proportional to a squared Brillouin function. Both scaling laws can be motivated straightforward from the model of spin-dependent Mott hopping. Conductivity in Mott’s original model (i.e.,

484


without taking into account the magnetic moments at the hopping sites), is given by [2]:

where R is the average hopping distance, the phonon frequency, the density of states at the Fermi energy, L the carrier localization length, and the effective hopping barrier between the and ions. Because the spin of the hopping electron is aligned parallel with the core spin of (at the initial as well as the final site of the hopping process due to Hund coupling), will be lowered in case of parallel spin orientation between the Mn ions and enhanced for antiparallel orientation. The difference

in hopping barriers might be considered as an extra energy required to accomplish a rotation of the electron spin during the hopping process in order to match into the spin configuration

of the hosting Mn ions. The modified barrier might be written as:

with

being a proportionality constant taking into account the overlap of the carrier wave

function with the host ions. Inserting the modified barrier into the conductivity formula

(Eq. (1)) and calculating the decrease of resistivity compared to a random relative orientation between Mn moments results in [7,10]:

where the temperature-dependent prefactor A is a measure for the amplitude of the negative magnetoresistance effect. The average scalar product between the magnetization vector at the initial and the final site of a hopping process can be calculated from the following concept: The total magnetization at each site is given as the sum of the uniform Weiss

magnetization (vanishing in the paramagnetic state) and a local correction term The absolute value of is the difference between and magnetic saturation, whereas it has a random orientation. External magnetic fields rotate the Weiss magnetization into

the main field direction and cause a common alignment of the independent vectors along the field axis. This problem is analogous to the magnetization of a paramagnet and the contribution of the vectors to the total magnetization can therefore be approximated by the Brillouin function. The resistivity decrease of Eq. (3) can be rewritten as:

It is evident that in the paramagnetic phase only, the third term will survive and the average scalar product is given by the product of two Brillouin functions. The correction terms are small compared to the Weiss magnetization in the FM state, and the resistivity

decrease will be governed by the first two terms of Eq. (4). The increasing below the Curie temperature causes even in the absence of any external field an increasingly parallel aligment of neighboring magnetic moments, which in turn causes a lowering of the hopping


Films

485

barrier and the corresponding resistivity as shown in Fig. 1. Applying external fields in the FM state results in an additional resistivity decrease, which is controlled by the second term of Eq. (4), which is linear in the Brillouin function. These results can be summarized as:

The exponent of the Brillouin function has either the value 1 (ferromagnet) or 2 (paramagnet), the argument of this function contains the gyromagnetic ratio the Bohr magneton, and the spin moment J at the hopping sites. The fit functions calculated on basis of Eq. (5) comply very well with the experimental results shown in Fig. 2, yielding correctly even the change in curvature observed for the CMR curves in the paramagnetic state. The temperature dependence of the two fitting parameters A(T) for the CMR amplitude and J(T) for the average spin moment at the individual hopping sites is given in Fig. 3. The amplitude strongly resembles the resistivity in zero field, shown as a thin solid line. The A(T) data are actually for all temperatures exactly 1 m below the zero-field resistivities, meaning that in the limit of magnetic saturation, a small, temperature-independent resistivity is obtained. The spin moments found from the fitting of resistivity data are considerably higher than the values expected for individual or ions, suggesting a kind of superparamagnetic behavior. The absolute J(T) values of up to 60 correspond to ferromagnetically ordered spin clusters extending over a diameter of 3 to 4 unit cells of the crystal lattice. This cluster size agrees closely with the polaron diameter found by De Teresa et al. [3] by means of magnetization measurements combined with neutron diffraction (1.2 nm), and also with the carrier localization length obtained by the quantitative analysis of the

Mott formula Eq. (1) [10]. In this picture, the charge carriers are localized in a ferromagnetically aligned environment and transport occurs through hopping (eventually also tunneling) of carriers between these small FM entities. The apparent decrease of J for cannot be interpreted directly in the sense of a shrinking cluster diameter because J(T) describes actually the magnetic behavior of the difference term between Weiss and saturation magnetization becoming identical in the limit

486


4. SUMMARY

In summary, we have generalized Mott’s hopping model to the CMR class of materials by introducing a barrier contribution depending on the relative magnetization orientation at the initial and the final site of individual hopping processes. The relative orientation was approximated by expressing the local magnetizations as a superposition of the Weiss magnetization and an external-field controlled correction term. By taking into account the presence or absence of spontaneous magnetic order, this model describes the considerably different behavior of resistivity and magnetoresistivity in the FM and in the paramagnetic state. The (squared) Brillouin scaling of the CMR effect is valid up to 50 T, corresponding to a magnetically saturated state. The fitting parameters of the Brillouin description point to the presence of FM spin clusters, which might be interpreted in terms of magnetic polarons. ACKNOWLEDGMENTS P. W. is a Marie Curie fellow of the European Union. This work has been supported by the Flemish GOA and FWO programmes. The authors are strongly indebted to M. J. Van Bael, D. Dierickx, and A. Vantomme for technical assistence. REFERENCES 1. P. G. de Gennes, Phys. Rev. 118, 141 (1960) and references therein. 2. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1979).

3. 4. 5. 6. 7.

J. M. De Teresa et al, Nature 386, 256 (1997). Guo-meng Zhao, K. Conder, H. Keller, and K. A. Miiller, Nature 381, 676 (1996). R. von Helmolt et al, Phys. Rev. Lett. 71, 2331 (1993). P. Wagner et al., Europhys. Lett. 41, 49 (1998). P. Wagner et al., Phys. Rev. B 55, 15 699 (1997).

8. H. Kawano et al., Phys. Rev. Lett. 78, 4253 (1997). 9. F. Herlach et al., Physica B 216, 161 (1996). 10. P. Wagner et al., Phys. Rev. Lett. 81, 3980 (1998).

Synthesis and Characteristics of the Indium-Doped Tl-1212 Phase R. Awad,1 N. Gomaa,1 and M. T. Korayem1

Superconductors of type for and 0.6 were synthesized at for 2 h. The characterization of these samples was made using the x-ray powder diffraction and the scanning electron microscope (SEM), whereas the composition of these samples was determined by microprobe analysis. The transition temperature for as-synthesized (Tl, In)-1212 compound, determined from the electrical resistance measurements, varied from 97 K for to 91 Kfor

The effect of the external applied magnetic field on the stage of transition is reported. Our data show that the transition width is increased by increasing the external applied magnetic field. The samples are more affected by the magnetic field with increasing the indium content.

1. INTRODUCTION

After the discovery of high-temperature superconductivity by Bednorz and Müller [1] in copper perovskites 1986, Bianconi and his group in Rome [2] provide a prototype for a

new heterogeneous material formed by a superlattice of quantum stripes at the atomic limit. The structure of most of the high-temperature superconductors is divided into two major structures. First is a superconducting layer, which acts as an electrical conduction band. Second is the metal-O isolation planes (also called charge reservoir), where the metal

isolation planes usually are Tl, Bi, and Hg [3–5]. It is clear that more of metal-O isolation planes solely having electronic configuration, such as Tl, Bi, Pb, and Hg, appear to play the key role to enhance the superconducting properties. In order to understand the effect of the metal-isolation plane, we substitute indium into the thallium sites in the Tl-1212. The electronic configuration of 1

Physics Department, Faculty of Science, Alexandria University, Egypt.


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Awad, Gomaa, and Korayem

In is where as the Tl is and the ionic radii of both Tl and In are identical. Thus, we expected similar tendencies for the In-doped materials as present for other (Tl, Hg)-1212 [6]. The experimental results of the (Tl, Hg)-1212 indicate that a small addition of Hg into the Tl site increases the transition temperature from 85 K to 117 K. Recently,

the addition of the In in the Tl-1223 phase was reported by Abou-Aly et al. [7]. They found that this addition reduced the transition temperature from 122 K to 100 K. In this work, we investigate the effect of In doping in the Tl-1212 superconductor by preparing a series of samples with final nominal composition with and Also, we report the effect of the weak applied magnetic field up to 4.9 kG on the transition stages. 2. EXPERIMENTAL

Samples with nominal composition of and were synthesized by two steps of solid-state reaction using stoichiometric amount of and The precursor was prepared by a solid-state reaction as previously reported [8J. The stoichiometric amount of and were ground in a glove box using an agate mortar under argon atmosphere in order to prevent absorption of moisture and Then the powder was pressed into disk that was wrapped in a silver foil to reduce the loss of thallium and indium that could eventually react with the quartz tube. Finally, the sample was inserted in a tube furnace of length 30 cm and 3 cm diameter. The sample was heated to 870°C by the rate of 400°C/h, held at this temperature for 2 h, and then furnace cooled to room temperature. Samples were characterized by x-ray powder diffraction. The x-ray diffraction (XRD) scans (0.1 º/sec) were carried out on Philips PW1729 powder diffractometer using radiation, The samples were examined in a Jeol scanning electron microscope JSM-5300, operated at 15 kV, with resolution power of 4 nm. The real composition of the samples’ content were determined using an Oxford x-ray microanalysis system (25 kV). The electrical resistance was measured by using a conventional four-probe technique in

the temperature range from 30 K to 227 K in a closed cryogenic system. The samples used for resistance measurements have dimensions of about The connections of the copper leads with the sample were made using silver paint. The effect of applied magnetic field up to 4.9 kG on the transition stages was made using an electromagnet. The external magnetic field was applied perpendicular to the driving current. 3. RESULTS AND DISCUSSIONS

The powder XRD patterns for In-doped Tl-1212 are shown in Fig. 1 with 0.4, and 0.6. All the major peaks correspond to the (Tl, In)-1212 phase. The minor impurity phases detected include and some unknown peaks. The lattice parameters of In-doped Tl-1212 are reported in Table 1, which shows that as the In-doping level increases, the lattice parameters a and c increase. The fact that the lattice parameters a and c increase monotonically with the increasing In content demonstrates that a continuous solid solution is formed between the Tl and In ions. The results of quantitative elemental analysis by energy dispersive x-ray spectroscopy (EDS) for the samples studied indicate that the indium ions were successfully doped in the structure of Tl-1212 phase.

Synthesis and Characteristics of the Indium-Doped Tl-1212 Phase

489

490


Figure 2 shows the temperature dependence of the normalized resistance for with and For all samples, the resistance exhibits linear temperature dependence in the range from 227 K to the transition temperature. The T1-1212 sample has the highest metallic behavior gradient, but the sample of has the lowest metallic gradient. The more metallic behavior is attributed to the best grain connection [9]. The samples with and show superconducting transition temperatures at 98.5, 96, 93.25 and 91.5 K, respectively, whereas they have zero resistance temperatures of 82.5, 76.75, 62.25, and 50.5 K, respectively. It is clear that, the sample with displays a superconducting transition temperature of 97 K, which is greater than that of

This result is probably due to the substitution of the low valant

by the higher valant in T1-1212 phase, which increases the hole concentration in the sheets and enhances the transition temperature. As Fig. 2 demonstrates, In doping can decrease the superconducting transition temperature. It is well known that for superconducting oxides is a sensitive function of the metal–oxygen layer. The number of charge carriers in the conduction band can be varied by modifying the chemistry of charge reservoir layer through charge transfer process. This means that the addition of In reduces the onset temperature on the contrary to the addition of Hg [6], and Pb [4] which increases the transition temperature. This result


491

suggests that the change from 6s to 5s metal states affects the energy level balance of the metal and oxygen, which could result in the depression of superconductivity.

The normalized resistance-temperature dependence for at different applied magnetic fields starting from 0 up to 4.9 kG is shown in Fig. 3. The data in this figure indicate that, the transition to the zero resistance has two stages of transition. The first stage (high-temperature region) starts at the onset temperature and consists of a rapid decrease in the resistance to about 86 K. In the second stage of transition (low temperature region), the resistance smoothly decreases to zero as the temperature approaches (zero resistance temperature). This is believed to be due to the anisotropy of the material and the random orientation of the grains [10], or due to the percolation effects across the grains [11]. Also, we notice that the magnetic field has no effect on the first stage of transition, and the curves obtained at different applied magnetic fields overlap in the high-temperature region. In the second stage we observe that for the higher value of the magnetic field, the transition temperature is enlarged. These data could contain some information concerning the spatial distribution of the “strong” superconducting grains and “weak” superconducting boundaries. The transition width determined from the difference between the onset temperature and zero resistance temperature, for is plotted as a function of the applied magnetic field in Fig. 4. It is clear that these curves have three features. The first is that the temperature width is enlarged by increasing the applied magnetic field. The enlargement in the temperature width could be explained as, while the sample is cooling in a magnetic field applied perpendicular to the driving current direction, a number of randomly oriented grains freeze and cluster in random positions. Such a situation leads to more defected specimen. Consequently, an amount of flux is trapped. As a result, the length of the vortex pinning is weakened and the specimen is in a more resistive state by the loss of complete superconducting current pass. The second feature is

492


that the samples are more affected by the external magnetic field with increasing the indium content. This means that the magnetoresistance of the samples containing indium is greater than that of the indium-free sample. This result suggests that a partial replacement of T1 with In does not produce any significant flux pinning in this system. The third is that the transition width sharply increases until and then almost a plateau in is observed for a field The transition width is well fitted to the formula

and the value of n was found to vary from 0.16 to 0.38. A similar equation for studying the variation of critical current density with the applied magnetic field was reported by Sun et al. [12], who found that n has a negative value. 4. CONCLUSIONS The superconducting compound with and was formed at temperature 870°C for 2 h. Both lattice parameters a and c were increased by increasing the In content. The transition temperatures are reduced by the In doping, indicating that the doping of the superconductors compound with elements reduces the transition temperature. On the contrary, the doping by enhances the transition temperature. External magnetic fields affect only the second stage of transition, but do not affect the first stage of transition. Also, the transition width is increased by the increase of both external magnetic field and In content. The Indium doping does not produce any significant flux pinning in this system. ACKNOWLEDGMENTS

The authors are very grateful to Prof. Dr. A. I. Abou-Aly and Prof. Dr. I. H. Ibrahim for useful discussion and suggestions. We also thank Mr. M. A. El hajji and Mr. A. Attia for their technical assistance.


493

REFERENCES 1. J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986). 2. A. Bianconi, International Conference on Superconductivity, edited by S. K. Joshi, C. N. Rao, and S. V.

Subramanyam (World Scientific Publishing Co., Singapore) (1990), p. 448. 3. Z. Z. Sheng and A. M. Hermann, Nature 332, 138 (1988).

4. H. Maeda, Y. Tanaka, M. Fukutomi, and T. Asano, Jpn. J. Appl. Phys. 27, L209 (1995). 5. A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott, Nature 363, 56 (1995). 6. R. Awad, G. A. Costa, M. Fenu, C. Ferdeghini, A. H. El-Sayed, A. I. Abou-Aly, and E. F. Elwahidy, Il Nuovo

Cimento 19D, No. 8–9, 1103 (1997). 7. A. I. Abou-Aly, R. Awad, and I. H. Ibrahim, unpublished, 1998. 8. R. Awad, G. A. Costa, M. Fenu, C. Ferdeghini, A. H. El-Sayed, A.I. Abou-Aly, and I. H. Ibrahim, unpublished, 1998. 9. L. Lechter, E. Toth, S. Osofsky, C. Kim, B. Qadri, and J. Soulen, Physica C 242, 21 (1995). 10. D. O. Welch, M. Suenaga, and T. Asano, Phys. Rev. Lett. 2386 (1988).

11. M. Celasco, G. Cone, V. Popescu, A. Masoero, and A. Stepanesou, Physica C 252, 375 (1995). 12. J. Z. Sun, C. B. Eom, B. Lairson, J. C. Bravman, T. H. Geballe, and A. Kapitulnik, Physica C 162, 687 (1989).


High-Frequency Optical Excitations in YBCO Measured from Differential Optical Reflectivity I. M. Fishman,1 W. R. Studenmund,1 and G. S. Kino1

YBCO differential optical reflectivity (DOR) responses are measured in the temperature range of 50–160 K at several optical frequencies. It is found that the background response relates to temperature variation of the scattering rate. Above the full DOR response may be attributed to a Drude plasma with a low mobility. Below accumulation of the superconducting condensate is accompanied

by appearance of a new optical spectrum in the range of 1.2–1.5 eV. We present

a model for a plasma having parameters close to that of a Mott transition that demonstrates qualitatively the features observed experimentally.

1. ANALYSIS OF EXPERIMENTAL DATA In an accompanying paper [1], we presented results of DOR measurements at five probe wavelengths varying from to We found that both normal and superconducting differential optical reflectivity (DOR) responses are strong functions of

the probe frequency. In Fig. 1, the DOR responses are shown for the a direction of one of our YBCO samples at five probe frequencies. At each frequency, the critical temperature is clearly indicated by a singularity at The responses measured on other samples or

within different domains of the same sample, although not identical, are very close to those of Fig. 1. All responses consist of two components, a smooth background, and temperaturedependent peaks in the vicinity of For all measurements the high-temperature background is smooth, and the total DOR response tends to disappear at low temperatures. Depending on the probe frequency, the background changes sign and the absolute value by over an order of magnitude. If normalized to one value at high temperature the DOR background responses coincide at t low temperatures 1

E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305-4085.



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We measured DOR only at five wavelengths, that is not adequate for spectral presentation of DOR responses. To overcome this deficiency, we made a correlation between DOR components and the frequency-dependent direct optical reflection coefficient that was carefully investigated earlier [2].

