(Antonio Bianconi, Naurang L. Saini) Stripes and R PDF
(Antonio Bianconi, Naurang L. Saini) Stripes and R PDF
(Antonio Bianconi, Naurang L. Saini) Stripes and R PDF
Phenomena
SELECTED TOPICS IN SUPERCONDUCTIVITY
Series Editor: Stuart Wolf
Naval Research Laboratory
Washington, D.C.
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
No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,
mechanical, recording, or otherwise, without written consent from the Publisher
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 con-
densed 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 interna-
tional conference on Stripes and Superconductivity from experts in the field of
stripes and related phenomena are selected for the publication. Apart from the relevant con-
tribution 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.
v
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Contents
INTRODUCTORY OVERVIEW
From Phase Separation to Stripes 1
K. A. Miiller
Lattice-Charge Stripes in the Superconductors 9
A. Bianconi, S. Agrestini, G. Bianconi, D. Di Castro, and N. L. Saini
vii
viii Contents
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 573
Yoshihiko Takano, Yoshihide Kimishima, Hiroyuki Taketomi, Shinji Ogawa,
Shigeru Takayanagi, and Nobuo Môri
Author Index 579
Subject Index 581
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From Phase Separation to Stripes
K. A. Müller1
The basic concept of Jahn-Teller polarons [ 1 ] was behind the search for supercon-
ductivity 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 ferromag-
netic 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.
1
2 Müller
that the superconducting phase was competing with the phase separation below a critical
temperature. That visit at Rome University made me more attentive to the topic of electronic
inhomogneities.
A year later, I gave a talk at the Brookhaven National Laboratory on something quite
different and came to know the work of Victor Emery and collaborators [5] on dynamic
metallic clusters in an antiferromagnetic lattice, which they concluded on the basis of
kinetic and magnetic energy calculations. This made me think, well, there are three different
theoretical groups talking about a phase separation of the doped holes, there might be
something to this. It would not be too surprising, after all, as we have a number of examples
in physics where phase separation exists. From this arose the idea to organize a workshop to
discuss these matters, and with the help of Giorgio Benedek, director of the School of Solid
State Physics in Erice, we decided to hold the workshop in May 1992 [6]. In this workshop,
on the experimental side, phase separation was shown to exist in oxygen-doped
where the dopants are mobile interstitial oxygens. The phase separation into oxygen-poor
antiferromagnetic domains and oxygen-rich metallic domains with was found
by several groups using electrochemical oxidation, susceptibility experiments, and nuclear
magnetic resonance (NMR). A point of discussion was whether these chemically separated
phases were driven by electronic forces or not. Experimental indications for an electronic
origin were given by A. Heeger using photo excitation experiments, and by the Orsay group
using Mössbauere experiments. A different phase separation due to ordering of polarons in
linear arrays in the plane was presented by Bianconi using EXAFS and diffraction on
This was assigned to an one-dimensional (ID) incommensurate charge
density wave (ICDW) that coexists with itinerant carriers moving like waves of water on a
corrugated shore. On the theoretical side, Victor Emery gave a quite convincing talk at the
beginning of the workshop which invoked several experiments that could be explained by
a phase separation of clusters of doped holes in the antiferromagnetic lattice, frustrated by
Coulomb interactions. In Erice, it was decided to hold a follow-up workshop on the same
topic, which was organized by Sigmund in Cottbus in 1993 [7].
In Cottbus there was an interesting experimental report from the group of Albert Furrer
on using inelastic neutron scattering. They could show that upon doping the
system with oxygen, it shows the evolution of three different local cluster types characterized
by crystalline electric field (CEF) lines of Er are shown. They also presented the evolution
of the fractal sizes of the clusters, derived from CEF line widths, as a function of oxygen
content. Thus, this was clear evidence of a chemical structural phase separation on a local
scale. We have to accept that there are clusters and, of course, we are then interested in what
happens inside the cluster, as in the high-energy physics, where, for example, first there are
protons and neutrons, and then one becomes interested in what the neutrons and protons
consist of.
David Johnston gave a report on in the insulating phase at very low hole
doping. His group measured the magnetic susceptibility as a function of temperature for
various dopings, and found that the more one dopes the system, the smaller the susceptibility
peak becomes. From the analysis of these measurements, they concluded that the holes were
forming linear metallic lines, increasing in widths with doping. Their results were evidence
for linear (1D) domain walls separating antiferromagnetic domains.
Antonio Bianconi reported on the measurement of the relative number of polaronic
charge carriers contributing to the one-dimensional incommensurate CDW (ICDW), and
the itinerant carriers based from the variation of the periodicity of the ICDW with Y doping
in the superconducting phase of the In his talk, he introduced the
From Phase Separation to Stripes 3
idea of a 2D electron gas in this doped superconducting cuprate, driven close to the Wigner
localization limit by a large electron-lattice interaction. Therefore, the 1D ICDW could be
assigned to a generalized Wigner CDW in a 2D electron gas with large Coulomb and lattice
interactions.
There was a report by Chris Hammel from Los Alamos on the NMR and Nuclear
Quadrupole Resonance (NQR) investigation of the La ions. In his experiment, he observed
two lines that were assigned to a distribution of tilted octahedra. This was an important
step because until then, NMR people had always been saying that there was a single line
supporting the homogeneous antiferromagnetic Fermi-liquid model of Pines et al. [8]. Now
we think that his findings indicate pinned stripes; otherwise, he could not have seen them.
Furthermore, these tilts were continuous, showing that in these stripes we have a distribution
of lattice distortions.
In Cottbus we decided to split up these topics and hold a meeting in Bled on
“Anharmonic Properties of Cuprates” in 1994 [9], followed by a conference on
“Stripes and Lattice Instabilities” in Rome at the end of 1996 [10]. In between there was a
NATO School entitled, Superconductivity. Ten Years after the Discovery,” orga-
nized 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 organiz-
ing 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 conduc-
tivity and also a quasi-localized one (he put a question mark regarding their polaronic
character).
On the theoretical side, there was a remarkable report by Alexandrov on the formation
of bipolarons at a temperature higher than which might be responsible for the
anomalous kinetics and thermodynamics observed in the cuprates. To me, this was the first
time when the bipolarons entered the picture. Next came a report by Julius Ranninger,
who emphasized that itinerant valence electrons coexist with bipolarons above while
condensation occurs at
In Delphi, many subjects were discussed in the review talks [11]. One of them was by
Wells, who summarized the results of MIT and Brookhaven on the system,
4 Müller
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 represen-
tation [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
From Phase Separation to Stripes 5
is the same as that shown by the EXAFS results [14]. I will come again back the mode
later.
There was also a report by Teplov and his group at Kazan University, who performed
extensive NMR and NQR measurements in the TmBaCuO compound, which also becomes
superconducting in the 123 and 124 phases. Basically, they found different copper centers
having orthorhombic and tetragonal distortions characterized by the NMR/NQR line widths
and spin relaxation rates. The relative content was about 2/3 for the orthorhombic and 1/3
6 Müller
for the tetragonal. Their interpretation was pinned stripes with antiferromagnetic ordering
in the plane along the ( 1 1 ) direction. However, they were not sure about the orientation of
the stripes, i.e., whether they occurred along the (11) or the (10) direction.
Then came the STRIPES 96 conference in Rome. The proceedings were published as
a special issue of the Journal of Superconductivity and edited by Bianconi and Saini [9].
This was a relatively large conference, even if not as large as this STRIPES 98, but still large
enough to indicate that the subject of stripes was gathering momentum. There were a number
of interesting contributions from several groups, so let me just give one or two examples that
particularly attracted my attention. Mook et al. of Oak Ridge performed inelastic neutron
scattering measurements and were able to see incommensurate fluctuations in the Bi2212
compound. They associated the results with a dynamic stripe phase. De Lozanne from
Houston used atomic force microscopy YBCO superconductors to show that there is an
incommensurate plane wave with coherence along the a-direction of about three lattices, as
in Mook’s experiments.
John Goodenough presented results on the temperature dependence of the resistance
and the thermoelectric power with hydrostatic pressure. In the proceedings paper [10], he
wrote
We conclude that electron–lattice coupling is the dominant
factor influencing the stabilisation of a dynamically
heterogeneous, thermodynamically distinguishable phase
and the transport properties exhibited by that phase. It is
probable that in the copper oxides a non-retarded elastic
coupling of cooper pairs replaces the BCS retarded-potential
coupling of conventional superconductors.
An elastic coupling, because if there are Jahn–Teller polarons, they deform the plane
in a quadrupolar manner and it is clear that one cannot have an abrupt discontinuity from one
stripe to the other. This issue will perhaps become clearer in the coarse of this conference,
i.e., up to what extent the elastic coupling is relevant or more relevant than the Coulomb
forces between the charges.
I am now moving to the colossal magneto-resistance (CMR) compounds. Let me
show you a figure which I modified from Khomskii and Sawatzky [15] (Fig. 4) and that is in
relation to the superconductivity in the cuprates and the ferromagnetism in the CMR systems.
What they have considered in their paper is the following: Normally transition metal oxides
there are antiferromagnets because of the super exchange between the transition metal ion
via the oxygen, as shown schematically in Fig. 4 [15]. Of course, this is the case for the
copper oxides and also for many other magnetic oxides. Now, if the transition metal ion has
a spin 1/2 and if a large Coloumb repulsion, U, is present, the hole’s main probability is on
the oxygens. Then, the two spins can couple and form a spin 0 state, as shown in the lower
part of the figure. This is the so-called Zhang–Rice singlet, which can become mobile in
the antiferromagnetic lattice. This is the case for the cuprate superconductors. However, if
the spin is larger than 1/2—suppose it is 1—then the magnetic moment will also couple as
well to the doped hole (U being large) and ferromagnetism results (upper part of Fig. 4).
In the Khomskii–Sawatzky scheme, you do not have that. Copper is the last in the row of
the 3D elements, and if one moves to the manganese, the U is smaller, and the hole resides
mainly on the transition metals, and an antiferromagnetic state is realized.
Finally, I want to show that in these systems the presence of stripes of polarons is
now becoming clear. Figure 5 shows an example of the ordering of alternated rows of
From Phase Separation to Stripes 7
8 Müller
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
We report a 2D plot for the doped perovskites, where is the doping and
is the mismatch between the layers and the rocksalt layers. We identify
for the first time the quantum critical point (QCP) at and
for the onset of the polaron stripes coexisting with itinerant
carriers. The plot for shows the highest at the critical
point The lattice mismatch drives the system to this QCP of the electron–
lattice interaction for local lattice deformations at metallic densities. The
incommensurate superlattice of charge-lattice (polaron) stripes are due to critical
fluctuations near this QCP. The solution of the mystery of the anomalous normal
phase of superconductors implies a solution for the pairing mechanism:
the attractive pseudo-Jahn–Teller polaron–polaron interaction and a particular
type of critical charge and spin fluctuations (forming a superlattice of quantum
stripes tuned at a “shape resonance”) near this QCP play a key role in the pairing
mechanism.
1. INTRODUCTION
A charge transfer Mott insulator is at in the conventional phase diagram of
cuprate superconductors The superconductivity with appears in the range
We show that the mystery of the normal phase of the superconductors
is solved by introducing a second variable, the electron–lattice interaction The electron
lattice interaction is driven by the compression acting on the lattice due to the lattice
mismatch of the metallic plane embedded in the perovskite crystalline lattice. In
1
Dipartimento di Fisica, and Unitá INFM, Università di Roma “La Sapienza,” P. le Aldo Moro 2, 00185 Roma,
Italy.
9
10 Bianconi, Agrestini, Bianconi, Di Castro, and Saini
1. A disordered phase in the limit of low density where the charges are in the localization
limit
2. an insulating ordered phase A where the charges are associated with vibronic co-
operative 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, form-
ing a superlattice of lattice-charge stripes (polaron stripes) with an associated orbital
density wave (ODW).
3. a phase B made of itinerant carriers forming a normal metallic phase.
4. 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 and the superconductivity occurs in the critical region in
the plane near this quantum critical point.
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 neigh-
boring 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 fluctu-
ations 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 ob-
served in infrared spectroscopy have provided evidence for the universal coexistence of
polarons and free carriers in superconducting cuprates [33].
measure of the charge density and the distance from the Mott Hubbard insulator at
By increasing the system goes through quite different states. At low doping, the doped
polaronic charges are pinned to impurities and form a disordered electron glass. At high
doping, a normal metal phase appears. The high superconducting phase appears between
these two phases. In 1993 at the Cottbus conference [16–20], we presented the phase diagram
for Bi2212, shown in Fig. 4, that is becoming quite robust at this second international
conference on “Stripes and High Superconductivity.”
14 Bianconi, Agrestini, Bianconi, Di Castro, and Saini
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.
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
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
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 ). For example, in manganites at the insulator-to-
metal transition from CPC to B, at an 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 com-
pression 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 tol-
erance 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 mis-
match Moreover, the mismatch induces also the decrease the polaronic bandwidth [61].
Lattice-Charge Stripes in the Superconductors 21
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 temper-
ature 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 dif-
ferent 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 inhomo-
geneous 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 po-
laron 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 superconduct-
ing 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).
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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. Anoma-
lous 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 super-
conductivity.
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.
27
28 Ashkenazi
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.
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 hy-
bridization 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 diagram-
matic technique where introduces a vertex connecting QE, stripon, and spinon propaga-
tors. 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, re-
spectively. 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 polaron-
like 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 ap-
proximately expressed as
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 and are, respectively, the contributions of QEs and stripons to the electrons
density of states at
where
and because
32 Ashkenazi
where
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).
By Eq. (7), the condition for the evaluation of the TEP becomes
Thus, by introducing
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
similarly to the TEP in normal metals. However, the stripon bandwidth is of order 0.02 eV.
Thus, one expects to saturate at to the narrow-band result [17]
36 Ashkenazi
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 mech-
anism 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 normal-
state 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
band maintains coherence between different stripe segments. Thus, an upper limit for 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
where in the stripons effective mass and is their density per unit area of a plane
(note that the stripons are spinless).
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.
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7. J. M. Tranquada et al., Phys. Rev. B 54, 7489, (1996); Phys. Rev. Lett. 78, 338 (1997).
8. C. M. Varma et al., Phys. Rev. Lett. 63, 1996 (1989).
9. D. B. Tanner, and T. Timusk, Physical Properties of High Temperature Superconductors III, edited by
D. M. Ginsberg (World Scientific, 1992), p. 363.
10. M. I. Salkola et al., Phys. Rev. Lett. 77, 155 (1996).
11. D. S. Marshall et al., Phys. Rev. Lett. 76, 4841 (1996).
12. T. R. Chien et al., Phys. Rev. Lett. 67, 2088 (1991).
13. Y. Kubo and T. Manako, Physica C 197, 378 (1992).
14. H. Takagi et al., Phys. Rev. Lett. 69, 2975 (1992); H. Y. Hwang et al., ibid. 72, 2636 (1994).
15. P. W. Anderson, Phys. Rev. Lett. 67, 2092 (1991).
16. H. D. Drew, S. Wu, and H.-T. S. Linh, preprint.
17. B. Fisher et al., J. Supercond. 1, 53 (1988); J. Genossar et al., Physica C 157, 320 (1989).
18. S. Tanaka et al., J. Phys. Soc. Jpn. 61, 1271 (1992).
19. K. Matsuura et al., Phys. Rev. B 46, 11923 (1992); S. D. Obertelli et al, ibid., p. 14928; C. K. Subramaniam
et al., Physica C 203, 298 (1992).
20. J. Kondo, Prog. Theor. Phys. 29, 1 (1963).
21. Y. J. Uemura et al., Phys. Rev. Lett. 62, 2317 (1989).
22. Ch. Niedermayer et al., Phys. Rev. Lett. 71, 1764 (1993).
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Charge Ordering and Stripe Formation
in Cuprates
Annette Bussmann-Holder1
The local structure of transition metal oxides and especially high-temperature supercon-
ducting (HTSC) copper oxides deviates strongly from their average crystallographic struc-
ture [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.
In the following, the same model and its nonlinear solutions for the ionic displace-
ments 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 simplic-
ity 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 dimer-
ized 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 tem-
perature 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 anti-
ferromagnetic 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 spin-
correlated regimes, do not dominate the dynamics but coexist with the antiferromagnetic
polarons. Preliminary results for a three-dimensional model for HTSC [9] yield convinc–
ing evidence that the dynamic polarons are related to effects stemming from the c axis,
which drives a self-consistent buckling of the planes and stabilizes an in-plane singlet
state [10].
The extra dynamics due to anharmonicity and higher-order electron–phonon effects
are accompanied by potential fluctuations. In real space (Fig. 2), these potential fluctuations
are very close to zero within a few lattice constants and small anharmonicity. Increasing
the depth of the double-well potential leads to short-range repulsive areas. It is important to
42 Bussmann-Holder
Charge Ordering and Stripe Formation in Cuprates 43
note here that a small degree of orthorhombicity is sufficient to always observe the repulsive
regimes. This specifically applies to the HTSC compounds.
In conclusion, it has been shown that stripe formation of the lattice is accompanied
by modulations of the electronic energies that form charge-rich and charge-poor areas on
the same length scale as the lattice. Although the charge-rich domains can be identified
with c axis–induced polarons, the charge-poor areas are dominated by antiferromagnetic
correlations and have a strong similarity to the Zhang–Rice singlet state [10].
REFERENCES
1. See, e.g., Lattice Effects in Superconductors, Y. Bar-Yam, T. Egami, J. Mustre-de-Leon, and
A. R. Bishop, eds. (World Scientific, Singapore, 1992).
2. A. Bianconi et al, Phys. Rev. Lett. 76, 3412 (1996); S. D. Conradson, D. Raistrich, and A. R. Bishop, Science
248, 1394 (1990); T. Egami et al., Proceedings of the International Workshop on Anharmonic Properties of
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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,
J. Supercond. 10, 289 (1997).
4. A. Bussmann, H. Bilz, R. Roenspiess, and K. Schwarz, Ferroelectrics 25, 343 (1980).
5. H. Bilz, G. Benedek, and A. Bussmann-Holder, Phys. Rev. B 35, 4840 (1987).
6. Y. Lepine, Phys. Rev. B 28, 2659 (1983).
7. 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).
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The Stripe-Phase Quantum-Critical-Point
Scenario for Superconductors
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.
The properties of the normal state are to some extent even more challenging, the stan-
dard 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 charac-
terized 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
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 obser-
vation 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 ex-
planation for the peculiar nature of this doping regime in the phase diagram of all SC copper
oxides.
There is an increasing amount of theoretical and experimental evidence in favor of
the presence of a QCP near optimum doping. Indeed, the instability with respect to phase
separation (PS) into hole-rich and hole-poor regions is a generic feature of models for
strongly correlated electrons with short-range interactions [25], which is turned into a
frustrated PS [26] or in an ICDW instability [22] when long-range Coulomb forces are
taken into account to guarantee large-scale neutrality. Close to PS (or ICDW), there is
always a region in parameter space where Cooper pair formation is present, pointing toward
a connection between PS and superconductivity.
The most compelling evidence for a QCP near optimum doping is provided by the
resistivity measurements. An insulator-to-metal transition is found when the SC phase is
suppressed by means of a pulsed magnetic field [27]. When extrapolated to zero temperature,
such a transition takes place near optimum doping, and at too high a doping to be associated
to the spin-glass region [28] characterized by the local moment formation as seen in the muon
experiments [29]. The spin-glass region should instead be a signature of the coexistence of
superconductivity with antiferromagnetism proper of the SO(5) theory [30]. Moreover, a
clear indication that this insulator-to-metal transition [27] is driven by some spatial charge
ordering is provided by its occurrence at a much higher temperature in samples near the
filling 1/8. Commensurability effects near this “magic” filling have repeatedly been reported
in related compounds [31].
Hints for a critical behavior of the charge susceptibility come from the study of the
chemical potential shift in PES and BIS experiments [32]. A dramatic flattening of the vs
curve starting at could be the signature of a divergent compressibility. Finally,
48 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 re-
pulsion that gives rise to the AFM state at low doping and the attraction giving rise to SC
through the (local) PS or ICDW.
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 reso-
lution function We take
according to numeric convenience. For the sake of definiteness, we choose
our parameters in Eq. (2) to fit the band structure and FS of Bi2212; namely, t = 200 meV,
and corresponding to a hole doping with respect
to half filling. The parameters appearing in the effective interaction in Eq. (1) were taken
as and
The quasi-particle spectra are characterized by a coherent quasi-particle peak at an
energy and by shadow peaks at energies produced by the interaction
with charge and spin fluctuations. The shadow peaks do not generally correspond to new
poles in the electron Green function and are essentially incoherent, although they follow the
dispersion of the shadow bands. Their intensity varies strongly with k and increases when
approaches the value . In particular, at the hot spots, where and
the non-FL inverse scattering time there is a suppression of the coherent
spectral weight at the Fermi energy.
We also study the k-distribution of low-laying spectral weight The
transfer of the spectral weight from the main FS to the different branches of the shadow
FS at produces features that are characteristic of the interaction with charge and
spin fluctuations and of their interplay. In particular, the symmetric suppression of spectral
weight at the M points of the Brillouin zone, which would be due to spin fluctuations
alone, is modulated by charge fluctuations (Fig. 2). This is also the case for the (weak) hole
pockets produced by spin fluctuations around the points The interference
50 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
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 un-
derlying 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 liq-
uid 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 opti-
mum 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 the-
ory of the QCP associated to the coexistence of antiferromagnetism and superconductiv-
ity, 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 phys-
ical 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
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
Lepes, C.N.R.S, 38042, Grenoble, France.
2
Indian Institute of Technology, New Delhi, 110016, India.
55
56 Chakraverty and Jain
which determines the amplitude of the gap and that this must be related to the phase proprties
of the complex order parameter. In general, we have where the phase
is blocked in the superconducting state at the same value everywhere spatially, indicating
infinite phase correlation or off-diagonal long-range order. It was pointed out for the first
time [4] that the energy scale associated with phase fluctuation is related to Josephson plasma
frequency. The resultant agrees quite well with the observed values in the cuprates [5]
and also quantum fluctuation of the phase gives the right trend as to the diminution of
with underdoping [6]. In this communication, we address the question of how and where the
pairing amplitude itself is lost. In particular, we indicate that the BCS meanfield transition
superconducting instability at is suppressed as a result of amplitude fluctuation if
we assume that there is no electron-hole symmetry in the cuprate oxide superconductors.
This fluctuation takes the form of a spinodal decomposition or gap stripes, which are
associated with charge density fluctuation (incidentally, a lattice distortion wave is expected
to accompany the charge density) [7]. Thus the so-called normal nonsuperconducting phase
above is to be envisioned as a phase of dynamic gap stripes.
We have the basic Hamiltonian given by
where cs are the electron operators, g is a local attractive interaction between the carriers
with momentum labels k, arrows are the spin indices, and is the electron kinetic energy
measured from the fermi level Applying a Hubbard-Stratanovitch transformation to this
Hamiltonian, we can obtain [8] the free energy as a sum of two parts: a homogeneous part
of the Ginzburg-Landau meanfield form and a quadratic term that contains quadratic
fluctuation at the saddle point of the superconducting gap amplitude its phase and
the electronic density n. The homogeneous part is written in the standard form to the fourth
order
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 Andreson-
Bogoliubov 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
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 Superconductors 59
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 are the pairing fields defined by and the gap is accordingly
given by We get the response as
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 fluc-
tuation, 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 is the quasi-particle exci-
tation energy given by Note the factor and so Eq. (13) vanishes
60 Chakraverty and Jain
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.
This gives
Eq. (18) has a spatially periodic solution for the the amplitude fluctuation given
by
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. J. W. Loram et al., Physica C 282, 1405(1997).
2. J. M. Harris et al., Phys. Rev. B 54, R15665 (1996).
3. H. Ding et al., Nature 382, 51 (1996).
4. 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).
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10. G. S. Boebinger et al., Phys. Rev. Lett. 77, 5417 (1996).
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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
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 con-
nection was established between this elastic and the more “microscopic” model [3].
The longitudinal transport in cuprate-stripes was considered, based on the Luttinger liq-
uid 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.
the collective (sliding) mode displays perfect conductivity. Later, Lee, Rice, and Anderson
(LRA) [8] analyzed the dynamics of the sliding mode in the presence of a lattice and argued
that pinning by the lattice and/or by impurities destroys the perfect conductivity. However,
LRA did not consider the effect of quantum fluctuations, which can compete with pinning
even at Hence, it remains to be verified whether the quantum fluctuations are able to
depin the stripes (or commensurable CDW), and whether this quantum depinning restores
the perfect conductivity of the sliding mode.
In order to investigate this problem, we study both the transversal (and the longitudinal)
dynamics of a completely (half) filled stripe, accounting for quantum fluctuations. By
performing a canonical transformation in the quantum Hamiltonian, we map both systems
onto a Josephson junction chain, which is known to exhibit a insulator/superconductor
transition. Then, we calculate the transport properties in two limitting cases that can be
solved explicitly, and clarify the meaning of the transition in the stripe contexte. We find
that the conducting properties of the stripe are determined from the competition between the
hopping amplitude t and the stiffness parameter J: in the weak-fluctuation limit,
the ground state (GS) of the system is a kink (soliton) vacuum. In this case, the stripe is
pinned by the lattice, the excitation spectrum exhibits a gap and the system is insulating.
In the opposite strong-fluctuation limit, the GS of the system is a kink (soliton)
condensate. Then, the excitation spectrum contains a gapless phonon-like mode and the
system is superconducting. The low t/ J insulating phase is separated from the large t/ J
superconducting phase by a Kosterlitz–Thouless (KT)-like phase transition at
corresponding to the unbinding of the kink/antikink (soliton/antisoliton) pairs.
We consider a single stripe consisting of N holes placed on a square lattice. In the case
of a completely filled stripe, we analyze only the transversal motion of the holes. It is the
quantum string model [2,3]. The representative states of the quantum string (flat state, kink,
antikink) are shown in Fig. l(a–c). In the case of a half-filled stripe, we analyze only the
longitudinal motion of the holes. It is the quantum spring model. The representative states
of the quantum spring (uniform state, soliton, antisoliton) are shown in Fig. l(d–f). In both
cases the holes are restricted to move in the x direction only. Assuming an elastic interaction
Stripe on a Lattice 65
between neighboring holes, one can write a common Hamiltonian for both models
The integer v assumes the values for the quantum string and for the quantum
spring. The lattice constant is taken as the unit of length and the index
numerates the holes. Together with the current operator and the canonical
commutation relations the Hamiltonian Eq. (1) contains all the conducting
properties of the system.
The local excitations of the stripe play an important role in the further analysis of
these properties: They are the kinks/antikinks (K-AK) for the quantum string model and
the solitons/antisolitons (S-AS) for the quantum spring model (see Fig. 1). It is convenient
to classify these states by the value of the topological charge K
and S are states with AK and AS are states with Here, we consider periodic
boundary conditions, Hence, the total topological charge of the stripe is zero,
and only creation of pairs K-AK or S-AS is allowed.
Next, we perform a duality transformation to new variables
These variables to not refer to separate holes, but to pairs of neighboring holes. They
are restricted to the interval and can be treated as a phase. As a result, the
Hamiltonian and the current operator acquire the form
which is known from the theory of superconducting chains. Equation (3) describes a
Josephson junction chain (JJC), with the Coulomb interaction taken into account. The
solution of this problem at was found by Bradley and Doniach [9]. Depending on
the ratio t/J, the chain is either insulating (small t/ J) or superconducting (large t / J ) .
At the Josephson chain undergoes a KT insulator/superconductor transition.
Because both the quantum string and the quantum spring models are described by exactly
the same Hamiltonian and current operator as the JJC, they also undergo an insulator/
superconductor transition at this point. Below, we analize the meaning of this transition for
the single stripe. It is instructive to do it in two limiting cases, which allow for an explicit
solution:
Let us start considering the limit of weak fluctuations, with In the absence
of hopping, the excitation spectrum of the stripe is discrete, with
The ground level corresponds to the vector representing a flat
state of the completely filled stripe (see Fig. la), or a uniform state of the half-filled stripe
(see Fig. 1d). In other words, it is the kink vacuum or the soliton vacuum, respectively.
The first elementary excitation corresponds to the creation of a pair K-AK or S-AS, i.e., to
the state are local creation operators of K or and
66 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 lo-
cal 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 above-
mentioned 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 superconduct-
ing 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/half-
filled 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 un-
dergoes 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, Taking and the lattice constant one
obtains Evidence of such a low-frequency and size-dependent response in
uniform monocrystallic samples would confirm both the phase coherence within one stripe
and the phase coherence between the stripes. Together, this would mean the kink/solitonic
origin of the supercurrent in the cuprates.
68 Dimashko and Morais Smith
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 par-
tially 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
1. INTRODUCTION
The high-temperature superconductors [1] are quasi-two-dimensional doped insula-
tors, obtained by chemically introducing charge carriers into a highly correlated antiferro-
magnetic 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
Dept. of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000.
2
Dept. of Physics, University of California at Los Angeles, Los Angeles, CA 90095.
69
70 Emery and Kivelson
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 super-
conductors 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 phe-
nomenological 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 con-
ventional 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 supercon-
ductors [ 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 ampli-
tude 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
72 Emery and Kivelson
(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 su-
perconductors 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 high-
temperature 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.
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 phenomenol-
ogy 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 fluctu-
ations [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].
74 Emery and Kivelson
4. MOMENTUM SPACE
So far, we have discussed the consequences of stripes in real space, but ARPES ex-
periments 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 degener-
ate 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].
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.
Microscopic Theory of High-Temperature Superconductivity 75
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Two Reasons of Instability
in Layered Cuprates
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 three-
band 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
Physics Department, Kazan State University, 420008 Kazan, Russia.
2
Corresponding Author: I. Eremin, Physics Department, Kazan State Univeristy, 420008 Kazan, Russian
Federation: E-mail: IIya.Eremin@ksu.ru; Tel: 007 8432 315116; Fax: 007 8432 380901.
77
78 Eremin, Eremin, and Varlamov
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].
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 short-
range 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 temper-
ature, 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 Eremin, Eremin, and Varlamov
been derived. It looks like Eq. (2) and can be now written as:
Two Reasons of Instability in Layered Cuprates 81
82 Eremin, Eremin, and Varlamov
Here,
and,
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 sub-
system 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. N. Furukawa, T. M. Rice, and M. Salmhofer, Phys. Rev. Lett. 81, 3195 (1998).
2. F. Becca, F. Bucci, and M. Grilli, Phys. Rev. B 57, 4382 (1998).
3. A. T. Zheleznyak, V. M. Yakovenko, and I. E. Dzyaloshinskii, Phys. Rev. B 55, 3200 (1997).
4. F. C. Zhang, and T. M. Rice, Phys. Rev. B 37, 3759 (1988).
5. S. V. Varlamov, M. V. Eremin, and I. M. Eremin, JETP Lett. 66, 569 (1997).
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7. I. Eremin, Physica (North-Holland) B 234–236, 792 (1997).
8. K. Yamada, C. L. Lee, Y. Endoh et al., Physica (North-Holland) C 282–287, 85 (1997).
9. M. V. Eremin, S. G. Solovjanov, S. V. Varlamov et al., JETP Lett. 60, 125 (1994).
10. Z.-X. Shen, and D. S. Dessau, Phys. Rep. 253, 1 (1995).
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Influence of Disorder and Lattice Potentials
on the Striped Phase
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 coher-
ent 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
Dept. of Physics, University of California, Riverside, CA, 92521, USA.
2
I. Institut für Theoretische Physik, Universität Hamburg, D-20355 Hamburg, Germany.
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 poten-
tials 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 interac-
tion 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 the action of the stripe can be
written as [14]
Influence of Disorder and Lattice Potentials on the Striped Phase 85
where is the stripe density and i is the replica index. The gaussian action is
given by
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
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
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
88 Hasselmann, Castro Neto, and Morais Smith
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
Influence of Disorder and Lattice Potentials on the Striped Phase 89
PROBRAL project no. 415. AHCN acknowledges support from the Alfred P. Sloan Foun-
dation and the U.S. Department of Energy.
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Stripe Liquid, Crystal, and Glass Phases
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 dom-
inant, 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 antiferromag-
nets, 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
Dept. of Physics, UCLA, Los Angeles, CA 90095.
2
Dept. of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000.
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.
3. Although Landau theory by its very character is relatively insensitive to the micro-
scopic 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 or-
dering 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 sep-
aration (and, indeed, experiments in the nickelates, manganates, and the appropriate
cuprates) imply that the ordering is, in fact, charge driven.
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
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 stripe-
glass transition temperature is greater than we await experimental input on this last
issue.
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-AC02-
98CH10886.
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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.
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Dynamical Mean-Field Theory of
Stripe Ordering
Charge localization has been observed in recent years in several doped transition metal
oxides, including nickelates, cuprates, and manganites. Localized holes organize into one-
dimensional (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]. Un-
fortunately, 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.
2
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.
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 in-
commensurability first appearing at temperatures above At present, it remains unclear
whether spatial segregation of the charges is associated with the observed incommensu-
rate 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 respec-
tively [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]
and longitudinal
susceptibility in Eq. (5) are found in random phase approximation (RPA) with renormalized
site-dependent interaction Here, the noninteracting susceptibilty is calculated from the
local DMFT Green function (4)
The renormalized interaction results from the screening by particle-particle diagrams [14],
with the scattering kernel
The self-energy (5) expresses the SF exchange interaction [16] with an effective potential
due to particle-particle scattering [14].
Equations (2), (4), (5), and (9) represent a solution for the one-particle Green functions
within the DMFT. They have been solved self-consistently, and the energetically stable
charge and magnetization densities were found. In the physically interesting regime of
the stable configuration at low temperature is found to be always
a stripe phase for . There is a competition between the horizontal (1,0) and diagonal
104 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, and The (1,0)
stripes (Fig. la) are more stable up to 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,
106 Lichtenstein, Fleck, Oles, and Hedin
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]
108 Lichtenstein, Fleck, Oles, and Hedin
used in Figs. 2b and 3b to show the magnetic structure. The phase factors adequate for stag-
gered 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.
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Tunneling and Photoemission in
an SO(6) Superconductor
1
Physics Department, Northeastern University, Boston, MA 02115, USA.
2
Barnett Institute, Northeastern University, Boston, MA 02115, USA.
3
On leave of absence from Inst. of Atomic Physics, Bucharest, Romania.
111
112 Markiewicz, Kusko, and Vaughn
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 pho-
toemission 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 predic-
tion, 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).
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 tunnel-
ing 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) generaliza-
tion, 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 tunnel-
ing 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
Tunneling and Photoemission in an SO(6) Superconductor 115
tunneling, whereas the gaps split into two separate features: B associated with supercon-
ductivity 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.
116 Markiewicz, Kusko, and Vaughn
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 su-
perspins, 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.
Tunneling and Photoemission in an SO(6) Superconductor 117
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 vec-
tors [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.
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 simpli-
fied 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.
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Spin, Charge, and Orbital Ordering in 3d
Transition-Metal Oxides Studied by Model
Hartree–Fock Calculation
We have spin, charge, and orbital ordered states in the perovskite-type lattice
models for by means of un-
restricted Hartree-Fock calculations. Present calculations show that, although the
vertical charge stripes along the (1,0) direction of the Cu-O square lattice are fa-
vored 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 sites are stabilized by
the Jahn–Teller distortion at the 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 diffrac-
tion 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
charge domain walls perpendicular to the (1,1,0) direction have been ob-
served, and in the 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.
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 The charge-transfer
energies for and are 4,2, and 4 eV, respectively [6].
the CO state with the oxygen-centered stripes is AFM for because each domain
has four rows of , those for and are ferrimagnetic. However, the CO state
with the magnetic polaron stripes is ferrimagnetic for and those for and
are AFM. When is decreased to 2.0 eV, the vertical magnetic polaron stripes become
lower in energy than the diagonal ones, indicating that the larger value of in the nickelates
than in the cuprates plays an important role to realize the diagonal stripes in the doped
system
3.2.
In the HF calculations for the cuprates, the CO solutions with the vertical charge stripes
running along the (1,0) direction are stable for various hole concentrations, whereas the CO
124 Mizokawa and Fujimori
states with the diagonal stripes are unstable. For and 1/3, the vertical
stripes or domain walls, which are centered at the metal sites, and those centered at the
oxygen sites are almost degenerate and are the lowest in energy. The energy difference
between the CO states with the vertical stripes and the AFM metallic state are
and 22 meV for and 1 /3, respectively. The CO states with the diagonal
stripes are unstable for the charge-transfer energy of 2 eV. However, when is increased
to 6 eV, the diagonal stripes become stable and are lower in energy than the vertical stripes,
indicating that the small charge-transfer energy in the cuprates is essential to stabilize the
vertical stripes compared to the diagonal ones. These domain walls have one hole per metal
site and are so-called filled domain walls [9], in which the period of the stripes is a/x, where
a is the in-plane lattice constant.
Zaanen and Oles [9] investigated the half-filled domain walls or stripes by using HF
calculations and found that the metal-centered half-filled stripes are moderately stabilized
by charge- and spin-ordering along the stripes, but are still higher in energy than the filled
stripes. In the metal-centered half-filled stripes, the charge- and spin-ordering along the
stripes causes quadrupling of the period along the stripes [9]. We studied the stability of
the half-filled stripes for and 1/8. For only a CO state with the oxygen-
centered half-filled stripes is obtained. However, for both the metal-centered and
oxygen-centered half-filled stripes are obtained. While the metal-centered stripes have the
quadrupling of the period along the stripes as predicted by Zaanen and Oles [9], the oxygen-
centered ones are not accompanied by the quadrupling (Fig. 2). In the present calculation
for the oxygen-centered half-filled stripes are by higher in energy than
the filled vertical stripes, as shown in Fig. 2. The magnitude of the band gap of the half-filled
stripes is calculated to be less than which is much smaller than that of the filled
stripes In this model calculation, the energy difference between the CO states
with the filled and half-filled stripes becomes smaller in going from to
indicating that the filled stripes become preferred to the half-filled ones with hole doping.
Very recently, the density matrix renormalization group calculation on the t-J model has
shown that the ground state has the half-filled vertical stripes for and the filled
ones for It has also been reported that the intersite Coulomb interaction
stabilizes the half-filled stripes compared to the filled ones [11]. In order to fully understand
the nature of the charge stripe, it is required to study more realistic models, including the
intersite Coulomb interaction and the electron-lattice interaction. In particular, the stripe
superstructure along the diagonal direction was observed in suggesting
that the coupling between the charge stripe and the structural modulation is important.
3.3.
In order to study how the stability of the CO states is affected by the orbital ordering in
the manganites, we have performed model HF calculations for Without the
JT distortion, the FM state is the lowest in energy and the A-type AFM state is the second
lowest. The FM and A-type AFM states are metallic and are not accompanied by charge
ordering. The CE-type AFM CO solution with a band gap of is obtained, but is
higher in energy than the FM and A-type AFM state. In the CE-type AFM CO state, the
and are interlaced like a checkerboard within the c plane as shown in Fig. 3a.
Along the c axis, the same in-plane arrangement of is stacked and the neigh-
boring planes are antiferromagnetically coupled. The sites are accompanied by the
orbital ordering even without the JT distortion [13]. This is because
Spin, Charge, and Orbital Ordering in 3d Transition-Metal Oxides 125
the orbital of the site tends to point the neighboring site in order to gain the
kinetic exchange energy. It is interesting to note that the orbital ordering in
is contrasted with that in which is a mixture of the -type and
the -type when the JT distortion is absent [13]. We have calculated the
energies of the A-type and CE-type AFM states relative to the FM state as functions of
the JT distortion that is consistent with the orbital arrangement.
The present calculation shows that the CE-type AFM state is stabilized by the JT distortion.
However, the tilting of the octaherda does not reduce the energy difference between
the AFM states and the FM state very much.
The checkerboard-type charge ordering couples with the in-plane breathing-type lattice
distortion in which the ion is expanded and the ion is contracted within the ab
plane, keeping the Mn-O bond distance along the c axis the same. The A-type AFM CO
state, the orbital ordering as shown in Fig. 3b is expected to be favored. Actually, with the
126 Mizokawa and Fujimori
breathing-type distortion, the A -type AFM CO state with the x2 – y2-type orbital ordering
is calculated to be the lowest in energy and to have a band gap of ~0.35 eV without the JT
distortion. The present model HF calculations indicate that the JT-type and breathing-type
distortions control the relative stability of the FM state and the CE- type and A -type AFM
states.
4. CONCLUDING REMARKS
Spin and charge ordered states with domain walls in two- and three-dimensional
perovskite-type 3d transition-metal oxides have been investigated using model HF cal-
culations. It has been found that the metal-centered and oxygen-centered stripes are nearly
degenerate in energy both in the cuprates and in the nickelates. The present HF calculations
show that the magnitude of the charge-transfer energy controles the relative stability of the
domain walls along the (1,0) and (1,1) directions. For the manganites, it has been found
that the (1,1,0)-type charge-ordered state with the orbital ordering at the sites are
stabilized by the JT distortion at the sites.
In the CO states with the oxygen-centered domain walls, the superexchange coupling
between the domain edges is affected by the doped holes sitting at the oxygen sites. The
coupling between the domain edges becomes FM, and therefore the neighboring AFM
domains become in antiphase. Let us denote the energy gain per metal-oxygen-metal bond
by the FM coupling between the domain edges as and that in the AFM domains as K.. In
the CO states with the metal-centered domain walls, the domain walls are formed by metal
ions with a different formal valence from that of the parent insulator. If the domain wall
has local magnetic moments, the superexchange coupling between the domain edge and
the domain wall and that within the domain wall (K ) are important. In this case, the
neighboring domains are in phase—namely, the spin arrangement of the parent AFM Mott
insulator is not disturbed.
In the two-dimensional lattice, the domain walls along (1,0) are favored for
K and those along (1,1) are favored for as shown in Fig. 4a. Although the
Spin, Charge, and Orbital Ordering in 3d Transition-Metal Oxides 127
ACKNOWLEDGMENTS
The authors would like to thank J. Zaanen, D. I. Khomskii, and G. A. Sawatzky for
useful discussions. The present work is supported by a Grant-in-Aid for Scientific Research
from the Ministry of Education, Science and Culture and by the New Energy and Industrial
Technology Development Organization (NEDO).
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Sliding Stripes in 2D Antiferromagnets
1. INTRODUCTION
There has been a crescent deal of interest concerning the charge and spin distribution in
doped antiferromagnets. Previously, the problem was addressed by assuming that the system
could be described by a gas of holes with uniform density. However, recent calculations
[1] and measurements [2–5] suggest that the holes cluster along lines (stripes) that separate
undoped antiferromagnetic domains.
The existence of stripes has been confirmed from several distinct experimental tech-
niques, like elastic [2] and inelastic [3] neutron scattering, muon spin resonance, nuclear
quadrupole resonance [4], and x-ray diffraction (XRD) [5].
Here, we suggest another simple experiment that can serve as a check for verifying the
stripe formation. By applying a small electrical bias field E to a sample, one can determine
the characteristics, which exhibit a different dependence on the bias, depending if
the system is a gas or a line of holes.
2. THE MODEL
We consider the limit of low doping, where the interaction between the domain walls
(chain of holes) can be neglected and we can concentrate on a single-stripe dynamics.
Moreover, we study a completely filled domain wall (DW), as found in the the nickelate
materials. For the sake of simplicity, we concentrate on the vertical configuration, with
the stripe located along the y direction, and allow for jumps of the holes only along the
perpendicular x direction (Fig. 1). The DW separates the antiferromagnetic plane into two
Néel phases, which we consider to be frozen.
The problem is treated in the frame of the t – J model,
Here, is the maximal relative displacement between two neighboring holes, and it can take
any integer value. A similar model accounting only for displacements, but allowing
for overhangs, was recently presented [7]. The configuration of the stripe is determined by
the projection of the spin,
Let us first calculate the quantum equation governing the motion of the operator,
Then, we consider the classical and the long wavelength
Sliding Stripes in 2D Antiferromagnets 131
(discrete continuous y) limits and find that the equation of motion acquires a wavelike
form,
3. RESULTS
Next, we account for the presence of randomly distributed impurities, which act to
pin the stripe. The pinning energy provided by an ensemble of quenched impurities was
shown to be where the pinning parameter Here,
denotes the Coulomb energy scale, is the doping concentration and L is the
stripe length.
In order to reduce the pinning barrier provided by the random impurities, we apply
and external electrical field perpendicular to the stripe formation. The holes dynamics then
arise from the competition between the pinning potential and the bias field. The threshold
field Ec is defined as the value for which the barriers vanish and the holes can freely flow
through the sample: this is the free flow (FF) or metallic regime, with ohmic dissipation.
Below the threshold field, a finite pinning barrier prevents the charge motion. However, the
holes can still jump over the barrier due to thermal activation. This is the thermally activated
flow (TAF) regime.
The thermal decay rate for a “particle” trapped into a metastable state is given by the
Arrhenius law e x p ( — U / T ) , where U is determined within the semiclassical approx-
imation by the free energy evaluated at the saddle point configuration
This rate corresponds to the escape of trapped charges, i.e., it is proportional to a current,
Hence, the conductivity in the thermally activated flow regime is expo-
nentially reduced in comparison with the conductivity in the free flow regime i.e.,
It is important to notice that the value of U and depend strongly on the considered
system. We denote the activation barrier and the conductivity for the gas of holes as and
respectively, whereas for the stripe configuration we use and
Next, we study each case in more detail. Let us start analyzing the stripe configuration.
The free energy describing an elastic line along the y direction, which tends to move due
to an external electrical field E competing against the pinning potential is
Collective Pinning Theory of Larkin and Ovchinnikov [9]. Deep in the insulating phase,
with we find that this length is of order and the threshold electrical
field
In the following, we apply the results that are known for describing the dynamics of
elastic manifolds in disordered media in order to predict the characteristics for the
stripe configuration.
The collective pinning barrier for an elastic line was estimated to be [6]
At low applied fields, the minimal barrier for creep displays a glassy behavior,
The conductivity in the thermally activated stripe flow (TASF) regime then reads
with given by Eqs. (5)–(7). Hence, the characteristics are highly nonlinear,
i.e., the barrier diverges and consequently the current vanishes in the limit of small applied
bias field (Fig. 2).
However, the gas of holes exhibits a different behavior. In this case, the threshold
field is still the same, whereas the pinning barrier felt by each hole is
substantially smaller, in agreement with experimental data [11]. As
a consequence, the holes can be thermally activated easier over the barriers. The low field
behavior is then the thermally activated hole flow regime (TAHF), which is ohmic, i.e.,
Sliding Stripes in 2D Antiferromagnets 133
In contrast to the stripe case [Eq. (8)], here does not depend on the bias field E[10]. The
characteristics then will be rather distinct than in the stripe case (Fig. 3).
4. CONCLUSIONS
We studied the sliding behavior of a stripe within a phenomenological elastic model,
which can be related to the t – J model. By accounting for the presence of randomly
distributed impurities, we evaluated the threshold electrical field that must be applied to
the system in order to release the stripes. We also considered the analogous problem for a
gas of holes. It was found that the behavior exhibited by the gas and the line of holes is
fundamentally different: In the limit of vanishingly small applied fields, the gas of holes
displays an ohmic behavior (non-zero slope of the current field dependence), whereas the
elastic line is in a glassy state characterized by a diverging barrier (zero slope). Therefore,
measurements of the characteristics of a doped antiferromagnet can serve as a test
to check the existence of the striped phase.
ACKNOWLEDGMENTS
We are indebted to H. Schmidt for fruitfull discussions. This work has been supported
by the DAAD-CAPES PROBRAL project number 415. NH acknowledges financial support
from the Gottlieb Daimler- und Karl Benz- Striftung and the Graduiertenkolleg “Physik
nanostrukturierter Festkörper,” Universität Hamburg. YD acknowledges financial support
from the Otto Benecke-Stiftung.
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1. J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989); H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990);
D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989); M. Inui and P. B. Littlewood, Phys. Rev. B 44,
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(1995).
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134 Morais Smith, Dimashko, Hasselmann, and Caldeira
S. I. Mukhin1
Recent experimental discovery of the coupled spin and charge ordering phase tran-
sitions (“stripe phase”) and/or dynamic stripe correlations in the various layered transi-
tion metal oxides [1] poses a challenging problem for the theory to discover the nature
of the multimode instabilities in the interacting electron system. Inelastic neutron scat-
tering experiments reveal a general property of the stripe spin-charge order. Namely,
the spin density wave-vectors are equal to in the two-
dimensional (2D) Brillouin zone while simultaneously the charge density wave-vectors
are equal to and with Here and below, the units of are
used, where a is the lattice constant. Geometrically, this means that spin orders nearly
antiferromagnetically with the wave vector , except for some slow mod-
ulation (“stripes”) with the period which is twice the period of the charge density
variation, Another basic observation is a dependence of on the (hole) doping con-
centration which measures deviation of the bare conduction band from the half-filling.
In particular, experimentally, for small enough and developes a shoulder at
greater
¹Moscow Institute for Steel and Alloys, Theoretical Physics Dept., Leninskii prospect 4, 117936 Moscow, Russia.
135
136 Mukhin
The simplest model Hamiltonian of the electron system in the presence of the spin and
charge mean fields, and has the following most general form
where and are electronic creation and annihilation fields, is the chemical potential
of electrons, 2t is the bare bandwidth, is the spin index and summation over repeated
indices is implied. A short-range repulsion between electrons is introduced that is
responsible for the SDW instability in the absence of a CDW. In general, the charge density
terms include both electron–electron and electron–lattice coupling energies as well as the
lattice deformation energy. The latter is incorporated in the K-term (compare, e.g., Ref. 6).
If the electron-lattice interaction dominates interelectron repulsion energy, then is the
density of the “lattice-charge,” g is the electron–lattice coupling, and the electronic density
deviation from the homogeneous value follows as: . In
the opposite case of a very weak electron–lattice coupling, the interelectron repulsion terms
are obtained in Eq. (1) after the changes: and + —in
front of the term. Then, the meaning of see for
example, Ref. 2.
The single-particle eigenstates of the Hamiltonian Eq. (1) can be written in the left and
right movers representation as
Here, the unit length is the lattice spacing a, and 2t is substituted below with which
has the meaning of the Fermi velocity of electrons. A single CDW corresponds to
A modulated spin density could be decom-
posed as the two incommensurate SDWs:
which in turn defines Fixing
of the mutual phase between the CDW and the SDWs is crucial for the quantum inter-
ference phenomenon described below. Now an explicit separation of the WKB part of the
eigenfunction in the slow CDW potential gives (compare Ref. 7)
where is the Bessel function of an integer order n, and the terms of the higher order
than are neglected provided that After a substitution of Eq. (3) into
138 Mukhin
Eq. (2), and with the “matching” condition for the CDW wave vector fulfilled, one
finds for the case of the “hole doping,”
The terms are neglected in Eq. (4) provided the doping concentration is not too
small, i.e., Solving Eq. (4), one finds the single-particle spectrum
where
In the “electron doping” case, the sign in front of and in Eqs. (5) and (6) should be
changed. The physical implication of Eqs. (5) and (6) is remarkable. Namely, the coupling
strength U, which causes SDW condensation, is renormalized and equals in the
presence of the CDW. Depending on the sign of the factor might be less
or greater than 1, which in turn means that coupling strength, , is either enhanced
or suppressed with respect to the bare value, U. This is a manifestation of the quantum
interference of the scattering amplitudes of electron in, for example, the and
in the combined plus CDW periodic potentials. Using Eqs. (1) and (5), it is
straightforward to derive the free energy of the system (per unit length), at a finite
temperature T:
where is the upper cutoff of the electron energy, and (see also Ref. 8). A
detailed description of the phase diagram following from Eq. (7) is given elsewhere. Here
we merely list the main results (see also Figs. 2-4).
1. Starting from the high temperature limit, the stripe phase condenses first
with depending on the sign of
for a hole (electron) doping . Thus, the spin density behaves as:
in the case and as :
case. While the CDW density is the same in the both cases:
Hence, the nodes of the spin density coincide with the maxima (minima) of the charge
density in the case of the hole (electron) doping. In the both cases
where This function of reaches its maximum
Quantum Interference Mechanism of the Stripe-Phase Ordering 139
where
4. In the low temperature limit, , the saturation value of the spin-order param-
eter, depends on
The dependences of the order parameters on the doping concentration at a low temper-
ature are presented in Fig. 4. In conclusion, a microscopic theory of the stripe-phase
ordering in the weak coupling limit is presented. It demonstrates the quantum interfer-
ence mechanism of the spin-charge coupling in the correlated electron system, which
leads to the stripe-phase instability. The stripe-ordering temperature in the underdoped
142 Mukhin
regime is enhanced with respect to the bare spin-ordering temperature. The mecha-
nism of a high-temperature condensation of the “matching” long-wavelength CDW in
the system, which is unstable with respect to the incommensurate antiferromagnetic
spin ordering, provides an important hint for the construction of the theory of the
high-temperature superconductivity in cuprates, where the role of the CDW may play
a space-modulated density of the Cooper-pairs condensate. A detailed analysis of this
possibility is now in progress.
ACKNOWLEDGMENTS
Useful discussions with Jan Zaanen, A. A. Abrikosov, and Wim van Saarloos are highly
appreciated. The work was supported in part by NWO and FOM (Dutch Foundation for
Fundamental Research) during the author’s stay at the Lorentz Institute in Leiden in the
January 1998.
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Rev. B 54, 12318 (1996);C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993).
2. J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7391 (1989); H. J. Schulz, J. Phys. (Paris) 50, 2833 (1989).
3. V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993); ibid., 235–240, 189 (1994); C. Castellani, C. Di
Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995).
4. B. Horovitz, H. Gutfreund, and M. Weger, Phys. Rev. B 12, 3174 (1975).
5. S.I. Mukhin, Self-Matching Property of Correlated Electrons: Charge-Density Wave Enhances Spin Ordering,
in the Proceedings of the Conference Spectroscopies in Novel Superconductors, Cape Cod, MA, September
14-18, 1997, to be published in J. Phys. Chem. Solids, 1998; S. I. Mukhin and Jan Zaanen, 1998 (unpublished).
6. H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990).
7. L. P. Gor’kov and A. G. Lebed’, J. Phys. Lett. 45, L-433 (1984).
8. It is important to note here that even in the CDW-free case a modulated spin-density state
has lower (free) energy than a single incommensurate state,
The reason is that despite the appearance of a factor 1/2 in front of the term in
Eq. (7) in the single-SDW case, the gap amplitude should be simultaneously devided by 2 in Eq. (5),
which finally makes the single SDW state energetically less favorable than the modulated spin-density state.
Spontaneous Orientation of a Quantum
Lattice String
Since the experimental discovery of the stripe phase [1] interest in this field has grown
rapidly. Many issues concerning stripes are discussed, ranging from their origin to their
relation to superconductivity, including the dynamical properties of the stripes. In
this contribution, we are concerned with the last subject. We focus on the problem of a
single-stripe/single-charged domain wall. We consider the domain wall to be a connected
trajectory (string) of particles, communicating with the lattice, whereas the precise nature
of these particles is not further specified: the quantum lattice string (QLS) model [2].
Studying this model numerically, we discovered a zero-temperature symmetry breaking:
Although the string can be quantum delocalized, it spontaneously picks a direction in space.
This symmetry breaking happens always in the part of parameter space that is of physical
relevance. At first sight, one might expect that the quantum fluctuations (kinetic energy)
would tend to disorder the string (i.e., to decrease the tendency for the string to be directed).
That the opposite effect happens can be seen as follows. A first intuition can be obtained
by considering the analogy with surface statistical mechanics. The quantum string problem
can be formulated as a classic problem of a two dimensional surface (worldsheet) in 2 + 1
dimension, where the third direction is the imaginary time direction. The larger the kinetic
1
Instituut-Lorentz, Universiteit Leiden, P.O.B. 9506, 2300 RA Leiden, The Netherlands.
term or the smaller the temperature, the further the worldsheet stretches out in the time
direction. At zero temperature, the worldsheet becomes infinite in this direction as well.
The statistical physics of a string is then equivalent to that of a fluctuating sheet in three
dimensions. Now, it is well known from studies of classical interfaces [5] that although a
one-dimensional (1D) classical interface in two dimensions does not stay directed due to
the strong fluctuations, for a two-dimensional (2D) sheet, the entropic fluctuations are so
small that interfaces can stay macroscopically flat in the presence of a lattice [6,7]. In other
words, even if microscopic configurations with overhangs are allowed, a classical interface
on a lattice in three dimensions can stay macroscopically flat or “directed.” In the present
context, we show that the directedness is a caused by an order-out-of-disorder mechanism:
In order to maximize the fluctuations transversal to the local string directions, overhangs
should be avoided on the worldsheet. It remains to be seen if this mechanism is of a more
general application.
In the QLS model the string configurations are specified by the position of the particles
Two consecutive particles i and l + 1 should either be nearest or next-nearest
neighbors, The set of all such configurations is the string Hilbert
space.
The Hamiltonian consists of a classical energy term and a quantum (hopping) term.
The classical energy is a sum of local interactions between nearest and next-nearest particles
in the string.
with
The quantum term allows the particles to hop to nearest-neighbor lattice positions,
giving rise to the meandering of the whole string. These hops should respect the string
constraint. To enforce the constraint, a projection operator
is introduced that ensures that the motion of particle l keeps the string intact. The string
is quantized by introducing conjugate momenta and the hopping
Spontaneous Orientation of a Quantum Lattice String 145
The above string model is invariant under rotation of the string in space. As is discussed
below, we find that for physical choices of the parameters, the invariance under symmetry
operations of the lattice is broken. The string acquires a sense of direction in space. This
occurs even when the string is critical (delocalized in space). The string’s trajectories, on
average, are such that they move forward in one direction while the string might delocalize
in the other direction.
The relation of the string problem to surface models is established by using Suzuki-
Trotter mapping, which maps a 2D quantum problem to a 2 + 1 D classical problem. A
classical model of two coupled RSOS (restricted solid-on-solid) surfaces results. These are
classical models for surface roughening [3] in which overhangs are not allowed. For the
quantum string case, the two RSOS surfaces describe the motion of the string in the x and
y spatial directions.
Skipping detailed calculation, the partition function of the quantum string can be
mapped to the following classical problem [2],
directed. The quantum string vacuum is a linear superposition of many string configura-
tions. When all configurations in the vacuum correspond to single-valued functions x(y)
or y(x), the string vacuum is directed. At zero temperature, the ground-state wave function
of the string is where every state in string configura-
tion space corresponds to a trajectory [ x ( t ) , y(t)]. Consider first the case of a
continuous string. For every configuration, the total string arclength is given by
Consider now an indicator function that equals 1 when the string is single-valued
when projected onto the x axis and zero otherwise, and analogously a function for
the y axis (Fig. 2). The total directed lengths in the x and y directions are defined as
and On the lattice, one measures the directedness in analogy with the above defi-
nition, except that we just count the number of directed bonds, irrespective of whether they
Spontaneous Orientation of a Quantum Lattice String 147
are oriented diagonally or horizontally. The finite temperature measure of the directedness
density is simply given by thermally averaging the above definition.
line is the result when all classical energies are zero (i.e., for optimally quantum string).
The dashed-dotted and dotted lines correspond to all potential energy parameters set to
zero except that and 1.8, respectively, corresponding to a string localized in the
(1,0) or (0,1) directions. Decreasing the parameter causes stronger local fluctuations.
The full line is the result for a classical string , where only flat segments and
corners are allowed (no diagonal segments). The same classical result is shown again in
Fig. 3b, together with the result at the point
corresponding to a free (critical) string.
The classical string would be flat at zero temperature, directed along (say)
a (1,0) direction. A local "corner" (Fig. la) would be an excitation with energy (al-
ternatively, one could consider two kinks). Clearly, a single corner suffices to destroy the
directedness of the classical ground state. At any finite temperature, the probability of the
occurrence of at least one corner is finite: Hence, directedness order
Spontaneous Orientation of a Quantum Lattice String 149
cannot exist at any non-zero temperature, for the same reason that any long-range order is
destroyed at any finite temperature in one dimension. In the simulations the string is of finite
length, and the infinite temperature limit of is therefore not zero, but rather a small
but non-zero value for a domain wall of length 50). is already close to
this value for all temperatures of order and larger. For an infinitely long domain wall,
drops very fast to zero with increasing temperature. For low T where
grows rapidly to 1. Again, because the string is of finite length, it becomes directed
already at a finite temperature: For all temperatures such that the string
configurations in our simulations are typically completely directed. An infinitely long clas-
sical string becomes directed only at , of course, because at any non-zero temperature
some corners always occur in a sufficiently long string.
The results for the quantum string always look similar to the classical one. For tem-
peratures higher than the kinetic scale, all curves approach each other and the
classical limit is reached. At low grows rapidly to 1. As in the classical
case, it reaches this value at a finite temperature for a finite length string. This is even valid
for the pure quantum string, where all classical energies are zero (dashed line in Fig. 3a).
