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
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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
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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
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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
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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.
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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
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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,
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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
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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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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
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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.
Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.
9
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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
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Bianconi, Agrestini, Bianconi, Di Castro, and Saini
Lattice-Charge Stripes in the
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.”
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Bianconi, Agrestini, Bianconi, Di Castro, and Saini
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
Lattice-Charge Stripes in the
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].
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Bianconi, Agrestini, Bianconi, Di Castro, and Saini
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.
Lattice-Charge Stripes in the
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
Bianconi, Agrestini, Bianconi, Di Castro, and Saini
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.
Lattice-Charge Stripes in the
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
Bianconi, Agrestini, Bianconi, Di Castro, and Saini
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].
Lattice-Charge Stripes in the
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
Bianconi, Agrestini, Bianconi, Di Castro, and Saini
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.
Lattice-Charge Stripes in the
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
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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.
Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.
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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].
Stripes, Electron-Like and Polaron-Like Carriers, and
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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
Stripes, Electron-Like and Polaron-Like Carriers, and
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
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Ashkenazi
The thermoelectric power (TEP) is given by
By Eq. (7), the condition Thus, by introducing
for the evaluation of the TEP becomes
Stripes, Electron-Like and Polaron-Like Carriers, and
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]
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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
Stripes, Electron-Like and Polaron-Like Carriers, and
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.
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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).
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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.
Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.
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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
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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).
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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.
Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.
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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,
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Caprara, Castellani, Di Castro, Grilli, Perali, and Sulpizi
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
The Stripe-Phase Quantum-Critical-Point Scenario
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
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Caprara, Castellani, Di Castro, Grilli, Perali, and Sulpizi
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
The Stripe-Phase Quantum-Critical-Point Scenario
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].
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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.
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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.
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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
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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
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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].
Phase and Amplitude Fluctuation in
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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
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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
Phase and Amplitude Fluctuation in
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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
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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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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
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Dimashko and Morais Smith
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.
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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
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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]
Microscopic Theory of High-Temperature Superconductivity
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).
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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:
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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
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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)
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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).
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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.
Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.
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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,
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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
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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
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Lichtenstein, Fleck, Oles, and Hedin
(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
Dynamical Mean-Field Theory of Stripe Ordering
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,
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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
Dynamical Mean-Field Theory of Stripe Ordering
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]
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Lichtenstein, Fleck, Oles, and Hedin
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
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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.
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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.
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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).
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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).
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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).
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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.
Stripes and Related Phenomena, edited by Bianconi and Saini. Kluwer Academic/Plenum Publishers, New York, 2000.
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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