Direct optical reflection coefficient

could be described by

a Drude formula

with suitable choices of and (high-frequency dielectric constant, plasma frequency, and electron collision rate, respectively). With considered a frequency-dependent function a rather good fit to the experimental data was obtained, as shown by the solid lines in Fig. 2. For the a direction, the best fit parameters were

Although the Drude model of Eq. (1) is not adequate for describing optical phenomena in YBCO, it is helpful as a numerical tool to establish qualitative relations between the direct reflection data and our DOR data. Using Eq. (1), one may express the temperature derivative

High-Frequency Optical Excitations in YBCO

497

of DC reflectivity as

In Eq. (2), in accordance with Drude’s Eq. (1), we assume that dA/dT, and are functions only of temperature and not of frequency. The frequency-dependent functions are

and These functions numerically calculated from Eq. (1) with the parameters and obtained from fitting the data of Fig. 2 are shown for the a direction in the insert of Fig. 2. For the b-direction, the shapes of the curves are similar with maximums and zeroes shifted along the frequency axis. In the absence of DOR data taken in a wide spectral range, using Eq. (2) allows us to present full DOR response as a sum of products of temperature-dependent and frequencydependent functions in which frequency dependence may be determined from the comparison with direct reflection data. This approach was further developed in [1] to prove that above the DOR response belongs to a Drude plasma. In the current work, we analyze different components of DOR response (background and transition-related responses) and identify their physical origin. To do so, we define the DOR background as a smooth function that is presumably a universal function of temperature and sample orientation. One might expect this kind of contribution, for example, from thermal expansion. At high and low temperatures, this function is well defined by a scaling process. To diminish the uncertainty involved in the definition of the temperature dependence of the background, we determined it as a smooth average of 10 datasets for 5 wavelengths and both a and b directions of the same sample; also, we compared the obtained result with DOR responses of samples without or almost without transition-related responses such as thin YBCO films. Data of Fig. 1 scaled at high temperatures all coincide with each other and automatically coincide at low temperatures In Fig. 3, a dotted line 1 shows the background frequency dependence at and From the comparison of this curve with the functions

498


and to

it is concluded that the experimental frequency dependence is proportional with very little, if any, contribution from other partial derivatives. We therefore

assume that the background temperature dependence is dominated by the scattering rate

term dA/dT with which is customary for the optical conductivity of metals. The transition-related response was determined as a difference between the experi-

mental curve and the background. In Fig. 3, the experimental frequency dependence of a transition-related DOR component is shown at (curve 3). The experimental curve has only five meaningful points corresponding to our probe frequencies; all other points are for as a guide to the eye. The frequency dependence of the curve 3 of Fig. 3 coincides with the frequency dependence of (curve 4). The mutual correlation between the experimental and theoretical data sets, especially the zero crossing point on the frequency axis, different from the crossing point for (curve 2) and respective experimental curve 1 is evident. The perturbation of dR/dT above frequency grows with temperature).

Below

corresponds to positive

is (plasma

the DOR frequency dependence is more complicated (Fig. 4). In the spectral

range from 1 to 2 eV, this DOR component crosses the zero line twice. The first zero occurs

at approximately 1.8 eV, far above the zeroes of and , which were calculated using the model in Eq. (1) and were found in good correlation with the experimental data above The second zero occurs at approximately 1 eV. There is no linear combination in Eq. (3)

that can fit curve 3. Furthermore, the DOR response of the superconducting condensate becomes very small at both 1310 nm and 1550 nm probe for both a and b directions. It seems that, below approximately 1 eV and below 90 K, the superconducting condensate does not contribute to the DOR signal at all. Data for b direction, which is presented in detail

elsewhere, also show that the superconducting condensate contributes to the DOR signal only at frequencies above

We conclude that the frequency dependence of the DOR

response of the superconducting condensate cannot be explained by the plasma resonance only. At frequencies above an additional reflection spectrum is associated with the condensate. Our failure to fit the DOR data below

using arbitrary functions

and confirms that we observe a new optical spectrum in the range of 1 eV associated with the superconducting phase. That explains why we were unable to find direct correlation between our data and the IR measurements conducted by others [3].

High-Frequency Optical Excitations in YBCO

499

2. IMPLICATIONS FOR THE MODEL OF A SUPERCONDUCTING TRANSITION Optical responses of superconductors, both conventional and high are customarily discussed using Kramers–Kronig relations for the real and imaginary parts of the dielectric constant at a frequency

A conductivity at which perturbs both real and imaginary parts of when the temperature falls below should result in a high-frequency response with only a weak dependence on the frequency Our DOR responses observed in the frequency range of 1–2 eV, especially the variation of the plasma frequency above and appearance of a new spectral line below shows that our results cannot be explained as “propagation” of this -perturbation to the optical frequencies. Correct theory of superconductivity should predict a new optical spectrum in the range of 1–2 eV.

The physical interpretation of the DOR data requires better understanding of the optical response of the normal phase, or an electron plasma with very low and frequency-dependent mobility. The difficulties involved in the theory of normal phase are well known. Our experimental results demonstrate that above the temperature derivatives dA/dT and have, by order of magnitude, similar power. That means that weak temperature variations of the scattering rate and weak electron–electron attraction may have a similar physical origin. Here we present a simple model that may be used for qualitative support of this observation. Above YBCO, and other materials behave like bad metals with very high collision rate, which may be understood as a result of a strong correlation between the mobile carriers and the negative ions. For a gas of free particles in thermal equilibrium at temperature T, the lower limit for the scattering time is The relaxation time, measured in YBCO in the IR range [3] and extrapolated to is an order of magnitude less: sec. Our fits of the reflection coefficient (Fig. 2), which are confirmed by our DOR results, suggest that the conductivity in materials may be due to hopping of carriers from cite to cite, and in the visible range The doping levels corresponding to the superconducting state at low temperatures are close to the Mott transition criterion [4], e.g.,

where

is the Bohr radius. The essence of the Mott transition is the disappearance of the electron binding energy in the screened Coulomb potential when the periodic system of atoms turns from an insulating to a metallic state. Periodicity is important because in a totally random potential the electron states should be localized, at least at low densities. The qualitative picture of “hopping” suggests that in YBCO, the carrier correlation with the ions is very strong, and each scattering

500


event tends to destroy the ordering caused by screening. In equilibrium, the carrier transport

scattering destroys the order on a time scale whereas the dielectric relaxation enhances the order on a time scale When the system reaches the critical density, and should be close to each other. If the transport scattering time is very small,

and the Coulomb screening is dynamically turned off, then in a certain range of densities and temperatures the system may stay random. With where the scattering time t is esimated as h/E and the hole binding energy The critical density n is

which practically coincides with the static Mott transition criteria in Eq. (4). If the ordered state is unavailable, the strongest correlation between the carriers will be exchange pairing, which may overwhelm the Coulomb repulsion. As a result, Bose condensation may occur in the system of strongly correlated hole pairs. Thus, we suggest

that a nontrivial state of matter may exist in a plasma having density exceeding the Mott transition density. No “quasi-particles with integrity” are theoretically identified in materials, yet though an intuitive model may be provided by resonant scattering states in shallow wells [5 ]. The major difference between the state of matter qualitatively described above and other disordered systems, such as heavily doped semiconductors, is relaxation of the order parameter. Similar situations may be expected under certain conditions in a highdensity exciton gas in semiconductors. REFERENCES 1. W. R. Studenmund, I. M. Fishman, and G. S. Kino, Differential Optical Reflectivity Measurements of YBCO, p. 529 in this volume.

2. S. L. Cooper, D. Reznik, A. Kotz, M. A. Karlow, R. Liu, M. V. Klein, W. C. Lee, J. Giapintzakis, D. M. Ginsberg, B. W. Veal, and A. P. Paulikas, Phys. Rev. B 47, 8233 (1993). 3. A. V. Puchkov, D. N. Basov, and T. Timusk, J. Phys.: Cond. Matt. 8, 10049 (1996). 4. J. M. Ziman, Principles of the Theory of Solids (Cambridge, University Press, 1964). 5. D. Bohm, Quantum Theory (New York, Prentice-Hall, 1951).

Low-Temperature Structural Phase Transitions and Suppression in Zn-Substituted Y. Inoue,1 Y. Horibe,2 and Y. Koyama2,3

Crystal structures between room temperature and 12 K in have been investigated by transmission electron microscopy to confirm the existence of the low-temperature structural phase transition. In electron-diffraction patterns of at 12 K, there exist 100-type forbidden spots

due to the Pccn/LTT structure in addition to the fundamental spots due to the LTO one. That is, the low-temperature transition was actually confirmed to occur in the Zn-substituted oxide. Dark-field images using the 100 forbidden spot also indicated that the Pccn/LTT phase appears not only along the twin boundary between the LTO domains, but also in the interior of the LTO domain, just as in the case of other La cuprates. The low-temperature transition must then play a certain role

in the

suppression found in the Zn-substituted

as well.

1. INTRODUCTION

It has been found that is strongly suppressed in around The suppression has been discussed in relation to the low-temperature transitions to the Pccn/low-temperature tetragonal (LTT) phase [3–12]. Tranquada et al. [13] interpreted from their neutron-diffraction data that the suppression originates from

the stripe ordering, which is expected to occur in a highly correlated electron system. Based on their interpretation, the Pccn/LTT phase plays an important role in the pinning of the stripe ordering. The pinning of the ordering can be also expected by introducing an impurity 1

Structural Analysis Section, Research Department, NISSAN ARC LTD., 1 Natsushima-cho, Yokosuka, Kanagawa 237, Japan.

2

3

Department of Materials Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan. Kagami Memorial Laboratory for Materials Science and Technology, and Advanced Research Center for Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.



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potential. The suppression has been actually reported in Zn-substituted [14]. Although the stripe ordering seems to stabilize the Pccn/LTT structure, the existence of the low-temperature transition has not be examined in Zn-substituted We then checked its existence and examined features of a microstructure between room temperature and 12 K in by transmission electron microscopy. Among Zn-substituted samples, we describe in particular experimental data obtained in a

sample in this paper.

2. EXPERIMENTAL

Powder samples of were prepared from CuO, and ZnO in a coprecipitation technique using citric acid and calcined at 1173 K for 15 hr in air, followed by the pelletization of the samples. The pellets were sintered at 1323 K for 100 hr in air and annealed at 773 K for 48 hr in In the present work, the observation was performed for a sample with The energy dispersive x-ray spectroscopy (EDX) measurement was carried out to check the Zn content of the sample made in the present work. A value of in the sample was also determined by a SQUID measurement and was then determined to be of about 8 K. In order to examine both crystal structures and related microstructures in lower temperatures, an in-situ observation was made in the temperature range between room temperature and 12 K. We used a H-800-type

transmission electron microscope equipped with a cooling stage liquid helium reservoir. In the present experiment electron-diffraction patterns and bright- and dark-field images were recorded on imaging plates in order to avoid a drift of a specimen during an exposure. Specimens for transmission- electron microscopy observation were prepared by an Ar-ion thinning technique. 3. EXPERIMENTAL RESULTS

The existence of the low-temperature transition was first checked by means of electron diffraction. Figure 1 shows two electron diffraction patterns obtained from an sample at room temperature. Electron incidences of Figs. 1a and b are parallel to the [001 ] and [111] directions, respectively. In addition to fundamental spots due to the high-temperature tetragonal (HTT) structure, only diffuse spots are seen in electrondiffraction patterns as indicated by an arrow in Fig. 1b. This means that the sample at room temperature has the HTT structure, and the HTT to low-temperature orthorhombic (LTO) transition should take place just below room temperature. Actually, it was confirmed experimentally that the transition occurs around 280 K. When the temperature is lowered to 85 K, two types of diffraction spots appear in electron-diffraction patterns in addition to the fundamental spots due to the LTO structure. Figure 2 is an electron-diffraction pattern at 85 K. An electron incidence of the pattern is parallel to the [001] direction. The 100type forbidden spots due to the Pccn/LTT structure are observed in Fig. 2, as indicated by A. Superlattice reflection spots are also detected at 1/2 1/2 0-type positions, as indicated by B. This clearly implies that the Zn-substituted La-Sr-Cu-O sample undergoes the lowtemperature transition. It should be remarked that in the present experiment, charge-density wave (CDW) spots found by Tranquada et al. [13] could not be detected in any electrondiffraction patterns at 85 K.

Low-Temperature Structural Phase Transitions and

Suppression

503

The details of a microstructure related to the low-temperature transition in the sample was examined by taking bright- and dark-field images. A bright-field image at 85 K was first shown in Fig. 3. A banded contrast due to the twin structure in the LTO phase is clearly seen in the image. One note here is that the dark-contrast line is observed along a twin boundary between two neighboring LTO domains. In order to understand an origin of the dark-contrast line, then dark-field images were taken by using the 100 forbidden spot. Figure 4a and b are, respectively, 100 dark-field images at 85 K and 12 K, which were taken from the same area as that in Fig. 3. The bright-contrast line indicated by A is observed along the twin boundary and corresponds to the dark-contrast line in the brightfield image of Fig. 3. Because the contrast should be diffraction contrast, the line along the

504


twin boundary has the Pccn/LTT structure and is called the line phase. When the specimen is cooled to 12 K, a spotty phase B also appears in the interior of the LTO domain in addition to the line phase A in Fig. 4b. In addition, an antiphase boundary with respect to the tilt of the octahedron is observed as a dark-line contrast C in the bright-contrast region with the Pccn/LTT structure near an edge of the sample, as was shown in our previous

work [9]. These features are therefore understood to be the same as those observed in other La-cuprates [5–7, 9–12].

4. DISCUSSION Now we discuss an origin of the suppression in Zn-substituted La-Sr-Cu-O. The suppression has so far been explained in terms of the ordering of the stripes, which was proposed by Tranquada et al. [13] as the basis of their neutron-diffraction experiment.

In the present work, however, the CDW spot related to the ordering was not detected in the electron-diffraction patterns of the sample. This clearly means that no stripe ordering exists in the oxide. However, the low-temperature structural transitions take place in and the features are entirely the same as those reported in other La cuprates. So, the suppression should be correlated with the low-temperature structural transition instead of the stripe ordering.

5. CONCLUSION From the present observation made by transmission electron microscopy, a lowtemperature transition exists in the sample. The 100 dark-field images also show that the line phase and spotty phase with the Pccn/LTT structure appear along the twin boundaries between the LTO domains and in the interior of the LTO

domains, respectively. On the basis of the present experimental data, it is suggested that

Low-Temperature Structural Phase Transitions and

Suppression

505

the low-temperature transition should play the certain role in the Tc suppression in Znsubstituted REFERENCES 1. J. D. Axe, A. H. Moudden. D. Hohlwein, D. E. Cox, K. M. Mohanty, A. R. Moodenbaugh, and Y. Xu, Phys. Rev. Lett. 62, 2751 (1989).

2. K. Kumagai, K. Kawano, I. Watanabe, K. Nishiyama, and K. Nagamine, J. Supercond. 7, 63 (1994). 3. A. R. Moodenbaugh, Y. Xu, M. Suenaga, T. J. Folkerts, and R. N. Shelton, Phys. Rev. B 38, 4596 (1988). 4. T. Suzuki and T. Fujita, J. Phys. Soc: Jpn. 58, 1883 (1989). 5. C. H. Chen, D. J. Werder, S.-W. Cheong, and H. Takagi, Physica C 183, 121 (1991).

6. C. H. Chen, S.-W. Cheong, D. J. Werder, and H. Takagi, Physica C 206, 183 (1993). 7. Y. Zhu, A. R. Moodenbaugh, Z. X. Cai, J. Tafto, M. Suenaga, and D. O. Welch, Phys. Rev. Lett. 73, 3026 (1994).

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8. M. K. Crawford, R. L. Harlow, E. M. McCarron, W. E. Farneth, J. D. Axe, H. Chou, and Q. Huang, Phys. 9. 10. 11. 12. 13.

Rev. B 44, 7749 (1991). Y. Inoue, Y. Horibe, and Y. Koyama, Phys. Rev. B 56, 14176 (1997). Y. Inoue, Y. Horibe, and Y. Koyama, 9th International Symposium on Superconductivity (1996). Y. Inoue, Y. Horibe, and Y. Koyama, J. Supercond. 10, 361 (1997). Y. Horibe, Y. Koyama, and Y. Inoue, J. Supercond. 10, 461 (1997). J. M. Tranquada, B. J. Stemlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995).

14. G. Xiao, M. Z. Cieplak, J. Q. Xiao, and C. L. Chien, Phys. Rev. B 42, 8752 (1990).

Josephson Nanostructures and the Universal Transport and Magnetic Properties of YBCO J. Jung,1 H. Van,1 H. Darhmaoui,1 and W. K. Kwok2

We have measured both the temperature dependence of the critical current

and the effective energy barrier for vortex motion vs current density different

of

for 1 1

YBCO thin films. The studies revealed the universal behavior

and

The results imply that

and vortex motion (pinning)

are determined by Josephson nanostructures (i.e., superconducting cells of a few nanometers in size), coupled by overdamped Josephson tunnel junctions in the ab planes of YBCO films. The presence of such nanodomains in YBCO has been confirmed by the high-resolution electron microscopy studies by Etheridge at

Cambridge University [Philos. Mag. A 73, 643 (1996)].

1. INTRODUCTION Recent studies of YBCO using high-resolution electron microscopy by Etheridge [1] at Cambridge University revealed the presence of structural perturbations at intervals of

the coherence length in the ab planes of this superconductor. The resulting nanostructure of cells has originated in a struggle to relieve internal stresses in the planes. These findings generated an important question: How can these perturbations influence electrical and magnetic properties of YBCO? One interesting possibility is that the nanostructure of cells in the ab planes could act as an array of Josephson tunnel junctions with the modulation of the Josephson coupling energy between the cells in the ab planes and along the c axis (Fig. 1). In this picture, the YBCO superconductor behaves like a granular system with the grain size of the order of a few nanometers. Nanometer-size granularity is known to occur, for example, in a conventional type-II superconductor such as NbN thin film of The grain sizes of the film range from 3 to 9 nm, as revealed by transmission electron microscopy (TEM). The film showed no preferential orientation, but 1 2

Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2J1. Materials Science Division, Argonne National Laboratory, Argonne, IL 64039 USA.


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Jung, Yan, Darhmaoui, and Kwok

the grains of similar orientation were frequently clustered together, with the cluster size approaching 20 nm. This granular structure is reflected in a temperature dependence of the critical current density which shows an Ambegaokar–Baratoff (AB) behavior (characteristic of Josephson tunnel junctions) at low temperatures with a relatively narrow plateau at temperatures below (with reaching at and a Ginzburg–Landau (GL) dependence close to at temperatures above Clem et al. [2] stated that the crossover from an AB to a GL form of occurs when the Josephson coupling energy of an intergrain junction is approximately equal to the superconducting condensation energy of a grain. At the crossover temperature, the GL coherence length is of the order of the grain size. The effective grain size inferred from measurement was calculated to be 22 nm, which corresponds to the cluster size of 20 nm. The effective grain size is larger than the average grain size of 6 nm measured by TEM. This was explained in terms of the expected spread of intergrain Josephson-coupling strengths, using the argument that the similarly oriented grains are among those that are strongly coupled together. Below the crossover temperature the coherence length (which is about 5 nm at is smaller than the grain cluster size, and is governed by the weak Josephson coupling between the clusters (an AB behavior). Above the coherence length is larger than the cluster size and the current does not “see” Josephson junctions. In this case, is governed by the ability of the current to suppress the order parameter (a GL behavior). During the course of this research, we were searching for the Josephson response in the electrical transport and magnetic properties of YBCO thin films. Regarding the temperature dependence of we were looking for similarities between the

Josephson Nanostructures and the Universal Transport

509

data for YBCO thin films and those obtained for a granular NbN film. We applied a different

experimental method that allowed us to measure both the magnitude of the critical current and the relaxation (decay) of the current from the

level due to the motion of vortices.