Again, this can be understood in terms of an effective corner or bend energy that is
produced by the quantum fluctuations. In analogy to the classical case, the probability for
the occurrence of a bend is proportional to At zero temperature, no bend is
present and the string becomes directed. A finite length string effectively becomes directed
already at a temperature such that At intermediate temperatures, where
the temperature is of the order of the kinetic term, the situation is less clear. Especially
in this region, all the various energies may play a role, and the interplay of these on the
directedness is rather complicated. Nevertheless, as is clear from the data of Fig. 3a, this
region connects the high and low temperature limits smoothly. Finally, by comparing the
results for the three quantum strings in this figure, it is also clear that when the string is
more quantum, mechanical is higher.
The spontaneous directedness of the quantum string for can be understood by
the following argument. The bends in strings block the propagation of links along the
chain. Close to the bend itself the particles in the chain cannot move as freely as in the rest
of the chain. This effect is shown in Fig. 4.
In space-time, the bend is like a straight rod in time. Therefore, the presence of
such kinks increase the kinetic energy. For the argument, it makes little difference whether
the bend consists of a single corner or two corners. This confirms that it is the
kinetic energy that keeps the strings oriented along one particular direction. In terms of a
150 Osman, van Saarloos, and Jan Zaanen
directedness order parameter, this result implies that such a quantity is always finite, except
when or when the hopping term vanishes (it is easy to see that in the classical case,
in many regions of parameter space the problem becomes that of a self-avoiding
walk on a lattice in the limit For the two equivalent RSOS surfaces, this means
that one of the two surfaces spontaneously orders while the other RSOS sheet can be either
ordered or disordered.
Our general conclusion, based also on Monte Carlo studies of the behavior in many
other points in the parameter space, is that apart from some extreme classical limits, the
general lattice string model at zero temperatures is a directed string. The qualitative picture
of corners blocking the propagation of kinks appears to be a natural explanation for
these numerical findings.
ACKNOWLEDGMENT
We are grateful to Henk Eskes for a collaboration from which this work is an outgrowth.
REFERENCES
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4. In the high temperature limit, the string becomes a self-avoiding walk on a two-dimensional lattice. For such
a walk of length N, the radius of gyration grows as where in This implies that
the directedness defined in this paper should go to zero as for in the high temperature
limit.
5. J. D. Weeks, J. Chem. Phys. 67, 3106 (1977); J. D. Weeks, Phys. Rev. Lett. 52, 2160 (1984).
6. J. D. Weeks, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste, Plenum, New
York, 1980, p. 293.
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Blanckenhagen (Springer, Berlin, 1987).
Domain Wall Structures in the
Two-Dimensional Hubbard Model
with Long-Range Coulomb Interaction
The structure and stability of partially filled domain walls is investigated in the
two-dimensional Hubbard model supplemented with a long-range Coulomb inter-
action. Using an unrestricted Gutzwiller variational approach, we show that the
strong local interaction favors charge segregation in stripe domain walls. This
approach supports the stabilization of half-filled walls for commensurate dop-
ing and large values of U even without the long-range part in contrast to results
of Hartree–Fock calculations. Inclusion of the long-range interaction favors the
formation of half-filled vertical stripes also in the intermediate U regime. These
walls are characterized by a period doubling due to the charge and a period qua-
drupling due to the spins along the wall. We find that, as well as the underlying
lattice structure, there is also an electronic stabilization mechanism for half-filled
vertical domain walls in
1. INTRODUCTION
The observation of charged domain walls in the high- superconductors presently
attracts a lot of interest also with regard to possible pairing scenarios [1–3]. Incommensurate
spin correlations have first been observed in (LSCO) by neutron scattering
experiments [4–6]. More recently, it was found that the incommensurate spin fluctuations
are pinned in Nd-doped LSCO and nickel oxide compounds [7–9], leading to spin- and
charge-stripe order in these materials [10]. However, whereas the domain walls in the
hole-doped system are oriented along the diagonals of the lattice, it turns
1
Institut fur Physik, BTU Cottbus, PBox 101344, 03013 Cottbus, Germany.
2
Istituto Nazionale di Fisica della Materia e Dipartimento di Fisica, Università di Roma “La Sapienza,” Piazzale
A. Moro 2, 00185 Roma, Italy.
out that in the orientation of the stripes is along the Cu-O bond
direction. Moreover, the hole concentration in the domain walls is one hole per Ni site in the
nickelates and one hole every second Cu site in the Nd-doped cuprates. A comparison of the
low-temperature orthorhombic and the low-temperature tetragonal structure suggests that
the vertical stripes in are pinned by the tetragonal lattice potential,
whereas the orthorombic phase in the nickel oxides favors a diagonal orientation. We show
here that the electronic interactions may also play an important role in establishing different
domain wall structures.
The stripe instability for doped antiferromagnets was predicted theoretically in [ 1 1 ]
within Hartree-Fock (HF) theory applied to the extended Hubbard model and confirmed
by a number of subsequent investigations [12,13]. For small values of the Hubbard on-
site repulsion U (generally smaller than 3t-4t) these calculations result in a striped phase
oriented along the (10) or (01) direction, whereas for higher values of U the orientation
is along the diagonals. Within HF theory the stripe solutions become unstable for
toward the formation of isolated spin polarons.
However, all stripe calculations performed so far within the HF approximation of
the Hubbard model predict one hole per site along the domain wall (filled stripe).
This contrasts the observation of half-filled stripes (i.e., with half a hole per site) in the
system, which is a yet unresolved problem of mean-field theory.
Zaanen and Oles [114] addressed the question of whether the inclusion of an additional
nearest-neighbor repulsion V in the Hubbard-Hamiltonian may favor the formation of
partially filled stripes. According to Ref. [14J, the half-filled stripe solution is stabilized
by a quadrupling of the charge or spin period along the stripe. However, although the
nearest-neighbor repulsion slightly enhances the stability of the half-filled wall, this never
corresponds to the HF ground state for realistic parameter values. Instead, the main effect
of V is to shift the crossover to isolated spin polarons to lower values of U.
In the present paper we show that a proper treatment of the strong local repulsion U
plays an indirect but crucial role in stabilizing half-filled vertical domain walls. Specifically,
we apply a slave-boson version of the Gutzwiller approach within an unrestricted variational
scheme. Contrary to the pure HF approach, which underestimates heavily the effective
attraction between the charge carriers and predicts repulsion for it was recently
shown that within the slave-boson scheme the attraction persists up to very large U [15]. As
a consequence of this more suitable treatment of the strong coupling limit, U greatly favors
the charge segregation in striped domains as opposed to spin polarons. In the absence of
long-range (LR) forces and values of (which is believed to be the physically
relevant regime for copper oxides [17]), completely filled diagonal stripes stay more stable
than half-filled vertical ones. However, due to their increased stability with respect to isolated
polarons, the stripe solutions now allow for a less disruptive introduction of stronger LR
forces, which affect the completely filled stripes more than the half-filled ones. Then, for
a sizable but still realistic LR repulsion, the half-filled vertical stripe may become the
ground-state configuration.
In the following we consider the two-dimensional (2D) Hubbard model on a square
lattice, with hopping restricted to nearest neighbors (indicated by the bracket ) and an
additional LR interaction:
Domain Wall Structures in the Two-Dimensional Hubbard Model 153
Figure 1 differs in various important aspects from corresponding results obtained via
the HF approximation [12]. First, there does not occur a crossover from the stripe to the
polaronic (Wigner) phase for all considered values of U. Within the HF approach, this
154 Seibold, Castellani, Di Castro, and Grilli
to the staggered structure indicated in Fig. 2b, although the difference in energy to the
CDW-type of Fig. 2a is rather small (the difference in energy per hole for is
To compare the stability of the various stripe textures, we report in Fig. 3 the energy
differences E(config) — E(polaron) as a function of the nearest-neighbor value of the
Coulomb repulsion for in Fig. 3a and for in Fig. 3b. Because we
are dealing with a finite size system, we cut the LR forces at “half-minus-one” the size of
the supercell in order to avoid double counting in the Coulomb interaction energies.
In case of half-filled vertical stripes the curves in Figs. 3a and 3b correspond to the
structures in Figs. 2a and 2b, respectively. Disregarding the eventual above-mentioned
instability of the single stripes with respect to isolated polarons by increasing length, various
features are worth noting. First, completely filled stripes increase their energy more rapidly
(have a larger slope) than half-filled ones on increasing the LR repulsion. Therefore, LR
forces favor half-filled stripes, which eventually become the most favorable wall textures.
However, the most relevant effect to be noticed here is the role of a large local repulsion
U affecting the energies of the various textures (cf. Fig. 1). In particular, a comparison
between Figs. 3a and Fig. 3b shows that U strongly reduces the energy of the half-filled
stripes with respect to the filled ones already at By combining this reduction with
the effects of LR forces on the stripes, it follows that increasing U makes the half-filled
vertical stripe the most stable among the wall solutions at smaller values of
only. To incorporate the repulsion between stripes, we calculated the energy of vertically
oriented stripes on a lattice. For a concentration of 1/8, this results in an array of
4 completely filled or 8 half-filled stripes. The energy of these arrays with respect to the
polaronic Wigner lattice are depicted in Fig. 4 (the Wigner lattice now corresponds to 16 spin
polarons with maximum distance). In this case we obtain a crossover to half-filled vertical
domain walls for It is interesting to observe that now the stripe solutions gain
in energy on switching on the LR part with respect to the polaron lattice. This enhances the
parameter range of stability for the stripes, which no longer become unstable toward the
decay into isolated polarons.
To assess the absolute stability of the half-filled vertical stripes, we should also compare
their energy with the filled diagonal stripes, which, in the single-stripe analysis, result to
be more stable than the filled vertical stripes. However, filled diagonal stripes are strongly
destabilized by the elongated shape of the supercell so that we do not included their
energy in Fig. 4. As an alternative to the direct calculations on the elongated cluster, to
extract informations about diagonal stripe configurations, we analyzed a 2D regular array
of charged wires with fixed global charge density. We found that the electrostatic potential
energy is lower for wires with higher linear charge density at a larger distance than for
less-charged wires more closely spaced. Therefore, diagonal stripes, which at given planar
density are closer by but less densely charged by the same factor, are less favorable than
the vertical stripes as far as the electrostatic Coulombic energy is concerned. However, our
single-stripe investigation already demonstrated that a proper treatment of the strong local
repulsion U opens the way to a stabilization of half-filled vertical stripes with respect to the
filled diagonal stripes. From the above purely electrostatic analysis and from the results of
Fig. 4, we can therefore safely conclude that half-filled vertical stripes are the ground-state
configuration for at large enough U.
To summarize, we have shown that a LR Coulomb interaction added to the 2D Hubbard
model gives rise to half-filled vertically oriented domain walls when treated within an
unrestricted Gutzwiller approach. This feature does not appear in semiclassical HF ap-
proximations in which the effective attraction between spin polarons is underestimated.
Depending on the value of the on-site repulsion U, we expect the domain wall structure
Domain Wall Structures in the Two-Dimensional Hubbard Model 157
in the Nd-doped LaCuO system to be of the type shown in Figs. 2a and 2b, respectively.
Moreover, the competition investigated here between completely filled diagonal and half-
filled vertical stripes can explain the different hole orderings in nickelates and in Nd cuprates
even without invoking a relevant role of lattice interactions. Specifically, our findings suggest
that nickelates could be characterized by a smaller and/or a smaller V accounting
for their filled diagonal stripes. However, although the half-filled vertical stripes in the Nd-
enriched LaSrCuO systems are to some extent likely fixed along the (1,0) direction by the
underlying lattice structure, we showed here that, if large values of U and sizable values
of V characterize these systems, then electronic correlations would also contribute to give
rise to vertical half-filled stripes.
In general, the filling and orientation of the stripes depend on the specificity of the
electronic forces and structures, and therefore inside the various oxide families different
textures of the stripe phase may prevail.
ACKNOWLEDGMENTS
GS acknowledges financial support from the Deutsche Forschungsgemeinschaft as
well as hospitality and support from the Dipartimento di Fisica of Università di Roma “La
Sapienza,” where part of this work was carried out. This work was partially supported by
INFM-PRA(1996).
REFERENCES
1. For a review, see, e.g., Proceedings of the IV International MMS-HTSC Conference, February 28–March 4,
1997, Beijing, China.
2. C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. Lett. 75, 4650 (1995); C. Castellani, C. Di Castro, and
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78,338(1997).
10. The existence of charge modulation in the CuO 2 planes of Bi22I2 was proposed early on by Bianconi and
coworkers: A. Bianconi, Proceedings of the workshop on Phase Separation in Cuprate Superconductors,
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17. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
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Boson-Fermion Mixtures, d-Wave Condensate,
and Tunneling in Cuprates
A. S. Alexandrov1
We argue that local pairs (bipolarons), formed by any short-range attraction are
localized and cannot give rise to no matter whether they are hybridized
with Fermions. However, a long-range Fröhlich electron-phonon interaction can
provide mobile intersite bipolarons in the plane condensing at high
The ground state of cuprates is thought to be a charged Bose-liquid of intersite
bipolarons with single polarons existing only as thermal excitations. We show that
some bipolaron configurations lead to a d wave charged Bose-Einstein condensate
in cuprates. It is the bipolaron energy dispersion rather than a particular pairing
interaction that is responsible for the d wave symmetry. Single-particle spectral
density is derived, taking into account realistic band structure and disorder. The
tunneling and photoemission (PES) spectra of cuprates are described.
PACS numbers: 74.20.-z, 74.65.+n, 74.60.Mj
1. INTRODUCTION
There is a fundamental problem with any theory involving real-space pairs (bosons)
tightly bound by a field of a pure electronic origin. As stressed by Emery et al. [1], such the-
ories are a priori implausible due to the strong short-range Coulomb repulsion between two
carriers. A direct (density–density) repulsion is usually much stronger than any exchange
interaction. The attraction potential generated by the electron-phonon interaction of the
Holstein model may overcome the short-range Coulomb repulsion, but inevitably involves
a huge carrier mass enhancement otherwise the phonon frequency would be extremely
high [2], Although we do not exclude a coexistence of degenerate Fermi and Bose carriers
in some systems (in fact, we discussed their mixture in 1986 [3]), we present reasoned
arguments [4] ruling out any role that their hybridization could play. Our study of localized
1
Department of Physics, Loughborough University, Loughborough LE 11 3TU, U.K.
bosons hybridized with propagating fermions [i.e., a boson–fermion model (BFM)] beyond
the mean-field approximation shows that:
Statements 1, 2, and 4, derived with the most divergent “ladder” approximation for the
boson self-energy and for the fermion vertex, are, in fact, exact.
However, we have shown [5] that the Fröhlich electron–phonon interaction can provide
intrinsically mobile intersite small bipolarons, which are condensed at high of the order
of 100 K. We believe that this interaction operating on a scale of the order of 1 eV can com-
pensate the intersite Coulomb repulsion allowing the deformation potential (together with
an exchange interaction of any origin) to bind two holes into an intersite mobile bipolaron in
the plane. The bipolaron mass renormalization appears to be smaller by several orders
of magnitude than in the Holstein model with the same value of the attraction potential. Al-
though the charged Bose liquid of bipolarons describes well the anomalous thermodynamic
and kinetic properties of cuprates [2], finite frequency/momentum response functions of
bipolaronic superconductors remain to be established. Moreover, it was claimed [6] that an
experimental evidence for the d-wave order parameter and a large Fermi-surface in several
cuprates are incompatible with bipolarons. In this paper, we establish a d-wave symme-
try of the bipolaronic condensate and a single-particle spectral function that quantitatively
describes the tunneling spectra and some photoemission features of cuprates ([7-9], and
references therein).
interaction and also from the symmetry of the excitation spectrum, which depends on the
bipolaron–bipolaron repulsion and the polaronic band dispersion. We show that the Bose-
Einstein condensate in cuprates is d wave owing to the bipolaron energy band structure
rather than to a particular pairing interaction.
The existence of the “parent” Mott insulators allows us to consider cuprates
as doped semiconductors with narrow electron bands. Therefore, different bipolaron con-
figurations can be found with computer simulation techniques based on the minimization
of the ground-state energy of an ionic lattice with two holes fully taking into account the
lattice deformation and the Coulomb repulsion [15,16]. The intersite pairing of the in-plane
oxygen hole with the apex one is energetically favorable in the layered perovskite structures
as established by Catlow et al. [17]. The apex or peroxy-like bipolaron can tunnel from
one cell to another via a direct single polaron tunneling from one apex oxygen to its apex
neighbor as shown in Fig. 1. The bipolaron band structure is derived by the use of the
generic Hamiltonian including the oxygen–oxygen and oxygen-copper hopping integrals,
the coupling of holes with phonons and their Coulomb repulsion [5].
The hole bipolaron energy spectrum in the tight binding approximation consists of two
bands formed by the overlap of and apex polaron orbitals, respectively,
Here and below we take the in-plane lattice constant t is twice of the renormalized
bipolaron hopping integral between p orbitals of the same symmetry elongated in the
direction of the hopping and is twice of the renormalized hopping integral in the
perpendicular direction The energy band minima are found at the Brillouin zone
boundary, and rather than at point owing to the opposite sign
of the and hopping integrals. Only their relative sign is important, so we choose
given by
where is the bipolaron (boson) annihilation operator in the k space that is a c number for
the condensate and N is the number of cells. Other combinations of four degenerate states
do not respect time reversal and (or) parity symmetry. Two solutions, Eq. (3), are physically
identical because one of them is expressed through another by the use of the translation,
They have d-wave symmetry, changing sign when the
plane is rotated by around (0,0) or around (0,1) for and respectively
(Fig. 2). We notice that the continuous (r) real space representation of the order parameter
is irrelevant for cuprates because of a very small coherence volume compared with the
unit cell one. The d-wave symmetry is entirely due to the bipolaron energy dispersion with
four minima at When the minima located at the point of the Brillouin zone the
condensate is s-like.
the matrix element and into a paired hole (bipolaronic) state B with the matrix element
Here, and are the quantum numbers describing an electron in the tip (Fig. 3) (an-
nihilation operator a hole polaron and a hole bipolaron in the plane
in the random field, respectively. If the eigenstates are known, the matrix el-
ements and are derived by the use of the site representation and the canonical
polaronic transformation as discussed in detail in Ref. [18]. They are almost indepen-
dent of in a wide voltage and binding energy range,
In general, B and P are different because the second hole in
a small coherence volume changes a potential barrier of the contact for the tunneling B
compared with P.
Calculating the injection and emission rates with the Fermi Golden Rule, one can
obtain the expression for the tunneling and PES spectra in the voltage (energy) region,
in which the high-frequency phonon shakeoff is forbidden by the energy conservation. In
particular, we find for
where is the density of states of a single polaron (unoccupied) band. The p-hole polaron
in cuprates is almost one-dimensional (1D) due to a large difference in the and
hopping integrals and an effective 1D localization by the random potential as described
in Ref. [5]. This is confirmed by the angle-resolved photoemission (ARPES) [20] with no
dispersion along certain directions of the 2D Brillouin zone. Because the amount of disorder
is high and the screening radius is about the lattice constant, we can describe the effect of
disorder and of the thermal fluctuations as “white Gaussian noise.” The relevant spectral
164 Alexandrov
density A(k, E) for a one dimensional particle in a random Gaussian potential was derived
by Halperin [21] and the density of states, by Frish and Lloyd [22]. The result is
ARPES can be described with the spectral function A(k, E) determined numerically [21].
Although such a feature of ARPES as the normal state gap is understood within the present
analysis, the k dispersion is presented elsewhere.
4. CONCLUSION
In summary, we would like to outline the main results: First, A simple estimate of
the polaronic level shift shows that the electron-phonon coupling is more than sufficient to
bind two polarons into a small mobile bipolaron. For the Fröhlich interaction, one estimates
the polaron level shift as The hole-hole coupling via phonons is much
stronger than the magnetic coupling,
2. The ground state of superconducting cuprates is the Bose-Einstein condensate of
peroxy bipolarons with the d-wave symmetry owing to the bipolaron band dispersion.
3. The single-particle spectral function is derived for cuprates that describes the spectral
features observed in tunneling and photoemission. In particular, the temperature independent
gap and the anomalous ratio, injection/emission asymmetry both in magnitude and
shape, zero-bias conductance at zero temperature, the spectral shape inside and outside
the gap region, temperature/doping dependence, and dip-hump structure of the tunneling
conductance are described.
ACKNOWLEDGMENTS
The author greatly appreciates enlightening discussions with A. R. Bishop, B. Brandow,
J. T. Devreese, O. Fisher, V. V. Kabanov, H. Kamimura, P. E. Kornilovitch, G. J. Kaye,
W. Y. Liang, Ch. Renner, S. G. Rubin, J. R. Schrieffer, A. Simon, Z.-X. Shen, G. Zhao, and
K. R. A. Ziebeck.
REFERENCES
1. V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B56, 6120 (1997).
2. A. S. Alexandrov and N. F. Mott, Rep. Prog. Phys. 57 1197; “High Temperature Superconductors and Other
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J. R. Kirtley et al., Nature 373, 225 (1995); C. C. Tsuei et al.. Science 272, 329 (1996).
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14. H. Kamimura et al., Phys. Rev. Lett. 77, 723 (1996).
15. X. Zhang and C. R. A. Catlow, J. Mater. Chem. 1, 233 (1991).
Boson-Fermion Mixtures, d-Wave Condensate, and Tunneling in Cuprates 167
16. N. L. Allan and W. C. Mackrodt, Advances in Solid-State Chemistry, Vol. 3, ed. C. R. A. Catlow (London:
JAI Press) (1993).
17. C. R. A. Catlow, M. S. Islam and X. Zhang, J. Phys.: Condens. Matter 10, L49 (1998).
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19. A. S. Alexandrov, invited talk at the Workshop on “Strongly Correlated Electrons” (Tellahassee, Florida,
11–14 March, 1998).
20. D. M. King et al., Phys. Rev. Lett. 73, 3298 (1994); K. Gofron et al., ibid. 3302 (1994).
21. B. I. Halperin, Phys. Rev. 139, A104 (1965).
22. H. L. Frisch and S. P. Lloyd, Phys. Rev. 120, 1175 (1960).
23. A. S. Alexandrov, V. V. Kabanov, and N. F. Mott, Phys. Rev. Lett. 77, 4796 (1996).
24. K. A. Müller et al., J. Phys.: Condens. Matter 10, L291 (1998).
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(JETP Lett. 45, 455 (1987)).
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The Small Polaron Crossover:
Role of Dimensionality
The crossover from quasi–free electron to small polaron in the Holstein model for a
single electron is studied by means of both exact and self-consistent calculations in
one dimension (1D) and on an infinite coordination lattice in order to understand
the role of dimensionality in such a crossover. We show that a small polaron
ground-state occurs when both strong coupling and multiphonon
1) conditions are fulfilled, leading to different relevant coupling constants
in adiabatic and anti-adiabatic region of the parameter’s space. We also
show that the self-consistent calculations obtained by including the first electron–
phonon vertex correction give accurate results in a sizeable region of the phase
diagram well separated from the polaronic crossover.
1. INTRODUCTION
Recent optical measurements of the insulating parent compounds of the high-
temperature superconductors show the presence of polaronic carriers [1], and evidence
for intermediate and strong lattice distortions has been given also for the colossal mag-
netoresistance manganites [2] and nickel compounds [3]. The recent observation of one-
dimensional (1D) stripes in the high-temperature superconductors [4] and in manganites
suggests a comprehensive study of the role of dimensionality in the polaronic crossover. A
detailed study of the small polaron crossover is demanded also by the recent experimental
results on manganites [5].
1
I.N.F.M. and International School for Advanced Studies, SISSA-ISAS, Trieste, Italy 34013.
2
Dipartimento di Fisica, Universitá de L’Aquila, via Vetoio, 67100 Coppito-L’Aquila, Italy and I.N.F.M., Unitá
de L’Aquila.
3
I.N.F.M., Unitá di Roma 1, Dipartimento di Fisica, Universitá di Roma “La Sapienza.” P.le A. Moro 2, 00185
Roma, Italy. Present address: École Polytechnique Fédérale de Lausanne, DMT-IPM, CH-1015 Lausanne,
Switzerland.
169
170 Capone, Ciuchi, and Grimaldi
where and are, respectively, the destruction (creation) operators for an electron
and for a dispersionless phonon of frequency on site i. The Hamiltonian (1) represents
a nontrivial many-body problem, and has been already studied in recent years by means of
numeric [6–9] and analytic [10–12] techniques.
Two dimensionless parameters are introduced to measure the strenght of electron–
phonon (el-ph) interaction: and where is the half-
bandwidth for the free electron and d is the system dimensionality.
is originally introduced in the weak coupling pertubation theory and is
the coupling parameter of a ME approach in the case of one electron. It can also be viewed
as the ratio between the small polaron energy and the free-electron energy
The parameter α is the relevant coupling in the atomic limit In this limit, α
measures the lattice displacement associated to the polaron and is the average num-
ber of phonons bound to the electron. According to the Lang–Firsov results followed
by the Holstein approximation, α also rules the reduction of the effective hopping
[9,13].
Besides and α , the el-ph system described by Eq. (1) is governed also by another
dimensionless parameter: It measures the degree of adiabaticy of the lattice motion
(lattice kinetic energy compared to the electron one (electron kinetic energy
In the adiabatic regime is a condition sufficient to give a polaronic
state because the electron is bound to the slowly moving lattice giving rise to a strong
enhancement of effective mass. In the antiadiabatic regime such a picture is
no longer true due to the fast lattice motion. In this case, polaronic features such as strong
local electron–lattice correlations arise only when the electron is bound to a large number of
phonons To summarize, in both adiabatic and antiadiabatic regimes, a polaronic
state is formed when both and inequalities are fulfilled [9]. This conclusion
is in contrast with ref. [10], in which it is argued that is the only condition for small
polaron formation.
The parameter influences also the dependence of the behavior of the el-ph coupled
system on the system dimensionality. We shall show that in the antiadiabatic regime the small
polaron formation does not depend on the system dimensionality. However, dimensionality
plays a crucial role in the adiabatic regime This can be traced back to the adiabatic
limit In fact, in the ground-state is localized for any finite value of
and a crossover occurs between large and small polaron at whereas for it
has been shown that a localization transition occurs at finite from free electron to small
polaron [14]. The different adiabatic behaviors between 1d and 2d systems could be relevant
The Small Polaron Crossover 171
2. RESULTS
We study the relevance of and of the lattice dimensionality d by using two
alternative exact calculations: exact diagonalization of small, 1D clusters (ED-1d) and
dynamical mean field theory (DMFT-3d). In the ED-1d approach, the infinite phonon Hilbert
space must be truncated to allow for a given maximum number of phonons per site In
order to properly describe the multiphonon regime (expecially in the adiabatic regime, in
which a large number of low-energy phonons can be excited), we chose a cutoff of
20. This high value forced us to restrict our analysis to a four-site cluster in the strong-
coupling adiabatic regime. In the weak-coupling regime and for larger phonon frequencies,
a lower value of is needed, allowing us to consider larger clusters up to 10 to 12 sites. We
checked that finite-size effects do not affect the crossover coupling because small-polaron
formation is a local, high-energy process.
The dynamical mean field theory approach can be seen as the exact solution of the
small polaron problem on an infinite coordination lattice [11]. The formulation of the
DMFT requires the knowledge of the free-particle DOS. A semicircular DOS can mimic
a three-dimensional (3D) case: In the following, we therefore refer to this approach to as
DMFT-3d.
We calculate the exact ground state energy obtained by means of ED-1d and
DMFT-3d and we compare the results with the self-consistent noncrossing (NCA) and
vertex corrected approximations (VCA). These two approximations are defined by the self-
consistent calculation of the electronic zero-temperature self-energy given below:
which will be determined self-consistently. From Eqs. (2,3), the ground-state energy is
given by the lowest energy solution of Re The NCA approach amounts to
compute by retaining only the 1 in the square brackets of Eq. (2). The VCA is instead given
by the inclusion also of the second term in square brackets of Eq. (2), which represents the
first vertex correction. This approach is formally similar to the approximation scheme used
in the formulation of the nonadiabatic theory of superconductivity [15], and a comparison
with exact results therefore provides also a test of reliability of such an approach for the
one-electron case.
In Fig. 1 we compare the ground-state energy obtained by ED-1d with the NCA
and VCA results. The same quantities evaluated in the DMFT-3d case are shown in Fig. 2.
We have chosen the same half-bandwidth D in both DMFT-3d and ED-1d. In the adiabatic
172 Capone, Ciuchi, and Grimaldi
The Small Polaron Crossover 173
regime, the agreement of both approximations with exact results strongly depends on the
system dimensionality as a result of the different low-energy behavior of the DOS. In fact,
moving from before the crossover the agreement of the self-
consistent calculations with the exact results is improved for the 1d case (Fig. 1), whereas it
becomes poorer for the 3d case (Fig. 2). However, the VCA approach represents a significant
improvement with respect to the NCA for every system dimensionality and over a significant
range of parameters.
As is seen from the comparison of Figs. 1 and 2, for large both approximate
and exact results tend to become independent of dimensionality. This can be understood by
realizing that in this regime, the system can be thought as a flat-band “atomic” system in
interaction with high-energy phonons. It is also clear from Figs. 1 and 2 that both the self-
consistent NCA and VCA calculations deviate from the exact results when the crossover
toward the small polaron regime is approached.
A complete comparison between the exact results and the VCA approach in the pa-
rameter space is shown in Fig. 3. We explicitly evaluated both in 1d (Fig. 3a) and
3d (Fig. 3b) the relative difference , where
and are the ground-state energies evaluated by exact techniques and the VCA, re-
spectively. To analyze the region in the parameter space in which the VCA agrees within a
given accuracy with the exact results, in Fig. 3 we report lines of constant
174 Capone, Ciuchi, and Grimaldi
3. CONCLUSIONS
In conclusion, we have shown that the crossover towards the small polaron state de-
pends strongly on the adiabaticity parameter In the antiadiabatic regime, the crossover
is ruled by and is independent of the system dimensionality. In the adiabatic regime
the relevant coupling is and the crossover occurs from large to small polaron in 1d,
whereas in 3d the crossover is from quasi–free electrons to small polarons. In the latter
case, self-consistent approximations work better than in 1d systems. We have also shown
that self-consistent calculations provide ground-state energies that agree well with exact
results outside the small and large polaron region of the phase diagram and that such an
agreement is increased when vertex corrections are taken into account.
ACKNOWLEDGMENTS
We thank M. Grilli, F. de Pasquale, D. Feinberg and L. Pietronero for stimulating
discussions. C. G. acknowledges the support of a I.N.F.M. PRA project.
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CDW Instability and Infrared Absorption
of an Interacting Large Polaron Gas
A simple approach to the many-polaron problem for both weak and intermedi-
ate electron–phonon coupling and valid for densities much smaller than those
typical of metals is presented. Within the model, the collective excitation spec-
trum is studied pointing out the presence of charge density wave instabilities in
a finite range of the electron densities. Finally, preliminary results on the optical
absorption of an interacting large polaron gas are presented.
Several experiments on the optical response of the cuprates have been reported since the
late 1980s, both in the normal and in the superconducting phases [1]. The large amount
of conductivity data in the normal phase has shown an important infrared absorption that
has been interpreted as an indication of the presence of small and/or large polarons, both
in electron-doped and in hole-doped compounds, with features depending strongly on the
doping and the temperature [2].
From a theoretic point of view, the formation of large polarons and bipolarons in polar
materials has been studied quite extensively [3]. However, a large amount of work has been
devoted to the simpler single-polaron and bipolaron problems neglecting the effects of the
polaron–polaron interaction that, instead, are expected to play an important role in high-
superconductors .
In the present paper, we focus our attention on a model of interacting Fröhlich large
polarons, showing the presence of charge density wave (CDW) instabilities in a finite range
of particle density when the electron–phonon (e-ph) interaction is sufficiently large [4].
However, for low-charge carrier densities and/or small values of the e-ph coupling constant
a gas of large polarons describes correctly the interacting electron–phonon system.
In this region, we present preliminary results for the normal state conductivity [5],
1
Dipartimento di Scienze Fisiche, Università di Napoli I-80125 Napoli, Italy.
2
Dipartimento di Scienze Fisiche, Università di Salerno I-84081 Baronissi (Salerno), Italy.
including the many-body effects, and we show that exhibits features similar to those
observed in the infrared spectra of the cuprates.
THE MODEL
We consider a system made of electrons interacting with nondispersive LO phonons
and repelling each other through the Coulomb potential screened by the background high
frequency dielectric constant [6]. The e-ph interaction is assumed to be
where
The phonon distribution functions are determined in a self-consistent way from a func-
tional variational procedure [9,10]. In particular, the Eulero–Lagrange equations for the
functions can be solved exactly and the parameters and of the pair envelope func-
tion fixed by imposing the total energy to be at a minimum. Within this variational ap-
proach, it is possible to obtain an effective e-e potential due to the exchange of virtual
phonons [10]
It contains a short-range attractive term and a long-range repulsive term screened at large
distances by the static dielectric constant and, in the opposite limit, by the background
high-frequency dielectric constant [9,10]. In particular, when tends to the
self-energy of two free polarons in the LLP approximation [8]. It is worthwhile to note
that the proposed approach can be, in principle, improved if one chooses better and better
estimates for the effective potential.
This procedure allows us to eliminate the phonon degrees of freedom from the system,
simplifying the treatment of many electron effects, and to investigate larger values of the
e-ph coupling constant with respect to the perturbative approach proposed by Mahan [6].
This happens because the LLP transformation gives rise to phonon corrections to the bare
e-ph vertex [11], corrections that can be neglected according to the Migdal theorem only
when the Fermi energy is much larger than (normal metals).
178 Cataudella, De Filippis, and Iadonisi
The model proposed is studied within the R.P. A. [12] and Hubbard [13] approximation
at T = 0 . Within these approximations the effective interaction between the electrons takes
the form
In Fig. 1, we present the numeric results for the collective energy mode in the Hubbard
approximation. goes from to the roots of an electron gas screened
by the background high-frequency dielectric constant for large values of q and softens
for a critical wave vector indicating strong correlation between the electrons. If the
attractive potential is sufficiently strong, the collective energy softens completely. This
softening, which is present in a finite range of densities indicates that the
system becomes unstable with respect to the formation of the CDW. The parameters chosen
are and
It is well known that the retarded dielectric function is analytic in the upper half of
the complex plane provided There are examples of physical systems for
which the linear response function violates the causal requirement: In this case, the linear
retarded dielectric function can no longer describe the behavior of the system correctly. It
has been shown that the temperature-dependent correlation function in the R.P.A. for an
interacting many-particle system, when the interaction is sufficiently attractive,
has forbidden zeros on the imaginary axis of the complex frequency plane [15,16]. In our
model, if the value of the coupling constant α is sufficiently large, the linear response
function possesses a pair of imaginary poles. It is interesting to note that the same type
of instability has been suggested by Di Castro et al. [17] for a large class of systems of
interacting electrons and phonons as due to very ineffective electron corrections to the e-ph
vertex.
In Fig. 2, we report the charge density and the strength of the attractive term of at
which we observe the softening of the collective energy mode We see that
there is a wide region where CDW instability sets in. Therefore, a gas of large polarons
CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas 179
provides a useful description of a system of interacting electrons and phonons only for small
values of the e-ph coupling constant and/or low-charge carrier densities.
OPTICAL PROPERTIES
In the range of values of α where a gas of large polarons is well defined, it is possible
to study the normal-state conductivity within the R.P.A. approximation, describing
the polarons through the spectral weight function of the Feynman model [18]
We note that the model proposed in this paper restores, when the well-known
results of the optical absorption of a single large polaron [20,21] and allows to introduce
within the R.P.A. approximation the effects of the polaron–polaron interaction.
RESULTS
In Fig. 3 is reported the optical absorption per polaron as a function of the frequency for
different values of the charge carrier density at Three different structures appear in
the normal state conductivity: (a) a zero-frequency delta function contribution; (b) a strong
band starting at that is the overlap of two components: a contribution from the
intraband process and a peak due to the polaron transition from the ground-state to the first
relaxed excited state; and (c) a smaller band at higher frequency due to the Frank–Condon
transition of the polaron.
Increasing the charge carrier density, we find that the large polaronic band due to the
excitation involving the relaxed states (b contribution) tends to move toward lower frequen-
cies, whereas its intensity decreases in favor of the rise of a Drude-like term around
This behavior is in agreement with the experimental data on the normal-state conductivity
of many cuprates both in the insulating and in the metallic phases [22,23].
CDW Instability and Infrared Absorption of an Interacting Large Polaron Gas 181
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182 Cataudella, De Filippis, and Iadonisi
We apply the dynamical mean field theory to the problem of charge ordering.
In the normal state as well as in the charge-ordered (CO) state, the existence of
polarons (i.e., electrons strongly coupled to local lattice deformation) is associ-
ated to the qualitative properties of the lattice polarization distribution function
(LPDF). At intermediate and strong coupling, a CO state characterized by a certain
amount of thermally activated defects arises from the spatial ordering of preex-
isting, randomly distributed polarons. Properties of this particular CO state give
a qualitative understanding of the low frequency behavior of optical conductivity
of Ni perovskites.
1. INTRODUCTION
There has been a renewed interest for the charge-ordering transition that has been
found associated to lattice displacements in cuprates nickelates and manganites.
Charge-stripes order has been detected in neodimium-doped cuprates
[1] and conjectured in LASCO [2]. X-ray studies of BISCO also shown a
modulated structure of CuO planes [3]. Commensurate charge order appears in doped
nickelates [2] also related with peculiar magnetic properties [4], and finally, large lattice
distortions have been found in manganites, which can be associated to either commensurate
or incommensurate charge ordering [5]. The amplitude of such a distortion increases from
cuprates to manganites, a fact that may support the hypothesis of an increasing charge-lattice
interaction. Another important observation supporting the presence of polaronic carrier,
1
Dipartimento di Fisica and INFM UdR l’Aquila, Università de L’Aquila, via Vetoio, I-67100 Coppito-L’Aquila,
Italy.
2
Dipartimento di Fisica and INFM UdR Roma 1, Università di Roma “La Sapienza,” piazzale Aldo Moro 5,
I-00185 Roma, Italy.
where creates (destroys) an electron at site i, and are the local oscillators
displacements and momentum. Electrons and phonons are coupled via the density fluctu-
ations (n being the average electron density). The electrons move on a bipartite lattice of
connectivity z and have a band of half-bandwidth t.
The main approximation we consider is the adiabatic approximation which can be
obtained in the limit In this limit we neglect the first term in Eq. (1), therefore
are constant of motion and can be replaced by c numbers. This approximation turns out to
be valid if the following two conditions hold:
1. As far as thermodynamic properties are concerned, temperatures must be greater than
the typical phonon energy scale
2. As far as spectral properties are concerned, energies must be greater than the typical
phonon energy scale
However, adiabatic limit allows us to solve the model with a little amount of numerics
giving the spectral properties of electrons in real frequencies and statistical properties of
the lattice. We consider also spinless electrons to account for a polaronic rather than a
bipolaronic ground-state at large couplings. This restriction, even if at a very rough level,
mimics the action of an on-site Coulomb repulsion.
We apply the machinery of the DMFT, which is the exact solution of local-type in-
teraction on an infinite coordination lattice (infinite dimensions) [14]. To have a
nontrivial limit, a scaling of the hopping, as in Eq. (1), t with the number of neighbors
is required. The DMFT approach maps the problem of locally interacting fermions on a
lattice into a single site equivalent problem [14]. A detailed study of the Holstein model
based on the Monte Carlo solution of the single site problem has been first carried out in
Ref. [15]. This analysis has been extended to the spectral properties of the normal state in
Ref. [ 1 1 ] by using the adiabatic limit to obtain an analytical solution of the single site
model. We extend this analytical approach to the study of the CO state. We consider here
alternate charge ordering in two interpenetrating sublattices, A and B. The quantity to be
determined self-consistently is the lattice polarization distribution function (LPDF) P(X).
The Charge-Ordered State from Weak to Strong Coupling 185
Different regimes are related to qualitative changes in the shape of P(X). Our main results
can be summarized by the self-consistent equations that determine the LPDF and the local
electron Green function in each sublattice
Equations (2) and (3) are obtained in the simple but nontrivial case of Bethe lattice of
bandwidth 2t. From Eq. (3), we see that the Green function is that of a particle propagating
in a randomly distorted sublattice, and sublattice A is coupled to B and vice versa. The real
frequency representation of the (retarded) Green function is simply obtained by substituting
in Eq. (3); therefore, the adiabatic limit allows us to obtain the spectral
properties in real frequencies.
2. RESULTS
We have solved the self-consistent scheme introduced in the previous section by nu-
meric iteration procedure. We consider the spinless electron half filled case, i.e., one electron
for each two sites . We start with an ansatz for the sublattice Green function, then
we get the function P at discrete points through Eq. (2) and through a numeric integration
[Eq. (3)], we obtain the new G.
In the adiabatic limit, only one relevant coupling parameter measures the electron–
lattice interaction It can be expressed as the ratio of self-trapping energy
(polaron energy to the electron kinetic energy energy (t). A crossover from
strong to weak coupling behavior is expected around 1 [12, 13]. These expectations are
confirmed by the the phase diagram at half-filling shown in Fig. 1. The continuous curve
represents the CO critical temperature as a function of the coupling strength. The dashed line
marks the normal to polaron crossover in the normal phase [11] and the crossover from weak
186 Ciuchi and de Pasquale
coupling CO (A) to strong coupling CO (B). In both normal and ordered states, a crossover
line separates the monomodal and the bimodal behavior of the LPDF. In the ordered state,
bimodality appears in the sublattice LPDF. Results are summarized in Fig. 2. The typical
weak coupling behavior of LPDF across the transition temperature is shown in Fig. 2a. Upon
decreasing the temperature, a uniform polarization of a given sublattice arises. The other
sublattice, whose LPDF is not shown in the figure, develops an opposite polarization so that
the net total polarization is zero as should be for the Hamiltonian Eq. (1), which couples
deformation and density fluctuations. The variance decreases with decreasing temperature.
Moving to polaronic nonordered state, the LPDF is clearly bimodal and symmetric at
half-filling, as seen from Fig. 2c (dashed lines). Upon decreasing the temperature below
the sublattice LPDF unbalances in favor of a net sublattice polarization but still remains
bimodal. The weight of the secondary peak decreases by a further decreasing of temperature.
We explain this secondary peak as due to temperature-activated defects in the CO state [9].
The bimodal behavior of LPDF, which is clear at very large coupling, becomes less
pronounced at intermediate coupling (see Fig. 2b). In this region in both the nonordered
and the ordered states near we observe that a nonnegligible amount of sites are nearly
undistorted. In this intermediate region of the coupling we observe a
strong non-Gaussian behavior of LPDF. Even if a secondary peak is present in the normal
The Charge-Ordered State from Weak to Strong Coupling 187
phase, it is not well pronounced in this region. As the temperature is lowered below it
may happen that this secondary peak is washed out in the ordered phase but a pronounced
shoulder remains in the LPDF. It eventually develops again a secondary peak upon a further
decrease in temperature.
The optical conductivity is obtained once the local Green functions of the two sub-
lattices are known by a generalization of the Kubo formula (details of the calculation are
presented elsewhere). We show in Fig. 3 the results obtained at half-filling for three different
values of the coupling constant characteristic of small, intermediate, and large couplings. We
see that at small coupling (Fig. 3a) the optical conductivity of the normal state shows a peak
at reminiscent of the classical Drude behavior. This peak is shifted in the CO state at
where is the CDW gap. The position of the peak depends on temperature and is
shifted toward higher energies as the temperature decreases following the enhancing of the
order parameter. As the coupling is increased (Fig. 3b), a peak at is still present also
at indicating the presence of polarons in the disordered phase. We notice, however,
a shift of the spectral weight from low to high energies as the temperature is decreased. This
effect is less evident at stronger couplings (Fig. 3c) because in this case almost all sites are
polarized (see. Fig. 2c), and consequently we have no appreciable spectral weight at low
energy. In any case, whenever polarons are present in the nonordered state, a shift toward
larger energies of the spectral weight and a change in the temperature dependence of the
amplitude of the peak is observed in the ordered state.
188 Ciuchi and de Pasquale
3. CONCLUSIONS
The crossover from weak to strong coupling CO have been studied in details by
introducing an LPDF. The qualitative change from monomodal to bimodal behavior of this
function is interpreted as existence of defects in the ordered phase. In terms of the defects
activation, we obtain a qualitative understanding of the shift from low to high frequency in
the spectral weight observed below the CO transition in Ni perovskites.
ACKNOWLEDGMENTS
The authors acknowledge useful discussions with D. Feinberg and P. Calvani.
REFERENCES
1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 563 (1995).
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Cheong, Phys.. Rev. B 54 R9592 (1996).
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The Charge-Ordered State from Weak to Strong Coupling 189
9. S. Aubry and P. Quemerais, “Breaking of analiticity in charge-density wave systems physical interpretation
and consequences,” in Low Dimensional Electronic Properties of Molybdenium Bronzes and Oxides, edited
by C. Schlenker (Kluwer Academic Publishers, Dordrecht, Boston, London, 1987).
10. T. Holstein, Ann. Phys. 8, 325 (1959), ibid. 343 (1959).
11. A. J. Millis, R. Mueller, and B. I. Shraiman, Phys. Rev. B 54, 5389 (1996).
12. S. Ciuchi, F. de Pasquale, S. Fratini, and D. Feinberg, Phys. Rev. B 56, 4418 (1997).
13. M. Capone, S. Ciuchi, and C. Grimaldi, Europhys. Lett. 42, 523 (1998).
14. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
15. J. K. Freericks, M. Jarrel, and D. J. Scalapino, Phys. Rev. B 48, 6302 (1993).
16. H. G. Reik, Z. Phys. B 203, 346 (1967).
17. D. Emin, Phys. Rev. B 48, 13692 (1993).
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Low-Temperature Phonon Anomalies
in Cuprates
1. INTRODUCTION
For some time, the electron–lattice coupling has been considered to play a negligi-
ble role in the superconductivity of cuprates, and most theories have focused on
magnetic mechanisms. However, the observation of the static spin-charge stripes in non-
superconducting opened up the possibility that the role of the lattice
is more important than has been previously thought [1]. Currently, the prevailing thought
is that the charge-spin stripes exist even in the superconducting systems, but are dynamic
and short range. The well-known dynamic incommensurate magnetic peaks observed by
inelastic neutron scattering are supposed to be the signature of magnetic stripes, but the
corresponding lattice signature of charge stripes has not been observed. The initial aim of
this work has been to observe the dynamic charge-spin stripes by studying high-energy
1
Department of Materials Science and Engineering and Laboratory for Research on the Structure of Matter,
University of Pennsylvania, Philadelphia, PA 19104.
2
Los Alamos National Laboratory, Los Alamos, NM 87545.
3
Department of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania,
Philadelphia, PA 19104.
4
Physics Department, Brookhaven National Laboratory, Upton, NY 11973.
5
Department of Physics, Tohoku University, Sendai, 980 Japan.
phonons, because for such phonons the stripes may look frozen in time. As it turned out,
we observed something related but decidedly different. Our data are not consistent with
the current idea of the charge-spin stripes, but suggest that the periodicity of the dynamic
charge stripes is half of what is currently believed. For the periodicity
of the magnetic correlation determined from the dynamic magnetic satellites is This
implies the periodicity of the charge stripes whereas our data are more consistent with
the charge periodicity of 2a.
2. EXPERIMENTAL RESULTS
We carried out inelastic neutron scattering measurements of the high-frequency bond-
stretching LO phonons of This phonon branch is known
to show strong softening with hole doping [2]. Two single crystals were comounted in an
aluminum can filled with He exchange gas. Both samples were grown by the floating zone
method and were obtained from the same batch. The total size of the sample is approximately
Previous measurements and characterizations attest to the high quality
of this sample [3].
Low-Temperature Phonon Anomalies in Cuprates 193
The experiments were made on the HB-3 triple axis spectrometer at the high flux isotope
reactor (HFIR) at the Oak Ridge National Laboratory. The spectrometer configuration used
the beryllium (002) and pyrolitic graphite (002) reflections as the monochromator and
analyzer, respectively. The analyzer angle was fixed when performing inelastic scans, giving
a fixed final energy of 14.87 meV. Soller collimators of angular divergence
were placed along the flight path from source to detector. To reduce higher-order Bragg
scattering contamination from the analyzer, a pyrolitic graphite filter was placed before the
analyzer.
Some of the results of constant-Q energy scans taken in the Brillouin
zone, in tetragonal notation are shown in Fig. 1 for and 0.2.
Because of the large incident energy required for measurements up to 90 meV, the flux
was low, and the count rates were The measurement in the Brillouin zone
around (5, 0, 0) [2, 4] suffers from spurious scattering consisting of the (6, 2, 0) Bragg
reflection from the sample scattering incoherently from the analyzer that obscured the main
phonon branch. Thus, we stayed in the zone, even though the intensity here
is significantly weaker than in the zone. In Fig. 1, the large peak at 58 meV
is due to the oxygen in-plane Cu-O bond-bending mode. The 70 meV peak is associated
with the oxygen Cu-O bond-stretching mode also in the plane. The energy scans at
various values of show that the frequency of the 70 meV branch remains constant from
(3.5, 0, 0) to (3.25, 0, 0), below which its intensity diminishes rapidly. At the same time,
some intensity appears at 85 meV, and becomes a strong peak below (3.25, 0, 0) down to
(3,0,0). The peak positions are shown in Fig. 2. Thus, it appears that bond-stretching phonon
branch has split into two nearly dispersionless subbranches, with the intensity crossover at
(3.25, 0, 0).
194 Egami et al.
3. DISCUSSION
The experimental results at T = 10 Kshown here are in conflict with the previous
measurement of the bond-stretching branch in at room temperature [2]. The
previous result shows a strongly dispersing, but continuous, single branch from (0, 0, 0)
to (0.5, 0, 0), in contrast to the two remarkably dispersionless subbranches shown in the
data presented here. The difference in these two measurements most likely arises from
the difference in temperature, according to our preliminary measurements of temperature
dependence. However, our results are similar to the ones observed for and
The two peaks seen in Fig. 1 for resemble those observed
for that provoked a dispute on the “extra” phonon branch [2, 6]. The claim
of the extra phonon branch was later withdrawn, citing the possibility of compositional
inhomogeneity, but such inhomogeneity is highly unlikely for the present sample, which
shows a very clean spin gap [3]. Thus, our observation revives the controversy over the
extra branch.
The dynamic magnetic satellites in this sample were observed at and
with indicating the wavelength of magnetic periodicity is 8a
[3]. Thus, we expected the lattice signature of the charge-spin stripes at and
Such dynamic superstructures create pseudo-Brillouin zone boundaries for
high-energy phonons at and The observed dispersion shown in Fig. 2 is consistent
Low-Temperature Phonon Anomalies in Cuprates 195
with the pseudo-Brillouin zone boundary at but not at and Indeed a spring model
created assuming the charge-spin stripes with the period of 4a shows a dispersion which is
split at and and is qualitatively very different from the observed one as shown in Fig. 3.
However, if we assume the charge-spin stripes with the periodicity of 2a (Fig. 4) instead of
4a, the calculated dispersion is in good agreement with the observation as shown in Fig. 5.
It is interesting to note that in this case the charge must be on oxygen ions. The model with
the charge on Cu showed poor agreement. Because we assumed a periodic structure in the
simulation, the branches have some dispersion. If we introduced disorder and localization,
the branches would have shown less dispersion. From the flatness of dispersion, we estimate
the correlation length of charge ordering to be about 20 Å (5a) along the stripes and 8 Å
(2a) across them. These correlation lengths define an area that contains just about one hole,
because the linear charge density in the stripe is 1/4 per unit cell. Our earlier search for
superlattice diffraction (elastic) at (0.5, 0, 0) was negative. Thus, we conjecture that the
charge periodicity with is dynamic and short range.
Our results cast a serious doubt on the conventional picture of charge-spin stripes
obtained merely by extrapolating from the static stripe structure of nonsuperconducting
compounds. Instead, our results suggest that although the dynamic charge stripes do indeed
exist, they have the periodicity of 2a and are centered on oxygen ions. Such charge stripes
interact strongly with the Cu bond-stretching half-breathing mode at (0.5, 0, 0). It is inter-
esting to note that Harashina et al. [7] reported strange phonon behavior in
for this mode in the vicinity of the (0.5, 0, 0) point around 31–33 meV. We observed that the
196 Egami et al.
corresponding mode in the present sample at 29 meV also shows anomalous temperature
dependence. This provides more evidence of the presence of the 2a charge periodicity.
It is still possible that our data are consistent with the stripe periodicity of 4a, if we
assume that the 2a charge periodicity exists within the stripe. Because the linear charge
density is 1/2 in the conventional stripe, the Peierls distortion in the spin-polarized state
produces the lattice distortion with the periodicity of 2a, and charges are localized. However,
our simulations for such a case failed to reproduce the strong crossover at and the
dispersion appeared similar to the result in Fig. 3. Furthermore, this picture is inconsistent
with the high conductivity of the system and the widely held assumption that spins are
unpolarized within the stripes. Thus, in our view, this possibility is very remote.
The static stripes observed earlier were associated with the antiferromagnetic (AFM)
domain boundaries [1, 8]. In order for the charge periodicity of 2a to be compatible with the
magnetic periodicity of 8a, the spin rotation through the charge stripe must be rather
than as in the static stripe, and the magnetic structure must be in a chiral AFM state with
a phase slip of at every other Cu-Cu bond. The average linear charge density is about
1/4 per unit cell, rather than 1/2 as in the static stripes. This provokes interesting thoughts
about the relationship among charge, spin of the charge, the magnitude of spin rotation, and
chirality. This subject deserves a very detailed study.
We expanded the model of Emery and Reiter [9] with the holes on oxygen ions,
and studied the effect of Cu half-breathing mode using the exact diagonalization method
[ 10] . For a perfectly periodic plane, the strongest hole-hole pairing occurs for oxygen
Low-Temperature Phonon Anomalies in Cuprates 197
ions separated by 2a along the Cu-O chain, followed by the pairs separated by a across the
square. This is already suggestive of the possible relation between the stripes and
pairing. If a static half-breathing Cu mode was introduced by the frozen phonon approach,
the 2a charge stripes are induced. Also, the strengths of the two kinds of pairs above are
exchanged, making the intrastripe pairing (separated by a across the square) more
favored compared to the interstripe pairing (separated by 2a along the Cu-O chain). These
results suggest that the presence of the 2a charge stripes may enhance superconductivity in
the plane.
ACKNOWLEDGMENTS
The authors are grateful to J. B. Goodenough, A. R. Bishop, J. Tranquada, V. Emery,
J. Zaanen, H. Mook, E. Mele, H. Kamimura and M. Tachiki for useful discussions. The
work at the University of Pennsylvania was supported by the National Science Foundation
through DMR96-28134. HFIR is operated by the U.S. Department of Energy.
REFERENCES
1. J. M. Tranquada et al., Nature 375, 561 (1995).
2. L. Pintschovius et al., Physica C 185–189, 156(1991).
3. K. Yamada et al., Phys. Rev. Lett. 75, 1626 (1995).
4. R. J. McQueeney et al., Phys. Rev. B 54, R9689 (1996).
5. M. Braden et al., J. Supercond. 8, 1 (1995).
6. L. Pintschovius and W. Reichardt, in Physical Properties ofHigh Temperature Superconductors IV, ed. D. M.
Ginsberg, World Scientific, Singapore (1994) p. 295.
7. H. Harashina et al., J. Phys. Soc. Jpn. 63, 1386 (1994).
8. J. Zaanen and O. Gunnarson, Phys. Rev. B 40, 7391 (1989).
9. V. J. Emery and G. Reiter, Phys. Rev. B 38, 4547 (1988).
10. Y. Petrov and T. Egami, unpublished, 1998.
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Enhanced Thermoelectric Power and Stripes
in Cuprate Superconductors
1. INTRODUCTION
Transition-metal oxides with perovskite-related structures provide an opportunity to
study the transition from localized to itinerant electronic behavior in a three-dimensional
array or a 2D sheet of corner shared octahedra in which the dominant
interactions of interest are M-O-M interactions between configurations on the
transition-metal atoms M. Transitions in single-valent systems must be distinguished from
those occurring in mixed-valent arrays or sheets. In this brief paper, attention
is focused on evidence for dynamic electron–lattice interactions that give rise to isotropic
1
Texas Materials Institute. ETC 9.102, University of Texas at Austin, Austin, TX 78712.
2. PEROVSKITE STRUCTURE
The ideal structures of perovskites, and related intergrowth oxides such as
are, respectively, cubic and tetragonal. In this paper, the larger A
cation is a lanthanide, yttrium, or an alkaline earth, and M is a first-long-period transition-
metal atom. In each structure, the deviation from unity of a tolerance factor
3. ELECTRONIC CONSIDERATIONS
Localized configurations are described by crystal-field theory; itinerant 3d elec-
trons by tight-binding band theory. If the lowest unoccupied orbitals of a configuration
lie at an energy above the top of the orbitals of an ionic model that is large enough
for electronic back transfer from the oxygen to the M atoms to be treated in second-order
perturbation theory, the basis d orbital wave functions of an M cation in an octahedral site
Enhanced Thermoelectric Power and Stripes in Cuprate Superconductors 201
may be formulated as
where and are, respectively, the ionic three-fold degenerate xy, orbitals
that only with the oxygen ligands and the two-fold degenerate
orbitals that only with the oxygen ligands. The and
are appropriately symmetrized and orbitals, and the covalent mixing
parameters
must be used, where is the angle between localized spins on neighboring atoms and
The tight-binding bandwidth is
parameters is no longer applicable to the basis wave functions associated with the holes;
the holes need to be treated in molecular-orbital theory either within a polaronic complex
or within a second phase. The system appears to represent a transition from
localized to itinerant electronic behavior where the parent compound has a gap.
The superconductive copper oxides, on the other hand, have sheets with a gap in
the ionic model of the antiferromagnetic parent compounds, but oxidation introduces holes
into molecular orbitals that, in the overdoped compositions, become transformed
into itinerant-electron states of an antibonding band.
Because and it is possible to have localized configurations in
the presence of itinerant electrons. This situation arises in mixed-valent, ferromagnetic
perovskites where the Hund intraatomic exchange field couples the itinerant
electron spins parallel to the localized configurations to give a tight-binding bandwidth
We have discussed elsewhere [5] the origin of the colossal magnetoresistance (CMR) found
in the manganese oxides; here, we draw attention to the consequences of strong, dynamic
electron–lattice interactions in some other oxides with perovskite-related structures.
4. SINGLE-VALENT SYSTEMS
4.1. Dynamic Jahn–Teller Effect
Stoichiometric undergoes a static, cooperative Jahn–Teller deformation to
the structure below a below it becomes a type A
antiferromagnet with ferromagnetic superexchange in the ab planes and antiferromagnetic
coupling between planes [6]. An antisymmetric superexchange term cants the spins from
collinear to give a weak ferromagnetic component. In 1961, one of us asked what would
happen if were substituted for in until the concentration of
ions became too small for a static, cooperative Jahn–Teller deformation to symme-
try [7]. At that time, and again subsequently [8], it was shown that at the to O transition,
the Mn-O-Mn interactions became isotropically ferromagnetic. According to the rules for
the sign of the superexchange interactions, this finding demonstrates that a dynamic Jahn–
Teller coupling of the -bonding e electrons to the two optical-mode lattice vibrations of
symmetry creates local “vibronic” states that correlate a dominant electron transfer from
half-filled e orbitals on one atom to an empty e orbital on the neighboring atoms.
clustering to minimize the elastic energy associated with dynamic Jahn–Teller interactions
was also appreciated at about that time [9].
for strongly correlated itinerant electrons breaks down with the onset of incoherent-state
fluctuations. As the on-site electrostatic energy U approaches the critical value the evo-
lution of the electronic state in a perovskite-related structure is not continuous, as envisaged
by Hubbard, but undergoes a first-order transition from itinerant to localized electronic
behavior.
Stoichiometric is a Pauli paramagnetic metal above a and becomes
a type G antiferromagnet below without exhibiting the cooperative Jahn–Teller de-
formation to be expected for a localized-electron collinear-spin antiferromagnet [14]. PES
data [15] have shown for this system also the coexistence of incoherent and coherent elec-
tronic states, which has suggested to us [16] that the measured [17] pressure dependence
does not signal a localized-spin configuration [18], as was originally in-
ferred [17], but a transfer of spectral weight from incoherent to coherent states and a
that increases with the density of states at the Fermi energy. This interpretation is
consistent with an observed decrease in with decreasing width of the band as the
smaller ion is substituted for in Significantly, as decreases to
zero with increasing x, a ferromagnetic Curie temperature increases with x, reaching a
204 Goodenough and Zhou
4.3.