2. EXPERIMENTAL PROCEDURE We used persistent current flowing in a ring-shaped YBCO thin film and the mea-

surement of its magnetic field with a scanning Hall probe to record the magnitude of and simultaneously the decay of the persistent current J(t) from the critical level. This technique is contactless and free of unwanted contributions to due to heating effects and normal currents. The studies have been performed for 1 1 c-axis oriented YBCO thin films deposited on two different substrates ( or ) with three different deposition methods (laser ablation and rf and dc magnetron sputtering) in six different laboratories.

The measurements were done for zero-field cooled samples in the remanent magnetization regime over a temperature range between 10 K and The persistent currents at different levels up to a critical value have been generated by applying and subsequently switching off an external magnetic field parallel to the ring’s axis (c axis of the film). The magnitude of was inferred from the maximum magnitude of the persistent current's self-field at the ring's center. The decay of the current’s self-field from the maximum level provided the

means to calculate the energy barrier for motion of vortices as a function of the current density The Maley’s procedure [3] has been used to calculate for a wide range of J. We measured the relaxation rates from level every 5 K between 10 K and 85 K, allowing evaluation of for different values of By eliminating the temperature

dependence from the data, a continuous curve of vs J was produced, which consisted of multiple segments, each representing relaxations of the persistent current's self-field at a fixed temperature. 3. EXPERIMENTAL RESULTS AND DISCUSSION The experimental results for in 11 YBCO thin films revealed that the temperature dependence of is a superposition of two separate contributions: (a) an Ambegaokar–

Baratoff behavior with a crossover to a GL behavior close to (at temperatures above ), and (b) a pure GL-like dependence. This is shown in Figs. 2 and 3. The data obtained on more than 30 different YBCO films indicated that these contributions can be observed either together or separately in the temperature dependence of A pure AB behavior at low temperatures with a crossover to a GL behavior close to has been observed on one of YBCO films (see Fig. 6 in Ref. 4). A pure GL-like dependence in at all temperatures up to is a characteristic of an underdoped YBCO or a YBCO under a high magnetic field. It was also found that the temperature dependence of is independent of magnetic flux pinning in YBCO [4]. An Ambegaokar–Baratoff component,

which converts into a GL component close to shows a striking similarity to measured on NbN thin film (Fig. 4). Because of the short coherence length in YBCO, this implies a nanometer size granularity in the ab planes of YBCO thin films, using the same arguments that explained the temperature dependence of in a NbN film [2]. According to our studies, high YBCO thin films of behave like a stack of two films having different temperature dependence of at low temperatures [i.e., an AB-like and a GL-like ].

510


Decays of the persistent current J ( t ) from level allowed us to calculate the effective energy barrier for the motion of vortices in the film as a function of current density J. The

calculation revealed a universal empirical form of a GL-like behavior dominates

(see Figs. 2d–f and 3d–f). When


where

is

extrapolated to

511

. When an AB-like behavior dominates

where and are constants. These formulas imply that the factor is the same for all YBCO thin films. This is based on systematic measurements of the persistent current’s

512

Jung, Van, Darhmaoui, and Kwok

relaxations on 11 films. A factor of in the exponents could be interpreted as a tilted washboard potential for an overdamped (resistively shunted) Josephson tunnel junction (RSJ) fixed at a phase of Dissipation in a single RSJ has been described by Tinkham [5]. In an RSJ model a phase of corresponds to a maximum voltage across the junction (maximum resistive dissipation). This suggests that for vortex motion in YBCO thin films depends on the properties of intrinsic Josephson junctions in the ab planes. From the point of view of vortex motion, YBCO film behaves like a single Josephson junction or a very coherent array of Josephson junctions. We interpret the dissipation of the persistent current as due to the vortex motion through the Josephson nanostructures in the ab planes of YBCO films. In this model, the ab planes consist of cells of a few nanometers in size, which are coupled by resistively shunted Josephson tunnel junctions. We postulate that the pinning arises from a variation of the order parameter within the cell structure. The change of the order parameter from cell to cell leads to the spatial variation of the Josephson coupling energy between the cells in the ab planes and between the cells on adjacent planes in the c direction. We made a preliminary investigation of an effect of irradiation with uranium ions of high dose on and the relaxation of the current J(t) from the level for YBCO film in which the magnitude of is dominated by an AB component at low temperatures.


513

This type of irradiation produces columnar tracks of density that can trap magnetic field of 0.5 T. Diameter of the tracks is of the order of 8–10 nm. Figure 5 shows the temperature dependence of the critical current and the effective energy barrier as a function of the current for YBCO film before and after irradiation. The results imply that there is a very

little change to both and due to this type of irradiation-induced defects. In the irradiated film, at low temperatures increased by about 10% and decreased by 2°. remained similar to that calculated for the unirradiated film. This suggests that the motion of vortices through this YBCO film is dominated by the Josephson junction array. In summary, we measured the temperature dependence of the critical current density and the decay of the current from the critical value in 1 1 YBCO thin films manufactured using various deposition methods. The results show the universal behavior of both and calculated from the relaxation data. Both quantities suggest that Josephson effects are responsible for the observed dependence of

on temperature and

on J.

An AB-like component in is similar to that measured in a NbN film of nanometer size granularity and implies the presence of coherent Josephson nanostructures in the ab planes of YBCO. We postulate (taking also the irradiation experiments into account) that the coherent Josephson junction array could dominate the pinning and flux motion in YBCO thin films. Recent theoretical work by Castro Neto [6] proposed a scenario for the problem of phase coherence and superconductivity in striped cuprates that is based on the assumption of a network of stripes and lakes of holes in the planes. In this model the stripes are pinned by impurities, and a Josephson current is transferred from stripe to stripe via the network.

514


ACKNOWLEDGMENTS We are grateful to J. Preston, R. Hughes, J. Z. Sun, A. Fife, and J. Talvacchio for supplying us with YBCO thin films. This work was supported by a grant from the Natural Sciences and Engineering Council of Canada. REFERENCES 1. 2. 3. 4. 5. 6.

J. Etheridge, Philos. Mag. A 73, 643 (1996). J. R. Clem, B. Bumble, S. I. Raider, W. J. Gallagher, and Y. C. Shih, Phys. Rev. B 35, 6637 (1987). M. P. Maley, J. O. Willis, H. Lessure, and M. E. McHenry, Phys. Rev. B 42, 2639 (1990). H. Darhmaoui and J. Jung, Phys. Rev. B 53, 14621 (1996). M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996), pp. 205–206. A. H. Castro-Neto, Phys. Rev. Lett. 78, 3931 (1997).

Evidence of Chemical Potential Jump at Optimal Doping in Z. G. Li1 and P. H. Hor1

We have studied the evolution of structure and electrochemical potential of at for We show that, at elevated temperature, the electrochemical intercalation is close to equilibrium. In a sudden increase of potential is observed when approaches 0.12 in the middle of a biphasic region This anomalous potential increase depends, instead of excess oxygen content only on the hole concentration Our results suggest that there is a sudden jump of chemical potential of holes at optimal doping concentration.

1. INTRODUCTION Oxygen-doped

exhibits rich and interesting physical properties such as

macroscopic chemical phase separation [1], microscopic inhomogeneous distribution of doped holes [2], graphite-like staging behavior of the interstitial oxygen [3,4], onedimensional (1D) modulation of in-plane ordering [5], and the change in doping efficiency at a critical hole concentration [6]. Among all the techniques used to introduce excess oxygen into the electrochemical intercalation [7] is the most attractive. It can achieve the highest possible excess oxygen content in a well-controlled way. However, we have found that room-temperature electrochemical oxidation produces

samples far from thermodynamic equilibrium [8,9]. Subsequently, we have shown that electrochemical doping at elevated temperature with proper annealing produces well-behaved samples and the doping efficiency (number of holes introduced per excess oxygen atom) is

established [6]. This system, therefore, provides us a unique opportunity to study the physical properties of with precise and continuous control of carrier concentration. Unfortunately, the oxygen doping is limited to a maximum hole concentration 1

Department of Physics and Texas Center for Superconductivity at University of Houston, Houston, Texas 77204.



515

516

Li and Hor

which severely restricts our study of this system to be only in the underdoped region. To understand the physical origin of this limitation and in the hopes of extending our studies to the overdoped region, we studied the evolution of electrochemical potential of strontium and oxygen codoped It is found that the limitation of oxygen doping in is due to a chemical potential jump of doped holes at optimal doping level.

2. EXPERIMENT The starting materials,

(x = 0, 0.025, and 0.05), were prepared by

standard solid-state reaction [8]. Electrochemical oxidation (intercalation) and reduction (deintercalation) of were carried out in a three-electrode electrochemical cell with the sample pellet as the working electrode, a gold foil as the counterelectrode, Ag/AgCl as the reference electrode, and 1M KOH aqueous solution as the electrolyte. The cell was kept at constant current was applied to the working electrode to increase (negative current) or decrease (positive current) the excess oxygen content of the sample. The potential difference between the working electrode and the reference electrode was monitored vs time during the electrochemical process. X-ray powder diffraction data were collected at 25ºC–110°C using a Rigaku D/MAX-B x-ray diffractometer. Intensity data for angles ranging from 22° to 81° were taken in steps of in fixed intervals of 6 s. 3. RESULTS AND DISCUSSION

At thermodynamic equilibrium, when oxygen atoms are intercalated into the relation among Gibbs energy change electrode potential change and chemical potential change can be expressed by

Here, z stands for the charge of an oxygen ion and e the elementary charge [10]. Thus, measuring the electrode potential at equilibrium as a function of the total charge passed through the electrode is equivalent to measuring the chemical potential of each intercalated oxygen atom as a function of

The potential data obtained at 70°C for intercalation and deintercalation are shown in Fig. 1. The potential of is plotted vs the composition calculated from where Q is the charge transferred per formula unit. It is seen that the intercalation potential see curve a) and deintercalation potential ( see curve b) are parallel to each other over a wide range of compositions and the oxygen atoms inserted on oxidation can be completely removed on reduction, indicating that the process of oxygen insertion into at 70°C is reversible in terms of intercalation [11]. Furthermore, the potential difference between the oxidation process and the reduction process at fixed value is only about 40 mV in the range of (Notice that this potential difference is equivalent to the sum of overpotentials and which are due to kinetic barriers for inserting and removing interstitial oxygen.) These results indicate that is very close to, if not at, thermodynamic equilibrium during the process of oxygen intercalation. As a consequence, the open circuit potential (OCV) at 70°C, measured under the

Evidence of Chemical Potential Jump at Optimal Doping in

517

condition that “no” current passing between working electrode and counterelectrode, is taken as the equilibrium electrode potential. OCV as a function of composition is also plotted in Fig. 1 (curve c). Although the data were measured directly for those for

were taken as the average of intercalation and deintercalation potentials assuming the absolute values of and are equal. The shape of OCV curve is very similar to the potential curves of intercalation and deintercalation. A potential plateau can especially be observed in all three curves when The potential plateau implies there is no Gibbs energy change during oxygen intercalation into (or deintercalation from) the system in this composition range. In other words, the sample is at multiphase equilibrium. Indeed, x-ray diffraction data (Fig. 2) taken at 25°C show, as we reported previously [ 12], that two distinct orthorhombic phases coexist

in the samples of with and 0.11. The volume fraction, extracted by profile fitting, of the phase with larger orthorhombicity increases proportionally to the value. Clearly, there is a miscibility gap over the range of The upper limit was determined by extrapolation because no more excess oxygen atoms can be intercalated into above under equilibrium process. In the two-phase region, one may reasonably expect that the intercalation potential stays constant till if the overpotential does not change. Surprisingly, it begins to increase at and increases abruptly at (see Fig. 1). Now, we first show that the increase of is not due to an overpotential change. The overpotential is a kinetic parameter rather than a thermodynamic parameter. A much higher overpotential at required to intercalate more oxygen atoms into implies that there is

518

Li and Hor

a kinetic barrier to be overcome. To explain this kind of barrier, one must assume that the distribution of interstitial oxygen is inhomogeneous as increases. This can happen if the rate of electrochemical reaction occurring at the electrode surface is much faster than the rate of oxygen diffusion into bulk, which results in the accumulation of excess oxygen atoms inside electrode surface. Consequently, a thin layer inside the electrode surface should be oxidized to a much higher oxygen content at However, this hypothesis is not consistent with our observation that the value at which the potential suddenly increases is independent of the magnitude of the charging current. For various applied currents ranging from 3.6 to 18 the potential increase begins at the same value. Because the rate of electrochemical reaction at the surface is directly proportional to the applied current, it is expected that the potential jump should appear at different values for different currents. The upper limit of biphasic region was determined by x-ray diffraction data collected at 25°C whereas the potential data were taken at 70°C. Therefore, another possible cause of the potential increase at is the shift of the upper boundary of the biphasic region from at at 70°C. This possibility can also be ruled out. As shown by the results of x-ray diffraction (see the inset of Fig. 2b), the volume fraction of the phase with higher value in a sample of is only changed about 1% at 70°C ( %) compared with that at 25°C ( %), indicating that upper limit biphasic region at 70°C is almost the same as at 25ºC.

Evidence of Chemical Potential Jump at Optimal Doping in

519

When an oxygen atom is inserted into it occupies the tetrahedral site between two LaO layers [13] and contributes 2 (when ) or 1.3 (when ) holes to planes [6], becoming an oxygen ion. Therefore, the chemical potential of intercalated atom ( ) is the sum of contributions of oxygen ions and doped holes

Here, is lattice contribution and is the electronic contribution to the chemical potential of intercalated atom. Finally, the potential for intercalation can be written into three components

Because there is no change in the overpotential ( ) and chemical potential of ions ( ) at we attribute the sudden increase of electrode potential to the chemical potential jump of doped holes. In Fig. 3, potential vs hole concentration p is plotted. Clearly, this potential jump happens at the optimal doping level of highcuprates. To further verify that this chemical potential jump at optimal doping level is an intrinsic characteristic of holes in planes, we intercalated two Sr-doped samples of

for x = 0.025 and 0.05. During the intercalation, all the parameters were controlled in the same manner as those in intercalation of pure As shown also in Fig. 3, electrode potentials increase abruptly at same doping level ( ) for all three samples of (x = 0, 0.025, and 0.05), although their excess oxygen contents are different. Again, it indicates that this chemical potential jump is due to doped holes rather than intercalated oxygen ions. Several issues are associated with the potential jump at The first question we must answer is how high the chemical potential increases at this particular point. Unfortunately, by the current electrochemical intercalation method, the potential level after the jump cannot be reached because it exceeds the oxygen evolution potential at the electrode surface.

520

Li and Hor

Therefore, when the sample reaches optimal hole doping level, oxygen atoms energetically favor the formation of molecules and their release at the surface. Consequently, no more holes can be doped into planes. That is exactly the reason why the maximal doping level of electrochemically intercalated is about 0.16, which was reported by several groups [6,14] for values even higher than 0.12. Because the potential difference

between the potential before the jump and the oxygen evolution potential is about 0.15V, the chemical potential jump of doped holes is estimated to be no less than 0.15 eV. Summarizing all of the above observations, we therefore concluded that the origin of the maximum doping of oxygen is due to an intrinsic chemical potential jump of doped holes in planes. 4. CONCLUSIONS We have studied the electrochemical potential change as a function of

in

for and 0.05. We find that the maximum doping level is limited by an intrinsic chemical potential jump of doped holes at optimal doping level of the highcuprates. Further studies of the implications of this chemical potential jump to the destroy of high is underway. ACKNOWLEDGMENTS

This work was funded in part by ARPA (MDA 972-90-J-1001) and the state of Texas through the Texas Center for Superconductivity at University of Houston.

REFERENCES 1. J. D. Jorgensen, B. Dabrowski, S. Pei, D. G. Hinks, L. Soderholm, M. Morasin, J. E. Schirber, E. L. Venturini, and D. S. Ginley, Phys. Rev. B 3 8 , 11377 (1988). 2. J. H. Cho, F. C. Chou, and D. C. Johnston, Phys. Rev. Lett. 70, 222 (1993). 3. B. O. Wells, R. J. Birgeneau, F. C. Chou, Y. Endoh, D. C. Johnston, M. A. Kastner, Y. S. Lee, G. Shirane, J. M. Tranquada, and K. Yamada, Z. Phys. B 100, 535 (1996).

4. X. Xiong, Q. Zhu, Z. G. Li, S. C. Moss, H. H. Feng, P. H. Hor, D. E. Cox, S. Bhavaraju, and A. J. Jacobson, J. Mater. Res. 11, 2121 (1996). 5. X. Xiong, P. Wochner, S. C. Moss, Y. Cao, K. Koga, and M. Fujita, Phys. Rev. Lett. 76, 2997 (1996). 6. Z. G. Li, H. H. Feng, Z. Y. Yang, A. Hamed, S. T. Ting, P. H. Hor, S. Bhavaraju, J. F. DiCarlo, and A. J.

Jacobson, Phys. Rev. Lett. 77, 5413 (1996). 7. J. C. Grenier, A. Wattiaux, N. Lagueyte, J. C. Park, E. Marquestaut, J. Etourneau, and M. Pouchard, Physica C 173, 139(1991).

8. H. H. Feng, Z. G. Li, P. H. Hor, S. Bhavaraju, J. F. DiCarlo, and A. J. Jacobson, Phys. Rev. B 51, 16499 (1995). 9. S. Bhavaraju, J. F. DiCarlo, I. Yadzi, A. J. Jacobson, H. H. Feng, Z. G. Li, and P. H. Hor, Mat. Res. Bull. 29, 735(1994). 10. W. R. McKinnon, in Solid State Electrochemistry, edited by P. G. Bruce (Cambridge University Press, Cambridge, 1995), p. 175. 11. M. S. Wittingham, in Intercalation Chemistry, edited by M. S. Whittingham and A. J. Jacobson (Academic Press, New York, 1982), p. 1. 12. P. H. Hor, H. H. Feng, Z. G. Li, J. F. DiCarlo, S. Bhavaraju, and A. J. Jacobson, J. Phys. Chem. Sol. 57, 1061 (1996).

13. C. Chaillout, S. W. Cheong, Z. Fisk, M. S. Lehmann, M. Marezio, B. Morosin, and J. E. Schirber, Physica C 158, 183(1989).

14. F. C. Chou, J. H. Cho, and D. C. Johnston, Physica C 197, 303 (1992).

Studies of the Insulator to Metal Transition in the Deoxygenated System

The deoxygenated

samples have been investigated. The

substitution of Ca in the Y places induces the insulator to metal transition and the superconductivity appears at The ac magnetic susceptibility measurements show that the effective magnetic moment increases with Ca content.

The electrical resistivity vs temperature measurements provided an experimental material to discuss various models responsible for electrical transport properties.

They are analyzed for different values of the x parameter in the insulator to metal transition. The Mott law fits best to our experimental data.

1. INTRODUCTION

The calcium substitution has been investigated by many researchers in both oxygenated [1,2] and deoxygenated [1,3–6] systems. It is known that Ca is substituted mostly to the Y places [2,7]. The substitution of divalent Ca in the places of trivalent Y increases the concentration of holes. Although Ca is substituted to fully

oxygenated Y-123 compound, the system becomes overdoped and decreases. If the samples are deoxygenated, it acts also as the electric carrier donor, induces insulator to metal transition and superconductivity [3–6]. However, superconductivity induced in such a way does not reach high critical temperatures because of impurity phases accompanying the doping process. With both oxygen and calcium doping we obtain similar phase diagrams, and transitions from antiferromagnetic (AF) to superconducting phase [6,8]. In the phase diagram with the changing Ca content, it is clearly visible that the intermediate region between AF and superconducting phases exists [6]. With Ca substitution, it is easier to 1 2

Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland. Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Kraków, Poland.



521

522 manipulate in the carrier concentration, it does not change the crystal symmetry as we induce superconductivity in the tetragonal samples.

Recently, the inelastic neutron-scattering experiments provided evidence for the dynamic stripe phases in The hole-rich regions and AF domains form alternating stripes. In the AF spin correlations become completely dynamic for The fluctuating stripes were detected in superconducting samples with higher oxygen content: and The fluctuations are not present in nonsuperconducting oxygen-deficient they appear only at higher carrier concentration. The modulation period of analogic stripes in decreases with hole doping [14]. By Ca substitution, we perform analogic hole doping, and in deoxygenated samples with higher Ca concentration, the occurrence of dynamic stripes is also possible. The aim of this work is to study the effect of Ca substitution in the deoxygenated samples and the metal-insulator transition induced by Ca doping. The different models of the transport properties are analyzed.

2. EXPERIMENTAL The samples have been prepared by the standard ceramic method. The powers and CaO were mixed in appropriate proportions, pressed into pellets, and sintered in air in 936°C for 24 hr. Then the samples were crushed and sintered in flowing oxygen twice, first in 942°C for 30 hr, then in 943 to 948°C for 62 hr, then kept 450°C and slowly cooled to room temperature. After confirmation of their superconducting properties, they were sintered in following argon in 730°C for 24 hr, which was followed by slow cooling to room temperature. The x-ray diffraction (XRD) measurements were carried out in room temperature using Phillips diffractometer type X’pert with PW 3020 goniometer. radiation was applied. The measurements were performed in the angle range of with the step of 0.02° and 8 s collecting data time per single point to obtain sufficiently accurate diffractograms and determine the amount of impurity phases quite precisely.

Iodometric titration was performed to obtain the average copper valency and oxygen content. ac magnetic susceptibility vs temperature curves were recorded using the mutual inductance method at a frequency of 21.5 Hz from 300 K down to 11 K. The rms value of the applied field was 30 Oe. The electrical resistivity measurements were performed by the four-point contact method with reversing current direction in the range of temperatures from 300 K to 4.2 K. 3. RESULTS The structural analysis performed by XRD showed that all the samples have the tetragonal symmetry. Most of Ca substitutes to Y positions and only a small amount substitutes to Ba positions. We detected the impurity phase of whose content increases with

Ca content (Table 1) and less than 0.08% weight fraction of CuO. Iodometric titration made possible to determine the average copper valency and the oxygen content. The results are in the Table 1. All the oxygen indexes are close to 6.1. We

Insulator to Metal Transition

523

also see that Ca substitution increases the average copper valency, which is well visible at the higher Ca concentrations. The results of ac magnetic susceptibility measurements are presented in the Fig. 1. In the system with the small superconducting part of the sample manifests as a weak diamagnetism with onset temperature equal to 14 K. The sample with exhibits strong diamagnetism related to superconductivity below the temperature of 35 K. The superconductivity in the sample with is not visible above 4.2 K in the resistivity curve, whereas the sample with has the onset in resistivity equal to 35 K

524

and the zero resistivity at 19.5 K (Fig. 2). Despite the zero resistivity, for the sample with the diamagnetic signal is too weak to indicate that the whole sample becomes superconducting in the ac magnetic field of 30 Oe rms value (inset a in Fig. 1). Superconductivity is very sensitive to the applied magnetic field here. In the temperature of is equal to emu/g for the value of applied field Oe and equals emu/g for Analogic measurement of the oxygenated sample gives emu/g for Oe in this temperature. The Curie–Weiss law was fitted to the paramagnetic regions of ac susceptibility curves. The Curie–Weiss behavior was also observed in the sample with regarded as antiferromagnetic, which has, however, paramagnetic contribution. The determined effective magnetic moment is presented in the Table 1 and drawn in the inset b of Fig. 1. The magnetic moment dependence on the Ca concentration seems to be linear. Some part of the effective magnetic moment originates from However, the effective magnetic moment of is equal to per Cu atom [15], and as was calculated and presented in Table 1 and in the Fig. 1 is too weak to be responsible for paramagnetism in these samples. The detected could give only small part of the total magnetic moment. The increasing effective magnetic moment in samples is probably related to hole doping into AF matrix. The electrical resistivity vs temperature curves are drawn in Fig. 2. We tested several models for (resistivity vs temperature) curves that have already been observed in the HTS related compounds. The curves were drawn in appropriate scales, functions were fitted to experimental data, and the test was applied. The tested


525

dependencies are described as follows. Relation

is the result of theoretical predictions for disordered metals [16]. This law was observed in with impurities [17]. Conductivity following the relation

indicates weak localization in two dimensions or electron–electron interactions in the presence of disorder in two dimensions [16], observed in NCCO in ab plane [18]. Equation

with exponents

and 1/4 is related to the following models:

—thermally activated conductivity, observed e.g., in

at high

temperatures [19]; —variable range hopping conductivity in the presence of a Coulomb gap [20],

found, e.g., in —variable range hopping conductivity in two dimensions (Mott's law) [22], observed, e.g., in

—variable range hopping conductivity in three dimensions (Mott's law) [22], observed, e.g., in The smallest values have been obtained for Eq. (3) with and not much bigger for We observed that fitting curves preferred in lower temperatures and in higher temperatures. This indicates the existence of hopping conductivity in two and three dimensions. The fitting results for are printed in Table 2. The fits do not include the lowest temperatures, where some little changes in the behavior were observed but still are not confirmed. In Fig. 3, the

are drawn vs temperature in

and

dependencies in logarithmic scale

Linear regions are the indications that Mott's

law is obeyed.

In the inset of Fig. 2, there are dependencies of resistivity vs Ca content at constant temperature. As

from

to

is in the logarithmic scale,

dependencies seem to be exponential

526


for

527

In the more metallic samples we have dependencies in high temperatures to 0.3, which indicates electron–phonon scattering (Fig. 4).

4. CONCLUSIONS Ca substitution in the deoxygenated system induces superconductivity at however, this superconductivity may be related to only small part of the sample and is easily reduced by magnetic field. The onset of the insulator to metal transition appears at The chosen criterion is the change from positive to negative slope of resistivity vs temperature curves at room temperature. The increase in Cu valency was observed for increasing Ca content. Low-temperature resistivity of the samples can be fitted to the law with in higher and in lower temperatures. This suggests variable range hopping mechanism of conductivity. In the samples with more Ca, we observe linear dependence of The linear increase of the effective magnetic moment with Ca concentration is observed. The impurity is not capable to be the origin of such a high effective magnetic moment. This moment should originate mainly from compound, and the increase is caused by Ca substitution.

528 ACKNOWLEDGMENT

This work was supported by the Polish Committee for Scientific Research under the Grant No. 2 P03B 024 13. REFERENCES 1. Y. Tokuru, J. B. Torrance, T. C. Huang, and A. I. Nazzal, Phys. Rev. B 38, 7156 (1988). 2. R. G. Buckley, D. M. Pooke, J. L. Tallon, M. R. Presland, N. E. Flower, M. P. Staines, H. L. Johnson,

M. Meylan, G. V. M. Williams, and M. Bowden, Physica C 174, 383 (1991). 3. E. M. McCarron III, M. K. Crawford, and J. B. Parise, J. Solid State Chem. 78, 192 (1989).

4. J . B . Parise and E. M. McCarron, J. Solid State Chem. 83, 188(1989). 5. R. S. Liu, J. R. Cooper, J. W. Loram, W. Zhou, W. Lo, P. P. Edwards, W. Y. Liang, and L. S. Chen, Solid State Commun. 76, 679 (1990). 6. H. Casalta, H. Alloul, and J.-F. Marucco, Physica C 204, 331 (1993). 7. P. Starowicz, S. Baran, A. and Yong Fan Dong, Mol. Phys. Rep. 15/16, 237 (1996). 8. H. D. C. Johnston, S. K. Sinha, A. J. Jacobson, and J. M. Newsam, Physica C 153–155, 572 (1988). 9. H. A. Mook, P. Dai, K. Salama, D. Lee, F. Dogan, G. Aeppli, A. T. Boothroyd, and M. E. Mostoller, Phys. Rev. Lett. 77, 370 (1996). 10. P. Dai, H. A. Mook, and F. Dogan, Phys. Rev. Lett. 80, 1738 (1998). 11. R. F. Kiefl, J. H. Brewer, J. Carolan, P. Dusanjh, W. N. Hardy, R. Kadono, J. R. Kempton, R. Krahn, P. Schleger,

B. X. Yang, Hu Zhou, G. M. Luke, B. Sternlieb, Y. J. Uemura, W. J. Kossler, X. H. Yu, E. J. Ansaldo,

H. Takagi, S. Uchida, and C. L. Seaman, Phys. Rev. Lett. 63, 2136 (1989). 12. P. Bourges, L. P. Regnault, J. Y. Henry, C. Vettier, Y. Sidis, and P. Burlet, Physica B 215, 30 (1995). 13. L. P. Regnault, P. Bourges, P. Burlet, J. Y. Henry, J. Rossat-Mignod. Y. Sidis, and C. Vettier, Physica B

213–214, 48 (1995) 14. K. Yamada, C. H. Lee, J. Wada, K. Kurahashi, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, and M. A. Kastner,J. Supercond. 10, 343 (1997). 15. R. Z. Bukowski, R. and J. Klamut, Phys. Lett. 125, 222 (1987).

16. P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985). 17. M. Z. Cieplak, S. Guha, H. Kojima, P. Lindenfeld, G. Xiao, J. Q. Xiao, and C. L. Chien, Phys. Rev. B 46, 5536 (1992). 18. A. Kussmaul, J. S. Moodera, P. M. Tedrow, and A. Gupta, Physica C 177, 415 (1991). 19. P. Mandal, A. Poddar, B. Ghosh, and P. Choudhury, Phys. Rev. B 43, 13102 (1991). 20. B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin, 1984). 21. B. Ellman, H. M. Jaeger, D. P. Katz, T. F. Rosenbaum, A. S. Cooper, and G. P. Espinosa, Phys. Rev. B 39, 9012 (1989). 22. N. F. Mutt and E. A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed. (Oxford University Press, London, 1979).

Differential Optical Reflectivity Measurements of W. R. Studenmund,1 I. M. Fishman,1 G. S. Kino,1 and J. Giapintzakis2

A pump-probe thermal modulation technique has been used to measure temperature-dependent changes in differential optical reflectivity (DOR) on (YBCO) as a function of sample temperature. Measurements have been made at five probe frequencies in the range of 0.8–1.96 eV. We have found distinctive evidence of the superconducting transition at all frequencies. In the normal-state, we are able to fit the DOR curves fairly well with a modified Drude model. The superconducting-state DOR frequency variation is inconsistent with trivial explanations, such as being due to thermal property discontinuities at the superconducting transition. We believe this frequency variation is evidence of new physical processes in YBCO.

A photothermal microscope has been used to measure the rate of change of optical reflectivity with temperature (DOR). Two laser beams, the pump and the probe, are focused on the sample; with the focused spots typically apart. The periodic modulation of the pump generates thermal waves that propagate across the sample, harmonically modulating the temperature under the probe beam, which in turn gives rise to an ac-reflected signal. A photodiode and lock-in amplifier are used to measure the probe signal. Details of the experiment are discussed elsewhere [1]. We examined YBCO and observed a sharp change in the DOR as a function of temperature at the superconducting critical temperature. This sharp change permits the determination of the critical temperature without touching the sample. The nature of the sharp change varies from sample to sample, with probe optical frequency, and with probe beam polarization. All samples are detwinned to clearly resolve the strong ab optical anisotropy. 1 2

Edward L. Ginzton Laboratory, Stanford CA 94305-4085. Department of Physics and Materials Research Laboratory, University of Illinois, 1 1 1 0 West Green Street, Urbana IL61801.



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The photothermal experiment yields output amplitude in arbitrary units that are calibrated to give absolute values of The calibration process is described elsewhere [1]. These curves include all calibrations except for conductivity variation. The major source of uncertainty in the calibration is the determination of the thermal modulation. We put the combined calibration uncertainty for most measurements. Unfortunately, the 1064 nm measurement was more ambiguous than normal, and that value could have been twice the value used. Figures 1 and 2 show the a axis DOR results obtained on two samples of Both samples had a

near 93 K.

These curves show a tremendous variation in DOR response both in terms of a normal and a superconductivity-onset response. All curves show a strong discontinuity at (visible

as the right edge of the feature at ). We believe the normal state response is primarily due to plasma effects. To test this hypothesis, we start with a standard Drude model,

Differential Optical Reflectivity Measurements of

with representing higher-energy oscillators, the plasma frequency, and tering rate. The optical power reflectivity is given by

531

the scat-

We then take the scattering rate to vary linearly with frequency over the range of measurement,

Obviously, this model is not causal for and is at best an approximation for the case where is small over the frequency range of interest (0.8–1.96 eV). Figure 3 shows direct (temperature-independent) reflectivity measurements for the two samples studied here, results from the literature [2], and fits using Eq. (1). The literature curve is fit very well with a curve taking and The quality of this fit indicates that Eq. (2) captures much of the physics. As this fit is fairly insensitive to the exact value of B for B small ( mEV did not noticeably differ from ), we take in the rest of this work. Given room-temperature values for and A, we can then calculate and from Eq. (1). Given DOR measurements at three wavelengths, we can write a system of equations relating these DOR measurements to temperature variations of the three parameters, and The beauty of this separation is that all of the frequency dependence is contained in the terms and the temperature dependence in the (T ) terms. If we have more measurements than model parameters, we can use the extracted parameter temperature variation to predict the measurements at wavelengths not used in the extraction process. We can thus test the self-consistency of the model. As we must add a scaling parameter for each experimental curve to compensate for the uncertainty in the calibration of each curve, we must have at least two more measurement wavelengths than model parameters. For this case, we have five measurements and three model parameters. Adding scaling parameters, we have free parameters. We also have 10 combinations of wavelengths (5 choose to obtain and (we can choose 1064, 780,

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and 633 nm; or 1310, 780, and 633 nm; or 1310, 1064, 633 n m . . . ) . For any one choice, we call the chosen wavelengths the fit wavelengths, as we fit the model to the DOR at these wavelengths. The other wavelengths we call the predicted wavelengths, as we can use the model parameter temperature variation to predict them. We can thus formulate a self-consistency test for a model of We say a model is self-consistent if (1) all combinations of fit wavelengths, the predictions from the model at the predicted wavelengths, agree with the experimental results for these wavelengths; (2) the scalings of the experimental curves needed for this fit are within the systematic error of the calibration; and (3) the model parameters initially used agree with the direct reflectivity of the material. Figures 4 and 5 show, respectively, the 1310 nm and 633 nm results of such a test for the of Eq. (1). The 1550 nm, 1064 nm, and 780 nm results are omitted for brevity. Remember that for each wavelength we choose to model, only three of the other four datasets are needed to fit the theory. We thus have four choices as to the curve to throw out.

Differential Optical Reflectivity Measurements of

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These plots show the results for ignoring each of the (four) possible wavelengths in turn. In these plots, the dots are the scaled experimental curve, and the four other curves represent

a modeling with one particular wavelength ignored. The solid line is for 1550 nm ignored, the dashed 1064 nm, the dot-dashed 780 nm, and the solid with crosses the 633 nm (for Fig. 4) or 1310 nm (for Fig. 5) data ignored. These fits were obtained with

and The experimental curves were scaled by 1.15 (1550 nm), 1.1 (1310 nm), 0.4 (1064 nm), 0.75 (780 nm), and 1.15 (633 nm). The 1064 nm scaling, although large, is consistent with the measurement for that curve. The 780 nm scaling is not consistent with the error we claim for the calibration. We thus must conclude that the of Eq. (1) comes close to but does not agree fully with the experimental results. Given that we have only taken Eq. (1) as an approximation, Figs. 4 and 5 indicate that it gives a very effective fit above We feel that the agreement between the theory and experiment is as close as could be reasonably expected from Eq. (1). We thus conclude that the normal-state reflectivity is given by a dielectric function similar to the Drude approximation of Eq. (1). Below the theoretical predictions agree neither with the experiment nor with each other. If the optical consequence of the onset of superconductivity were modelable as (potentially drastic) changes in the three model parameters then the prediction curves should match the experiment below In addition we have attempted to model the below- part using Eq. (1) and any model and scaling parameters, neglecting the above- DOR, without success. As no fitting works, we must conclude that the superconductivity response is not modelable by a Drude plasma approximation. The superconducting condensate is typically modeled as a function in the real part of the conductivity, and is considered the dominant consequence of superconductivity. Causality requires a term in the imaginary part of the conductivity, This term generates a very weak reflectivity change that should be indistinguishable from that due to the normal carriers. We do not know the origin of the superconducting DOR response, but we believe it represents either a new consequence of the superconducting condensate, such as changes in optical matrix elements, or some new feature in the excitation spectrum. These ideas are discussed further in Ref. [3). In conclusion, we present a axis DOR results for two. YBCO samples at five measurement wavelengths. We fit a simple extension of the Drude model for to the roomtemperature optical reflectivity data, and use the parameter values thus obtained to selfconsistently model the DOR results. We have successfully modeled the normal-state DOR using this extended Drude model. The superconducting state DOR are not explainable with this model. ACKNOWLEDGMENT This work was supported by DOE contract DE-FGO3-90ER14157.

REFERENCES 1. W. R. Studenmund, I. M. Fishman, G. S. Kino, and J. Giapintzakis, unpublished, 1998. 2. S. L. Cooper et al., Phys. Rev. B 47, 8233 (1993). 3. I. I. Fishman, W. R. Studenmund, and G. S. Kino, p. 495 in this volume.