The metallic perovskite system contains a half-filled band. In the
absence of localized spins, and the factor becomes unity in Eq. (8).
Introduction of a smaller ion increases which decreases and narrows
whereas pressure increases without introducing any additional perturbation of the pe-
riodic potential. We have measured under different hydrostatic pressures for
0.25, and 0.50 [20]. In exhibited a negative phonon-drag component with
a maximum at a typical of a conventional metal. However, the phonon-drag
component was enhanced by the application of pressure as in The sample
showed a stronger suppression of the phonon-drag component, undoubtedly partly due to
the perturbation of the periodic potential by different A cations, but it was enhanced by the
application of pressure. Moreover, this sample showed an increase with pressure of
as in and opposite to Pt, which is indicative of an increase in m* with pressure. A
similar situation was found for the sample, but the phonon-drag component was
more strongly suppressed. These results indicate that the perovskite system
has a narrow o band with an electronic heterogeneity similar to that observed in
The low-spin ions would be strong Jahn–Teller ions like the high-spin ions
of were the single e electron per ion localized. Therefore, it is interesting to
compare the isotope shift found in with that found for the Curie temperature
in the mixed-valent, ferromagnetic manganese oxides exhibiting a CMR [27,28]. In our
experiments [28], we chose an sample that
was at the threshold of the to O-orthorhombic transition and applied pressure to see how
the shift changes on going from the static to the dynamic Jahn–Teller regime.
We found a change from a second-order magnetic transition with no isotope shift of in the
phase to a first-order transition in the O phase. In the sample, the to O transition
occurs at a whereas in the sample it is shifted to a
Moreover, the isotope-shift parameter has a maximum value of about
5 at 10.5 kbar and decreases with further increase in pressure. Zhao et al. [27] also noted a
decrease in with increasing in the O-orthorhombic phase. Here also is evidence for
strong electron coupling to dynamic, cooperative oxygen displacements associated with a
first-order magnetic transition and a tolerance factor that depends sensitively on
6. COPPER-OXIDE SUPERCONDUCTORS
The copper oxides have intergrowth structures in which sheets have a partially
occupied, antibonding band. The simplest superconductive copper-oxide system is
206 Goodenough and Zhou
that includes the spin degeneracy. The parameter k1 corresponds to the mean number of
copper centers in a polaron; the fit in Fig. 2 is to Below 240 K, we anticipate
a dynamic segregation into antiferromagnetic and superconductive phases in the absence
of a magnetic field. However, Boebinger et al. [34] showed an increase in the resistivity
with decreasing temperature below 50 K in single crystals of with
in which superconductivity was suppressed by a magnetic field
It is tempting to interpret this result as a manifestation of polaronic motion in these high fields
with a only at lowest temperatures. The multicenter polarons can be considered
a segregated hole-rich phase in which the holes occupy molecular orbitals within a matrix
of localized electrons at
Enhanced Thermoelectric Power and Stripes in Cuprate Superconductors 207
measurements by Norman et al. [40] of the temperature dependence of the Fermi surface. As
Coleman [41] illustrated, the data show a dramatic transfer of spectral weight into directions
of the Cu-O bonds in the sheets. Such a transfer of spectral eight requires a stabili-
zation of itinerant-electron states propagating along the [100] and [010] axes. A strong
coupling of itinerant-electron states to the optical phonon modes traveling parallel to the
propagation vector Q of a mobile stripe would increase with greater ordering of the stripes
on lowering the temperature below The electron–phonon coupling would create itinerant
vibronic states having a stabilization energy proportional to or, taking account of
perpendicular stripes in alternate planes, to where
is the angle between k and Q in a particular plane. An exceptional flatness of near
the M point of the Brillouin zone was already noted by Dessau et al. [42] in 1993, and
we emphasized that the enhancement in the below indicates an increase in the
asymmetry of the curve about The increasing transfer of spectral weight with de-
creasing temperature accounts well for the increase in with decreasing temperature in
the range We therefore conjecture that the decrease in with decreasing
temperature in the range is due to the onset of vibronic Cooper-pairs from
itinerant vibronic states.
ACKNOWLEDGMENT
We wish to thank the NSF and the Robert A. Welch Foundation, Houston, Texas, for
financial assistance.
REFERENCES
1. R. D. Shannon and C. T. Prewitt, Acta Crystallogr. B 25, 725 (1969).
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A Refined Picture of the
Structure: Sequence of Dimpling-Chain
Superstructures, 1D-Modulation of the
Planes, Phase Separation Phenomena
1. INTRODUCTION
It is well known that O-doping at the normal state changes the occupancy of the chains
and the carriers concentration in the planes. In the past, high-resolution electron microscopy
1
Laboratorium für Festkörperphysik ETH, CH-8093 Zurich, Switzerland.
2
National Techn. University of Athens, Physics Department, Athens, Greece.
(HREM) investigations with nearly atomic resolution have shown in microcrystallites the
existence of oxygen–vacancy chain superstructures with increasing O-doping: 2ao (Ortho
II, every second chain empty), 3ao (every third chain empty), etc. [1a]. Some other super-
structures have been also observed by x-ray diffraction (XRD) in some single crystals [2].
The importance of the chain superstructures for superconductivity was manifested by the
plateau of the curve, which was considered to be the transition from the Ortho I
to the Ortho II superconducting phase [1b]. This plateau does not
appear as expected at but is smeared between 6.60 and 6.80.
This idea of the transition has been unanimously adopted with-
out any clear understanding how this takes place structurally and thermodynamically. Not
so many possibilities are existing from the point of view of the thermodynamics of the
Gibbs phase rule: In temperature-composition (T-x) or pressure-composition (P-x) phase
diagrams of nonstoichiometric systems, variation of the nonstoichiometry (x) leads to a
sequence of phases of the same components with different structures, separated by misci-
bility gaps (two-phase regions). A simple phase diagram of this type has been discussed in
the past for 123 [3]. In two-phase regions, the intensive properties (e.g., lattice parameters,
magnetic moment) of each phase remain constant, and only their ratio changes with the
overall composition. The mechanism is diffusive and the transformation (miscibility gap)
is second order. The coexistence of two phases has never been proved for underdroped
with any structural method except for the indications of HREM. The latter did
not show a two-phase region with sharp concentration boundaries but always mixtures of
phases with the ordered superstructure of the matrix. Therefore, from the point of
view of the phase rule, the I transformation leaves several open questions.
In the case that diffusion between the two phases of the miscibility is not possible
due to lattice strains, the alternative is a displacive first-order phase transition (e.g., a
martensitic transformation that very often appears in LT superconducting phases [4]). Two
such transitions have been found for 123: at . (tetragonal to orthorhombic) [4]
and recently at (underdoped to overdoped) [5]. No transition has been found for
with increasing doping
More ambiguous was the information about the structure of the planes in which a
dimpling of the O2 and O3 has been observed very early [6].
Triggered from our observation of the nonlinearity of the lattice parameters as a func-
tion of the carrier’s concentration (O-doping) and an anomaly of the c lattice parameter at
we have performed a systemic study of a great number of oxygen compo-
sitions.
Neutron diffraction confirmed the minimum of the c lattice parameter and showed that
it is due to a phase transition in the dimpling at the onset of the overdoped range [3,10].
Raman scattering as a function of doping showed a strong softening of the in-phase O2/O3
phonon at the same composition [8].
As we discuss later in this paper, magnetization measurements show a splitting of the
diamagnetic transition in the overdoped range
2. EXPERIMENTAL
The oxygen concentration of more than 100 polycrystalline samples in the
range of has been determined with an extremely accurate volumetric
method at the third decimal point 10–20 times more accurate [11] than
A Refined Picture of the Structure 213
all other methods discussed in the literature. Thus, a unique opportunity is given to study with
very high resolution the trends of structure and physical properties with increasing carriers
concentration induced by oxygen doping. The samples analyzed for oxygen with the above
method have been, therefore, characterized with dc magnetization and neutron diffraction
[3,9,10], EXAFS [12], and micro-Raman scattering [8,10] as a function of doping.
We note that the samples used in these investigations are near to the thermodynamic
equilibrium (very slowly cooled) [9], in contrast to the usually used quenched samples
[1b]. Quenching of the samples was used to avoid the freezing of the oxygen mobility that
was assumed to take place at Investigations of the diffusion coefficient of 123
showed that thermodynamic equilibrium is reached at least down to and kinetics
controlled mobility exists at much lower temperatures. Earlier experiments have shown that
even at much lower temperaturers, oxygen is still mobile [14]. In La 214, it has been shown
that the mobility of oxygen is kept down to 200 K! [15].
The selection of polycrystalline samples instead of single crystals is for two reasons.
The determination of oxygen needs 100 mg of the material, a rather large weight for single
crystals of cuprates, and the homogeneity of the polycrystalline material is much better
than that of single crystals. Generally, in single crystals in addition to the incorporation of
traces of the flux and impurities from the crucible walls, the long diffusion paths for oxygen
hinder the attainment of equilibrium.
Experimental details about the micro-Raman measurements have been given else-
where [8].
microscopy ( and probably ). Near the optimal doping the picture becomes
less clear, due to the coexistence of several phases. In the overdoped range two phases
appear (A, B). Thus, we can conclude that instead of a two-phase sequence
as the macroscopic structural evidence indicates, a sequence of at least four phases
exists in mesoscopic scale, separated by miscibility gaps (phase separation).
These results show for the first time the correspondence existing between the dimpling
in the planes and the one-dimensional ordering of the chains. It seems, therefore, reasonable
to assume that a ID-ordering could also exist in the superconducting planes of
The interaction between dimpling and chains could possibly take place via the apical bond.
Characteristic of these superstructures is the stepwise increase of the distance between the
oxygens (O2, O3) and the copper (Cu2) of the planes with doping, as the Cu2 is moving
away of the O2/O3 with doping. The above results show that this change cannot take place
continuously, but a “quantisation” along the c axis exists, associated with the changes of
the chain ordering. We note that in the underdoped region with increasing distance of the
oxygen (O2,O3) from the copper (Cu2) of the planes (dimpling) and distortion of the
pyramids, the also increases. This picture of 123 (near equilibrium samples) has some
interesting similarities with the phase diagram of proposed recently [15].
In this compound the O-atoms are mobile down to 200 K, in contrast to
where the Sr atoms freeze at the melting point (1400 K). Thus, the O-doped 214 system is
much nearer to the thermodynamic equilibrium, and we expect that it will show much more
structural details. Indeed, similar to a series of miscibility gaps is found, dividing
superconducting 214 phases with different stages (superstructures) of oxygen intercalation.
Therefore, the question arises that if a pattern of superconducting regions divided by phase
separated regions is a general one, necessary for superconductivity. An interesting
difference between the two-phase diagrams is that the phase separation in underdoped Y-
appears in mesoscopic scale, whereas in La-124 in macroscopic scale. The authors
A Refined Picture of the Structure 215
[15] investigated single crystals with elastic and inelastic neutron diffraction. From the
elastic scattering stripe superstructures, appears to be similar to some degree to those of
the Tranquada model [16]. From the incommensurate spin fluctuations they measure, they
conclude that possibly two kinds of stripes exists: (a) Charge and spin stripes fluctuating in
time and therefore giving inelastic but not elastic neutron scattering peaks. These stripes are
oriented along the Cu-O bond, (b) The static stripes resulting from the in-plane ordering of
the 214 stages, taking place at temperatures above These form a 45° angle to the Cu-O
direction, and cannot be responsible for the measured spin modulations [15].
We note that an important difference between the stripes in the La-cuprates and the 1D
modulation that could be expected in is that the former appears in the ab planes,
whereas the later appears along the c axis and probably influences the apical bond. The
importance of this bond for superconductivity has been discussed very early after
its discovery [17]. Recently, this discussion has been revived after the finding of near-edge
x-ray absorption fine structure (NEXAFS) investigations [18], that the apical site plays an
important role for the superconductivity in This is supported also from the
increasing consensus that infinite layer compounds do not so bulk superconductivity [19],
except if some apical bonds are formed by doping [20].
We note that in underdoped 123 the dimpling/chain superstructure phases appear only
in mesoscopic scale, invoking phase-separation scenario. The sequence of phases separated
by miscibility gaps seems to obey the Gibbs phase rule as in macroscopic phase diagrams
(chemical phase separation). However, the mesoscopic length scale indicates a picture of
physical phase separation as could be resulting from the hole doping of the antiferromag-
netic insulator. Near optimal doping and in the overdoped region, several phases coexist
in mesoscopic scale. A splitting of the diamagnetic transition [3,5,10] supports the phase
separation in the overdoped region.
4. CONCLUSIONS
The main messages emanating from the above investigations are:
• The intrinsic inhomogeneity of 123 and the existence in mesoscopic scale of a staircase
of dimpling/chain superstructures separated by minute miscibility gaps. With increas-
ing unit length of these superstructures, the Cu2 of the planes move away (along the c
axis) from the oxygens (O2, O3) of the planes and increases.
• SQUID measurements show a splitting of the diamagnetic transition at
indicating the coexistence of two superconducting phases in the overdoped region, the
one having a near that of the optimally doped and the other a few degrees lower.
Unfortunately, the 123 overdoped range is not wide enough to reveal possibly existing
trends. Doping with Ca to extend the overdoped range shows more than one phase and
a strong splitting of the diamagnetic transition [21].
• The correspondence between distortion of the planes (dimpling) and the superstructures
of the chain leads to the assumption that the distortion of the planes may have 1D-
symmetry.
• This 1D distortion of the planes would be significantly different from the stripes found
in the La-124 compounds. Whereas the later are in the ab plane, the former is a
distortion along the c axis. It is not clear at present how this vertical distortion can
influence the copper–oxygen bonds in the ab plane.
216 Kaldis, Liarokapis, Poulakis, Palles, and Conder
ACKNOWLEDGMENT
Many thanks are due to the NFP30 program of the Swiss Nat. Fonds for supporting
the work at ETH.
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A Refined Picture of the Structure 217
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1. INTRODUCTION
Femtosecond photo-induced optical modulation experiments of high-temperature su-
perconductors can give very detailed information about quasi-particle (QP) recombination
dynamics across the energy gap. With the aid of a recently developed theoretical model,
the data on photo-induced absorption can give direct information about the temperature
dependence of the gap and its symmetry and magnitude.
After photo-excitation of a metal or superconductor by a short laser pulse, an electron
and a hole are created with a relative kinetic energy, which is equal to the energy of the
1
Jozef Stefan Institute, Jamova 39, 1001 Ljubljana, Slovenia.
incident photon. These charge carriers loose their excess energy very rapidly, settling within
10–30 fs of photo-excitation into QP states near the Fermi energy. In a superconductor,
further energy relaxation of the QP into pairs is much slower because the energy gap
limits the total number of final states available and further energy loss is inhibited. The
recombination time across the gap is determined by high-energy phonon emission whose
energy is greater that the gap and is in the range of 400–3000 fs, depending on the size of
the gap and temperature. The QPs therefore form a quasi-steady-state distribution due to
this relaxation bottleneck and can be probed by a suitably delayed weak laser pulse. The
initial states for the optical transitions of the probe pulse are the occupied photo-populated
QP states, whereas the final states are in a band well above the plasma frequency. The form
of the temperature dependence is determined by the balance between phonon emission
and their reabsorption across the gap. Measurement of the temperature-dependence of the
photoinduced absorption, transmission, or reflection can thus give direct information on the
QP population as a function of time and temperature, and consequently also a great deal
of information about the gap itself. The model calculation for the temperature dependence
of the QP population recently proposed [1] gives very distinct predictions regarding the
temperature dependence of the photoinduced change in optical constants for different gap
symmetries and for different temperature dependences of the gap.
Here we show photoinduced transmission data through thin films of
for a large range of doping and as a function of temperature, from which we deduce the
temperature and doping dependence of the gap as well its symmetry.
2. EXPERIMENTAL DATA
For these measurements thin films of substrates were used
with O concentration adjusted by annealing at high temperatures. The experimental set-up
consists of a Ti: sapphire laser giving 100 fs pulses at 800 nm. The details of the experimental
setup as well as time-domain data have been shown in detail in ref. [1].
The photoinduced transmission amplitude, as a function of temperature is
shown in Fig. 1 for three different doping levels. In the underdoped sample data, the fall of
the photoinduced transmission amplitude is asymptotic and occurs at progressively higher
temperatures as increases. The temperature T* where falls to approximately 10%
of its maximum value is shown by the arrows for three different in Fig. 1. For small
however, the amplitude of the photoinduced transmission falls much more rapidly and close
to (Although many samples with different were measured, only three such temperature
dependences are shown in Fig. 1 for reasons of clarity.)
3. DISCUSSION
The theoretical model for the photoinduced transmission amplitude [1] predicts very
different temperature dependences, depending on the temperature dependence of the gap
itself. In the case of a temperature independent gap,
Evolution of the Gap Structure 221
where is the gap, is the effective number of phonons per unit cell emitted in the QP
recombination process, is the typical phonon frequency, and N ( 0 ) is the density of states
at A fit to the data for the two samples with using this formula is shown in
Fig. 1. The predicted photoinduced transmission amplitude falls to zero exponentially at
high temperatures in good agreement with the data, and one clearly cannot speak of a gap
opening at some specific temperature. The criterion for choosing an onset temperature T* is
therefore somewhat arbitrary and signifies the point when (Here we have chosen
T* at the point where the amplitude of the signal falls to 10% of maximum.)
If the gap closes at a well-defined temperature due to a collective effect—i.e.,
as as in the BCS scenario—the relaxation bottleneck dissappears at this temper-
ature and the formula is somewhat modified [1], and one obtains a sharp drop of the
photoinduced transmission amplitude at as shown by the fit in Fig. 1 for The
temperature dependence data clearly imply a distinction between the underdoped case with
and samples near optimum doping with We emphasize that the two cases
are qualitatively different. A T-dependent BCS-like collective gap, which closes at
cannot be used to describe the asymptotic behaviour at high temperatures, whereas in the
T-independent gap model, falls far too slowly at high T to be able to describe
the data near optimum doping. The fact that no change in is observed at in the
underdoped phase implies that the DOS and gap structure are also unchanged at from
which we can conclude that phase coherence is established at with no change in pairing
amplitude (via the sum rule).
An important aspect of the phase diagram is the crossover region between the two
regions of doping. As already discussed, falls very rapidly near and the BCS
222 Mihailovic, Demsar, and Podobnik
gap model fits the data very well (including the slight maximum, which is observed in
the data below However, close examination of the plot for in Fig. 2 above
shows that some photoinduced transmission is present up to temperatures as high as
130 K. This cannot be described by either model, and is well beyond the temperature at
which pair amplitude fluctuations are expected to play a role. The fit to the data over the
entire temperature range is now made using a two-component model, with both gaps present
simultaneously. The values of the two gaps are found to be very similar in this doping region.
This observation is consistent with the fact that only a single exponential recombination
time is observed in the 0–3 ps timescale in the time-resolved data.
A further independent signature of the coexistence of two gaps near optimum doping
comes from the anomaly in the QP recombination time observed in the time-evolution
of the photoinduced transmission in our experiments. The recombination time has been
shown to behave as [1,2]. For a BCS-like collective gap,
so is expected to diverge at We observe no such divergence for which
is consistent with a T-independent gap. However, near optimum doping a divergence at
temperatures has been observed by a number of authors [1,3,4], corroborating
the evidence from the temperature dependence of In Fig. 3, we plot and
from an exponential fit to our time-resolved data for different clearly showing the
evolution of two gaps with doping near
A somewhat unexpected feature of the data is that over the whole region of doping,
the gap appears to be described very accurately using an isotropic gap [1]. It is easy to see
qualitatively why the data are not consistent with a d-wave gap model. Because a d-wave
gap implies the existence of QP excitations down to the lowest temperatures, we would
expect to keep increasing down to the lowest temperatures with no real bottleneck
in the QP recombination above The model calculation [1] for the QP recombination
Evolution of the Gap Structure 223
dynamics in the case of a d-wave gap (calculated for 2D and 3D) gives a very distinct form
of temperature dependence, as shown in Fig. 4. The data, which fits very well to the isotropic
model, clearly cannot be fitted by the d-wave model. A comparison of the predictions for
the T-dependence of the induced transmission recombination time intensity
224 Mihailovic, Demsar, and Podobnik
dependence, and time-dependence of between the models with the data for s and d
gap symmetries is shown in Table 1.
It should be noted that the experimental evidence for d-wave behavior is strong in
where many of the tunneling and photoemission measurements have been
performed. YBCO, however, has an orthorhombic structure in which the d-wave repre-
sentation is—strictly speaking—not allowed by symmetry and an s-wave component is
necessary to satisfy symmetry requirements. One possible reason for the different gap sym-
metry found here compared to other experiments is that the order parameter may be different
in the bulk of the sample than on the surface. The possibility that the order parameter varies
as where s and d are the amplitudes of the s- and d-wave components and
and describe the changing magnitude of the two components at a distance z away from the
surface was suggested by K. A. Müller in connection with the results of muon penetration
depth measurements [5]. The present optical measurements are performed in transmission,
which means that we are mainly probing the bulk of the sample. (The absorption length of
light in YBCO is In contrast, many of the techniques that have shown the pres-
ence of a d-wave order parameter—for example, tri-crystal tunneling experiments—probe
the order parameter on the surface [6].
It is possible that the surface has an intrinsically different electronic structure from the
bulk. Surface effects become important when the characteristic length of the experimental
probe is comparable to the unit cell, as in tunneling experiments [6] or photoemission [7].
Another possible experimental problem is that in some experiments, YBCO samples are
cleaved and that these surfaces are not the same as the bulk. We have found that photoin-
duced optical reflection experiments (which are also surface sensitive, albeit much less
so than photoemission or tunneling) on cleaved YBCO crystals often show rather irrepro-
ducible behavior, from which we deduce that possibly cleavage occurs in regions where the
stoichiometry is different from the bulk. In contrast, we have found that polished YBCO
samples show sample-independent and reproducible behavior, which is essentially identical
to the thin-film transmission experiments.
Finally, we mention the important possibility that the d-like gap behavior arises be-
cause of the existence of localized states in the gap in YBCO and is not, in fact, intrinsic.
Experiments like tunneling and photoemission cannot distinguish between such localized
states and QP states because only a spectral density is obtained. However, time-domain
measurements by Stevens et al. [8] show that in YBCO temperature-activated excitations
exist whose activation energy is significantly smaller than the QP gap
In fact, the values found are consistent with activation from intragap
states. However their time-dynamics is qualitatively different than the QP recombination
Evolution of the Gap Structure 225
dynamics. It is very slow and has a very different T-dependence, consistent with localized
state relaxation. Further time-resolved investigations of the systematics of the localized
intragap states will hopefully elucidate this issue.
4. CONCLUSION
The presented data show that at low doping the energy “pseudogap” in is
large and T-independent. Upon further doping it decreases, according to an inverse law
1/ p , where p is the hole concentration [ 1 ]. Near optimum doping, another gap with a mean-
field-like temperature dependence becomes apparent and coexists with the “pseudogap,” the
two being comparable in magnitude. Its temperature dependence is suggestive of a collective
excitation (e.g., Bogolyubov mode) or collective BCS-like gap. Apart from exhibiting a
BCS-like temperature dependence the T-dependent gap has one
further characteristic of BCS phenomenology, namely that it appears at In contrast, the
underdoped state exhibits a gap, which shows no change at and is better viewed as a
splitting between energy levels [9].
Whereas it is clear that the coexistence of the two gaps near optimum doping is difficult
to understand in a homogeneous medium, the data can be easily understood by invoking the
two-component paradigm with the existence of stripes or clusters as shown schematically
in Fig. 5. In this scenario, the two gap regions are separated in real space and correspond
to high carrier density and low carrier density in adjacent regions a few unit cells apart.
The seemingly alternative viewpoint—where the QPs are separated in momentum space,
in which one part of the Fermi surface (FS) has a collective gap whereas another has a
T-independent gap—is in fact quite compatible with the stripe picture, provided the momen-
tum relaxation time between such regions in k-space is much shorter than the QP recombina-
tion time. This is indeed the case, because the measured QP recombination rates of
are slow compared to the momentum scattering rates of
226 Mihailovic, Demsar, and Podobnik
inferred from infrared reflectivity measurements [10]. In other words, the excited QPs do
not travel very far in real space before they are scattered elastically, but show recombination
dynamics according to the region in space where they find themselves. This explains the
common FS in ARPES and shows why its appearance is consistent with the stripe scenario.
ACKNOWLEDGMENTS
We acknowledge V. V. Kabanov for valuable discussions, funding from the EC Ultrafast
network, and also J. E. Evetts, G. A. Wagner, and L. Mechin for sending us YBCO samples
used in the present investigation.
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1. V. V. Kabanov, J. Demsar, B. Podobnik, and D. Mihailovic, Phys. Rev. B 59, 1497 (1998).
2. A. Rothwarf and B. N. Taylor, Phys. Rev. Lett. 19, 27 (1967).
3. D. Mihailovic, B. Podobnik, J. Demsar, G. Wagner, and J. Evetts, J. Phys. Chem. Sol. 59, 1937 (1998).
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10. D. Basov et al., Phys. Rev. Lett. 77, 4090 (1996).
Local Lattice Distortions in
Doping Dependence
1. INTRODUCTION
A theory on the basis of spin-charge separation has been proposed as a possible mech-
anism of superconductivity in the early years of superconductivity (HTSC) re-
search [1]; however, the lattice was assumed to be homogeneous on the basis of
available crystallographic studies. Recently, a number of experimental techniques have
shown that the lattice is rather inhomogeneous. In fact, the new models, based on
phase separation in carrier-rich and carrier-poor regions are coming up to review the situa-
tion [2]. At the experimental front, a combination of extended x-ray absorption fine struc-
ture (EXAFS) and x-ray diffraction (XRD) has demonstrated that there are lattice stripes
of undistorted and distorted local structure [3] alternating with a mesoscopic length scale
comparable to the coherence length in HTSC. Structural evidence for two-component elec-
tronic system was also provided by neutron pair distribution function (see, e.g., Egami et al.
1
Electrotechnical Laboratory, Umezono 1-1-4 Tsukuba Ibaraki 305, Japan.
2
Max-Planck Institute fur Festkorperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany.
in Ref. [4]), and other techniques have also indicated a temperature T* [5], below which the
planes may have ordered stripes of carrier-rich and carrier-poor domains [6]. As the
inhomegeneous structure became evident, renewal of interests in an alternative mechanism,
i.e., phonon scattering, the lattice effects on superconductivity have attracted much
attention recently [7–9].
Spin susceptibility obtained from the NMR relaxation rate [10] and inelastic
neutron scattering [ 11] have shown anomalies well above Including EXAFS, these are
the techniques that provide snapshot of lattice. Among these, EXAFS is an ideal means that
reflects the radial distribution of atoms with a time scale of sec. In-plane lattice anoma-
lies observed above are one of the general features of HTSC materials: lattice distortions
have been observed for and
Bianconi et al. [15] attributed the local distortion to the charge ordering into
stripes of locally distorted low temperature tetragonal (LTT) and undistorted low tempera-
ture orthorhombic (LTO) regions. The stripe structure has been proposed to a mechanism
of enhancement.
In case of much attention has been paid to out-of-plane oxygen motions,
e.g., EXAFS studies observed a large apical oxygen displacement around attributed to the
bipolaron tunneling [16]. However, the origin of the out-of-plane phonon anomaly is still a
controversial problem, because the frequency shift in infrared (IR) and Raman experiments
are interpreted as an anomalous change in phonon self-energy [17] or fluctuations associated
with superconductivity [18] is smaller than the reported change in tunneling frequency [16].
However, our polarized EXAFS study of optimally doped have shown the
in-plane local lattice anomaly below a characteristic temperature that lies above
and close to the characteristic temperature for opening of a spin gap T*. Whether the
in-plane lattice anomaly is related to the charge stripe or spin–phonon interaction is an
interesting problem. Spin gap observed in various experiments such as NMR [10], neutron
scattering [11], and transport properties [19,20] have been related to short-range ordering
of spin singlets [21]. In case of however, the T* is found to coincide with
the in an optimally doped sample [22]. In this communication, we report the doping
dependence of in-plane lattice anomalies in over a wide range of oxygen doping
and discuss the results in relation to the lattice-charge stripes [3] and spin
and charge excitations.
2. EXPERIMENTAL
Highly oriented thin-film samples were prepared by a pulsed laser ablation
technique using a KrF excimer laser [23] onto single crystal substrates of
at 750°C. Oxygen stoichiometry (y) was carefully controlled by an oxygen partial
pressure after the growth; samples were slowly cooled down from the growth temperature
under oxygen pressure of 0.1–200 mbar, yielding samples with ranging from 31.3 to
90 K. Temperature dependence of resistivity was measured prior to the experiment for all
samples. As-grown samples (100-nm thick) showed a sharp superconducting transition with
a typical transition width of about 5 K indicating a high degree of oxygen ordering along
the Cu1-O1 chain and homogeniety. Three samples with a sharp transition width have been
chosen to study doping dependence of local distortions in over a wide range in
In this work, we focus our attention on two samples
Local Lattice Distortions in 229
3. RESULTS
Normalized EXAFS oscillations are Fourier-transformed (FT) after multiplying k
over the range Upper and lower columuns of Fig. 1 compare the magni-
tude and imaginary part of the Fourier transform (FT) for the EXAFS data for
taken with the electrical field vector parallel to the ab plane at
and 40 K. For comparison, the results for undistorted phase are
also shown. Shaded area indicates the difference in the FT magnitude between the two
data. In Fig. 2, the magnitude and imaginary part of the FT for the data for
thin-film measured at and 20 K are shown where the
FT results for are also compared. In Figs. 1 and 2, a prominent peak observed
at around consists of square planar oxygen atoms (O2, O3) of the pyramid
and a small contribution from pair correlation. Theoretical Cu-O phase shift func-
tions obtained by FEFF6 [26] were corrected in the FT. As can be seen in Figs. 1 and 2,
the site-averaged Cu-O pair correlation around 2.0 A shows asymmetry and broadens on
decreasing temperature. In Fig. 3, the magnitude of the FT for is plotted as a function
of T over a wide temperature range (20–300 K) for the two samples and
55 K). Figure 4 shows the same plot as a function of normalized T, i.e., A constant
increase of FT magnitude with the decrease of T is due to the thermal vibration term. At
however, the magnitude increases its intensity rapidly and a sharp
drop giving rise to a minimum at
4. DISCUSSION
In the unit cell of optimally doped copper atoms (Cu2) in the
plane are coordinated by square-planar oxygen atoms (O2 and O3) and an apical
oxygen atom (O4), whereas Cu1 atoms form a linear chain (Cu1-O1) along the b axis. The
in-plane polarized Cu K-EXAFS probes the Cu-O pair correlation averaged over all Cu-O
pairs parallel with the ab plane. The contribution of Cu1-O1 to the average coordination
(2y — 4)/3 in a twinned sample is for sample and
for sample As can be seen in Figs. 1 and 2, the Cu-O peak in the FT
magnitude becomes asymmetric with the decrease of T. Inspecting the imaginary part that
230 Oyanagi, Zegenhagen, and Haage
grows only at the large R side, one can find that the asymmetry is caused by an additional
component with a slightly longer Cu-O bond, which indicates that the local distortion occurs
at low temperature.
One can imagine various models of local lattice distortions involving in-plane Cu-O
bonds. One of such models is a pseudo-Jahn–Teller distortion in which two opposing
Cu-O bonds are shortened while the other two perpendicular bonds are elongated [27].
This model would result in the symmetric variation (decrease) of the first peak because
of the interference between the two closely separated Cu-O bond lengths, which is in
disagreement with the observation. Other possible distortions are a LTO-like and LTT-
like tilts of pyramids. In the former distortion, where all square-planar Cu-O bonds
are elongated, the site-averaged Cu-O peak would be symmetric and shift to a large R
direction. In the latter case, in which two Cu-O square-planar bonds (Cu2-O3) are elon-
gated while other two bonds (Cu2-O2) are kept constant in length, an asymmetric and
broadened pair correlation is expected. Thus we conclude that the observed asymmetric
pair distribution indicates the phase separation into the distorted domain with an elon-
gated Cu-O bond and undistorted domains. Here we omitted the effect of Cu1-O1 linear
Local Lattice Distortions in 231
chain because the Cu1-O1 distance (1.94 Å) is close to the Cu2-O2/O3 distances (1.93–
1.96 Å).
In the following, we describe the qualitative results of FT as a function of T. In Fig. 1,
in the upper column, we show the highest temperature data which shows an asymmetry
in the FT peak Comparing the low temperature data for the two samples
it seems that the tilt angle is almost the same. A detail study on
this is an object of future publications [28].
The FT magnitude is a sensitive measure of the change in the pair correlation function
either due to local distortions and/or phonons. As illustrated in Figs. 3 and 4, the FT
magnitude for both samples show anomalous T-dependence above that can be described
as a universal behavior if T is normalized to i.e., onset around mamimumat
and a minimum at We define as an onset temperature of local lattice anomaly.
A remarkable feature is that as seen in Fig. 4, the maximum around 1.3 exactly coincides
for the two samples. Two major factors which affect the FT magnitude are considered, i.e.,
the degree and fraction of distortion and magnitude of relative displacement of the Cu-O
distance. Because the lattice distortion evidenced by an asymmetric FT peak is observed
232 Oyanagi, Zegenhagen, and Haage
over a wide T-range below , the anomalous variation of FT peak magnitude indicates
local phonon anomalies observed by other techniques. Arai et al. [11] reported that T-
dependence of S(Q, E) obtained by an inelastic neutron scattering technique for optimally
doped shows an anomalous increase around 120 K close to
(123 K) for A sharp increase of S(Q, E) maximizing at is
ascribed to an expansion of dynamical correlation length associated with a local structural
distortion [11]. A sharp increase in the FT magnitude at is explained by the increased
correlation in the Cu-O stretching vibration, which would decrease the mean-square relative
displacement (MSRD) and thus sharpen the radial distribution as observed as a maximum
at Similar anomalies have been reported in internal friction at The
present observation that is consistent with Raman experiments reported
by Ruani and Ricci [30] who observed anomalies in the electronic peak at 1.6 It is
interesting to note that the T-range for a possible increased correlation in Cu-O stretching
coincides with the anomalous S(Q, E) variation around
As shown in Fig. 5, for slightly underdoped agree
well with defined as an onset of deviation from T-linear resistivity associated with
a pseudogap opening of a spin excitation [19]. Because of insufficient number of data
points, for underdoped the relation between the local lattice
distortion and pseudogap opening is not clear but the formation of distorted domain is
consistent with the phase diagram of pseudogap [31]. Thus we conclude that the signatures
Local Lattice Distortions in 233
234 Oyanagi, Zegenhagen, and Haage
of local phonon anomalies below occur after the opening of pseudogap in spin and
charge excitation. Mihailovic et al. [31] reported that below T*, in-plane optical conductivity
changes from a single-component to two-component carrier regime. Such a two-component
carrier picture below T* is consistent with a charge stripe of a distorted (localized) and
undistorted (itinerant) domains [3]. A variety of interpretations are possible for T*; in the
Bose–Einstein condensation picture, T* is interpreted as the onset of pairing which can
be separate with condensation [31]. However, T* observed in and Raman
frequency shifts is related to the onset of short range order of Zhang–Rice singlets [21].
Bianconi et al. [15] proposed a amplification mechanism due to the shape resonance
of charge stripes. It is still a long way from distinguishing the mechanism of HTSC, but
the structural phase separation into distorted and undistorted domains below T* and local
phonon anomalies below seem to be a universal feature of underdoped HTSC materials.
5. CONCLUSION
We have reported the doping dependence of the local lattice distortions in epitaxial
thin films using ab plane polarized EXAFS. The results indicate that the in-
plane Cu-O pair correlation becomes asymmetric on lowering T associated with the local
distortion of units below which can be explained by LTT-like tilting of
units. The FT magnitude shows a universal feature, i.e., maximum and minimum, at certain
characteristic temperature normalized by These anomalous changes in
the FT magnitude above suggest that there exist local phonon anomalies in ab plane Cu-O
bonds. Comparison of the present results with c axis polarized experiments suggests that the
MSRD of apical oxygen correlates with the in-plane lattice anomalies. Future experiments
on an untwinned single crystal would provide us details of the anomalies in local lattice
dynamics and hopefully the role of lattice dynamics and charge stripes in HTSC mechanism.
ACKNOWLEDGMENTS
The authors are thankful to A. Bianconi, N. L. Saini, A. Lanzara, D. Mihailovic,
M. Arai, T. Ito, C. H. Lee, J. Ranninger, and K. Yamaji for valuable discussions.
REFERENCES
1. P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett. 58, 2790 (1987).
2. V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990); C. Di Castro and M. Grilli, in Phase
Separation in Cuprate Superconductors, ed. by K. A. Muller and G. Bendek (World Scientific, Singapore,
1992), p. 85.
3. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi, and
Y. Nishihara, Phys. Rev. B 54, 12018 (1996); A. Bianconi, M. Lusignoli, N. L. Saini, P. Bordet, A. Kvick,
and P. G. Radaelli, Phys. Rev. B 54, 4310 (1996).
4. T. Egami and S. J. L. Billinge, Prog. Mater. Sci. 38, 359 (1994).
5. T* is defined as an onset temperature of pseudogap opening in spin or charge excitation spectra, whereas
is defined as an onset of local phonon anomalies and The onset temperature of local lattice
distortion is close to T* rather than
6. C. Berthier et al., Physica C 235–240, 67 (1994); M. A. Teplov et al., in ., Superconductivity 1996: Ten
Years after the Discovery, ed. by E. Kaldis, E. Liarokapis, and K. A. Muller (Kluwer Academic Publishers,
Dordecht, 1997) p. 531; Y. Wu, S. Pradhan, and P. Boolchand, Phys. Rev. Lett. 67, 3184 (1991).
Local Lattice Distortions in 235
1. INTRODUCTION
Angle-resolved photoemission has been a useful tool to study single particle properties
of superconducting materials [1]. There are two main approaches of angle-resolved
photoemission (ARPES) that are used to study the Fermi surface features of the
superconductors. The standard method is based on the measurement of energy distribution
curves (EDC) in all high-symmetry directions of the Brillouin zone for determination of the
points in which the quasi-particle peaks cross the Fermi level. The second approach is based
on measuring the photointensity within a narrow energy window at the Fermi energy
defined by the spectrometer resolution to get the distribution of spectral weight near the
Fermi level in the k space [2]. The second approach has an advantage over the standard EDC
method because it provides a global view of the Fermi surface, whereas the standard EDC
1
LURE, Bat 209D Universite Paris-Sud, F-91405 Orsay, France, & Instituto de Ciencia de Materiales de Madrid,
CSIC, 28049 Madrid, Spain.
2
U n i t à INFM and Dipartimento di Fisica, Università di Roma “La Sapienza” P. A. Moro 2, 00185 Roma, Italy.
3
Superconductivity Research Laboratory, ISTEC, Shinonome 1-10-13 Koto-ku, Tokyo 135, Japan.
method may suffer from the extrapolation in the process of constructing the Fermi surface
image using dispersion curves. However, the EDC method has an advantage over the new
angle-scanning approach because the EDC method also provides important information on
the identification of the quasi-particle bands and their dispersion below the Fermi level. In
addition, the angle-scanning method is constrained by the spectrometer resolution, and it
may not be straightforward to differentiate the states representing the true Fermi surface and
the occupied electronic states, below the Fermi surface, lying within the resolution. More-
over, the measured Fermi surface image is affected by the matrix element effects depending
on the angles between the polarization vector of the photon beam and the wavevectors of
the initial state and the final state [3].
In this paper, we report the Fermi surface of Bi2212 system measured
by angle-scanning photoemission using high intensity of polarized synchrotron radiation to
address the mentioned points. We have made the measurements in two different polarization
geometries to identify the matrix element effects.
2. EXPERIMENTAL
A single crystal of size grown by floating zone method was used for the
experiments. The crystal was well characterized for its transport and structural properties. It
is as grown at optimum doping with a sharp superconducting transition of
91 K [4]. Its structure, studied by synchrotron radiation diffraction, shows the characteristic
features of superconducting samples with satellites due to the incommensurate modulations
of both BiO and plane [5]. The crystal was aligned by standard method using specular
laser reflection from the crystal surface. The clean and flat surface was obtained by cleaving
the crystal at room temperature. The experiment was repeated with different cleavage. Each
cleavage gave the same Fermi surface, proving the high quality of the crystal, and the surface
was stable for several days.
The experiments were carried out at the Laboratoire pour l’Utilisation du Rayon-
nement Electromagnetique (LURE) (Orsay-France) on the SU6 beamline. The experiments
were performed in an ultra-high vacuum (UHV) chamber
equipped with an angle-resolved hemispherical analyzer and a high-precision manipulator
that permits rotation in the full 360° azimuth emission angle and 90° polar emission
angle relative to the surface normal [6]. The photoelectron intensity at the Fermi level
and bellow was recorded along a series of azimuth scans. The sample was rotated around
its normal and the intensity was recorded every 1.5° at fixed theta, with an absolute angular
precision better than 0.5°.
The Fermi surface map was obtained by centering the electron energy window at the
and collecting the electrons within an energy window of the order of spectrometer resolution
using a photon energy of 32 eV. The polarization vector of the synchrotron light,
the direction of the photon beam, and the surface normal were kept in the same horizontal
plane, called the scattering plane, for all the measurements. The mirror plane is defined by
the sample normal and the direction of the emitted photoelectron selected by the detector
position. The detector is moved in the fixed mirror plane by changing the polar angle in
order to select different values of the wave vector in the superconducting plane
The direction of the initial state is selected by rotating the sample around its normal,
which is collinear with the crystallographic c axis.
Fermi Surface of Superconductor 239
We mapped the Fermi surface in two different experimental geometries, shown picto-
rially in Fig. 1. The upper picture shows the geometry for the “even” symmetry, whereas the
lower picture represents the geometry used for the “odd” symmetry. In the even symmetry,
the scattering plane is coplanar with the mirror plane, whereas in the odd symmetry the
mirror plane is orthogonal to the scattering plane.
The initial states of symmetry (formed by a mixing of and or-
bitals) are even with respect to the mirror plane for parallel to the Cu-O-Cu bonds whereas
they are odd with respect to the mirror plane for at 45° to the Cu-O-Cu bonds [1,7,8].
In the even experimental geometry, the transition from these states forming the conduction
band is fully allowed for in the and equivalent directions and forbidden for
in the and equivalent directions. On the contrary, in the odd geometry, the
transition from these states is fully allowed for in the and equivalent directions
and forbidden for in the and equivalent directions.
All the experiments were repeated with different cleavage in several experimental runs
and a perfect reproducibility of the experimentally observed Fermi surface features was
achieved ascertaining intrinsic nature of the features and high quality of the crystal used.
panel shows the one measured in the odd experimental geometry. The brighter regions
indicate higher intensity of emitted photoelectrons excited from the initial state having
constant energy and in-plane wavevector spanned over reciprocal space of the
two-dimensional (2D) plane.
In the Fermi surface map recorded using the experimental even geometry (upper panel),
the photointensity along the direction in the first Brillouin zone is absent due to matrix
element effects. In this experimental geometry, the transitions from initial states of (Cu
symmetry are allowed having even symmetry with respect to the mirror
plane in the direction, and hence the Fermi surface features around the and
directions can be clearly identified in this map.
However, in the Fermi surface map recorded using the experimental odd geometry
(lower panel of Fig. 2), the photointensity along the and direction in the first
Fermi Surface of Superconductor 241
Brillouin zone is absent due to matrix element effects. In this experimental geometry, the
transitions are allowed from initial states having odd symmetry with respect to the mirror
plane in the and direction, and hence the Fermi surface features around these
directions can be clearly identified in the map. Thus the combination of the two maps
completes the information on the Fermi surface features of the Bi2212 superconductor.
One can obtain precise Fermi surface parameters by using the fact that, within the Fermi
liquid framework, the sum rule for ARPES is that relates the
energy-integrated ARPES intensity to the momentum distribution The n(k)
shows discontinuities at each Fermi wave-vector These discontinuities are smeared out
by the finite angular resolution. The discontinuities at show singularities in the modules
of the gradient corresponding to the Fermi surface crossings. Thus the gradient of
the Fermi surface measured in angle-scanning mode is a useful method to define precise
Fermi surface parameters [9]. We have used this approach and calculated the gradient of
the two Fermi surface maps.
The modules of the Fermi surface gradient is shown in Fig. 3. All Fermi surface
features could be seen clearly in the gradient however with photointensity doublets. The
242 Asensio et al.
outer part of the gradient map represents the real Fermi surface and unaffected by the spectral
photointensity below the Fermi level [9]. A careful observation of the photointensity maps
(Fig. 2) and their gradients reveals global information about the topological features on the
Fermi surface of the Bi2212 system. The even symmetry image reproduces most of the
features observed in the earlier studies (e.g., the umklapp bands in the direction due to
superstructure in the Bi-O plane), the small pockets around and
expected in lightly doped Mott insulators, absence of photointensity around the and
and asymmetry of the and directions [10–14]. The shadow bands
associated with the formation of small pockets could be seen clearly in the odd symmetry
image (lower panel of Fig. 2) in which the transition along the are fully allowed
by the selection rules.
The Fermi surface in Fig. 2 clearly shows the shadow features around
locations. It should be mentioned that the observation of the shadow band features has
been controversial due to the fact that presence of umklapp reflections along the di-
rection complicates their identification on the Fermi surface. However, the present mea-
surements made in the even and odd symmetries allow us to identify the presence of
these bands along the and directions ascertaining their presence. The present
Fermi surface maps confirm previous results obtained at high k values that have shown
shadow band features to appear with even higher photointensity around lo-
cations [15]. The shadow features can be associated with coupling of electrons with spin den-
sity waves in the plane of superconductors [13, 16, 17] or to structural origin
[18,19].
We observe a strong suppression of photointensity around the M points. The suppres-
sion of spectral weight has been argued to be due to spin density wave background [21],
quasi-particle decay into holons and spinons [20], and several other related reasons. We
argue that a phase segregation giving quasi-one-dimensional charge ordering in stripes
along direction [5,22] is the reason for the same. It is now being established that the
charge segregation in stripes within the plane plays important role in the electronic
structure of the cuprate superconductors [23]. In the Bi2212 system, the charge ordering in
stripes within the plane along the direction has been shown by Cu K-edge
extended x-ray absorption fine structure (EXAFS) and anomalous diffraction measure-
ments [5,22]. In fact, the photointensity suppression around the M points is asymmetric
with a well-defined nesting vector in the diagonal direction that is the second
harmonic of the main lattice modulation.
In summary, a complete k space mapping of the Fermi surface features of Bi2212
superconductor at the optimum doping is done by angle-scanning photoemission in the
even and odd symmetry, allowing us to provide a clear identification to the Fermi surface
features. The results show absence of photointensity at the Fermi surface around the M
points. In conclusion, the Fermi surface reported in the present work indicate that electrons
moving in a superlattice of quantum stripes are strongly interacting with spin and charge
collective excitations.
ACKNOWLEDGMENTS
The experiment was done at the Spanish–French beam line of LURE with the sup-
port of the large-scale Installation Program and the Spanish agency DGICYT under grant
Fermi Surface of Superconductor 243
PB-94-0022-C02-01. This work was partially funded by Istituto Nazionale di Fisica della
Materia (INFM) and Consiglio Nazionale delle Ricerche (CNR) of Italy.
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J. C. Campuzano, A. F. Bellman, T. Yokoya, M. R. Norman, M. Randeria, T. Takahashi, H. Katayama-Yoshida,
T. Mochiku, K. Kadowaki, and G. Jennings, Phys. Rev. Lett. 74, 2784 (1995); M. R. Norman, M. Randeria.
H. Ding, and J. C. Campuzano, Phys. Rev. B 52, 615 (1995); M. R. Norman, M. Randeria, H. Ding, J. C.
Campuzano, and A. F. Bellman, Phys. Rev. B 52, 15107 (1995).
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273, 325 (1996); A. G. Loeser, D. S. Dessau, and Z. X. Shen, Physica C 263, 208 (1996).
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Phys. A 60, 247 (1995); P. Aebi, J. Osterwalder, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and
K. Kadowaki, Phys. Rev. Lett. 72, 2757 (1994).
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Rev. B 57, R11101 (1998).
15. N. L. Saini, J. Avila, A. Bianconi, A. Lanzara, M. C. Asensio, S. Tajima, G. D. Gu, and N. Koshizuka, Phys.
Rev. Lett. 79, 3464 (1997).
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20. R. B. Laughlin, Phys. Rev. Lett. 79, 1726 (1997).
21. J. Schmalian, D. Pines, and B. Stojkovic, Phys. Rev. Lett. 80, 3839 (1998).
22. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito,
Phys. Rev. Lett. 76, 3412 (1996).
23. See, for example, the special issue on Stripe Lattice Instabilities and Superconductivity, edited by
A. Bianconi and N. L. Saini [J. Supercond. 10, No. 4 (1997)].
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Local Lattice Fluctuations and the Incoherent
ARPES Background
¹Centre de Recherches sur les Très Basses Temperatures, Laboratoire Associé á l’ Université Joseph Fourier, Centre
National de la Recherche Scientifique, BP 166, 38042, Grenoble Cédex 9, France.
2
Dipartimento di Scienze Fisiche “E. R. Caianiello,” Università di Salerno, I-84081 Baronissi (Salerno), Italy-Unità
I.N.F.M. di Salerno.
fluctuations. Such a situation can be described in terms of the Boson–Fermion model (BFM),
which was proposed [6] well before the discovery of HTS, in order to overcome the inherent
difficulty of the scenario of a bipolaronic superconductivity [7]. In this article, we introduce
a generalization of this model [8], which takes into account the internal structure of the
localized bipolarons. Introducing deformed harmonic oscillator states having
equilibrium positions shifted by (M is the mass of the oscillators), a
bipolaron localized on a site i is represented in terms of the state
where are hard-core boson operators associated with self-trapped electron pairs,
are phonon operators describing local lattice deformations, and are undisplaced
oscillator states. Because of the small overlap of the electron and bipolaron wave functions,
we may, to within a first approximation, consider the boson and fermion operators as
commuting with each other. We are then led to a generalization of the original BFM, in
which only the bosonic component is coupled to the lattice fluctuations. The corresponding
Hamiltonian is
where are fermionic operators describing itinerant electrons with spin and t,
and v denote the bare hopping integral for the electrons, their half bandwidth, the boson
energy level, and the boson–fermion pair-exchange coupling constant, respectively. The
chemical potential is taken to be common to fermions and bosons (with a factor 2 for the
bosons, which are made out of two charge carriers) in order to guarantee charge neutrality
in the system. The indices i denote effective sites corresponding to the molecular clusters
susceptible of local lattice deformations.
As far as the low-energy properties of the above model are concerned, we can consider
the internal degrees of freedom of the bipolarons to be frozen out. This leads us back to the
original BFM, which is recovered upon neglecting the coupling to the phonons
and replacing the boson–fermion coupling constant v by some renormalized value. Our
previous studies of the original BFM permitted us to predict the temperature variation of
the pseudogap and the corresponding manifestations of it in thermodynamic, transport,
and magnetic properties [9]. In order to study the high-energy sector associated with the
Hamiltonian (2), as required for the interpretation of the angle-resolved photo-emission
spectroscopy (ARPES) results, we must deal with the indirect polaronic nature of the
electrons that couple to the phonons only via a charge exchange with the bipolarons.
ARPES has now clearly established that a pseudogap opens up in the normal state
below some characteristic temperature T* that, depending on doping, can be well above the
superconducting transition temperature This pseudogap develops predominantly near
the of the Brillouin zone and is accompanied by a large incoherent background
in the quasi-particle spectrum, extending over a regime in frequency of typically half the
Local Lattice Fluctuations and the Incoherent ARPES Background 247
bandwidth [10]. The pseudogap smoothly evolves into the true superconducting gap upon
either decreasing the temperature below or crossing the boundary of the vortex
cores in the mixed phase at a fixed temperature below This strongly suggests
that the pseudogap and the superconducting gap are of the same origin, both being due
to strong pair fluctuations rather than antiferromagnetic fluctuations. This does not mean,
however, that the pseudogap is due to preformed pairs, what is clearly ruled out by its
non-s-wave symmetry [13]. The presence of preformed pairs is relevant here essentially
because it leads, via charge exchange with the conduction electrons, to pair fluctuations in
the electronic subsystem, which in general are anisotropic in the Brillouin zone. The opening
of the pseudogap is thus due to the fact that, close to the Fermi surface, the single-particle
states are diminished in favor of two-particle states.
It has invariably been observed that in the regions of the Brillouin zone where the
pseudogap opens, ARPES shows the appearance of a strong incoherent background in the
single-particle spectrum. It has been suggested earlier that this feature might come from
a coupling of the electrons to some collective modes such as spin fluctuations [14]. The
explanation for the incoherent spectrum that we shall put forward here is different from
that, and is directly linked to the excitations of the quantum coherent deformed oscillators
in terms of which small bipolarons are described. In an ARPES experiment, such coherent
states are broken up, leading to multiphonon shake-off processes that cover a wide regime of
frequencies. Contrary to the pseudogap phenomenon, the incoherent part of the spectrum,
extending over an energy region of about 0.5 eV, is a high-frequency phenomenon for
which a study of the Hamiltonian in Eq. (2) in the atomic limit makes sense. In this case,
the eigenstates are given by
The above eigenstates of the local BFM problem fully determine its spectral function,
which is given by
decreasing temperatures give rise to an increase in the number of bosons relative to that of
fermions (Fig. 3).
A crucial test whether the incoherent part of the quasi-particle spectrum is indeed
related to phonon shake-off effects would be an independent verification of the existence
of coherent state excitations of polaronic nature. One such possible test could be the inves-
tigation of the temperature dependence of the local intracluster deformations, which can be
measured by EXAFS and pulsed neutron scattering techniques. The measured quantity is
250 Ranninger and Romano
The behavior of the PDF, illustrated in Fig. 4 for different temperatures, is characterized by
a single sharp peak at low T due to the predominance of sites occupied by bipolarons (see
Fig. 3). As T* is approached from below, we observe a distinct splitting of this feature into
two well-separated peaks that, upon further increase of temperature, get broadened such
Local Lattice Fluctuations and the Incoherent ARPES Background 251
that at high temperature, the PDF is again characterized by a single-peak structure, although
now very smeared. The two peak positions characterize the two deformations of the local
lattice environment in which a given site is occupied alternatively by a pair of electrons
or by a bipolaron with comparable proability, as expected on the basis of Fig. 3. Recent
EXAFS [15, 16] and pulsed neutron scattering [17] experiments give direct evidence for
such dynamical local lattice fluctuations.
In conclusion, we have shown that a scenario of fluctuating deformable molecular
clusters, triggering alternate occupation by quasi-free electrons and bipolarons, can account
for the anomalous quasi-particle features of the electrons seen in ARPES. In particular, the
occurrence of a broad incoherent background associated with the formation of a pseudogap
in the normal phase is ascribed to temperature-dependent phonon shake-off effects. An
independent check for the lattice-driven origin of the background is represented by the
behavior of the pair distribution function, which allows for the detection of the existence of
such dynamical local lattice fluctuations. Taking into account long-range Coulomb forces
between the various charge carriers (so far neglected in our studies) would lead to instabilities
such as stripe formation of the deformable clusters. The essential physics reported here,
however, would remain essentially the same, except, of course, for specific momentum-
dependent features, such as those recently examined in experimental studies [18].
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1. C. Taliani et al., in Electronic Properties of Superconductors and Related Compounds, eds. H.
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16. J. Röhler et al., in Superconductivity 1996: Ten Years after the Discovery, E. Kaldis et al., eds.
(Kluwer Academic Publishers, Dordrecht, 1997), p. 469.
17. T. Egami and S. J. L. Billinge, in Physical Properties of High-Temperature Superconductors, ed. V. D. M.
Ginsberg (World Scientific, 1996), p. 265.
18. Z.-X. Shen et al., Science 280, 259 (1998).
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Evidence for Strongly Interacting Electrons
with Collective Modes at
and in the Normal Phase of
Superconductors
1. INTRODUCTION
Anomalous electronic properties of cuprates have been a point of wide discus-
sion in recent years. Angle resolved photoemission (ARPES) is one of the few experimental
tools that has been used to take up this task as the technique has the advantage of being
resolved both in energy and momentum space [1]. Study of the evolution of anomalous
single-particle properties of superconductors could be possible due to availability
1
Unita’ INFM and Dipartimento di Fisica, Università di Roma “La Sapienza” P. A. Moro 2, 00185 Roma, Italy.
2
LURE, Bat 209D Universite Paris-Sud, F-91405 Orsay, France, and LURE and Instituto de Ciencia de Materiales
de Madrid, CSIC, 28049 Madrid, Spain.
3
Superconductivity Research Laboratory, ISTEC, Shinonome 1-10-13 Koto-ku, Tokyo 135, Japan.
253
254 Saini et al.
of better quality samples with a wide range of doping and new developments of infrastructure
for performing ARPES experiments.
There is growing experimental evidence for the breakdown of the one-electron picture
in superconductors. The breakdown occurs for a strongly interacting electron gas
where the Fermi liquid model of quasi-particles is not valid. In this situation, one expects that
the energy indetermination of charge carriers in the conduction band is of the order of the
bandwidth, i.e., in the photoemission spectra, the uncertainty in the energy determination
is of the same order of magnitude as its energy dispersion. The well-known example of
a strongly interacting electron gas, where the Fermi liquid model breaks down, is a one-
dimensional conductor where charge current can flow through collective excitations with
separation of spin and charge modes.
Recent experiments are uncovering the fact that the doped perovskites are complex
materials with segregation of localized and itinerant charge carriers in stripes [2]. It has been
found by combined analysis of EXAFS and diffraction studies [3–7] that super-
conductivity coexists with an incommensurate and anharmonic lattice modulation, forming
stripes in the plane along the diagonal direction. These lattice fluctuations co-
exist with incommensurate spin fluctuations with slow dynamics in the horizontal direction
(the direction of Cu-O-Cu bonds) [8]. Based on the magnetic and charge superstructure peaks
observed in insulating doped nikelates and 1/8 doped cuprates, a model of charge and spin
fluctuating horizontal stripes has been proposed in inelastic neutron scattering experiments
[9].
In the stripe scenario the dimensionality of the system is lower than two due to reduced
hopping between the stripes. Therefore, the Fermi liquid picture for a quasi-particle is not
valid anymore to describe the electronic structure of these materials, that is close to a one-
dimensional Luttinger liquid or a marginal Fermi liquid [10]. To investigate these aspects
we have measured the k distribution of the electronic states near the Fermi level in an energy
range of 50 meV, which is of the order of expected energy cutoff for the interactions involved
in the superconducting pairing mechanism. This is done by an unconventional mode ARPES
based on constant initial state angle scanning [11]. We have selected Bi2212 system at the
optimum doping as representative material due to its good suitability for such measurements.
Because there is not a two-dimensional Fermi surface for this strongly interacting electron
gas, we call the measured constant energy contour marginal Fermi surface map. The result-
ing marginal Fermi surface (MFS) provides a clear identification to the key features related
with the collective excitations in the superconductors. The measurements are com-
bined with conventional mode of ARPES based on energy distribution curves (EDC) [12].
In this contribution, we discuss two main features of the MFS: (1) asymmetric suppres-
sion of spectral weight around the M points and (2) identification of one-dimensional set of
electronic states. We argue that the asymmetric suppression of the spectral weight around
the M points is due to coupling of electrons with dynamical charge fluctuations along the
diagonal direction with a wavevector whereas the one-dimensional set of
states in the direction might be at the origin of the dynamical incommensurate spin
fluctuations [8].
2. EXPERIMENTAL
The ARPES measurements were carried out at the Laboratoire pour l’Utilisation du
Rayonnement Electromagnetique (LURE) (Orsay-France) on the SU6 undulator beamline.
Evidence for Strongly Interacting Electrons with Collective Modes 255
3. RESULTS
In Fig. 1 we plot the well-known and well-established band dispersion (E(k) curves)
in the and directions obtained by plotting the energy positions of the dispersing
peaks of EDC. Similar band dispersions on the Bi2212 system have been measured by many
groups and interpreted in term of dispersing quasi-particles and fitted with band structure
models. The ratio of the width and the binding energy of the peaks (obtained by fitting the
EDC curve with an asymmetric Gaussian function and a Fermi step function) is shown as
insets. The ratio is close to 1 indicating the breakdown of the single particle picture.
The measured EDC are shown in Fig. 2 as a two-dimensional picture, where the inten-
sity is plotted in a two-dimensional plane (E,k) [16]. In the picture, it could be seen that the
EDCs are quite complex. The dispersing maxima (peaks) disappear for
in the and in the direction and the
energy dispersion for is of the order of 300 meV. Apart from a large peak width
that is of the same order of magnitude as the total dispersion, the EDC curves show long
tails extending up to about 0.6 eV energy below the chemical potential This kind of
tail is observed in the one-dimensional ladder systems and interpreted in term of dispersion
of spinons and holons [17]. The shape of these curves indicate directly that the electrons in
the conduction band are strongly interacting.
Figure 3 shows the global view of the MFS of the Bi2212 superconductor. We now
come directly to the point of missing segments of the MFS around the M points. To have a
better view of the suppression of the spectral weight, we enlarged the parts around the M
and and they are shown in Fig. 3. It is clear that the photointensity is suppressed
asymmetrically [16]. The asymmetric topology of the missing segments could be seen
in Fig. 4, which shows measured photointensity along azimuthal curvature from
direction to direction across the direction on the MFS. From this figure, it
256 Saini et al.
is clear that the suppression of spectral weight around the M points is asymmetric. Moving
away from the M direction to the direction we find no evidence of a peak (the
so-called quasi-particle peak), whereas along the direction we clearly observe such
a peak, as shown in Fig. 4b.
Evidence for Strongly Interacting Electrons with Collective Modes 257
258 Saini et al.
After a careful analysis, we identified two well-defined wavevectors on the MFS con-
necting the points where a suppression of the spectral weight occurs. These points in the
first Brillouin zone are and
They are connected by the vectors and as shown in Fig. 5.
These vectors are the wavevectors of collective excitations that suppress the spectral weight
of the quasi-particles at We notice that the wavevector is diagonal and is
the second harmonic of the incommensurate and anharmonic lattice fluctuation; therefore, it
can be associated with diagonal charge density waves CDW. However, the vector is
the antiferromagnetic wavevector and it is the wavevector of SDW that have been observed
in the superconducting phase [18].
We now turn to the next observation. We identified a new set of electronic states with a
one-dimension-like dispersion in the to direction crossing the Fermi
level at one point The observed electronic states are beyond the
expected one electron-like Fermi surface, and we could identify these states by combining
the angle-scanning mode with the conventional EDC mode. Figure 6 compares the energy
distribution curves measured in the (solid line) and (dotted
line) directions at the same k locations. There are dispersing features in both directions;
Evidence for Strongly Interacting Electrons with Collective Modes 259
however, the EDC in the two directions show clear differences in their line shapes. The
main spectral band is clearly visible in both directions for polar angles above 5° off the
point and disperses toward the Fermi energy.
Although the main band appears quite similar in the two orthogonal directions, the
EDC along shows a second dispersive spectral feature at lower angles with a
smaller intensity. This feature crosses the Fermi level at around with a
total energy dispersion However, we do not see this new set of states in the
EDCs measured along the orthogonal direction. The direct spectral differences
are plotted in Fig. 6 (right panel), showing clearly the new set of electronic states. The second
band, appearing only in one direction, was reproducibly observed in different runs performed
on different cleaved surfaces ascertaining its intrinsic nature. Thus this result not only shows
the anisotropy of the MFS, but also provides a direct evidence for a set of electronic states at
260 Saini et al.
the MFS having one-dimensional character along the Cu-O-Cu direction. The differences
in the and were further ascertained by measuring phtotointensity at the
MFS using scanning the polar angle along and The direct
measurement of the photointensity scans at the MFS demonstrated that the observed band
exists only along the direction, whereas it is absent along the direction [19].
ACKNOWLEDGMENTS
The authors are happy to acknowledge stimulating and useful discussions with
A. Perali. This work was partially funded by Istituto Nazionale di Fisica della Materia
(INFM) and Consiglio Nazionale delle Ricerche (CNR) of Italy. The experiment was done
at the Spanish–French beam line of LURE with the support of the large-scale Installation
Program and the Spanish agency DGICYT under grant PB-94-0022-C02-01.
REFERENCES
1. Z.-X. Shen and D. S. Dessau, Phys. Rep. 253, 1 (1995), and references therein.
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A. Bianconi and N. L. Saini [J. Supercond. 10, No. 4 (1997)].
3. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguci, K. Oka, and T. Ito,
Phys. Rev. Lett. 76, 3412 (1996).
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5. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi,
Y. Nishihara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996).
6. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55, 12759
(1997).
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12. The conventional method of Fermi surface measurements is based on measurement of energy distribution
curves (EDC) in all high-symmetry directions of the Brillouin zone and find the locations of the Fermi surface
by following the dispersion of peaks in the EDC (see, e.g., Ref. 1).
11. J. Osterwalder, P. Aebi, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and K. Kadowaki, Appl.
Phys. A 60, 247 (1995); P. Aebi, J. Osterwalder, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku, and
K. Kadowaki, Phys. Rev. Lett. 72, 2757 (1994); M. Lindroos and A. Bansil, Phys. Rev. Lett. 77, 2985 (1996).
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M. Randeria, H. Ding, J. C. Campuzano, and A. F. Bellman, Phys. Rev. B 52, 15107 (1995).
16. N. L. Saini, J. Avila, A. Bianconi, A. Lanzara, M. C. Asensio, S. Tajima, G. D. Gu, and N. Koshizuka, Phys.
Rev. Lett. 79, 3467(1997).
17. C. Kim, Z. X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa, Phys. Rev. B 56,
15589 (1997).
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262 Saini et al.
19. N. L. Saini, J. Avila, M. C. Asensio, S. Tajima, G. D. Gu, N. Koshizuka, A. Lanzara, and A. Bianconi, Phys.
Rev. B 57, R11101 (1998).
20. R. B. Laughlin, Phys. Rev. Lett. 79, 1726 (1997).
21. J. Schmalian, D. Pines, and B. Stojkovic, Phys. Rev. Lett. 80, 3839 (1998).
22. Z. X. Shen and J. R. Schrieffer, Phys. Rev. Lett. 78, 1771 (1997).
23. B. O. Wells, Y. S. Lee, M. A. Kastner, R. J. Christianson, R. J. Birgeneau, K. Yamada, Y. Endoh, and
G. Shirane, Science 277, 1067 (1997) and references therein.
24. A. Bianconi et al., p. 9 in this volume.
Angle-Resolved Photoemission Study of 1D
Chain and Two-Leg Ladder
1. INTRODUCTION
The interacting one-dimensional (1D) system referred as Tomonaga–Luttinger (TL)
liquid shows many anomalous physical properties different from those of a 3D system
(Fermi liquid). It has been predicted theoretically that spin and charge degrees of free-
dom are separated in TL liquid and behave as if they are two independent quasi particles
called spinon and holon [1]. Being stimulated by the discovery of high-temperature su-
perconductivity in 2D cuprates, many theoretical and experimental studies have also been
directed to 1D systems because the spin-charge separation has been regarded as one of
possible driving forces in 2D curprates [2]. Theoretically, possible superconduc-
tivity has been predicted in a quasi-1D spin ladder system with even legs [3] and recently
1
Department of Physics, Tohoku University, Sendai 980-8578, Japan.
2
Department of Physics, Aoyama–Gakuin University, Tokyo 157-0071, Japan.
3
Institute for Solid State Physics, University of Tokyo, Tokyo 106-0032, Japan.
263
264 Sato et al.
2. EXPERIMENT
Single crystals of and were prepared with the traveling solvent
floating zone method. ARPES measurements have been performed at Tohoku University
using a home-built ARPES spectrometer equipped with a He discharging lamp with total
energy and angular resolutions of 100 meV and respectively. The samples were cleaved
and kept at 300 K or 130 K during photoemission measurements.
around These spectral changes suggest the existence of two different bands
near that have different energy dispersions but have a common maximum point closest
to is further increased from to the 1 eV peak at
disperses to higher binding energy and at the same time loses its intensity. In contrast to the
first half of the Brillouin zone, we do not find a steplike structure in the spectrum at the zone
boundary in the second half of Brillouin zone. Instead, the ARPES spectra near
the zone boundary have a weaker and broader structure on the tail from the main peak. We
found that this small structure around is extrinsic because this structure showed a
gradual growth with time. It is noted here that the present experimental result on is
essentially the same as that observed for having double Cu-O chains [8,9], except
for the relative spectral intensity at
In order to map out the “band dispersion,” we took the second derivative of ARPES
spectra in Fig. 2 and plotted the contour map of intensity with gradual shading as a function of
wavevector and binding energy. The result is shown in Fig. 3a. The dark parts correspond to
“bands.” Broken lines are used as a guide to the eyes. Taking the second derivative of spectra
diminishes the effect of background and determines the peak position more accurately. This
method is particularly effective in the present case, in which the strong main peak located
around 2.6 eV has a wide and strong tail spreading to forming a sizable background
to the small structures near All the characteristic features of band dispersions observed
in the raw ARPES spectra are more clearly visible in Fig. 3a. We find again that there are
two dispersive bands in the first half of the Brillouin zone and they have
different energy dispersions with a common maximum point closest to One
of new findings in Fig. 3a is that there is a continuous distribution of finite intensity between
these two bands, which is not clear in the raw ARPES spectra. It is also clear again in Fig. 3a
that one of the two bands with a larger energy dispersion looks symmetric with respect to
but has a much stronger intensity in the second half of the Brillouin zone.
Angle-Resolved Photoemission Study 267
To compare the obtained band dispersion with the theoretical calculation, we show the
spectral function for different momenta k calculated with a 1D Hubbard model at
in Fig. 3b. We find that is not symmetric with respect to
According to the theory, the flat band located at appearing only in the first half of
the Brillouin zone represents the spinon, whereas the symmetric dispersive
band corresponds to the holon. The finite intensity distributing between these two bands in
the first half of the Brillouin zone indicates a mixture of spinon and holon excitations. We
find that all these characteristic features indicative of the spin-charge separation are clearly
observed in the experiment (Fig. 3a). However, the “shadow band” resulting from spin
fluctuation [11,12], which is theoretically predicted to appear in the energy range of
to (not shown in Fig. 3b), was not observed in the present study, probably due to over-
lapping from a large main peak located at 2.6 eV binding energy. In the t-J model with a
finite U, the dispersive feature of spinons and holons is scaled with J (exchange coupling)
and t (hopping parameter), the band width being calculated to be and 2t, respec-
tively [13]. Because the experimental spinon and holon bands show the band width of 0.2–
0.25 eV and 1 eV, respectively, we obtain and although there
remains ambiguity due to the broad feature of bands, in particular for spinon. The obtained
value of J seems consistent with the value (0.13–0.2 eV) reported by the magnetic sus-
ceptibility measurement [14,15], but is smaller than the value (0.26 eV) from the optical
measurement [16]. Although there is no direct experimental estimate of t to be compared
with the present value the ratio t / J obtained in the present study (3–4) is
comparable to the value of cuprate high-temperature superconductors.
at 1 eV, it may originate in the chain because it is expected that the holes are localized on
the chain due to negligible transfer integral (t) between two nearest-neighbor Cu atoms via
nearly 90° Cu-O-Cu bond angle.
Next we compare the present ARPES result of with the t-J calculations
[17–19]. The calculation predicts that the one-particle excitation spectrum at a half filling and
posses a quasi-particle band near which consists of two degenerated
bands due to two Cu-O chains (legs) in the two-leg ladder. The quasi-particle band is roughly
symmetric with respect to where the band has the lowest binding energy and the
spectral intensity of the quasi-particle band is weaker in the second half of the Brillouin zone
than in the first half of the Brillouin zone This prediction
is qualitatively consistent with the present ARPES result in Fig. 5 showing highest energy
at and intensity reduction in the region from to This suggests that the
observed band is a quasi-particle band in which spinon and holon are confined due to the
existence of finite spin gap.
5. COMPARISON OF AND
Finally, we compare the electronic structure between a 1D chain and two-leg ladder. For
we observe two dispersive bands near one exhibits the dispersion only from
and the other is symmetric with respect to (Fig. 3a). Because these features
show a good agreement with the Hubbard model as well as the t-J model calculation they
are ascribed to the spinon and holon bands, respectively. However, for only
one band with the periodicity of ladder sublattice is observed near (Fig. 5). Because
the observed band with ladder origin is qualitatively understood within the result of t-J
calculations, the band is assigned to the quasi-particle band. These observations clearly
illustrate the difference in character of the electronic states between the two compounds.
Electrons in are regarded to form a TL liquid. In contrast, the existence of finite
spin gap in prevents the spin-charge separation and tends to confine the both
into one quasi-particle located near as shown by the exact diagonalization of the t-J
ladder.
6. CONCLUSIONS
We have performed ARPES on and to study the difference in
the electronic structure near between the 1D chain cuprate and the two-leg ladder. We
observed two dispersive bands near ascribable to spinon and holon due to the spin-
charge separation in whereas we found only one dispersive band in
that is ascribed to a quasi-particle in which spin and charge freedoms are confined. These
results indicate a clear difference in the electronic structure between two different quasi-1D
cuprates.
ACKNOWLEDGMENT
This work was supported by grants from Core Research for Evolutional Science and
Technology Corporation (CREST) and the Ministry of Education, Science and Culture of
Japan.
270 Sato et al.
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1. S. Tomonaga, Prog. Theor. Phys. 5, 349 (1950); J. M Luttinger, J. Math. Phys. 4, 1154 (1963).
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Optical Study of Spin/Charge Stripe Order
Phase in
1. INTRODUCTION
The spin and charge stripe-ordering discovered by neutron scattering [1,2] has become
a hot issue in the research of doped antiferromagnetic (AF) Mott insulators, such as Mn-,
Ni-, and Cu-oxides. Although there are some common properties in these perovskite oxides,
the nature of stripes in the cuprates seems to be considerably different from that in the other
two materials. For example, in resistivity in the static ordered
state is not large [2], whereas in the manganites and nickelates it becomes extremely large
1
Superconductivity Research Laboratory, ISTEC, Tokyo 135-0062, Japan.
2
Dept. of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan.
3
Dept. of Basic Science, The University of Tokyo, Tokyo 153-0041, Japan.
271
272 Tajima et al.
below the phase transition temperature [3]. Therefore, it is of great importance to investigate
the charge dynamics of stripe-ordered state peculiar to the curates. However, so far there
has been little direct study of charge excitation in LNSC, although a charge ordering was
observed indirectly as a lattice distortion in the neutron diffraction experiments [4].
In this work, we investigated charge dynamics of LNSC by measuring optical reflec-
tivity spectra and microwave conductivity. Comparing the spectra of LNSC and Nd-free
we found a dramatic effect of Nd substitution on the in-plane optical
conductivity at room temperature. In this high-temperature phase, a strong stripe fluctuation
seems to exist, and T- and -dependencies of optical conductivity are metallic, which per-
sists even below the phase transition temperature. In the superconducting state, a substantial
reduction of superfluid density and/or reduction of Josephson coupling was observed in the
out-of-plane spectra of Nd substituted crystals.
2. EXPERIMENT
Large single crystals of and with various Sr
contents (x) were grown by a traveling solvent floating zone method. The measurement
samples were cut out along the c axis from as grown crystals. The ac surfaces were polished
by using powder for optical measurements. LNSC crystals show the phase transition
at from the low-T orthorhombic (LTO) to the low-T tetragonal (LTT)
phase. In the LTT phase, the static stripe order has been observed by the neutron diffraction
measurements [4].
Optical reflectivity spectra were measured by using a standard FTIR spectrome-
ter for and grating-type spectrometer for
with the light polarization and The samples were mounted in a
He-gas flow-type cryostat together with Au-evaporated mirror, which enables us to mea-
sure reflectivity accurately from 300 to 6 K. Conductivity in the microwave region was
determined by measuring surface impedance of the samples in a microwave cavity.
the observed spectral change is a result of strong fluctuation of the stripe order far above
the static ordering temperature. The reduced conductivity by the Nd substitution up to
high of order of 1 eV is reminiscent of the effect of lowering the doping level. This
may give a support for the 1D charge stripe model proposed by Tranquada et al. [4].
Assuming that a stripe direction rotates 90° in the adjacent layers, an optical spectrum is
expected to be a mixture of the two components. One is a metallic spectrum along the
stripes and the other is a presumably nonmetallic component perpendicular to the stripes.
If the carriers could not hop between stripes or the spectrum were a completely insulating
one for perpendicular polarization like that for the resulting spectrum would have
shown much lower reflectivity or conductivity for to Therefore, the
present result indicates that the hopping between stripes is not necessarily prohibited, or
that carriers with 2D character may coexist with 1D carriers confined to the stripes.
Next, temperature dependence of the reflectivity spectrum is shown for LNSC with
in Fig. 2. Because of the strong stripe fluctuation effect at high temperatures,
there is no dramatic effect on the spectrum when we cool the sample across the phase
transition temperature With lowering temperature, the far-infrared reflectivity increases
monotonously, which results from a reduction of the carrier damping. This metallic
temperature dependence as well as the metallic dependence of reflectivity does not change
274 Tajima et al.
remarkably, even below This is in contrast to the dc resistivity behavior, which shows
a small jump at and a following upturn, reflecting a static charge ordering.
In order to know at which frequency the crossover from a semiconducting to a metallic
behavior takes place, we investigated the in-plane conductivity in the microwave region by
measuring surface impedance. Figure 3 shows the temperature dependence of resistivity
for LNSC at 50 GHz, that was calculated from the surface impedance. The observed tem-
perature dependence is in agreement with the dc behavior. We also measured the resistivity
at 100 GHz Its T dependence is qualitatively the same as the result in Fig. 3.
Therefore, we can expect a crossover at frequency between 3 and
One of the plausible explanations for the difference between the low- and the
behaviors is that the charge stripes are not ideally long straight lines but are broken into
short segments due to defects or something else. In such a case, the low conductivity
would be affected by the disorder, whereas conductivity becomes metallic at frequencies
higher than the hopping energy between the segments.
Optical Study of Spin/Charge Stripe Order Phase 275
4. OUT-OF-PLANE SPECTRA
As is similar to the spectra of the Nd-free LSC, the c axis spectrum of LNSC is
dominated by phonons. As temperature decreases, the phonon peaks become sharp, but
there is no remarkable change at except for an appearance of a small peak around
This is in a sharp contrast to another typical stripe phase in where
a clear phonon split due to the lattice distortion is observed [6]. In the case of LNSC, the
lattice distortion accompanied by the charge ordering is very small.
As is well known, below a sharp plasma edge for the superconducting carriers
appears in the c axis spectrum of LSC, which indicates that a superconducting gap energy is
larger than a screened plasma energy [7]. Because the temperature limit in our measurement
system is about 6 K, we could not examine the superconducting state in LNSC for
and 0.12 with lower than 4 K. For and 0.20, no reflectivity edge was observed,
even at the lowest temperature (Fig. 4). The values are 12 K and 16 K for
and 0.20, respectively. There are some possible explanations for this phenomenon. One
possibility is that a plasma frequency becomes lower than our frequency limit
276 Tajima et al.
5. SUMMARY
The optical conductivity and the microwave conductivity of LNSC with various Sr
contents have been investigated, compared with the conductivity of Nd-free LSC. For
the Nd substitution substantially suppresses conductivity spectral weight below 1 eV at all
measurement temperatures. This is suggestive of a strong fluctuation of the stripe order that
makes the CuO plane insulating for polarization perpendicular to the stripes, resulting in a
relative enhancement of the mid-infrared absorption. Compared with the spectra of other
typical materials, such as and with the same doping
level, the conductivity spectrum of LSC is enhanced in the mid-infrared region, as shown
in Fig. 5. It may indicate that even the Nd-free LSC is affected by the stripe fluctuation,
which could be an origin of the disordered nature of the electronic state in LSC.
The and the T-dependencies of conductivity are dominated by the metallic com-
ponent, which presumably corresponds to the conductivity along the stripes. Being incon-
sistent with the behavior in the microwave region and at dc, the metallic behavior at
Optical Study of Spin/Charge Stripe Order Phase 277
high remains even below This supports a segment model of charge stripes in which
the disorder affects the charge dynamics up to the frequency of the order of millimeter
wavelength.
In the c axis spectra, a striking effect of the Nd substitution on the superconducting
plasma is observed, namely, a sharp reflectivity edge for the superconducting plasma disap-
pears in the studied LNSC crystals. It indicates that the stripe-order fluctuation substantially
reduces the number of superconducting carriers and/or the interlayer Josephson coupling.
This can be understood if the stripe order destroys the superconducting phase coherence
between the layers.
ACKNOWLEDGMENT
This work was partially supported by NEDO for the Research and Development of
Industrial Science and Technology Frontier Program.
REFERENCES
1. C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993); J. M. Tranquuda et al., Phys.
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5. J. Tranquada et al., provate communication.
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Vibrational Pseudo-Diffusive Motion of the
Oxygen Octahedra in from
Anelastic and Quadrupolar Relaxation
1. INTRODUCTION
Evidence is accumulating that the local structure of many, if not all, the cuprate super-
conductors differs from the average structure that is extracted from traditional diffraction
experiments. In some cases, the local inhomogeneities can be put in close relationship with
the separation of the carriers into antiferromagnetic insulating domains and conducting
domains, often observed as parallel stripes [1]. In spite of the extensive investigation on
this subject, it is only recently that the issue of the dynamics of such inhomogeneities has
been addressed by showing that the anelastic relaxation spectrum of nearly stoichiomet-
ric exhibits intense peaks of thermally activated type that are due to intrinsic
1
CNR, Area di Ricerca di Tor Vergata, Istituto di Acustica “O.M. Corbino,” Via del Fosso del Cavaliere 100,
I-00133 Roma, and INFM, Italy.
2
Universidade Estadual Paulista, Departamento de Fisica, 17.033-360, Bauru, SP, Brazil.
3
Università di Roma “La Sapienza,” Dipartimento di Fisica, P.le A. Moro 2, I-00185 Roma, and INFM, Italy.
4
Dipartimento di Fisica “A. Volta,” Università di Pavia, Pavia, and INFM, Italy.
fluctuations of the lattice [2]. Here, we present combined anelastic and NQR relaxation
measurements of undoped which we interpret in terms of the collective dynam-
ics of the tilts of the octahedra in the LTO phase.
where x is the atomic fraction of relaxing entities, each producing a change of the strain
when its state changes; is times the measuring frequency. The curve has a maximum
at the temperature where so allowing the determination of the microscopic
relaxation rate at the peak temperature.
Figure 1 shows two spectra of sample 2 in the as-prepared state
and after extracting excess oxygen by annealing in vacuum at 750°C for 1 hr. All the peaks
are shifted to higher temperature at the higher frequency, indicating that they are due to
thermally activated relaxation processes with decreasing with temperature. The peaks
in the as-prepared state are due to the diffusive hopping of interstitial oxygen atoms (peak
O 1 with and activation energy of 5600 K) and to pairs or other complexes of excess oxygen
(peak O2) [2]. The outgassing treatment apparently did not completely remove the excess
oxygen because a trace of peak O1 is still present both in the anelastic and NQR spectra; from
its residual intensity, we estimate that has been lowered of at least 20 times, and therefore
In this condition, the curve is not flat, as expected from a defect-free
stoichiometric lattice, but develops an intense peak with an apparent activation energy of
2800 K, labeled T, which we attribute to relaxational dynamics of the octahedra [2].
In the same sample 2, relaxation measurements have been carried out, and
from the recovery plots of the NQR echo signals at and
the relaxation mechanism has been identified [3,4]. It was found that, at least
for the relaxation mechanism is quadrupolar, namely due to the time-dependent
electric field gradient at the La site (Fig. 2a, inset).
Two main contributions to the quadrupolar relaxation are present. One
contribution, corresponding to peaks O1 and O2, is due to the diffusion of the extra
stoichiometric oxygen [5], and for small it is sizeable only for
Vibrational Pseudo-Diffusive Motion 281
For relatively large (sample 5 oxygenated by slowly cooling from 620°C to 310°C
in 820 torr O2), the contribution from the diffusion of the oxygen becomes dominant
also at T 200 K (Fig. 2a). The relaxation mechanism related to oxygen diffusion is
discussed elsewhere [6]. Here, we focus our attention only on the results that refer to
the contribution from the phonon-like motions of the oxygen octahedra (T < 400 K) for
sample 2.
One should remark that, for harmonic local potential, namely underdamped phonon
modes, Eq. (2) would give a negligible contribution to the relaxation. For purely phonon
modes in fact, only the second-order Raman process, with no maxima as a function of
T and no frequency dependence in the relaxation rate, would be present [7]. The mere
observation of the maxima in (Fig. 3) and in the anelastic relaxation spectrum
implies that a strongly anharmonic local potential characterizes the oxygen motions. Such
motions can be described in terms of a 1D model of interacting atoms in a potential of
Vibrational Pseudo-Diffusive Motion 283
where is of the order of the distance between the two minima in the double well local
potential.
The imaginary part of the elastic compliance can also be expressed in terms of
the strain correlation function
V being the sample volume is inversely proportional to the volume over which
it is averaged). The strain is directly related to the displacements s (t) of the ions
284 Cordero et al.
of the octahedra, although not in a trivial way, and therefore Eq. (4) differs from that
for the NQR relaxation rate, Eq. (2) below, only for a factor and the constants expressing
the dependence of strain and NQR frequency on the atomic displacements [compare also
Eqs. (3) and (1)].
The interstitial oxygen atoms prevent the relaxational dynamics of the neighbour-
ing octahedra, as demonstrated by the fact that the anelastic relaxation peak T is readily
suppressed by the introduction of small amounts of excess oxygen (Fig. 1 and Ref. [2]).
Therefore, the clusters of octahedra free to build up the cooperative relaxational dynamics
are limited by the interstitial oxygen atoms, and this implies in turn that the cluster average
of the interaction strength depends on the cluster size and shape. For this reason, we
distributed . according to a gaussian, which results in a distribution of effective energy
barriers (the effect on was included but is negligible). A feature of the anelastic peak
that cannot be accounted for by the above formulas is the increase of its height at higher
temperature instead of a decrease as 1/T. Such a temperature dependence is observed in
the case of relaxation among states that differ in energy by it can be shown [11]
that the relaxation strength between the two states 1 and 2 differing in energy by must
be multiplied by a factor containing the product of their equilibrium occupation numbers,
In addition, by writing the rate equation for the relaxation
between the inequivalent states, one obtains
Although the correction to the rate does not affect much the relaxation curves, the cor-
rection to the intensity produces a maximum at in the relaxation strength,
which again falls off as at higher temperature. These corrections are valid for relaxation
between definite levels without cooperative effects, and their extension to the above model
of cluster dynamics is not obvious; nonetheless, the temperature dependence of the anelastic
relaxation strength imposes the consideration of relaxation occurring among states that are
somehow energetically inequivalent.
Figure 3 presents a fit of both the anelastic and NQR relaxation curves with the above
expressions, namely,
The values of the potential parameters were chosen in order to obtain a single particle energy
barrier with (as theoretically estimated from a self-consistent ana-
lysis of the temperature dependence of the soft mode and of the elastic constant [12]). The
mean value of the coupling constant was and its distribution width was
and 0.25 for the NQR and anelastic data (a temperature-dependent width may result from
the ordering of interstitial O); such values of result in a mean effective energy barrier
with a distribution width of The mean value of is
The asymmetry energy is 10 times smaller than and therefore does not
change the overall picture much.
Vibrational Pseudo-Diffusive Motion 285
REFERENCES
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(1998).
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1 1 . F. Cordero, Phys. Rev. B 47, 7674 (1993).
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512(1991).
13. M. Corti, A. Rigamonti, F. Tabak, P. Carretta, F. Licci, and L. Raffo, Phys. Rev. B 52, 4226 (1995).
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B 55, 9120(1997).
16. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito,
Phys. Rev. Lett. 76, 3412 (1996).
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Charge and Spin Dynamics of Cu-O Chains in
Cuprates An
NMR/NQR Study
Spin and charge fluctuations are investigated along the three lattice directions
on the Cu(1) site in a nonsuperconducting sample using
NMR/NQR. Our data give evidence for charge instability (CDW) involving charge
fluctuations along the O(4)-Cu(l)-O(4) apical axis. In view of these findings, we
reanalyzed the situation of chains and planes in using Cu( 1) and Cu(2)
NQR. We confirm the important role of charge/lattice effects in these cuprates
and we discuss the consequences of these mechanisms on the interaction between
Cu-O chains and planes.
1. INTRODUCTION
There is an increasing interest in the physics of Cu-O chains in rare earth (RE)-123
systems, as they may play an important role in the normal and superconducting states of these
compounds. Although many NMR results are available for the properties ofthe planes,
little attention has been paid to the intermediate CuO layers. The study of the intrinsic physics
of these chains is not straightforward, as they are surrounded by superconducting
planes in most of the RE-123 cuprates. An exception occurs with the insulating compound
in which antiferromagnetic order in the planes is present below 285 K.
The holes are localized in these planes [1], whereas the chains are expected to remain metallic
on a local scale as in In a previous study [2], we show that a charge instability
occurred at 120 K preceded by charge fluctuations in the temperature range 120–180 K. The
1
Laboratoire de Spectrométrie Physique UMR 5588 CNRS, Université Joseph-Fourier Grenoble-1, B.P. 87, 38402
Saint Martin d’Hères, France.
2
Laboratoire Léon Brillouin, CEA-Saclay, 91191 Gif sur Yvette, France.
3
LPS URA2 CNRS, Université Paris Sud, 91405 Orsay Cedex, France.
key problems, therefore, are the origin of this transition, the range of the order parameter,
and the possible interaction with the planes when they are superconducting, as in
In order to give some insight into these fundamental questions, we present a
comparative NMR/NQR study of the copper sites in these two compounds.
the relaxation between 200 K and 120 K. Below this temperature, a gap
opens in the low energy excitation. Furthermore, no discontinuity occurs
between 200 K and 100 K on the spin part of the magnetic hyperfine shift measured on
Cu(1), as shown in Fig. 3c. All these results clearly indicate that charge fluctuations are
involved in the mechanisms of this transition.
2.2 Analysis and Discussion
One of the key problems raised by our results concerns the nature of the mechanisms
responsible for the charge instability observed at The comparison between the
relaxation rates measured in NQR and NMR with parallel to the c axis strongly suggests
that the relaxation mechanism involves charge fluctuations along the c axis. Indeed NMR
probes only fluctuations perpendicular to the c axis, whereas NQR—in the particular
case of the Cu(1) site [where the EFG is asymmetric ]—is sensitive to thefluctuations
in the three directions [3,4]. This accounts very well for the stronger critical fluctuations
observed in NQR.
The temperature dependence of the NMR spin–lattice relaxation rate suggests a si-
multaneous condensation of charge and spin degrees of freedom in the same energy gap
below a temperature that is quite high. This indicates that the electronic correlations are in a
moderately strong regime (i.e., no spin–charge separation). Another feature of the transition
is revealed by the large value These two points are in favor of a CDW driven
by a rather strong electron–phonon coupling.
The change in the NQR lineshape between high and low temperatures is well simulated
with the hypothesis of an EFG modulation induced by a CDW transition. Details of this
simulation are the subject of a later paper.
A last question concerning the range of the low-temperature order (long range or short
range) is beyond the possibility of our NMR/NQR investigations alone. At this stage, it
290 Grévin, Berthier, Collin, and Mendels
should be noted that our results agree quite well with some aspects of the microscopic
model proposed by Fehrenbacher [5]. Following this author, a short-range CDW order is
expected if one assumes polaron–polaron interactions in the chains. This assumption is
consistent with our preceding conclusion.
We also emphasize that experiments on Y-123 and Y-124 compounds have shown
the absence of long-range order in the chains at low temperatures [6,7]. In particular, pair
distribution function (PDF) analysis of pulsed neutron data in Y-124 [8] has been inter-
preted in terms of polarized microdomains linked to the pseudogap phenomena observed
in underdoped cuprates.
is still controversial [11,12]. This issue, which implies a lattice effect accompanying the
superconducting transition, underlines the consequences of such a result. We have revisited
this crucial issue by making accurate measurements.
measured on the two copper isotopes showed that, for temperatures between 300 and 170 K,
charge fluctuations partially contribute to the relaxation of Cu(1) site.
In addition to the overall temperature dependence of on the Cu(1) site, new
features come out from our measurement both in the normal state and near the critical
temperature. A marked discontinuity occurs at 240 K and a crossover starts below 170 K. A
linear T dependence is observed between 80 and 140 K and, contrary to previous reports,
the opening of the low-temperature gap seems to occur below This important point is
discussed in detail in a forthcoming paper.
CDW order parameter. Such an ordered state is possible only below the superconducting
transition, as the plane carriers condense in the superconducting gap. In this context, the
crossovers detected below 170 K and 180 K in and respectively,
could be attributed to short-range CDW fluctuations in the chains. In this case, long-range
CDW order should be present at low temperatures. Such a situation does not appear clearly
from experimental results, as PDF and tunneling microscopy (STM) [6,7] only indicate
short-range correlation, even at low temperatures.
The presence of a crossover in the underdoped compound as well as in
overdoped place a severe constraint on the models assuming a link between this
mechanism and the pseudogap phenomenon. If we assume that the crossover in
is effectively connected to the pseudogap, can we imagine the same link in
It is important to point out that the existence of a pseudogap at in overdoped
opening at very near is still controversial. Moreover, in Bi-2212 sys-
regime. Furthermore, the oxygen order in the chains strongly affects in underdoped
4. CONCLUSION
In summary, our NMR/NQR study has shown that charge fluctuations along the O(4)-
Cu(1)-O(4) axis play an important role in the physics of chains in
cuprates
The comparison between and clearly demonstrates that a
complex interaction exists between chains and planes in in the normal state
and at In particular, the dips observed on the quadrupolar frequencies on Cu(1)
and Cu(2) give some arguments for charge/lattice effects occurring above and at the super-
conducting transition.
REFERENCES
1. R. Fehrenbacher and T. M. Rice, Phys. Rev. Lett. 70, 3471 (1993).
2. B. Grévin, Y. Berthier, G. Collin, and P. Mendels, Phys. Rev. Lett. 80, 2405 (1998).
3. J. Chepin and J. H. Ross, J. Phys.: Cond. Matt. 3, 8103 (1991).
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Solids (1978).
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7. H. L. Edwards et al., Phys. Rev. Lett. 75, 1387 (1995).
8. T. R. Sendyka et al., Phys. Rev. B 51, 6747 (1995).
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10. M. Mali et al., Phys. Lett. A 124, 112 (1987).
11. D. Brinkmann, Appl. Magn. Res. 3, 483 (1992).
12. H. Riesmeier, S. Gärtner, V. Müller, and K. Lüders, Appl. Magn. Reson. 3, 641 (1992).
13. T. Imai et al., J. Phys. Soc. Jpn. 57, 2280 (1988).
14. A. Suter et al., Phys. Rev. B 56, 5542 (1997).
15. I. Eremin et al., Phys. Rev. B 56, 11305 (1997).
16. P. Hertel, J. Appel, and J. C. Swihart, Phys. Rev. B 39, 6708 (1989).
17. Ch. Renner et al., Phys. Rev. Lett. 80, 149(1998).
18. P. Carretta, Physica C 292, 286 (1997).
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Mobile Antiphase Domains in Lightly Doped
Lanthanum Cuprate
Light hole doping of lanthanum cuprate strongly suppresses the onset of antiferro-
magnetic (AF) order. Surprisingly, it simultaneously suppresses the extrapolated
zero temperature sublattice magnetization. results in-lightly doped
demonstrate that these effects are independent of the details of
the mobility of the added holes. We propose a model in which doped holes phase
separate into charged domain walls that surround “antiphase” domains. These do-
mains are mobile down to at which point they either become pinned to the
lattice or evaporate as their constituent holes become pinned to dopant impurities.
1. INTRODUCTION
A fundamental issue in the normal state of the superconducting cuprates is the behavior
of holes doped into a two-dimensional (2D) lattice of spins with strong antiferromagnetic
(AF) interactions. Even for lightly doped, single-layer lanthanum cuprate many important
issues remain poorly understood. Long-range AF order occurs at in undoped
lanthanum cuprate, but is rapidly suppressed by the addition of a small density, p of
holes per Cu. This rapid suppression is clearly related to the disruptive effects of mobile
holes: is sufficient to suppress to zero, whereas isovalent substitution of
Zn or Mg for Cu is required [1] to produce the same effect. A range of studies [2] including
measurements [3] in lightly doped have demonstrated that
the suppression of and in fact, all the magnetic properties of lightly doped lanthanum
cuprate are essentially invariant without regard for the means of hole doping and consequent
variations in hole mobility.
It is unlikely that a collection of individual holes can lead to magnetic behavior that
is entirely independent of compositional variation that leads to substantial variations in
1
Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545.
2
National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306.
resistivity (at constant doping). We argue, instead, that this is strong evidence that holes
form collective structures. An important and well-documented aspect of doped cuprates is
their tendency toward inhomogeneous charge distribution [4]. Segregation of doped holes
into charged stripes separating hole-free domains has been predicted [5–11] and recently
observed directly in lanthanum cuprate [12]. It was proposed earlier that phase segregation of
holes could be responsible for the unusual magnetic properties of lightly Sr-doped lanthanum
cuprate [13–15]. We make a related proposal that holes form charged domain walls that
form closed loops with the important differences that these walls form antiphase domain
walls (so the phase of the AF order inside these domains is reversed) and that the walls and
hence the enclosed domains are mobile, and the charged walls have the density of 1 hole
per 2 Cu sites in agreement with neutron scattering results [12]. The antiphase character
means that mobile (above 30 K) domains suppress the time-averaged static moment, thus
suppressing as well as These domain structures have contrasting interactions with
in-plane vs. out-of-plane dopants (e.g., stronger scattering by in-plane impurities), which
explains the different transport behaviors, whereas the universal magnetic properties can
be understood as long as the domains are sufficiently mobile that they move across a given
site rapidly compared to a measurement time.
is much reduced compared to the Sr-doping case [3]. Comparing La2–ySryCuO4 (LSCO)
and at one finds that the room temperature
resistivity of LCLO [2,17] exceeds that of LSCO [18] by over an order of magnitude.
Furthermore, unlike LSCO, the resistivity of LCLO always increases monotonically with
decreasing temperature. With increasing doping, the contrast becomes more dramatic as
LSCO becomes metallic and superconducting whereas LCLO becomes ever more insulating
with doping above
In spite of this, we find that the magnetic behavior of the two materials is essentially
identical [3]. In addition to the similarly strong suppression of by doping [2], we find
that is also suppressed, and the correspondence between the suppression of and
by doping is identical to that observed in LSCO [15]. In Fig. 2, for both
LCLO [3] and LSCO [15] is plotted against Here, is the value of
obtained by extrapolating the data for i.e., the value of
298 Hammel, Suh, Sarrao, and Fisk
the solid lines shown in Fig. 1a. The solid line through the data is due to a theory of Neto and
Hone [16] (see also van Duin and Zaanen [19]). The strong peak in 2W occurs at the same
temperature and exhibits the same binding energy (as extracted from the T dependence
on the high temperature side of the peak) [3], Finally, the temperature dependence of
the low-energy dynamical susceptibility [obtained from measurements of 2W(T)] exhibits
the same finite-size effects [3] as were observed in the static susceptibility by Cho et al. [13].
Scalapino [29] observed that charged domain walls form loops. They point out that this
is favorable at low doping in the case in which the coupling between planes is significant.
Because the walls constitute antiphase domain walls, the coupling between two planes is
disrupted, in general, by domain walls. Hence, interplane coupling would favor domain
walls forming closed loops so that most of each plane would be in the dominant AF phase.
In the event that these antiphase domains are mobile, passage of such a domain over a
given site reverses the orientation of a particular ordered Cu moment. The splitting of the
line (shown in Fig. 1) is proportional to local hyperfine field due to the ordered
moment on the neighboring Cu site. If this moment is time varying, the splitting will be
proportional to the time-averaged local moment. In the absence of antiphase domains, the
hyperfine field will be constant, giving the value of observed in undoped lanthanum
cuprate. If the motion of the antiphase domains is rapid compared to the NQR measurement
time, the net local hyperfine field will be proportional to the fraction of time the moment is
in the dominant AF phase minus the time it is in an antiphase domain, and hence proportional
to the area of the dominant phase minus the area of the antiphase domain. We can estimate
the doping dependence of the size and spacing of the antiphase domains from the known
behavior of If we define the data [15]
for is well described by with For simplicity, we assume that a (1,0) or
(0,1) domain wall orientation is preferred, and so consider square domains.
If a region of size L contains, on average, one antiphase domain of size l (Fig. 3; all
lengths are in units of the lattice parameter), then
and is the number of sites in the dominant AF phase. The number of holes in the region
of size L is the domain wall that bounds the antiphase domain contains 1 hole per
2 Cu sites, so From Eq. 1,
and
the antiphase domains becomes slow compared to the NQR time scale time
averaging of the reversed spin directions ceases and the full ordered moment is observed.
This could arise either from pinning of the antiphase domain to the lattice or evaporation
of the domain walls due pinning of the constituent holes to the charged donor impurities;
in either case, the coincidence of the recovery of and the freezing of spin degrees of
freedom evidenced by the low T peak in 2 W is naturally explained. The correspondence
between suppression of and is natural in this case because interlayer coupling is
hampered wherever an antiphase domain is present, thus impeding the development 3D AF
ordering (see the discussion in Ref. [29] in this regard).
This model also explains the finite-size effects revealed by the susceptibility analysis
of Cho et al. [13] if we consider that the appropriate length scale between domain walls
is The variation of with p is shown in Fig. 4b and compared with the
variation of the square of the characteristic length scale obtained by Cho et al. [13] (scaled
vertically to obtain the best agreement). Finally, we note from Fig. 4a that L and l converge
with increasing p, and we expect that loops will cease to be stable when L approaches l.
For the parameterization of R(p), we have chosen, when near
the doping at which the metal-insulator transition and spin–glass behavior are found. We
speculate, then, that these are related to the transition in the configuration of the charged
domain walls from loops to parallel stripes.
In conclusion, we presented a model that explains the range of unusual magnetic
phenomena observed in lightly doped lanthanum cuprate. In particular, we can understand
the insensitivity of magnetic properties to materials variations that substantially increase the
resistivity. This indicates that mobile antiphase domains play a central role in determining
the magnetic properties of lightly doped lanthanum cuprate. It may point to an explanation
of the poorly understood “spin–glass” regime of the phase diagram in terms of a crossover
in domain wall topology from loops to parallel stripes. More generally, it suggests that the
development of stripe order may play a determining role in the phase diagram of the cuprates
(see, e.g., Ref. [30]). Rather than requiring mobile domain walls, superconductivity may
more sensitively depend on the nature of the ordering of the walls into parallel stripes.
ACKNOWLEDGMENTS
We gratefully acknowledge stimulating conversations with J. Zaanen, who suggested
the idea behind the model presented here. Work at Los Alamos was performed under the
auspices of the U.S. Department of Energy. The NHMFL is supported by the NSF and the
state of Florida through cooperative agreement DMR 95-27035.
REFERENCES
1. S. W. Cheong et al., Phys. Rev. B 44, 9739 (1991).
2. J. L. Sarrao et al., Phys. Rev. B 54, 12014 (1996).
3. B. J. Suh et al., cond-mat/9804200 (unpublished).
4. Proceedings of the Workshop on Phase Separation in Cuprate Superconductors, edited by K. A. Müller and
G. Benedek (World Scientific, Singapore, 1993).
5. J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989).
6. D. Poilblanc and T. M. Rice, Phys. Rev. B 39, 9749 (1989).
7. H. J. Schulz, J. Phys. (Paris) 50, 2833 (1989).
8. V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990).
302 Hammel, Suh, Sarrao, and Fisk
By using NQR and new NMR methods, we isolated the A and B site signals in
for at 300 K. This enabled us to measure lineshapes,
relaxation rates, and the magnetic shifts separately for the Cu A and B sites.
Trapped holes cause a substantial magnetic linewidth that is experienced by both
Cu sites in a similar way. An axially symmetric lattice modulation is responsible
for the quadrupolar broadening, but also affects both sites. The B sites do not
cluster, and differ from the A sites mainly by a small change in the quadrupole
frequency. Combining our results with literature data, we conclude that the B
sites represent regular Cu sites in slightly contracted octahedra.
1. INTRODUCTION
The understanding of the normal state properties of the Sr or O doped
compounds remains a challenge. More evidence accumulates [1–4] that local deviations
from the average structure may be of importance in understanding the properties of these
materials. In this context, the structural assignment of the secondary Cu site, which was
detected some years ago with NQR [5,6], must be accomplished. This so-called B site
appears on doping of by Sr or Ba. A similar site also occurs for O doping.
NMR and NQR studies [7,8] on revealed that the number of B-site Cu
atoms increases approximately linearly with the doping level x above Various
models have been used to explain the B site in connection with particular Sr environments
1
Department of Physics and Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana,
Illinois 61801-3080.
2
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439.
3
Author to whom correspondence should be addressed: Charles P. Slichter, Department of Physics, University of
Illinois at Urbana–Champaign, 1110 W. Green Street, Urbana, IL 61801-3080; Tel.: (217) 333 3834; Fax: (217)
333-9819; email: cps@physics.uiuc.edu.
near the Cu atoms. Such approaches were questioned by Hammel et al. [9], who argued that
because the oxygen-doped materials also show a similar B line, it might be a consequence
of the doped holes rather than the dopant itself. This idea is supported by quantum chemical
cluster calculations [10]. In order to account for the observed number of B sites, which is
much larger for the O-doped material, Hammel [1] suggests that holes become localized
at Cu sites by a local charge of two (one doped O, or two doped Sr in close proximity).
He proposes that one O or two neighboring Sr localize one hole, which stays centered on
a certain copper site and its four neighbor oxygens. The NMR of this central Cu has not
yet been observed. Hammel assigns the B site to the 4 Cu sites next to the trapped hole.
With this model, one can explain the experimental intensities for the B site. For a random
Sr distribution on La sites and for this results in 0.02 localized holes per Cu.
3. RESULTS
Our ordinary Cu NQR line positions and widths (data not shown) agree with
reported literature data [8,14,15]; decomposing the spectra with 4 Gaussian lines (two
isotopes and two sites), we find for for the A site and 38.3 MHz for the
B site. The linewidths are about 1.9 MHz for the A site and 1.5 MHz for the B site. The
intensity of the B line is about of the total intensity.
The ordinary NMR spectrum of the central transition for both isotopes and alignments
consists of overlapped A and B lines, and yields for both sites an equal magnetic shift of
1.23% for and 0.70% for (c is the crystal c axis and B the external magnetic field).
The NMR satellite transition spectrum for Fig. 1a, was recorded and fitted
with 4 Gaussians using the measured magnetic shifts,
with the two widths and the B site intensity as variables.
The fit results are and
Because there is a slight discrepancy between the NMR and NQR linewidths for both
lines, we used our new technique to transfer spins from the central transition into the
satellite transitions. With the central transition transfer pulse fixed, we recorded the satellite
transition lineshape. The results, single isotope satellite lineshapes, are shown in Fig. 1b.
On the Structure of the Cu B Site 305
For we fit linewidths of 2.0 MHz and 1.7 MHz for the A and B line, and conclude
that NQR and NMR linewidths agree quite well. The isotopic dependence of the satellite
linewidths shows that it arises from a spread in the electric field gradient. Comparison of
Fig. 1a with Fig. 1b shows that the low frequency tail as well as part of the linewidths are
due to misaligned grains.
NMR lineshapes for are extremely sensitive to the size of the asymmetry param-
eter because for this orientation the crystal a and b axis are at random angles with respect
to the magnetic field. The 300 K data are shown in Fig. 2. We estimate for both sites that
Thus, both the A and the B sites are axially symmetric about the c axis.
From the values for and we expect for very small linewidths of the central
transitions; however, experimentally one observes a rather large total width [16]. To clarify
this point, we again used our new technique for the A and B site separation in the central
transition: The central transitions of the different isotopes are well separated, whereas for
a given isotope the central transitions of A and B are not resolved. However, because the
satellites of the two sites A and B of a given isotope do not overlap (cf. Fig. 1) we can
306 Haase, Stern, Hinks, and Slichter
transfer spins from and separately into the central transition, and thus obtain
their NMR parameters one at a time. Holding the transfer pulse at fixed frequency at the
satellite, we plotted out the central transition lineshape (in add/subtract mode). The result
for orientation is shown in Fig. 3a. We find that both sites have very similar shifts and
linewidths, at From isotope comparisons,
we conclude that the linebreadth is magnetic in origin. In Fig. 3b, we show the spectra for
which confirm the similarity of their shapes. The discrepancy in the apparent shifts
for this orientation arises from different quadrupolar couplings.
We only briefly mention our data for the spin–lattice relaxation time and the Gaussian
component of the spin echo decay By employing the new technique, we measured for
both sites selectively and found agreement with NQR results, in that the B site relaxation
is slower but has a similar anisotropy as the A line. Next, we determined With the
knowledge of all we corrected the raw data for effects [17]. We find at the line
maxima for the (for A and B, for the central an satellite transitions, for both isotopes,
for NMR and NQR), when normalized to the a unique value of at
300 K. Thus, the of both sites are the same.
4. DISCUSSION
First, we would like to discuss the possible origin of the observed magnetic width. It
is known from experiment [18] that Ni substitution for Cu causes a magnetic width that is
well understood by theory [19]. Magnetic impurities like Ni couple to distant nuclei via the
electron spin susceptibility. Estimates based on the same mechanism for our material [20]
show that we would need on the order of a few percent of a spin–1/2 impurity to induce our
observed width. Because such impurity levels are well above any level of foreign atoms in our
material, we suggest that localized holes are responsible for the magnetic line broadening.
Randomly distributed trapped holes broaden our NMR lines effectively. Assuming a striplike
ordering would increase the necessary amount of trapped holes substantially.
If the B sites were close to a trapped hole, their magnetic linewidth should be sub-
stantially bigger than that of the A sites, and one would expect that their differ
On the Structure of the Cu B Site 307
from that of the A site. Also, it seems not very likely that the B sites would maintain
an axially symmetric field gradient in such a position. Finally, one would not expect the
NMR satellite and NQR linewidths of the B line to be even smaller than those for the
A site.
It is known from NQR (see, e.g., Ref. [21]) that small doping levels cause a drastic
increase in the Cu NQR linewidths
However, above the NQR line widths remain nearly constant as x is increased.
Such a nonlinear x dependence cannot be understood in terms of local distortions induced
by the Sr atoms alone. More likely, additional static lattice modulations of axial symmetry
appear already above and affect both A and B sites similarly.
Recent XAFS and EXAFS data [2,3,22,23] show two Sr—apical oxygen and Cu-apical
oxygen distances, respectively, at all temperatures. This observation suggests that the Cu B
sites could be formed by contracted octahedra that have an apical oxygen connected
to a Sr atom. Such a conclusion would be consistent with our results.
5. CONCLUSIONS
We have resolved and studied the A and B Cu NMR/NQR sites in
Our results show that the A and B site Cu nuclei are very similar, at least in the optimally
doped material. This is inconsistent with the current interpretation as A being the main,
undisturbed Cu site whereas B arises from Cu neighbors to trapped holes. We find that
neither of the two sites shows a particular relation with the lattice modulations or the
location of the trapped holes. Rather, the subtle differences between A and B are caused by
local lattice distortions of axial symmetry that create a contracted B site octahedron.
ACKNOWLEDGMENTS
This work was supported by the Science and Technology Center for Superconductivity
under NSF Grant No. DMR 91-200000 and the U.S. DOE Division of Materials Research
under Grant No. DEFG 02-91ER45439. J.H. acknowledges support from the Deutsche
Forschungsgemeinschaft.
REFERENCES
1. P.C. Hammel, Phys. Rev. B 57, R712 (1998).
2. D. Haskel, E. A. Stern, D. G. Hinks, A. W. Mitchell, and J. D. Jorgensen, Phys. Rev. B 56, R521 (1997).
3. A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito,
Phys. Rev. Lett. 76, 3412 (1996).
4. N. L. Saini, A. Lanzara, H. Oyanagi, H. Yamaguchi, K. Oka, T. Ito, and A. Bianconi, Phys. Rev. B 55, 12759
(1997).
5. K. Yoshimura, T. Imai, T. Shimizu, Y. Ueda, K. Kosuge, and H. Yasuoka, J. Phys. Soc. Jpn. 58, 3057 (1989).
6. K. Kumagai and Y. Nakamura, Physica C 157, 307 (1989).
7. K. Yoshimura, T. Uemura, M. Kato, K. Kosuge, T. Imai, and H. Yasuoka, Hyper. Interact. 79, 867 (1993).
8. M. A. Kennard, Y. Song, K. R. Poeppelmeier, and W. P. Halperin, Chem. Mater. 3, 672 (1991).
9. P. C. Hammel, A. P. Reyes, S.-W. Cheong, and Z. Fisk, Phys. Rev. Lett. 71, 440 (1993).
10. R. L. Martin, Phys. Rev. Lett. 75, 744 (1995).
11. J. Haase, N. J. Curro, and C. P. Slichter, unpublished, 1998.
12. P. G. Radaelli, D. G. Hinks, A. W. Mitchell, B. A. Hunter, J. L. Wagner, B. Dabrowski, K. G. Vandervoort,
H. K. Viswanathan, and J. D. Jorgensen, Phys. Rev. B 49, 4163 (1994).
308 Haase, Stern, Hinks, and Slichter
1
Department of Physics “A. Volta,” Unitá INFM di Pavia, 27100 Pavia, Italy.
with the resonance frequency and the hyperfine coupling constant. We remark that
there is a factor 4 difference with respect to the equation reported by Troyer et al. [17] that
is related to a different definition of the hyperfine Hamiltonian and of the dispersion curve.
The values of the hyperfine constants are for for
In the case of a general form for the dispersion relation, by considering that
the low-energy processes are the ones corresponding to an exchanged momentum
and one can write the contribution related to 2-magnon Raman processes in the
form [17]
where is the dispersion relation for the triplet spin excitations, normalized to the gap
value, whereas For a 2-leg ladder, a general form describing is
In Fig. 2 we report the results obtained on the basis of Eqs. 3 and 4 for
for different values of the superexchange anisotropy r. One observes that whereas for the
dimerized chains, corresponding to the limit follows an activated behavior as
the one given in Eq. 2, for the 2-leg ladders with r ~ 1 one observes some differences with
respect to the simple activated behavior already at temperatures This analysis
points out that for a 2-leg ladder with r of the order of unity it is not correct to estimate
the gap from by using Eq. 2, at least for In fact, it is noticed that the
quadratic approximation for the dispersion curve becomes valid for a more restricted range
of around as r decreases (see Fig. 1). This seems to contradict the results reported in
Fig. 2a, where the departure from the quadratic approximation is found more pronounced for
than for However, this artifact is related to the choice of the horizontal scale,
namely to have reported vs. because increases with In fact, if we report
(Fig. 2b), with independent of r, one immediately notices that the deviation
from the quadratic approximation starts at lower temperatures for the lowest value of r.
One can then analyze the experimental data on the basis of Eq. 3 by taking the value
for the gap estimated by other techniques and check if there is an agreement. We have fit
On the Estimate of the Spin-Gap 313
the experimental data for (Fig. 3b) and (Fig. 3a) in by taking
and respectively, as estimated from susceptibility or NMR shift
data [7, 10]. In both cases, we find a good agreement between theory and experiment by
taking for the ladder site and for the chain site. If the data for
were fitted according to Eq. 2 one would derive a value for the gap around 650 K, a factor
1.5 larger than the actual value (see Table 1).
For also a quantitative agreement with the experimental data for is
found. However, this fact seems to be at variance with the estimates by Johnston [21]
based on the analysis of DC susceptibility data and with the recent findings by Imai
et al. [14] based on the study of NMR shift anisotropy, in which a value for
was derived. If we take this value for r we find that the experimental data are a factor
larger than expected. This disagreement could originate, at least partially, from having
considered for the processes the values for the matrix elements
estimated by Troyer et al. [17] for the case One must also mention that the estimate
314 Melzi and Carretta
of the hyperfine coupling constants could suffer from some uncertainties, particularly the
contribution from the transferred hyperfine interaction with the neighboring spins.
This contribution should be particularly relevant for the nuclei, whereas it should
be small for However, it must be recalled that because depends quadratically
on the hyperfine coupling constant even for sizeable corrections can be exepected.
Finally, it must be observed that in these systems the low-frequency divergence of
is cut because of the finite coupling among the ladders (or chains), introducing another
correction to the absolute value of
The low-frequency divergence of was found to follow the logarithmic behavior
reported by Troyer et al. [17] (see also Eq. 2) and does not change on varying the anisotropy
factor r, for . In fact, the form of this divergence is related to the shape of the dispersion
curve close to where it is always correctly approximated by a quadratic form for
ACKNOWLEDGMENTS
We would like to thank D. C. Johnston for useful discussions. The research was carried
out with the financial support of INFM and of INFN.
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Magnetic and Charge Fluctuations
in Superconductors
Neutron scattering has been used to study the spin fluctuations in the
and materials. Evidence is found for both incommensurate fluc-
tuations and a commensurate resonance excitation. Measurements on the lattice
dynamics for show incommensurate structure that appears to stem
from charge fluctuations that are associated with the spin fluctuations.
1. INTRODUCTION
Neutron scattering measurements continue to provide information of direct relevance
to some of the most important issues in the cuprate superconductors. The magnetic
excitations of these materials are the spin fluctuations, and recent measurements have shown
that the low-energy spin fluctuations in are incommensurate in
nature whereas a commensurate excitation that is relatively sharp in energy called a res-
onance is found at about 35 meV [1]. The incommensurability was originally discovered
by the filter integration technique [2] that integrates over the outgoing neutron energy in
a direction along and thus provides a high data collection rate for the study of lower
dimensional excitations. The disadvantage of the technique is that no discrete energy infor-
mation is available. Thus when a discovery is made by the integration technique, further
measurements are made by triple-axis or time-of-flight techniques to determine the energy
spectrum. Figure 1a shows the direction of the integrating scan that is made through the
point to observe the incommensurate fluctuations shown by the dots at the posi-
tion from the commensurate position. Such a scan uses high resolution along the scan
direction but coarse resolution perpendicular to the scan direction, and thus cannot deter-
mine the exact wave-vector position of the incommensurate peaks. The result of the scan
1
Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393.
2
Department of Materials Sciences and Engineering, University of Washington, Seattle, Washington 98195, USA.
for YBCO6.6 is shown in Fig. 1b. The position is at (0.5, 0.5) in reciprocal lattice
units (rlu).
A recent measurement using a pulsed spallation source with time-of-flight energy de-
termination has used a two-dimensional (2D) position-sensitive detector bank to determine
the wave-vector and energy of the YBCO6.6 incommensurate peaks [3]. The peaks are
found to be along the and directions as shown in Fig. 1a. The peaks are there-
fore not exactly on the scan direction shown in Fig. 1a, but are above and below it, being
observed in the integrated scan through the relaxed vertical resolution. The peaks in the
scan in Fig. 1b are found at about 0.055 on either side of the commensurate point. The
wave-vector of the incommensurate scattering is then times 0.055 from the geometry in
Fig. 1, and this value must be multiplied by again as the scan is in units of (h, h), giving
The number determined from the time-of-flight measurement is
This incommensurability is essentially identical to that observed in simi-
larly doped [4] It was found that the intensities and the correlation
lengths are also very similar in the (214) single-layer and YBCO (123) bilayer materials,
thus the low-energy spin fluctuations appear to be universal for the cuprate materials
measured to date.
aligned direction of the crystal. One must then set the spectrometer to sample a point off
the [110J direction, which means that only some of the values desired fall within the
resolution volume of the spectrometer, considerably reducing the magnetic signal.
One might expect that a resonance excitation might exist in BSSCO in a similar way
as for YBCO, and one can search for it in the same way [5] using a triple-axis spectrometer.
Our sample of BSSCO has an oxygen composition near optimum doping and we expect that
the resonance will not be observable much above in this case. The simplest experiment
is thus to scan energy at the momentum value where the resonance is expected and take the
difference between data taken well below and data taken above This works in YBCO,
but fails in BSCCO for two related reasons. The first is that the signal is small relative to the
phonons as we cannot achieve the full magnetic intensity that would be available at a fixed
value for The second problem is the phonons are much more temperature sensitive in
BSSCO than they are in YBCO, so that the difference in data at high and low temperatures
strongly reflects the phonon differences. The only way to circumvent these difficulties is to
use a polarized neutron beam to isolate the magnetic scattering. Unfortunately, this results
in even less intensity because polarized beams are much weaker then unpolarized ones.