On Some Common Features in Highand Low- Superconducting Perovskites I. Vobornik,1 D. Ariosa,1 H. Berger,1 L. Forró,1 R. Gatt,2 M. Grioni,1 G. Margaritondo,1 M. Onellion,2 T. Schmauder,1,2 and D. Pavuna1

Recent photoemission evidence for non-zero gap in highly overdoped along the directions in the Brillouin zone where the d-wave gap exhibits nodes, is in accordance with more conventional behavior in strongly overdoped highcuprates. This differs from underdoped regime, where departures from Fermi-liquid (FL) behavior are found. Observed non-FL linear resistivity up to in low- perovskite is reminiscent of cuprates and implies some similarities between high- and low- perovskites.

1. INTRODUCTION Understanding of electronic properties of highcuprates [1] and their complex electronic phase diagram (Fig. 1) still presents a major challenge, despite thousands of research papers and remarkable progress [1,2] in both sample preparation and advanced experimental techniques. In this paper, we briefly discuss some anomalies found in highcuprates as well as the anomalous transport [5] recently found in a low- perovskite ruthanate, 2. ELECTRONIC PHASE DIAGRAM AND SPECTRAL FUNCTION EVOLUTION IN HTSC OXIDES Figure 1 illustrates simplified phase diagram of high temperature superconductors.

Two predominant features in the underdoped region are shown in number of experiments. Those are the existence of the pseudogap [ 1,7] and of the stripe correlations [7] of spins and 1

2

Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.

Physics Department, University of Wisconsin, Madison, WI 53589, USA.


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charges [7]. In transport, pseudogap is often seen as a change of slope at

in otherwise linear temperature resistivity dependence. Although the origin of the stripe correlations is discussed by number of authors at this conference, there are two major lines of thought [8,9] about the origin of The first approach considers that (possibly bipolaronic) pairs are preformed at and in an outcoming two-component scenario, the s-wave component of superconducting coherence gap should appear in most optimally and overdoped samples. On the contrary, models that emphasize the role of antiferromagnetic (AF) correlations require

pure d-wave scenario [ 1 ] . On further doping, in a highly overdoped regime, cuprates seem to behave as Fermi liquids (FL) and the superconducting state resembles the BCS state. Generally, FL behavior is expected in oxides whose in-plane resistivity exhibits usual dependence at low temperatures. However, as we argue in Section 3, the growth-induced disorder and oxygen deficiency can induce the non-FL behavior and produce anomalous linear resistivity [5], even in the case of low- perovskite-like So far, the stripe

phase was not found in ruthanates, although it was reported in nickelates, manganates, and cuprates [7]. Figure 2 shows the normal and superconducting state angle-resolved photoemission spectra (ARPES) obtained for highly overdoped (a) and highly underdoped (b) high-temperature superconductor. The spectra are taken at the location in Brillouin zone where d-wave gap exhibits maximum. Although in the overdoped regime

Common Features in High- and Low-

Superconducting Perovskites

the gap vanishes once the temperature is higher than

537

the “normal” state spectrum in

underdoped regime remains gapped. Furthermore, in the underdoped system, the absence of coherent features in both the normal and the superconducting state spectra indicates strong departure from the FL behavior, as viewed by photoelectron spectroscopy [3]. The superconducting gap of the underdoped and optimally doped cuprates exhibits dominant d-wave symmetry [1,3]. Nevertheless, ARPES data on highly overdoped indicate a finite gap along the

direction in the Brillouin zone [4], where

the d-wave gap has nodes. This point is still under active investigation. Due to badly defined surface termination of most cuprate samples, almost all ARPES studies since the late 1980s were done on single crystals, even though a large fraction of other physical studies and electronic applications was reported on various

families of superconducting compounds. In order to overcome this limitation, we have built a pulsed-laser ablation system in which superconducting films are grown and transferred under ultra-high vacuum to the photoemission chamber. First ARPES studies are in progress. 3. ANOMALOUS LINEAR RESISTIVITY IN

Finally, we note a very recent result (Fig. 3) on transport studies of the first noncuprate perovskite superconductor, single crystals grown by flux technique [5]. Although the temperature dependence of the Hall coefficient is similar to results reported in cuprates, the linear resistivity persists up to ~ 1050 K, with superconductivity being confined This suggests that even in which normally does exhibit typical (low

temperature) FL behavior, the non-FL behavior can be induced by growth-induced disorder (and oxygen vacancies). Furthermore, the linear temperature dependence of resistivity is obviously not an exclusive signature of the anomalous normal state of highcuprates, but rather of layered oxides in general, especially layered perovskites, possibly even independently of the magnitude of the superconducting critical temperature [5]. We note that the reflectance, ellipsometric, and Raman spectra obtained on thin films of isotropic metal-

lic oxides [6] and closely resemble the spectra of highcuprates, thus indicating that the “anomalous” dielectric response could also not be the sole root of high- superconductivity; the resistivity of nonsuperconducting is

also anomalously linear [6]. The exact role of the stripe phase in anomalous transport of layered oxides remains to be clarified.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support by Ecole Polytechnique Fédérale de Lausanne, the University of Wisconsin, and the Swiss National Science Foundation.

REFERENCES 1. The Gap Symmetry and Fluctuations in High- Superconductors, Proceedings of the NATO Advanced Study Institute, Cargese, September 1997, edited by J. Bok, G. Deutscher, D. Pavuna, and S. A. Wolf (Plenum Press, 1998). 2. D. Pavuna and I. Bozovic, eds., Oxide Superconductor Physics and Nano-Engineering I, II & III, 2058, 2697, 3481—SPIE (Bellingham, USA, 1994, 1996, 1998). 3. H. Ding el al., Phys. Rev. B 54, R9678 (1996). 4. C. Kendziora et al., Phys. Rev. Lett. 77, 727 (1996); I. Vobornik et al., unpublished. 5. H. Berger, L. Forró, and D. Pavuna, Europhys Lett. 41(5), 531 (1998). 6. I. Bozovic et al., Phys. Rev. Lett. 73(10), 1436 (1994). 7. M. Randeria and J. C. Campuzano, Varenna Lectures 1997, Report No. LANL cond-mat/9709107, and referenced therein; J. M. Tranquada et al., Nature 375, 561 (1995); N. L. Saini et al., Phys. Rev. B 57, Rl1101 (1998), and references therein. 8. K. A. Müller, p. 1 in this volume; D. Mihailovic et al., Phys. Rev. B 57, 6116 (1998). 9. D. Pines, see the contribution in Ref. 1; P. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994).

Pinning Mechanisms in a-Axis-Oriented and Multilayers E. M. González,1 J. M. González,2 Ivan K. Schuller,3 and J. L. Vicent1

a-axis-oriented planes perpendicular to the substrates) and multilayers have been grown by dc sputtering on (100) susbstrates and characterized by the refinement of the structure from x-ray spectra. The results show that interfacial structure (thickness fluctuations, interdiffusion, etc.) of the a axis are similar to the c-axis oriented superlattices. The angular dependence of the resistivity in the mixed state allows us to study the interplay among different types of dissipation mechanisms, as, for example, intrinsic, surface, microscopic defects, and superlattice-induced pinning mechanisms. These mechanisms are relevant in different angular, magnetic field, and

temperature regimes.

1. INTRODUCTION One of the most recent [1] and active topics (see the whole issue in Ref. [2]) in highsuperconductors is the development of the charge stripes in the planes of some families of cuprates. The study of samples with the possibility of changing at will the length of the planes could be crucial in this stripes scenario. Very recently, several authors [3] and [4] have fabricated a-axis-oriented 123 films and superlattices. These a-axis-oriented systems allow us to study experimental situations that are impossible in single crystals and in the usual c-axis-oriented films. 1

2

Dept. Física de Materiales, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain. Instituto de Ciencia de Materiales, Dept. Propiedades Opticas, Magnéticas y de Transporte (C.S.I.C.), 28049

3

Madrid, Spain. Dept. Physics, University of California San Diego, La Jolla, CA 92093-0319, USA.


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González, González, Schuller, and Vicent a-axis films grown on the usual cubic substrates have a microstructure with 90°

microdomain (average size 20 nm) [5]. a-axis superlattices are very peculiar in comparison with the c-axis multilayers because in a-axis, the planes are perpendicular to the substrate and the effects due to the natural anisotropy planes) and the artificial anisotropy (insulating layers) are uncoupled. An additional important aspect in a-axis superlattices of 123 superconductor/123 nonsuperconductor (for instance is

that planes can be locally superconducting or insulating depending on whether the rare-earth neighbors of the plane are Eu or Pr. In this paper, we present a carefully structural characterization of two types of superlattices, and , The former with the planes (superconducting/nonsuperconducting) running along the whole sample, and the

latter with the

planes interrupted by the

layers. However, in these artificially

structured systems, different dissipation mechanisms could compete. We show experimental results that could clarify the anisotropy, magnetic field, and temperature requirements

needed to enhance different kinds of dissipation mechanisms. 2. EXPERIMENTAL Superlattices of the so-called a-axis orientation (Cu-O planes perpendicular to the susbtrate) of and (EBCO/STO)

were grown by dc magnetron sputtering from stoichiometric targets. The multilayers were grown by alternately depositing and layers using two independent targets and stopping the substrates in front of the EBCO and PBCO or STO cathodes by a computer-controlled stepping motor. The samples were fabricated on (100) (STO) and (100) (LAO) substrates, and with a total thickness of 250 nm. A commercial cryostat with a 90 kOe superconducting magnet, a temperature controller, and a rotatable sample holder computer controlled by a stepping motor allows us to take angular

dependence resistivity measurements with different values of the applied magnetic field. The structural and superconducting characterization of the multilayers have been reported elsewhere [6]. A powerful technique (SUPREX program) [7] of structural refinement of

x-ray diffraction (XRD) profiles from superlattices has been used to characterize the quality of these a-axis EBCO/PBCO and EBCO/STO multilayers. In a-axis superlattices, it is not possible to subtract the diffraction maxima coming from the substrate as it is usually done in

the c-axis-oriented superlattíces. These substrate maxima have to be included in the fitting data. The substrate peaks are at (STO) and (LAO), the (200) peak is at (a-axis-oriented sample), and the (005) peak is at (c-axis-oriented sample). Figure 1 shows the refinement result for both kind of superlattices. This method gives the actual thickness of the layers and parameters related with the interface. In these samples the refinement implies that the interface step disorder is around 1 unit cell and

the interface diffusion 20%, values very similar to the values obtained in c-axis-oriented YBCO/PBCO multilayers [8]. 3. RESULTS AND DISCUSSION The angular dependence of the resistivity with an applied magnetic field is a good tool to study the anisotropy of the dissipation mechanisms. The measurements have been done in the

Pinning Mechanisms in a-Axis-Oriented Multilayers

541

vortex liquid region close to Velez et al. [3] reported in a-axis-oriented 123 superlattices two minima in the angular dependence of the resistivity with constant temperature and applied magnetic field. These minima occur (a) when the magnetic field is applied parallel to the substrate or (b) with the applied field perpendicular to the substrate. These authors suggest that these two resistivity minima (critical current maxima) are due to two different pinning mechanisms. When the magnetic field is applied parallel to the planes, the intrinsic anisotropy leads to the pinnig due to the depression of the superconducting order

parameter between the planes. Otherwise when the magnetic field is applied parallel to the substrate (perpendicular to the planes) the magnetic field could be pinned by the artificial anisotropy due to the superlattice modulation. In this case, the PBCO insulating layers play a similar role to the areas between the superconducting planes in the intrinsic anisotropy case. In this artificially induced effect, Velez et al. [3] found that, when the modulation length of the multilayers and the vortex lattice parameter [given by the value of the applied magnetic field, have similar values an enhancement of the critical current occurs for a a range of applied magnetic field around this matching

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field. Figure 2 shows these two minima in an a-axis-oriented EBCO (50 unit cells)/PBCO (5 unit cells) superlattice (matching field 5 T). This minimum still remains at temperature very close to and in an interval of applied magnetic field around the matching field value (see Fig.2B). The role of the artificially induced structure (in this case, the PBCO layers) could be better understood studying another artificially layered system. a-axis-oriented EBCO (250 u. c.)/STO (5 u. c.). Figure 3, A and B, shows the data at the same reduced temperature respectively) than in the EBCO/PBCO sample. In this EBCO/STO multilayer, the matching field is 0.2 T. However, in addition to the minima due to the intrinsic pinning, a clear “second” minimun appears in the high magnetic field region. This

“second” minimum vanishes when the temperature is increased (see Fig. 3B), whereas the actual minima due to the artificially induced anisotropy remains up to temperatures very close to (see Fig. 2B). The experimental behavior of this pinning mechanism seems to indicate that it is due to the defects in the EBCO layers and that is not related to the artificially structure. Prouteau et al. [4] recently reported the same kind of minima in pure a-axis-oriented films. The very peculiar microstructure in a-axis-oriented films with microdomains (see Ref. [5]) could pin the vortices when the magnetic field is parallel to

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543

the substrate. Another experimental fact that is interesting to point out is shown in detail in Fig. 4. A new minimun develops in the EBCO/STO multilayer at high temperature, close to

and low values of the magnetic field when the field is applied parallel to the substrate. These are the footprints of the surface pinning effect (see, for instance, Ref. [9]).

4. CONCLUSIONS In summary, a-axis-oriented 123 superlattices are the ideal system to study the competition among different dissipation processes.

The angular dependence of the resistivity shows two minima: (1) when the magnetic fieldis applied parallel to the planes (perpendicular to the substrate), and (2) when the applied magnetic field is parallel to the substrate (perpendicular to the planes) and parallel to the artificial structure. The former is due to the intrinsic pinning, and in the case of the latter, three different origins were experimentally detected: (a) microstructure defects

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(90° microdomains) in the superconducting layers, which were observed at high magnetic

fields and low temperatures; (b) surface and interface pinning, which was observed at low applied magnetic fields and temperatures close to and (c) pinning due to the artificially induced anisotropy (superlattice structure), a very effective mechanism that could be tuned with the value of the modulation length of the multilayer, and is relevant in a wide magnetic field interval, around a matching field, which is given by the modulation of the artificially layered structure. The insulating layers seem to be very effective pinning centers, and they could enhance the critical current when the applied magnetic field is around the matching field value and it is applied close to or parallel to the substrate. ACKNOWLEDGMENTS

This work was supported by Spanish CICYT grant MAT96-0904, Universidad Complutense, and by the U.S. Air Force MURI program at UCSD.

REFERENCES 1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995). 2. 3. 4. 5. 6.

J. Supercond. 10(4), (1997). M. Velez, E. M. Gonzalez, J. I. Martin, and J. L. Vicent, Phys. Rev. B 54, 101 (1996). C. Prouteau, F. Warmont, Ch. Goupil, J. F. Hamet, and Ch. Simon, Physica C 288, 243 (1997). C. B. Eom, A. F. Marshall, S. S. Laderman, R. D. Jacowitz, and T. H. Geballe, Science 249, 1549 (1990). J. I. Martin, M. Velez, and J. L. Vicent, Phys. Rev. B 52, R3872 (1995); M. Velez, E. M. Gonzalez, J. M.

Gonzalez, A. M. Gomez, and J. L. Vicent, J. Alloys Comp. 251, 218 (1997). 7. E. E. Fullerton, I. K. Schuller, H. Vanderstraeten, and Y. Bruynseraede, Phys. Rev. 45, 9292 (1992). 8. E. E. Fullerton, J. Guimpel, O. Nakamura, and 1. K. Schuller, Phys. Rev. Lett. 69, 2859 (1992). 9. J. Z. Wu and W. K. Chu, Phys. Rev. B 49, 1381 (1994).

Angular Dependence of the Irreversibility Line in Irradiated a-Axis-Oriented Films J. I. Martín,1 W.-K. Kwok,2 and J. L. Vicent1

a-axis-oriented thin films have been grown by dc magnetron sputtering on substrates. The samples have been irradiated with heavy ions in the GeV energy range in order to induce columnar detects at an angle relative to the planes. The angular dependence of the resistivity and the irreversibility line indicate a reduction in the dissipation and therefore an enhanced pinning

force when the field is applied parallel to the columnar tracks. This effect becomes more relevant for fields such that the number of vortices per unit area is comparable to the defect density.

1. INTRODUCTION High-temperature superconductors are very anisotropic materials, as the superconductivity is mainly localized in the planes. Recently, it has been shown that these planes present spin and charge stripes structure [1], and it has been suggested that an amplification in the critical temperature could be associated with these stripes [ 2 ]. Films grown

with different orientations, a axis or c axis, are very useful to get a full understanding of the anisotropic behavior of these materials. In a-axis films, the planes are perpendicular to the substrate; therefore, they allow studies in different experimental geometries from the usual c-axis films. The possible stripes in the planes can be affected by the material microstructure and defects [3]. Then, it is very interesting to characterize the behavior of a-axis films with artificially controlled defects as those created by heavy ion irradiation, which, so far as we know, has only been performed in c-axis films [4] and single crystals [5,6]. 1 2

Dept. Física de Materiales, F. Físicas, Universidad Complutense, 28040 Madrid, Spain. Materials Science Division, Argonne National Laboratory, IL 60439 USA.



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Martín, Kwok, and Vicent The most structure sensitive property of the material is the critical current density

for example, when defects are created in the planes, they could affect the periodicity of the stripe structure inducing a local reduction in and, therefore, a pinning effect. In

the case of a-axis films, the samples grow on the usual cubic substrates with a peculiar microstructure of domains separated by 90° boundaries [7]. As a consequence, the critical current density in these a-axis films is in the range of that is, lower than in c-axis films. Also, the irreversibility line lies lower in the H-T plane for a-axis films [8]. However, in the presence of a magnetic field, it has been shown that in a-axis films is not limited by weak links, but rather it is determined by flux motion and vortex pinning [9]. In this work, we have irradiated these a-axis films with heavy ion irradiation. It creates artificial columnar tracks that crosses the planes structure to act as good pinning centers and, therefore, produce an enhancement of their irreversibility region.

2. EXPERIMENTAL

Pure a-axis-oriented thin films have been grown by dc magnetron sputtering on (100) substrates as reported elsewhere [10]. The as-grown samples were patterned into a 500 wide bridge, using photolitography and wet etching. The irradiation was performed with with a current of 45 pA for 28 min. Under these experimental conditions, the high-energy heavy ions create amorphous columnar tracks, parallel to the irradiation direction and threading the whole sample thickness of The average defect density corresponding to this dose is the same as the number of vortices per unit area with an applied field of Figure 1 shows the resistivity transition of an a-axis film before and after the irradiation process. In this particular sample, the irradiation direction is perpendicular

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547

to the patterned bridge (i.e., to the transport current) and makes an angle with the film normal (see sketch in Fig. 1). The critical temperature and the metallic behavior

have not been esentially affected by the irradiation process. There is 15% increase in the normal-state resistivity, which can be attributed to a reduction in the effective crossection for the transport current due to the amorphous tracks created by the heavy ions.

Transport measurements were performed in a helium cryostat with a 9 T superconducting solenoid and a rotatable sample holder. The magnetic field was always applied in the plane perpendicular to the transport current. 3. RESULTS AND DISCUSSION

The angular dependence of the resistivity at different values of the magnetic field is shown in Fig. 2 for this irradiated a-axis film. All the curves present a minimum at (i.e., with the field parallel to the planes) that is due to the natural anisotropy of these compounds. However, curves are not symmetric with respect to the origin; whereas the resistivity increases continously as a function of angle in the negative range, the dissipation is reduced for positive angles and it presents a nonmonotonous behaviour. This reduction

in the resistivity is specially relevant for magnetic field values in the range of where a minimum in the resistivity at is clearly observed. A similar behavior is found in the angular dependence of the irreversibility line (Fig. 3).

This line has been defined by the onset of the dissipation in the curves meassured at different constant angles. Two maxima appear in this graph, one with field parallel to the

planes and the other with B parallel to the direction of the heavy ion irradiation. The results indicate the important pinning role played by the artificial tracks, so that the vortex motion is specially reduced when the field corresponds to similar vortex density

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values to that of the number of amorphous tracks and when the field is parallel to the direction of the irradiation. A comparison between positive and negative values shows a

reduction of the irradiation. A comparison between positive and negative values shows a reduction in the dissipation due to artificial pinning as high as a factor of 3 for Also, a more-detailed examination of the curves reveals that the field and angle ranges where this pinning mechanism is significant is not very narrow, but it develops over more than 1 Tesla and 20° around the matching conditions. This wide range of enhancement in the irreversibility region can be attributed to the soft vortex lattice of these materials, which can easily be deformed so that the vortex lines become pinned by the columnar defects. In summary, the pinning centers created by the irradiation process have proved to be very effective in improving the transport behavior of a-axis films so that the irreversibility line moves to higher field values. ACKNOWLEDGMENTS

This work has been supported by the Spanish CICYT (grant MAT96/904) and Universidad Complutense. The irradiation at ATLAS was supported by the U.S. Department

of Energy, BES, Materials Science under contract No. W-31-109-ENG-38. REFERENCES 1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe. Y. Nakamura, and S. Uchida, Nature 375, 561 (1995); P. Dai, H. A. Mook, and F. Dogan, Phys. Rev. Lett. 80, 1738 (1998). 2. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, Y. Nishihara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996).

3. M. A. Teplov, Y. A. Sakhratov, A. V. Dooglav, A. V. Egorov, E. V. Krjukov, and O. P. Zaitsev, JETP Lett. 65, 821 (1997).

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4. R. A. Doyle, W. S. Seow, J. D. Johnson, A. M. Campbell, P. Berghuis, R. E. Somekh, J. E. Evetts, G. Wirth, and J. Wiesner, Phys. Rev. B 51, 12763 (1995). 5. L. Civale, A. D. Marwick, T. K. Worthington, M. A. Kirk, J. R. Thompson, L. Krusin-Elbaum, Y. Sun, J. R.

Clem, and F. Holtzberg, Phys. Rev. Lett. 67, 648 (1991). 6. L. M. Paulius, J. A. Fendrich, W.-K. Kwok, A. E. Koshelev, V. M. Vinokur, G. W. Crabtree, and B. G. Glagola, Phys. Rev. B 56, 913 (1997), and references therein. 7. C. B. Eom, A. F. Marshall, S. S. Laderman, R. D. Jacowitz, and T. H. Geballe, Science 249, 1549 (1990). 8. J. [. Martín, M. Vélez, J. Colino, M. A. González, and J. L. Vicent, Physica C 235–240, 3123 (1994).

9. M. Vélez, J. I. Martín, and J. L. Vicent, Appl. Phys. Lett. 65, 2099 (1994). 10. J. I. Martín, M. Vélez, and J. L. Vicent, Thin Solid Films 275, 119 (1996).


Defect-Modulated Long Josephson Junctions as Source of Strong Pinning in Superconducting Films E. Mezzetti,1 E. Crescio,1 R. Gerbaldo,1 G. Ghigo,1 L. Gozzelino,1 and B. Minetti1

A model based on long Josephson junction (LJJ) modulated by columnar defects shows that the modulated depth of Josephson potential wells, the length scale of disorder, and the defect density, as well as the variance of their sizes are responsible of the plateau-like features in critical current density vs field logarithmic curves. The fitting of the defect structure to the “matching” fields is also discussed. The ordered structure represented by a LJJ with “stripes” alternately penetrated by a quantum flux and not penetrated, is the reference feature for controlled disorder in LJJ. This feature drives the best accommodation of the matching fields in a

given detect lattice.

I. INTRODUCTION AND MODEL

High- superconducting (HTS) optimized films have critical current densities higher than single crystals. This fact depends on the pinning center distribution, shape, and size. However, the main source of flux flow is the excitation of Josephson vortices inside the planar structure of the film at the boundaries between islands of different shape and size. Applied to the case of extremely good quality HTS films, the interfaces between granular domains become a critical path representing an inhomogeneous macroscopic long Josephson junction (LJJ), whose length scale is comparable to the sample size. The excitation energy of Josephson vortices is an order of magnitude lower than that of Abrikosov vortices. The appealing fact is that, as screening currents maintain zero field in the junction interior for “small” applied fields, suitable defects implanted as columnar trenches into the 1

I.N.F.M., I.N.F.N., Dept. of Physics–Politecnico di Torino, Torino, Italy.


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LJJ percolating across the film serve to maintain commensurate field values in the junction interior [ 1–3 ]. It occurs when the applied field is close to the one of these values. The aim of this work is to show, by means of a model already introduced by

Fehrenbacher [4] and suitably modified to the specific problem dealt with, how the static behavior of the critical current density in a high-quality film is driven by the behavior of its basic component, the average LJJ. This LJJ is modulated by random columnar defects. An important issue is to extract the main parameters in order to plan a particular performance in the applied field. As pointed out in [4], the Owen–Scalapino (OS) theory, modified to simulate columnar defects, can be used to evaluate the critical currents in an external magnetic field. The basic equations of LJJ [which involve the tunnelling supercurrent j ( x ) and the relative pair phase are

is the maximum Josephson current density, with the London penetration depth, and the insulator thickness). Equation (1) yields to the stationary sine-Gordon equation:

where the penetration depth is constant in a uniform LJJ, whereas columnar defect yields to a variable penetration depth The boundary conditions we consider are different from that of OS, but found to be suitable to take into account the columnar defects:

where

is a critical field corresponding to the usual first critical field

of a type-II

superconductor and is the pair phase at the end of the junction. The total current density through the LJJ is given by and from sine-Gordon equation The critical current density can be obtained by the maximum value of with respect to Due to the periodic behavior of the internal current j ( x ) , the total current through the uniform junction is always restricted to the junction surface and it cannot reach large val-

ues. The net current carried by a complete vortex vanishes, for a junction without defects, because every half-wavelength of positive internal current is compensated by a half-length of negative current. The current crossing the junction must be attributed to a single uncompensated half-wavelength. This means that the total current is due to incomplete vortices setting up only at the “surface.” There is a small but finite surface current due to the pinning of the vortex lattice (VL) at the surface. The surface acts as a defect. However, if defects are present in the inner part of the junction, they can block some negative oscillations of the internal current, leaving a large positive current crossing the junction [ 1–4 ].

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The large values of critical current densities in disordered LJJ can be attributed to

the presence of the defects acting as pinning centers. The presence of a single defect can be described in terms of a local change of the Josephson coupling energy through a suitable pinning potential which is identically zero for the uniform case. The corresponding change in the maximum Josephson current density yields to the desired negative oscillations blocking. The simplest model to simulate a defect of size s, located at the point with a pinning strength q, is the square-well model. The inhomogeneous coupling energy becomes

where the pinning potential is simulated by

In order to have positive coupling energy inside a defect, the strength q must satisfy the inequality The Josephson penetration depth and the other variables becomes a piecewise constant function of the position x, where the sine-Gordon equation has analytical solutions. The phase can be evaluated starting from the initial value at the edge At any boundary between two regions with different values of

the phase

and its

derivative, proportional to H(x), must be continuous, whereas the current density jumps. The solution is then obtained by propagating across regions with constant up to the edge The columnar character of the defects is taken into account by considering the energy modulation only along the x direction. In the following, we consider mainly a random-defect distribution. Random defects can be characterized by their random position their random size

and their random strength q. In our model, the defects are supposedly by uniformly distributed into the LJJ, their size and strength are chosen normally distributed around a suitable mean value. The size and position randomness make crucial to understand and simulate the eventual

defect overlapping. In the general theory, the j th defect lowers the coupling energy by a factor In our model, the j th defect, if overlapping the i th one, lowers by a factor the already lowered coupling energy. In the overlapping region, the new value of is lowered by the global factor In this way, any random defect simulation make the coupling energy a piecewise constant function, and analytical solutions can be obtained from the general theory. For each simulation of the random disorder, a value

of the critical current is evaluated. The statistical average of is then taken over many configurations, yielding to the result. This procedure essentially models a superconducting film as a disordered array of LJJ, separating the islands of single crystal-like regions. Each junction has a different defect distribution. The overall is the mean value of the critical current taken over all the LJJ. The statistical error depend on the number N of realizations, decreasing like In our calculations, we choose to have the computing time reasonably small in spite of some statistical fluctuations in the behavior of

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curve, in a log–log scale, generally exhibits a plateau where the average

value of is nearly independent on At the end of the plateau a large drop of occurs, followed by a series of minima located at fields with m integer. The oscillatory behavior is evident for random defects with constant size. It decreases and it is smoothened as the standard deviation of increases. 2. RESULTS

Among the types of disorder investigated, the random case is the most interesting. In fact, it explains the pinning behavior of as-grown films with suitable defects [5] as well as the behaviors of films where the underlying intrinsic defect network is modulated by implanted columnar defect. In the present paper, we consider fully penetrated LJJ. The dependence of on the temperature is implicitly taken into account through the temperature dependence of diverging as vanishes near The potential well depth, represented by the value of q (Fig. 1) and the length scale of disorder (Fig. 2), are the chief parameters in determining the main drop of the critical current density as a function of the external field. Above this drop, some plateau-like features can set up. Figure 2 shows an increase of the

width of the plateau as decreases. The more interesting topic in film fabrication and optimization technologies is the handling of the disorder to provide plateau-like features for a large and programmable range of “matching” fields. Two different but correlated conditions are meaningful, the first one concerning the defect size and distribution, the second one the fluxon lattice accommodation. Among the different possible defect configurations, the more relevant is corresponding to the equation

With this topology, the plateau exhibits its maximum value. Any further increase of the defect concentration leads to a decrease of but does not affect the plateau shape (Figs. 3 and 4).


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In order to understand the meaning of this result, one must refer to a asymmetric

junction in which the local critical Josephson current density is reduced to zero in periodic “stripes” by making the junction barrier thick there. This procedure is the practical way to obtain periodic modulation of the LJJ coupling energy. The relevant fact is that this

configuration can be thought of as a periodic array of columnar pins, or simply as a periodic array of separate subjunctions within the overall junction area [6].

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In such a configuration the main relevant parameter is f , the fraction of a flux quantum per unit cell, also referred as the frustration index. Theoretical treatment has shown that novel phenomena arise when the frustration index takes on a rational value. For a more important example of the superlattice of fluxons relative to the underlying lattice is a simple checkerboard pattern, with cells not occupied by a flux quantum and cells occupied by one flux quantum represented by the black and white squares. When a current is applied to this ground-state, the vortex pattern is effectively pinned in place by its commensurate “fit” to the underlying lattice: an individual vortex center cannot jump into an adjacent cell without creating configurations with much higher energy.


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In the case of a random array of defects, the topology for which all the defects are filled by a fluxon (maximum matching field) corresponds to the ordered reference pattern with In the literature concerning pseudoperiodic or random columnar defects, the matching field is usually considered the maximum field we can accommodate inside a given distribution of columnar defect [6]. From the results of the present work, it can be deduced

that the accomodation of a wide range of fields lower than the maximum is optimized in

a disordered defect distribution only when Eq. (4) is fulfilled. The two conditions—one vortex, one defect and therefore be considered as the best equivalent to the checkerboard pattern in the random case. Up to now, we did not consider the existence of minima. For a wide spectrum of ap-

plications, the variance or the defect size ascertains a smoothening of the resonant behavior at high interference orders (Fig. 5). In Fig. 6, the Thompson experimental results [5] are

interpreted in the framework of this theory. As a conclusion, we emphasize that optimal performances of films in a given range of field and temperatures can be achieved by overlapping size randomness of defects and controlled disorder. The control is achieved, by means of the parameter

as well as the optimized filling of a given average LJJ with

defects. In such a way extrinsic trenches of columnar defects could either create or modulate intrinsic LJJ.

ACKNOWLEDGMENT

This work was partially supported by the I.N.F.M.–PRA PROJECT “HTCS electronic devices.” REFERENCES 1. M. A. Itzler and M. Tinkham, Phys. Rev. B 51, 435 (1995). 2. E. Mezzetti, S. Colombo, R. Gerbaldo, G. Ghigo, L. Gozzelino, B. Minetti, and R. Cherubini, Phys. Rev. B 54, 3633 (1996).

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3. J. C. Martinez, B. Dam, B. Stauble-Pumpin, G. Doornbos, R. Surdrdeanu, U. Poppe, and R. Griessen, Low. Temp. 105, 1017 (1996).

4. R. Fehrenbacher, V. B. Geshkenbein, and G. Blatter, Phys. Rev. B 45, 5450 (1992). 5. J. R. Thompson, L. Krusin-Elbaum, and D. K. Christen et al., Appl. Phys. Lett. 71(4), 28, July 1997. 6. M. Tinkham, M. A. Itzler, and J. M. Hergenrother, Macroscopic Quantum Phenomena and Coherence in Superconducting Networks, edited by C. Giovannella and M. Tinkham (World Scientific, 1995), pp. 3–14. 7. J. R. Thompson, L. Krusin-Elbaum, L. Civale, G. Blatter, and C. Feild, Phys. Rev. Lett. 78, 3181 (1997), and references therein.

Bulk Confinement of Fluxons by Means of Surface Patterning of Columnar Defects in BSCCO Tapes E. Mezzetti,1 R. Gerbaldo,1 G. Ghigo,1 L. Gozzelino,1 B. Minetti,1 P. Caracino,2 L. Gherardi,2 L. Martini,3 G. Cuttone,4 A. Rovelli,4 and R. Cherubini5

We have studied the effect of surface columnar defects on the vortex dynamics inside the whole sample. Trenches of columnar defects along about 5% of the sample thickness were created by means of 0.25 GeV Au ions in Ag/BSCCO2223 high-quality tapes. Strong phenomena of vortex localization inside the bulk were revealed by notable shifts of the irreversibility lines (ILs) as well as their after-irradiation shape. The enhanced ILs exhibit their own particular characteristics, such as a Bose-glass-like behavior up to quite high magnetic fields, with a dose-dependent onset point. Moreover, the irreversible regime expands with decreasing defect density. The results are consistent with the setting up of vortex morphologies confined in the bulk. The central achievement of this work is that a surface patterning with nanometric defects provides boundary conditions for mesoscopic domains in the bulk where vortices are confined. In particular, the lateral wandering is allowed inside tubes whose lateral cross section is dose de-

pendent and whose length scale is comparable with the distance between defects for each dose.

I. INTRODUCTION

The effects of correlated disorder due to heavy ion irradiation have recently received considerable attention. The shape of the induced columnar defects can match the fluxon I

I.N.F.N., I.N.F.M., Dept. of Physics–Politecnico di Torino, Torino, Italy. Pirelli Cavi S.p.A., Milano, Italy. 3 ENEL–SRI, Milano, Italy. 4 L.N.S.–I.N.F.N., Catania, Italy. 2

5

L.N.L.–I.N.F.N., Legnaro (Padova), Italy.


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shape, and the radial column dimensions are close to the coherence length [1]. Due to the small range of penetration of heavy ions in the material, only a limited part of the sample bulk can be affected, so that heavy ions were so far used only for irradiations of films and thin single crystals. However, surface columnar defects act as trenches where vortices are confined in such a way that the bulk properties are also affected, as already demonstrated for melt-textured materials [2]. In this work, we show a possibility to control the vortex flux dynamics inside BSCCO2223 materials by surface ion implantation. Columnar defects were created on a surface layer of about 5% of the total thickness. Measurements were performed on well-characterized Ag/BSCCO-2223 samples coming from two different batches prepared independently by two staffs: Pirelli Cavi S.p.A. [3] (samples labelled GHx) and ENEL-SRI [4](samples labelled CLx), in order to verify the important issue of the sample independence of the experimental findings. These samples were irradiated with 0.25 GeV Au ions to have five defect densities corresponding to equivalent fields (matching fields, [5] ranging from 1 to 5 Tesla. Irradiations were performed at the 15 MV Tandem facility of the L.N.S–I.N.F.N., Catania and at the 15 MV Tandem facility of the L.N.L.–I.N.F.N., Legnaro (Padova) [2,6]. Irreversibility lines (ILs) were determined by the onset of magnetic susceptibility third harmonic [7], at 1 kHz. 2. EXPERIMENTAL RESULTS AND DISCUSSION Figures 1 and 2 show vs. T curves for five GHx twin samples and one CIx sample, respectively, irradiated at five different fluences. The magnetic field was applied

parallel to the ion tracks. Unlike what happens for tracks crossing the whole sample [8], the irreversible regime expands with decreasing defect density in the range of the investigated fluences, from The after-irradiation ILs exhibit features usually related to fluxon localization [9]. However, in our defect topology, it must be stressed that the irradiated samples show localization phenomena only above an onset phase point [10]. As the field increases above this point, all the curves show a kink, which is a

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signature of a Bose-glass-like transition. We focus our analysis on the region between the onset and such kink.

The crossover fields, between the two regimes corresponding to the higher localization and weaker localization regions, respectively, and the onset field, can be better highlighted in the log–log plot of Fig. 3. In fact, all the ILs measured before and after irradiation follow, in the considered range of fields, the law with different values of the exponent for different dynamical regimes. In the logarithmic scales of Fig. 3, the different regimes result in different linear slopes. The kink is found as the crossing point between the curves with exponents In Fig. 3, the onset and the kink of the sample irradiated with are indicated. The position of the kink is shifted toward lower reduced temperature, and reduced field, as the density of tracks increases.