Nevertheless, after considerable counting, reasonable results were obtained. Momentum
values near (0.5, 0.5) and (1.5, 1.5,) were both tried, and similar patterns were
with the magnetic scattering near (1.5, 1.5) being considerably weaker because of the
magnetic form factor. The results are shown in Fig. 2. A peak in energy about 10 meV wide,
which is equal to the energy resolution of the experiment, is observed at 10 K, whereas the
result at 100 K appears rather featureless. The peak is only found in the spin flip channel
guaranteeing that it is magnetic. The peak is observed at about 37 meV, which is near the
value expected if the of the BSSCO of 84 K is scaled to that of YBCO for the same
doping level. The results strongly suggest that BSSCO has a resonance excitation rather
similar to that of YBCO. However, the results should be checked with single crystals when
they become available.
The same BSSCO sample was used to search for magnetic incommensurate fluctua-
tions. The integration technique was used in the same way as for YBCO6.6. The experiment
works in a similar manner except that the integration now takes place over the directions
perpendicular to the [1,1,0] direction and thus is only partly along For 2D scattering
from bilayers, this results in an intensity loss in the magnetic signal. However, we see from
Fig. 1 that the magnetic signal in the integration technique is substantial so that an intensity
loss may be tolerated. The results of the measurement are shown in Fig. 3, which shows data
presented in the same way as for the YBCO6.6 in Fig. 1. The results suggest the possibility
of small incommensurate peaks, although the counting errors are larger than desired. The
data shown are from a number of runs averaged together. Fig. 3b–d show one of the satellite
peaks measured at different temperatures. In this case, the background was obtained by a
30° rotation of the sample relative to the position where the scan is performed. The magnetic
signal is expected to be small in the 30° rotation case, which samples reciprocal space well
removed from (0.5, 0.5). The signal decreases with temperature, as would be expected for
a magnetic excitation. The peak is broad so the center is hard to determine accurately with
the errors involved; however, the peak appears to be centered at about 0.42 rlu or 0.08 rlu
units from the (0.5,0.5) position. If the magnetic satellites are arranged as in Fig. 1, would
be about 0.32. The value of for fully doped 214 materials is about 0.25, so the value for
BSSCO appears to be somewhat larger than for the 214 materials assuming the same type
of incommensurability. However, the BSCCO measurement has sizable counting errors and
318 Mook, Dogan, and Chakoumakos
measurement at 300 K as a background. As the sample is cooled, distinct peaks form on both
sides of the (1,0) peak that we assume reflects a dynamic incommensurate mass fluctuation
that can be considered to stem from an incommensurate charge fluctuation. However, we
have not completely ruled out magnetic effects. We note the peaks are small, being an order
of magnitude smaller than the magnetic satellite peaks shown in Fig. 1. The scan is along
the (h, 0) reciprocal lattice or the direction, and thus is along the direction of the
magnetic incommensurate scattering. The charge fluctuation peaks are about 0.22 in rlu
units from the commensurate position so that the value for them is 0.22, or twice the
wave-vector of the incommensurate magnetic satellites. However, the absolute direction of
the incommensurate wave-vector cannot be determined with the integration technique, and
the peaks could be at a wave-vector off the direction. Figure 4d shows an identical
measurement for YBCO6.35 that has only commensurate magnetic order. No indications
of incommensurate charge fluctuations are found for this material.
Work has been underway with triple-axis spectrometry to determine the energy spectra
of the charge fluctuations, but that work is still incomplete. It has been noted, however, that
certain phonon branches show anomalies at the wave-vector of the charge fluctuations.
The origin of the charge fluctuations is not clear. It would seem extremely likely that the
320 Mook, Dogan, and Chakoumakos
magnetic and charge fluctuations stem from the same source. Obviously, the observation
of charge fluctuations strongly suggests a dynamic striped phase in YBCO6.6. However,
other possibilities exist, including Fermi surface effects or dynamic charge density waves
(CDW). The next step is to determine the energy spectra and absolute wave-vector of the
incommensurate charge scattering.
4. CONCLUSION
We have shown new neutron scattering results for the cuprate superconductors. Mea-
surements on a BSCCO sample of crystals with a [110] direction aligned show strong
evidence for a resonance excitation and indications of incommensurate magnetic fluctua-
tions. It would be good to have these results confirmed by a high-quality single crystal. For
YBCO6.6, clear dynamic incommensurate peaks are observed at low temperatues on either
side of the (1, 0) reciprocal lattice peak. Because these are found at positions relative to
Magnetic and Charge Fluctuations 321
the crystal reciprocal lattice, they are assumed to stem from mass fluctuations driven by
charge fluctuations. No such peaks are found for a YBCO6.35 sample. The wave-vector
of the charge fluctuation peaks is twice that of the magnetic fluctuations if we assume the
charge peaks are on the direction. It would seem likely the magnetic and charge
excitations are related. The results give support to a dynamic striped phase model for the
cuprate superconductors.
ACKNOWLEDGMENTS
The submitted manuscript has been authored by a contractor of the U.S. Government
under contract No. DE-AC05-96OR22464. Accordingly, the U.S. Government retains a
nonexclusive, royalty-free license to publish or reproduce the published form of this con-
tribution, or allow others to do so, for U.S. Government purposes.
Prepared by Solid State Division Oak Ridge National Laboratory; Managed by Lock-
heed Martin Energy Research Corp. under Contract No. DE-AC05-96OR22464 with the
U.S. Department of Energy Oak Ridge, Tennessee, September 1998.
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6. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995).
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Neutron Scattering Study of the
Incommensurate Magnetic Fluctuation
in
1. INTRODUCTION
The persistence of antiferromagnetic fluctuations in the metallic state of
cuprates probably has an important role for the superconductivity mechanism. In order
to elucidate the details of the magnetic fluctuations, intensive experimental works on
have been performed by
using neutron scattering techniques, which have an crucial role in characterizing the wave-
vector and energy dependence of the imaginary part of the dynamical susceptibility. In
LSCO, the magnetic scattering is incommensurate, and four sharp magnetic rods locate
1
Institute of Materials Structure Science, KEK, 1-1 Oho, Tsukuba 305, Japan.
2
Department of Physics, Tohoku University, Aoba, Sendai, 980, Japan.
3
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK.
4
Superconductivity Research Laboratory, ISTEC, Koto-ku, Tokyo 135, Japan.
2. EXPERIMENTAL
Experiments were performed on the chopper spectrometers MARI and HET at the
ISIS pulsed spalation neutron source of Rutherford Appleton Laboratory. The sample used
in the present experiments was an assembly of single crystals of in
total amount. The samples were synthesized by SRL-CP method at the Superconductivity
Research Laboratory [7]. The optimum-doped sample was prepared by annealing at
for 39 days and susceptibility measurements revealed with the transition width
of Underdoped samples were prepared with the same crystals annealed at
640°C for 3 weeks after the neutron scattering experiment on HET, and susceptibility
measurements revealed with a somewhat broad transition of The c*
axis was aligned to the incident neutron beam so that the (HHL) plane was the scattering
plane. The large number of detectors can cover a wide range of the energy—momentum
space in the (HHL) plane simultaneously.
3. RESULTS
Figure 1 shows the intensity contour map of the dynamical structure factor
on the plane for the underdoped sample observed on
the MARI spectrometer, where stands for the 2D momentum transfer and E the en-
ergy transfer. As shown in Fig. 1, two magnetic rods were clearly observed from about
15 meV to 40 meV, and these merge at 41 meV above the AF zone center of (1/2, 1 /2, 0),
corresponding to It can be interpreted that the magnetic incommensurate peaks
locate at with the same symmetry of that of LSCO;
the counter length along (–h, h, 0) of the spectrometer can cover the two incommensurate
peaks simultaneously, as shown in Fig. 2. The constant-energy slices of at 24,
32, 40, and 52 meV show q dependence of the magnetic spectrum in Fig. 3. The q depen-
dence at 24 and 32 meV shows by two peaks around the AF zone center at . The
magnetic signal was fitted by double gaussians with the incoherent phonon background of
Although the profile at 40 meV looks like a single peak, it can be fit by double
gaussians. The peak separation of two peaks seems to make a shorter distance at 40 meV.
Actually, just below 40 meV the peaks make a flat-top structure and are well fitted by double
gaussians. The energy dependence of the peak separation, which is defined by in Fig. 2,
is shown in Fig. 4. In the low-energy region, the peak separation, incommensurability, is
about rlu and decreases with increasing energy up to 40 meV. Above this energy
the peaks split again, with a somewhat longer distance and a much broader peak profile.
Incommensurate Magnetic Fluctuation in 325
For the optimum-doped sample, the magnetic resonance peak was observed at 41 meV,
resonance energy at the (1/2, 1/2, 0) position by using HET spectrometer, as was already
reported [5]. The spectrum at 10 K and 100 K around the resonance energy was shown in
Fig. 5a and the subtracted data (10–100 K) was shown in Fig. 5b. As shown in Fig. 5b,
326 Nishijima et al.
the large enhancement of magnetic signal at the commensurate position was observed.
There seems to be shoulder structures around commensurate resonance peak. Hence the
peak profile was fit by three gaussians, although the physical meaning is unknown, with
a constraint that the two gaussians for the shoulders should have the same width and the
Incommensurate Magnetic Fluctuation in 327
same splitting from (1/2, 1/2, 0). The fitting results indicate that the incommensurability
is about
4. DISCUSSION
We could confirm that the magnetic excitation spectrum in the underdoped YBCO has
incommensurate peaks with the symmetry as the same as that of LSCO. One theoretical
explanation about incommensurate magnetic peaks in LSCO suggests that the dynamical
spin susceptibility is enhanced at incommensurate position, which is related to a nesting
wave-vector of the Fermi surface [8,9]. However, the theory does not predict any incom-
mensurability in YBCO due to the lack of Fermi-surface nesting. Hence this scenario
should be reconciled by our experimental results. Another viewpoint on the incommensu-
lability is stripe domain structure based on the recent neutron scattering experiments on
by Tranquada et al. [10, 11]—i.e., dopant-induced holes segregate
into periodically spaced stripe structure that separates AF domains with double periodic-
ity of the former. Emery et al. [12] pointed out a possible emergence of the dynamical
microphase separation in the plane by taking into account the long-range Coulomb
force in the t-J model, which expects that the incommensurate magnetic fluctuation can be
a common feature in cuprate superconductor.
Recently, Yamada et al. [13] reported a linear relation between and up to the op-
timum doping regime revealed by systematic neutron scattering studies on
Incommensurability scaled by where (max) stands for transition tem-
perature at optimum doping in each system, is depicted in Fig. 6 together with LSCO and
YBCO. It is clearly recognized that there is a similarity in the diagram for
LSCO and YBCO. However, it is noted that the incommensurability has a strong energy
dependence and makes a sudden dip at around 40 meV, which has never been observed
in LSCO. Therefore, further analysis and additional experiments are needed to discuss the
328 Nishijima et al.
ACKNOWLEDGMENT
Quite recently, we were informed that H. Mook et al. also observed the incommensurate
magnetic peaks in YBCO with the same symmetry as ours. This work was supported by a
Grant-in-Aid on Scientific Research on Priority Areas “Anomalous Metallic State near the
Mott Transition” (07237102) of the Ministry of Education, Science, Sports and Culture,
Japan, and done under collaboration with NEDO. The authors acknowledge J. W. Jang,
A. I. Rykov, M. Kusao, S. Koyama, and K. Tomimoto for preparing the sample.
REFERENCES
1. S. M. Hayden et al., Phys. Rev. Lett. 76,1344(1996).
2. K. Yamada et al., Phys. Soc. Japan 64, 2742 (1996).
3. H. A. Mook et al., Phys. Rev. Lett. 70, 3490 (1993).
4. P. Dai et al., Phys. Rev. Lett. 77, 5425 (1996).
5. H. F. Fong et al., Phys. Rev. Lett. 78, 713 (1997).
6. P. Dai et al., Science 284, 1344 (1999).
7. Y. Yamada et al., Physica C 217, 182 (1993).
8. Q. Si et al., Phys. Rev. B 47, 9055 (1993).
9. T. Tanamoto et al., J. Phys. Soc. Jpn. 63, 2739 (1994).
10. J. M. Tranquada et al., Nature 375, 561 (1995).
11. J. M. Tranquada et al., Phys. Rev. Lett. 73, 338 (1997).
12. V. J. Emery et al., Physica C 209, 597 (1993).
13. K. Yamada et al., Phys. Rev. B 57, 6165 (1998).
Rare Earth Spin Dynamics in the Nd-Doped
Superconductor
1. INTRODUCTION
The investigation of magnetic correlations in superconducting and related materials
is essential to understand the mechanisms leading to high-temperature superconductivity.
In this paper, we present inelastic magnetic neutron scattering experiments on Nd-doped
at various temperatures. In LSCO, the rare earth (RE) doping causes
a further structural phase transition from the low-temperature orthorhombic (LTO) to the
low-temperature tetragonal (LTT) phase [1]. In a certain composition range, this LTT phase
1
II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, D-50937 Köln, Germany.
2
Laboratoire Leon Brillouin, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France.
3
Hahn-Meitner-Institut, Glienicker Str. 100, D-14109 Berlin.
is not superconducting, but antiferromagnetic (AF) order occurs at finite Sr, i.e., charge
carrier concentration. Tranquada et al. [2] show from elastic neutron diffraction experiments
on that the influence of the structural transition on the electronic
properties is due to a pinning of stripe correlations of spins and holes, i.e., due to a formation
of static antiphase AF domains in the planes that are separated by quasi-1D stripes
containing the doped charge carriers.
We investigated the magnetic response of the RE 4f moments at low-energy transfers
(typical neutron incident energy ) in order to obtain information about the Cu
magnetism in the layers via the RE–Cu interaction. Such an interaction was suggested
from previous experiments on that show an anomalous behavior of the 4f
magnetic scattering response (see, e.g., Ref. [3]). RE–Cu interaction is also well established
in the electron-doped compound [4].
2. EXPERIMENTAL
We performed temperature-dependent studies on
and using the time-of-flight (TOP) spectrometers V3 NEAT [5]
(HMI, Berlin) and G6.2 MIBEMOL (LLB, Saclay) with a cold neutron source. We chose
incoming energies between and
resulting in energy resolutions between and
respectively. The experiments were performed on well-characterized powder samples [6].
For details concerning the data analysis, see Ref. [7].
spectra are observed that correlate with the electronic properties of the layers. In the
insulating compounds with long-range AF order of the Cu
moments occurs below In both samples, the QE line vanishes below 80 K, and
instead an inelastic excitation is observed [19] (Fig. 3). This excitation can be interpreted
as lifting of the degeneracy of the ground-state Kramers doublet due to the Nd-Cu–
exchange interaction, in quantitative agreement with the observed Schottky anomaly of the
low-temperature specific heat [20]. The increase of the energy excitation from meV
at 80 K to meV at 3.3 K (see inset, Fig. 3) indicates an increase of the staggered
magnetization in the layers. The fact that the INE in occurs in both
samples at the same temperature (independent of the Nd concentration) shows that the Cu
spin reorientation [21] at does not influence the exchange field at the Nd site
markedly.
The findings for the Sr-doped compounds differ from those for the sample
with due to the change of the electronic properties. In with
superconductivity is strongly suppressed in the LTT phase [1] and a formation
of AF domains separated by stripes containing the charge carriers is expected [2]. In both
samples we observe a strongly enhanced line width of the QE line (Fig. 4) below
In contrast to the high-temperature region (where the QE line is a Lorentzian), the spectra
are consistent with a gaussian QE line shape. The line width is meV
and meV for 30 K and 20 K, respectively, and remains roughly constant for
lower temperatures. Because there is a coupling between the Cu and Nd moments
see above), these INE neutron data are most probably due to a freezing of the Cu spin
fluctuations [22] in the samples with The differences in the data for and
i.e., the observation of a strongly enhanced QE line width instead of a well-defined
INE line, suggests a distribution of different energy splittings over different Nd sites in the
Rare Earth Spin Dynamics 333
Sr-doped compound. It is obvious that the inhomogeneity due to pattern of stripe correlation
of spins and holes [2] is the reason for this observation. Finally, comparing the width of
the gaussian line at with the energy of the INE line reveals a reduced
(average) splitting, which is related to a reduced zero temperature staggered magnetization
in the planes.
No difference of the temperature behavior of the QE line widths is observable in
with (Fig. 2). In both samples the line width decreases
with decreasing temperature, as expected by the Orbach relaxation process. No hints for
an INE excitation or a QE broadening are found at lowest tempera-
ture (2.8 K and 1.6 K for and respectively). This is not surprising for
which is (bulk) superconducting below In contrast, the
tilt angle of the octahedra exceeds the critical value
and thus superconductivity is strongly suppressed [1]. Therefore, magnetic order of the Cu
moments and hence a broadening of the QE line below is expected. The absence of such
334 Roepke et al.
4. CONCLUSION
To summarize, we presented INE magnetic neutron scattering experiments on Nd-
doped LSCO. In all samples at high temperatures, a QE line is observed, with a line width that
decreases with decreasing temperature. In the absence of RE–Cu interaction, the relaxation
of the 4f moments is dominated by the Orbach relaxation process via the coupling of
phonons and CEF excitations. The low-temperature behavior clearly correlates with the
electronic properties. For the undoped samples below about 80 K an INE excitation
occurs that shows the splitting of the Kramers ground-state due to the exchange field
at the Nd site. For the samples with stripe order of spins and holes a broad QE
line infers a distribution of different energy splittings over different Nd sites. In
no indication for a RE–Cu interaction has been found, i.e., follows the temperature
dependence as expected by the Orbach relaxation process down to the lowest temperature.
ACKNOWLEDGMENT
Our work is supported by the BMBF under contract number 03-HO4KOE.
REFERENCES
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Static Incommensurate Magnetic Order in the
Superconducting State of
1. INTRODUCTION
Although the mechanism of pairing interaction of superconductivity is not
fully understood at present, the interplay between the magnetic correlation and
superconductivity is one of the central issues in the basic physics of the lamellar
1
Institute for Chemical Research, Kyoto University, Uji 611 -0011, Japan.
2
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
3
Department of Physics Tohoku University, Aramaki Aoba, Sendai 980-77, Japan.
4
Center for Neutron Research, NIST, Gaithersberg, Washington, USA.
5
Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA.
energy gap in the underdoped superconducting state can be connected with the Zn-doping
effect.
Stimulated by the observation of incommensurate elastic peak in the Sr-doped
Lee et al. [17] searched for the elastic peak in the electrochemically oxygen-doped super-
conducting As a result, they found similar incommensurate peaks. In this case,
the onset of the magnetic order almost coincides with which is higher than that
for the optimally Sr-doped sample. More surprisingly, the doping rate of the oxygenated
sample was estimated to be about 0.12, where an anomalous • suppression appears for
the Sr-doped system.
In order to elucidate the interplay of such long-range magnetic order with the supercon-
ductivity, extension of the doping region is quite important. Wakimoto et al. [18] observed
a mixture of commensurate and incommensurate elastic spin correlation in the sample at
exhibiting no superconductivity but a spin–glass behavior in the uniform mag-
netic susceptibility. We note that a recent measurement suggests a coexistence of the
spin–glass and superconducting phases [19]. However, the momentum-sensitive neutron
scattering revealed the more surprising fact that the observed peak position differs by 45°
from that of the superconducting phase [18]. We remark that such a change in the direction
of spin–charge density modulation was previously predicted by theoretical calculations
based on the simple Hubbard model on a 2D square lattice [20].
Due to the sharp q spectrum of the elastic peak, the precise peak position around
was determined by Lee et al. [17] for the oxygen-doped Originally, the four
peaks were believed to locate on the corner of a square as shown in Fig. 4. However, they
found the peaks are on a rectangule. Although the reason for such shift of peak position is
340 Yamada et al.
not understood, same deviation is also observed in the Sr-doped sample with much smaller
orthorhombicity. Therefore, the origin of the shift may not relate with the orthorhombic
distortion or twinned structure.
ACKNOWLEDGMENT
The neutron experiments have been dominantly performed by using thermal and cold
neutron 3-axis spectrometers of both reactors in JAERI and NIST. Single crystals were
grown in Tohoku University and Yamanashi University and Kyoto University by using
lamp-image furnaces for TSFZ method. The authors acknowledge K. Nemoto and M.
Onodera for their technical assistance at JAERI and Tohoku University. We also thank Y.
Kojima, I. Tanaka, and S. Hosoya in Yamanashi University for their helpful discussions of
crystal growth. We wish to thank M. Greven, M. A. Kastner, Y. M. Kim, Y. S. Lee, T. Suzuki,
and T. Fukase for their valuable discussions. Work at Brookhaven National Laboratory was
carried out under contract no. DE-AC02-98-CH10886, Division of Material Science, U.S.
Department of Energy. The research at Massachusetts Institute of Technology was supported
by the National Science Foundation under grant no. DMR97-04532 and by MRSEC Program
of the National Science Foundation under award no. DMR94-00334. The present work in
NIST and Brookhaven National Laboratory was supported by a US–Japan collaboration
program on neutron scattering. The present work in part was also supported by a grant-in-aid
for scientific research from the Ministry of Education, Science, Culture and Sports of Japan
and by a grant for the promotion of science from the Science and Technology Agency and
by CREST.
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(1998).
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J. Phys. Soc. Jpn. 69, 1170 (2000).
8. K. Yamada, Advances in Superconductivity X (1998) 37.
9. For a review, see N. P. Ong, In Ginsberg, D. M. (ed.) Physical Properties of High-Temperature Superconductors
II(World Scientific, Singapore, 1990, pp. 459).
10. A. Fujimori, A. Ino, T. Mizokawa, C. Kim, Z.-X. Shen, T. Sasagawa, T. Kimura, K. Kishio, M. Takaba,
K. Tamasaku, H. Eisaki, and S. Uchida, 1997 International Conference on Spectroscopies in Novel Super-
conductors.
1 1 . K. Kumagai, K. Kawano, I. Watanabe, K. Nishiyama, and K. Nagamine, J. Supercond. 7, 63 (1994).
342 Yamada et al.
12. G. M. Luke, L. P. Le, B. J. Sternlieb, W. D. Wu, Y. J. Uemura, J. H. Brewer, T. N. Riseman, S. Ishibashi, and
S. Uchida, Physica C 185–189, 1175 (1991).
13. T. Goto, S. Kazama, K. Miyagawa, and T. Fukase, J. Phys. Soc. Jpn. 63, 3494 (1994).
14. H. Kimura, K. Hirota, H. Matsushita, K. Yamada, Y. Endoh, S. H. Lee, C. H. Majkrzak, R. Erwin, G. Shirane,
M. Greven, Y S. Lee, M. A. Kastner, and R. J. Birgeneau, Phys. Rev. B. 59, 6517 (1999).
15. K. Hirota, K. Yamada, I. Tanaka, and H. Kojima, Physica B 241–243, 880 (1998).
16. M. Matsuda, R. J. Birgeneau, H. Chou, Y. Endoh, M. A. Kastner, H. Kojima, K. Kuroda, G. Shirane, I. Tanaka,
and K. Yamada, J. Phys. Soc. Jpn. 62, 443 (1993).
17. Y. S. Lee et al., unpublished, 1999.
18. S. Wakimoto, S. Ueki, K. Yamada, and Y. Endoh, in preparation.
19. Ch. Niedermayer, C. Bernhard, T. Blasius, A. Golnik, A. Moodenbaugh, and J. I. Budnick, Phys. Rev. Lett.
80, 3843 (1998).
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Soc. Jpn. 59, 1047 (1990).
21. T. Suzuki and T. Fukase et al., unpublished data.
Marginal Stability of d-Wave
Superconductor: Spontaneous P
and T Violation in the Presence
of Magnetic Impurities
The point of this note is to emphasize the recently recognized new aspect of the high-
temperature superconductors: a marginal stability of the wave superconductor to-
ward secondary ordering in the presence of the symmetry perturbing field. Namely, in the
presence of the perturbing field the wave superconductor generates the secondary su-
perconducting component of the order parameter, likely to be in our case, to maximize
the coupling to this field and hence lower the total energy.
This instability can occur in many different ways. Recently, the surface scattering-
induced s wave component in materials has been observed [1] and the model
explaining the effect was proposed [2]. The existence of the secondary gap in the external
magnetic field was suggested to explain the anomalies in thermal transport in Bi2212 [3,4].
In both of these cases the superconductor was subjected to the perturbing fields: the surface
scattering or the external magnetic field. The above examples can be thought of as a specific
realizations of the general phenomena of marginal stability of wave superconductor.
1
T-Div and MST-Div, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
The finite induced xy component of the order parameter also leads to the xy gap:
ACKNOWLEDGMENTS
This work was done in collaboration with M. A. Hubbard (UIUC), M. B. Salamon
(UIUC), R. Yoshizaki (Univ. of Tsukuba), J. Sarrao (LANL), and M. Jaime (LANL). The
useful discussions with E. Abrahams, L. Greene, R. Laughlin, D. H. Lee, M. Salkola, and
J. Sauls are gratefully acknowledged. This work was supported by the U.S. Department of
Energy.
REFERENCES
1. M. Covington et al., Phys. Rev. Lett. 79, 277 (1997).
2. M. Fogelstrom et al., Phys. Rev. Lett. 79, 281 (1997).
3. K. Krishana et al., Science 277, 83 (1997). See also H. Aubin, K. Behnia, S. Ooi, T. Tamegai, K. Krishana,
N. P. Ong, Q. Li, G. Gu, and N. Koshizuka, Science 280, 5360:11 (1998) (in Technical Comments).
4. R. B. Laughlin, preprint, cond-mat/9709004.
5. A. V. Balatsky, Phys. Rev. Lett. 80, 1972 (1998). Also cond-mat/9710323.
6. R. Movshovich et al., Phys. Rev. Lett. 80, 1968 (1998). Also cond-mat/9709061.
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Skyrmions in 2D Quantum Heisenberg
Antiferromagnet Static Magnetic
Susceptibility
Two-dimensional (2D) antiferromagnet (AF) has been the subject of intensive research for
the last few years. It is well established that the properties of superconducting cuprates are
strongly influenced by 2D critical fluctuations and the undoped materials can be modeled by
a nearest neighbor quantum Heisenberg antiferromagnet (QHAF) on a square lat-
tice with a large isotropic exchange constant (J = 1580 K in La2CuO4). In this connection,
numerous efforts have been made to study 2D QHAF Most of them are based
on some modifications of the perturbation expansion near a homogeneous ground state.
Chakravarty, Halperin, and Nelson (CHN) succeeded in a renormalization group analysis
of a nonlinear -model which is assumed to represent QHAF in a continuum limit [1,2].
They were able to obtain the correlation length, local order parameter, and static magnetic
susceptibility for using no adjustable parameters. Their results were later improved
by Hasenfratz and Niedermayer [3] with the chiral perturbation theory. Chubukov et al. [4]
developed a theory of QHAF using the 1/N-expansion method on AF with an N-component
order parameter. They were able to consider both renormalized classical and
quantum critical regimes. However, their results for and do
not agree with each other in the intermediate interval of temperatures. In recent papers [5,6]
1
Department of Physics, Kazan State University, 420008, Kazan, Russian Federation.
there has been proposed another method based on the picture of thermally excited skyrmions
and antiskyrmions. The local order parameter, energy spectrum of elementary spin exci-
tations above the skyrmion background, the skyrmion-averaged radius and renormalized
by quantum fluctuations energy were calculated by the Green function method [5]. It has
been found that the temperature dependence of the skyrmion radius maps very well the
corresponding results for the spin correlation length of 2D QHAF calculated by CHN and
Chubukov et al. [4] in the absence of skyrmions. In [6] the nuclear spin relaxation rate
was obtained in the temperature region Its behavior at is almost iden-
tical with the results based on the renormalizazation group approach and 1/N—expansion
method. To make possible a comparative analysis of all proposed theories and reveal pecu-
liarities of properties of 2D QHAF with skyrmions, it is desirable to calculate their static and
dynamical characteristics that could be measured by well-developed experimental methods
and easily compared.
In this report we present our study of uniform magnetic susceptibility of 2D QHAF
using the results obtained in [5]. We have calculated directly all components of
susceptibility tensor and found that it is almost isotropic as it should be for the rotationally
invariant system without long range order. The temperature dependence of susceptibility
is close to linear in RC regime and quadratic in QC one. At low temperatures our result
qualitatively agrees with Chubukov et al. [4], although the slope of the line is different.
Besides, the skyrmion approach allows to extrapolate the function to giving the
remarkably well agreement with rather complicated two-loop spin-wave calculations [7].
It was found previously that the number of thermally excited topological excitations
is large at temperatures where is the renor-
malized skyrmion energy, is the local order parameter, and L/a is a linear size of a
sample in lattice units [5]. We expect that the nearest neighbors of every skyrmion are an-
tiskyrmions and vice versa, and the total staggered magnetizations equal to zero. Although
the long-range order is absent, the local order inside the skyrmion (antiskyrmion) still takes
place. The spin excitations above the inhomogeneous ground state are described by the
Hamiltonian presented in the local coordinate axes:
Here
where
Using Eq. (5) it is easy to calculate the longitudinal and transverse magnetization of
the system:
where
The local order parameter and the averaged skyrmion radius are defined by the
set of nonlinear equations [5]:
with
As one can see, the longitudinal and transverse components of susceptibility tensor
are close to each other so that the system is almost isotropic. This result is obtained by
the straightforward calculations as against one usually should do. To find one must
consider the local nonzero magnetization and define the susceptibility tensor as a response to
magnetic field in the direction perpendicular to magnetization vector (see, for example [2,4]).
A response in parallel direction is put to zero with the following rotational averaging of the
susceptibility tensor components. Our calculation gives an isotropic susceptibility with no
auxiliary ideas.
Using Eq. (7), we can obtain the expressions for in RC and QC regimes.
ACKNOWLEDGMENTS
We are grateful to Profs. K.-A. Müller and H. Keller for valuable discussions. We
thank also Zürich University, where the part of this work was carried out, for its hospitality.
The work has been supported by Swiss National Science Foundation (under grant no. 7 IP
051830) and Russian Foundation for Basic Research (under grant no. 98-02-17974).
Skyrmions in 2D Quantum Heisenberg Antiferromagnet 353
REFERENCES
1. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. Lett. 60, 1057 (1988).
2. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. B 39, 2344 (1989).
3. P. Hasenfratz and F. Niedermayer, Phys. Lett. B 268, 231 (1991).
4. A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994).
5. S. I. Belov and B. I. Kochelaev, Solid State Commun. 103, 249 (1997).
6. S. I. Belov and B. I. Kochelaev, Solid State Commun. 106, 207 (1998).
7. J. Igarashi, Phys. Rev. B 39, 9760 (1989).
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Spin Peierls Order and d-Wave
Superconductivity
Partha Bhattacharyya1
We show that a spin Peierls (SP) state in which a Peierls order Q is modulated as
is a magnetic localization wave number is manifestly
scale invariant when the fermions maintaining single-site occupancy are creation
operators for fermions forming Cooper pairs. A charge-ordered stripes state is
a state of modulated chirality being evidence for superconductivity.
Then a quantum mode of gauge symmetry restoration, like the Josephson mode
in SNS junctions, is a possibility.
1. INTRODUCTION
Among the most important ideas in the theory of electronic correlations in low dimen-
sional systems is the possibility of d-wave superconductivity. An electronic mechanism
of superconductivity implies the possibility of a symmetry restoring Josephson effect by a
single electron transition (SET) in a manner similar to superfluidity in Our proposal
is that in these circumstances the ground state has an additional quantum number, chirality,
similar to the kinematic quantum number that determines whether the electron is on the left
or right side of a normal-state barrier. This can be ensured in a spin Peierls (SP) state, in which
the Peierls order Q is modulated to is a magnetic localization
wave number. The SP order parameter for the chiral electrons having chirality
are charges placed on the diameter of a unit sphere corresponding to single electron
occupancy on a unit cell connected by a Dirac string. A longitudinal mode measures a Peierls
charge discommensuration at preferred regions on the Fermi surface. A transition occurs
when a monopole changes chirality, allowing a Josephson transition to take place amongst
chirally equivalent configurations. Charge ordering is seen in superconductors [1] with Ising
1
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai
400 005, India.
355
356 Bhattacharyya
Scale invariance requires that a single electron transition maintaining a steady state
has a relaxation rate
3.1 NMRandNQRin
Asayama et al. [5] give an excellent survey of lot of the work done on knight shifts and
the relaxation rates and in cuprate superconductors. is related to the transverse
Spin Peierls Order and d-Wave Superconductivity 359
susceptibility Eq. (5), and the temperature dependence in the normal state is easily seen as
The electromagnetic penetration length is obtained from the transverse response, where
In that limit, where implies
REFERENCES
1. D. Wollmann, D. J. Van Harlingen, W. Lee, D. M. Guisberg, and A. J. Leggett, Phys. Rev. Lett. 11, 2134 (1993);
J. M. Tranquada, Physica B 241–243, (1998), for modulated spin and charge ordering in cuprates.
2. S. K. Sarker and P. W. Anderson, cond-mat/9704123.
3. C. Itzykson, Int. J. Mod. Phys. A 1, 65 (1986).
4. M. Klein and R. Dierker, Phys. Rev. B 29, 4976 (1994). Raman scattering cross-sections calculated in second
order vector potentials is shown in T. P. Deverena and D. Einzel, cond-mat/9408019.
5. K. Asayama, Y. Kitaoka, G.-q. Zheng, and K. Ishida, Prog. NMR Sped. 28, 221 (1996).
6. D. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994).
7. M. Franz, C. Kallin, A. J. Berlinsky, and M. I. Salkola, Phys. Rev. B 56, 7882 (1997-I); A. Andreone et al.
Phys. Rev. B 56, 7874 (1997-I).
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On Localization Effects in Underdoped
Cuprates
1. INTRODUCTION
1
Istituto di Fisica della Materia e Dipartimento di Fisica, Università “La Sapienza,” piazzale A. Moro 2, 00185
Roma, Italy.
361
362 Castellani, Schwab, and Grilli
Standard localization effects have been discussed as source of the increasing resis-
tance since the first experiments [1,6]. Evidence against this interpretation has arised from
measurements of the Hall resistance [7], which is nearly temperature independent. How-
ever, because the mechanism that dominates the Hall effect in the cuprates is not clear, it is
hard to make conclusions. So far, a detailed analysis of the temperature dependence of the
resistivity versus the predictions of localization theory has not been given. This is subject
of the present paper.
In the next section we briefly recall standard localization theory in two-dimensional
(2D) systems, discussing both coherent backscattering and interaction effects. Then we
apply localization theory to LSCO. We demonstrate that the crossover from 3d to 2d local-
ization in LSCO is expected near optimal doping. However, the amplitude of the increase
of resistance in that material in the underdoped region does not fully agree with standard 2d
localization theory. We argue that anomalous localization effects are indeed to be expected
in that region in the presence of a disordered stripe phase.
2. LOCALIZATION IN 2D SYSTEMS
In two dimensions, arbitrarily weak disorder can localize all electronic states. This
famous result follows from the single parameter scaling theory of localization [8]. This
theory is justified when interactions are negligible. In the weak disorder limit
the conductivity at low temperature is given by
where is the “classical” Drude conductivity, and the logarithmic term is due to quantum
corrections, with the (elastic) scattering time and the dephasing time. These “weak
localization” corrections are due to quantum interferences for electrons that are diffusing
along paths containing closed loops (coherent backscattering). Dephasing is due to inelastic
processes, with leading to a correction to conductivity, which is logarithmic in
temperature,
However, single-parameter scaling fails in presence of interactions, where a scaling
theory including electron interactions is needed. Such a theory has been put forward by
Finkel’stein [9]. In perturbation theory, new singular contributions to the conductivity are
found, which are—in 2d—proportional to log T. These singular corrections to the con-
ductivity are due to the interplay between disorder and interaction, and arise because on
distances that are larger than the mean free path, electrons move slowly and have more time
to interact with each other.
The correction to the resistivity due to this mechanism is
where 1 is due to interactions in the singlet channel and 3[· · ·] are due to the triplet chan-
nels. The universal value of the singlet amplitude is due to the long-range nature of the
Coulomb interaction because after screening the dimensionless interaction equals in the long
On Localization Effects in Underdoped Cuprates 363
3. APPLICATION TO CUPRATES
In standard studies of localization the magnetoresistance is a main probe for extract-
ing important informations on both backscattering and interaction effects. Unfortunately,
in cuprates many different contributions to the magnetoresistance have been observed re-
lated to the superconducting fluctuations [11] and to the peculiar behavior of the Hall
conductance [12]. These effects may mask the localization contributions and thus make
an interpretation difficult. Therefore, we concentrate in the following on the temperature
dependence of the resistance under conditions, where possibly the above complications are
not present.
We refer to two types of experiments. Extremely underdoped LSCO which
is nonsuperconducting even in the absence of magnetic fields, and the system near-optimum
doping, but large magnetic field In both cases log T behavior in the re-
sistivity or conductivity has been observed, suggesting that the physics of disorder and
interaction in two dimensions is relevant. However, despite a substantial anisotropy, the
LSCO materials are bulk systems of weakly coupled layers. This raises the relevant issue
of the effective dimensionality of LSCO with respect to localization. Weak localization in
a nearly 2D metal has been considered by Abrikosov [13]. We performed a similar cal-
culation for the interaction contribution. We generalized the model of c-axis transport of
Ref. [13], incorporating interplanar disorder as discussed in Ref. [14]. We found that the
system behaves in a 2D fashion when the tunneling time between layers is larger than
the time of the slowest processes that are contributing to localization. For weak localiza-
tion, the relevant time scale is the phase coherence time whereas the time scale for the
interaction contribution, which is the relevant one in high magnetic field, is Therefore,
the crossover from 3d to 2d is defined by The tunneling rate is
hard to estimate directly because it is not clear if processes that conserve in-plane momen-
tum dominate, or momentum nonconserving processes dominate, for
which are tunneling amplitudes, the quasi-particle
364 Castellani, Schwab, and Grilli
lifetime, and N(0) is the density of states. More conveniently, the tunneling rate is deter-
mined from the c conductivity, because Inserting the free-electron
value for the 2d density of states with we find for
cm a tunneling rate of the c resistivity at 20 K
was between and , By compar-
ing to the temperature at 20 K, we conclude that the samples near optimal doping at
are near the dimensional crossover from 3d to 2d. The underdoped samples are pre-
sumably still 2D at 20 K, whereas the overdoped samples may be three dimensional. In this
case, a correction to the conductivity at low temperature instead of log T is expected.
Preyer et al. [6] reported the conductivity in highly underdoped For a
sample with they found a logarithmic correction to the conductivity
where and are sample dependent. Analyzing the amplitude of the log T near the onset
calculating the “interaction constant” according to
which is of order 1, but never larger than 2, because stability of the Fermi liquid requires
Apparently, are of the same order of magnitude.
To the first view the experiments seem to be in reasonable agreement with theory. There
are, however, a number of problems: (a) Although theory predicts log T in the conductivity,
it is seen experimentally in a large range of resistivity. (b) The ratio does not depend
on temperature (i.e., the function in ab and c direction is the same),
This is predicted for anisotropic, but 3D localization [15]. In the temperature
region of 2D localization a logarithmic correction to the c conductivity is expected due to
the corrections to the tunneling density of states, N. Explicitly working out the theory, we
found which in general differs from the correction to the
resistivity in ab direction, (c) A third problem arises from a quantitative analysis of the
amplitude of the log T, which is of the right order of magnitude, but nevertheless is too
large.
Further investigating problem (c), we found an intriguing relation between the exper-
imentally measured amplitude of the log corrections and the amount of disorder as
obtained from the absolute value of resistivity at some fixed temperature
In Table 1, we report for a number of samples, comparing “clean” and “dirty”
samples of the same material and dopant concentration [1,7,16]. Whereas is indepen-
dent of disorder, the experimentally determined value decreases with increasing dirtiness:
As shown in the table, the product is nearly disorder independent. Moreover,
seems to decrease with increasing doping. For the four LSCO samples we report in Table 1,
the product is roughly independent from disorder and doping.
A second observation is that various features of the anomalous localization can be
described phenomenologically by using the Drude formula for the conductivity and taking
the scattering rate from the ansatz
This logarithmically enhanced scattering rate appears directly in the resistivity, not in the
366 Castellani, Schwab, and Grilli
conductivity, and is therefore consistent with the property outlined in point (a). Moreover
the constant ratio of can be reconciled with a log T correction that is typical of 2D
systems [problem (b) outlined above] by the assumption in Eq. (7) if tunneling between
planes is dominated by momentum conserving processes. Finally, if dirtiness only affects
but not the logarithmic term, the amplitude of the log T as a function of disorder
behaves according to the experimental observation discussed above,
4. DISCUSSION
We discussed some features of transport experiments in LSCO compounds in the
normal state. On the one hand, the log T correction in strongly underdoped LSCO
appears to be consistent with standard localization theory, although an explanation for the
magnetoresistance is still lacking. On the other hand, for higher doping,
standard theory is not able to explain the experiments.
There are several reasons why the conventional “old-fashioned” localization theory
is not expected to work well in strongly correlated anisotropic systems like the cuprates.
One first possibility is that the cuprates cannot be described by the Fermi liquid theory as
already pointed out in the introduction [17]. If this is the case, a new localization theory
starting from a clean non-Fermi liquid system should be devised [2,5]. Alternatively, a
singular interaction could be responsible for both the disruption of the Fermi liquid and of
the anomalous localization effects. Mirlin and Wölfle [18] reported anomalous localization
effects within a gauge field theory [19], where particles interact via a singular transverse
gauge field propagator At low temperature, a log T correction has been
found with an amplitude that depends on resistance itself,
In the quantum critical point (QCP) scenario of superconductivity, a QCP exists
near optimum doping, with an “ordered” stripe phase in the underdoped regime [20–22].
In this context, possible sources of singular interactions are soft modes from dynamical
stripes, or critical fluctuations near the QCP. Specifically, the interaction near the stripe
critical wave-vector may be parameterized as [20]
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Castro, and M. Grilli, Phys. Rev. B 54, 16216 (1996).
21. For experimental evidence of a stripe phase in the cuprates, see the contributions to this conference by
A. Bianconi and by J. M. Tranquada. For theoretical aspects, see instead the contributions by C. Di Castro,
V. J. Emery, and J. Zaanen.
22. Different QCPs have been suggested by C. M. Varma, Physica C 263, 39 (1996); P. Montoux and D. Pines,
Phys. Rev. B 50, 16015 (1994), and references therein.
23. C. Castellani, C. Di Castro, and M. Grilli, cond-mat/9709278.
24. J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, Phys. Rev. B 54, 7489
(1996).
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Interpolative Self-Energy Calculation for the
Doped Emery Model in the Antiferromagnetic
and in the Paramagnetic State
Electronic properties of the doped Emery model for the planes in HTSC
materials, including magnetic phases, are calculated. The self-energy of the prob-
lem is approximated by an interpolation between the weak coupling second-order
perturbation theory limit and the strong coupling atomic limit. The self-consistent
calculation of the grand canonical potential for zero temperature and all magnetic
phases gives a purely antiferromagnetic (AF) phase in the neutral and nearly neu-
tral case. Upon doping phase separation shows up between an AF and a doped
paramagnetic (PM) regime, whereas for larger doping only the PM phase exists.
1. INTRODUCTION
The discovery of the high-temperature superconductivity (HTSC) enforced the interest
in the highly correlated models of electronic structure, especially for the superconducting
copper oxygen planes common to all HTSC materials. An adequate description of the
electronic state needs at least three orbitals in the plane, the copper dx2– y2 and the oxygen
and orbitals (Fig. 1). For sake of simplicity, this three-band model is mostly mapped
to simpler models as the one-band Hubbard model or the t-J model, respectively. As
experiments on oxygen isotope effects in the last years showed the importance of the
oxygen orbitals for the coupling to the lattice, we believe the full three-band model to
give a more adequate description of the electronic structure. The large values of the on-
site Coulomb repulsion on the copper sites requires some treatment that covers strong
coupling aspects, and the antiferromagnetic (AF) state in the neutral case must be included
for a realistic picture of the interesting electronic state at low doping. We apply some
1
Institut für Theoretische Physik, Universität Kiel, 24098 Kiel, Germany.
interpolation method to the self-energy that we previously used for the paramagnetic (PM)
state only [1]. Comparison with available QMC results for the one-band Hubbard model
proved the method to give appropriate results in this low-doping regime.
2. MODEL
We formulate the Emery (or three-band Hubbard) Hamiltonian for the A-B unit cell
with sublattices A and B
where these Fermi operators (in short notation) are related to the real space Fermi operators
Interpolative Self-Energy Calculation for the Doped Emery Model 371
which depends on the spin index and on the wave vector of the A-B Brillouin zone
(ABBZ) via the quantities
3. METHOD
We repartition the Hamiltonian into two parts by introducing four effective atomic
levels such that perturbational treatment of the second order in
These effective levels, of which at least two are independent due to the para-, ferro- and
antiferromagnetic symmetries, are to be fixed later.
First we have to determine the self-energy in second order perturbation theory, which,
using the only relevant Feynman graph in this order, gives thefollowing standard expression
372 Fritzenkotter and Dichtel
The copper spectral density matrices denoted are determined by the unper-
turbed Hamiltonian matrix
expressed in our representation. Of course the para- and ferromagnetic case with identical
A and B lattice are included in the formulae.
The interpolation of the self-energy to the strong coupling limit is performed
as in Refs. 2 and 3 and was studied later [4] for a different self-consistency condition. The
interpolation is applied only to the diagonal parts of the self-energy matrix
with
and
because the off-diagonal spectral densities already vanish as for vanishing hop-
ping amplitude [see Eq. (3)], whence the off-diagonal part of the self-energy
Interpolative Self-Energy Calculation for the Doped Emery Model 373
yields
The copper occupancy and the “fictitious” [4] copper occupancy are determined by
the corresponding unperturbed
Assuming that the chemical potential is given, for a complete determination of all free
parameters a sufficient number of conditions must be chosen in order to fix the effective
atomic levels (i.e., one and two additional conditions for the para- and nonparamagnetic
cases, respectively). Reference 5 discusses several conditions. We choose the condition from
Refs. 2 and 3
with the full spectral density and normalized per paramagnetic unit cell and spin direction.
For the internal energy is equal to the free energy, which allows us to write down
the grand canonical potential
4. RESULTS
We solve the problem for several couplings and chemical potentials with a charge
transfer energy Figure 2 shows the grand canonical potentials for and
Interpolative Self-Energy Calculation for the Doped Emery Model 375
that is, the low-doped AF and high-doped PM phase coexist in the system. The FM phase
is always thermodynamically unstable, and so it disappears from the phase diagram as
it should and can be seen from Fig. 3. This must be compared with pure Hartree–Fock
calculations where a FM phase still exists [7]. Up to now, we could solve the problem only
for relatively large couplings because for lower values of Coulomb interaction the
region of phase separation becomes more difficult to determine numerically. The PM/AF
phase separation is mainly due to the correlation effects in the PM phase only, because the
spectral density of the self-energy matrix is positive semidefinite and for the FM and AF
phase turns out to have a small trace near half filling, so that the correlation is predominantly
of Hartree–Fock type in the non-PM phases.
The densities of states (DOS) in Fig. 4 finally show the characteristic behavior of the
chemical potential in the paramagnetic state, passing a correlation-induced van Hove peak
with increasing carrier concentration from half filling (A) up to large doping (F). From the
phase separation at however, we conclude that the PM states with
are metastable and not of real relevance. In the interval the experimentally
relevant DOS is a superposition with doping-dependent weight of both the DOS of the AF
insulator (D) and the PM state at (E). Upon further doping, the DOS remains
purely paramagnetic [see (F)]. For the parameters used in (E), the chemical potential
happens to lie very near to the peak maximum; for other values, this is also
approximatively the case.
376 Fritzenkötter and Dichtel
5. CONCLUSION
An approximative calculation of the DOS covering both weak and strong coupling sit-
uations and different magnetic phases shows that in the interesting doping regime the phase
diagram differs qualitatively from the Hartree-Fock result. Contrary to the former, no ferro-
magnetic phase occurs and phase separation between an AF nearly neutral phase and a PM
phase at slightly overdoped carrier concentration takes place. Electronic phase separation is
a well-known result in models with large on-site Coulomb correlations. Whether this effect
supports the formation of electronically inhomogeneous regions, say stripes or clusters of
carriers, or is suppressed by the neglected long-range Coulomb effects remains an open
question. We obtained the result in a realistic DOS calculation of the three-band model.
Thus, we show that it is an effect to bring the Fermi level already for small doping values to
the van Hove singularity and pin it near this singularity over a quite large doping interval.
REFERENCES
1. J. Fritzenkötter and K. Dichtel, J Supercond. 9, 449 (1996).
2. A. Martin-Rodero and F. Flores, Sol. Slate Commun. 44, 911 (1982).
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7. X. Y. Chen, W. P. Su, C. S. Ting, and D. Y. Ying, Sol. State Comm. 67, 349 (1988).
The Quasi-Particle Density of States of
Optimally Doped Bi 2212: Break-Junction
vs. Vacuum-Tunneling Measurements
PACS numbers:
1. INTRODUCTION
The importance of tunnel spectroscopy in giving essential indications on the micro-
scopical pairing mechanism in high superconductors is clear. Until a short time ago, all
the best measures of tunnel spectroscopy in single crystals of
gave a gap around 20–25 me V, and therefore a ratio of the order of 6 [1–4]. Among
the most recent and reproducible measures on optimally doped we
remember our break-junction ones that allowed us to reproducibly determine the spectral
function of electron–boson interaction by means of the solution of Eliashberg equations in
1
INFM—Dipartimento di Fisica, Politecnico di Torino, 10129 Torino, Italy.
2
P.N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia.
377
378 Gonnelli, Ummarino, and Stepanov
s-wave symmetry and in presence of a normal density of states (DOS), depending on the
energy [5–7]. More recently, some remarkable measures in optimally doped Bi 2212 (espe-
cially of STM type) appeared [8–11] that have shown more peculiar features: peaks of the
conductance at approximately 35–40 meV, values of the ratio of the order of
8–10, and presence of conductance peaks also at (quasi-particle pseudogap). These
characteristics have never been observed before also in the best crystals and, therefore, it
comes spontaneous to ask for which reason the various measures on apparently similar
samples turn out to be in apparent contrast. In this article, we discuss and compare the
most recent tunnel measures in optimally doped Bi 2212 crystals in order to understand
the origin of the differences. Due to the rather strong evidence of the d-wave symmetry
of the wave function in quasi-2D layered superconductors [12], we analyze the experi-
mental curves by using the Eliashberg equations [13–15] for the strong electron–boson
interaction in such a symmetry and compare the results to those we obtained previously in
s-wave.
The conductances of Fig. 1 show some similarities but also large differences. The
most striking one is the position of the conductance peak that varies between 25 mV of our
break-junction data to 35 mV of STM data of Ref. [9], to 41 mV of STM data of Ref. [8].
The ratios calculated from these values are between 6.25 and 10.3.
These differences are really difficult to understand. All the three experiments are done
on high-quality single crystals with similar critical temperatures that are practically at
the top of the curve of as function of doping, so it is difficult to believe that doping
differences can be the cause of the different values. One could argue that in the break-
junction technique, the operation of breaking the crystal at low temperature can produce
a structural modification of the surfaces of fracture (due to the mechanical stress) with a
consequent reduction of of the junction. This is contradicted by the fact that we have
observed very broaden conductance peaks up to 92.8 K in these junctions, as it should
occur in a superconductor with in presence of strong fluctuation phenomena
[17].
A question now arises: Is a DOS with
compatible with the known models for a strong boson-mediated coupling both
in s- and in d-wave pair symmetry? In order to investigate the previous point we tried to fit
both our break-junction and STM data by Renner et al. [8] by means of a direct solution of
the equations for the retarded electron–boson coupling (Eliashberg equations), both in pure
s- and d-wave pair symmetries. As far as concern our recent Bi 2212 data, we reproducibly
obtained the electron–boson spectral function by inverse solution of the
s-wave Eliashberg equations. Subsequently, we numerically calculated the direct solution
of these equations in presence of a normal DOS depending on energy, finding a very good
agreement with experiments as far as is concerned, the shape of the
and the critical temperature
The extension of these results to the d-wave case is obtained by numerically solving the
generalized Eliashberg equations for the order parameter and the renormalization
function in a single-band approximation, having expanded
in terms of basisfunctions where is the azimuthal angle of the wave-vector
k in the ab plane. The expansion is truncated to the first-order term
and therefore we have
where and on one side and and on the other are the isotropic and
anisotropic parts of the electron–boson spectral function and of the Coulomb pseudopoten-
tial, respectively. Here, for simplicity, we pose and where
is a constant [18,19]. As a consequence, two coupling constants and are defined
and the final symmetry of the solution [i.e., the symmetry of the gap function is
determined by their relative values, or if by a specific constraint on
the initial values of the isotropic gap. In practice, if the solution has a pure d-wave
symmetry in the sense that the gap function keeps only its anisotropic part whereas,
at the same time, the renormalization function shows only its isotropic one The role
of this isotropic part of the renormalization function is very important in the comparison
with experimental data because it has a broadening effect on the quasi-particle DOS similar
380 Gonnelli, Ummarino, and Stepanov
to that one produced by a lifetime broadening: the DOS peaks become less sharp and thus
very similar to the low-temperature conductance curves in Bi 2212. Details on this d-wave
strong coupling model and its direct solution in the real energy axis formulation can be
found elsewhere [20,21].
Fig. 2b. The great surprise occurs when we go to calculate the theoretical of these fitting
curves as the temperature where In the case of no impurities, we have
whereas in presence of the impurities we obtain These values
are completely different from the critical temperature of the junction mentioned in Ref. [8],
In pure s-wave symmetry and by using the same spectral function, we obtain
quite similar results as far as is concerned. In this case, the proper fit of is
obtained for which corresponds to
The impossibility to contemporary fit the gap value and the critical temperature of these
STM data on Bi 2212 in the framework of an electron–boson strong-coupling model both in
s- and d-wave symmetry is confirmed by the calculation of the ratios
382 Gonnelli, Ummarino, and Stepanov
The conclusion is that, according to our model, the experimental is too low to be
consistent with a conductance peak at 41 mV (or vice versa).
One can argue that the present results are strongly dependent on the particular form
of the spectral function used in the numerical calculations. This is not the case, as shown
in Fig. 3b, where the same d-wave ratio of Fig. 3a as function of for various values
is shown, for a spectral function made by a single Einstein peak at meV, which
corresponds roughly to the equivalent phonon energy of the The
behavior of the ratio as function of is practically unchanged, and the modest increase of
its maximum value as function of cannot justify the observed . and values.
In conclusion, provided that an appropriate spectral function is used, the real-axis
direct solution of the equations for the strong electron–boson coupling in d-wave symmetry
demonstrated effectively to fit well the tunneling conductance in all the energy range and the
critical temperature of Bi 2212 break-junction tunneling experiments. The same thing does
not occur for more recent STM data on the same material that exhibit features (particularly
quite incompatible with the d-wave strong coupling model. The reasons for these
differences are still under investigation, but due to the generality of the strong electron–
boson approach, we believe that explanations related to intrinsic properties of the Bi 2212
samples are rather improbable. In the near future, this d-wave strong electron–boson model
will be used to investigate the effect on the tunneling conductance of a peak in the normal
DOS around the Fermi energy that has been predicted as a consequence of the presence of
stripes in Bi 2212 [24].
REFERENCES
1. D. Mandrus et al., Nature 351, 460 (1991).
2. R. S. Gonnelli et al., in Advances in High-Temperature Superconductivity, eds. D. Andreone, R. S. Gonnelli,
and E. Mezzetti (World Scientific, Singapore, 1992); R. S. Gonnelli et al., Physica C 235–240, 1861 (1994).
3. S. I. Vedeneev et al., Physica C 235–240, 1851 (1994).
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6. R. S. Gonnelli et al., Physica C 282–287, 1473 (1997).
7. G. A. Ummarino et al., Physica C 282–287, 1501 (1997).
8. Ch. Renner et al., Phys. Rev. Lett. 80, 149 (1998).
9. Y. DeWilde et al., Phys. Rev. Lett. 80, 153 (1998).
10. M. Oda et al., Physica C 281, 135 (1997).
11. H. Hancotte et al., Phys. Rev. B 55, R3410 (1997).
12. D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995).
13. G. M. Eliashberg, Sov. Phys. JETP 3, 696 (1963).
14. J. P. Carbotte, Rev. Mod. Phys. 62, 1028 (1990).
15. P. B. Allen and B. Mitrovich, Theory ofSuperconducting Tc , in Solid State Physics 37 (Academic Press, New
York, 1982).
16. R. S. Gonnelli et al., J. Phys. Chem. Solids 59, 10–12, 2058 (1998).
17. A. A. Varlamov et al., cond-mat 9710175.
18. K. Sakai et al., Physica C 279, 127 (1997).
19. H. J. Kaufmann et al., cond-mat 9805108.
20. G. A. Ummarino and R. S. Gonnelli, p. 407 in this volume.
21. G. A. Ummarino and R. S. Gonnelli, unpublished, 1998.
22. C. Jang et al., Phys. Rev. B 47, 5325 (1993).
23. H. Chi and J. P. Carbotte, Phys. Rev. B 49, 6143 (1994).
24. A. Bianconi et al., Physica C 296, 269 (1997).
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Long-Range Terms in the Dynamically
Screened Potential of
We show that for almost all directions, the dynamically screened potential con-
tains contributions of longer range than ordinary Friedel oscillations
instead of , if an unusually strong Kohn anomaly of the dielectric function
appears. We show that such anomalies exist for the theoretical band structure of
if the Van Hove singularity lies near the Fermi level (i.e., in the
vicinity of an electronic topological transition).
1. INTRODUCTION
The dielectric function of metals has singularities or “anomalies” due to the inho-
mogeneous distribution of electrons in the momentum space. At such anomalies, is
continuous but behaves nonanalytically. These anomalies cause some effects in metals like
anomalies in the phonon dispersion relation (Kohn effect [1]) and long-range oscillating
terms in the screened potential of impurity ions (Friedel oscillations, [2]). We
discuss here the influence of such anomalies for high superconductors because even if
there are discussions about existence of a real Fermi-liquid, experimental results on band
structure [3] support such studies.
1
Institut für Festkörperphysik, Friedrich–Schiller–Universität Jena, Helmholtzweg 5, D-07743 Jena, Germany.
385
386 Grassme and Seidel
which are also denoted as auxiliary surfaces, touches the Fermi surface
This construction is shown in Fig. 1. In general, these critical wave-vectors form a frequency-
dependent surface in the space, denoted as surface of anomalies (SOA).
The evaluation of the integral in Eq. (1) similar to [7] gives the singular part of the
dielectric function
There is the coordinate perpendicular to the SOA, is only a cutoff parameter, is the
Heavyside function and is a sign that does not depend on The weight function
of the anomaly
The matrix in Eq. (5) determines the type of the anomalies, O type: sgn and
X type: sgn and furthermore the sign of the O type via Therefore,
the anomaly in Eq. (3) depends on local charakteristics of the endpoints of the critical wave-
vector, lying at the auxiliary and the Fermi surface.
the Fermi surfaces 2 and 4 (Table 1), one can conclude that there lies an isolated pole line
between at
For the critical wave-vector there is at its top (near the line greater than
at its origin (see Fig. 3). However, the Fermi velocity at the top of can be arbitrarily
small (by convenient choice of near the topological transition of
For such frequencies is and between these wave-vectors lies a pole line at
(see Fig. 3b). Because is far from in Fig. 3c, it must appear isolated.
From these considerations, we can conclude that the dielectric function of
has extraordinarily strong anomalies for a convenient choice of frequency
and doping (what determines , see [3,12]).
Long-Range Terms in the Dynamically Screened Potential 389
(6)
using only the scalar potential is allowed for nonrelativistically moving particles
with energies up to MeV [13]. Now we consider the charge distribution
of an oscillating dipole
which can be used as a model for a vibrating ion in the crystal latice (as a sum of a point
charge and a time-dependent dipole). The asymptotic solution (6) with the
charge distribution in Eq. (7) is similar to the static case [7]. For a given direction of
one gets a sum about “stationary points” in the space (lying at the dynamic SOA), which
tangential plane is perpendicular to
and the phase shift of these Friedel oscillations depend similar to Eq. (4) on local
characteristics (Fermi velocities and curvatures) of the isoenergetic surfaces at the endpoints
of the stationary wavevectors
For isolated pole lines with the solution in Eq. (8) is not valid. Developing
in the vicinity of the pole line (t and are coordinates at the SOA), one
gets from Eq. (6) asymptotic potential contributions with a longer range
with
The solution is similar to Eq. (8), but the potential behaves and the sum runs about
stationary points at the pole line. It can be shown that such potential terms exist
390 Grassme and Seidel
for almost all directions (i.e., they fill out a 3D volume area). These potential contributions
also have a longer range than that of parabolic lines [14]. Therefore,
they can dominate the screened potential.
REFERENCES
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Phys. Rev. Lett. 76, 1075 (1996).
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Science Publishers, 1989).
5. M. I. Kaganov and A. G. Plyavenek, Zh. Éksp. Teor. Fiz. 88, 249 (1985).
6. M. I. Kaganov, A. G. Plyavenek, and M. Hietschold, Zh. Éksp. Teor. Fiz. 82, 2030 (1982).
7. L. M. Roth, H. J. Zeiger, and T. A. Kaplan, Phys. Rev. 149, 519 (1966); A. Blandin, J. Phys. Rad. 22, 507
(1961).
8. W. E. Pickett, R. E. Cohen, and H. Krakauer, Phys. Rev. B 42, 8764 (1990).
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H. Rietschel, and H. Mutka, Z. Phys. B 80, 193 (1988).
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A. I. Liechtenstein, O. Rodriguez, I. I. Mazin, O. Jepsen, V. P. Antropov, O. Gunnarsson, and S. Gopalan,
Physica C 185–189, 147 (1991).
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Chemical Analysis of the Superconducting
Cuprates by Means of Theory
Itai Panas1
1. INTRODUCTION
A traditional approach to properties of macroscopic objects has been to focus on
assumed basic macroscopic key features, from which the phenomenology has been deduced.
We believe that such approaches will never, in a satisfactory way, articulate the origin of
superconductivity in the cuprates. Rather, the cause must be sought in the chemical
composition, crystal structure, and unique local spectroscopic properties of the material. As
crucial properties of an extremely inhomogeneous gap field are sought, we propose
the satisfactory approach to comprise an atomistic cluster model, which displays essences
of all key properties of the plane simultaneously. Such an unbiased tool is currently
used to extract information regarding which bands provide the charge carrier channels;
what is the role of a fluctuating AF order; how phonons affect the electronic and magnetic
1
Department of Inorganic Environmental Chemistry, Chalmers University of Technology, S-412 96 Göteborg,
Sweden; E-mail: itai@inoc.chalmers.se; Fax:
391
392 Panas
properties; what is the origin of holes segregation and stripes formation; what causes the
order parameter symmetry; what is the mechanism for Cooper pairs formation; and
how to understand off-diagonal long-range order (ODLRO) in the cuprates.
The nonlocal form of BCS theory is the main source of inspiration for any effort that
seeks to connect chemistry and superconductivity in the cuprates. A cluster model
is sought, which is able to mimic a resulting self-consistent gap quality [1] of the form
and explicit wave function based ab initio quantum chemistry is used similar to the
Bogoliubov equations, only we are able to address the chemical morphology of the lo-
cal gap field.
Presently, increasingly more experimental observations of electronic and lattice insta-
bilities are made in the cuprate materials [2]. Neutron diffraction data are analyzed in terms
of dynamic and static stripe phases [3], and theories are developed based on these features
being of crucial importance to superconductivity [4]. In this context, the objective
of this presentation is twofold as it seeks to demonstrate how a quantum chemical model
Hamiltonian for the cuprate superconductors is constructed, how it is evaluated, and outline
the microscopic results that emerge from a stripes perspective.
2. CONCEPTUAL BASIS
Modeling in quantum chemistry involves choosing electron correlation descriptions
and cluster models. These two components define the adequate quantum chemical model
Hamiltonian and are described below. The essences of all the sought qualities necessary
for successful modeling are included simultaneously and treated at the same level of the-
ory by the model Hamiltonian. The reg-CASSCF method (vide infra) is chosen because
near-degeneracy effects are expected to play crucial roles in a system that displays Cooper
instability in the charge-carrier channel and superexchange coupling in the magnetic de-
gree of freedom. Similarly, a cluster model is chosen with structure based on the unit cell
because crystal symmetry and strong correlation are believed to be essential features in the
phenomenology of the cuprates. The particular choice of cluster is a key element in the
modeling because the gap field in the cuprates is believed to be strongly inhomogeneous.
and are the occupation numbers of the spin orbitals The total energy expression for
Chemical Analysis of the Superconducting Cuprates 393
where and are the spin-orbital representations of the one-electron operator (kinetic
energy and nuclear attraction) and two-electron (electron repulsion) operators, respectively,
i.e.,
and are the reduced one-particle and two-particle density matrices where the spin
and space parts of the spin-orbitals are assumed to factorize
and
where is a specific simple increasing function of the four exponents of the primitive
gaussians that span Eq. (6). Particularly, it is noted that
394 Panas
where and
Cluster orbitals
Chemical Analysis of the Superconducting Cuprates 395
Effects of local ligand fields from the closest ions are accounted for by including such
ions explicitly in the model.
where and are the probability amplitudes for the two contributions. The dependencies
on and are not shown.
Pairing of charge carriers can be understood as holes cluster states formed by hy-
bridization of and bands (see Fig. 1 and [7,8]). Again, the pair-state is
easiest seen if expressed in independent particle states with molecular orbitals symmetrized
in accord with the crystal structure
where and are the probability amplitudes for the two contributions. Thus the Cooper
instability is realized by mixing near-degenerate independent-particle states of anisotropic
S and D symmetries. The dependencies on and are not shown in Eq. (15).
Local crystal fields from buffer ions are sensitive parameters controlling the pairing
stability (0–60 meV), and this stability increases with increasing lattice field from buffer
ions. It is emphasized that only pairing resonances in the energy range of the AF fluctuations
contribute to the superconducting ground-state, and that fields that are too strong localize
the holes in the system [12], producing the Zhang–Rice scenario [13].
Local signatures of nonadiabaticity in both spin and space symmetry descriptors of
the quantum mechanical ground-state are found in the charge carrier channel and magnetic
degree of freedom. This results from excitations in the former contributing to the correlated
many-body ground state when phase-coherent excitations in the a priori disjoint local mag-
netic degree of freedom are made. The depairing resonances are of
symmetry in Eq. (15), and the magnetic excitation is of symmetry (14). Cou-
pling to the remaining electrons in each of the and subspaces is necessary for
396 Panas
these excitations to become resonances that contribute to the correlated ground-state. The
local wave function that emphasizes the additional nonadiabatic coupling can be written
schematically as
ACKNOWLEDGMENTS
Support by grants from the Swedish Natural Science Research Council and the Swedish
Consortium for Superconductivity are gratefully acknowledged.
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1. A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Method of Quantum Field Theory in Statistical
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Superconductivity with Antiferromagnetic
Background in a Hubbard Model
1. INTRODUCTION
Many superconducting materials, cuprates, bismuthates, organics, and heavy fermion
systems have been discovered that might be dominated by strong correlations between
electrons. However, on the theoretical side, we still have no consensus about mechanisms
for these systems. In particular, we do not have a definite solution as to whether the Hubbard
model in higher dimensions, one of the simplest models, has a superconducting phase [1].
Difficulties arise in taking into account various possible instabilities of spin and charge,
coming from thermal and/or quantum fluctuations.
1
Department of Physics, Waseda University, 3-Okubo, Shinjuku-ku, Tokyo 169-8555, Japan.
2
NTT Basic Research Laboratories, Atsugi 243-0198, Japan.
399
400 Saito, Kurihara, and Suzuki
which shows that a superconducting phase exists in the vicinity of the Mott transition. The
first and second terms describe the nearest and next-nearest neighbor hopping, respectively.
We take the average band width as an energy unit [2,3]. We assume
in numerical evaluations.
In describing a Fermi-liquid phase, one of the most successful wave functions is the
Gutzwiller wave function [5]. Using it, Brinkman and Rice [6] succeeded in qualitatively
understanding the mass enhancement by approaching the metal-insulator transition. Al-
though improved wave function shows that the Brinkman–Rice metal-insulator transition is
absent for the perfect nesting case [2,7], it does not lose its classical meanings in the under-
standing of various experimental systems such as vanadates [8], a normal liquid 3He [9,10],
and titanates [11]. However, it does not describe kinetic exchange processes that lead the
system to an antiferromagnetic (AF) state in the strong coupling.
3. MOTT TRANSITION
The variational calculations are carried out by using the Metnzer–Vollhardt method [2],
which becomes exact in limit. The staggered magnetic moment at half-filling
is shown in Fig. 1. Other characteristic results are as follows: (i) near half-filling,
the system exhibits a second-order quantum phase transition from a Fermi liquid to an AF
metal; (ii) at half-filling it becomes an insulator, where the Hubbard gap opens up; (iii) there
exists the critical interaction strength for ordering and (iv) electron effective mass
Superconductivity with Antiferromagnetic Background 401
remains finite, in contrast to the Brinkman–Rice theory. These results seem to be consistent
with recent experiments on titanates by the Tokura et al. [11] and Katsufuji et al. [12].
However, a linear stability analysis on the AF metal shows that the compressibility
takes negative which, in a naive interpretation, would lead to phase separation [7].
This is the same as that of Landau, if we regard sublattice indices as spin indices. We
phenomenologically obtain the staggered spin susceptibility and the charge susceptibility
402 Saito, Kurihara, and Suzuki
as
has a lower energy than phase-separated normal state, where is a creation operator of
a quasi-particle for X sublattice. It describes a coexistence phase of superconductivity and
commensurate antiferromagnetism. The superconducting order parameter depends on the
sign of wave for and d wave for
By taking into account the superconductivity, we have calculated the charge suscep-
tibility χ c, as shown in Fig. 6. As a result, the compressibility restores a positive value
and the phase separation in a normal phase turns out to be an artifact.
This kind of restoration was first pointed out by Nozières et al. [14] in the context of a
crossover from a BCS state to a Bose–Einstein condensed state of tightly bound pairs,
bipolarons.
7. COEXISTENCE PHASE
In Fig. 7, we show the phase diagram thus obtained. Near half-filling, we find a
coexistence phase of superconductivity and commensurate antiferromagnetism. At exactly
half-fiiling or for stronger U, the superconductivity is absent, and we have also found the
Superconductivity with Antiferromagnetic Background 405
quantum phase transition between a pure AF phase and a coexistence phase. The coexistence
of superconductivity and AF order such as stripes or incommensurate magnetic order seems
to be recently observed in cuprate superconductors [15,16]. Our phase diagram may be
qualitatively related to the organic and/or cuprate superconductors [15,16].
8. SUMMARY
In summary, we made a variational analysis of a Hubbard model and found
a strong evidence for the coexistence of superconductivity with antiferromagnetism. We
show that the phase separation in the normal phase turns out to be an artifact by ignoring
superconductivity.
ACKNOWLEDGMENT
We are grateful to Prof. I. Terasaki for various useful discussions. In particular, we
appreciate his pointing out a possible relevance to organic systems.
REFERENCES
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d-Wave Solution of Eliashberg Equations
and Tunneling Density of States in Optimally
Doped Superconductors
In this work we discuss the results of the direct solution of equations for the
retarded electron–boson interaction (Eliashberg equations) in the case of d-wave
symmetry for the pair-wave function and in presence of scattering from impurities.
In order to obtain these results, we used the Eliashberg theory and a spectral func-
tion containing an isotropic part and an anisotropic one:
For appropriate values of the isotropic
electron–boson coupling constant and the anisotropic one solutions are
obtained with only d-wave symmetry for the order parameter and only s-wave
symmetry for the renormalization function. The results of our numerical sim-
ulations are able to explain the shape of the density of states, the value of
the gap, and the critical temperature of optimally doped superconduc-
tor as recently determined in our break-junction tunneling
experiments.
1. INTRODUCTION
Since the work of Allen and Dynes [ 1 ], it is well known that there is no formal limitation
to the critical temperature one can reach in the framework of Migdal–Eliashberg theory
[2–4] provided the electron–boson coupling strength is sufficiently strong. In the past few
years, many attempts have been made to fit some of the superconducting properties
using this theory, but the great most of the works discussed only the solution in s-wave
pair symmetry. However, there are now many experimental results that strongly suggest the
presence of d-wave pair symmetry, at least in layered cuprates.
1
INFM-Dipartimento di Fisica, Politecnico di Torino, 10129 Torino, Italy.
2. MODEL
Our starting point are the well-known generalized Eliashberg equations for the renor-
malization function and the order parameter whose kernels depend on the
retarded interaction the Coulomb interaction and the effective
band [8–10]. Here we assume for simplicity that k and lie in the ab plane (
plane). Thus we neglect the relatively small band dispersion and the gap in the c direction.
The solutions of the full equations for a tight-binding band show that the Green’s functions
are sharply peaked at the Fermi surface. Thus it is a good approximation to integrate over
normal to the Fermi surface from We use a single-band approximation where
the Fermi line is nearly a circle and is the azimuthal angle of k in the ab plane.
We expand and in terms of basis functions where
the first few functions of lowest order are and
Obviously‚
where and are the Fermi and Bose functions, respectively, whereas is a cutoff
energy. If we want to obtain an solution, we need only to replace the denominator of
and with
The equation for which we have not written, is homogeneous in In the
weak-coupling case its only solution is In principle, in the strong-coupling case
there is a chance that above some threshold a solution exists with a nonzero We
do not consider this rather exotic possibility and instead assume that the stable solution
corresponds to for all couplings [9].
In our numerical analysis, we put for simplicity and
where is a constant [12]. As a consequence, we define
and
410 Ummarino and Gonnelli
3. RESULTS
By solving in a direct way the real-axis Eliashberg equations we can now check if a
couple of and values exist that, together with a proper electron–boson spectral function
can reproduce the experimental density of states For doing this,
we compare the to the quantity
where is calculated from Eq. (6) by using Here,
is the electron–boson spectral function we previously determined by the
inverse solution of the s-wave Eliashberg equations applied to the same Bi 2212 break-
junction tunneling data [6] and is the corresponding coupling constant. Of course, this is
not a fully consistent procedure, but in absence of a program for the numerical inversion of
the d-wave Eliashberg equations, it can be regarded as a first-order approach to the d-wave
modelization of tunneling curves.
Figure 1 shows our tunneling experimental data (open circles) and the best fit curve
at 4 K (solid line) that is obtained for and and yields almost the exact
With a small amount of impurities in the unitary limit ( and
d-Wave Solution of Eliashberg Equations 411
ACKNOWLEDGMENTS
We deeply acknowledge the very useful discussions with O. V. Dolgov and S. V.
Shulga.
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Enhancement of Electron–Phonon Coupling
in Exotic Superconductors near a
Ferroelectric Transition
1. INTRODUCTION
Conventional superconductivity is attributed to a pairing interaction originating from a
phonon-mediated interaction. The motivation for the search of high-temperature supercon-
ductivity in the cuprates was this mechanism, based on estimates of a particularly strong
electron–phonon interaction [1]. Nevertheless, almost all the current work in this field at-
tributes the pairing to other interactions, such as Coulomb interactions, mediated by param-
agnons, for example [2]. Some of the reasons for the rejection of the phonon mechanism are:
1. The d-wave symmetry of the superconducting gap parameter.
2. The maximum value of due to the phonon-mediated mechanism was estimated at
about 30 K [3], and 30 years of research seem to confirm McMillan’s estimate.
1
Racah Institute of Physics, Hebrew University, Jerusalem, Israel.
2
Department of Physics, Geneva University, Geneva, Switzerland.
Here, is the Bohr radius in the c direction (perpendicular to the plane of the 2D electron
gas), m is the band mass of the electrons (assumed constant), and M is the mass of the ion.
For the cuprates, is about 200 K.
This expression for takes into account the screening of the very strong electron–
electron interaction by the ionic dielectric constant; Empirically, a negligi-
ble is derived from tunneling measurements on BKBO [15].
Conceptually, this derivation differs from more “conventional” models in that a very
strong effective coupling is considered. By BCS–McMillan theory,
For the phonon-mediated mechanism, the cutoff frequency is and
. This gives a maximum of about 30 K. To obtain higher values, mechanisms with
a higher cutoff frequency are considered, such as excitons, paramagnons, or even the Fermi
416 Weger and Peter
energy. With such a high cu toff, can be moderate, or even smaller than 1. We consider here
an opposite scenario; the cutoff frequency is very small whereas
the effective coupling constant is very large As a result, is increased by
about a factor of 2.5. The elimination of provides another factor of 3, thus we get a total
increase from about 30 K to about 200 K.
5. DISCUSSION
The discovery of stripes [23], their very wide occurrence, and the near constancy of
the stripe wavelength gives some weight to our conjecture that the stripe wavevec-
tor is the dressed Debye screening vector, and is a single parameter
Enhancement of Electron-Phonon Coupling 419
responsible for many of the unusual properties of the cuprates. There are a dozen or so
serious objections to the validity of the phonon pairing mechanism in the cuprates, involv-
ing many diverse physical phenomena (Section 1). If the causes for the objections were
independent, the phonon mechanism would be a very unlikely candidate to account for
high-temperature superconductivity. However, this is not the case. We see that replacing
removes the various objections simultaneously. The various
objections depend on one factor: The screening length being the smallest length in
the problem, as it is in normal superconductivity. When the screening length is
larger than the Bohr radius the inverse Fermi momentum and the
interatomic spacing radical qualitative changes occur in the physical properties. In
the present work, we briefly discuss some changes in the physical properties—the increase
in the d-wave symmetry of the gap parameter, the EVHS, the change in the analytic
properties of the Eliashberg equations, etc.—and hint at some other properties—the cutoff
in at causing a pseudogap, for example. Therefore, the objections to the
phonon mechanism as the source of high-temperature superconductivity are much weaker
than is generally believed.
In this interpretation, the stripes are not the cause of superconductivity. A high
is observed in optimally doped samples, where no stripes are seen. Rather, the stripes
are an indicator of some unusual behavior in the normal state; and this unusual behavior is
(in our opinion) the cause of high
The replacement of by is a frequency-dependent renormalization. It causes
the coupling constant to be renormalized as well, This is a new feature that does
not exist in conventional Eliashberg theory. Transition from weak-coupling BCS theory to
strong-coupling Eliashberg–McMillan theory involves the replacement of the bare mass m
by the renormalized frequency-dependent mass We claim that the transi-
tion from the well-known strong-coupling theory to an extremely strong-coupling scenario
requires us to introduce a renormalization of the coupling constant for empirical physical
(rather than mathematical) reasons. The failure of the phonon-mediated mechanism to gain
serious consideration is probably due to neglect of this factor.
A point emphasized by Pietronero et al. [24] is the inapplicability of Migdal’s theorem
because the phonon frequency is relatively high and the effective Fermi energy in the
neighborhood of the Van Hove singularity is small. Here, the cutoff frequency enters in
an essential way into the electronic properties, which are therefore highly anomalous at very
low energies. Therefore the condition for the applicability of Migdal’s theorem does not hold,
and the applicability of the Eliashberg equations is somewhat questionable. Nevertheless,
it is illuminating to see how the Eliashberg formalism works when the dielectricity is
introduced into this formalism, admittedly in an ad-hoc way.
ACKNOWLEDGMENTS
We benefited greatly from discussions with D. J. Scalapino, W. Kohn, M. Onellion,
and V. Z. Kresin.
REFERENCES
1. J. G. Bednorz and K. A. Muller, Rev. Mod. Phys. 60, 685 (1988).
2. P. Monthoux, A. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992); D. J. Scalapino, Phys. Rep. 250, 329
(1995).
3. W. L. McMillan, Phys. Rev. 167, 331 (1968).
420 Weger and Peter
4. J. P. Franck, in Physical Properties of High Temperature Superconductors IV (World Scientific, 1994), p. 189.
5. S. Massidda et al., Physica C 176, 159 (1991); C. O. Rodriguez et al., Phys. Rev. B 42, 2692 (1990).
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15. Q. Huang et al., Nature 347, 369 (1990).
16. F. Marsiglio, M. Schossmann, and J. P. Carbotte, Phys. Rev. B 37, 4965 (1988).
17. M. Peter, M. Weger, and L. P. Pitaevskii, Ann. Physik 7, 174 (1998).
18. M. Weger, A. Nowack, and D. Schweitzer, Synth. Met. 42, 1885 (1992); G. Ernst et al., Europhys. Lett. 31,
411 (1995).
19. C. A. Balseiro and L. M. Falicov, Phys. Rev. Lett. 45, 662 (1980).
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23. A. Bianconi, Phys. Rev. Lett. 76, 3412 (1996).
24. L. Pietronero, S. Strassler, and C. Grimaldi, Phys. Rev. B 52, 10516 (1995).
Features of the Structural Phase
Transitions in
1. INTRODUCTION
It is known that (M: 3d transition metal) with the simple perovskite
structure exhibits a metal-insulator (M-I) transition and its electronic state is characterized by
the highly correlated-electron system. Origins of the M-I transition, including the formation
of a charge stripe, have attracted attention. Very recently, the stripe ordering related to the
charge and orbital orderings was found in [1].
Among is a paramagnetic metal in and an
antiferromagnetic (AF) insulator in [2,3]. That is, the M-I transition occurs
around in this oxide. As for the crystal structure in only a crystal
system was reported to be cubic in and tetragonal in from the
analysis of x-ray powder diffraction profiles [2].
We so far examined crystal structures in by means of electron diffraction
and found that the structures at room temperature are divided into three groups [4]. That
is, the cubic structure with the space group of Pm3m appears only in and two
different structures characterized by the rotational displacements of the oxygen octahedron
1
Kagami Memorial Laboratory for Materials Science and Technology, and Department of Materials Science and
Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.
2
NISSAN ARC, Ltd., Yokosuka, Kanagawa 237, Japan.
2. EXPERIMENTAL PROCEDURE
samples in a whole composition range were prepared from initial pow-
ders of and by means of an arc-melting method in an Ar/20%
atmosphere. A x-ray powder diffraction profile from each sample made in the present
work confirmed that no impurity phase is involved in the sample. Features of its crystal
structure were examined by taking electron diffraction patterns and bright- and dark-field
images from a single-crystal region. The observation was made by using both H-700H
and H-800 transmission electron microscopes. In particular, the latter microscope with a
helium-cooling holder was used for in-situ observation. Specimens for the observation were
flakes obtained by crushing the sample.
In order to understand the details of diffraction patterns in these types of the crystal
structure, we examined an extinction rule of superlattice spots in each structure by taking
diffraction patterns with a lot of different incidences. As a result, there is no superlattice
spot in whereas the superlattice reflection spots exist only at in
and at and in
From a careful examination of the extinction rule for these superlattice reflection spots, a
space group was eventually determined to be Pm3m for for
0.6, and Pbnm for In addition, there coexist the and Pbnm structures
in Both the existence of these superlattice spots and their extinction rule
indicates that the and Pbnm structures involve the rotational displacement, as pointed
out in our previous paper.
From the determined crystal structures mentioned just above, the struc-
tural phase transition is expected around in Then we examined a
change in the crystal structure in a sample with on cooling from room temperature.
Figure 2a and b are, respectively, two electron diffraction patterns at room temperature
and 85 K that were taken from the sample. An electron incidence of both patterns
is parallel to the [301] direction. In the pattern at room temperature, Fig. 2a, the R-type
424 Arao, Miyazaki, Inoue, and Koyama
superlattice reflection spots indicating the structure are observed, in addition to the
fundamental spots due to the Pm3m structure. When the sample is cooled to 85 K, the
M-type superlattice spots appears in the pattern of Fig. 2b. In order to understand features
of a microstructure just after the transition, we then took darkfield images by using the
M-type superlattice spot.
The darkfield images at 85 K are shown in Fig. 3. An electron incidence is almost
parallel to the [301] direction. Note that no change in a microstructure could be detected in
brightfield images. In the image, we can see dark-line contrasts in a bright-contrast region
that are perpendicular to the [010] direction. Because the contrast is due to diffraction
one, the bright-contrast region should have the Pbnm structure. The dark-line contrasts
can be then identified as an antiphase boundary with respect to the rotational displacement
of the octahedron. From the existence of the large number of the antiphase boundaries,
it is further understood that a large number of the Pbnm-structure regions are nucleated
and grow in the transition. Eventually, actually undergoes the
structural phase transition, just as in other 3d transition metal oxides such as
and
4. CONCLUSION
The present experimental data shows that when the La content increases, the crystal
structure in changes as follows: Pm3m in in
0.6, and Pbnm in The and Pbnm phases coexist in
In particular, the and Pbnm structures were confirmed to be characterized by the
rotational displacement of the octahedron. In addition, the structural
phase transition was found to occur in on cooling.
ACKNOWLEDGMENTS
The present work was supported by grant-in-aid for Research on Specific Subject from
Waseda University (No. 97A-312).
Features of the Structural Phase Transitions in 425
REFERENCES
1. S. Mori, C. H. Chen, and C.-W. Cheong, Nature 392, 473 (1998).
2. A. V. Mahajan, D. C. Johnston, D. R. Torgeson, and F. Borsa, Phys. Rev. B 46, 10973 (1992).
3. F. Inaba, T. Arima, T. Ishikawa, T. Katsufuji, and Y. Tokura, Phys. Rev. B 52, R2221 (1995).
4. M. Arao, S. Miyazaki, Y. Inoue, and Y. Koyama, in Proceedings of the 10th International Symposium on
Superconductivity (ISS ‘97), A. Tanaka and M. Kojima, eds. (Springer-Verlag, Tokyo, 1998), pp. 219–222.
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Infrared Signatures of Charge Density Waves
in Manganites
1. INTRODUCTION
The substitution of ions by ions in produces several related
effects: (i) x holes per formula unit are injected into the lattice, leading to conversion of x
ions into ones and providing carriers for magnetic double exchange; (ii) the
oxygen octahedra around the ions lose their Jahn–Teller distortion; and (iii) the
hybridization of the Mn and O orbitals changes with x due to the smaller size of Ca ions
with respect to that of La ions. The combination of these effects results in a remarkably
complicated phase diagram for this manganite [1]. Although the high-temperature state is
paramagnetic (PM) at any x, the ground-state of the system changes from an antiferromag-
netic (AF) insulator (at ) to a ferromagnetic (FM) metal (at ). According
to a widely accepted point of view [2], this transition (as well as the related “colossal
1
Istituto Nazionale di Fisica della Materia and Dipartimento di Fisica, Università di Roma “La Sapienza,” Piazzale
A. Moro 2, I-00185 Roma, Italy.
2
Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974, USA, and Department of Physics,
Rutgers University, Piscataway, New Jersey 08855, USA.
2. EXPERIMENTAL PROCEDURE
The infrared spectra have been collected on polycrystalline prepared
as described in Ref. [ 1]. The material has been finely milled, diluted in CsI (1:100 in weight),
and pressed into pellets under a vacuum, as discussed in Ref. [6]. The infrared intensity
transmitted by the pellet containing the oxide and that transmitted by a pure CsI pellet
have been measured at the same T. One thus obtains a normalized optical density
that, as shown in Refs. [6,7], is proportional to the optical conductivity of the pure
perovskite over the frequency range of interest here:
The magnetic susceptibility of the pure powders has been measured in zero static
field at the frequency of 127 Hz in a commercial device based on the Hartshorn method.
if its shape is influenced by the random orientation of the Mn-O conducting planes [9],
confirms the polaronic behavior of the carriers pointed out earlier on the basis of reflectivity
measurements [10]. One may also notice that, as the temperature is lowered below
the mid-infrared absorption increases and the phonon peaks are increasingly shielded by
the mobile charges. Both these observations show increasing metallization of the sample.
The spectral weight transfer in Fig. 1 can be evaluated by considering the effective number
of carriers
430 Calvani, Dore, De Marzi, Lupi, Fedorov, Maselli, and Cheong
where is the real part of the optical conductivity and is a suitable cutoff frequency.
By taking into account Eq. (1), one can here define a similar quantity
as shown by the observation that all the absorption curves cross each other at
Indeed, the absence of any coherent peak at is confirmed by the monotonic decrease
in the dc conductivity (0) [ 1 ] of between room temperature and
The evolution with temperature of the spectral weight in the gap region is shown in
Fig. 4, where obtained by Eq. (3) is compared with the real part of the magnetic sus-
ceptibility . When cooling the sample, starts decreasing around whereas no
abrupt change is observed at the Néel temperature Moreover, this process is completed
well below when has reached its minimum value and an AF phase is established
in the whole sample. Like the incommensurability parameter and the dc resistivity
is also sensitive to the thermal history of the sample. When heating the sample,
starts increasing around 120 K, again well below The difference between the
infrared absorption spectra measured at 160 K when cooling and heating the sample can be
evaluated in the inset of Fig. 4a.
The behavior of the infrared absorption of in Fig. 3 at temperatures
where incommensurate charge ordering has been reported [5] suggests the formation of a
charge density wave (CDW). One can then extract from Fig. 3 the optical gap and
compare its behavior with that expected for a CDW. This can be done by considering that
at is similar to that of a semiconductor in the presence of direct band-to-
band transitions [12]. By remembering again Eq. (1), one can then fit to the experimental
432 Calvani, Dore, De Marzi, Lupi, Fedorov, Maselli, and Cheong
The curves thus obtained are reported as dashed lines in Fig. 3. They describe well the
resolved part of the gap profile at all temperatures with the same The resulting values
for are plotted in Fig. 4c. BCS-like fits for have been successfully applied to the
optical behavior of CDW in polar systems [13], even if the resulting values for
are much higher than that (3.52) expected in the weak coupling approximation. An analytic
expression for between and [14], which also holds under moderately strong
coupling [15], can be written as
Unlike for an ordinary CDW, here the AF background fully localizes the charges at a finite
which moreover depends on the thermal history of the sample. We
find that these effects can be taken into account by simply rescaling the temperature in
Eq. (5)
As shown in Fig. 4c, Eq. (6) fits well both series of data by using the same values
and One thus obtains
and are introduced into Eq. (6) for the cooling and heating cycles, respectively.
An interesting comparison can be done between the gap value measured here and the dc
resistivity reported for Therein, it follows the exponential
behavior The activation energy
(marked by the asterisk in Fig. 4c) is in excellent agreement with the present determination
of However, a refinement of the present infrared data would allow the investigation of
the nature of the gap (or possibly pseudogap) that appears slightly above
The close correspondence between charge ordering and antiferromagnetism, pointed
out in Fig. 4, is predicted by the double-exchange model. The present confirmation that the
charges are partially localized in the FM phase is rather surprising. The results of Fig. 4,
in connection with those of electron diffraction in the same powder, may
help to find an explanation. As already mentioned, the behavior of the incommensurability
parameter (reported in Fig. 2 of Ref. [5]) is quite similar to that of in Fig. 4a. At
240 K, where starts decreasing, weak peaks from a charged superlattice appear in
the electron diffraction spectra, corresponding to As T is lowered, decreases
to 0.01 at and vanishes below If a closed thermal cycle is performed,
exhibits a hysteretic behavior that is quite similar to that of in Fig. 4a.
The above-reported observations in the FM phase can point either to an incommen-
surate and homogeneous CDW or to discommensurations suggestive of phase separation
in the charge system. The former possibility seems to be unlikely because at the
Infrared Signatures of Charge Density Waves in Manganites 433
charge density is intrinsically commensurate with the lattice and because the electron–lattice
coupling here is strong enough to create small polarons. We remain therefore with the latter
assumption, which implies a phase separation scenario. One may expect that in the CDW
regions an AF phase is established by the superexchange interaction (evidence for such
coexistence is provided below for the sample). By assuming that is due
to discommensurations between charge-ordered domains, their average size should be [5]
where nm is the lattice parameter in the Mn-O planes and is
the order of commensuration. This gives at at If this is true, the
FM observed below could only take place in the disordered regions that separate the
AF clusters. In this context, the similar behavior of and between and
can be explained in a simple way. Indeed, all those quantities should be sensitive to
the average size of the AF clusters, which increases at the expense of the FM regions as T
approaches
It should be stressed that the above phase-separation scenario for at
is consistent with the present optical data. Indeed, the infrared waves average
out any inhomogeneity in the sample on a scale so that the mobile holes in the
FM regions may appear as excited states of the ordered charges confined within the AF
clusters. In Fig. 3, the mobile holes may produce the Drude-like absorption that partially
fills the gap of the CDW condensate.
However, in order to confirm that interpretation, one must show that the regions with
commensurate charge ordering may produce an infrared gap that behaves according to
Eq. (6). For this purpose, we show in Fig. 5 the mid-infrared spectra of
This sample exhibits commensurate charge ordering below with wave-vector
The spectra of Fig. 5 show the formation of a gap in the infrared
background, starting around By extracting as done for the sample with
one obtains the plot of Fig. 6. No appreciable effects around the PM-AF transition at
are found to influence the gap. is again well fitted by Eq. (6) with
(in good agreement with the above value of and
434 Calvani, Dore, De Marzi, Lupi, Fedorov, Maselli, and Cheong
4. CONCLUSION
The following conclusions can now be drawn. First, the present spectra of
show the optical response of CDWs interacting with a magnetic background. Such
effects can hardly be observed in low-dimensional systems where CDWs are usually de-
tected. As a consequence, we deal here with the optical response of a CDW that undergoes
a hysteretic transition. The temperature dependence of the gap is the same as pre-
dicted for a CDW under moderately strong coupling, provided that one renormalizes the
temperature scale by introducing a finite at which the AF order has “frozen in”
all the charges. This represents a novel generalization of the ordinary CDW model and
provides information on the strength of the couplings among the three systems involved:
charges, lattice, and spins. In the carriers are strongly coupled with the
lattice, as suggested by a good fit to Eq. (6) and by the result However, a weak
coupling between the CDW and the spin system is suggested by the behavior of the infrared
gap through the FM-AF transition. Both when cooling and heating the sample through the
FM-AF transition, continues to follow a BCS-like law as expected for an ordinary
CDW. However, the magnetic hysteresis reflects into the existence of one and of two
different “freezing temperatures”
Second, the present data confirm that the intriguing coexistence of FM and charge
localization observed in at intermediate temperatures does not contradict a
(polaronic) double-exchange mechanism, provided that one introduces a phase-separation
scenario. AF clusters (where the CDW is commensurate as in the whole sample)
are expected to coexist with disordered FM domains. Even in these latters, however, most
charges are confined within a few cells by the Hubbard repulsion at the sites. At
the long wavelengths typical of infrared radiation, these poorly mobile holes appear as the
excited states of the CDW condensate of the AF clusters.
Infrared Signatures of Charge Density Waves in Manganites 435
ACKNOWLEDGMENTS
We are indebted to Denis Feinberg and Marco Grilli for many helpful discussions.
REFERENCES
1. P. Schiffer, A. P. Ramirez, W. Bao, and S.-W. Cheong, Phys. Rev. Lett. 75, 3336 (1996).
2. A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett. 74, 5144 (1995), and references therein.
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3336(1996).
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5. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996).
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R9592(1996).
7. A. Paolone, P. Giura, P. Calvani, S. Lupi, and P. Maselli, Physica B 244, 33 (1998).
8. C. H. Rüscher, M. Götte, B. Schmidt, C. Quitmann, and G. Güntherhodt, Physica C 204, 30 (1992).
9. J. Orenstein and D. H. Rapkine, Phys. Rev. Lett. 60, 968 (1988).
10. K. H. Kim, J. H. Jung, and T. W. Noh, preprint.
11. Y. Okimoto, T. Katsufuji, T. Ishikawa, A. Urushibara, T. Arima, and Y. Tokura, Phys. Rev. Lett. 75, 109
(1995).
12. P. A. Lee, T. M. Rice, and P. W. Anderson, Solid State Commun. 14, 703 (1974).
13. T. Katsufuji, T. Tanabe, T. Ishikawa, Y. Fukuda, T. Arima, and Y. Tokura, Phys. Rev. B 54, R14230 (1996).
14. G. Burns, Solid State Physics (Academic Press, London, 1985), p. 649.
15. D. J. Thouless, Phys. Rev. 117, 1256 (1960).
16. A. P. Ramirez, P. Schiffer, S.-W. Cheong, C. H. Chen, W. Bao, T. T. M. Palstra, P. L. Gammel, D. J. Bishop,
and B. Zegarski, Phys. Rev. Lett. 76, 3188 (1996).
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Recent Results in the Context of Models
for Ladders
Recent calculations in the context of ladder systems, but with implications also for
two-dimensional (2D) systems, are described here. In particular, efforts
are concentrated on the influence of a hole–hole electrostatic repulsion on hole–
pair binding in the t–J model on ladders. It is concluded that the pairs are very
robust because a nearest-neighbor hole–hole repulsion ofstrength is needed
to break the pairs. In the second part of the paper, the spectral function of holes is
provided on up to clusters using a new numerical technique. A gap in the
spectrum caused by hole binding is observed, as well as flat regions near
Implications for studies of materials are discussed.
1. INTRODUCTION
The purpose of this paper is to summarize the recent efforts of our group in the study of
electronic models for copper–oxide ladder systems. Electrons moving on ladder geometries
present very interesting properties that have attracted the attention of the condensed matter
community [1]. In addition, several analogies with the two-dimensional (2D) cuprates exist,
and in many respects learning about ladders contribute to our understanding of In
Section 2, recent work going beyond the simple t–J model introducing hole–hole repulsions
is described. This is of importance for theories of ladders and _ because calculations
in this context are usually carried out without such a repulsion. In Section 3, the spectral
function of holes in doped ladders is shown using clusters as large as A gap is
observed in the spectrum as well as flat bands near in close analogy with results for
the 2D cuprates. The large clusters can be studied using a novel technique briefly described
here.
1
Department of Physics, and National High Magnetic Field Lab, Florida State University, Tallahassee, FL 32306,
USA.
437
438 Dagotto, Martins, Gazza, and Malvezzi
space recently introduced [13]. The Hamiltonian used is the t–J model at a realistic cou-
pling supplemented by a hole–hole repulsion where
is the hole number operator at site i. The range R of the repulsion was restricted to 1
and lattice spacings in Ref. [11] because the speed of convergence of the numerical
techniques (both variational) decreases as R grows. References to previous results in this
context can be found in Ref. [11].
Figure 1 contains the hole–hole correlation obtained on clusters
with open-boundary conditions (OBC) using DMRG (with up to states per block
and truncation error for the case where acts only at a distance of one lattice
spacing (i.e., 0 is a site at the center of the cluster, and the figure shows the hole–hole
correlation along the leg opposite where 0 is located (results for the other leg are similar).
C(j) intuitively is related with the probability of finding one hole at site j for the case where
there is already one at 0. Figures 1a-b corresponds to 2J, and 4J. Here, the results
change only slightly as the lattice grows, and the two holes remain close to each other,
indicating the existence of a bound state. Apparently a NN repulsion already larger than J
is not enough to destroy the bound state, although it weakens it. This repulsion does not
seem to cover the full spatial range of the effective attraction regulated by On the other
hand, Fig. 1c-d shows similar results, but now for and 10J where a substantial
change in the hole distribution is observed as the cluster grows. The spreading of the hole
over the whole lattice suggests either the breaking of the pair or a weak bound state. In the
large-V regime, one of the holes may act as a sharp “wall” to the other, which tries to spread
its wave function in an effective square-well potential.
A better estimation of the critical value at which pairs are no longer formed can be
found using the binding energy defined as where E(nh)
is the lowest energy in the subspace of n holes. A negative implies that a bound state
exists. In the absence of intersite Coulomb repulsion and at there is pairing of
440 Dagotto, Martins, Gazza, and Malvezzi
holes on 2-leg ladders, and we want to analyze what happens as V grows. Figure 2a shows
that for an NN repulsion, the binding effect continues even up to large values of V/ J.
Actually the region corresponds to a weakly bound state. The same figure
shows results for range repulsion. In this case, the critical coupling at which the
bound state is lost is between 3 and 4.
Figure 2b shows the critical coupling for two ranges of Coulomb interactions [11].
Supplementing this information with the known existence of hole pairs in the pure t–J
model at . as well as when independently of the range, allow us
to obtain a rough estimation of the region of hole pair formation by simple interpolation.
Figure 2b is a qualitative plot that summarizes the main result of the paper—namely, the
stability of the hole-bound states in t–J-like models depends on the value and range of the
electrostatic Coulomb interaction between holes. When the Coulombic term is restricted to
a realistic range a repulsion as large as weakens but does not destroy
the pair, implying that the effective range of attraction caused by spin polarization is larger
than one lattice spacing. Retardation effects (fully considered in the present calculation)
due to the different energy scales of spin and charge excitations (J vs t) likely contribute to
the strong stability of the bound states [ 1 1 ] . Note also that using, for example, the
pair size is This result is close to estimations of the coherence length
and for optimally doped La – 214 and YBCO, respectively [14] (using To
the extent that are similar on planes and ladders, apparently a realistic NN hole–hole
repulsion can actually improve quantitatively the predictions of the t–J model, without
destroying the pairs.
Summarizing, in this section and Ref. [ 1 1 ] it has been shown that the bound-state of two
holes in ladders is more stable than naively expected upon the introduction of a NN Coulomb
repulsion among holes. This result provides support to theories of ladders that predict hole
pairing based on electronic mechanisms that describe holes as immersed in spin-liquid
Recent Results in the Context of Models for Ladders 441
ladders. The reason is that in the basis, one of the states with the highest weight in
the ground-state is still the Néel state, in spite of the existence of a short AF correlation
length A small basis built up around the Néel state incorrectly favors long-range spin
order. However, if the Hamiltonian of the problem is exactly rewritten in, for example, the
rung-basis (9 states/rung for the t–J model) before the expansion of the Hilbert space is
performed, then the tendency to favor a small is natural because one of the dominant
states in this basis for the undoped case corresponds to the direct product of singlets in each
rung, which has along the chains. Fluctuations of the resonant-valence-bond
(RVB) variety around appear naturally in this new representation of the Hamiltonian,
leading to a finite Note that is just one state of the rung basis, whereas in the
basis it is represented by states, with the number of rungs of the 2-leg ladder.
In general, a few states in the rung basis are equivalent to a large number of states in the
basis. Expanding the Hilbert space [20] in the new representation is equivalent to working
in the basis with a number of states larger than can be reached directly with present-
day computers. For simplicity, this technique is referred to as the optimized reduced basis
approximation (ORBA).
In Ref. [19] it was shown that by calculating equal-time correlations (in particular, the
energy), a good agreement was found between DMRG results and those coming from ORBA
(details can be found in that reference). Then, we concentrate directly on the main results.
An interesting advantage of the method proposed here is that having a good approximation
to the ground-state expressed in a simple enough basis allows us to obtain dynamical
information without major complications. The actual procedure is simple: Consider that an
operator (which could be a spin, charge, or current operator) is applied to the ground-
state in the reduced basis denoted by If all states of the subspace generated by
the operation were kept in the process, typically one would exceed the memory
capabilities of present-day workstations if the truncated ground-state has about
states. Then, it is convenient to work with just a fraction of say keeping about 10%
of the states. In this way the subspace under investigation typically has a similar size as the
original reduced-basis ground-state, namely approximately rung-basis states. The
state constructed by this procedure is now used as the starting configuration for a
standard continued fraction expansion generation of the dynamical response associated to
Test of this approach are documented in Refs. [13,19].
In Figs. 3 and 4, the main results of Ref. [19] are reproduced. Only data for the
bonding band are discussed here. The -functions are given a width 0.1t throughout the
paper. Figure 3a corresponds to the undoped limit. A sharp quasi-particle (qp) peak is
observed at the top of the PES spectra, maximized at momenta that is, close
to the Fermi momentum for noninteracting electrons The qp has a very small
bandwidth, because it occurs in 2D models, due to the interaction of the injected holes
with the spin background [4]. Note that all peaks at momenta carry a similar
weight and the dispersion in this regime is almost negligible. This unusual result is caused
by strong correlation effects. The PES weight above (e.g., at is induced by the
finite but robust and its existence resembles the antiferromagnetically induced features
discussed before in 2D models. Figure 3b contains results at a low but finite hole density.
Several interesting details are observed: (i) the PES band near continues being very
flat; (ii) PES has lost (gained) weight compared with (iii) the total
PES qp bandwidth has increased; and (iv) the IPES band is intense near and it is
separated from the PES band by a gap. The observed gap is and is caused by
Recent Results in the Context of Models for Ladders 443
hole pairing. Actually, the binding energy calculated with DMRG/PBC for the same cluster
and density is truncation error In the overall energy scale of
the ARPES spectra, this difference is small and does not affect the study of the evolution
of the quasi-particle dispersion shown here. Note that the results of Fig. 3b are similar to
those observed experimentally near in the 2D cuprates’ normal state using ARPES
methods [18].
Figure 4a contains the weight of the qp peaks in the PES band vs density. Size effects
are small. The weight at diminishes rapidly with x, following the strength of the spin
correlations of Fig. 1c. This result and those in previous figures clearly show that the region
affected the most by spin correlations is approximately Figure 4b summarizes the
main result of Ref. [19], providing to the reader the evolution of the ladder qp band with x.
The areas of the circles are proportional to the peak intensities. At small x a hole pairing–
induced gap centered at is present in the spectrum, both the PES and IPES spectra are
flat near and the qp band is narrow. The PES flat regions at high momenta exist also
in the undoped limit, where they are caused by the short-range spin correlations. Actually,
the undoped and lightly doped regimes are smoothly connected. As x grows to the
flat regions rapidly loose intensity near and the gap collapses.
The similarities between ladders and planes imply that our results are also of relevance
for 2D systems along the line For instance, the abnormally flat regions near
induced by hole pairing (Fig. 3b) are in good agreement with ARPES experiments for
the 2D cuprates [18], and they should appear in high-resolution photoemission experiments
for ladders as well. Note that in the regime studied here with pairs in the ground-state, the
flat bands do not cross with doping but simply melt. When x is between 0.3 and 0.4, a
quasi-free dispersion is recovered. The results of Fig. 4b resemble a Fermi level crossing at
and beyond, whereas at small hole density no crossing is observed. It is remarkable
that these same qualitative behavior appeared in the ARPES results observed recently in
underdoped and overdoped LSCO [21]. These common trends on ladders and planes suggest
444 Dagotto, Martins, Gazza, and Malvezzi
that the pseudogap of the latter may be caused by long-lived tight hole pairs in the normal
state (as for the doped ladders studied here). However, note that the hole pairs themselves
are caused by the spin–liquid RVB character of the ladder ground state. Then, this idea
brings together the “preformed-pair” and “magnetic” scenarios for the high- pseudogap.
We propose that the presence of d-wave hole-pairs in the normal state induces an ARPES
gap at but these pairs exist as long as the is nonnegligible. In this mixed scenario,
the pseudogap and the short-range spin fluctuations are correlated.
Summarizing this section, the predicted ladder ARPES spectra along
are remarkably similar to experimental results for the 2D cuprates along the same line.
A common explanation for these features was proposed. ARPES experiments for ladders
should observe flat bands and gap features near in the normal state. Finally, note that
the novel numerical method discussed in Refs. [13,19] introduces a new way to calculate
dynamical properties of spin and hole models on intermediate-size clusters.
ACKNOWLEDGMENTS
E.D. is supported by NSF under grant DMR-95-20776. Additional support is provided
by the National High Magnetic Field Lab and MARTECH.
REFERENCES
1. B. Levy, Phys. Today, October 1996, p. 17. See also “Physics News in 1996,” supplement to APS News,
May 1997, p. 13.
2. M. Uehara et al., J. Phys. Soc. Jpn. 65, 2764 (1996).
3. E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B 45, 5744 (1992); T. Barnes et al., Phys. Rev. B 47, 3196
(1993); E. Dagotto and T. M. Rice, Science 271, 618 (1996), and references therein.
Recent Results in the Context of Models for Ladders 445
4. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994); and references therein.
5. D. Poilblanc et al., Phys. Rev. B 49, 12318 (1994), and references therein; H. Tsunetsugu, M. Troyer, and
T. M. Rice, Phys. Rev. B 49, 16078 (1994).
6. S. White and D. Scalapino, Phys. Rev. B 55, 6504 (1997), and references therein.
7. L. H. Tjeng, H. Eskes, and G. A. Sawatzky, in Strong Correlation in Superconductivity (Vol. 89), ed. H.
Fukuyama, S. Maekawa, and A. P. Malozemoff, Springer Series in Solid-State Sciences, (1989). See also
D. K. G. de Boer et al., Phys. Rev. B 29, 4401 (1984).
8. M. S. Hybertsen et al., Phys. Rev. B 39, 9028 (1989).
9. F. Barriquand and G. A. Sawatzky, Phys. Rev. B 50, 16649 (1994).
10. V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 56, 6120 (1997), and references therein.
11. C. Gazza, G. Martins, J. Riera, and E. Dagotto, preprint, cond-mat/9803314.
12. S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
13. E. Dagotto, G. Martins, J. Riera, A. Malvezzi, and C. Gazza, preprint.
14. M. Cyrot and D. Pavuna, Introduction to Superconductivity and High- Materials (World Scientific, 1992).
15. T. Takahashi et al., Phys. Rev. B 56, 7870 (1997); T. Sato et al., to appear in J. Phys. Chem. Solids 59, 1918
(1998).
16. T. Mizokawa et al., Phys. Rev. B 55, R13373 (1997).
17. T. Mizokawa et al., preprint.
18. D. S. Dessau et al., Phys. Rev. Lett. 71, 2781 (1993); K.Gofron et al., J. Phys. Chem. Solids 54, 1193 (1993).
19. G. Martins, C. Gazza, and E. Dagotto, preprint.
20. J. Riera and E. Dagotto, Phys. Rev. B 47, 15346 (1993); ibid. B 48, 9515(1993); and references therein.
21. A. Ino, C. Kim, T. Mizokawa, Z.-X. Shen, A. Fujimori, M. Takaba, K. Tamasaku, H. Eisaki,and S. Uchida,
preprint.
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Charge-Ordered States in Doped AFMs:
Long-Range “Casimir” Attraction
and Instability
1. INTRODUCTION
The extensive recent interest in doped quantum antiferromagnets (AF), particularly in
two dimensions, has been provoked in large part by the drive to understand the evolution
with hole doping of the layered oxide materials from AF Mott insulators to high-temperature
superconductors [1]. However, the behavior of these and other antiferromagnets, including
coupled-chain materials, are also of intrinsic interest, apart from their potential for under-
standing high- superconductivity. One class of proposals [2–4] for the ground-state of
the doped antiferromagnets involves spatially inhomogeneous “charge ordering.” Unfortu-
nately, numerical analysis of the stability of such states is often inconclusive because the
typical energy differences between states are small and the Goldstone modes (spin waves)
produce finite-size effects that decrease slowly with system size. The necessary limitation
1
Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106.
2
Department of Physics & Astronomy, University of California, Los Angeles, CA 90095.
2. MODEL
The number of spin degrees of freedom changes with doping, and therefore the Hilbert
spaces appropriate to the doped and undoped AF are different. For mathematical conve-
nience and to make possible a simple perturbation Hamiltonian formalism, it is preferable
to treat a model with a spin S operator on every site and treat any system with localized
holes as a limiting case in which the coupling between a set of “impurity” sites and its
neighbors goes to zero. Effects of virtual hole hopping can be included with a larger set
of modified exchange interactions in the neighborhood of the holes. Then, the spin Hamil-
tonian of the doped system differs from that of the pure AF only in the strength of some
Charge-Ordered States in Doped AFMs 449
exchange couplings:
where is the Hamiltonian for the perfect antiferromagnet, which, for concreteness, we
take to have only nearest-neighbor interactions on a hypercubic lattice,
and the perturbation Hamiltonian specifies a set of pairwise exchange interactions such
that, in the limit a spin near which the hole is localized is disconnected from the rest
of the system. Clearly, the interaction energy between hole clusters is obtained correctly
in this limit, although the cluster self-energy could depend on the interactions between the
fictitious disconnected spins. One exceptional geometry that we treat differently is a stripe
that is simultaneously an antiphase domain wall in the AF order. Such a stripe can be treated,
as shown in Fig. 2, effectively as a wall of bonds with altered exchange coupling so that we
work in the proper Hilbert space from the beginning.
integer multiples of so the distance dependent part of the energy per unit hyperarea is
which is proportional to the spin-wave velocity, c (in linear spin wave approximation [6]
where the subscript on the expectation value is a reminder that it is to be taken in the full
ground-state appropriate to that value of the coupling constant. However, that is practical
for giving analytic results only for an independent magnon Hamiltonian. The simplest of
these is the linear spin wave (LSW) approximation, which is quantitatively accurate for
large S, but which we also expect to be a reliable method for extracting the long-distance
physics for even for because already in AF order is very robust [7].
Within this approximation, the perturbation Hamiltonian is quadratic in the magnon creation
and destruction operators, so the coupling constant integrand can be expressed directly
in terms of one-magnon Green’s functions. These can, in turn, be calculated exactly by
standard methods for spatially localized perturbations. Symbolically, for a perturbation V
with nonzero matrix elements in a spatial basis for a limited number of sites, the Dyson
equation for suitable magnon Green’s functions G is
where m, n are summed only over those sites for which V has finite matrix elements, so
that G can be found by the inversion of the finite matrix over this limited space.
Clearly, the simplest example algebraically is that of two isolated holes at a separation
r with finite matrix elements of the perturbation only on the space spanned by each of the
two hole sites and its z nearest neighbors. Then the matrix inversion must be done in a
subspace of size
The Dyson Eq. (5) describes propagation between sites i and j as a series of terms of the
form free propagation to one of the two perturbation centers, followed
Charge-Ordered States in Doped AFMs 451
by a series of multiple scatterings from that and the other such center, with free propagation
between them and finally to the site j. For the multiple scattering from one of the centers,
the problem block diagonalizes in a representation based on the point symmetry of that
center (generalization of a “partial wave” analysis) [8], in this case the point symmetry of
the lattice. The totally symmetric representation (“s-wave scattering”) plays a special role
in that. The isolated artificial spin at the hole site is free to rotate, a zero energy bound state
that implies that the determinant vanishes at zero frequency. By continuity, there
is a scattering resonance at zero energy in the only channel with amplitude on the hole site,
namely the s-wave. This term dominates the long-distance Casimir attraction, for which we
find, in arbitrary dimension d,
Although the general finite cluster of holes does not share the full high symmetry of
this special case, we assert that the asymptotic distance dependence of the Casimir attraction
will be of the same form. The simplest argument is that the interaction should have the same
form as the sum of the interactions between each pair of holes that would obtain if that pair
were isolated as above. We point out that the same reasoning does correctly lead to the
Casimir form for the interaction between stripes in d dimensions. However, it is both
more rigorous and also suitable for making generalizations to less-restrictive models and
approximations to generate this result by the following perturbative calculation.
The artificial spins on the isolated cluster are again free to rotate, there are zero energy
bound states, and there will be magnon-scattering resonances from the cluster at zero energy.
However, we can remove that resonant behavior by adding a large magnetic field only to
the fictitious spins on the hole sites, removing them dynamically from the problem at the
start. Because these spins are fictitious, physically they must have no dynamical impact on
the results; it cannot matter how we impose these local fields on them. Then the multiple
scattering from a single cluster has no singular behavior at low energies. It is useful to resum
the Dyson equation partially so as to display explicitly the full scattering matrix from each
cluster. Within the two perturbation clusters (labeled 1 and 2), we denote the various Green’s
functions by the obvious notation between two sites in cluster 1, between sites one
of which lies in each cluster, etc. The ground-state energy shift is given by Eq. (4) as an
integral over correlation, or Green’s, functions within a single cluster only, for which
we write the Dyson equation in the form
where we have suppressed all specific site indices and the sums over them, and
is the full multiple scattering matrix of the single cluster The pure
crystal Green’s functions fall off rapidly with distance r between the clusters, so the
leading r-dependent term for clusters separated by a large distance r is the one involving
only two such factors, giving as the leading term in the interaction (“Casimir”) energy
452 Hone, Kivelson, and Pryadko
In the last line, we have taken advantage of the assumption that the single cluster scat-
tering matrix is nonsingular, and in fact featureless at low frequency. (Note also that
we have taken advantage of an imaginary time formalism to keep the denominators of
the free propagators positive definite.) These arguments continue to hold when some of
our simplifying assumptions are relaxed. Beyond the LSW approximation, the magnons
are renormalized, but that only modifies the magnon velocity c for the long wavelength
excitations that determine the above behavior.
We have also considered the effects of virtual hopping of the spins to unoccupied
(hole) sites. If, for example, in Fig. 3 the hole at the center virtually hops to the site above
it and back, the displaced spin in the intermediate state interacts antiferromagnetically with
the three remaining near neighbors to the center. This effectively introduces additional AF
exchange coupling between the “moving” spin and each of those three neighbors. Those
are indicated by two diagonal and one vertical dashed bond lines in the figure. Other dashed
lines indicate some of the other bonds in the neighborhood of the hole that are modified. In
all cases, the tendency is to weaken the AF order. Again, however, the asymptotic distance
dependence of the intercluster attraction is unchanged.
We discussed above the case of insulating stripes that isolate the AF region between
them. However, another situation of possible relevance is unsaturated extended hole con-
figurations, like stripes, along which the holes are mobile. Their effect may be modeled
by a reduced but nonvanishing AF exchange between the spins on either side, as shown in
Fig. 2. Within the above model, this amounts to a final value of coupling constant less
than unity. In the general case (where, for example, the exchange extends beyond nearest
neighbors) the above analysis leads again to the asymptotic behavior (see also the
discussion in Ref. [2]).
Charge-Ordered States in Doped AFMs 453
5. CONCLUSIONS
We have considered Casimir interactions between well-separated hole clusters in AFs.
For hole clusters or stripes in a uniform AF, this energy is uniformly attractive and gen-
erally falls off with distance as and respectively. The interaction is
quantitatively weak; for two holes in the AF, the interaction between next-nearest
neighbor holes is less than However, because the interaction falls slowly with dis-
tance, it is important for an analysis of the stability of static charge-ordered structures in
systems lacking long-range Coulomb repulsion. It has been conjectured [3,10] that phase
separation is a ubiquitous feature of lightly doped antiferromagnets and that consequently
there is always a first-order transition separating the undoped and doped states. Evidence in
support [11] of and in conflict [4,12] with this conjecture has been obtained from numerical
studies of small-size systems. Phase separation has been shown to occur [13,14] in the large
d limit of the Hubbard and t–J models and in the mean-field spiral states of the large N
t–J model [15]. The present results offer strong additional support for the validity of this
conjecture. Specifically, we claim that because of this Casimir-like interaction, any static-
ordered state of neutral holes will be thermodynamically unstable with respect to phase
separation at small-enough doping.
ACKNOWLEDGMENTS
We thank A. H. Castro Neto for informative conversations. This work was supported
in part by NSF grants DMR93-12606 at UCLA and PHY94-07194 at ITP-UCSB.
REFERENCES
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11. V. Emery, S. Kivelson, and H. Lin, Phys. Rev. Lett. 64, 475 (1990); C. Hellberg and E. Manousakis, Phys.
Rev. Lett. 78, 4609 (1997), and references therein.
12. H. Viertio and T. Rice, J. Phys. (Cond. Matt.) 6, 7091 (1994).
13. P. van Dongen, Phys. Rev. Lett. 74, 182 (1995).
14. E. W. Carlson, S. A. Kivelson, Z. Nussinov, and V. J. Emery, cond-mat/9709112 (unpublished).
15. A. Auerbach and B. E. Larson, Phys. Rev. B 43, 7800 (1991).
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Features of the Modulated Structure in the
Layered Perovskite Manganate
1. INTRODUCTION
In with the -type structure, there exists a modulated structure
around and its features have been investigated so far both experimentally and the-
oretically [1,2]. According to the previous work on the neutron-diffraction experiment,
superlattice reflection spots characterizing the modulated structure appear at
and which were due to the nuclear and magnetic reflections, respectively. Based
on these experimental data, they interpreted that the modulated structure is ascribed to
the static ordering of doped carriers. However, Bao et al. [3] indicated on the basis of
their electron-diffraction data that the superlattice reflection at can be iden-
tified as the nuclear scattering and the reflection at is just the second-order
harmonic.
1
Department of Materials Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.
2
Kagami Memorial Laboratory for Materials Science and Technology, and Advanced Research Center for Science
and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.
3
Structural Analysis Section, Research Department, NISSAN ARC LTD., 1 Natsushima-cho, Yokosuka, Kanagawa
237, Japan.
2. EXPERIMENTAL PROCEDURE
Ceramic samples used in the present work were prepared by standard procedure as
mentioned in the previous work [5]. The features of the modulated structure were examined
by taking electron-diffraction patterns. The observation was carried out in the tempera-
ture range between room temperature and 85 K by an H-800 type transmission electron
microscope equipped with a cooling stage with a liquid He reservoir. The He stage pro-
vides stability of temperature during the observation. Specimens for the observation were
prepared by an Ar-ion thinning technique.
The origin of modulated structure is simply discussed on the basis of the present
experimented data. From the existence of the second-order harmonics, the modulation
mode is identified as a phase modulation. Then the modulated structure should involve an
transverse atomic displacement in the crystal structure. Because a transverse wave with a
large wavelength is not basically interacted with an electron, the charge-ordering model
seems to be inappropriate for the origin of the structure. As a simple interpretation of these
experimental data, we believe that a polarization wave due to the metal ions would be
responsible for the 1q modulated structure.