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We conjecture that for the vortices are localized on tracks in the irradiated part of the sample, whereas in the remaining part they are allowed to wander, roughly keeping their center of mass in place. In order to check the above hypothesis and to compare our phase diagram with the deeply investigated diagrams for percolating tracks, we follow the analysis suggested in [11]. Such analysis is based on the assumption that the vortices are allowed to lay into columnar tubes of a given size [12]. An analytic expression of the is derivated by

where is the Bose-glass reduced temperature and is the reduced melting temperature; is the defect diameter, d the average distance between the tracks, and Gi the Ginzburg number [ 12]. In order to take into account the experimental findings related to the existence of an onset point, we assumed a Boseglass-like behavior with a dose-dependent onset. As a consequence, we modified Eq. (1) by introducing a new parameter A. Upon renaming the irreversibility temperature in this confinement regime as we have

The IL of the irradiated sample is coincident with the curve for fields lower than and temperatures higher than i.e., the point belongs to both and curves (see Fig. 3 and the inset of Fig. 1). This condition finally gives

We fitted our data by means of Eq. (2), with as fitting parameters. We assume [11] the preirradiation irreversibility line as a good estimate of (B). Figures 2

and 4 show the fit for the one and one GHx sample respectively, whereas in Table 1, the fitting parameters for all the samples are reported. The parameter in our case is crucial, because it relates the interdefect distance d and the confinement


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diameter We do not assume a priori that is equal to the defect diameter. In order to obtain from we assumed and The experimental results show that a single value of for the different doses cannot be found. In Table 1, the values of for different values of are reported. The magnitude order of the fluence-dependent is much higher than the diameter of gold-ion columnar defect crosssection reported by Zhu [1]. In Table 1, the average distances between defects at the respective irradiation doses are reported for comparison. It emerges that roughly scales with the irradiation dose as the interdefect distance. We interpret the fitted values of as a diameter of the vortex “confinement” crosssection, which now takes the place of the columnar defect crosssection. It represents the transverse size of the vortex bundle with a part pinned into columnar defects near the IL. Different sizes are allowed for different doses, and the smaller ones correspond to higher fluences. In order to perform a deeper investigation of the main characteristics of the vortex phase induced by the surface defect topology, we analyze the results by means of a further comparison with the before-mentioned analysis. In [13] the IBM group formalized the pinning efficiency for YBCO as where is the track cluster area, is the flux quantum, and is the accommodation field, first introduced by Nelson and Vinokur [12]. Our hypothesis is that the field at the kink, is also proportional to and, as a consequence, that

where c is a constant. We fitted our data by means of this expression, with and c as fitting parameters. Results are reported in Fig. 5, where the values of the parameters are also indicated. From this analysis, a confinement diameter nm, of the same magnitude order of the parameter evaluated above, is found. As previously outlined, in our case 2r does not represent the average diameter of track clusters, but rather represents the average transversal dimension of a confined vortex line. In this case, the value of r obviously is the result of a spatial averaging as well as an averaging on different The magnitude order of the obtained value confirms this interpretation in relation to the main output of the analysis

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with Eq. (2). Also the lower limit of the confinement region, i.e., dependence. In Fig. 5 is reported a fit of the data with the expression

fits the same dose

similar to Eq. (3). The value of the parameter , i.e., the value of the confinement radius, is, within errors, the same as in the previous case. We can therefore conclude that, in our case,

3. CONCLUSIONS The central achievement of this work is that surface columnar defects affect the dynamics of the underlying unaffected bulk layer by inducing local confinement of the fluxon

lattice, starting at a dose-dependent point phase. In particular, the lateral wandering is allowed inside tubes whose lateral corsssection bo is dose dependent. In summary, the particular strategy involved in surface irradiation de facto uses in the bulk the very low elastic energy surviving at high temperatures, with just a little external energy supply due to surface columnar defects, to obtain someway ordered vortex morphologies [14]. REFERENCES 1. Y. Zhu, R. C. Budhani, Z. X. Cai, D. O. Welch, M. Suenaga, R. Yoshizaki, and H. Ikeda, Phyl. Mag. Lett. 67 125 (1993).

2. E. Mezzetti, R. Gerbaldo, G. Ghigo, L. Gozzelino, B. Minetti, and R. Cherubini, J. Appl. Phys. 82, 6122 (1997); lEEETrans. Appl. Supercond. 7, 1993 (1997). 3. L. Gherardi, P. Caracino, G. Coletta, and S. Spreafico, Mat. Sri. Eng. B 39, 66 (1996). 4. L. Martini, A. Gandini, L. Rossi, V. Ottoboni, and S. Zannella, Physica C 196 (1996). 5. The dose equivalent field is the magnetic field that would be ideally required to fill each track with a flux quantum, i.e., where ø0 is the flux quantum and n is the track density.


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6. R. Gerbaldo, G. Ghigo, L. Gozzelino, E. Mezzetti, B. Minetti, P. Caracino, L. Gherardi, and R. Cherubini,

Mat. Sci. Eng. B 55, 109 (1998); E. Mezzetti, R. Gerbaldo, G. Ghigo, L. Gozzelino, B. Minetti, P. Caracino, L. Gherardi, L. Martini, F. Curcio, S. Zannella, G. Cuttone, A. Rovelli, and R. Cherubini, Proc. CIMTEC’98, Symposium VI, Florence, June 14–19, 1998.

7. E. R. Yacoby, A. Shaulov, Y. Yeshurun, M. Konczykowski, and F. Rullier-Albenque, Physica C 199, 15 (1992). 8. L. Civale, Supercond. Sci. Technol. 10, A 11 (1997). 9. D. Zech, S. L. Lee, H. Keller, G. Blatter, P. H. Kes, and T. W. Li, Phys. Rev. B 54, 6129 (1996). 10. D. Giller, A. Shaulov, R. Prozorov, Y. Abulafia, Y. Wolfus, L. Burlachkov, Y. Teshurun, E. Zeldov, V. M. Vinokur, L. J. Peng, and R. L. Greene, Phys. Rev. Lett. 79, 2542 (1997).

11. L . Krusin-Elbaum, L. Civale, G. Blatter, A. D. Marwick, F. Holtzberg, and C. Feild, Phys. Rev. Lett. 72, 1914 (1994). 12. D.R. Nelson and V. M. Vinokur, Phys. Rev. B 48, 13060 (1993). 13. L. Krusin-Elbaum, L. Civale, J. R. Thompson, and C. Feild, Phys. Rev. B 53, 11744 (1996).

14. F. J. Nédélec, T. Surrey, A. C. Maggs, and S. Leiber, Nature 385, 305 (1997).


A Finite-Size Cluster Study of M. Cuoco,1 C. Noce,1 and A. Romano1

We investigate the magnetic properties of the compound which is the only known copper-free layered-perovskite compound becoming superconductor The analysis is performed by means of a microscopic model describing the electron dynamics in ruthenium-oxygen planes, which is solved via Lanczos algorithm on a finite-size cluster. From the behavior of the the spin correlation functions we determine the conditions for the formation of a local triplet state on ruthenium ions, also finding that for realistic choices of the parameter values the system is close to a ferromagnetic instability.

Experimental constraints on models describing the physical properties of the highsuperconductors (HTS) can come not only from studies of the cuprates, but also from the analysis

of other related layered perovskites, sharing with them a quasi-two-dimensional (2D) structure. Of fundamental importance in this context has been the discovery of a superconducting phase at very low temperatures in the compound which has the same crystal structure as the parent compound with planes replacing the planes. Actually, represents at present the only known example of a layered pervoskite material that exhibits superconductivity without the presence of copper. However, in spite

of the close structural similarity, high- materials and show important differences as far as their physical properties are concerned. At room temperature, is an insulator compound showing enhanced Pauli paramagnetism [2], which, as T is decreased, becomes first a metal and then a superconductor without doping [HTS are instead mostly antiferromagnetic (AF) insulators becoming superconducting only after chemical doping]. For LDA-based calculations have shown [3] that antibonding bands cross the Fermi energy, whereas in high-

originate from

and O

cuprates, the bands at EF orbitals. For ruthenium ions, an on-site Coulomb

repulsion of 2.4 eV is estimated from photoemission spectra [4]. Being the single-particle 1

Dipartimento di Scienze Fisiche “E.R. Caianiello,” Università di Salerno, I-84081 Baronissi (Salerno), Italy–Unità I.N.F.M. di Salerno.



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bandwidth equal to 1.4 eV, one gets a ratio which indicates that ruthenium electrons in are less strongly correlated than copper holes in HTS. de Haas-van Alphen experiments [5] show the existence of three almost-cylindrical Fermi surface sheets with an essentially 2D topology that agrees well with band structure calculations. In agreement with the Luttinger’s theorem prediction that the Fermi volume is conserved even in presence of strong electron interactions, the volume enclosed by the Fermi surface corresponds to exactly four electrons in the Brillouin zone. This points out the importance of the Hund rule coupling in the ruthenium orbitals, confirmed by the experiments performed on showing that the spin on ions is equal to 1. Specific features seem also to characterize the superconducting phase that develops at about 1 K. In there are at low temperatures many-body enhancements of the specific heat and the magnetic susceptibility quantitatively similar to that of In addition, the compound which is the 3d analog of exhibits a metallic ferromagnetic (FM) phase [6]. These findings suggest that superconductivity is likely to develop in an odd-parity state, consistently with the lack of a Hebel-Schlichter peak in in NQR experiments [7]. We formulate in this paper a microscopic model for describing the combined effect on the physical properties of the system of the interorbital Hund coupling and the intraorbital Coulomb repulsion. The Hamiltonian is written down neglecting the orbital, which, according to band structure calculations, is lower in energy and does not undergo significant charge fluctuations. Confining our anlaysis to the electron dynamics in the ruthenium–oxygen planes, we introduce the Hamiltonian [8]

Here the fermionic operators refer to electrons with spin on the i site, belonging to (along the x and y axis), and orbitals, respectively; and are the corresponding bare energies, is the direct hopping between orbitals and is the hybridization coupling between p and d orbitals. U is the usual on-site Coulomb repulsion and J denotes the intraatomic Hund coupling between electrons on and orbitals. Hopping processes are restricted to nearest-neighbor sites. The ground-state properties of the above Hamiltonian are here investigated by applying the Lanczos technique to the planar cluster (made out of two adjacent units). The site energies and have been chosen such that (all energies are from now on expressed in eV), in agreement with the experimental evidence that p orbitals are lower in energy than d orbitals. To take into account that the hybridization between Ru and O orbitals is much stronger than the direct O-O hopping, and have been assumed to be equal to –0.15 and 0.85, respectively. These values ensure that the volumes enclosed by the Fermi surface coincide with that measured in quantum oscillation experiments [9].

A Finite-Size Cluster Study of

The electron filling has been chosen to be

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which gives configurations in which

the oxygen orbitals, lower in energy, are almost fully occupied and the contain about two electrons per ruthenium atom.

and

orbitals

In general, the value of the total spin on each ruthenium atom is strongly dependent on the strength of the Hund coupling J and the Coulomb repulsion U. The transitions between different spin states induced by variations of J and U have been determined from the behavior of the Ru spin–spin local and nonlocal correlation functions, defined respectively as and and are the spin operators for electrons on the site i in the and orbitals, respectively, and is the total spin on site i). The results reported in Fig. 1 give the behavior of as a function of U for three fixed values of J. We can see that threshold values of U are needed to have local triplet states

associated with electrons in different d orbitals. This means that for the above Hamiltonian, the Hund coupling cannot induce by itself a spin alignment. Finite values of the on-site repulsion are also required in order to avoid intraorbital double occupation. From the curves

of Fig. 1, we observe that for a rather low value of 7, equal to 0.3, the increase of U causes a transition from a configuration of local singlets to a configuration of local triplets, occurring at a critical value of U approximately equal to 2.9. For higher values of shows a qualitatively different behavior. The increase of U causes a first transition from a configuration of local singlets to an equally weighted superposition of local singlets and triplets, followed by a second transition from this latter state to one with local triplets only. As expected, the corresponding critical values of U are decreasing functions of 7. We want also to point out that the fact that threshold values of U must be exceeded in order to have local triplet states, is a consequence of the high degree of hybridization existing in between ruthenium and oxygen orbitals. Figure 2 clearly shows that for a lower value of the increase of J makes tend to zero the critical value of U at which the transition to

the local triplet state occurs.

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The dependence on U of the nonlocal spin correlation function is reported in Fig. 3 for

the same choice of parameters as in Fig. 1. Comparing these curves with those for we notice that for both choices of J the transition to a local triplet spin state is accompanied by a transition to a global singlet spin state (when the total spin quantum number takes the allowed values 0, 1,2, becomes equal to –2, –1, and 1, respectively). This means that, when locally spins are ferromagnetically coupled, the ground-state of the whole system of ruthenium electrons is a singlet state. This result is consistent with the experiments performed in Ref. [2], showing that is a paramagnetic compound.

A Finite-Size Cluster Study of

571

As a final consideration, let us notice that calculations performed by Baskaran [10] give an estimate for the Hund coupling of about 0.5 eV. From an inspection of Fig. 3, one can see that this value, in conjunction with the experimental estimate is such that the system is close to a region where and thus FM fluctuations are expected to be important. That seems to indicate, in agreement with previous studies [11,12], that superconductivity is likely to exhibit a p-wave symmetry for the order parameter, rather than s or d wave.

REFERENCES 1. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Nature 372, 532 (1994). 2. R. J. Cava, B. Batlogg, K. Kiyono, H. Takagi, J. J. Krajewski, W. F. Peck Jr., L. W. Rupp Jr., and C. H. Chen, Phys. Rev. B 49, 11890 (1994). 3. T. Oguchi, Phys. Rev. B 51, 1385 (1995).

4. K. Inoue, Y. Aiura, Y. Nishihara, Y. Haruyama, S. Nishizaki, Y. Maeno, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Physica B 223–224, 516 (1996). 5. A. P. Mackenzie, S. R. Julian, A. J. Diver, G. J. McMullan, M. P. Ray, G. G. Lonzarich, Y. Maeno, S. Nishizaki, and T. Fujita, Phys. Rev. Lett. 76, 3786 (1996).

6. T. C. Gibb, R. Greatrex, N. N. Greenwood, D. C. Puxley, and K. G. Snowden, J. Solid State Chem. 11, 17 (1994).

7. Y. Kitaoka, K. Ishida, K. Asayama, S. Ikeda, S. Nishizaki, Y. Maeno, K. Yoshida, and T. Fujita, Physica C 282–287,210 (1997). 8. M. Cuoco, C. Noce, and A. Romano, Phys. Rev. B 57, 11989 (1998). 9. C. Noce and M. Cuoco, Phys. Rev. B, submitted.

10. G. Baskaran, Physica B 223–224, 490 (1996). 11. D. F. Agterberg, T. M. Rice, and M. Sigrist, Phys. Rev. Lett. 78, 3374 (1997). 12. I. I. Mazin and D. Singh, Phys. Rev. Lett. 79, 733 (1997).


New Copper-Free Layered Perovskite Superconductors: and Related Compounds Yoshihiko Takano,1 Yoshihide Kimishima,2 Hiroyuki Taketomi,3 Shinji Ogawa,3 Shigeru Takayanagi,4 and Nobuo Môri 5

We have discovered three new copper-free superconductors in the Li intercalated triple-layered perovskite niobate. The Li intercalation was carried out also to the double- and quadruple-layered analogs. Contrary to our expectation, the insulator–metal transition was observed without a corresponding superconducting transition. Two- and three-dimensional variable range hopping were found in the double and quadruple perovskite niobates, respectively, at low temperatures.

1. INTRODUCTION

All of the high-

cuprate superconductors are layered perovskites, and the existence

of Cu-O planes is regarded to be essential for superconductivity. To investigate the role of Cu-O layers for the occurrence of high- superconductivity, discovery of new copper-free

layered perovskite superconductors are required. Recently, we succeeded in synthesizing the Li intercalated as a new copper-free superconductor [ 1 – 4 ] . In this paper, we present the transport and superconducting properties of and related

compounds. A schematic of the crystal structure of and its related compounds [5–6] is summarized in Fig. 1. Each compound has a layered perovskite structure characterized by alternate stacking of Nb-O planes and an alkali–metal layer. The Nb-O plane and the 1

National Research Institute for Metals, Tsukuba, Japan. Dept. of Physics, Yokohama National Univ., Yokohama, Japan. 3 Dept. of Appl. Sci., Tokyo Denki Univ., Tokyo, Japan. 2

4

Dept. of Physics, Hokkaido Univ. of Education Sapporo, Sapporo, Japan. Institute for Solid State Physics, Univ. of Tokyo, Tokyo, Japan.

5


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New Copper-Free Layered Perovskite Superconductors

575

alkali–metal layer in the niobate system seem to correspond to the Cu-O plane and blocklayer in highsuperconductors, respectively. According to the number of Nb-O planes and the structural properties of the alkali–metal layer, the samples summarized in Fig. 1 can be divided into two groups: group and group and The compounds in group A have triple Nb-O planes as a common structure, although the crystal structure of the alkali–metal layer differs slightly between each compound. However, for group and have double, triple, and quadruple Nb-O planes, respectively; nevertheless, the crystal structures around the alkali–metal layers are almost the same. 2. EXPERIMENTAL

All the polycrystalline samples were prepared by a solid-state reaction. An appropriate mixture of starting materials, and were pelletized and sintered at 1100–1200°C in the air until the samples became single phase. The sintered samples were white in color and electrically insulating. To introduce charge carriers, Li was intercalated into the samples around the alkali–metal layer by nButyllithium n-hexane solution. The temperature dependence of the electronic resistivities was measured by a standard four probe method using a refrigerator. Magnetic susceptibility measurements were performed down to 2 K under 100 Oe after zero-field cooling in a SQUID magnetometer. 3. RESULTS AND DISCUSSION Figure 2a–c shows the temperature dependence of the magnetic susceptibilities of the samples in group A. At low temperatures, the susceptibility drops rapidly corresponding to the onset of superconductivity. All the samples with a triple-layered perovskite structure were found to exhibit superconductivity. Superconducting transition temperatures estimated from magnetic measurements are summarized as follows:

The superconducting volume fraction of at 2 K is much larger than that of although is the lowest. The temperature dependence of the electrical resistivities of

and are shown in Fig. 3a–c, respectively, for different periods of Li intercalation. Although initially semiconducting, the resistivity behavior become more metallic after each period of intercalation. For the double-layered perovskite Li intercalation induces a insulator-metal transition within a month, although a superconducting transition was not observed down to 0.5 K. The logarithm of resistivity varied as on the insulator side. This can be explained by a two-dimensional variable range-hopping conduction model. Incidentally, the electron energy-band structure calculation for which is similar in crystal structure to has revealed that the conduction band is two-dimensional in nature [7]. However, for the quadruple-layered analog, the logarithm

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of resistivity qualitatively follows implying that a three-dimensional VRH conduction model can describe the low-temperature transport properties. Increasing the number of Nb-O layers appears to increase the dimensionality of the electronic properties. In summary, we have discovered three new superconductors having a triple-layered perovskite structure. The double-layered can exhibit metallic behavior but does not go superconducting. The VRH conduction suggests that the existence of a random potential around the alkali–metal layer, generated by inhomogeneous Li intercalation, may localize electron carriers on the Nb-O plane. In the case of the triple-layered analog, the central Nb-O planes are protected on both sides from this random potential by the additional

Nb-O planes, and thus superconductivity can appear in the homogeneous Nb-O plane. Two-dimensional and three-dimensional electron transfer were observed in the doubleand quadruple-layered perovskite analogs, respectively. Investigation of the dimensionality and homogeneity of the Nb-O plane is important for identifying the conditions leading to superconductivity in this niobate system. REFERENCES 1. Y. Takano, S. Takayanagi, S. Ogawa, T. Yamadaya, and N. Môri, Solid State Commun. 103, 215 (1997). 2. Y. Takano, H. Taketomi, H. Tsurumi, T. Yamadaya, and N. Môri, Physica B 237–238, 68 (1997). 3. Y. Takano, Solid State Physics (in Japanese) 32, 737 (1997). 4. Y. Takano, Y. Kimishima, T. Yamadaya, S. Ogawa, S. Takayanagi, and N. Môri, Rev. High Press. Sci. Technol. 7, 589 (1998). 5. M. Dion, M. Ganne, and M. Tournoux, Mater. Res. Bull. 16, 1429 (1981). 6.

M. Dion, M. Ganne, and M. Tournoux, Revue de Chimie Minerale t. 23, 61 (1986).

7.

I. Hase and Y. Nishihara, Phy. Rev. B 58, R1707 (1998).

Author Index Agrestini, S., 9

Akimitsu, J., 263 Alexandrov, A.S., 159

Arai, M., 323 Arao, M.,421

Cordero, F., 279 Corti, M., 279 Crescio, E., 551 Cuoco, M., 567 Cuttone, G., 559

Gonzalez, E.M., 539 Gonzalez, J.M., 539 Goodenough, J.B., 199 Gordon, I., 481

Gozzellino, L., 551, 559 Grandini, C.R., 279 Grassme, R., 385

Ariosa, D., 535

Asensio, M.C., 237, 253

Dagotto, E., 437

Ashkenazi, J., 27 Avila, J., 237, 253 Awad, R., 487

Darhamaoui, H., 507 Darie, C., 465 DeFilippis, G., 175 De Marzi, G., 427 Demsar, J., 219 dePasquale, F., 183 Di Castro, C., 45, 151 Di Castro, D., 9

Grèvin, B., 287 Grilli, M.,45, 151, 361 Grimaldi, C., 169 Grioni, M., 535 Gu, G.D., 237, 253

Dichtel, K., 369 Dimashko, Y.A., 63, 129

Hammel, P.C., 295 Hanfland, M., 465

Bianconi, A., 9, 237, 253

Dogan, F., 315

Hasselmann, N., 83, 129

Bianconi, G., 9 Birgeneau, R.S., 335

Dore, P., 427

Hatton, P.D., 473

Du, C.H., 473

Hedin, L., 101

Eccleston, R.S., 323

Hinks, D.G., 303 Hirota, K., 335

Balatsky, A., 343 Belov, S.I., 349

Bennington, S.M., 323 Berger, H., 535 Berthier, Y., 287 Bhaltacharyya, P. , 355

Bordet, P., 465

Borowski, R., 329 Brown, S., 473

Bruynseraede, Y., 481 Bussmann-Holder, A., 39

Büchner, B., 329 Caldeira, A.O., 129

Calvani, P., 427 Campana, A., 279 Cantelli, R., 279 Capone, M., 169 Caprara, S., 45 Caracino, P., 559 Carretta, P., 309

Egami, T., 191 Eisaki, H., 271 Emery, V.J., 69, 91 Endoh, Y. , 191, 323, 335 Eremin, I., 77

Haage, T., 227 Haase, J., 303

Holland-Moritz, E., 329 Hone, D.W., 447 Hor, PH., 515 Horibe, Y, 455, 501

ladonisi, G., 175 Ichikawa, N., 271 Inoue, Y, 421, 455, 501

Erernin, M., 77 Fedorov, I., 427

Fishman, I.M., 495, 529

Fisk, Z., 295

Jain, K.P., 55 Jung, J., 507

Fitter, J., 329 Fleck, M., 101 Forro, L., 535

Kahn, R., 329 Kaldis, E., 211 Kibune, M., 263 Kimishima, Y, 573 Kimura, H., 335

Castellani, C., 45, 151, 361

Fritzenkoetter, J., 369

Castro Neto, A.H., 83 Cataudella, V., 175 Cava, R.J., 465

Fujimori, A., 121 Fujisawa, H., 263 Fujita, M., 335

Chaillout, C., 465 Chakoumakos, B.C., 315 Chakraverty, B.K., 55 Cheong, S.-W., 427, 473 Cherubini, R., 559 Ciuchi, S., 169, 183 Collin, G., 287

Gatt, R., 535 Gazza, C., 437 Gerbaldo, R., 551, 559 Gherardi, L., 559 Ghigo, G., 551, 559

Kitano, H., 271

Giapintzakis, J., 529

Koshizuka, N., 237, 253

Collins, S.P., 473

Gomaa, N., 487

Koyama, Y. , 335, 421, 455, 501

Conder, K., 211

Gonnelli, R.S., 377, 407, 535

Kurihara, S., 399

Kino, G.S., 495, 529

579

Kivelson, S.A., 69, 91, 447 Kochelaev, B.I.,349 Komine, N., 455 Korayem, M.T., 487

580

Author Index

Kwok, W.K., 507, 545

Nishijima, T., 323 Noce, C., 567

Studenmund, W.R., 495, 529 Su, Y., 473 Suh, B.J., 295

Lanzara, A., 237, 253

Ogawa, S., 573

Sulpizi, M., 45

Lechner, R.E., 329

Oles, A.M., 101

Suzuki, Y.Y., 399

Lee, C.H., 335 Lee, S.H., 335 Li, Z., 515

Onellion, M., 535 Osman, O.Y., 143 Oyanagi, H., 227

Szytula, E., 521

Kusko, C., 1 1 1

Pachot, S., 465

Tajima, S., 237, 253, 271, 323 Takaba, M., 271

Palles, D., 211

Takagi, H., 263, 465

Panas, I., 391

Takahashi, T., 263

Paul, D.F., 473

Takano, Y., 573

Maeda, A., 271 Malvezzi, A., 437 Marezio, M., 465

Pavuna, D., 535 Perali, A., 45 Peter, M., 413

Margaritondo, G., 535 Markiewicz, R.S., 1 1 1 Martin, J.I., 545 Martini, L., 559

Petrov, Y., 191 Podobnik, B., 219 Poulakis, N., 211 Pryadko, L.P., 447

Takayanagi, S., 573 Taketomi, H., 573 Tanner, B.K., 473 Trappeniers, L., 481

Martins, G., 437 Maselli, P., 427 Matsushita, H., 335

Ranninger, J., 245 Rigamonti, A., 279

Liarokapis, E., 211 Lichtenstein, A.I., 101 Longeville, S., 329 Lupi, S., 427

McQueeney, R.J., 191 Melzi, R., 309 Mendels, P.. 287 Mezzetti, E., 551, 559 Mihailovic, D., 219 Minetti, B., 551,559 Miyasaka, S., 263

Roepke, M., 329 Romano, A., 245, 567 Rovelli, A., 559

Miyazaki, S.,421 Mizokawa, T., 121 Mook, H.A., 315 Morais Smith, C., 63, 83, 129 Moreo, A., 459 Moshchalkov, V.V.,481

Sato, T., 263 Schmauder, T., 535

Movshovich, R., 343 Mukhin, S.I., 135

Saini, N.L., 9, 237, 253

Uchida, S., 271 Uehara, M., 263

Ueki, S., 335 Ummarino, G.A., 377, 407 van Saarloos, W., 143 Varlamov, S., 77 Vaughn, M.T., 1 1 1 Vicent.J.L., 539, 545 Vobornic, I., 535

Saito, S., 399 Sarrao, J.L., 295

Wagner, P., 481 Wakimoto, S., 335

Schuller, I.K., 539

Wang, N.L., 271 Weger, M., 413

Schwab, P., 361 Seibold, G., 151

Yamada, K., 335

Seidel, P., 385

Yan, H., 507

Shirane, G., 191, 335

Yokoyo, T., 263 Yunoki, S., 459

Môri, N., 573 Müller, K.A., 1

Slichter, C.P., 303 Sokolowski, J., 521 Starowicz, P., 521 Stepanov, V.A., 377

Nagata, T., 263

Stern, R., 303

Zaanen, J., 143

Zegenhagen, J., 227 Zhou, J.-S., 199

Subject Index Amplitude fluctuations, 55 Anelastic spectra, 279 Angle resolved photoemission (ARPES), 70, 237, 245, 246, 253, 263, 443, 536 Angle scanning photoemission, 237, 253 Antiferromagnetic background, 1,391,399

Copper free layered perovskite, 567, 573 Coulomb interactions, 2 Critical current, 507, 546, 553, 554 Critical fluctuations, 46 Dark-field image, 505 Dielectric constant, 499

correlations, 438 coupling, 7, 465

Dielectric function, 385 Differential optical reflectivity, 495, 496, 529 Diffuse x-ray scattering, 3 Disorder effect, 83, 366

domains, 1, 129 Fermi liquid, 3, 401

fluctuations, 323 lattice, 2 ordering, 6, 287, 295, 330, 395, 401, 459 phase, 399 state, 6, 369 Antiphase domain, 295, 298, 330 Atomistic cluster model, 391

Dimensional crossover, 364

Domain wall, 63, 91, 104, 130, 143, 151, 449 Doped Heisenberg antiferromagnet, 447 Double exchange, 459 Dynamical mean-field theory, 101, 171, 184

Dynamically screened potential, 385 Elastic energy, 280, 281 Electrical conductivity, 31, 364 Electrical resistivity, 524, 526, 527, 537, 542, 543,

BCS many body theory, 70 Bipolaron, 3, 159, 160,245 Bose–Einstein condensation, 72, 160 Boson–Fermion model (BFM), 160, 245

546, 577, 578 Electrochemical intercalation, 515

Electron–boson coupling, 379, 407 Electron diffraction, 423, 457, 501, 503 Electron-lattice interaction, 9, 20, 185, 199, 202, 433

Bragg glass, 96

Break-junction, 377, 381 Brinkman–Rice Fermi liquid, 399, 402 relationship, 203

Electron liquid crystal, 94

Bright held image, 504

Electron–phonon coupling, 413 Electron–phonon interaction, 39

Casimir effect, 447

Eliashberg equation, 377, 407, 408, 413, 416, 417 Emery hamiltonian, 370

Chain superstructure, 211,212

Energy distribution curves (EDC), 256

Charge density wave (CDW), 63, 77, 83, 92, 112, 113, 135, 175, 200, 253, 289, 320, 357, 427, 431, 502 Charge dynamics, 272, 287 Charge fluctuations, 315, 318 Charge ordering, 15, 39, 121, 183, 195, 271, 355, 447, 459, 474 Chemical potential, 515 Cluster orbitals, 394 Collective mode, 253 Colossal magneto-resistance (CMR), 6, 16, 202, 427, 481,482 Columnar defects, 545, 551, 557, 559, 564

EXAFS, 2, 10, 227, 249

Excessions, 28 Femtosecond spectroscopy, 219

Fermi surface, 237, 240, 253, 257, 259, 388 instability, 77, 83 Ferrolectric transition, 413 Ferrolectricity, 414 Ferrons, 1 Finite size cluster study, 567 Flux dynamics, 560

Flux phase, 113

Columnar tracks, 545

Fluxons, 559

Commensurate polaron crystal (CPC), 10, 14, 15, 16

Friedel oscillations, 385

Complete active space self-consistent field method, 391,392

Frustrated phase separation, 15

Cooper instability, 402

Gap density wave, 61

581

582 Gap parameter, 51 Gap stripes, 55, 60 Hall constant, 31 Hartree–Fock calculation, 121 Holon, 28, 263 Hubbard–Holstein model, 48 Incommensurate charge density wave (ICDW), 2, 10, 47 Incommensurate fluctuations, 6 Incommensurate magnetic order, 335 Inelastic neutron scattering, 2, 6, 191, 323 Infrared spectra, 428 Irreversibility line, 545, 559, 561, 563 Jahn–Teller cooperative, 203 deformation, 202 distortion, 4, 122,213,427 polaron, 1,6, 10,428 OT mode, 4, 5 O.3 mode, 4, 5 stripes, 8 van Hove, 112 Josephson coupling, 507, 508 Josephson junction, 551 Josephson tunnel junction, 507 Kohn anomaly, 356, 385 Kondo lattice, 459, 461 Korringa law, 330 Kosterlitz–Thouless transition, 64 Ladders systems, 437, 465 Landau parameter, 363 Lattice annarmonicity, 40 charge stripes, 9, 17 deformation, 246 distortion, 3 fluctuations, 245, 254 instabilities, 3, 228 mismatch, 9, 21, 200 polaron, 3 potential, 83 Local lattice distortions, 227 Local lattice fluctuations, 245 Local structure, 39 Localization, 361, 362 LTO, 4 LIT, 4, 501 Magnetic fluctuations, 316, 323 Magnetic impurity, 343, 344 Magnetic susceptibility, 523, 576

Subject Index Manganites, 455, 459, 473, 481 Marginal Fermi liquid, 30 Marginal Fermi surface, 254 Mesoscopic stripes, 3, 14 Metal to insulator transition, 1, 47, 363, 421, 521, 573 Metnzer–Vollhardt method, 400 Microwave resistivity, 274 Migdal–Eliashberg, 170 Mirror plane, 238 Miscibility gap, 212, 214 Modulated structure, 455, 473 Mott transition, 400 Multilayers, 539 Muon spin resonance 15 Nanostructures, 507 Nematic phase, 95 Neutron scattering, 3, 315, 323, 329, 336 Nickelates, 473 Nuclear magnetic resonance (NMR), 2, 3, 5, 11, 287, 288, 303, 304, 309, 358 Nuclear quadruple resonance (NQR), 3, 5, 280, 287, 288, 295, 296, 303, 304, 358 Optical absorption, 429 Optical conductivity, 3, 30, 271 Optical excitations, 495 Optical reflectivity, 495, 529, 531 Optical spectroscopy, 4, 219 Orbital density wave (ODW), 10, 11 Orbital ordering, 121 Oxygen doped La2CuO4, 2, 17, 515 Pair distribution function (PDF), 11, 250 Paramagnetic Meissner effect, 356 Perovskite structure, 200 Phase diagram, 4, 9, 19, 46, 57, 92, 94, 114, 127, 173,

185, 233, 347, 375, 404, 460, 461, 534 Phase fluctuations, 55 Phase separation, 1, 2, 15, 47, 91, 112, 211, 214, 369, 402, 459 Phase stiffness, 56, 71 Phonon anomaly, 191 Phonon dispersion, 191 Photo-excitation, 2, 3, 219 Photoinduced transmission, 221 Pinning centers, 548 Pinning force, 545 Pinning mechanism, 539 Polaron carrier, 183 crossover, 169, 185 formation, 170 gas, 175, 199 lattice liquid, 199

Subject Index Polaron (cont.) magnetic, 1 spin, 155 Pole lines, 387 Pseudogap, 225, 246 Pseudo Jahn–Teller, 9, 206, 230 Pump-probe thermal modulation, 529

583 Spinon, 28, 29, 263 STM, 378 Stripe charge dynamics, 271 correlations, 330 crystal, 91, 94

dynamics, 129

formation, 39, 93 Quadrupolar relaxation, 279 Quantum critical point (QCP), 9, 19, 20, 47, 113 AFM, 47, 447 ICDW, 47 interference, 135 stripe phase, 45 Quantum fluctuations, 55, 64, 73, 143, 350, 399 Quantum Heisenberg antiferromagnet, 309, 349 Quantum interference, 356

Quantum lattice string, 143 Quantum phase transition (QPT), 16 Quantum spring, 64 Quantum stripes, 9, 237, 487 Quasi particles, 28, 219, 377, 441

glass, 91,96

liquid, 91,96

ordering, 101, 135, 191, 271 pattern, 196 pinning, 129 stiffness, 84

superstructure, 215 wave-vector, 413 Stripons, 27, 29 Superfluid density, 71, 271

Surface of anomalies, 386 Svivons, 27, 30 Symmetry perturbing field, 343

Quasi particles recombination, 222, 224

Tc amplification, 3 Thermal conductivity, 346

Scanning electron microscope, 487 Scattering plane, 238 Self-energy, 369

Thermoelectric power, 6, 33, 199 Time-resolved spectroscopy, 219

Shadow bands, 30, 49

Two-leg ladder, 263, 267, 309, 310

Shape resonance, 3, 9, 14 Single electron transition, 355 Skyrmion, 349

Variational wave function, 4(X)

Tunneling, 111, 159, 377, 407

Van Hove singularity (VMS), 1 1 1 , 387

Slave-boson, 48, 152 Slave-fermion, 27

Vibrational dynamics, 279

Smectic phase, 95 SO(6) symmetry, 111, 116 Soliton, 63, 69 Spin charge separation, 28, 263 Spin charge stripes, 15 Spin correlations, 335, 336, 337 Spin density modulation, 336

Virtual spin excitation

Spin density wave (SOW), 83, 92, 135, 253

Ward identity, 357

Spin dynamics, 287, 329

Wigner crystal, 57, 91 Wigner glass, 97 Wigner lattice, 156 Wigner localization, 3, 15

Spin fluctuations, 315, 336 Spin gap, 309 Spin gap proximity effect, 69, 70, 72

Spin glass, 98 Spin lattice relaxation, 306, 309, 312 Spin ordering, 15, 121,271 Spin Peierls order, 355

Vibronic state, 199,200 Vortex corrected approximation, 171 Vortex dynamics, 559 Vortex lattice, 552 Vortex motion, 507 Vortex pinning, 551

Wigner phase, 153

X-ray diffraction (XRD), 10, 11, 18, 465, 488, 518, 522,541

Spin relaxation, 5

Spin waves, 401, 447

Zhang–Rice singlet, 6, 438

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