458 Horibe, Komine, Koyama, and Inoue
4. CONCLUSION
In the present work, the features of the modulated structure in were
examined by transmission electron microscopy with the temperature range between room
temperature and 85 K. It was found that the modulated structure is characterized as 1q-
incommensurate structure and its modulation mode is simply due to the transverse wave
with the incommensurate wave-vector of
REFERENCES
1. Y. Moritomo, Y. Tomioka, A. Asamitu, Y. Tokura, and Y. Matsui, Phys. Rev. B 51, 3297 (1995).
2. B. J. Sternlieb, J. P. Hill, U. C. Wildgruber, G. M. Luke, B. Nachumi, Y. Moritomo, and Y. Tokura, Phys. Rev.
Lett. 76, 2149(1996).
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(1997).
5. Y. Horibe, N. Komine, Y. Koyama, and Y. Inoue, unpublished, 1998.
Numerical Studies of Models for Manganites
The Kondo lattice Hamiltonian with ferromagnetic Hund coupling and antiferro-
magnetic (AF) interaction between the localized spins is investigated as a model
for manganites. The phase diagram has been obtained using Monte Carlo simu-
lations in the limit where the localized spins are classical. At low temperatures,
three dominant regions were found: (i) a ferromagnetic phase, (ii) phase sep-
aration between hole-poor AF and hole-rich ferromagnetic (FM) domains, and
(iii) a phase with incommensurate spin correlations. An AF interaction between
classical spins enhances the tendency to phase separation. Comparisons of these
results with recent neutron scattering experiments are made, including phononic
degrees of freedom phase separation between two phases with spin-ferro order is
observed. One has staggered-orbital order, and the other is orbitally disordered.
1
Department of Physics and National High Magnetic Field Lab, Florida State University, Tallahassee, FL 32306,
USA.
459
460 Moreo and Yunoki
more fundamental Kondo model in which the (localized) electrons are ferromagnetically
(Hund) coupled with the (mobile) electrons. More precisely, the presence of phase
separation between hole-poor AF and hole-rich FM regions in the low temperature phase
diagram of the FM Kondo model has recently been observed [8]. Upon the inclusion of long-
range Coulombic repulsion, charge ordering in the form of nontrivial extended structures
may be stabilized.
The FM Kondo Hamiltonian [4,9] is defined as
where are destruction operators for one species of -fermions at site i with spin and
is the total spin of the electrons, assumed localized. The first term is the electron
transfer between nearest-neighbor Mn ions, is the Hund coupling, the number of
sites is L, and the rest of the notation is standard. The density is adjusted using a chemical
potential In this paper, the spin is considered classical (with rather than quan-
tum mechanical, unless otherwise stated. Phenomenologically, but here was
considered an arbitrary parameter (i.e., both large and small values for were studied).
The results of Ref. [8] are summarized in the phase diagram shown in Fig. 1. The
diagram in the figure corresponds to 1D but similar results were obtained for 2D and
infinite dimension [8]. At low temperatures, clear indications of (i) strong ferromagnetism,
(ii) incommensurate (IC) correlations, and (iii) a regime of phase separation were identified.
For and 2, finite size effects were found to be small for the lattice sizes used in this
study, although the PS-IC boundary in 2D was difficult to identify accurately. Results are
also available in small 3D clusters and qualitatively they agree with those in Fig. 1. In the
small region, IC correlations were observed, but in a region of parameter space not
realized in the manganites.
The addition of an AF coupling between the localized spins produces interesting
modifications to the phase diagram [10]. At large Hund coupling, phase separation continues
Numerical Studies of Models for Manganites 461
where is the spin of the mobile electrons. The rest of the notation
is as in Eq. (1) and the indices a, b indicate the orbitals and take values 1 and 2. The
electron–phonon interaction is given by
The variables Q1 and Q3 are classical and allowed to vary between In Fig. 4, typical
results are shown at a density of -electrons equal in average to one per site and using
S(0) and are the Fourier transforms at momentums 0 and respectively,
Numerical Studies of Models for Manganites 463
of the real-space correlation functions among the classical spins. At a clear change
from spin-ferro to spin-antiferro order is observed. This is reasonable because at large it is
expected that symmetry breaking will occur and the one orbital results of the single orbital
Kondo model will be recovered. T(0) and are Fourier transform at momenta 0 and
respectively, of the real-space orbital–orbital correlations. For details, the reader can consult
Ref. [13] and references therein. Here, it is sufficient to know that when T(0) is large, the
orbitals are uniformly ordered (i.e., the same orbital dominates in all sites), whereas if
is large, then the orbital with the lowest energy (dynamical effect) alternates from site to
site. A transition from one to the other is observed at , together with the change in
the character of the spin order. At a transition from an orbital disordered regime to
a staggered orbital ground-state is observed. Both sides of are spin FM.
It is very important to remark that phase separation is also present in the model with
Jahn–Teller phonons. It appears in three regimes:
1. At low electronic density phase separation is observed between electron-poor
AF and electron-rich FM regions, similar to the way it occurs in the one-orbital Kondo
model after the addition of a direct coupling between the classical spins.
2. At large near phase separation occurs in the regime where at the
spin AF, order is stabilized. This region, again, is the analog of the phase separation
observed in the one orbital model.
3. A novel regime of phase separation has been observed near and at intermediate
values of In this case, the phase separation occurs between two spin-ferro phases.
One has antiferro orbital order and the other has weak orbital correlations (similar to
the way it occurs on both sides of in Fig. 4).
In Fig. 5, the density vs chemical potential obtained with a Monte Carlo simulation on
a 10-site cluster is presented at At close to both 1 and 0, a region of unstable
densities is observed. This regime may be of relevance for the real manganites, and its
importance is discussed in Ref. [13].
464 Moreo and Yunoki
SUMMARY
In this paper and Refs. [8,10, and 13], the phase diagrams of models for manganites
have been presented using computational techniques. Some of these models were formulated
decades ago, but only now they can be analyzed using a variety of numerical many-body
algorithms. Contrary to previous expectations, no indications of canting order have been
observed. This tendency is replaced instead by phase separation, which occurs for a variety of
manganites models at both low- and high-electronic density. The experimental implications
of this new regime have been discussed in previous publications. It is reasonable to assume
that including Coulomb interactions explicitly in the problem will transform the phase
separation regime between electron-rich and electron-poor regions (or hole-rich and hole-
poor) into clusters of one phase immersed into the other. The stabilization of more exotic
structures such as stripes is a clear possibility in this context. It may occur that the charge-
ordering regimes observed experimentally in manganites may be related with the phase
separation discussed here, again, when proper Coulomb interactions are incorporated into
the problem.
ACKNOWLEDGMENTS
A.M. is supported by NSF under grant DMR-95-20776. Additional support is provided
by the National High Magnetic Field Lab and MARTECH.
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E. Dagotto, S. Yunoki, A. Malvezzi, A. Moreo, J. Hu, S. Capponi, D. Poilblanc, and N. Furukawa, preprint,
cond-mat/9709029.
9. N. Furukawa, J. Phys. Soc. Jpn. 63, 3214 (1994); ibid. 64, 2754 (1995).
10. S. Yunoki and A. Moreo, preprint, cond-mat/9712152.
1 1 . T. G. Perring, G. Aeppli, Y. Moritomo, and Y. Tokura, Phys. Rev. Lett. 78, 3197 (1997); ibid. 80, 4359 (1998).
12. S. Mori, C. H. Chen, and S.-W. Cheong, Nature 392, 473 (1998); Y. Murakami, H. Kawada, H. Kawata,
M. Tanaka, T. Arima, Y. Moritomo, and Y. Tokura, Phys. Rev. Lett. 80, 1932 (1998).
13. S. Yunoki, A. Moreo, and E. Dagotto, Phys. Rev. Lett. 81, 5612 (1998).
Pressure-Induced Structural Phase Transition
in the Spin-Ladder Compounds
with
The evolution at high pressure of the title compounds cell parameters has been
investigated up to GPa by synchrotron x-ray diffraction (XRD) using a dia-
mond anvil cell and an imaging plate 2D detector. The compressibility is found
to be highly anisotropic, with easy compressibility along the stacking axis of the
structure. The effect of applied pressure appears to be similar to the internal pres-
sure effect brought about by Ca substitution. A strong anomaly in the a parameter
variation is observed at pressures increasing with Ca content in the 6–8 GPa
range. This anomaly could be related to the disappearance of superconductivity
observed in the same pressure range.
1. INTRODUCTION
Following the discovery of high- superconducting cuprates, the theoretical investiga-
tion of the magnetic and transport properties in low-dimensional copper oxide systems has
become increasingly active. Recently, the so-called spin-ladder systems have drawn much
of the attention due to their predicted striking properties. For example, Dagotto and Rice [1]
predicted the existence of a spin gap when the antiferromagnetic (AF) coupling along rungs
is stronger than along legs, and Rice et al. [2] showed later that this property would exist
1
Laboratoire de Cristallographie CNRS, BP166, 38042 Grenoble Cedex 9, France.
2
Department of Chemistry and Materials Institute, Princeton University Princeton NJ 08540, USA.
3
ESRF, BP220, 38043 Grenoble, France.
4
MASPEC-CNR, via Chiavari 18/A, 43100 Parma, Italy.
5
Institute for Solid State Physics, University of Tokyo, Roppongi Minato-ku, Tokyo 106, Japan.
for even-leg ladders only. They also predicted the appearance of superconductivity when
holes are introduced in this system.
Superconductivity was detected for the first time in a compound containing spin-
ladders by Uehara et al. [3], who measured the resistivity as a function of pressure and
temperature in the compounds. They reported the appearance of a
superconducting transition in the 3–5 GPa range, with a maximum transition temperature
of about 12 K in samples with a large calcium concentration The structure
of can be described as the alternate stacking of layers having the
two-leg ladder arrangement and composition and layers made of isolated chains of
edge-sharing squares separated by planes containing the alkaline earth cations. The
ladder planes are formed by the corner-sharing connection of double chains of edge-sharing
squares. Within the double chains, the Cu cations are linked through 90° Cu-O-Cu
bonds, leading to a very weak magnetic coupling, and the ladders may be considered to be
magnetically quasi-isolated objects. The squares belonging to the chains above and
below are oriented at 45° with respect to the squares belonging to the ladders. The
incommensurate ratio between the Cu-Cu separation in both types of layers resulting
from this arrangement leads to a composite type of structure, which in first approximation
can be described by using two orthorhombic unit cells with parameters
and for the ladder-containing slabs, and
and for the chain layers. The a direction is in the plane of and
perpendicular to the chains and ladders, the b direction is perpendicular to the stacking, and
the c direction is along the chains and ladders. However, this schematic description of the
structure may be oversimplified, and thorough crystallographic studies [4,5] pointed out
the presence of buckling in both ladder and chain layers, and of interlayer Cu-O bonds that
may play an important role in the doping mechanism.
Optical measurements have shown [6] that the intrinsic hole doping due to the 2.25+
average copper valence is gradually transferred from the chains to the ladders on increasing
calcium concentration. Because this substitution is isovalent, the doping modification is
induced by altering the average size of the (Sr,Ca) mixed site, which leads to changes of the
Cu-O bond lengths in both chains and ladders. Due to the smaller ionic size of Ca cations,
the effect of Ca substituting for Sr can be viewed as a chemical pressure effect. Because
the superconducting properties of the compound appear only at high
pressure and for specific calcium concentrations, it is of the highest relevance to investigate
the evolution of the structure with pressure and composition. Therefore, we have carried out
a high-pressure synchrotron diffraction experiment at room temperature in the 0–10 GPa
range for a set of samples with and x between 0 and
13.6 in order to investigate the combined effects of applied pressure and modification of
the average ionic size and/or valence of the (Sr,M) mixed site.
2. EXPERIMENTAL
Five different samples of general formula were used for this exper-
iment, hereafter denoted as
(superconducting composition); The
first four samples were used to investigate the effect of pressure as a function of the average
size of the mixed site, whereas the data comparison between and allows us to
Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds 467
compare the effect of changing only the average valence of this site and cations
have very close ionic sizes). They were previously characterized for cationic stoichiometry
and phase purity by EDS analysis and laboratory powder x-ray diffraction (XRD).
The high-pressure synchrotron diffraction experiment was carried out at the ID9 beam-
line of the European Synchrotron Radiation Facility (ESRF) using angle-dispersive powder
diffraction with image plates as detectors. Samples were loaded into a diameter and
high gasket hole of a membrane diamond anvil cell (MDAC). Pressures were mea-
sured using the ruby fluorescence method [7]. Silicon oil was used as a pressure transmitting
medium. Although it is known to be a rather poorly hydrostatic medium above 6 GPa, it
was chosen because of its inert character with respect to the samples. In order to check the
effect of pressure gradients induced by this medium at higher pressures, two additional ex-
periments were carried out on samples and by using a 4:1 methanol/ethanol
mixture known to remain a better pressure transmitter to 10 GPa. The results obtained were
similar for the sample, but the sample started to decompose above 3 GPa in
the methanol/ethanol mixture, leading to the appearance of calcium- and copper-containing
phases. Nevertheless, the similar results obtained for the sample with both trans-
mitting media indicate that the observed effects described below are not due to pressure
gradients brought about by silicon oil at high pressures.
The monochromatic beam was selected by a horizontally focusing
asymmetrically cut bent Si(111) Laue monochromator [8]. It was vertically focused by a
curved Pt-coated Si mirror. The beam size on the sample was Diffraction rings
from the powder samples were recorded on an A3-size Fuji image plate located 441 mm
from the sample. The plates were scanned on a molecular dynamics image plate reader,
and processed by using the Fit2D software developed at the ESRF [9]. The images were
corrected for spatial distortion effects. Corrections for the image plate tilt with respect to
the direct beam were applied by using images from a standard silicon powder, which were
also used to calibrate the wavelength and the sample-to-detector distance. The corrected
images were averaged over 360° about the direct beam position, yielding spectra similar to
those obtained by classical diffraction techniques.
Due to the complexity of the structure, we concentrated our analysis on the evolution
of the cell parameters with pressure. For this purpose, we selected for each sample a set of
diffraction lines (generally at low angles) that were well isolated in the whole pressure
range, and obtained their angular positions by least-square refinement using a pseudo-Voigt
line-shape function. The cell parameters were obtained by transforming the positions
into d values using the Bragg's law. Note that in most cases, it was not possible to identify
isolated peaks from the “chain” unit cell, so only the parameters and
from the “ladder” unit cell could be obtained.
inside the ladder and chain layers. A similar anisotropy is to be expected for the effect of
pressure. It is interesting to point out that the a- and b-cell parameters of the sample
follow the regular evolution due to size effects, whereas the c parameter value is markedly
higher. This indicates that the charge effect brought about by the replacement of by
cations leads to modifications of the Cu-O bonding scheme mainly along the chain
and ladder directions.
The compressibility is the smallest in the a axis direction, with values ranging from
to The compressibility along c is about twice as large as that along
a, with values ranging from to These values are close to those
reported for the in-plane compressibilities of superconducting cuprates [10]. The b axis
compressibility is much larger than the in-plane ones, with values ranging between 6.3 and
These values are twice as large as the largest c axis compressibilities
reported for the superconducting cuprates.Moreover, the b axis compressibility markedly
increases with increasing average ionic radius of the mixed cation site. This could indicate
an increase of interlayer interactions when the size of the mixed site is decreased (i.e., when
the ladder and chain layers come closer to each other).
In Fig. 4, we present the relative evolution of the a cell parameters as a function of
pressure. The most striking feature observed is the presence of a lattice anomaly consisting in
a strong increase of the a cell parameter above 6–8 GPa, depending on sample composition.
The pressure value at which the anomaly takes place seems to increase with increasing
Ca content, going from GPa for to GPa for and to GPa for
For and the anomaly appears at GPa. For the latter sample,
above the a axis anomaly pressure value the c parameter seems to become stable and the b
axis compressibility starts to increase again. The former effect seems to exist also for the
other samples, although less pronounced. The diffraction spectra recordered at pressures
above the lattice anomaly do not present noticeable differences, such as peak splitting or
superlattice reflections, from those recordered below, indicating that no major structural
rearrangement is occuring at the transition.
Although the present data do not allow us to draw definitive conclusions about the
nature of this lattice anomaly, a possible origin could be the abrupt decrease of the ladder
and chain layers buckling induced by the applied pressure. Such an effect could indeed lead
to an increase or stabilization of the in-plane cell parameters a and c and allow the restoration
Pressure-Induced Structural Phase Transition in the Spin-Ladder Compounds 471
of the compressibility in the b axis stacking direction. The effect of applied pressure and
lattice anomaly on the physical and superconducting properties of the
compounds may be discussed on the basis of the present results and already reported
structural and physical measurements [4–6]. The main effect of the substitution of Ca for
Sr and of applied pressure is a strong decrease of the b-cell parameter, which corresponds
to an increased coupling between the ladder and chain layers and to a charge transfer
from the chains to the ladders. The similarity between the substitution and pressure effects
on the cell parameter indicates that the cation substitution acts as an internal chemical
pressure. However, the substitution effect alone is not sufficient to induce superconductivity
at low temperature, and an additional external pressure must be applied, with appearance
of the superconducting state at for the compound, which
corresponds to a cell parameter The disappearance of superconductivity at
might be related to the lattice anomaly, even though it seems to appear at higher
pressures for the sample) at room temperature. Low temperature and
high-pressure diffraction experiments are needed to confirm this model.
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Publishers, Inc., 1996), vol. 16, p. 1.
X-Ray Scattering Studies of Charge Stripes
in Manganites and Nickelates
1. INTRODUCTION
Charge and spin ordering into stripes in direct space have recently attracted inten-
sively attention due to their role in manganite colossal magnetoresistence (CMR) [ 1 ] and
cuprate superconductivity [2]. Since the discovery of CMR in the compounds
1
Department of Physics, University of Durham, Durham, DH1 3LE, UK.
2
Daresbury Laboratory, CCLRC, Warrington, WA4 4AD, UK.
3
XMaS CRG, ESRF, 38043 Grenoble Cedex, France.
4
Department of Physics & Astronomy, Rutgers University, Piscataway, NJ 08854, USA, and Bell Laboratories,
Lucent Technologies, Murray Hill, NJ 07974-0636, USA.
473
474 Su, Du, Tanner, Hatton, Collins, Brown, Paul, and Cheong
sample sizes and high wave-vector resolution. By using laboratory high-brilliance rotating
anode sources, and particularly synchrotron radiation, it is possible to obtain count rates
similar to those obtained by neutron techniques on far larger samples. This is important
not only to avoid the difficulties in growing large crystals but also because the quality (as
evidenced by the inverse FWHM of Bragg reflections) is often higher in smaller crystals.
This means that x-ray scattering is a useful probe of long-range correlations of charge
ordering.
2. EXPERIMENTAL PROCEDURE
Single crystals of and were grown by the flux
method at Bell Laboratories. These samples had been previously characterized by a number
of techniques [7]. The in-house x-ray experiments were performed at the University of
Durham. The crystal was mounted on the cold finger of a Displex closed-circle cryostat,
where the temperature was monitored with an Si diode to an accuracy of The
whole cryostat was mounted on a four-circle triple-axis diffractometer, which employed a
high-brilliance rotating anode generator operated at 2.8 kW with a Cu anode. The Cu
x-ray beam was selected and collimated by two flat (0001) pyrolytic graphite crystals used
as the monochromator and analyzer. Such an arrangement gives a relatively poor resolution
but very high intensity, and avoids any multiple scattering from higher energy harmonics.
The synchrotron experiments were performed at beamline 16.3 at the SRS, Daresbury
Laboratory, and at the XMaS CRG (BM28), ESRF. Double-axis geometry without using a
crystal analyzer was employed on both beamlines. Both antiscatter and detector slits were
carefully adjusted to the minimum possible size to reduce the background and increase the
wave-vector resolution.
stripes in manganites. With the higher resolution and count rate achieved with SR, the
charge-ordering peaks were checked on warming up and cooling down, and hysteresis of
the integrated intensity was clearly observed as Fig. 2a. A first-order structural phase transi-
tion in the vicinity of was observed as well, as shown in Fig. 2b. The Bragg peak (800) in
the low-temperature monoclinic phase was monitored on warming up across the region
when, at 243 K, the intensity abruptly dropped and eventually disappeared because of the
formation of the high-temperature orthorhombic phase. This can be seen as clear evidence
for the strong interaction between charge stripes and the structural phase transition.
Synchrotron radiation measurements were also undertaken on a single crystal
at station 16.3 at the SRS. A solid-state detector was used for detecting charge-
stripe scattering. A wavelength of 0.8Å was chosen to optimize the beam intensity from
the Wiggler beamline, increase the number of observable Bragg and charge-stripe peaks,
and reduce harmonic interference from and The face-centered tetragonal notation
was used, and all measurements were done in the (hhl) zone in reciprocal space. At 20 K,
a systematic search for charge-stripe reflections was done by scanning along four differ-
ent directions. Relatively sharp and strong charge-stripe peaks could be indexed as (3.33,
3.33, 1), (3.67, 3.67, 2), and (4.33, 4.33, 2), etc. These peaks all had an intensity approx-
imately that of Bragg reflections. A consistent wave-vector relationship in the (hhl)
zone is shown in Fig. 3, in which charge-ordered satellites are related to neighboring Bragg
peaks by a modulation wave-vector (2/3, 2/3, 1). We also observed a distinct broadening
of the charge-ordered satellites above 200 K, well below the charge-ordering temperature
of 240 K. More detailed measurements on the melting of the charge stripes were undertaken
X-Ray Scattering Studies of Charge Stripes 477
478 Su, Du, Tanner, Hatton, Collins, Brown, Paul, and Cheong
using XMaS at the ESRF, which has a peak flux almost 500 times higher than station 16.3
at the SRS. Strong and very clean singlet charge-ordering peaks were observed that could
be well fitted by a Voigt function, and accurate FWHM can be evaluated by considering the
convolution of instrumental resolution function. The correlation length along the longitu-
dinal direction was more than 7 times longer than the value of 250 Å observed
along the transverse direction. This is strong evidence that the charge stripes have a much
longer correlation within the Ni-O plane compared to that of out of the plane [i.e., the
charge stripes in have a quasi-2D-like nature]. Further evidence for 2D
behavior was obtained by critical scattering measurements. Measurements of the intensity
and width of the charge-stripes satellites were undertaken in the high-temperature regime
measuring the melting of the charge stripes under increasing temper-
ature. The following power laws, as shown in Eqs. (1) and (2), were chosen to fit the order
parameter (integrated intensity) and the inverse correlation length in both longitudinal and
transverse directions:
ACKNOWLEDGMENTS
We thank the Director of Daresbury Laboratory for access to facilities at Daresbury,
and to Profs. W. G. Stirling and M. J. Cooper for access to XMaS during the early stages of
commissioning. We acknowledge the help received from Dr. I. Pape, Mr. B. D. Fulthorpe,
and Mr. J. Clarke during the experiment at the ESRF. This work was supported by a grant
from the Engineering and Physical Sciences Research Council. Y. S. would like to thank the
CVCP for the award of an ORS studentship and the Department of Physics at the University
of Durham for financial support during his doctoral studies.
REFERENCES
1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995); J. M.
Tranquada, J. D. Axe, N. Ichikawa, A. R. Moodenbaugh, Y. Nakamura, and S. Uchida, Phys. Rev. Lett. 78, 338
(1997).
2. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 (1996).
3. A. R. Ramirez, J. Phys. Cond. Matt. 9, 8171 (1997).
4. C. H. Chen, S.-W. Cheong, and A. S. Cooper, Phys. Rev. Lett. 71, 2461 (1993).
5. J. M. Tranquada, D. J. Buttrey, V. Sachan, and J. E. Lorenzo, Phys. Rev. Lett. 73, 1003 (1994).
6. J. Zannen, Phys. Rev. B 40, 7391 (1989); V. J. Emery and S. A. Kivelson, Physica C 209, 597 (1993).
7. W. Bao, J. D. Axe, C. H. Chen, and S.-W. Cheong, Phys. Rev. Lett. 78, 543 (1997); S.-H. Lee and S.-W. Cheong,
Phys. Rev. Lett. 79, 2514 (1997).
Colossal Negative Magnetoresistivity of
Films in Fields up to 50 T
The low carrier mobility found in Mn perovskites implies that the dominant con-
ductivity mechanism is related to Mott hopping. As a modification to the original
model, we take into account that the hopping barrier is influenced by the relative
spin orientation at the two Mn ions involved in an elemental hopping process.
From this, we deduce a scaling behavior of the colossal negative magnetoresis-
tivity according to the Brillouin function in the ferromagnetic–quasi-metallic state
and proportional to a squared Brillouin function in the paramagnetic–semiconduct-
ing phase. Both predictions could be verified by pulsed magnetic field measure-
ments up to 50 T.
1. INTRODUCTION
Rare earth manganites with divalent substitution at the rare-earth site are character-
ized by a transition from a paramagnetic–semiconducting to a ferromagnetic–quasi-metallic
state at the Curie temperature Conductivity arises from the simultaneous presence of
and ions and the transfer of the electrons between them. The charge
transfer in the ferromagnetic (FM) phase is usually ascribed to the double-exchange mech-
anism [1], and it is essential to note that the apparently metallic conductivity is orders of
magnitude lower than in case of real metals. The charge carriers are therefore characterized
by strong localization, suggesting that a type of hopping transport in the framework of
Mott’s formalism should be taken into account [2]. Hopping of Jahn–Teller-type polarons
is meanwhile widely accepted to explain the conductivity on the paramagnetic side of the
phase diagram [3,4]. Both magnetic phases exhibit a colossal negative magnetoresistance
1
Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D,
B-3001 Leuven.
481
482 Wagner, Gordon, Trappeniers, Moshchalkov, and Bruynseraede
(CMR) effect [5], indicating that the charge transfer can be strongly activated by de-
creasing the misorientation between the local magnetization vectors at the different Mn
sites.
The origin of CMR is indeed a field-induced mobility enhancement, because Hall
measurements on magnetic perovskites gave no indication for any field dependence of the
charge-carrier density [6]. This article tries to extend the hopping description of conductiv-
ity, established for the paramagnetic phase, also to the FM regime by taking into account
the misorientation angle between neighboring magnetic moments as the main difference
between the two magnetic states. As a side effect, this description also predicts the correct
field-scaling of the CMR effect, which is phenomenologically parabolic for the paramag-
netic phase and hyperbolic-like for the ferromagnet.
2. EXPERIMENTAL
films (thickness 300 nm) were prepared by de-magnetron sputtering
onto heated substrates [7]. The epitaxial growth was confirmed by x-ray diffraction
(XRD) and the magnetic properties were investigated by a SQUID magnetometer, yielding
a Curie temperature The samples remain ferromagnetic–quasi-metallic down
to the lowest temperatures and the transition to an antiferromagnetic (AF), charge-ordered
state, as found for single crystals [8], could not be observed. The reason is
most probably the slight excess of compared to resulting in a deviation from the
exact 1:1 ratio of and The composition analysis on the films was performed by
Rutherford backscattering, which is accurate within 1 % of the relative concentration. After
characterization, the samples were cut into stripes and gold pads, evaporated
and annealed, served as low-ohmic current and voltage probes. Resistive measurements
were performed with the magnetic field orientation parallel to the film surface and the
current direction. The temperature dependence of the resistivity at fixed external fields was
studied in a superconducting coil (up to 12 T) with a variable temperature insert. The
negative magnetoresistivity at fixed temperatures was investigated in a pulsed magnetic
fields installation [9], reaching up to 50 T with a pulse duration of 20 ms. Heating effects
on the sample were found to be negligible.
in external fields of 50 T. The solid lines for the FM part of Fig. 2 are fit curves based on the
Brillouin function B and in case of the paramagnetic part we used fit functions proportional
to a squared Brillouin function. Both scaling laws can be motivated straightforward from
the model of spin-dependent Mott hopping. Conductivity in Mott’s original model (i.e.,
484 Wagner, Gordon, Trappeniers, Moshchalkov, and Bruynseraede
without taking into account the magnetic moments at the hopping sites), is given by [2]:
where R is the average hopping distance, the phonon frequency, the density of
states at the Fermi energy, L the carrier localization length, and the effective hopping
barrier between the and ions. Because the spin of the hopping electron
is aligned parallel with the core spin of (at the initial as well as the final site of
the hopping process due to Hund coupling), will be lowered in case of parallel spin
orientation between the Mn ions and enhanced for antiparallel orientation. The difference
in hopping barriers might be considered as an extra energy required to accomplish a rotation
of the electron spin during the hopping process in order to match into the spin configuration
of the hosting Mn ions. The modified barrier might be written as:
with being a proportionality constant taking into account the overlap of the carrier wave
function with the host ions. Inserting the modified barrier into the conductivity formula
(Eq. (1)) and calculating the decrease of resistivity compared to a random relative orientation
between Mn moments results in [7,10]:
where the temperature-dependent prefactor A is a measure for the amplitude of the negative
magnetoresistance effect. The average scalar product between the magnetization vector at
the initial and the final site of a hopping process can be calculated from the following
concept:
The total magnetization at each site is given as the sum of the uniform Weiss
magnetization (vanishing in the paramagnetic state) and a local correction term
The absolute value of is the difference between and magnetic saturation, whereas
it has a random orientation. External magnetic fields rotate the Weiss magnetization into
the main field direction and cause a common alignment of the independent vectors
along the field axis. This problem is analogous to the magnetization of a paramagnet and the
contribution of the vectors to the total magnetization can therefore be approximated
by the Brillouin function. The resistivity decrease of Eq. (3) can be rewritten as:
It is evident that in the paramagnetic phase only, the third term will survive and the average
scalar product is given by the product of two Brillouin functions. The correction
terms are small compared to the Weiss magnetization in the FM state, and the resistivity
decrease will be governed by the first two terms of Eq. (4). The increasing below the
Curie temperature causes even in the absence of any external field an increasingly parallel
aligment of neighboring magnetic moments, which in turn causes a lowering of the hopping
Colossal Negative Magnetoresistivity of Films 485
barrier and the corresponding resistivity as shown in Fig. 1. Applying external fields in the
FM state results in an additional resistivity decrease, which is controlled by the second term
of Eq. (4), which is linear in the Brillouin function. These results can be summarized as:
The exponent of the Brillouin function has either the value 1 (ferromagnet) or 2 (para-
magnet), the argument of this function contains the gyromagnetic ratio the Bohr
magneton, and the spin moment J at the hopping sites. The fit functions calculated on basis
of Eq. (5) comply very well with the experimental results shown in Fig. 2, yielding correctly
even the change in curvature observed for the CMR curves in the paramagnetic state.
The temperature dependence of the two fitting parameters A(T) for the CMR ampli-
tude and J(T) for the average spin moment at the individual hopping sites is given in Fig. 3.
The amplitude strongly resembles the resistivity in zero field, shown as a thin solid line. The
A(T) data are actually for all temperatures exactly 1 m below the zero-field resis-
tivities, meaning that in the limit of magnetic saturation, a small, temperature-independent
resistivity is obtained. The spin moments found from the fitting of resistivity data are con-
siderably higher than the values expected for individual or ions, suggesting a
kind of superparamagnetic behavior. The absolute J(T) values of up to 60 correspond to
ferromagnetically ordered spin clusters extending over a diameter of 3 to 4 unit cells of the
crystal lattice.
This cluster size agrees closely with the polaron diameter found by De Teresa et al. [3]
by means of magnetization measurements combined with neutron diffraction (1.2 nm),
and also with the carrier localization length obtained by the quantitative analysis of the
Mott formula Eq. (1) [10]. In this picture, the charge carriers are localized in a ferro-
magnetically aligned environment and transport occurs through hopping (eventually also
tunneling) of carriers between these small FM entities. The apparent decrease of J for
cannot be interpreted directly in the sense of a shrinking cluster diameter because
J(T) describes actually the magnetic behavior of the difference term between Weiss and
saturation magnetization becoming identical in the limit
486 Wagner, Gordon, Trappeniers, Moshchalkov, and Bruynseraede
4. SUMMARY
In summary, we have generalized Mott’s hopping model to the CMR class of materials
by introducing a barrier contribution depending on the relative magnetization orientation
at the initial and the final site of individual hopping processes. The relative orientation
was approximated by expressing the local magnetizations as a superposition of the Weiss
magnetization and an external-field controlled correction term. By taking into account the
presence or absence of spontaneous magnetic order, this model describes the considerably
different behavior of resistivity and magnetoresistivity in the FM and in the paramagnetic
state. The (squared) Brillouin scaling of the CMR effect is valid up to 50 T, corresponding
to a magnetically saturated state. The fitting parameters of the Brillouin description point to
the presence of FM spin clusters, which might be interpreted in terms of magnetic polarons.
ACKNOWLEDGMENTS
P. W. is a Marie Curie fellow of the European Union. This work has been supported by
the Flemish GOA and FWO programmes. The authors are strongly indebted to M. J. Van
Bael, D. Dierickx, and A. Vantomme for technical assistence.
REFERENCES
1. P. G. de Gennes, Phys. Rev. 118, 141 (1960) and references therein.
2. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford,
1979).
3. J. M. De Teresa et al, Nature 386, 256 (1997).
4. Guo-meng Zhao, K. Conder, H. Keller, and K. A. Miiller, Nature 381, 676 (1996).
5. R. von Helmolt et al, Phys. Rev. Lett. 71, 2331 (1993).
6. P. Wagner et al., Europhys. Lett. 41, 49 (1998).
7. P. Wagner et al., Phys. Rev. B 55, 15 699 (1997).
8. H. Kawano et al., Phys. Rev. Lett. 78, 4253 (1997).
9. F. Herlach et al., Physica B 216, 161 (1996).
10. P. Wagner et al., Phys. Rev. Lett. 81, 3980 (1998).
Synthesis and Characteristics of the
Indium-Doped Tl-1212 Phase
1. INTRODUCTION
After the discovery of high-temperature superconductivity by Bednorz and Müller [1]
in copper perovskites 1986, Bianconi and his group in Rome [2] provide a prototype for a
new heterogeneous material formed by a superlattice of quantum stripes at the atomic limit.
The structure of most of the high-temperature superconductors is divided into two ma-
jor structures. First is a superconducting layer, which acts as an electrical conduction
band. Second is the metal-O isolation planes (also called charge reservoir), where the metal
isolation planes usually are Tl, Bi, and Hg [3–5].
It is clear that more of metal-O isolation planes solely having electronic
configuration, such as Tl, Bi, Pb, and Hg, appear to play the key role to enhance the
superconducting properties. In order to understand the effect of the metal-isolation plane,
we substitute indium into the thallium sites in the Tl-1212. The electronic configuration of
1
Physics Department, Faculty of Science, Alexandria University, Egypt.
487
488 Awad, Gomaa, and Korayem
In is where as the Tl is and the ionic radii of both Tl and In are identical.
Thus, we expected similar tendencies for the In-doped materials as present for other (Tl,
Hg)-1212 [6]. The experimental results of the (Tl, Hg)-1212 indicate that a small addition
of Hg into the Tl site increases the transition temperature from 85 K to 117 K. Recently,
the addition of the In in the Tl-1223 phase was reported by Abou-Aly et al. [7]. They found
that this addition reduced the transition temperature from 122 K to 100 K.
In this work, we investigate the effect of In doping in the Tl-1212 superconductor by
preparing a series of samples with final nominal composition
with and Also, we report the effect of the weak applied magnetic
field up to 4.9 kG on the transition stages.
2. EXPERIMENTAL
Samples with nominal composition of
and were synthesized by two steps of solid-state reaction using stoichiometric amount
of and The precursor was pre-
pared by a solid-state reaction as previously reported [8J. The stoichiometric amount of
and were ground in a glove box using an agate mortar
under argon atmosphere in order to prevent absorption of moisture and Then the pow-
der was pressed into disk that was wrapped in a silver foil to reduce the loss of thallium and
indium that could eventually react with the quartz tube. Finally, the sample was inserted
in a tube furnace of length 30 cm and 3 cm diameter. The sample was heated to 870°C
by the rate of 400°C/h, held at this temperature for 2 h, and then furnace cooled to room
temperature.
Samples were characterized by x-ray powder diffraction. The x-ray diffraction (XRD)
scans (0.1 º/sec) were carried out on Philips PW1729 powder diffractometer using
radiation, The samples were examined in a Jeol scanning electron microscope
JSM-5300, operated at 15 kV, with resolution power of 4 nm. The real composition of the
samples’ content were determined using an Oxford x-ray microanalysis system (25 kV).
The electrical resistance was measured by using a conventional four-probe technique in
the temperature range from 30 K to 227 K in a closed cryogenic system. The samples used for
resistance measurements have dimensions of about The connections
of the copper leads with the sample were made using silver paint. The effect of applied
magnetic field up to 4.9 kG on the transition stages was made using an electromagnet. The
external magnetic field was applied perpendicular to the driving current.
suggests that the change from 6s to 5s metal states affects the energy level balance of the
metal and oxygen, which could result in the depression of superconductivity.
The normalized resistance-temperature dependence for
at different applied magnetic fields starting from 0 up to 4.9 kG is shown in Fig. 3. The data
in this figure indicate that, the transition to the zero resistance has two stages of transition.
The first stage (high-temperature region) starts at the onset temperature and
consists of a rapid decrease in the resistance to about 86 K. In the second stage of transi-
tion (low temperature region), the resistance smoothly decreases to zero as the temperature
approaches (zero resistance temperature). This is believed to be due to the anisotropy of
the material and the random orientation of the grains [10], or due to the percolation effects
across the grains [11]. Also, we notice that the magnetic field has no effect on the first
stage of transition, and the curves obtained at different applied magnetic fields overlap in
the high-temperature region. In the second stage we observe that for the higher value of
the magnetic field, the transition temperature is enlarged. These data could contain some
information concerning the spatial distribution of the “strong” superconducting grains and
“weak” superconducting boundaries.
The transition width determined from the difference between the onset temper-
ature and zero resistance temperature, for
is plotted as a function of the applied magnetic field in Fig. 4. It is clear that these
curves have three features. The first is that the temperature width is enlarged by increasing
the applied magnetic field. The enlargement in the temperature width could be explained as,
while the sample is cooling in a magnetic field applied perpendicular to the driving current
direction, a number of randomly oriented grains freeze and cluster in random positions. Such
a situation leads to more defected specimen. Consequently, an amount of flux is trapped.
As a result, the length of the vortex pinning is weakened and the specimen is in a more
resistive state by the loss of complete superconducting current pass. The second feature is
492 Awad, Gomaa, and Korayem
that the samples are more affected by the external magnetic field with increasing the indium
content. This means that the magnetoresistance of the samples containing indium is greater
than that of the indium-free sample. This result suggests that a partial replacement of T1
with In does not produce any significant flux pinning in this system. The third is that the
transition width sharply increases until and then almost a plateau in is
observed for a field The transition width is well fitted to the formula
and the value of n was found to vary from 0.16 to 0.38. A similar equation for studying
the variation of critical current density with the applied magnetic field was reported by Sun
et al. [12], who found that n has a negative value.
4. CONCLUSIONS
The superconducting compound with
and was formed at temperature 870°C for 2 h. Both lattice parameters a and c were
increased by increasing the In content. The transition temperatures are reduced by the In
doping, indicating that the doping of the superconductors compound with elements
reduces the transition temperature. On the contrary, the doping by enhances the
transition temperature. External magnetic fields affect only the second stage of transition,
but do not affect the first stage of transition. Also, the transition width is increased by
the increase of both external magnetic field and In content. The Indium doping does not
produce any significant flux pinning in this system.
ACKNOWLEDGMENTS
The authors are very grateful to Prof. Dr. A. I. Abou-Aly and Prof. Dr. I. H. Ibrahim
for useful discussion and suggestions. We also thank Mr. M. A. El hajji and Mr. A. Attia
for their technical assistance.
Synthesis and Characteristics of the Indium-Doped Tl-1212 Phase 493
REFERENCES
1. J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986).
2. A. Bianconi, International Conference on Superconductivity, edited by S. K. Joshi, C. N. Rao, and S. V.
Subramanyam (World Scientific Publishing Co., Singapore) (1990), p. 448.
3. Z. Z. Sheng and A. M. Hermann, Nature 332, 138 (1988).
4. H. Maeda, Y. Tanaka, M. Fukutomi, and T. Asano, Jpn. J. Appl. Phys. 27, L209 (1995).
5. A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott, Nature 363, 56 (1995).
6. R. Awad, G. A. Costa, M. Fenu, C. Ferdeghini, A. H. El-Sayed, A. I. Abou-Aly, and E. F. Elwahidy, Il Nuovo
Cimento 19D, No. 8–9, 1103 (1997).
7. A. I. Abou-Aly, R. Awad, and I. H. Ibrahim, unpublished, 1998.
8. R. Awad, G. A. Costa, M. Fenu, C. Ferdeghini, A. H. El-Sayed, A.I. Abou-Aly, and I. H. Ibrahim, unpublished,
1998.
9. L. Lechter, E. Toth, S. Osofsky, C. Kim, B. Qadri, and J. Soulen, Physica C 242, 21 (1995).
10. D. O. Welch, M. Suenaga, and T. Asano, Phys. Rev. Lett. 2386 (1988).
11. M. Celasco, G. Cone, V. Popescu, A. Masoero, and A. Stepanesou, Physica C 252, 375 (1995).
12. J. Z. Sun, C. B. Eom, B. Lairson, J. C. Bravman, T. H. Geballe, and A. Kapitulnik, Physica C 162, 687 (1989).
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High-Frequency Optical Excitations in YBCO
Measured from Differential Optical
Reflectivity
YBCO differential optical reflectivity (DOR) responses are measured in the tem-
perature range of 50–160 K at several optical frequencies. It is found that the
background response relates to temperature variation of the scattering rate. Above
the full DOR response may be attributed to a Drude plasma with a low mobil-
ity. Below accumulation of the superconducting condensate is accompanied
by appearance of a new optical spectrum in the range of 1.2–1.5 eV. We present
a model for a plasma having parameters close to that of a Mott transition that
demonstrates qualitatively the features observed experimentally.
1
E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305-4085.
We measured DOR only at five wavelengths, that is not adequate for spectral pre-
sentation of DOR responses. To overcome this deficiency, we made a correlation between
DOR components and the frequency-dependent direct optical reflection coefficient that was
carefully investigated earlier [2].
Direct optical reflection coefficient could be described by
a Drude formula
Although the Drude model of Eq. (1) is not adequate for describing optical phenomena
in YBCO, it is helpful as a numerical tool to establish qualitative relations between the direct
reflection data and our DOR data. Using Eq. (1), one may express the temperature derivative
High-Frequency Optical Excitations in YBCO 497
of DC reflectivity as
In Eq. (2), in accordance with Drude’s Eq. (1), we assume that dA/dT, and
are functions only of temperature and not of frequency. The frequency-dependent
functions are and These functions numerically calculated from
Eq. (1) with the parameters and obtained from fitting the data of Fig. 2 are shown
for the a direction in the insert of Fig. 2. For the b-direction, the shapes of the curves are
similar with maximums and zeroes shifted along the frequency axis.
In the absence of DOR data taken in a wide spectral range, using Eq. (2) allows us to
present full DOR response as a sum of products of temperature-dependent and frequency-
dependent functions in which frequency dependence may be determined from the compar-
ison with direct reflection data. This approach was further developed in [1] to prove that
above the DOR response belongs to a Drude plasma.
In the current work, we analyze different components of DOR response (background
and transition-related responses) and identify their physical origin. To do so, we define
the DOR background as a smooth function that is presumably a universal function of
temperature and sample orientation. One might expect this kind of contribution, for example,
from thermal expansion. At high and low temperatures, this function is well defined by a
scaling process. To diminish the uncertainty involved in the definition of the temperature
dependence of the background, we determined it as a smooth average of 10 datasets for
5 wavelengths and both a and b directions of the same sample; also, we compared the
obtained result with DOR responses of samples without or almost without transition-related
responses such as thin YBCO films. Data of Fig. 1 scaled at high temperatures
all coincide with each other and automatically coincide at low temperatures
In Fig. 3, a dotted line 1 shows the background frequency dependence at
and From the comparison of this curve with the functions
498 Fishman, Studenmund, and Kino
event tends to destroy the ordering caused by screening. In equilibrium, the carrier transport
scattering destroys the order on a time scale whereas the dielectric relaxation enhances
the order on a time scale When the system reaches the critical density, and should
be close to each other. If the transport scattering time is very small,
and the Coulomb screening is dynamically turned off, then in a certain range of densities
and temperatures the system may stay random. With where the scattering
time t is esimated as h/E and the hole binding energy The critical density n
is
which practically coincides with the static Mott transition criteria in Eq. (4).
If the ordered state is unavailable, the strongest correlation between the carriers will
be exchange pairing, which may overwhelm the Coulomb repulsion. As a result, Bose
condensation may occur in the system of strongly correlated hole pairs. Thus, we suggest
that a nontrivial state of matter may exist in a plasma having density exceeding the Mott
transition density.
No “quasi-particles with integrity” are theoretically identified in materials,
yet though an intuitive model may be provided by resonant scattering states in shallow
wells [5 ]. The major difference between the state of matter qualitatively described above
and other disordered systems, such as heavily doped semiconductors, is relaxation of the
order parameter. Similar situations may be expected under certain conditions in a high-
density exciton gas in semiconductors.
REFERENCES
1. W. R. Studenmund, I. M. Fishman, and G. S. Kino, Differential Optical Reflectivity Measurements of YBCO,
p. 529 in this volume.
2. S. L. Cooper, D. Reznik, A. Kotz, M. A. Karlow, R. Liu, M. V. Klein, W. C. Lee, J. Giapintzakis, D. M.
Ginsberg, B. W. Veal, and A. P. Paulikas, Phys. Rev. B 47, 8233 (1993).
3. A. V. Puchkov, D. N. Basov, and T. Timusk, J. Phys.: Cond. Matt. 8, 10049 (1996).
4. J. M. Ziman, Principles of the Theory of Solids (Cambridge, University Press, 1964).
5. D. Bohm, Quantum Theory (New York, Prentice-Hall, 1951).
Low-Temperature Structural Phase
Transitions and Suppression
in Zn-Substituted
1. INTRODUCTION
It has been found that is strongly suppressed in around
The suppression has been discussed in relation to the low-temperature
transitions to the Pccn/low-temperature tetragonal (LTT) phase [3–12]. Tranquada et al.
[13] interpreted from their neutron-diffraction data that the suppression originates from
the stripe ordering, which is expected to occur in a highly correlated electron system. Based
on their interpretation, the Pccn/LTT phase plays an important role in the pinning of the
stripe ordering. The pinning of the ordering can be also expected by introducing an impurity
1
Structural Analysis Section, Research Department, NISSAN ARC LTD., 1 Natsushima-cho, Yokosuka, Kanagawa
237, Japan.
2
Department of Materials Science and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.
3
Kagami Memorial Laboratory for Materials Science and Technology, and Advanced Research Center for Science
and Engineering, Waseda University, Shinjuku-ku, Tokyo 169, Japan.
2. EXPERIMENTAL
Powder samples of were prepared from CuO,
and ZnO in a coprecipitation technique using citric acid and calcined at 1173 K for 15 hr
in air, followed by the pelletization of the samples. The pellets were sintered at 1323 K for
100 hr in air and annealed at 773 K for 48 hr in In the present work, the observation was
performed for a sample with The energy dispersive x-ray
spectroscopy (EDX) measurement was carried out to check the Zn content of the sample
made in the present work. A value of in the sample was also determined by a SQUID
measurement and was then determined to be of about 8 K. In order to examine both crystal
structures and related microstructures in lower temperatures, an in-situ observation was
made in the temperature range between room temperature and 12 K. We used a H-800-type
transmission electron microscope equipped with a cooling stage liquid helium reservoir.
In the present experiment electron-diffraction patterns and bright- and dark-field images
were recorded on imaging plates in order to avoid a drift of a specimen during an exposure.
Specimens for transmission- electron microscopy observation were prepared by an Ar-ion
thinning technique.
3. EXPERIMENTAL RESULTS
The existence of the low-temperature transition was first checked by means of electron
diffraction. Figure 1 shows two electron diffraction patterns obtained from an
sample at room temperature. Electron incidences of Figs. 1a and b are
parallel to the [001 ] and [111] directions, respectively. In addition to fundamental spots due
to the high-temperature tetragonal (HTT) structure, only diffuse spots are seen in electron-
diffraction patterns as indicated by an arrow in Fig. 1b. This means that the sample at
room temperature has the HTT structure, and the HTT to low-temperature orthorhombic
(LTO) transition should take place just below room temperature. Actually, it was confirmed
experimentally that the transition occurs around 280 K. When the temperature is lowered
to 85 K, two types of diffraction spots appear in electron-diffraction patterns in addition to
the fundamental spots due to the LTO structure. Figure 2 is an electron-diffraction pattern
at 85 K. An electron incidence of the pattern is parallel to the [001] direction. The 100-
type forbidden spots due to the Pccn/LTT structure are observed in Fig. 2, as indicated
by A. Superlattice reflection spots are also detected at 1/2 1/2 0-type positions, as indicated
by B. This clearly implies that the Zn-substituted La-Sr-Cu-O sample undergoes the low-
temperature transition. It should be remarked that in the present experiment, charge-density
wave (CDW) spots found by Tranquada et al. [13] could not be detected in any electron-
diffraction patterns at 85 K.
Low-Temperature Structural Phase Transitions and Suppression 503
twin boundary has the Pccn/LTT structure and is called the line phase. When the specimen
is cooled to 12 K, a spotty phase B also appears in the interior of the LTO domain in
addition to the line phase A in Fig. 4b. In addition, an antiphase boundary with respect to
the tilt of the octahedron is observed as a dark-line contrast C in the bright-contrast region
with the Pccn/LTT structure near an edge of the sample, as was shown in our previous
work [9]. These features are therefore understood to be the same as those observed in other
La-cuprates [5–7, 9–12].
4. DISCUSSION
Now we discuss an origin of the suppression in Zn-substituted La-Sr-Cu-O. The
suppression has so far been explained in terms of the ordering of the stripes, which
was proposed by Tranquada et al. [13] as the basis of their neutron-diffraction experiment.
In the present work, however, the CDW spot related to the ordering was not detected
in the electron-diffraction patterns of the sample. This clearly
means that no stripe ordering exists in the oxide. However, the low-temperature structural
transitions take place in and the features are entirely the same
as those reported in other La cuprates. So, the suppression should be correlated with the
low-temperature structural transition instead of the stripe ordering.
5. CONCLUSION
From the present observation made by transmission electron microscopy, a low-
temperature transition exists in the sample. The 100 dark-field
images also show that the line phase and spotty phase with the Pccn/LTT structure ap-
pear along the twin boundaries between the LTO domains and in the interior of the LTO
domains, respectively. On the basis of the present experimental data, it is suggested that
Low-Temperature Structural Phase Transitions and Suppression 505
the low-temperature transition should play the certain role in the Tc suppression in Zn-
substituted
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(1994).
506 Inoue, Horibe, and Koyama
1. INTRODUCTION
Recent studies of YBCO using high-resolution electron microscopy by Etheridge [1]
at Cambridge University revealed the presence of structural perturbations at intervals of
the coherence length in the ab planes of this superconductor. The resulting nanostructure
of cells has originated in a struggle to relieve internal stresses in the planes. These
findings generated an important question: How can these perturbations influence electrical
and magnetic properties of YBCO? One interesting possibility is that the nanostructure
of cells in the ab planes could act as an array of Josephson tunnel junctions with the
modulation of the Josephson coupling energy between the cells in the ab planes and
along the c axis (Fig. 1). In this picture, the YBCO superconductor behaves like a granular
system with the grain size of the order of a few nanometers. Nanometer-size granularity is
known to occur, for example, in a conventional type-II superconductor such as NbN thin
film of The grain sizes of the film range from 3 to 9 nm, as revealed by
transmission electron microscopy (TEM). The film showed no preferential orientation, but
1
Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2J1.
2
Materials Science Division, Argonne National Laboratory, Argonne, IL 64039 USA.
the grains of similar orientation were frequently clustered together, with the cluster size
approaching 20 nm. This granular structure is reflected in a temperature dependence of
the critical current density which shows an Ambegaokar–Baratoff (AB) behavior
(characteristic of Josephson tunnel junctions) at low temperatures with a relatively narrow
plateau at temperatures below (with reaching at and a
Ginzburg–Landau (GL) dependence close to at temperatures above
Clem et al. [2] stated that the crossover from an AB to a GL form of occurs
when the Josephson coupling energy of an intergrain junction is approximately equal to
the superconducting condensation energy of a grain. At the crossover temperature, the GL
coherence length is of the order of the grain size. The effective grain size inferred from
measurement was calculated to be 22 nm, which corresponds to the cluster size of
20 nm. The effective grain size is larger than the average grain size of 6 nm measured by
TEM. This was explained in terms of the expected spread of intergrain Josephson-coupling
strengths, using the argument that the similarly oriented grains are among those that are
strongly coupled together. Below the crossover temperature the coherence
length (which is about 5 nm at is smaller than the grain cluster size, and is
governed by the weak Josephson coupling between the clusters (an AB behavior). Above
the coherence length is larger than the cluster size and the current does not “see” Josephson
junctions. In this case, is governed by the ability of the current to suppress the order
parameter (a GL behavior). During the course of this research, we were searching for the
Josephson response in the electrical transport and magnetic properties of YBCO thin films.
Regarding the temperature dependence of we were looking for similarities between the
Josephson Nanostructures and the Universal Transport 509
data for YBCO thin films and those obtained for a granular NbN film. We applied a different
experimental method that allowed us to measure both the magnitude of the critical current
and the relaxation (decay) of the current from the level due to the motion of vortices.
2. EXPERIMENTAL PROCEDURE
We used persistent current flowing in a ring-shaped YBCO thin film and the mea-
surement of its magnetic field with a scanning Hall probe to record the magnitude of
and simultaneously the decay of the persistent current J(t) from the critical level. This
technique is contactless and free of unwanted contributions to due to heating effects and
normal currents. The studies have been performed for 1 1 c-axis oriented YBCO thin films
deposited on two different substrates ( or ) with three different deposition
methods (laser ablation and rf and dc magnetron sputtering) in six different laboratories.
The measurements were done for zero-field cooled samples in the remanent magnetization
regime over a temperature range between 10 K and The persistent currents at different
levels up to a critical value have been generated by applying and subsequently switching
off an external magnetic field parallel to the ring’s axis (c axis of the film). The magnitude
of was inferred from the maximum magnitude of the persistent current's self-field at
the ring's center. The decay of the current’s self-field from the maximum level provided the
means to calculate the energy barrier for motion of vortices as a function of the current den-
sity The Maley’s procedure [3] has been used to calculate for a wide range
of J. We measured the relaxation rates from level every 5 K between 10 K and 85 K,
allowing evaluation of for different values of By eliminating the temperature
dependence from the data, a continuous curve of vs J was produced, which
consisted of multiple segments, each representing relaxations of the persistent current's
self-field at a fixed temperature.
Decays of the persistent current J ( t ) from level allowed us to calculate the effective
energy barrier for the motion of vortices in the film as a function of current density J. The
calculation revealed a universal empirical form of (see Figs. 2d–f and 3d–f). When
a GL-like behavior dominates
Josephson Nanostructures and the Universal Transport 511
where and are constants. These formulas imply that the factor is the same for
all YBCO thin films. This is based on systematic measurements of the persistent current’s
512 Jung, Van, Darhmaoui, and Kwok
This type of irradiation produces columnar tracks of density that can trap magnetic field of
0.5 T. Diameter of the tracks is of the order of 8–10 nm. Figure 5 shows the temperature
dependence of the critical current and the effective energy barrier as a function of the
current for YBCO film before and after irradiation. The results imply that there is a very
little change to both and due to this type of irradiation-induced defects. In
the irradiated film, at low temperatures increased by about 10% and decreased by 2°.
remained similar to that calculated for the unirradiated film. This suggests that the
motion of vortices through this YBCO film is dominated by the Josephson junction array.
In summary, we measured the temperature dependence of the critical current density
and the decay of the current from the critical value in 1 1 YBCO thin films manu-
factured using various deposition methods. The results show the universal behavior of both
and calculated from the relaxation data. Both quantities suggest that Josephson
effects are responsible for the observed dependence of on temperature and on J.
An AB-like component in is similar to that measured in a NbN film of nanometer
size granularity and implies the presence of coherent Josephson nanostructures in the ab
planes of YBCO. We postulate (taking also the irradiation experiments into account) that
the coherent Josephson junction array could dominate the pinning and flux motion in YBCO
thin films.
Recent theoretical work by Castro Neto [6] proposed a scenario for the problem of
phase coherence and superconductivity in striped cuprates that is based on the assumption
of a network of stripes and lakes of holes in the planes. In this model the stripes are
pinned by impurities, and a Josephson current is transferred from stripe to stripe via the
network.
514 Jung, Yan, Darhmaoui, and Kwok
ACKNOWLEDGMENTS
We are grateful to J. Preston, R. Hughes, J. Z. Sun, A. Fife, and J. Talvacchio for
supplying us with YBCO thin films. This work was supported by a grant from the Natural
Sciences and Engineering Council of Canada.
REFERENCES
1. J. Etheridge, Philos. Mag. A 73, 643 (1996).
2. J. R. Clem, B. Bumble, S. I. Raider, W. J. Gallagher, and Y. C. Shih, Phys. Rev. B 35, 6637 (1987).
3. M. P. Maley, J. O. Willis, H. Lessure, and M. E. McHenry, Phys. Rev. B 42, 2639 (1990).
4. H. Darhmaoui and J. Jung, Phys. Rev. B 53, 14621 (1996).
5. M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996), pp. 205–206.
6. A. H. Castro-Neto, Phys. Rev. Lett. 78, 3931 (1997).
Evidence of Chemical Potential Jump
at Optimal Doping in
1. INTRODUCTION
Oxygen-doped exhibits rich and interesting physical properties such as
macroscopic chemical phase separation [1], microscopic inhomogeneous distribution of
doped holes [2], graphite-like staging behavior of the interstitial oxygen [3,4], one-
dimensional (1D) modulation of in-plane ordering [5], and the change in doping efficiency
at a critical hole concentration [6]. Among all the techniques used to introduce
excess oxygen into the electrochemical intercalation [7] is the most attrac-
tive. It can achieve the highest possible excess oxygen content in a well-controlled
way. However, we have found that room-temperature electrochemical oxidation produces
samples far from thermodynamic equilibrium [8,9]. Subsequently, we have shown that elec-
trochemical doping at elevated temperature with proper annealing produces well-behaved
samples and the doping efficiency (number of holes introduced per excess oxygen atom) is
established [6]. This system, therefore, provides us a unique opportunity to study the phys-
ical properties of with precise and continuous control of carrier concentration.
Unfortunately, the oxygen doping is limited to a maximum hole concentration
1
Department of Physics and Texas Center for Superconductivity at University of Houston, Houston, Texas 77204.
which severely restricts our study of this system to be only in the underdoped region. To
understand the physical origin of this limitation and in the hopes of extending our studies
to the overdoped region, we studied the evolution of electrochemical potential of strontium
and oxygen codoped It is found that the limitation of oxygen doping in
is due to a chemical potential jump of doped holes at optimal doping level.
2. EXPERIMENT
The starting materials, (x = 0, 0.025, and 0.05), were prepared by
standard solid-state reaction [8]. Electrochemical oxidation (intercalation) and reduction
(deintercalation) of were carried out in a three-electrode electrochemical
cell with the sample pellet as the working electrode, a gold foil as the counterelectrode,
Ag/AgCl as the reference electrode, and 1M KOH aqueous solution as the electrolyte. The
cell was kept at constant current was applied to the working electrode to
increase (negative current) or decrease (positive current) the excess oxygen content of the
sample. The potential difference between the working electrode and the reference electrode
was monitored vs time during the electrochemical process. X-ray powder diffraction data
were collected at 25ºC–110°C using a Rigaku D/MAX-B x-ray diffractometer. Intensity
data for angles ranging from 22° to 81° were taken in steps of in fixed intervals
of 6 s.
Here, z stands for the charge of an oxygen ion and e the elementary charge [10]. Thus,
measuring the electrode potential at equilibrium as a function of the total charge passed
through the electrode is equivalent to measuring the chemical potential of each intercalated
oxygen atom as a function of
The potential data obtained at 70°C for intercalation and deintercalation are shown in
Fig. 1. The potential of is plotted vs the composition calculated from
where Q is the charge transferred per formula unit. It is seen that the intercalation potential
see curve a) and deintercalation potential ( see curve b) are parallel to each other
over a wide range of compositions and the oxygen atoms inserted on
oxidation can be completely removed on reduction, indicating that the process of oxygen
insertion into at 70°C is reversible in terms of intercalation [11]. Furthermore,
the potential difference between the oxidation process and the reduction process at fixed
value is only about 40 mV in the range of (Notice that this potential differ-
ence is equivalent to the sum of overpotentials and which are due to kinetic barriers
for inserting and removing interstitial oxygen.) These results indicate that is
very close to, if not at, thermodynamic equilibrium during the process of oxygen interca-
lation. As a consequence, the open circuit potential (OCV) at 70°C, measured under the
Evidence of Chemical Potential Jump at Optimal Doping in 517
condition that “no” current passing between working electrode and counterelectrode, is
taken as the equilibrium electrode potential. OCV as a function of composition is also
plotted in Fig. 1 (curve c). Although the data were measured directly for those for
were taken as the average of intercalation and deintercalation potentials
assuming the absolute values of and are equal.
The shape of OCV curve is very similar to the potential curves of intercalation and
deintercalation. A potential plateau can especially be observed in all three curves when
The potential plateau implies there is no Gibbs energy change during oxygen
intercalation into (or deintercalation from) the system in this composition range. In other
words, the sample is at multiphase equilibrium. Indeed, x-ray diffraction data (Fig. 2) taken
at 25°C show, as we reported previously [ 12], that two distinct orthorhombic phases coexist
in the samples of with and 0.11. The volume fraction,
extracted by profile fitting, of the phase with larger orthorhombicity increases proportionally
to the value. Clearly, there is a miscibility gap over the range of The
upper limit was determined by extrapolation because no more excess oxygen
atoms can be intercalated into above under equilibrium process.
In the two-phase region, one may reasonably expect that the intercalation potential
stays constant till if the overpotential does not change. Surprisingly, it begins
to increase at and increases abruptly at (see Fig. 1). Now, we first
show that the increase of is not due to an overpotential change. The overpotential is a
kinetic parameter rather than a thermodynamic parameter. A much higher overpotential at
required to intercalate more oxygen atoms into implies that there is
518 Li and Hor
a kinetic barrier to be overcome. To explain this kind of barrier, one must assume that the
distribution of interstitial oxygen is inhomogeneous as increases. This can happen if the
rate of electrochemical reaction occurring at the electrode surface is much faster than the
rate of oxygen diffusion into bulk, which results in the accumulation of excess oxygen atoms
inside electrode surface. Consequently, a thin layer inside the electrode surface should be
oxidized to a much higher oxygen content at However, this hypothesis is not
consistent with our observation that the value at which the potential suddenly increases is
independent of the magnitude of the charging current. For various applied currents ranging
from 3.6 to 18 the potential increase begins at the same value. Because the rate
of electrochemical reaction at the surface is directly proportional to the applied current, it
is expected that the potential jump should appear at different values for different currents.
The upper limit of biphasic region was determined by x-ray diffraction data collected
at 25°C whereas the potential data were taken at 70°C. Therefore, another possible cause of
the potential increase at is the shift of the upper boundary of the biphasic region
from at at 70°C. This possibility can also be ruled out. As shown
by the results of x-ray diffraction (see the inset of Fig. 2b), the volume fraction of the phase
with higher value in a sample of is only changed about 1% at 70°C ( %)
compared with that at 25°C ( %), indicating that upper limit biphasic region at 70°C is
almost the same as at 25ºC.
Evidence of Chemical Potential Jump at Optimal Doping in 519
Therefore, when the sample reaches optimal hole doping level, oxygen atoms energetically
favor the formation of molecules and their release at the surface. Consequently, no more
holes can be doped into planes. That is exactly the reason why the maximal doping
level of electrochemically intercalated is about 0.16, which was reported by
several groups [6,14] for values even higher than 0.12. Because the potential difference
between the potential before the jump and the oxygen evolution potential is about 0.15V,
the chemical potential jump of doped holes is estimated to be no less than 0.15 eV.
Summarizing all of the above observations, we therefore concluded that the origin of
the maximum doping of oxygen is due to an intrinsic chemical potential jump of doped
holes in planes.
4. CONCLUSIONS
We have studied the electrochemical potential change as a function of in
for and 0.05. We find that the maximum doping level is limited by
an intrinsic chemical potential jump of doped holes at optimal doping level of the high-
cuprates. Further studies of the implications of this chemical potential jump to the destroy
of high is underway.
ACKNOWLEDGMENTS
This work was funded in part by ARPA (MDA 972-90-J-1001) and the state of Texas
through the Texas Center for Superconductivity at University of Houston.
REFERENCES
1. J. D. Jorgensen, B. Dabrowski, S. Pei, D. G. Hinks, L. Soderholm, M. Morasin, J. E. Schirber, E. L. Venturini,
and D. S. Ginley, Phys. Rev. B 3 8 , 11377 (1988).
2. J. H. Cho, F. C. Chou, and D. C. Johnston, Phys. Rev. Lett. 70, 222 (1993).
3. B. O. Wells, R. J. Birgeneau, F. C. Chou, Y. Endoh, D. C. Johnston, M. A. Kastner, Y. S. Lee, G. Shirane,
J. M. Tranquada, and K. Yamada, Z. Phys. B 100, 535 (1996).
4. X. Xiong, Q. Zhu, Z. G. Li, S. C. Moss, H. H. Feng, P. H. Hor, D. E. Cox, S. Bhavaraju, and A. J. Jacobson,
J. Mater. Res. 11, 2121 (1996).
5. X. Xiong, P. Wochner, S. C. Moss, Y. Cao, K. Koga, and M. Fujita, Phys. Rev. Lett. 76, 2997 (1996).
6. Z. G. Li, H. H. Feng, Z. Y. Yang, A. Hamed, S. T. Ting, P. H. Hor, S. Bhavaraju, J. F. DiCarlo, and A. J.
Jacobson, Phys. Rev. Lett. 77, 5413 (1996).
7. J. C. Grenier, A. Wattiaux, N. Lagueyte, J. C. Park, E. Marquestaut, J. Etourneau, and M. Pouchard, Physica C
173, 139(1991).
8. H. H. Feng, Z. G. Li, P. H. Hor, S. Bhavaraju, J. F. DiCarlo, and A. J. Jacobson, Phys. Rev. B 51, 16499 (1995).
9. S. Bhavaraju, J. F. DiCarlo, I. Yadzi, A. J. Jacobson, H. H. Feng, Z. G. Li, and P. H. Hor, Mat. Res. Bull. 29,
735(1994).
10. W. R. McKinnon, in Solid State Electrochemistry, edited by P. G. Bruce (Cambridge University Press,
Cambridge, 1995), p. 175.
11. M. S. Wittingham, in Intercalation Chemistry, edited by M. S. Whittingham and A. J. Jacobson (Academic
Press, New York, 1982), p. 1.
12. P. H. Hor, H. H. Feng, Z. G. Li, J. F. DiCarlo, S. Bhavaraju, and A. J. Jacobson, J. Phys. Chem. Sol. 57, 1061
(1996).
13. C. Chaillout, S. W. Cheong, Z. Fisk, M. S. Lehmann, M. Marezio, B. Morosin, and J. E. Schirber, Physica C
158, 183(1989).
14. F. C. Chou, J. H. Cho, and D. C. Johnston, Physica C 197, 303 (1992).
Studies of the Insulator to Metal Transition
in the Deoxygenated
System
1. INTRODUCTION
The calcium substitution has been investigated by many researchers in both oxy-
genated [1,2] and deoxygenated [1,3–6] systems. It is known that
Ca is substituted mostly to the Y places [2,7]. The substitution of divalent Ca in the places
of trivalent Y increases the concentration of holes. Although Ca is substituted to fully
oxygenated Y-123 compound, the system becomes overdoped and decreases. If the sam-
ples are deoxygenated, it acts also as the electric carrier donor, induces insulator to metal
transition and superconductivity [3–6]. However, superconductivity induced in such a way
does not reach high critical temperatures because of impurity phases accompanying the
doping process. With both oxygen and calcium doping we obtain similar phase diagrams,
and transitions from antiferromagnetic (AF) to superconducting phase [6,8]. In the phase
diagram with the changing Ca content, it is clearly visible that the intermediate region
between AF and superconducting phases exists [6]. With Ca substitution, it is easier to
1
Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland.
2
Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Kraków, Poland.
manipulate in the carrier concentration, it does not change the crystal symmetry as we
induce superconductivity in the tetragonal samples.
Recently, the inelastic neutron-scattering experiments provided evidence for the dy-
namic stripe phases in The hole-rich regions and AF domains form
alternating stripes. In the AF spin correlations become completely dynamic
for The fluctuating stripes were detected in superconducting samples
with higher oxygen content: and The fluctuations
are not present in nonsuperconducting oxygen-deficient they appear
only at higher carrier concentration. The modulation period of analogic stripes in
decreases with hole doping [14].
By Ca substitution, we perform analogic hole doping, and in deoxygenated
samples with higher Ca concentration, the occurrence of dynamic stripes is also
possible. The aim of this work is to study the effect of Ca substitution in the deoxygenated
samples and the metal-insulator transition induced by Ca doping.
The different models of the transport properties are analyzed.
2. EXPERIMENTAL
The samples have been prepared by the standard ceramic
method. The powers and CaO were mixed in appropriate propor-
tions, pressed into pellets, and sintered in air in 936°C for 24 hr. Then the samples were
crushed and sintered in flowing oxygen twice, first in 942°C for 30 hr, then in 943 to 948°C
for 62 hr, then kept 450°C and slowly cooled to room temperature. After confirmation of
their superconducting properties, they were sintered in following argon in 730°C for 24 hr,
which was followed by slow cooling to room temperature.
The x-ray diffraction (XRD) measurements were carried out in room temperature
using Phillips diffractometer type X’pert with PW 3020 goniometer. radiation was
applied. The measurements were performed in the angle range of with the
step of 0.02° and 8 s collecting data time per single point to obtain sufficiently accurate
diffractograms and determine the amount of impurity phases quite precisely.
Iodometric titration was performed to obtain the average copper valency and oxygen
content. ac magnetic susceptibility vs temperature curves were recorded using the mutual
inductance method at a frequency of 21.5 Hz from 300 K down to 11 K. The rms value of
the applied field was 30 Oe.
The electrical resistivity measurements were performed by the four-point contact
method with reversing current direction in the range of temperatures from 300 K to 4.2 K.
3. RESULTS
The structural analysis performed by XRD showed that all the samples have the tetra-
gonal symmetry. Most of Ca substitutes to Y positions and only a small amount substitutes
to Ba positions. We detected the impurity phase of whose content increases with
Ca content (Table 1) and less than 0.08% weight fraction of CuO.
Iodometric titration made possible to determine the average copper valency and the
oxygen content. The results are in the Table 1. All the oxygen indexes are close to 6.1. We
Insulator to Metal Transition 523
also see that Ca substitution increases the average copper valency, which is well visible at
the higher Ca concentrations.
The results of ac magnetic susceptibility measurements are presented in the Fig. 1.
In the system with the small superconducting part of the sample manifests as
a weak diamagnetism with onset temperature equal to 14 K. The sample with
exhibits strong diamagnetism related to superconductivity below the temperature of 35 K.
The superconductivity in the sample with is not visible above 4.2 K in the re-
sistivity curve, whereas the sample with has the onset in resistivity equal to 35 K
524
and the zero resistivity at 19.5 K (Fig. 2). Despite the zero resistivity, for the sample with
the diamagnetic signal is too weak to indicate that the whole sample becomes
superconducting in the ac magnetic field of 30 Oe rms value (inset a in Fig. 1). Super-
conductivity is very sensitive to the applied magnetic field here. In the temperature of
is equal to emu/g for the value of applied field Oe and
equals emu/g for Analogic measurement of the oxygenated
sample gives emu/g for Oe in this temperature.
The Curie–Weiss law was fitted to the paramagnetic regions of ac susceptibility curves.
The Curie–Weiss behavior was also observed in the sample with regarded as an-
tiferromagnetic, which has, however, paramagnetic contribution. The determined effective
magnetic moment is presented in the Table 1 and drawn in the inset b of Fig. 1. The magnetic
moment dependence on the Ca concentration seems to be linear. Some part of the effective
magnetic moment originates from However, the effective magnetic moment of
is equal to per Cu atom [15], and as was calculated and presented in
Table 1 and in the Fig. 1 is too weak to be responsible for paramagnetism in these sam-
ples. The detected could give only small part of the total magnetic moment. The
increasing effective magnetic moment in samples is probably related
to hole doping into AF matrix.
The electrical resistivity vs temperature curves are drawn in Fig. 2.
We tested several models for (resistivity vs temperature) curves that have already
been observed in the HTS related compounds. The curves were drawn in appropriate
scales, functions were fitted to experimental data, and the test was applied. The tested
Insulator to Metal Transition 525
is the result of theoretical predictions for disordered metals [16]. This law
was observed in with impurities [17].
Conductivity following the relation
The smallest values have been obtained for Eq. (3) with and not much
bigger for We observed that fitting curves preferred in lower temperatures
and in higher temperatures. This indicates the existence of hopping conductivity in
two and three dimensions. The fitting results for are printed in Table 2. The fits do
not include the lowest temperatures, where some little changes in the behavior were
observed but still are not confirmed. In Fig. 3, the dependencies in logarithmic scale
are drawn vs temperature in and Linear regions are the indications that Mott's
law is obeyed.
In the inset of Fig. 2, there are dependencies of resistivity vs Ca content at constant
temperature. As is in the logarithmic scale, dependencies seem to be exponential
from to
526
Insulator to Metal Transition 527
4. CONCLUSIONS
Ca substitution in the deoxygenated system induces supercon-
ductivity at however, this superconductivity may be related to only small
part of the sample and is easily reduced by magnetic field.
The onset of the insulator to metal transition appears at The chosen criterion
is the change from positive to negative slope of resistivity vs temperature curves at room
temperature.
The increase in Cu valency was observed for increasing Ca content.
Low-temperature resistivity of the samples can be fitted to the
law with in higher and in lower temperatures. This suggests variable range
hopping mechanism of conductivity.
In the samples with more Ca, we observe linear dependence of
The linear increase of the effective magnetic moment with Ca concentration is ob-
served. The impurity is not capable to be the origin of such a high effective mag-
netic moment. This moment should originate mainly from compound,
and the increase is caused by Ca substitution.
528
ACKNOWLEDGMENT
This work was supported by the Polish Committee for Scientific Research under the
Grant No. 2 P03B 024 13.
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213–214, 48 (1995)
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Differential Optical Reflectivity
Measurements of
A photothermal microscope has been used to measure the rate of change of optical reflec-
tivity with temperature (DOR). Two laser beams, the pump and the probe, are focused on
the sample; with the focused spots typically apart. The periodic modulation of the
pump generates thermal waves that propagate across the sample, harmonically modulating
the temperature under the probe beam, which in turn gives rise to an ac-reflected signal.
A photodiode and lock-in amplifier are used to measure the probe signal. Details of the
experiment are discussed elsewhere [1].
We examined YBCO and observed a sharp change in the DOR as a function of tem-
perature at the superconducting critical temperature. This sharp change permits the deter-
mination of the critical temperature without touching the sample. The nature of the sharp
change varies from sample to sample, with probe optical frequency, and with probe beam
polarization. All samples are detwinned to clearly resolve the strong ab optical anisotropy.
1
Edward L. Ginzton Laboratory, Stanford CA 94305-4085.
2
Department of Physics and Materials Research Laboratory, University of Illinois, 1 1 1 0 West Green Street, Urbana
IL61801.
The photothermal experiment yields output amplitude in arbitrary units that are cal-
ibrated to give absolute values of The calibration process is described elsewhere [1].
These curves include all calibrations except for conductivity variation. The major source
of uncertainty in the calibration is the determination of the thermal modulation. We put
the combined calibration uncertainty for most measurements. Unfortunately, the
1064 nm measurement was more ambiguous than normal, and that value could have
been twice the value used.
Figures 1 and 2 show the a axis DOR results obtained on two samples of
Both samples had a near 93 K.
These curves show a tremendous variation in DOR response both in terms of a normal
and a superconductivity-onset response. All curves show a strong discontinuity at (visible
as the right edge of the feature at ).
We believe the normal state response is primarily due to plasma effects. To test this
hypothesis, we start with a standard Drude model,
Differential Optical Reflectivity Measurements of 531
with representing higher-energy oscillators, the plasma frequency, and the scat-
tering rate. The optical power reflectivity is given by
We then take the scattering rate to vary linearly with frequency over the range of
measurement,
Obviously, this model is not causal for and is at best an approximation for the case
where is small over the frequency range of interest (0.8–1.96 eV).
Figure 3 shows direct (temperature-independent) reflectivity measurements for the two
samples studied here, results from the literature [2], and fits using Eq. (1). The literature
curve is fit very well with a curve taking and The
quality of this fit indicates that Eq. (2) captures much of the physics. As this fit is fairly
insensitive to the exact value of B for B small ( mEV did not noticeably differ from
), we take in the rest of this work.
Given room-temperature values for and A, we can then calculate
and from Eq. (1). Given DOR measurements at three wavelengths, we can
write a system of equations relating these DOR measurements to temperature variations
of the three parameters, and The beauty of this separation is that
all of the frequency dependence is contained in the terms and the temperature
dependence in the (T ) terms. If we have more measurements than model parameters,
we can use the extracted parameter temperature variation to predict the measurements at
wavelengths not used in the extraction process. We can thus test the self-consistency of the
model. As we must add a scaling parameter for each experimental curve to compensate for
the uncertainty in the calibration of each curve, we must have at least two more measurement
wavelengths than model parameters.
For this case, we have five measurements and three model parameters. Adding scaling
parameters, we have free parameters. We also have 10 combinations of wave-
lengths (5 choose to obtain and (we can choose 1064, 780,
532 Studenmund, Fishman, Kino, and Giapintzakis
and 633 nm; or 1310, 780, and 633 nm; or 1310, 1064, 633 n m . . . ) . For any one choice,
we call the chosen wavelengths the fit wavelengths, as we fit the model to the DOR at
these wavelengths. The other wavelengths we call the predicted wavelengths, as we can
use the model parameter temperature variation to predict them. We can thus formulate a
self-consistency test for a model of We say a model is self-consistent if (1) all com-
binations of fit wavelengths, the predictions from the model at the predicted wavelengths,
agree with the experimental results for these wavelengths; (2) the scalings of the experi-
mental curves needed for this fit are within the systematic error of the calibration; and (3)
the model parameters initially used agree with the direct reflectivity of the material.
Figures 4 and 5 show, respectively, the 1310 nm and 633 nm results of such a test for
the of Eq. (1). The 1550 nm, 1064 nm, and 780 nm results are omitted for brevity.
Remember that for each wavelength we choose to model, only three of the other four
datasets are needed to fit the theory. We thus have four choices as to the curve to throw out.
Differential Optical Reflectivity Measurements of 533
These plots show the results for ignoring each of the (four) possible wavelengths in turn. In
these plots, the dots are the scaled experimental curve, and the four other curves represent
a modeling with one particular wavelength ignored. The solid line is for 1550 nm ignored,
the dashed 1064 nm, the dot-dashed 780 nm, and the solid with crosses the 633 nm (for Fig.
4) or 1310 nm (for Fig. 5) data ignored. These fits were obtained with
and The experimental curves were scaled by 1.15 (1550 nm), 1.1 (1310 nm),
0.4 (1064 nm), 0.75 (780 nm), and 1.15 (633 nm). The 1064 nm scaling, although large, is
consistent with the measurement for that curve. The 780 nm scaling is not consistent
with the error we claim for the calibration. We thus must conclude that the of
Eq. (1) comes close to but does not agree fully with the experimental results.
Given that we have only taken Eq. (1) as an approximation, Figs. 4 and 5 indicate that
it gives a very effective fit above We feel that the agreement between the theory and
experiment is as close as could be reasonably expected from Eq. (1). We thus conclude
that the normal-state reflectivity is given by a dielectric function similar to the Drude
approximation of Eq. (1).
Below the theoretical predictions agree neither with the experiment nor with each
other. If the optical consequence of the onset of superconductivity were modelable as
(potentially drastic) changes in the three model parameters then the pre-
diction curves should match the experiment below In addition we have attempted to
model the below- part using Eq. (1) and any model and scaling parameters, neglect-
ing the above- DOR, without success. As no fitting works, we must conclude that the
superconductivity response is not modelable by a Drude plasma approximation.
The superconducting condensate is typically modeled as a function in the real part
of the conductivity, and is considered the dominant consequence of superconductivity.
Causality requires a term in the imaginary part of the conductivity, This term
generates a very weak reflectivity change that should be indistinguishable from that due to
the normal carriers.
We do not know the origin of the superconducting DOR response, but we believe it
represents either a new consequence of the superconducting condensate, such as changes
in optical matrix elements, or some new feature in the excitation spectrum. These ideas are
discussed further in Ref. [3).
In conclusion, we present a axis DOR results for two. YBCO samples at five mea-
surement wavelengths. We fit a simple extension of the Drude model for to the room-
temperature optical reflectivity data, and use the parameter values thus obtained to self-
consistently model the DOR results. We have successfully modeled the normal-state DOR
using this extended Drude model. The superconducting state DOR are not explainable with
this model.
ACKNOWLEDGMENT
This work was supported by DOE contract DE-FGO3-90ER14157.
REFERENCES
1. W. R. Studenmund, I. M. Fishman, G. S. Kino, and J. Giapintzakis, unpublished, 1998.
2. S. L. Cooper et al., Phys. Rev. B 47, 8233 (1993).
3. I. I. Fishman, W. R. Studenmund, and G. S. Kino, p. 495 in this volume.
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On Some Common Features in High-
and Low- Superconducting Perovskites
1. INTRODUCTION
Understanding of electronic properties of high- cuprates [1] and their complex
electronic phase diagram (Fig. 1) still presents a major challenge, despite thousands of
research papers and remarkable progress [1,2] in both sample preparation and advanced
experimental techniques. In this paper, we briefly discuss some anomalies found in high-
cuprates as well as the anomalous transport [5] recently found in a low- perovskite
ruthanate,
1
Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.
2
Physics Department, University of Wisconsin, Madison, WI 53589, USA.
the gap vanishes once the temperature is higher than the “normal” state spectrum in
underdoped regime remains gapped. Furthermore, in the underdoped system, the absence of
coherent features in both the normal and the superconducting state spectra indicates strong
departure from the FL behavior, as viewed by photoelectron spectroscopy [3].
The superconducting gap of the underdoped and optimally doped cuprates exhibits
dominant d-wave symmetry [1,3]. Nevertheless, ARPES data on highly overdoped
indicate a finite gap along the direction in the Brillouin zone [4], where
the d-wave gap has nodes. This point is still under active investigation.
Due to badly defined surface termination of most cuprate samples, almost all ARPES
studies since the late 1980s were done on single crystals, even though a
large fraction of other physical studies and electronic applications was reported on various
families of superconducting compounds. In order to overcome this limitation, we have built
a pulsed-laser ablation system in which superconducting films are grown and transferred
under ultra-high vacuum to the photoemission chamber. First ARPES studies are in progress.
ACKNOWLEDGMENTS
We gratefully acknowledge financial support by Ecole Polytechnique Fédérale de
Lausanne, the University of Wisconsin, and the Swiss National Science Foundation.
REFERENCES
1. The Gap Symmetry and Fluctuations in High- Superconductors, Proceedings of the NATO Advanced Study
Institute, Cargese, September 1997, edited by J. Bok, G. Deutscher, D. Pavuna, and S. A. Wolf (Plenum Press,
1998).
2. D. Pavuna and I. Bozovic, eds., Oxide Superconductor Physics and Nano-Engineering I, II & III, 2058, 2697,
3481—SPIE (Bellingham, USA, 1994, 1996, 1998).
3. H. Ding el al., Phys. Rev. B 54, R9678 (1996).
4. C. Kendziora et al., Phys. Rev. Lett. 77, 727 (1996); I. Vobornik et al., unpublished.
5. H. Berger, L. Forró, and D. Pavuna, Europhys Lett. 41(5), 531 (1998).
6. I. Bozovic et al., Phys. Rev. Lett. 73(10), 1436 (1994).
7. M. Randeria and J. C. Campuzano, Varenna Lectures 1997, Report No. LANL cond-mat/9709107, and ref-
erenced therein; J. M. Tranquada et al., Nature 375, 561 (1995); N. L. Saini et al., Phys. Rev. B 57, Rl1101
(1998), and references therein.
8. K. A. Müller, p. 1 in this volume; D. Mihailovic et al., Phys. Rev. B 57, 6116 (1998).
9. D. Pines, see the contribution in Ref. 1; P. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994).
Pinning Mechanisms in a-Axis-Oriented
and
Multilayers
1. INTRODUCTION
One of the most recent [1] and active topics (see the whole issue in Ref. [2]) in high-
superconductors is the development of the charge stripes in the planes of some
families of cuprates. The study of samples with the possibility of changing at will the length
of the planes could be crucial in this stripes scenario. Very recently, several authors [3]
and [4] have fabricated a-axis-oriented 123 films and superlattices. These a-axis-oriented
systems allow us to study experimental situations that are impossible in single crystals and
in the usual c-axis-oriented films.
1
Dept. Física de Materiales, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain.
2
Instituto de Ciencia de Materiales, Dept. Propiedades Opticas, Magnéticas y de Transporte (C.S.I.C.), 28049
Madrid, Spain.
3
Dept. Physics, University of California San Diego, La Jolla, CA 92093-0319, USA.
a-axis films grown on the usual cubic substrates have a microstructure with 90°
microdomain (average size 20 nm) [5]. a-axis superlattices are very peculiar in comparison
with the c-axis multilayers because in a-axis, the planes are perpendicular to the sub-
strate and the effects due to the natural anisotropy planes) and the artificial anisotropy
(insulating layers) are uncoupled. An additional important aspect in a-axis superlattices of
123 superconductor/123 nonsuperconductor (for instance is
that planes can be locally superconducting or insulating depending on whether the
rare-earth neighbors of the plane are Eu or Pr.
In this paper, we present a carefully structural characterization of two types of super-
lattices, and , The former with the
planes (superconducting/nonsuperconducting) running along the whole sample, and the
latter with the planes interrupted by the layers. However, in these artificially
structured systems, different dissipation mechanisms could compete. We show experimen-
tal results that could clarify the anisotropy, magnetic field, and temperature requirements
needed to enhance different kinds of dissipation mechanisms.
2. EXPERIMENTAL
Superlattices of the so-called a-axis orientation (Cu-O planes perpendicular to the susb-
trate) of and (EBCO/STO)
were grown by dc magnetron sputtering from stoichiometric targets. The multilayers were
grown by alternately depositing and layers using two
independent targets and stopping the substrates in front of the EBCO and PBCO or STO
cathodes by a computer-controlled stepping motor. The samples were fabricated on (100)
(STO) and (100) (LAO) substrates, and with a total thickness of 250 nm. A
commercial cryostat with a 90 kOe superconducting magnet, a temperature controller, and
a rotatable sample holder computer controlled by a stepping motor allows us to take angular
dependence resistivity measurements with different values of the applied magnetic field.
The structural and superconducting characterization of the multilayers have been reported
elsewhere [6]. A powerful technique (SUPREX program) [7] of structural refinement of
x-ray diffraction (XRD) profiles from superlattices has been used to characterize the quality
of these a-axis EBCO/PBCO and EBCO/STO multilayers. In a-axis superlattices, it is not
possible to subtract the diffraction maxima coming from the substrate as it is usually done in
the c-axis-oriented superlattíces. These substrate maxima have to be included in the fitting
data. The substrate peaks are at (STO) and (LAO), the (200) peak is at
(a-axis-oriented sample), and the (005) peak is at (c-axis-oriented
sample). Figure 1 shows the refinement result for both kind of superlattices. This method
gives the actual thickness of the layers and parameters related with the interface. In these
samples the refinement implies that the interface step disorder is around 1 unit cell and
the interface diffusion 20%, values very similar to the values obtained in c-axis-oriented
YBCO/PBCO multilayers [8].
vortex liquid region close to Velez et al. [3] reported in a-axis-oriented 123 superlattices
two minima in the angular dependence of the resistivity with constant temperature and
applied magnetic field. These minima occur (a) when the magnetic field is applied parallel
to the substrate or (b) with the applied field perpendicular to the substrate. These authors
suggest that these two resistivity minima (critical current maxima) are due to two different
pinning mechanisms. When the magnetic field is applied parallel to the planes, the
intrinsic anisotropy leads to the pinnig due to the depression of the superconducting order
parameter between the planes. Otherwise when the magnetic field is applied parallel
to the substrate (perpendicular to the planes) the magnetic field could be pinned by the
artificial anisotropy due to the superlattice modulation. In this case, the PBCO insulating
layers play a similar role to the areas between the superconducting planes in the
intrinsic anisotropy case. In this artificially induced effect, Velez et al. [3] found that, when
the modulation length of the multilayers and the vortex lattice parameter [given by the
value of the applied magnetic field, have similar values an enhancement
of the critical current occurs for a a range of applied magnetic field around this matching
542 González, González, Schuller, and Vicent
field. Figure 2 shows these two minima in an a-axis-oriented EBCO (50 unit cells)/PBCO
(5 unit cells) superlattice (matching field 5 T). This minimum still remains at temperature
very close to and in an interval of applied magnetic field around the matching field value
(see Fig.2B).
The role of the artificially induced structure (in this case, the PBCO layers) could
be better understood studying another artificially layered system. a-axis-oriented EBCO
(250 u. c.)/STO (5 u. c.). Figure 3, A and B, shows the data at the same reduced temperature
respectively) than in the EBCO/PBCO sample. In this EBCO/STO
multilayer, the matching field is 0.2 T. However, in addition to the minima due to the
intrinsic pinning, a clear “second” minimun appears in the high magnetic field region. This
“second” minimum vanishes when the temperature is increased (see Fig. 3B), whereas the
actual minima due to the artificially induced anisotropy remains up to temperatures very
close to (see Fig. 2B). The experimental behavior of this pinning mechanism seems
to indicate that it is due to the defects in the EBCO layers and that is not related to the
artificially structure. Prouteau et al. [4] recently reported the same kind of minima in pure
a-axis-oriented films. The very peculiar microstructure in a-axis-oriented films with
microdomains (see Ref. [5]) could pin the vortices when the magnetic field is parallel to
Pinning Mechanisms in a-Axis-Oriented Multilayers 543
the substrate. Another experimental fact that is interesting to point out is shown in detail in
Fig. 4. A new minimun develops in the EBCO/STO multilayer at high temperature, close to
and low values of the magnetic field when the field is applied parallel to the substrate.
These are the footprints of the surface pinning effect (see, for instance, Ref. [9]).
4. CONCLUSIONS
In summary, a-axis-oriented 123 superlattices are the ideal system to study the com-
petition among different dissipation processes.
The angular dependence of the resistivity shows two minima: (1) when the magnetic
fieldis applied parallel to the planes (perpendicular to the substrate), and (2) when the
applied magnetic field is parallel to the substrate (perpendicular to the planes) and
parallel to the artificial structure. The former is due to the intrinsic pinning, and in the case
of the latter, three different origins were experimentally detected: (a) microstructure defects
544 González, González, Schuller, and Vicent
(90° microdomains) in the superconducting layers, which were observed at high magnetic
fields and low temperatures; (b) surface and interface pinning, which was observed at low
applied magnetic fields and temperatures close to and (c) pinning due to the artificially
induced anisotropy (superlattice structure), a very effective mechanism that could be tuned
with the value of the modulation length of the multilayer, and is relevant in a wide magnetic
field interval, around a matching field, which is given by the modulation of the artificially
layered structure. The insulating layers seem to be very effective pinning centers, and they
could enhance the critical current when the applied magnetic field is around the matching
field value and it is applied close to or parallel to the substrate.
ACKNOWLEDGMENTS
This work was supported by Spanish CICYT grant MAT96-0904, Universidad Com-
plutense, and by the U.S. Air Force MURI program at UCSD.
REFERENCES
1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995).
2. J. Supercond. 10(4), (1997).
3. M. Velez, E. M. Gonzalez, J. I. Martin, and J. L. Vicent, Phys. Rev. B 54, 101 (1996).
4. C. Prouteau, F. Warmont, Ch. Goupil, J. F. Hamet, and Ch. Simon, Physica C 288, 243 (1997).
5. C. B. Eom, A. F. Marshall, S. S. Laderman, R. D. Jacowitz, and T. H. Geballe, Science 249, 1549 (1990).
6. J. I. Martin, M. Velez, and J. L. Vicent, Phys. Rev. B 52, R3872 (1995); M. Velez, E. M. Gonzalez, J. M.
Gonzalez, A. M. Gomez, and J. L. Vicent, J. Alloys Comp. 251, 218 (1997).
7. E. E. Fullerton, I. K. Schuller, H. Vanderstraeten, and Y. Bruynseraede, Phys. Rev. 45, 9292 (1992).
8. E. E. Fullerton, J. Guimpel, O. Nakamura, and 1. K. Schuller, Phys. Rev. Lett. 69, 2859 (1992).
9. J. Z. Wu and W. K. Chu, Phys. Rev. B 49, 1381 (1994).
Angular Dependence of the Irreversibility
Line in Irradiated a-Axis-Oriented
Films
1. INTRODUCTION
High-temperature superconductors are very anisotropic materials, as the superconduc-
tivity is mainly localized in the planes. Recently, it has been shown that these
planes present spin and charge stripes structure [1], and it has been suggested that an ampli-
fication in the critical temperature could be associated with these stripes [ 2 ]. Films grown
with different orientations, a axis or c axis, are very useful to get a full understanding of the
anisotropic behavior of these materials. In a-axis films, the planes are perpendicular
to the substrate; therefore, they allow studies in different experimental geometries from the
usual c-axis films. The possible stripes in the planes can be affected by the material
microstructure and defects [3]. Then, it is very interesting to characterize the behavior of
a-axis films with artificially controlled defects as those created by heavy ion irradiation,
which, so far as we know, has only been performed in c-axis films [4] and single crystals
[5,6].
1
Dept. Física de Materiales, F. Físicas, Universidad Complutense, 28040 Madrid, Spain.
2
Materials Science Division, Argonne National Laboratory, IL 60439 USA.
The most structure sensitive property of the material is the critical current density
for example, when defects are created in the planes, they could affect the periodicity
of the stripe structure inducing a local reduction in and, therefore, a pinning effect. In
the case of a-axis films, the samples grow on the usual cubic substrates with a peculiar
microstructure of domains separated by 90° boundaries [7]. As a consequence, the critical
current density in these a-axis films is in the range of that is, lower
than in c-axis films. Also, the irreversibility line lies lower in the H-T plane for a-axis
films [8]. However, in the presence of a magnetic field, it has been shown that in a-axis
films is not limited by weak links, but rather it is determined by flux motion and vortex
pinning [9]. In this work, we have irradiated these a-axis films with heavy ion irradiation.
It creates artificial columnar tracks that crosses the planes structure to act as good
pinning centers and, therefore, produce an enhancement of their irreversibility region.
2. EXPERIMENTAL
Pure a-axis-oriented thin films have been grown by dc magnetron sput-
tering on (100) substrates as reported elsewhere [10]. The as-grown samples were
patterned into a 500 wide bridge, using photolitography and wet etching. The irradia-
tion was performed with with a current of 45 pA for 28 min. Under these
experimental conditions, the high-energy heavy ions create amorphous columnar tracks,
parallel to the irradiation direction and threading the whole sample thickness of
The average defect density corresponding to this dose is the same as the number of vortices
per unit area with an applied field of
Figure 1 shows the resistivity transition of an a-axis film before and after
the irradiation process. In this particular sample, the irradiation direction is perpendicular
Angular Dependence of the Irreversibility Line 547
to the patterned bridge (i.e., to the transport current) and makes an angle with
the film normal (see sketch in Fig. 1). The critical temperature and the metallic behavior
have not been esentially affected by the irradiation process. There is 15% increase in the
normal-state resistivity, which can be attributed to a reduction in the effective crossection
for the transport current due to the amorphous tracks created by the heavy ions.
Transport measurements were performed in a helium cryostat with a 9 T supercon-
ducting solenoid and a rotatable sample holder. The magnetic field was always applied in
the plane perpendicular to the transport current.
values to that of the number of amorphous tracks and when the field is parallel to the
direction of the irradiation. A comparison between positive and negative values shows a
reduction of the irradiation. A comparison between positive and negative values shows a
reduction in the dissipation due to artificial pinning as high as a factor of 3 for
Also, a more-detailed examination of the curves reveals that the field and angle ranges
where this pinning mechanism is significant is not very narrow, but it develops over more
than 1 Tesla and 20° around the matching conditions. This wide range of enhancement in
the irreversibility region can be attributed to the soft vortex lattice of these materials, which
can easily be deformed so that the vortex lines become pinned by the columnar defects.
In summary, the pinning centers created by the irradiation process have proved to be
very effective in improving the transport behavior of a-axis films so that the irreversibility
line moves to higher field values.
ACKNOWLEDGMENTS
This work has been supported by the Spanish CICYT (grant MAT96/904) and
Universidad Complutense. The irradiation at ATLAS was supported by the U.S. Department
of Energy, BES, Materials Science under contract No. W-31-109-ENG-38.
REFERENCES
1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe. Y. Nakamura, and S. Uchida, Nature 375, 561 (1995); P. Dai,
H. A. Mook, and F. Dogan, Phys. Rev. Lett. 80, 1738 (1998).
2. A. Bianconi, N. L. Saini, T. Rossetti, A. Lanzara, A. Perali, M. Missori, H. Oyanagi, H. Yamaguchi,
Y. Nishihara, and D. H. Ha, Phys. Rev. B 54, 12018 (1996).
3. M. A. Teplov, Y. A. Sakhratov, A. V. Dooglav, A. V. Egorov, E. V. Krjukov, and O. P. Zaitsev, JETP Lett. 65,
821 (1997).
Angular Dependence of the Irreversibility Line 549
1
I.N.F.M., I.N.F.N., Dept. of Physics–Politecnico di Torino, Torino, Italy.
LJJ percolating across the film serve to maintain commensurate field values in the junction
interior [ 1–3 ]. It occurs when the applied field is close to the one of these values.
The aim of this work is to show, by means of a model already introduced by
Fehrenbacher [4] and suitably modified to the specific problem dealt with, how the static
behavior of the critical current density in a high-quality film is driven by the behavior of its
basic component, the average LJJ. This LJJ is modulated by random columnar defects. An
important issue is to extract the main parameters in order to plan a particular performance
in the applied field.
As pointed out in [4], the Owen–Scalapino (OS) theory, modified to simulate columnar
defects, can be used to evaluate the critical currents in an external magnetic field. The
basic equations of LJJ [which involve the tunnelling supercurrent j ( x ) and the relative pair
phase are
where the penetration depth is constant in a uniform LJJ, whereas columnar defect yields
to a variable penetration depth The boundary conditions we consider are different from
that of OS, but found to be suitable to take into account the columnar defects:
where is a critical field corresponding to the usual first critical field of a type-II
superconductor and is the pair phase at the end of the junction.
The total current density through the LJJ is given by and from
sine-Gordon equation The critical current density can be
obtained by the maximum value of with respect to
Due to the periodic behavior of the internal current j ( x ) , the total current through the
uniform junction is always restricted to the junction surface and it cannot reach large val-
ues. The net current carried by a complete vortex vanishes, for a junction without defects,
because every half-wavelength of positive internal current is compensated by a half-length
of negative current. The current crossing the junction must be attributed to a single uncom-
pensated half-wavelength. This means that the total current is due to incomplete vortices
setting up only at the “surface.” There is a small but finite surface current due to the pinning
of the vortex lattice (VL) at the surface. The surface acts as a defect. However, if defects
are present in the inner part of the junction, they can block some negative oscillations of
the internal current, leaving a large positive current crossing the junction [ 1–4 ].
Defect-Modulated Long Josephson Junctions 553
The large values of critical current densities in disordered LJJ can be attributed to
the presence of the defects acting as pinning centers. The presence of a single defect can
be described in terms of a local change of the Josephson coupling energy through
a suitable pinning potential which is identically zero for the uniform case. The
corresponding change in the maximum Josephson current density yields to the desired
negative oscillations blocking.
The simplest model to simulate a defect of size s, located at the point with a
pinning strength q, is the square-well model.
The inhomogeneous coupling energy becomes
In order to have positive coupling energy inside a defect, the strength q must satisfy the
inequality The Josephson penetration depth and the other variables becomes a
piecewise constant function of the position x, where the sine-Gordon equation has analytical
solutions.
The phase can be evaluated starting from the initial value at the edge
At any boundary between two regions with different values of the phase and its
derivative, proportional to H(x), must be continuous, whereas the current density jumps.
The solution is then obtained by propagating across regions with constant up to the
edge
The columnar character of the defects is taken into account by considering the energy
modulation only along the x direction. In the following, we consider mainly a random-defect
distribution.
Random defects can be characterized by their random position their random size
and their random strength q. In our model, the defects are supposedly by uniformly
distributed into the LJJ, their size and strength are chosen normally distributed around a
suitable mean value.
The size and position randomness make crucial to understand and simulate the eventual
defect overlapping. In the general theory, the j th defect lowers the coupling energy by a
factor In our model, the j th defect, if overlapping the i th one, lowers by a factor
the already lowered coupling energy. In the overlapping region, the new value of
is lowered by the global factor In this way, any random defect
simulation make the coupling energy a piecewise constant function, and analytical solutions
can be obtained from the general theory. For each simulation of the random disorder, a value
of the critical current is evaluated. The statistical average of is then taken over many
configurations, yielding to the result. This procedure essentially models a superconducting
film as a disordered array of LJJ, separating the islands of single crystal-like regions. Each
junction has a different defect distribution. The overall is the mean value of the critical
current taken over all the LJJ. The statistical error depend on the number N of realizations,
decreasing like In our calculations, we choose to have the computing time
reasonably small in spite of some statistical fluctuations in the behavior of
554 Mezzetti, Crescio, Cerbaldo, Ghigo, Gozzelino, and Minetti
The curve, in a log–log scale, generally exhibits a plateau where the average
value of is nearly independent on At the end of the plateau a large drop of occurs,
followed by a series of minima located at fields with m integer.
The oscillatory behavior is evident for random defects with constant size. It decreases
and it is smoothened as the standard deviation of increases.
2. RESULTS
Among the types of disorder investigated, the random case is the most interesting. In
fact, it explains the pinning behavior of as-grown films with suitable defects [5] as well as the
behaviors of films where the underlying intrinsic defect network is modulated by implanted
columnar defect. In the present paper, we consider fully penetrated LJJ. The dependence of
on the temperature is implicitly taken into account through the temperature dependence
of diverging as vanishes near The potential well depth, represented by the value
of q (Fig. 1) and the length scale of disorder (Fig. 2), are the chief parameters in
determining the main drop of the critical current density as a function of the external field.
Above this drop, some plateau-like features can set up. Figure 2 shows an increase of the
width of the plateau as decreases. The more interesting topic in film fabrication and
optimization technologies is the handling of the disorder to provide plateau-like features for a
large and programmable range of “matching” fields. Two different but correlated conditions
are meaningful, the first one concerning the defect size and distribution, the second one the
fluxon lattice accommodation. Among the different possible defect configurations, the more
relevant is corresponding to the equation
With this topology, the plateau exhibits its maximum value. Any further increase of the
defect concentration leads to a decrease of but does not affect the plateau shape (Figs. 3
and 4).
Defect-Modulated Long Josephson Junctions 555
In order to understand the meaning of this result, one must refer to a asymmetric
junction in which the local critical Josephson current density is reduced to zero in periodic
“stripes” by making the junction barrier thick there. This procedure is the practical way
to obtain periodic modulation of the LJJ coupling energy. The relevant fact is that this
configuration can be thought of as a periodic array of columnar pins, or simply as a periodic
array of separate subjunctions within the overall junction area [6].
556 Mezzetti, Crescio, Gerbaldo, Ghigo, Gozzelino, and Minetti
In such a configuration the main relevant parameter is f , the fraction of a flux quantum
per unit cell, also referred as the frustration index. Theoretical treatment has shown that
novel phenomena arise when the frustration index takes on a rational value.
For a more important example of the superlattice of fluxons relative to the
underlying lattice is a simple checkerboard pattern, with cells not occupied by a flux quantum
and cells occupied by one flux quantum represented by the black and white squares. When
a current is applied to this ground-state, the vortex pattern is effectively pinned in
place by its commensurate “fit” to the underlying lattice: an individual vortex center cannot
jump into an adjacent cell without creating configurations with much higher energy.
Defect-Modulated Long Josephson Junctions 557
In the case of a random array of defects, the topology for which all the defects are
filled by a fluxon (maximum matching field) corresponds to the ordered reference pattern
with
In the literature concerning pseudoperiodic or random columnar defects, the matching
field is usually considered the maximum field we can accommodate inside a given dis-
tribution of columnar defect [6]. From the results of the present work, it can be deduced
that the accomodation of a wide range of fields lower than the maximum is optimized in
a disordered defect distribution only when Eq. (4) is fulfilled. The two conditions—one
vortex, one defect and therefore be considered as the best equivalent to
the checkerboard pattern in the random case.
Up to now, we did not consider the existence of minima. For a wide spectrum of ap-
plications, the variance or the defect size ascertains a smoothening of the resonant behavior
at high interference orders (Fig. 5). In Fig. 6, the Thompson experimental results [5] are
interpreted in the framework of this theory. As a conclusion, we emphasize that optimal
performances of films in a given range of field and temperatures can be achieved by over-
lapping size randomness of defects and controlled disorder. The control is achieved, by
means of the parameter as well as the optimized filling of a given average LJJ with
defects. In such a way extrinsic trenches of columnar defects could either create or modulate
intrinsic LJJ.
ACKNOWLEDGMENT
This work was partially supported by the I.N.F.M.–PRA PROJECT “HTCS electronic
devices.”
REFERENCES
1. M. A. Itzler and M. Tinkham, Phys. Rev. B 51, 435 (1995).
2. E. Mezzetti, S. Colombo, R. Gerbaldo, G. Ghigo, L. Gozzelino, B. Minetti, and R. Cherubini, Phys. Rev. B 54,
3633 (1996).
558 Mezzetti, Crescio, Gerbaldo, Ghigo, Gozzelino, and Minetti
We have studied the effect of surface columnar defects on the vortex dynamics
inside the whole sample. Trenches of columnar defects along about 5% of the
sample thickness were created by means of 0.25 GeV Au ions in Ag/BSCCO-
2223 high-quality tapes. Strong phenomena of vortex localization inside the bulk
were revealed by notable shifts of the irreversibility lines (ILs) as well as their
after-irradiation shape. The enhanced ILs exhibit their own particular character-
istics, such as a Bose-glass-like behavior up to quite high magnetic fields, with
a dose-dependent onset point. Moreover, the irreversible regime expands with
decreasing defect density. The results are consistent with the setting up of vortex
morphologies confined in the bulk. The central achievement of this work is that
a surface patterning with nanometric defects provides boundary conditions for
mesoscopic domains in the bulk where vortices are confined. In particular, the
lateral wandering is allowed inside tubes whose lateral cross section is dose de-
pendent and whose length scale is comparable with the distance between defects
for each dose.
I. INTRODUCTION
The effects of correlated disorder due to heavy ion irradiation have recently received
considerable attention. The shape of the induced columnar defects can match the fluxon
I
I.N.F.N., I.N.F.M., Dept. of Physics–Politecnico di Torino, Torino, Italy.
2
Pirelli Cavi S.p.A., Milano, Italy.
3
ENEL–SRI, Milano, Italy.
4
L.N.S.–I.N.F.N., Catania, Italy.
5
L.N.L.–I.N.F.N., Legnaro (Padova), Italy.
Stripes and Related Phenomena, edited by Bianconi and Saini.
Kluwer Academic/Plenum Publishers, New York, 2000.
559
560 Mezzetti et al.
shape, and the radial column dimensions are close to the coherence length [1]. Due to the
small range of penetration of heavy ions in the material, only a limited part of the sample
bulk can be affected, so that heavy ions were so far used only for irradiations of films and
thin single crystals. However, surface columnar defects act as trenches where vortices are
confined in such a way that the bulk properties are also affected, as already demonstrated
for melt-textured materials [2].
In this work, we show a possibility to control the vortex flux dynamics inside BSCCO-
2223 materials by surface ion implantation. Columnar defects were created on a surface layer
of about 5% of the total thickness. Measurements were performed on well-characterized
Ag/BSCCO-2223 samples coming from two different batches
prepared independently by two staffs: Pirelli Cavi S.p.A. [3] (samples labelled GHx) and
ENEL-SRI [4](samples labelled CLx), in order to verify the important issue of the sample
independence of the experimental findings. These samples were irradiated with 0.25 GeV
Au ions to have five defect densities corresponding to equivalent fields (matching fields,
[5] ranging from 1 to 5 Tesla. Irradiations were performed at the 15 MV Tandem facility
of the L.N.S–I.N.F.N., Catania and at the 15 MV Tandem facility of the L.N.L.–I.N.F.N.,
Legnaro (Padova) [2,6]. Irreversibility lines (ILs) were determined by the onset of magnetic
susceptibility third harmonic [7], at 1 kHz.
signature of a Bose-glass-like transition. We focus our analysis on the region between the
onset and such kink.
The crossover fields, between the two regimes corresponding to the higher
localization and weaker localization regions, respectively, and the onset field, can be
better highlighted in the log–log plot of Fig. 3. In fact, all the ILs measured before and
after irradiation follow, in the considered range of fields, the law with
different values of the exponent for different dynamical regimes. In the logarithmic scales
of Fig. 3, the different regimes result in different linear slopes. The kink
is found as the crossing point between the curves with exponents In Fig. 3, the
onset and the kink of the sample irradiated with are indicated. The position of
the kink is shifted toward lower reduced temperature, and reduced field,
as the density of tracks increases.
562 Mezzetti et al.
The IL of the irradiated sample is coincident with the curve for fields lower than
and temperatures higher than i.e., the point belongs to both
and curves (see Fig. 3 and the inset of Fig. 1). This condition finally gives
diameter We do not assume a priori that is equal to the defect diameter. In order to
obtain from we assumed and The experimental results
show that a single value of for the different doses cannot be found. In Table 1, the values
of for different values of are reported. The magnitude order of the fluence-dependent
is much higher than the diameter of gold-ion columnar defect crosssection reported by
Zhu [1]. In Table 1, the average distances between defects at the respective irradiation doses
are reported for comparison. It emerges that roughly scales with the irradiation dose
as the interdefect distance. We interpret the fitted values of as a diameter of the vortex
“confinement” crosssection, which now takes the place of the columnar defect crosssection.
It represents the transverse size of the vortex bundle with a part pinned into columnar defects
near the IL. Different sizes are allowed for different doses, and the smaller ones correspond
to higher fluences.
In order to perform a deeper investigation of the main characteristics of the vortex
phase induced by the surface defect topology, we analyze the results by means of a further
comparison with the before-mentioned analysis. In [13] the IBM group formalized the pin-
ning efficiency for YBCO as where
is the track cluster area, is the flux quantum, and is the accommodation field, first
introduced by Nelson and Vinokur [12]. Our hypothesis is that the field at the kink,
is also proportional to and, as a consequence, that
where c is a constant. We fitted our data by means of this expression, with and c as fitting
parameters. Results are reported in Fig. 5, where the values of the parameters are also
indicated. From this analysis, a confinement diameter nm, of the same magnitude
order of the parameter evaluated above, is found. As previously outlined, in our case 2r
does not represent the average diameter of track clusters, but rather represents the average
transversal dimension of a confined vortex line. In this case, the value of r obviously is the
result of a spatial averaging as well as an averaging on different The magnitude order of
the obtained value confirms this interpretation in relation to the main output of the analysis
564 Mezzetti et al.
with Eq. (2). Also the lower limit of the confinement region, i.e., fits the same dose
dependence. In Fig. 5 is reported a fit of the data with the expression
similar to Eq. (3). The value of the parameter , i.e., the value of the confinement radius, is,
within errors, the same as in the previous case. We can therefore conclude that, in our case,
3. CONCLUSIONS
The central achievement of this work is that surface columnar defects affect the dy-
namics of the underlying unaffected bulk layer by inducing local confinement of the fluxon
lattice, starting at a dose-dependent point phase. In particular, the lateral wandering is
allowed inside tubes whose lateral corsssection bo is dose dependent.
In summary, the particular strategy involved in surface irradiation de facto uses in
the bulk the very low elastic energy surviving at high temperatures, with just a little ex-
ternal energy supply due to surface columnar defects, to obtain someway ordered vortex
morphologies [14].
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1. Y. Zhu, R. C. Budhani, Z. X. Cai, D. O. Welch, M. Suenaga, R. Yoshizaki, and H. Ikeda, Phyl. Mag. Lett. 67
125 (1993).
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5. The dose equivalent field is the magnetic field that would be ideally required to fill each track with a flux
quantum, i.e., where ø0 is the flux quantum and n is the track density.
Bulk Confinement of Fluxons by Means of Surface Patterning 565
Experimental constraints on models describing the physical properties of the high- super-
conductors (HTS) can come not only from studies of the cuprates, but also from the analysis
of other related layered perovskites, sharing with them a quasi-two-dimensional (2D) struc-
ture. Of fundamental importance in this context has been the discovery of a superconducting
phase at very low temperatures in the compound which has the same crystal
structure as the parent compound with planes replacing the planes.
Actually, represents at present the only known example of a layered pervoskite
material that exhibits superconductivity without the presence of copper. However, in spite
of the close structural similarity, high- materials and show important differences
as far as their physical properties are concerned. At room temperature, is an in-
sulator compound showing enhanced Pauli paramagnetism [2], which, as T is decreased,
becomes first a metal and then a superconductor without doping [HTS are instead mostly
antiferromagnetic (AF) insulators becoming superconducting only after chemical doping].
For LDA-based calculations have shown [3] that
antibonding bands cross the Fermi energy, whereas in high- cuprates, the bands at EF
originate from and O orbitals. For ruthenium ions, an on-site Coulomb
repulsion of 2.4 eV is estimated from photoemission spectra [4]. Being the single-particle
1
Dipartimento di Scienze Fisiche “E.R. Caianiello,” Università di Salerno, I-84081 Baronissi (Salerno), Italy–Unità
I.N.F.M. di Salerno.
bandwidth equal to 1.4 eV, one gets a ratio which indicates that ruthenium
electrons in are less strongly correlated than copper holes in HTS.
de Haas-van Alphen experiments [5] show the existence of three almost-cylindrical
Fermi surface sheets with an essentially 2D topology that agrees well with band structure
calculations. In agreement with the Luttinger’s theorem prediction that the Fermi volume
is conserved even in presence of strong electron interactions, the volume enclosed by the
Fermi surface corresponds to exactly four electrons in the Brillouin zone. This points out
the importance of the Hund rule coupling in the ruthenium orbitals, confirmed by the
experiments performed on showing that the spin on
ions is equal to 1.
Specific features seem also to characterize the superconducting phase that develops
at about 1 K. In there are at low temperatures many-body enhancements of
the specific heat and the magnetic susceptibility quantitatively similar to that of In
addition, the compound which is the 3d analog of exhibits a metallic
ferromagnetic (FM) phase [6]. These findings suggest that superconductivity is likely to
develop in an odd-parity state, consistently with the lack of a Hebel-Schlichter peak
in in NQR experiments [7].
We formulate in this paper a microscopic model for describing the combined
effect on the physical properties of the system of the interorbital Hund coupling and the
intraorbital Coulomb repulsion. The Hamiltonian is written down neglecting the
orbital, which, according to band structure calculations, is lower in energy and does not
undergo significant charge fluctuations. Confining our anlaysis to the electron dynamics in
the ruthenium–oxygen planes, we introduce the Hamiltonian [8]
The electron filling has been chosen to be which gives configurations in which
the oxygen orbitals, lower in energy, are almost fully occupied and the and orbitals
contain about two electrons per ruthenium atom.
In general, the value of the total spin on each ruthenium atom is strongly dependent on
the strength of the Hund coupling J and the Coulomb repulsion U. The transitions between
different spin states induced by variations of J and U have been determined from the
behavior of the Ru spin–spin local and nonlocal correlation functions, defined respectively
as and and are the spin operators
for electrons on the site i in the and orbitals, respectively, and is the
total spin on site i).
The results reported in Fig. 1 give the behavior of as a function of U for three fixed
values of J. We can see that threshold values of U are needed to have local triplet states
associated with electrons in different d orbitals. This means that for the above Hamiltonian,
the Hund coupling cannot induce by itself a spin alignment. Finite values of the on-site
repulsion are also required in order to avoid intraorbital double occupation. From the curves
of Fig. 1, we observe that for a rather low value of 7, equal to 0.3, the increase of U
causes a transition from a configuration of local singlets to a configuration of local triplets,
occurring at a critical value of U approximately equal to 2.9. For higher values of
shows a qualitatively different behavior. The increase of U causes a first transition from a
configuration of local singlets to an equally weighted superposition of local singlets and
triplets, followed by a second transition from this latter state to one with local triplets only.
As expected, the corresponding critical values of U are decreasing functions of 7. We want
also to point out that the fact that threshold values of U must be exceeded in order to have
local triplet states, is a consequence of the high degree of hybridization existing in
between ruthenium and oxygen orbitals. Figure 2 clearly shows that for a lower value of
the increase of J makes tend to zero the critical value of U at which the transition to
the local triplet state occurs.
570 Cuoco, Noce, and Romano
The dependence on U of the nonlocal spin correlation function is reported in Fig. 3 for
the same choice of parameters as in Fig. 1. Comparing these curves with those for we
notice that for both choices of J the transition to a local triplet spin state is accompanied
by a transition to a global singlet spin state (when the total spin quantum number takes
the allowed values 0, 1,2, becomes equal to –2, –1, and 1, respectively). This
means that, when locally spins are ferromagnetically coupled, the ground-state of the whole
system of ruthenium electrons is a singlet state. This result is consistent with the experiments
performed in Ref. [2], showing that is a paramagnetic compound.
A Finite-Size Cluster Study of 571
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New Copper-Free Layered Perovskite
Superconductors: and
Related Compounds
1. INTRODUCTION
All of the high- cuprate superconductors are layered perovskites, and the existence
of Cu-O planes is regarded to be essential for superconductivity. To investigate the role of
Cu-O layers for the occurrence of high- superconductivity, discovery of new copper-free
layered perovskite superconductors are required. Recently, we succeeded in synthesiz-
ing the Li intercalated as a new copper-free superconductor [ 1 – 4 ] . In this
paper, we present the transport and superconducting properties of and related
compounds.
A schematic of the crystal structure of and its related compounds [5–6]
is summarized in Fig. 1. Each compound has a layered perovskite structure characterized
by alternate stacking of Nb-O planes and an alkali–metal layer. The Nb-O plane and the
1
National Research Institute for Metals, Tsukuba, Japan.
2
Dept. of Physics, Yokohama National Univ., Yokohama, Japan.
3
Dept. of Appl. Sci., Tokyo Denki Univ., Tokyo, Japan.
4
Dept. of Physics, Hokkaido Univ. of Education Sapporo, Sapporo, Japan.
5
Institute for Solid State Physics, Univ. of Tokyo, Tokyo, Japan.
alkali–metal layer in the niobate system seem to correspond to the Cu-O plane and block-
layer in high- superconductors, respectively. According to the number of Nb-O planes
and the structural properties of the alkali–metal layer, the samples summarized in Fig. 1 can
be divided into two groups: group and group
and The compounds in group A have triple Nb-O planes
as a common structure, although the crystal structure of the alkali–metal layer differs
slightly between each compound. However, for group and
have double, triple, and quadruple Nb-O planes, respectively; nevertheless,
the crystal structures around the alkali–metal layers are almost the same.
2. EXPERIMENTAL
All the polycrystalline samples were prepared by a solid-state reaction. An appropri-
ate mixture of starting materials, and
were pelletized and sintered at 1100–1200°C in the air until the samples became single
phase. The sintered samples were white in color and electrically insulating. To introduce
charge carriers, Li was intercalated into the samples around the alkali–metal layer by n-
Butyllithium n-hexane solution. The temperature dependence of the electronic resistivities
was measured by a standard four probe method using a refrigerator. Magnetic suscep-
tibility measurements were performed down to 2 K under 100 Oe after zero-field cooling
in a SQUID magnetometer.
REFERENCES
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2. Y. Takano, H. Taketomi, H. Tsurumi, T. Yamadaya, and N. Môri, Physica B 237–238, 68 (1997).
3. Y. Takano, Solid State Physics (in Japanese) 32, 737 (1997).
4. Y. Takano, Y. Kimishima, T. Yamadaya, S. Ogawa, S. Takayanagi, and N. Môri, Rev. High Press. Sci. Technol.
7, 589 (1998).
5. M. Dion, M. Ganne, and M. Tournoux, Mater. Res. Bull. 16, 1429 (1981).
6. M. Dion, M. Ganne, and M. Tournoux, Revue de Chimie Minerale t. 23, 61 (1986).
7. I. Hase and Y. Nishihara, Phy. Rev. B 58, R1707 (1998).
Author Index
579
580 Author Index
581
582 Subject Index